The Density-Matrix Renormalization Group

REVIEWS OF MODERN PHYSICS, VOLUME 77, JANUARY 2005

The density-matrix renormalization group

U. Schollwöck RWTH Aachen University, D-52056 Aachen, Germany ͑Published 26 April 2005͒

The density-matrix renormalization group ͑DMRG͒ is a numerical algorithm for the efﬁcient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the ﬁelds of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.

CONTENTS IV. Zero-Temperature Dynamics 281 A. Continued-fraction dynamics 282 B. Correction vector dynamics 284 I. Introduction 259 C. Dynamical DMRG 285 II. Key Aspects of DMRG 262 V. Bosons and DMRG: Many Local Degrees of Freedom 286 A. Real-space renormalization of Hamiltonians 262 A. Moderate number of degrees of freedom 286 B. Density matrices and DMRG truncation 263 B. Large number of degrees of freedom 287 C. Infinite-system DMRG 265 VI. Two-Dimensional Quantum Systems 287 D. Infinite-system and finite-system DMRG 266 VII. Beyond Real-Space Lattices: Momentum-Space E. Symmetries and good quantum numbers 267 DMRG, Quantum Chemistry, Small Grains, and 1. Continuous Abelian symmetries 267 Nuclear Physics 289 2. Continuous non-Abelian symmetries 267 A. Momentum-space DMRG 289 3. Discrete symmetries 267 B. Quantum chemistry 292 4. Missing symmetries 268 C. DMRG for small grains and nuclear physics 295 F. Energies: Ground states and excitations 268 VIII. Transfer-Matrix DMRG: Classical and Quantum G. Operators and correlations 269 Systems 295 H. Boundary conditions 270 A. Classical transfer-matrix DMRG in two dimensions: I. Large sparse-matrix diagonalization 270 TMRG 296 1. Algorithms 270 B. Corner transfer-matrix DMRG: An alternative 2. Representation of the Hamiltonian 270 approach 297 3. Eigenstate prediction 271 C. Quantum statistical mechanics in one dimension: J. Applications 272 Quantum TMRG 299 III. DMRG Theory 273 IX. Systems Out of Equilibrium: Non-Hermitian and A. Matrix-product states 274 Time-Dependent DMRG 303 1. Matrix-product states 274 A. Transition matrices 303 2. Correlations in matrix-product states 274 B. Stochastic transfer matrices 304 3. DMRG and matrix-product states 275 C. Time-dependent DMRG 305 4. Variational optimization in matrix-product 1. Static time-dependent DMRG 305 states 276 2. Adaptive time-dependent DMRG 306 B. Properties of DMRG density matrices 277 3. Time-dependent correlations 307 1. DMRG applied to matrix-product states 277 D. Finite temperature revisited: Time evolution and 2. DMRG applied to generic gapped dissipation at TϾ0307 one-dimensional systems 277 X. Outlook 309 3. DMRG applied to one-dimensional systems Acknowledgments 309 at criticality 278 References 309 4. DMRG in two-dimensional quantum systems 278 5. DMRG precision for periodic boundary I. INTRODUCTION conditions 279 C. DMRG and quantum information theory—A new Consider a crystalline solid; it consists of some 1026 or perspective 279 more atomic nuclei and electrons for objects on a human

0034-6861/2005/77͑1͒/259͑57͒/$50.00 259 ©2005 The American Physical Society 260 U. Schollwöck: Density-matrix renormalization group scale. All nuclei and electrons are subject to the strong, 1 ˆ ͑ † ͒ ͩ ͪ long-range Coulomb interaction. While this system HtJ =− ͚ tij ci␴cj␴ + H.c. + ͚ Jij Si · Sj − ninj , ͗ ͘ ␴ ͗ ͘ 4 should a priori be described by the Schrödinger equa- ij , ij tion, its explicit solution is impossible to find. Yet, it has ͑2͒ become clear over the decades that in many cases the 2 physical properties of solids can be understood to a very in which the spin-spin interaction Jij=4tij/U is due to a good approximation in the framework of some effective second-order virtual hopping process possible only for one-body problem. This is a consequence of the very electrons of opposite spin on sites i and j. At half-filling, 1 efficient interplay of nuclei and electrons in screening the model simplifies even further to the spin-2 isotropic the Coulomb interaction, except on the very shortest Heisenberg model, length scales. ˆ ͑ ͒ This comforting picture may break down as various HHeisenberg = ͚ JijSi · Sj, 3 ͗ ͘ effects invalidate the fundamental tenet of weak effec- ij tive interactions, taking us into the field of strongly cor- placing collective ͑anti͒ferromagnetism, which it de- related quantum systems, in which the full electronic scribes, into the framework of strongly correlated sys- many-body problem has to be considered. Of the routes tems. These and other model Hamiltonians have been to strong correlation, let me mention just two, because it modified in multiple ways. is at their meeting point that, from the point of view of In various fields of condensed-matter physics, such as physical phenomena, the density-matrix renormalization high-temperature superconductivity, low-dimensional group ͑DMRG͒ has its origin. magnetism ͑spin chains and spin ladders͒, low- On the one hand, for arbitrary interactions, the dimensional conductors, and polymer physics, theoreti- Fermi-liquid picture breaks down in one-dimensional cal research is focussing on the highly nontrivial proper- solids and gives way to the Tomonaga-Luttinger liquid ties of these seemingly simple models, which are picture: essentially due to the small size of phase space, believed to capture some of the essential physics. Recent scattering processes for particles close to the Fermi en- progress in experiments on ultracold atomic gases ergy become so important that the ͑fermionic͒ quasipar- ͑Greiner et al., 2002͒ has allowed the preparation of ticle picture of Fermi-liquid theory is replaced by strongly correlated bosonic systems in optical lattices ͑bosonic͒ collective excitations. with tunable interaction parameters, attracting many On the other hand, for arbitrary dimensions, screen- solid-state physicists to this field. On the conceptual ing is typically so effective because of strong delocaliza- side, important old and new questions are at the center tion of valence electrons over the lattice. In transition of the physics of strongly correlated quantum systems, metals and rare earths, however, the valence orbitals are e.g., topological effects in quantum mechanics, the wide inner d or f orbitals. Hence valence electrons are much field of quantum phase transitions, and the search for more localized, though not immobile. There is now a exotic forms of order stabilized by quantum effects at strong energy penalty for placing two electrons in the low temperatures. same local valence orbital, and the motion of valence The typical absence of a dominant, exactly solvable electrons becomes strongly correlated on the lattice. contribution to the Hamiltonian, about which a pertur- To study strongly correlated systems, simplified model bative expansion as in conventional many-body physics Hamiltonians have been designed that try to retain just might be attempted, goes a long way towards explaining the core ingredients needed to describe some physical the inherent complexity of strongly correlated systems. phenomenon and methods for their treatment. Localiza- This is why, apart from some exact solutions like those tion suggests the use of tight-binding lattice models,in provided by the Bethe ansatz or certain toy models, which local orbitals on one site can take N =4 differ- most analytical approaches are quite uncontrolled in site their reliability. While these approaches may yield im- ent states of up to two electrons ͉͑0͘,͉↑͘,͉↓͘,͉↑↓͒͘. portant insight into the nature of the physical phenom- The simplest model Hamiltonian for just one valence ena observed, it is ultimately up to numerical ap- orbital ͑band͒ with a kinetic-energy term ͑electron hop- proaches to assess the validity of analytical ping between sites i and j with amplitude t ͒ and Cou- ij approximations. lomb repulsion is the on-site Hubbard model ͑Hubbard, ͒ Standard methods in the field are the exact diagonal- 1963, 1964 , in which just the leading on-site Coulomb ization of small systems, the various quantum Monte repulsion U has been retained: Carlo methods, and, less widely used, coupled-cluster methods and series expansion techniques. Since its introduction by White ͑1992͒, the density-matrix renor- † Hˆ =− ͚ t ͑c c + H.c.͒ + U͚ n ↑n ↓. ͑1͒ malization group has quickly achieved the status of a Hubbard ij i␴ j␴ i i ͗ij͘,␴ i highly reliable, precise, and versatile numerical method in the field. Its main advantages are the ability to treat unusually large systems at high precision at or very close ͗͘ designates bonds. In the limit U/tӷ1 double occu- to zero temperature and the absence of the negative- ͉↑↓͘ pancy can be excluded, resulting in the Nsite=3 sign problem that plagues the otherwise most powerful state t-J model: method, quantum Monte Carlo, making the latter of

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 261 limited use for frustrated spin or fermionic systems. A mary. All the sections that come after these key concep- first striking illustration of DMRG precision was given tual sections can then, to some degree, be read indepen- by White and Huse ͑1993͒. Using modest numerical dently. means to study the S=1 isotropic antiferromagnetic Among the various branches of applied DMRG, the Heisenberg chain, they calculated the ground-state applications to dynamical properties of quantum sys- ͑ ͒ energy at almost machine precision, E0 tems are presented first Sec. IV . Next, I discuss at- =−1.401 484 0389 71͑4͒J, and the ͑Haldane͒ excitation tempts to use DMRG for systems with potentially large gap as ⌬=0.410 50͑2͒J. The spin-spin correlation length numbers of local degrees of freedom such as phononic was found to be ␰=6.03͑2͒ lattice spacings. More ex- models or Bose-Hubbard models ͑Sec. V͒. amples of the versatility and precision that have made Moving beyond the world of T=0 physics in one di- the reputation of the DMRG method will be found mension, DMRG as applied to two-dimensional quan- throughout the text. The major drawback of DMRG is tum Hamiltonians in real space with short-ranged inter- that it displays its full force mainly for one-dimensional actions is introduced and various algorithmic variants systems; nevertheless, interesting forays into higher di- are presented in Sec. VI. mensions have been made. By now, DMRG has, despite Major progress has been made by abandoning the its complexity, become a standard tool of computational concept of an underlying real-space lattice, as various physics that many research groups in condensed-matter authors have developed DMRG variants for momentum physics will want to have at hand. In the simulation of space ͑Sec. VII.A͒, for small-molecule quantum chemis- model Hamiltonians, it is part of a methodological triad try ͑Sec. VII.B͒, and for mesoscopic small grains and consisting of exact diagonalization, quantum Monte nuclei ͑Sec. VII.C͒. All these fields are currently under Carlo, and DMRG. very active development. Most DMRG applications still treat low-dimensional, Early in the history of DMRG it was realized that its strongly correlated model Hamiltonians, for which the core renormalization idea might also be applied to the method was originally developed. However, the core renormalization of transfer matrices that describe two- idea of DMRG, the construction of a renormalization- dimensional classical ͑Sec. VIII.A͒ or one-dimensional group flow in a space of suitably chosen density matri- quantum systems ͑Sec. VIII.C͒ in equilibrium at finite ces, is quite general. In fact, another excellent way of temperature. Here, algorithmic details undergo major conceptualizing DMRG is in the very general terms of changes, such that this class of DMRG methods is often quantum information theory. The versatility of the core referred to as TMRG (transfer-matrix renormalization idea has allowed the extension of DMRG applications group). to fields quite far from its origins, such as the physics of Yet another step can be taken by moving to physical ͑three-dimensional͒ small grains, equilibrium and far- systems that are out of equilibrium ͑Sec. IX͒.DMRG from-equilibrium problems in classical and quantum sta- has been successfully applied to the steady states of non- tistical mechanics, and the quantum chemistry of small equilibrium systems, leading to a non-Hermitian exten- to medium-sized molecules. Profound connections to ex- sion of the DMRG in which transition matrices replace actly solvable models in statistical mechanics have Hamiltonians. Various methods for time-dependent emerged. problems are under development and have recently The aim of this review is to provide both the algorith- started to yield results unattainable by other numerical mic foundations of DMRG and an overview of its main methods, not only at T=0, but also for finite tempera- and by now quite diverse fields of applications. ture and in the presence of dissipation. I start with a discussion of the key algorithmic ideas In this review, details of computer implementation needed to deal with the most conventional DMRG have been excluded. Rather, I have tried to focus on problem, the study of T=0 static properties of a one- details of algorithmic structure and their relation to the dimensional quantum Hamiltonian ͑Sec. II͒. This in- physical questions to be studied using DMRG and to cludes standard improvements to the basic algorithm give the reader some idea of the power and the limita- that should always be used to obtain a competitive tions of the method. In the more standard fields of DMRG application, a discussion of how to assess the DMRG, I have not been ͑able to be͒ exhaustive even in numerical quality of DMRG output, and an overview of listing applications. In the less established fields of applications. Having read this first major section, the DMRG, I have tried to be more comprehensive and to reader should be able to set up a standard DMRG with provide as much discussion of their details as possible, in the major algorithmic design problems in mind. order to make these fields better known and, hopefully, I move on to a section on DMRG theory ͑Sec. III͒, to stimulate thinking about further algorithmic progress. discussing the properties of the quantum states gener- I have excluded, for lack of space, any extensive discus- ated by DMRG ͑matrix-product states͒ and the proper- sions of the analytical methods whose development has ties of the density matrices that are essential for its suc- been stimulated by DMRG concepts. Last but not least, cess; the section is closed by a reexamination of DMRG it should be mentioned that some of the topics of this from a quantum information theory point of view. At review have been considered by other authors: White least some superficial grasp of the key results of this ͑1998͒ gives an introduction to the fundamentals of section will be useful in the remainder of the review, DMRG; a very detailed survey of DMRG as it was un- while a more thorough reading might serve as a sum- derstood in late 1998 has been provided in a collection

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 262 U. Schollwöck: Density-matrix renormalization group of lectures and articles ͑Peschel, Hallberg, et al., 1999͒, One obtains a simplified ͑“renormalized”͒ effective which also contains an account of DMRG history by Hamiltonian that should catch the essential physics of White. More recently, the application of TMRG to the system under study. The success of this approach quantum systems and two-dimensional DMRG has been rests on scale separation: for continuous phase transi- reviewed by Shibata ͑2003͒. Hallberg ͑2003͒ gives a tions, the diverging correlation length sets a natural rather complete overview of DMRG applications. long-wavelength low-energy scale which dominates the Dukelsky and Pittel ͑2004͒ focus on DMRG applications physical properties, and fluctuations on shorter length to finite Fermi systems such as small grains, small mol- scales may be integrated out and summed up into quan- ecules, and nuclei. titative modifications of the long-wavelength behavior. A word on notation: All state spaces considered here In the Kondo problem, the width of the Kondo reso- can be factorized into local state spaces ͕͉␴͖͘ labeled by nance sets an energy scale such that the exponentially greek letters. DMRG forms blocks of lattice sites; I de- decaying contributions of energy levels far from the note ͑basis͒ states ͉m͘ of such blocks by latin letters. resonance can be integrated out. This is the essence of These states depend on the size of the block; when nec- the numerical renormalization group ͑Wilson, 1975͒. essary, I indicate the block length by subscripts ͉mᐉ͘. Following up on the Kondo problem, it was hoped ͉␴ ͘ Correspondingly, i is a local state on site i. Moreover, that thermodynamic-limit ground-state properties of DMRG typically operates with two blocks and two sites, other many-body problems such as the one-dimensional which are referred to as belonging to a “system” or “en- Hubbard or Heisenberg models might be treated simi- vironment.” Where this distinction matters, it is indi- larly, with lattice sites replacing energy levels. However, cated by superscripts, ͉mS͘ or ͉␴E͘. results of real-space renormalization schemes turned out to be poor. While a precise analysis of this observation is hard for many-body problems, White and Noack ͑1992͒ II. KEY ASPECTS OF DMRG identified the breakdown of the real-space renormaliza- Historically, DMRG has its origin in the analysis by tion group for the toy model of a single noninteracting White and Noack ͑1992͒ of the failure of real-space particle hopping on a discrete one-dimensional lattice ͑ ͒ renormalization-group ͑RSRG͒ methods to yield quanti- the “particle in a box” problem . For a box of size L, tatively acceptable results for the low-energy properties the Hilbert space spanned by ͕͉i͖͘ is L dimensional; in of quantum many-body problems. Most of the DMRG state ͉i͘, the particle is on site i. The matrix elements of algorithm can be formulated in standard ͑real-space͒ the Hamiltonian are band diagonal, ͗i͉Hˆ ͉i͘=2; renormalization-group ͑RG͒ language. Alternative ͗i͉Hˆ ͉i±1͘=−1 in some units. Consider now the following points of view in terms of matrix product states and real-space renormalization-group procedure: quantum information theory will be taken up later ͑Sec. III.A and Sec. III.C͒. Before moving on to the details, ͑1͒ Describe interactions on an initial sublattice let me mention a debate that has been going on among ͑ ͒ ᐉ ˆ “block” A of length by a block Hamiltonian HA DMRG practitioners on whether calling the DMRG al- acting on an M-dimensional Hilbert space. gorithm a renormalization-group method is a misnomer. The answer depends on what one considers the essence ͑2͒ Form a compound block AA of length 2ᐉ and the ͑ ͒ ˆ of a renormalization-group RG method. If this essence Hamiltonian HAA, consisting of two block Hamilto- is the systematic thinning out of degrees of freedom nians and interblock interactions. Hˆ has dimen- leading to effective Hamiltonians, DMRG is an RG AA sion M2. method. However, it does not involve an ultraviolet or infrared energy cutoff in the degrees of freedom, which ͑ ͒ ˆ 3 Diagonalize HAA to find the M lowest-lying eigen- is at the heart of traditional RG methods and hence is states. often considered as part of the core concept. ͑ ͒ ˆ 4 Project HAA onto the truncated space spanned by ˆ → ˆ tr A. Real-space renormalization of Hamiltonians the M lowest-lying eigenstates, HAA HAA. ͑5͒ Restart from step ͑2͒, with doubled block size: 2ᐉ The set of concepts grouped under the heading of →ᐉ → ˆ tr → ˆ “renormalization” has proven extremely powerful in ,AA A, and HAA HA, until the box size is providing succinct descriptions of the collective behavior reached. of systems of size L with a diverging number of degrees The key point is that the decimation procedure of the L of freedom Nsite. Starting from some microscopic Hamil- Hilbert space is to take the lowest-lying eigenstates of tonian, degrees of freedom are iteratively integrated out the compound block AA. This amounts to the assump- and accounted for by modifying the original Hamil- tion that the ground state of the entire box will essen- tonian. The new Hamiltonian will exhibit modified as tially be composed of energetically low-lying states liv- well as new couplings, and renormalization-group ap- ing on smaller blocks. The outlined real-space proximations typically consist of physically motivated renormalization procedure gives very poor results. The truncations of the set of couplings newly generated by breakdown can best be understood visually ͑Fig. 1͒: as- the a priori exact elimination of degrees of freedom. suming an already rather large block size, where dis-

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 263

FIG. 1. Lowest-lying eigenstates of ͑dashed line͒ blocks A and ͑solid line͒ AA for the problem of a single particle in a box in FIG. 2. System meets environment: Fundamental density- the continuum limit. matrix renormalization-group ͑DMRG͒ construction of a superblock from two blocks and two single sites. cretization can be neglected, the lowest-lying states of A all have nodes at the lattice ends, such that all product Because the ﬁnal goal is the ground state in the ther- states of AA have nodes at the compound block center. modynamic limit, one studies the best approximation to The true ground state of AA has its maximum ampli- it, the ground state of the superblock, obtained by nu- tude right there, such that it cannot be properly approxi- merical diagonalization:

mated by a restricted number of block states. S E M Nsite Nsite M Merely considering isolated blocks imposes wrong S S E E ͉␺͘ = ͚ ͚ ͚ ͚ ␺ S␴S␴E E͉m ␴ ͉͘m ␴ ͘ boundary conditions, and White and Noack ͑1992͒ could m m S ␴S ␴E E obtain excellent results by combining Hilbert spaces m =1 =1 =1 m =1 from low-lying states of block A by assuming various NS NE ͑ ␺ ͉ ͉͘ ͘ ͗␺͉␺͘ ͑ ͒ combinations of fixed and free boundary conditions i.e., = ͚ ͚ ij i j ; =1, 4 enforcing a vanishing wave function or a vanishing i j wave-function derivative, respectively, at the bound- S S E E S S where ␺ S␴S␴E E =͗m ␴ ;␴ m ͉␺͘. ͕͉m ␴ ͖͘ϵ͕͉i͖͘, and aries͒. They also realized that combining various bound- m m ͕͉mE␴E͖͘ϵ͕͉j͖͘ are the orthonormal product bases of ary conditions for a single particle would translate to system and environment ͑subscripts have been dropped͒ accounting for fluctuations between blocks in the case of S S E E many interacting particles. Keeping this observation in with dimensions N =M Nsite and N =M Nsite, respec- ͑ S Þ E͒ mind, we now return to the original question of a many- tively for later generalizations, we allow N N . S S Ͻ S body problem in the thermodynamic limit and formulate Some truncation procedure from N to M N states the following strategy: In order to analyze which states must now be implemented. Let me present three lines of have to be retained for a finite-size block A, A has to be argument on the optimization of some quantum- embedded in some environment, mimicking the mechanical quantity, all leading to the same truncation thermodynamic-limit system in which A is ultimately prescription focused on density-matrix properties. This embedded. is to highlight different aspects of the DMRG algorithm and to give confidence in the prescription found. ͑1͒ Optimization of expectation values (White, 1998): If B. Density matrices and DMRG truncation the superblock is in a pure state ͉␺͘ as in Eq. ͑4͒, Consider, instead of the exponentially fast growth pro- statistical physics describes the physical state of the ␳ cedure outlined above, the following linear growth pre- system through a reduced density matrix ˆ, scription ͑White, 1992͒: Assume that for a system ͑a ␳ˆ =Tr ͉␺͗͘␺͉, ͑5͒ block in DMRG language͒ of length ᐉ we have an E S S M -dimensional Hilbert space with states ͕͉mᐉ͖͘. The where the states of the environment have been S S traced out, Hamiltonian Hˆ ᐉ is given by matrix elements ͗mᐉ͉Hˆ ᐉ͉m˜ ᐉ͘. Similarly we know the matrix representations of local * ͗i͉␳ˆ͉iЈ͘ = ͚ ␺ ␺ . ͑6͒ ͗ S͉ ͉ S͘ ij iЈj operators such as mᐉ Ci m˜ ᐉ . j ˆ For linear growth, we now construct Hᐉ+1 in the prod- ␳ S ͕͉ S␴͖͘ϵ͕͉ S͉͘␴S͖͘ ͉␴S͘ ˆ has N eigenvalues w␣ and orthonormal eigen- uct basis mᐉ mᐉ , where are the Nsite lo- states ␳ˆ͉w␣͘=w␣͉w␣͘, with ͚␣w␣ =1 and w␣ ജ0. We cal states of a new site added. ജ The thermodynamic limit is now mimicked by embed- assume the states are ordered such that w1 w2 ജ ജ ding the system in an environment of the same size, as- w3 . The intuition that the ground state of the ¯ S sumed to have been constructed in analogy to the sys- system is best described by retaining those M states tem. We thus arrive at a superblock of length 2ᐉ+2 ͑Fig. with largest weight w␣ in the density matrix can be 2͒, in which the arrangement chosen is typical, but not formalized as follows. Consider some bounded op- mandatory. erator Aˆ acting on the system, such as the energy

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 264 U. Schollwöck: Density-matrix renormalization group

ʈ ˆ ʈ ͦ͗␾͉ ˆ ͉␾͘ ͗␾͉␾ͦ͘ϵ described by their wave function. It is thus a reason- per lattice bond; A =max␾ A / cA. able demand for a truncation procedure that the ap- The expectation value of Aˆ is found to be, using ͉␺˜ ͘ Eqs. ͑4͒ and ͑6͒, proximative wave function in which the system space has been truncated to be spanned by only MS ͗ ˆ ͘ ͗␺͉ ˆ ͉␺͘ ͗␺͉␺͘ ␳ ˆ ͑ ͒ ͉␣͘ ͚ ͉ ͘ A = A / =TrS ˆA. 7 orthonormal states = iu␣i i , Expressing Eq. ͑7͒ in the density-matrix eigenbasis, MS NE ͉␺˜ ͘ ͉␣͉͘ ͘ ͑ ͒ one finds = ͚ ͚ a␣j j , 12 ␣=1 j=1 NS

͗Aˆ ͘ = ͚ w␣͗w␣͉Aˆ ͉w␣͘. ͑8͒ minimize the distance in the quadratic norm ␣=1 ʈ͉␺͘ − ͉␺˜ ͘ʈ. ͑13͒ Then, if we project the system state space down to S the M dominant eigenvectors ͉w␣͘ with the largest This problem finds a very compact solution in a eigenvalues, singular-value decomposition, which was the origi-

S nal approach of White: Singular-value decomposi- M tion will be considered in a slightly different setting ͗ ˆ ͘ ͗ ͉ ˆ ͉ ͘ ͑ ͒ A approx = ͚ w␣ w␣ A w␣ , 9 in the next section. The following is an alternative, ␣ =1 more pedestrian approach. Assuming real coeffi- and the error for ͗Aˆ ͘ is bounded by cients for simplicity, one has to minimize S ␺ 2 ͑ ͒ N 1−2͚ ija␣ju␣i + ͚ a␣j 14 ␣ij ␣j ͉͗ ˆ ͘ ͗ ˆ ͉͘ ഛ ͩ ͚ ␣ͪ ϵ ⑀␳ ͑ ͒ A approx − A w cA cA. 10 ␣ϾMS with respect to a␣j and u␣i. For the solution to be ͚ ␺ This estimate holds in particular for energies. Sev- stationary in a␣j, we must have i iju␣i =a␣j, finding eral remarks are in order. Technically, I have ne- that glected to trace the fate of the denominator in Eq. ͚ ␳ ͑ ͒ ͑7͒ upon projection; the ensuing correction of ͑1 1− u␣i iiЈu␣iЈ 15 −1 ␣iiЈ −⑀␳͒ is of no relevance to the argument here, as ⑀␳ →0. The estimate could be tightened for any spe- must be stationary for the global minimum of the ͑ ͒ cific operator, since we know ͗w␣͉Aˆ ͉w␣͘, and a more distance 13 , where we have introduced the density- efficient truncation procedure could be named. For matrix coefficients arbitrary bounded operators acting on the system, ␳ = ͚ ␺ ␺ . ͑16͒ the prescription to retain the state spanned by the iiЈ ij iЈj j MS dominant eigenstates is optimal. For local quantities, such as energy, magnetization, or density, er- Equation ͑15͒ is stationary, according to the rors are of the order of the truncated weight Rayleigh-Ritz principle, for ͉␣͘ being the eigenvec- ͑ ͒ S tors of the density matrix. Expressing Eq. 15 in the M density-matrix eigenbasis, the global minimum is ⑀ ͚ ͑ ͒ S ␳ =1− w␣, 11 given by choosing ͉␣͘ to be the M eigenvectors ͉w␣͘ ␣=1 to the largest eigenvalues w␣ of the density matrix, which emerges as the key estimate. Hence a fast de- as they are all non-negative, and the minimal dis- cay of density-matrix eigenvalues w␣ is essential for tance squared is, using Eq. ͑11͒, the performance of this truncation procedure. The MS truncation error of Eq. ͑10͒ is the total error only if ʈ͉␺͘ ͉␺˜ ͘ʈ2 ⑀ ͑ ͒ the system had been embedded in the final and ex- − =1−͚ w␣ = ␳. 17 ␣=1 actly described environment. Considering the iterative system and environment growth and their ap- The truncation prescription now appears as a varia- proximate representation at each step, we see that tional principle for the wave function. additional sources of an environmental error have to ͑3͒ Optimization of entanglement (Gaite, 2001, 2003; be considered. In practice therefore errors for ob- Galindo and Martín-Delgado, 2002; Osborne and servables calculated by DMRG are often much Nielsen, 2002; Latorre et al., 2004): Consider the su- larger than the truncated weight, even after addi- perblock state ͉␺͘ as in Eq. ͑4͒. The essential feature tional steps to eliminate environmental errors have of a nonclassical state is its entanglement, the fact been taken. Careful extrapolation of results in MS ͑ ⑀ ͒ that it cannot be written as a simple product of one or ␳ is therefore highly recommended, as will be system and one environment state. Bipartite en- discussed below. tanglement as relevant here can best be studied by ͑2͒ Optimization of the wave function (White, 1992, representing ͉␺͘ in its form after a Schmidt decom- 1993): Quantum-mechanical objects are completely position ͑Nielsen and Chuang, 2000͒: Assuming

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without loss of generality NS ജNE, consider the While one may easily construct density-matrix ͑ S ϫ E͒ ␺ N N -dimensional matrix A with Aij= ij. spectra for which upon normalization the trun- Singular-value decomposition guarantees A cated state produced by DMRG no longer maxi- =UDVT, where U is ͑NS ϫNE͒ dimensional with or- mizes entanglement, for typical density-matrix thonormal columns, D is an ͑NE ϫNE͒-dimensional spectra the optimization statement still holds in diagonal matrix with non-negative entries D␣␣ practice. ͱ T E E = w␣, and V is an ͑N ϫN ͒-dimensional unitary matrix; ͉␺͘ can be written as C. Infinite-system DMRG NS NE NE ͉␺͘ ͱ T ͉ ͉͘ ͘ = ͚ ͚ ͚ Ui␣ w␣V␣j i j Collecting the results of the last two sections, the so- ␣ i=1 =1 j=1 called infinite-system DMRG algorithm can now be for- NE NS NE mulated ͑White, 1992͒. Details of efficient implementa- ͱ ͩ ͉ ͪͩ͘ ͉ ͪ͘ ͑ ͒ tion will be discussed in subsequent sections; here, we = ͚ w␣ ͚ Ui␣ i ͚ Vj␣ j . 18 ␣=1 i=1 j=1 assume that we are looking for the ground state. The orthonormality properties of U and VT ensure ͑1͒ Consider a lattice of some small size ᐉ, forming the ͉ S͘ ͚ ͉ ͘ ͉ E͘ ͚ ͉ ͘ S that w␣ = iUi␣ i and w␣ = jVj␣ j form orthonor- system block S. S lives on a Hilbert space of size M S S mal bases of system and environment, respectively, with states ͕͉Mᐉ͖͘; the Hamiltonian Hˆ ᐉ and the op- in which the Schmidt decomposition erators acting on the block are assumed to be

NSchmidt known in this basis. At initialization, this may still ᐉ ͱ S E ͑ ഛ S͒ ͉␺͘ = ͚ w␣͉w␣͉͘w␣ ͑͘19͒ be an exact basis of the block Nsite M . Similarly, ␣=1 form an environment block E. S E ␺ ͑ ͒ Ј holds. N N coefficients ij are reduced to NSchmidt 2 Form a tentative new system block S from S and ഛ E ͱ ജ ജ ജ ͑ ͒ Ј N nonzero coefficients w␣, w1 w2 w3 . one added site Fig. 2 .S lives on a Hilbert space of S E ¯ S S Relaxing the assumption N ജN , one has size N =M Nsite, with a basis of product states S S ͕͉Mᐉ␴͖͘ϵ͕͉Mᐉ͉͘␴͖͘. In principle, the Hamiltonian N ഛ min͑NS,NE͒. ͑20͒ Schmidt ˆ S Ј Hᐉ+1 acting on S can now be expressed in this basis The suggestive labeling of states and coefficients in ͑which will not be done explicitly for efficiency; see Eq. ͑19͒ is motivated by the observation that upon Sec. II.I͒. A new environment EЈ is built from E in tracing out environment or system the reduced den- the same way. sity matrices for system and environment are found ͑3͒ Build the superblock of length 2ᐉ+2 from SЈ and to be EЈ. The Hilbert space is of size NSNE, and the ma- NSchmidt NSchmidt ˆ ␳ ͉ S͗͘ S͉ ␳ ͉ E͗͘ E͉ trix elements of the Hamiltonian H2ᐉ+2 could in ˆ S = ͚ w␣ w␣ w␣ ; ˆ E = ͚ w␣ w␣ w␣ . ␣ ␣ principle be constructed explicitly, but this is avoided for efficiency reasons. ͑21͒ ͑4͒ Find by large sparse-matrix diagonalization of Hˆ ᐉ Even if system and environment are different ͑Sec. 2 +2 the ground state ͉␺͘. This is the most time- II.D͒, both density matrices would have the same consuming part of the algorithm ͑Sec. II.I͒. number of nonzero eigenvalues, bounded by the ͑ ͒ ␳ ͉␺͗͘␺͉ smaller of the dimensions of system and environ- 5 Form the reduced density matrix ˆ =TrE as in ment, and an identical eigenvalue spectrum. Quan- Eq. ͑6͒ and determine its eigenbasis ͉w␣͘ ordered by tum information theory now provides a measure of descending eigenvalues ͑weight͒ w␣. Form a new entanglement through the von Neumann entropy, ͑reduced͒ basis for SЈ by taking the MS eigenstates Ј NSchmidt with the largest weights. In the product basis of S , S S ␳ ␳ ͑ ͒ their matrix elements are ͗mᐉ␴͉mᐉ ͘; taken as col- SvN =−Trˆ ln2 ˆ =− ͚ w␣ln2w␣. 22 +1 S S ␣=1 umn vectors, they form an N ϫM rectangular matrix T. Proceed likewise for the environment. Hence our truncation procedure to MS states pre- tr serves a maximum of system-environment entangle- ͑6͒ Carry out the reduced basis transformation Hˆ ᐉ S +1 ment if we retain the first M states ͉w␣͘ for ␣ † S =T Hˆ ᐉ T onto the new M -state basis and take =1,…,MS,as−x ln x grows monotonically for 0 +1 2 ˆ tr → ˆ Ͻxഛ1/e, which is larger than typical discarded Hᐉ+1 Hᐉ+1 for the system. Do the same for the en- ͑ ͒ eigenvalues. Let me add that this optimization vironment and restart with step 2 with block size ᐉ statement holds strictly only for the unnormalized +1 until some desired final length is reached. Op- ͑ truncated state. Truncation leads to a wave- erator representations also have to be updated see ͱ Sec. II.G͒. function norm 1−⑀␳ and enforces a new normal- ͑ ͒ ͑ ization, which changes the retained w␣ and SvN. 7 Calculate desired ground-state properties energies

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FIG. 3. Finite-system DMRG algorithm: block growth and shrinkage.

͒ ͉␺͘ and correlators from ; this step can also be car- FIG. 4. Currents on rung r of a t-J ladder induced by a source ried out at each intermediate length. term on the left edge, after various sweeps of finite-system If the Hamiltonian is reflection symmetric, one may con- DMRG. From Schollwöck et al., 2003. sider system and environment to be identical. One is not restricted to choosing the ground state for ͉␺͘; any state tions of finite-system DMRG͒. When the system block accessible by large sparse-matrix diagonalization of the superblock is allowed. Currently available algorithms for reaches some minimum size and becomes exact, growth this diagonalization limit us, however, to the lowest-lying direction is reversed. The system block now grows at the excitations ͑see Sec. II.I͒. expense of the environment. All basis states are chosen while system and environment are embedded in the final system and in the knowledge of the full Hamiltonian. If D. Infinite-system and finite-system DMRG the system is symmetric under reflection, blocks can be mirrored at equal size, otherwise the environment block For many problems, infinite-system DMRG does not is shrunk to some minimum and then regrown. A com- yield satisfactory answers: The idea of simulating the fi- plete shrinkage and growth sequence for both blocks is nal system size cannot be implemented well by a small called a sweep. environment block in the early DMRG steps. DMRG is One sweep takes about two ͑if reflection symmetry usually canonical, working at fixed particle numbers for holds͒ or four times the CPU time of the starting ͑ ͒ a given system size Sec. II.E . Electronic systems in infinite-system DMRG calculation. For better perfor- which the particle number is growing during system mance, there is a simple, but powerful prediction algo- growth to maintain particle density approximately con- rithm ͑see Sec. II.I͒, which cuts down calculation times stant are affected by a lack of “thermalization” of the in finite-system DMRG by more than an order of mag- particles injected during system growth; t-J models with nitude. In fact, it will be seen that the speedup is larger a relatively small hole density or Hubbard models far the closer the current ͑ground͒ state is to the final, fully from half-filling or with complicated filling factors are converged result. In practice, one therefore starts run- particularly affected. The strong physical effects of im- ning the infinite-system DMRG with a rather small purities or randomness in the Hamiltonian cannot be number of states M , increasing it while running through accounted for properly by infinite-system DMRG, as the 0 the sweeps to some final M ӷM . The resulting slow- total Hamiltonian is not yet known at intermediate final 0 ing down of DMRG will be offset by the increasing per- steps. In systems with strong magnetic fields or close to a first-order transition one may be trapped in a metastable formance of the prediction algorithm. While there is no state favored for small system sizes, e.g., by edge effects. guarantee that finite-system DMRG is not trapped in Finite-system DMRG manages to eliminate these con- some metastable state, it usually finds the best approxi- cerns to a very large degree and to reduce the error mation to the ground state, and convergence is gauged ͑almost͒ to the truncation error. The idea of the finite- by comparing results from sweep to sweep until they system algorithm is to stop the infinite-system algorithm stabilize. This may take from a few to several dozen at some preselected superblock length L which is kept sweeps, with electronic problems at incommensurate fill- fixed. In subsequent DMRG steps ͑Fig. 3͒, one applies ings and random potential problems needing the most. the steps of infinite-system DMRG, but instead of simul- In rare cases it has been observed that seemingly con- taneous growth of both blocks, growth of one block is verged finite-system results are again suddenly improved accompanied by shrinkage of the other block. Reduced after some further sweeps without visible effect have basis transformations are carried out only for the grow- been carried out, showing a metastable trapping. It is ing block. Let the environment block grow at the ex- therefore advisable to carry out additional sweeps and pense of the system block; to describe it, system blocks to judge DMRG convergence by carrying out runs for of all sizes and operators acting on this block, expressed various M. Where possible, choosing a clever sequence in the basis of that block, must have been stored previ- of finite-system DMRG steps may greatly improve the ously ͑at the infinite-system stage or previous applica- wave function. This has been successfully attempted in

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 267 momentum-space DMRG ͑Sec. VII.A͒ and quantum- formations preserve these block structures if one fixes chemistry DMRG ͑Sec. VII.B͒. total magnetization and/or total particle number for the To show the power of finite-system DMRG, consider superblock. Assume that block and site states can at a the calculation of the plaquette currents induced on a given DMRG step be labeled by a good quantum num- t-J ladder by imposing a source current on one edge of ber, say, particle number N. This is an essential prereq- the ladder ͑Schollwöck et al., 2003͒. Figure 4 shows how uisite ͑cf. translational invariance leading to momentum the plaquette currents along the ladder evolve from conservation; see below͒. As the total number of par- sweep to sweep. While they are perfectly converged af- ticles is fixed, we have Ntot=NmS +N␴S +N␴E +NmE. ter six to eight sweeps, the final result, a modulated ex- Equation ͑6͒ implies that only matrix elements ͗ S␴S͉␳͉ S␴S͘ ponential decay, is far from what DMRG suggests after m ˆ m˜ ˜ with NmS +N␴S =Nm˜ S +N␴˜ S can be non- the first sweep, let alone in the infinite-system algorithm zero. The density matrix thus has block structure, and its ͑not shown͒. eigenvectors from which the next block’s eigenbasis is Despite the general reliability of the finite-system al- formed can again be labeled by particle number N S ͑ ͒ m gorithm there might be relatively rare situations in +N␴S. Thus, for all operators, only dense blocks of non- which its results are misleading. It may, for example, be zero matrix elements have to be stored and considered. doubted whether competing or coexisting types of long- For superblock wave functions, a tensorial structure range order are well described by DMRG, as we shall emerges as total particle number and/or magnetization see that it produces a very specific kind of wave func- dictate that only product states with compatible quan- tion, the so-called matrix-product states ͑Sec. III.A͒. tum numbers ͑i.e., adding up to the fixed total͒ may have These states show either long-range order or, more typi- nonzero coefficients. cally, short-range correlations. In the case of competing The performance gains from implementing just the forms of long-range order, the infinite-system algorithm simple additive quantum numbers magnetization and might preselect one of them incorrectly, e.g., due to edge particle number are impressive. Both in memory and effects, and the finite-system algorithm would then be CPU time significantly more than an order of magnitude quite likely to fail to “tunnel” to the other, correct kind will typically be gained. of long-range order due to the local nature of the improvements to the wave function. Perhaps such a prob- 2. Continuous non-Abelian symmetries lem is at the origin of the current disagreement between state-of-the-art DMRG ͑Jeckelmann, 2002b, 2003b; Non-Abelian symmetries that have been considered ͑ ͒͑ Zhang, 2004͒ and quantum Monte Carb results ͑Sandvik are the quantum group symmetry SUq 2 Sierra and et al., 2003, 2004͒ on the extent of a bond-ordered wave Nishino, 1997͒,SU͑2͒ spin symmetry ͑McCulloch and phase between a spin-density-wave phase and a charge- Gulacsi, 2000, 2001, 2002; Wada, 2000; Xiang et al., density-wave phase in the half-filled extended Hubbard 2001͒, and the charge SU͑2͒ pseudospin symmetry ͑Mc- model, but no consensus has as yet emerged. Culloch and Gulacsi, 2002͒, which holds for the bipartite Hubbard model without field ͑Yang and Zhang, 1990͒: + ͚ ͑ ͒i † † − ͚ its generators are given by I = i −1 c↑ic↓i , I = i E. Symmetries and good quantum numbers ͑ ͒i z ͚ 1 ͑ ͒ −1 c↓ic↑i, and I = i 2 n↑i +n↓i −1 . A big advantage of DMRG is that the algorithm con- Implementation of non-Abelian symmetries is much serves a variety of, but not all, symmetries and good more complicated than that of Abelian symmetries, the ͑ ͒ quantum numbers of the Hamiltonians. They may be best-performing one McCulloch and Gulacsi, 2002 exploited to reduce storage and computation time and building on Clebsch-Gordan transformations and the to thin out Hilbert space by decomposing it into a sum elimination of quantum numbers via the Wigner-Eckart of sectors. DMRG is optimal for studying the lowest- theorem. It might be crucial for obtaining high-quality lying states of such sectors, and refining any such decom- results in applications with problematically large trun- position immediately gives access to more low-lying cated weight such as in two dimensions, where the trun- states. Hence the use of symmetries is standard practice cated weight is cut by several orders of magnitude com- in DMRG implementations. Symmetries used in DMRG pared to implementations using only Abelian fall into three categories, continuous Abelian, continu- symmetries. The additional increase in performance is ͑ ͒ ous non-Abelian, and discrete. comparable to that due to use of U 1 symmetries over using no symmetries at all.

1. Continuous Abelian symmetries 3. Discrete symmetries The most frequently implemented symmetries in ͑ ͒ I shall formulate these for a fermionic Hamiltonian of DMRG are the U 1 symmetries leading to total magne- 1 z ͑ spin-2 particles. tization Stot and total particle number Ntot as good con- ͒ served quantum numbers. If present for some Hamil- ͑a͒ Spin-flip symmetry. If the Hamiltonian Hˆ is invari- tonian, all operators can be expressed in matrix form as ant under a general spin flip ↑↔↓, one may intro- dense blocks of nonzero matrix elements with all other ˆ ͟ ˆ matrix elements zero. These blocks can be labeled by duce the spin-flip operator P= iPi, which is imple- ˆ ͉ ͘ ͉ ͘ good quantum numbers. In DMRG, reduced basis trans- mented locally on an electronic site as Pi 0 = 0 ,

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ˆ ͉↑͘ ͉↓͘ ˆ ͉↓͘ ͉↑͘ ˆ ͉↑↓͘ ͉↑↓͑͘ cause momentum does not exist as a good quantum Pi = , Pi = , Pi =− fermionic ͒ ͓ ˆ ˆ ͔ number at the block level. Other DMRG variants with sign .As H,P =0, there are common eigenstates momentum as a good quantum number will be consid- of Hˆ and Pˆ with Pˆ ͉␺͘=±͉␺͘. ered in Sec. VII.A. ͑b͒ Particle-hole symmetry.IfHˆ is invariant under a particle-hole transformation, ͓Hˆ ,Jˆ͔=0 for the ˆ ͟ ˆ F. Energies: Ground states and excitations particle-hole operator J= iJi, where the local op- eration is given by Jˆ ͉0͘=͉↑↓͘, Jˆ ͉↑͘=͑−1͒i͉↓͘, i i As a method working in a subspace of the full Hilbert ˆ ͉↓͘ ͑ ͒i͉↑͘ ˆ ͉↑↓͘ ͉ ͘ Ji = −1 , Ji = 0 , introducing a distinc- space, DMRG is variational in energy. It provides upper tion between odd and even site sublattices, and ei- bounds for energies E͑M͒ that improve monotonically genvalues Jˆ͉␺͘=±͉␺͘. with M, the number of basis states in the reduced Hil- bert space. Two sources of errors have been identified, ͑c͒ Reflection symmetry (parity). In the case of the environmental error due to inadequate environment reflection-symmetric Hamiltonians with open blocks, which can be amended using the finite-system boundary conditions, parity is a good quantum DMRG algorithm, and the truncation error. Assuming number. The spatial reflection symmetry operator that the environmental error ͑which is hard to quantify ˆ C2 acts globally. Its action on a DMRG product theoretically͒ has been eliminated, i.e., finite-system state is given by DMRG has reached convergence after sufficient sweeping, the truncation error remains to be analyzed. Rerun- ˆ ͉ S␴S␴E E͘ ͑ ͒␩͉ E␴E␴S S͑͒͘ C2 m m = −1 m m 23 ning the calculation of a system of size L for various M, ␩ ͑ with a fermionic phase determined by = NmS one observes for sufficiently large values of M that to a ͒͑ ͒ +N␴S NmE +N␴E . Again, eigenvalues are given by good approximation the error in energy per site scales ˆ ͉␺͘ ͉␺͘ linearly with the truncated weight, C2 =± . Parity is not a good quantum number accessible to finite-system DMRG, except for the ͓ ͑ ͒ ͔ ϰ ⑀ ͑ ͒ DMRG step with identical system and E M − Eexact /L ␳, 25 environment. All three symmetries commute, and an arbitrary nor- with a nonuniversal proportionality factor typically of malized wave function can be decomposed into eigen- order 1–10, sometimes more ͓this observation goes back states for any desired combination of eigenvalues ±1 by to White and Huse ͑1993͒; Legeza and Fáth ͑1996͒ give a −10 successively calculating careful analysis͔.As⑀␳ is often of order 10 or less, ͑ ͒ 1 DMRG energies can thus be extrapolated using Eq. 25 ͉␺ ͘ ͉͑␺͘ ˆ ͉␺͒͘ ͑ ͒ ± = ± O , 24 quite reliably to the exact M=ϱ result, often almost at 2 machine precision. The precision desired imposes the ˆ ˆ ˆ ˆ size of M, which for spin problems is typically in the where O=P,J,C2. Parity may be easily implemented by starting the su- lower hundreds, for electronic problems in the upper perblock diagonalization from a trial state ͑Sec. II.I͒ that hundreds, and for two-dimensional and momentum- has been made ͑anti͒symmetric under reflection by Eq. space problems in the lower thousands. As an example ͑ ͒ ͓ ˆ ˆ ͔ of DMRG precision, consider the results obtained for 24 .As H,C2 =0, the final eigenstate will have the the ground-state energy per site of the S=1 antiferro- same reflection symmetry. Spin-flip and particle-hole magnetic Heisenberg chain in Table I. symmetries may be implemented in the same fashion, Experiments relate to energy differences or relative provided Pˆ and Jˆ are generated by DMRG; as they are ordering of levels. This raises the question of calculating products of local operators, this can be done along the excitations in DMRG. Excitations are easiest to calcu- lines of Sec. II.G. Another way of implementing these late if they are the ground state in some other symmetry two local symmetries is to realize that the argument sector of the Hamiltonian and are thus algorithmically given for magnetization and particle number as good no different from true ground states. If, however, we are density-matrix quantum numbers carries over to the interested in some higher-lying state in a particular Hil- spin-flip and particle-hole eigenvalues such that they can bert space sector, DMRG restricts us to the lowest-lying also be implemented as block labels. such states because of the restrictions of large sparse- matrix diagonalizations ͑Sec. II.I͒. Excited states have to be “targeted” in the same way as the ground state. This 4. Missing symmetries means that they have to be calculated for the superblock Momentum is not a good quantum number in real- at each iteration and to be represented optimally, i.e., space DMRG, even if periodic boundary conditions reduced basis states have to be chosen such that the ͑Sec. II.H͒ are used and translational invariance holds. error in the approximation is minimized. It can be This is because the allowed discrete momenta change shown quite easily that this amounts to considering the during the growth process, and, more importantly, be- eigenstates of the reduced density matrix

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 269

␳ ␣ ͉␺ ͗͘␺ ͉ ͑ ͒ TABLE I. Ground-state energies per site E0 of the S=1 iso- ˆ S =TrE͚ i i i , 26 i tropic antiferromagnetic Heisenberg chain for M block states kept and associated truncated weight ⑀␳͑M͒. Adapted from ͉␺ ͑͘ White and Huse ͑1993͒. where the sum runs over all targeted states i ground ͒ ͚ ␣ and a few excited states and i i =1. There is no known ␣ ME͑M͒ ⑀␳͑M͒ optimal choice for the i, but it seems empirically to be 0 most reasonable to weigh states roughly equally. To 36 −1.40148379810 5.61ϫ10−8 maintain a good overall description for all targeted 72 −1.40148403632 3.42ϫ10−10 states at a ﬁxed M, typically fewer than ﬁve or so excited ϫ −11 states are targeted. Best results are of course obtained 110 −1.40148403887 1.27 10 ϫ −13 by running DMRG for each energy level separately. 180 −1.401484038970 1.4 10

G. Operators and correlations In the case of two-point correlators, two cases have to be distinguished, whether the locations i and j of the In general, we shall also be interested in evaluating contributing one-point operators act on different blocks ͑ ͒ ͑ ͒ ͑ ͒ static n-point correlators Oˆ n =Oˆ 1 Oˆ 1 with respect or on the same block at the last step. This expression is i1¯in i1 ¯ in ͑ 3͒ to some eigenstate of the Hamiltonian. The most rel- again evaluated efficiently by splitting it into two O M evant cases are n=1 for density or local magnetization matrix-matrix multiplications. and n=2 for two-point density-density, spin-spin, or If they act on different blocks, one follows through ͗ ͘ ͗ + −͘ ͗ † ͘ the procedure for one-point operators, yielding creation-annihilation correlators, ninj , Si Sj ,or ci cj . ͗ S͉ ˆ ͉ S͘ ͗ E͉ ˆ ͉ E͘ Let us first consider the case n=1. The iterative m Oi m˜ and m Oj m˜ . The evaluation is done by growth strategy of DMRG imposes a natural three-step the following modification of the one-point case: procedure of initializing, updating, and evaluating corr- ͗␺͉ ˆ ˆ ͉␺͘ ͗␺͉ S␴S␴E E͗͘ S͉ ˆ ͉ S͘ elators. OiOj = ͚ m m m Oi m˜ mSm˜ S␴S␴EmEm˜ E ͑1͒ Initialization. Oˆ acts on site i. When site i is added i ϫ͗ E͉ ˆ ͉ E͗͘ S␴S␴E E͉␺͘ ͑ ͒ ᐉ ͗␴͉ ˆ ͉␴͘ m Oj m˜ m˜ m˜ . 30 to a block of length −1, Oi ˜ is evaluated. With ͕͉mᐉ͖͘ being the reduced basis of the new block in- If they act on the same block, it is wrong to obtain ͕͉ ͖͘ corporating site i and mᐉ−1 that of the old block, ͗m͉Oˆ Oˆ ͉m˜ ͘ through one has i j ͗ ͉ ˆ ˆ ͉ ͘ ͗ ͉ ˆ ͉ ͗͘ ͉ ˆ ͉ ͑͘ ͒ ͑ ͒ m OiOj m˜ = ͚ m Oi mЈ mЈ Oj m˜ false , 31 ͗ ᐉ͉ ˆ ͉ ˜ ᐉ͘ ͚ ͗ ᐉ͉ ᐉ ␴͗͘␴͉ ˆ ͉␴˜ ͗͘ ᐉ ␴˜ ͉ ˜ ᐉ͘ m Oi m = m m −1 Oi m −1 m . mЈ ␴␴ mᐉ−1 ˜ ͑27͒ where an approximate partition of unity has been in- serted. This partition of unity is close to 1 to a very good ͗ ͉ ␴͘ mᐉ mᐉ−1 is already known from the density- approximation only when it is used to project the tar- matrix eigenstates. geted wave function, for which it was constructed, but ͑2͒ Update. At each further DMRG step, an approxi- not in general. mate basis transformation for the block containing Instead, such operators have to be built as a compound object at the moment when they live in a product ˆ ͕͉ ͖͘ ͕͉ ͖͘ the site in which Oi acts from mᐉ to mᐉ+1 oc- Hilbert space, namely, when one of the operators acts on ˆ ͑ ᐉ ͒ curs. As Oi does not act on the new site, the opera- a block of length −1 , the other on a single site that tor transforms as is being attached to the block. Then we know

͗mᐉ ͉Oˆ ͉m˜ ᐉ ͘ and ͗␴͉Oˆ ͉␴˜ ͘ and within the reduced ͗ ͉ ˆ ͉ ͘ ͗ ͉ ␴͗͘ ͉ ˆ ͉ ͘ −1 i −1 j mᐉ+1 O m˜ ᐉ+1 = ͚ mᐉ+1 mᐉ mᐉ O m˜ ᐉ bases of the block of length ᐉ mᐉm˜ ᐉ␴ ϫ͗ ␴͉ ͘ ͑ ͒ ͗ ͉ ˆ ˆ ͉ ͘ ͗ ͉ ␴͗͘ ͉ ˆ ͉ ͘ m˜ ᐉ m˜ ᐉ+1 . 28 mᐉ OiOj m˜ ᐉ = ͚ mᐉ mᐉ−1 mᐉ−1 Oi m˜ ᐉ−1 ␴␴ mᐉ−1m˜ ᐉ−1 ˜ This expression is evaluated efficiently by splitting it 3 ϫ ͗␴͉ ˆ ͉␴͗͘ ␴͉ ͑͒͘ into two O͑M ͒ matrix-matrix multiplications. Oj ˜ m˜ ᐉ−1 ˜ m˜ ᐉ 32 ͑3͒ Evaluation. After the last DMRG step is exact. Updating and ﬁnal evaluation for compound ͗ S␴S␴E E ͉␺͘ ͗ ˆ ͘ operators proceed as for a one-point operator. m m is known and Oi reads, assuming One-point operators show similar convergence behav- Oˆ to act on some site in the system block, i ior in M to that of local energy, but at reduced precision. While there is no exact variational principle for two- ͗␺͉Oˆ ͉␺͘ = ͚ ͗␺͉mS␴S␴EmE͗͘mS͉Oˆ ͉m˜ S͘ i i point correlations, derived correlation lengths are mono- S ˜ S␴S␴E E m m m tonically increasing in M, but always underestimated. ϫ ͗m˜ S␴S␴EmE͉␺͘. ͑29͒ The underestimation can actually be quite severe, of the

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 270 U. Schollwöck: Density-matrix renormalization group order of several percent, while the ground-state energy doubles, computation time quadruples. Numerical sta- has already converged almost to machine precision. bility is assured because the density matrix remains Her- As DMRG always generates wave functions with ex- mitian. This generalization has been used on a ring with ponentially decaying correlations ͑Sec. III.A͒, power-law interactions and impurities to determine the current decays of correlations are problematic. Andersson et al. I͑␾͒, which is neither sawtoothlike nor sinusoidal ͑1999͒ show that for free fermions the resulting correla- ͑Meden and Schollwöck, 2003a͒, and thus to obtain the tion function mimics the correct power law on short- conductance of interacting nanowires ͑Meden and length scales increasing with M, but is purely exponen- Schollwöck, 2003b͒. For open boundary conditions, tial on larger scales. However, the derived correlation complex-valued DMRG has been used to introduce in- length diverges roughly as M1.3, such that for M→ϱ ﬁnitesimal current source terms for time-reversal sym- criticality is recovered. metry breaking in electronic ladder structures ͑Scholl- wöck et al., 2003͒.

H. Boundary conditions I. Large sparse-matrix diagonalization From a physical point of view, periodic boundary conditions are normally highly preferable to the open 1. Algorithms boundary conditions used so far for studying bulk prop- The key to DMRG performance is the efficient diago- erties, as surface effects are eliminated and finite-size nalization of the large sparse-superblock Hamiltonian. extrapolation works for much smaller system sizes. In All large sparse-matrix diagonalization algorithms itera- particular, open boundaries introduce charge or magne- tively calculate the desired eigenstate from some ͑ran- tization oscillations not always easily distinguishable ͒ ͓ dom starting state through successive costly matrix- from true charge-density waves or dimerization see vector multiplications. In DMRG, the two algorithms White et al. ͑2002͒ for a thorough discussion on using ͑ ͔ typically used are the Lanczos method Cullum and Wil- bosonization to make the distinction . loughby, 1985; Golub and van Loan, 1996͒ and the However, it was observed early in the history of Jacobi-Davidson method ͑Sleijpen and van der Vorst, DMRG that ground-state energies for a given M are 1996͒. The pleasant feature of these algorithms is that much less precise in the case of periodic boundary con- for an NϫN-dimensional matrix it takes only a much ditions than for open boundary conditions, with differ- ˜ Ӷ ences in the relative errors of up to several orders of smaller number N N of iterations, so that iterative ap- magnitude. This is reflected in the spectrum of the re- proximations to eigenvalues converge very rapidly to the duced density matrix, which decays much more slowly maximum and minimum eigenvalues of Hˆ at machine ͑see Sec. III.B͒. However, it has been shown by Verstra- precision. With slightly more effort other eigenvalues at ete, Porras, and Cirac ͑2004͒ that this is an artifact of the the edge of the spectrum can also be computed. Typical conventional DMRG setup and that, at some algorith- values for the number of iterations ͑matrix-vector mul- mic cost, essentially the same precision for a given M tiplications͒ in DMRG calculations are of the order of can be achieved for periodic as for open boundary con- 100. ditions ͑Sec. III.A͒. To implement periodic boundary conditions in the infinite-system DMRG algorithm, the block-site struc- 2. Representation of the Hamiltonian ture is typically changed as shown in Fig. 5; other setups Naively, the superblock Hamiltonian is a however, are also feasible and used. For finite-system 2 2 DMRG, the environment block grows at the expense of M Nsite-dimensional matrix. As matrix-vector multipli- ͑ ͒2 the system block, then the system block grows back, till cations scale as dimension , DMRG would seem to be ͑ 4͒ ͑ 3͒ the configuration of the end of the infinite-system algo- an algorithm of order O M . In reality, it is only O M , rithm is reached. This is repeated with changed roles as typical tight-binding Hamiltonians act as sums over ͑unless translational invariance allows identification of two-operator terms: Assuming nearest-neighbor interac- system and environment at equal size͒. A minor techni- tions, the superblock Hamiltonian decomposes as cal complication arises from the fact that blocks grow at both ends at various steps of the algorithm. ˆ ˆ ˆ ˆ ˆ ˆ ͑ ͒ H = HS + HS• + H•• + H•E + HE. 33 Beyond the usual advantages of periodic boundary conditions, combining results for periodic and antiperi- ˆ ˆ odic boundary conditions allows the calculation of re- HS and HE contain all interactions within the system and sponses to boundary conditions such as spin stiffness, environment blocks, respectively, and are hence of di- phase sensitivity ͑Schmitteckert and Eckern, 1996͒,or mension M. Multiplying them to some state ͉␺͘ is of ͑ ͒ 3 2 ˆ ˆ superfluid density Rapsch et al., 1999 . Periodic and an- order M Nsite. HS• and H•E contain interactions between † † tiperiodic boundary conditions cL+1=±c1 are special blocks and the neighboring sites and hence are of di- † + − cases of the general complex boundary condition cL+1 mension MNsite. Consider a typical interaction SᐉSᐉ+1, i␾ † ᐉ ᐉ =e c1. Implementing the latter is a tedious but straight- where is the last site of the block and +1 a single site. forward generalization of real-valued DMRG; memory Then

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 271

3. Eigenstate prediction

Providing a good guess for the final eigenstate as starting state of the iterative diagonalization allows for arbitrary cutdowns in the number of iterations in the iterative diagonalization procedures. For both the infinite-system and the finite-system DMRG it is possible to provide starting states that often have overlap Ϸ1 with the final state, leading to a dra- matic reduction of iterations, often down to fewer than 10, speeding up the algorithm by about an order of magnitude. In infinite-system DMRG, the physical system FIG. 5. Typical system and environment growth for periodic changes from step to step. It seems intuitive that for a boundary conditions in the infinite-system algorithm. very long system, the composition of the ground state from block and site states may only weakly depend on ͗ S␴S͉ + − ͉ S␴S͘ ͗ S͉ +͉ S͗͘␴S͉ − ͉␴S͑͒͘ m SᐉSᐉ+1 m˜ ˜ = m Sᐉ m˜ Sᐉ+1 ˜ 34 its length, such that the ground-state coefficients remain almost the same under system growth. One might there- ͉␺͘ and multiplying this term to is best carried out in a fore simply use the old ground state as a prediction two-step sequence: the expression state. This fails; while the absolute values of coefficients hardly change from step to step in long systems, the ͗ ␴␶ ͉␾͘ ͗ ␴͉ + −͉ Ј␴Ј͗͘ Ј␴Ј␶ ͉␺͘ ͑ ͒ m n = ͚ m S S m m n , 35 block basis states are fixed by the density-matrix diago- Ј␴Ј m nalization only up to the sign, such that the signs of the ͑ 3 3 ͒ coefficients are effectively random. Various ways of fix- which is of order O M Nsite for the determination of all state coefficients, is decomposed as ing these random signs have been proposed ͑Scholl- wöck, 1998; Qin and Lou, 2001; Sun et al., 2002͒. ͗mЈ␴␶n͉␯͘ = ͚ ͗␴͉S−͉␴Ј͗͘mЈ␴Ј␶n͉␺͘, ͑36͒ In the case of the finite-system DMRG, the physical ␴Ј system does not change from DMRG step to DMRG step, just the structure of the effective Hilbert space ͑ 2 3 ͒ of order O M Nsite , and changes. White ͑1996b͒ has given a prescription for how to predict the ground state expressed in the block-site ͗ ␴␶ ͉␾͘ ͚ ͗ ͉ +͉ Ј͗͘ Ј␴␶ ͉␯͘ ͑ ͒ m n = m S m m n , 37 structure of the next DMRG step. If basis transforma- Ј m tions were not incomplete, one could simply transform ͑ 3 2 ͒ the ground state from one basis to the next to obtain a of order O M Nsite , where an order of Nsite is saved, prediction state. The idea is to do this even though the important for large Nsite. The Hamiltonian is never explicitly constructed. Such a decomposition is crucial transformation is incomplete. The state obtained turns ˆ out to be an often excellent approximation to the true when block-block interactions HSE appear for longer- ranged interactions. Considering again S+S−, with S− ground state. now acting on the environment block, factorization of Let us assume that we have a system with open the Hamiltonian again allows us to decompose the origi- boundary conditions, with a current system block of nal term length ᐉ and an environment block of length L−ᐉ−2. ͗ ␴ ␴ ͉␺͘ The target state is known through mᐉ ᐉ+1 ᐉ+2mL−ᐉ−2 . ͗m␴␶n͉␾͘ = ͚ ͗mn͉S+S−͉mЈnЈ͗͘mЈ␴␶nЈ͉␺͘, ͑38͒ Now assume that the system block is growing at the ex- mЈnЈ pense of the environment block, and we wish to predict the coefficients ͗mᐉ ␴ᐉ ␴ᐉ m ᐉ ͉␺͘, where in both ͑ 4 2 ͒ +1 +2 +3 L− −3 which is of inconveniently high-order O M Nsite for the DMRG steps ͉␺͘ is describing the same quantum- determination of all state coefficients, as ͗ ␴ ͉ ͘ mechanical state. As we also know mᐉ ᐉ+1 mᐉ+1 from ͗ ␴ ͉ ͘ ͗mЈ␴␶n͉␯͘ = ͚ ͗n͉S−͉nЈ͗͘mЈ␴␶nЈ͉␺͘, ͑39͒ the current DMRG iteration and mL−ᐉ−3 ᐉ+3 mL−ᐉ−2 nЈ from some previous DMRG iteration, this allows us to carry out two incomplete basis transformations: First, we ͑ 3 2 ͒ of order O M Nsite , and transform system block and site to the new system block, giving ͗m␴␶n͉␺͘ = ͚ ͗m͉S+͉mЈ͗͘mЈ␴␶n͉␯͘, ͑40͒ mЈ ͗ ␴ ͉␺͘ ͗ ͉ ␴ ͘ of order O͑M3N2 ͒, such that an order of M is saved mᐉ+1 ᐉ+2mL−ᐉ−2 = ͚ mᐉ+1 mᐉ ᐉ+1 site m ␴ and the algorithm is generally of order O͑M3͒. This is ᐉ ᐉ+1 ϫ͗ ␴ ␴ ͉␺͘ ͑ ͒ essential for its performance. mᐉ ᐉ+1 ᐉ+2mL−ᐉ−2 ; 41

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 272 U. Schollwöck: Density-matrix renormalization group second, we expand the current environment block into a to the spin-spin correlations were determined by Hall- product-state representation of a block with one site less ͑ ͒ 1 berg et al. 1996 . Quasiperiodic S= 2 chains were con- and a site: sidered by Hida ͑1999b, 2000͒ and the case of transverse fields by Hieida et al. ͑2001͒. ͗mᐉ ␴ᐉ ␴ᐉ m ᐉ ͉␺͘ = ͚ ͗m ᐉ ␴ᐉ ͉m ᐉ ͘ +1 +2 +3 L− −3 L− −3 +3 L− −2 Bilinear-biquadratic S=1 spin chains have been exten- mL−ᐉ−2 sively studied ͑Bursill et al., 1994; Schollwöck, Jolicœur, ϫ͗mᐉ ␴ᐉ m ᐉ ͉␺͘. ͑42͒ +1 +2 L− −2 and Garel, 1996; Sato, 1998͒ as well as the effect of The calculation involves two O͑M3͒ matrix multiplica- Dzyaloshinskii-Moriya interactions ͑Zhao et al., 2003͒. tions, and works independently of any assumptions for Important, experimentally relevant generalizations of the underlying model. It relies on the fact that by con- the Heisenberg model are obtained by adding frustrat- struction our incomplete basis transformations are al- ing interactions or dimerization, the latter modeling ei- most exact when applied to the target state͑s͒. ther static lattice distortions or phonons in the adiabatic limit. The ground-state phase diagrams of such systems have been extensively studied.1 DMRG has been instru- J. Applications mental in the discovery and description of gapped and gapless chiral phases in frustrated spin chains ͑Kaburagi In this section, I want to give a very brief overview of et al., 1999; Hikihara, Kaburaga, et al., 2000a; Hikihara, applications of standard DMRG, which will not be cov- ͒ ered in more detail in later sections. 2001, 2002; Hikihara et al., 2001a, 2001b . Critical expo- From the beginning, there has been a strong focus on nents for a supersymmetric spin chain were obtained by ͑ ͒ one-dimensional Heisenberg models, in particular the Senthil et al. 1999 . S=1 case, in which the Haldane gap ⌬=0.41052J was As a first step towards two dimensions and due to the determined to five-digit precision by White and Huse many experimental realizations, spin ladders have been ͑1993͒. Other authors have considered the equal-time among the first DMRG applications going beyond structure factor ͑Sørensen and Affleck, 1994a, 1994b; simple Heisenberg chains, starting with White et al. Sieling et al., 2000͒ and focused on topological order ͑1994͒. In the meantime, DMRG has emerged as a stan- ͑Lou et al., 2002, 2003; Qin et al., 2003͒. There has been dard tool for the study of spin ladders ͑White, 1996a; a particular emphasis on the study of the existence of Legeza and Sólyom, 1997; Wang, 2000; Fath et al., 2001; 1 ͑ Hikihara and Furusaki, 2001; Trumper and Gazza, 2001; free S= 2 end spins in such chains White and Huse, 1993; Batista et al., 1998, 1999; Polizzi et al., 1998; Hall- Zhu et al., 2001; Wang et al., 2002; Kawaguchi et al., berg et al., 1999; Jannod et al., 2000͒, in which the open 2003͒. A focus of recent interest has been the effect of boundary conditions of DMRG are very useful; for the cyclic exchange interactions on spin ladders ͑Sakai and study of bulk properties authors typically attach real S Hasegawa, 1999; Honda and Horiguchi, 2001; Nunner et 1 ͒ = 2 end spins that bind to singlets with the effective ones, al., 2002; Hikihara et al., 2003; Läuchli et al., 2003 . removing degeneracies due to boundary effects. Soon, Among other spin systems, ferrimagnets have been studies were carried over to the S=2 Heisenberg chain, studied by Pati, Ramesha, and Sen ͑1997a, 1997b͒, Tone- in which the first reliable determination of the gap ⌬ gawa et al. ͑1998͒, Hikihara, Tonegawa, et al. ͑2000͒, and =0.085͑5͒ and the correlation length ␰Ϸ50 was provided Langari et al. ͑2000͒. One-dimensional toy models of the by Schollwöck and Jolicœur ͑1995͒ and confirmed and kagomé lattice have been investigated by Pati and Singh enhanced in other works ͑Schollwöck, Golinelli, and ͑1999͒, Waldtmann et al. ͑2000͒, and White and Singh Jolicœur, 1996; Schollwöck and Jolicœur, 1996; Qin, ͑2000͒, whereas supersymmetric spin chains have been Wang, and Yu, 1997; Aschauer and Schollwöck, 1998; considered by Marston and Tsai ͑1999͒. Wang et al., 1999; Wada, 2000; Capone and Caprara, Spin-orbit chains with spin and pseudospin degrees of ͒ ͑ ͒ 2001 . The behavior of Haldane integer spin chains in freedom are the large-U limit of the two-band Hubbard ͑staggered͒ magnetic fields was studied by Sørensen and model at quarter filling and are hence an interesting and Affleck ͑1993͒, Lou et al. ͑1999͒, Ercolessi et al. ͑2000͒, important generalization of the Heisenberg model. and Capone and Caprara ͑2001͒. In general, DMRG has DMRG has allowed full clarification of the rich phase been very useful in the study of plateaus in magnetiza- ͑ tion processes of spin chains ͑Tandon et al., 1999; Citro diagram Pati et al., 1998; Yamashita et al., 1998, 2000a, ͒ et al., 2000; Lou, Qin, Ng, et al., 2000; Lou, Qin, and Su, 2000b; Itoi et al., 2000; Pati and Singh, 2000 . 2000; Yamamoto et al., 2000; Kawaguchi et al., 2002; DMRG in its finite-system version has also been very Silva-Valencia and Miranda, 2002; Hida, 2003͒. successful in the study of systems with impurities or ran- The isotropic half-integer spin Heisenberg chains are domness, in which the true ground state can be found critical. The logarithmic corrections to the power law of 1 spin-spin correlations in the S= 2 chain were first consid- 1 ͑ ͒ ͑ ͒ Such studies include those of Bursill et al. 1995 ; Kolezhuk ered by Hallberg et al. 1995 , later by Hikihara and Fu- ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ et al. 1996, 1997 ; Pati et al. 1996 ; White and Affleck 1996 ; rusaki 1998 , Tsai and Marston 2000 , Shiroishi et al. Pati, Chitra, et al. ͑1997͒; Uhrig et al. ͑1999a, 1999b͒; Maeshima ͑ ͒ ͑ ͒ 3 2001 , and Boos et al. 2002 . For the S= 2 case, the and Okunishi ͑2000͒; Itoi and Qin ͑2001͒; Kolezhuk and central charge c=1 and again the logarithmic corrections Schollwöck ͑2002͒.

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 273 with reasonable precision.2 There has also been work on lene ͑Race et al. ͑2001, 2003͒, polyparaphenylene ͑Anu- edges and impurities in Luttinger liquids ͑Qin et al., sooya et al., 1997; Barford et al., 1998; Bursill and Bar- 1996; Qin, Fabrizio, et al., 1997; Bedürftig et al., 1998; ford, 2002͒, or polyenes ͑Bursill and Barford, 1999; Schönhammer et al., 2000͒. Barford et al., 2001; Zhang and George, 2001͒. Polymer The study of electronic models is somewhat more phonons have also been considered in the nonadiabatic complicated because of the larger number of degrees of case ͑Barford, Bursill, and Lavrentiev, 2002͒. There is freedom and because of the fermionic sign. However, closely related work on the Peierls-Hubbard model DMRG is free of the negative-sign problem of quantum ͑Pang and Liang, 1995; Jeckelmann, 1998; Otsuka, 1998; Monte Carlo and hence the method of choice for one- Anusooya et al., 1999͒. dimensional electronic models. Hubbard chains3 and Since the early days of DMRG history ͑Yu and White, Hubbard ladders4 have been studied in the entire range 1993͒ interest has also focused on the Kondo lattice, ge- of interactions as the precision of DMRG is found to neric one-dimensional structures of itinerant electrons depend only weakly on the interaction strength. The and localized magnetic moments, for both the 6 ͑ three-band Hubbard model has been considered by one-channel and two-channel cases Moreno et al., ͒ Jeckelmann et al. ͑1998͒ and Nishimoto, Jeckelmann, 2001 . Anderson models have been studied by Guerrero ͑ ͒ ͑ ͒ and Scalapino ͑2002͒. Similarly, authors have studied and Yu 1995 and Guerrero and Noack 1996, 2001 . both the t-J model on chains ͑Chen and Moukouri, 1996; The bosonic version of the Hubbard model has been ͑ ͒ White and Scalapino, 1997b; Mutou et al., 1998a; Dou- studied by Kühner and Monien 1998 , Kühner et al. ͑ ͒ ͑ ͒ blet and Lepetit, 1999; Maurel et al., 2001͒ and on lad- 2000 , and Kollath et al. 2004 , with emphasis on disor- ͑ der effects by Rapsch et al. ͑1999͒. Sugihara ͑2004͒ has ders Haywood et al., 1995; White and Scalapino, 1997a, ϩ 1998c; Rommer et al., 2000; Siller et al., 2001, 2002͒. used DMRG to study a bosonic 1 1-dimensional field The persistent current response to magnetic fluxes theory. Going beyond one dimension, two-dimensional elec- through rings has been studied by Byrnes et al. ͑2002͒ tron gases in a Landau level have been mapped to one- and Meden and Schollwöck ͑2003a͒; also it has been dimensional models suitable for DMRG ͑Shibata and studied in view of possible quasiexact conductance cal- Yoshioka, 2001, 2003; Bergholtz and Karlhede, 2003͒; culations ͑Meden and Schollwöck, 2003b; Molina et al., DMRG was applied to molecular iron rings ͑Normand et 2003͒. al., 2001͒ and has elucidated the lowest rotational band Even before the advent of the Hubbard model, a very of the giant Keplerate molecule Mo Fe ͑Exler and similar model, the Pariser-Parr-Pople model ͑Pariser and 72 30 Schnack, 2003͒. More generic higher-dimensional appli- Parr, 1953; Pople, 1953͒, accommodating both dimeriza- cations will be discussed later. tion effects and longer-range Coulomb interaction, had Revisiting its origins, DMRG can also be used to pro- been invented in quantum chemistry to study conjugated vide high-accuracy solutions in one-particle quantum polymer structures. DMRG has therefore been applied ͑ ͒ 5 mechanics Martín-Delgado et al., 1999 . As there is no to polymers by many authors, moving to more and entanglement in the one-particle wave function, the re- more realistic models of molecules, such as polydiacaty- duced basis transformation is formed from the MS lowest-lying states of the superblock projected onto the ͑ ͒ 2Impurities have been studied by Schmitteckert and Eckern system after reorthonormalization . It has been applied ͑1996͒, Wang and Mallwitz ͑1996͒, Martins et al. ͑1997͒, to an asymptotically free model in two dimensions ͑ ͒ Mikeska et al. ͑1997͒, Zhang et al. ͑1997͒, Laukamp et al. Martín-Delgado and Sierra, 1999 and modified for up ͑1998͒, Lou et al. ͑1998͒, Schmitteckert et al. ͑1998͒, Zhang, to three dimensions ͑Martín-Delgado et al., 2001͒. Igarashi, and Fulde ͑1998͒,Nget al. ͑2000͒, whereas other authors have focused on random interactions or fields: Hida ͑1996, 1997a, 1997b, 1999a͒; Hikihara et al. ͑1999͒; Juozapavi- III. DMRG THEORY cius et al. ͑1999͒; Urba and Rosengren ͑2003͒. 3Studies include Sakamoto and Kubo ͑1996͒; Daul and Noack DMRG practitioners usually adopt a quite pragmatic ͑1997, 1998, 2000͒; Lepetit and Pastor ͑1997͒; Zhang ͑1997͒; approach when applying this tool to the study of some ͑ ͒ ͑ ͒ ͑ ͒ Aligia et al. 2000 ; Daul 2000 ; Daul and Scalapino 2000 ; physical system. They consider the convergence of ͑ ͒ ͑ ͒ Lepetit et al. 2000 ; Maurel and Lepetit 2000 ; Nishimoto et DMRG results upon tuning the standard DMRG con- al. ͑2000͒; Aebischer et al. ͑2001͒; Jeckelmann ͑2002b͒. 4See, for example, Noack et al. ͑1994, 1995a, 1995b, 1997͒; trol parameters, system size L, size of the reduced block Vojta et al. ͑1999, 2001͒; Bonca et al. ͑2000͒; Daul, Scalapino, Hilbert space M, and the number of finite-system and White ͑2000͒; Weihong et al. ͑2001͒; Hamacher et al. sweeps, and judge DMRG results to be reliable or not. ͑2002͒; Marston et al. ͑2002͒; Scalapino et al. ͑2002͒; Schollwöck Beyond empiricism, in recent years a coherent theoreti- et al. ͑2003͒. 5These include Anusooya et al. ͑1997͒; Lepetit and Pastor ͑1997͒; Ramasesha et al. ͑1997, 2000͒; Shuai, Bredas, et al. 6See the work of Carruzo and Yu ͑1996͒; Shibata, Nishino, et ͑1997͒, Shuai, Pati, Bredas, et al. ͑1997͒; Shuai, Pati, Su, et al. al. ͑1996͒, Shibata, Ueda, et al. ͑1996͒, Caprara and Rosengren ͑1997͒; Boman and Bursill ͑1998͒; Kuwabara et al. ͑1998͒; ͑1997͒; Shibata, Sigrist, and Heeb ͑1997͒; Shibata, Tsvelik, and Shuai et al. ͑1998͒; Yaron et al. ͑1998͒; Bendazzoli et al. ͑1999͒; Ueda ͑1997͒; Shibata, Ueda, and Nishino ͑1997͒; Sikkema et al. Barford and Bursill ͑2001͒; Barford, Bursill, and Smith ͑2002͒; ͑1997͒; Wang ͑1998͒; McCulloch et al. ͑1999͒; Watanabe et al. Raghu et al. ͑2002͒. ͑1999͒; Garcia et al. ͑2000, 2002͒.

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 274 U. Schollwöck: Density-matrix renormalization group cal picture of the convergence properties and the algo- ˆ ͓␴ ͔ ˆ †͓␴ ͔ ͑ ͒ ͚ Ai i Ai i = I. 45 rithmic nature of DMRG has emerged, and it is fair to ␴ say that we now have good foundations of a DMRG i theory: DMRG generically produces a particular kind of A position-dependent unnormalized matrix-product ansatz state, known in statistical physics as a matrix- state for a one-dimensional system of size L is then product state; if it well approximates the true state of the given by system, DMRG will perform well. In fact, it turns out to be rewarding to reformulate DMRG in terms of varia- L tional optimization within classes of matrix-product ͉␺͘ ͩ͗␾ ͉ ˆ ͓␴ ͔͉␾ ͉ͪ͘␴͘ ͑ ͒ = ͚ L ͟ Ai i R , 46 states ͑Sec. III.A͒. In practice, DMRG performance is ͕␴͖ i=1 best studied by considering the decay of the eigenvalue ͗␾ ͉ ͉␾ ͘ spectrum of the reduced density matrix, which is fast for in which L and R are left and right boundary states one-dimensional gapped quantum systems, but generi- in the auxiliary state spaces located in the above visual- cally slow for critical systems in one dimension and all ization to the left of the ﬁrst and to the right of the last systems in higher dimensions ͑Sec. III.B͒. This renders site. They are used to obtain scalar coefﬁcients. Position- DMRG applications in such situations delicate at best. independent matrix-product states are obtained by mak- A very coherent understanding of these properties is ͑ ͒ ˆ ͓␴ ͔→ ˆ ͓␴ ͔ ing Eq. 44 position independent, Ai i A i . For now emerging in the framework of bipartite entangle- simplicity, we shall consider only those in the following. ment measures in quantum information theory. For periodic boundary conditions, boundary states are replaced by tracing the matrix product:

A. Matrix-product states L ͉␺͘ ͚ ͫ͟ ˆ ͓␴ ͔͉ͬ␴͘ ͑ ͒ Like conventional RG methods, DMRG builds on = Tr Ai i . 47 ͕␴͖ Hilbert-space decimation. There is, however, no Hamil- i=1 tonian flow to some fixed point and no emergence of The best-known matrix-product state is the valence- relevant and irrelevant operators. Instead, there is a flow bond-solid ground state of the bilinear-biquadratic S=1 to some fixed point in the space of the reduced density Affleck-Kennedy-Lieb-Tasaki Hamiltonian ͑Affleck et matrices. As has been pointed out by various authors al., 1987, 1988͒, in which M=2. ͑Östlund and Rommer, 1995; Martín-Delgado and Si- erra, 1996; Rommer and Östlund, 1997; Dukelsky et al., 1998; Takasaki et al., 1999͒, this implies that DMRG 2. Correlations in matrix-product states generates position-dependent matrix-product states ͑Fannes et al., 1989; Klümper et al., 1993͒ as block states. Consider two local bosonic ͑for simplicity; see Ander- However, there are subtle but crucial differences be- sson et al., 1999͒ operators Oˆ and Oˆ , acting on sites j tween DMRG states and matrix-product states j j+l ͑ and j+l, applied to the periodic boundary condition Takasaki et al., 1999; Verstraete, Porras, and Cirac, matrix-product state of Eq. ͑47͒. The correlator C͑l͒ 2004͒ that have important consequences regarding the ͗␺͉ ˆ ˆ ͉␺͘ ͗␺͉␺͘ variational nature of DMRG. = OjOj+l / is then found, with Tr X Tr Y =Tr͑X Y͒ and ͑ABC͒ ͑XYZ͒=͑A X͒͑B Y͒͑C Z͒, to be given by 1. Matrix-product states These are simple generalizations of product states of Tr O¯ ¯l−1O¯ ¯L−l−1 ͑ ͒ jI j+lI ͑ ͒ local states, which we take to be on a chain, C l = , 48 Tr I¯L ͉␴͘ ͉␴ ͘ ͉␴ ͘ ͉␴ ͘ ͑ ͒ = 1 2 ¯ L , 43 ͑ ˆ ͓␴ ͔ where we have used the following mapping Rommer obtained by introducing linear operators Ai i depend- and Östlund, 1997; Andersson et al., 1999͒ from an op- ing on the local state. These operators map from some ˆ M-dimensional auxiliary state space spanned by an or- erator O acting on the local state space to an thonormal basis ͕͉␤͖͘ to another M-dimensional auxil- M2-dimensional operator O¯ acting on products of auxiliary state space spanned by ͕͉␣͖͘: iary states ͉␣␤͘=͉␣͘ ͉␤͘: ˆ ͓␴ ͔ ͑ ͓␴ ͔͒ ͉␣͗͘␤͉ ͑ ͒ Ai i = ͚ Ai i ␣␤ . 44 ¯ ͗␴Ј͉ ˆ ͉␴͘ ˆ *͓␴Ј͔ ˆ ͓␴͔ ͑ ͒ ␣␤ O = ͚ O A A . 49 ␴␴Ј One may visualize the auxiliary state spaces to be lo- ͑ ͒ ͑ ͒ cated on the bonds i,i+1 and i−1,i . The operators Note that Aˆ * stands for Aˆ complex-conjugated only as are thus represented by MϫM matrices ͑A ͓␴ ͔͒␣␤; M i i ˆ † ͑ ͒ will be seen later to be the number of block states in opposed to A . Evaluating Eq. 48 in the eigenbasis of DMRG. We further demand for reasons explained be- the mapped identity, I¯, we find that in the thermody- low that namic limit L→ϱ

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2 M ␭ l block states for very short blocks is less than M. For C͑l͒ = ͚ c ͩ i ͪ exp͑− l/␰ ͒͑50͒ N˜ i ͉␭ ͉ i simplicity, I assume that Nsite=M and stop the recursion i=1 i at the shortest block of size N˜ that has M states, such ␰ ͉␭ ͉ ␭ ¯ that with i =−1/ln i . The i are the eigenvalues of I, and ˆ the ci depend on O. This expression holds because, due ͉mᐉ͘ = ͚ ͚ ͑Aᐉ͓␴ᐉ͔ A ˜ ͓␴ ˜ ͔͒ ˜ ¯ N+1 N+1 mᐉmN to Eq. ͑45͒, ͉␭ Ͼ ͉ഛ1 and ␭ =1 for the eigenstate m ˜ ␴ ˜ ,…,␴ᐉ i 1 1 N N+1 ͗␣␤͉␭ ͘=␦␣␤. Equation ͑45͒ is thus seen to ensure nor- 1 ϫ ͉ ͘ ͉␴ ␴ ͘ ͑ ͒ malizability of matrix-product states in the thermody- mN˜ N˜ +1 ¯ ᐉ , 54 namic limit. Generally, all correlations in matrix-product ͉ ˜ ͘ϵ͉␴ ␴ ˜ ͘ ͑ Þ ␭ ͒ where we have boundary-site states mN 1 ¯ N ; states are either long ranged if ci 0 for a i =1 or hence purely exponentially decaying. They are thus not suited for describing critical behavior in the thermodynamic ͉ ͘ ͑ ͓␴ ͔ ˜ ͓␴ ˜ ͔͒ mᐉ = ͚ Aᐉ ᐉ AN+1 N+1 mᐉ,͑␴ ␴ ˜ ͒ ␴ … ␴ ¯ 1¯ N limit. Even for gapped one-dimensional quantum sys- 1, , ᐉ tems their utility may seem limited, as the correlators ϫ͉␴ ␴ᐉ͘. ͑55͒ C͑l͒ of these systems are generically of a two- 1 ¯ dimensional classical Ornstein-Zernike form, A comparison to Eq. ͑46͒ shows that DMRG generates position-dependent MϫM matrix-product states as e−l/␰ C͑l͒ϳ , ͑51͒ block states for a reduced Hilbert space of M states; the ͱl auxiliary state space to a local state space is given by the Hilbert space of the block to which the local site is the ͑ ͒ whereas exponential decay as in Eq. 50 is typical of latest attachment. Combining Eqs. ͑4͒ and ͑55͒,wefind one-dimensional classical Ornstein-Zernike forms. How- that the superblock ground state of the full chain is the ever, matrix-product states such as the Affleck- variational optimum in a space spanned by products of ͑ ͒ Kennedy-Lieb-Tasaki AKLT ground state arise as two local states and two matrix-product states, quantum disorder points in general Hamiltonian spaces ͉␺͘ ␺ as the quantum remnants of classical phase transitions in = ͚ ͚ mS␴ ␴ mE ͕␴͖ L/2 L/2+1 two-dimensional classical systems; these disorder points mSmE are characterized by dimensional reduction of their cor- ϫ ͑A ͓␴ ͔ A ˜ ͓␴ ˜ ͔͒ S ͑␴ ␴ ˜ ͒ relations and typically characterize the qualitative prop- L/2−1 L/2−1 ¯ N+1 N+1 m , 1¯ N erties of subsets of the Hamiltonian space, which turns ϫ ͑A ͓␴ ͔ A ˜ ͓␴ ˜ ͔͒ E ͑␴ ˜ ␴ ͒ them into most useful toy models, as exemplified by the L/2+2 L/2+2 ¯ L−N L−N m , L+1−N¯ L AKLT state ͑Schollwöck, Jolicœur, and Garel, 1996͒. ϫ͉␴ ␴ ͘ ͑ ͒ 1 L , 56 Away from the disorder points, choosing increasingly ¯ large M as dimension of the ansatz matrices allows us to which I have written for the case of the two single sites model the true correlation form as a superposition of at the chain center; an analogous form holds for all exponentials for increasingly large l; even for power-law stages of the finite-system algorithm. correlations, this modeling works for not too long dis- For gapped quantum systems we may assume that for ᐉӷ␰ tances. blocks of length the reduced basis transformation becomes site independent, such that the ansatz matrix A generated by DMRG is essentially position independent 3. DMRG and matrix-product states in the bulk. At criticality, the finite-dimensional matrix- product state generated introduces some effective corre- To show that a DMRG calculation retaining M block lation length ͑growing with M͒. In fact, this has been ϫ states produces M M matrix-product states, Östlund verified numerically for free fermions at criticality, ͑ ͒ and Rommer 1995 considered the reduced basis trans- where the ansatz matrices for bulk sites converged ex- ᐉ formation to obtain a block of size , ponentially fast to a position-independent ansatz matrix, but where this convergence slowed down with M ͗mᐉ ␴ᐉ͉mᐉ͘ϵ͑Aᐉ͒ ␴ ϵ͑Aᐉ͓␴ᐉ͔͒ , ͑52͒ −1 mᐉ;mᐉ−1 ᐉ mᐉ;mᐉ−1 ͑Andersson et al., 1999͒. such that The effect of the finite-system algorithm can be seen from Eq. ͑56͒ to be a sequence of local optimization steps of the wave function that have two effects: on the ͉mᐉ͘ = ͚ ͑Aᐉ͓␴ᐉ͔͒ ͉mᐉ ͘ ͉␴ᐉ͘. ͑53͒ mᐉ;mᐉ−1 −1 ␴ ␺ S␴S␴E E mᐉ−1 ᐉ one hand, the variational coefficients m m are optimized, and on the other, a new, improved ansatz matrix The reduced-basis transformation matrices Aᐉ͓␴ᐉ͔ auto- is obtained for the growing block, using the improved matically obey Eq. ͑45͒, which here ensures that ͕͉mᐉ͖͘ is variational coefficients for a new reduced basis transfor- ͕͉ ͖͘ an orthonormal basis provided mᐉ−1 is one as well. mation. We may now use Eq. ͑53͒ for a backward recursion to In practical applications one observes that, even for ͉ ͘ ͉ ͘ ͑ express mᐉ−1 via mᐉ−2 and so forth. There is a con- translationally invariant systems with periodic boundary ceptually irrelevant͒ complication as the number of conditions and repeated applications of finite-system

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sweeps, the position dependency of the matrix-product 4. Variational optimization in matrix-product states state does not go away completely as it strictly should, ͑ ͒ ͑ ͒ indicating room for further improvement. Dukelsky et If we compare Eq. 46 to Eq. 56 , we see that the al. ͑1998͒ and Takasaki et al. ͑1999͒ have pointed out and DMRG state differs from a true matrix-product state in ͉␺͘ numerically demonstrated that finite-system DMRG its description of : The A matrices link auxiliary state ͑and TMRG; see Sec. VIII͒ results can be improved and spaces of a bond on the right of a site to those on the left better matrix-product states for translationally invariant for sites in the left block, but vice versa in the right Hamiltonians can be produced by switching, after con- block. This may be mended by a transposition. This ͉␴ ␴ ͘ vergence is reached, from the S••E scheme for the done, one may write the prefactors of 1 ¯ L as a true product of matrices by rewriting ␺ S␴ ␴ E as an finite-system algorithm to an S•E scheme and carrying m L/2 L/2+1m ϫ ͑⌿͓␴ ␴ ͔͒ out some more sweeps. The rationale is that the varia- M M matrix L/2 L/2+1 mS;mE. The remaining tional ansatz of Eq. ͑56͒ generates ͑after the Schmidt anomaly is, as pointed out above, that the formal translational invariance of this state is broken by the indexing decomposition and before truncation͒ ansatz matrices of of ⌿ by two sites, suggesting the modification of the dimension MN at the two local sites due to Eq. ͑20͒, site S••E scheme for the finite-system algorithm to an S•E whereas they are of dimension M at all other sites; this scheme as discussed above. However, as Verstraete, Por- introduces a notable position dependence, deteriorating ras, and Cirac ͑2004͒ have demonstrated, it is conceptu- the overall wave function. In the new scheme, the new ally and algorithmically worthwhile to rephrase DMRG ansatz matrix, without truncation, also has dimension M consistently in terms of matrix-product states right from ͓ ͑ ͔͒ the dimension of the environment in Eq. 20 , so that the beginning, thereby also abandoning the block con- the local state is not favored. The variational state now cept. approaches its global optimum without further trunca- To this end, they introduce ͑with the exception of the tion, just improvements of the ansatz matrices. first and last sites; see Fig. 6͒ two auxiliary state spaces The observation that DMRG produces variational of dimension M, aᐉ to the left and bᐉ to the right of site states formed from products of local ansatz matrices has ᐉ, such that on bond ᐉ one has auxiliary state spaces bᐉ inspired the construction of variational ansatz states for →H and aᐉ+1. They now consider maps from aᐉ bᐉ ᐉ the ground state of Hamiltonians ͑the recurrent varia- from the product of two auxiliary state spaces to the tional approach; see Martín-Delgado and Sierra in Pe- local state space, which can be written using matrices schel, Hallberg, et al., 1999͒ and the dominant eigenstate ͑ ͓␴͔͒ ͚ ͚ ͑ ͓␴͔͒ ͉␴ ͗͘␣ ␤ ͉ ͉␣͘ Aᐉ ␣␤:Aᐉ = ␴ᐉ ␣ᐉ␤ᐉ Aᐉ ␣␤ ᐉ ᐉ ᐉ . The and of transfer matrices ͑the tensor-product variational ap- ͉␤͘ are states of the auxiliary state spaces. On the first proach; Nishino et al., 2000, 2001; Maeshima et al., 2001; and last site, the corresponding schemes map only from Gendiar and Nishino, 2002; Gendiar et al., 2003͒. one auxiliary state space. These maps can now be used Closer in spirit to the original DMRG concept is the to generate a matrix-product state. To this purpose, Ver- product wave-function renormalization group ͑Nishino straete, Porras, and Cirac ͑2004͒ apply the string of maps and Okunishi, 1995; Hieida et al., 1997͒. This has been A1 A2 ¯ AL to the product of maximally en- applied successfully to the magnetization process of spin ͉␾ ͉͘␾ ͘ ͉␾ ͘ ͉␾ ͘ tangled states 1 2 ¯ L−1 , in which i chains in an external field where infinite-system DMRG =͚␤ ␣ ͉␤ ͉͘␣ ͘. The maximal entanglement, in the i= i+1 i i+1 is highly prone to metastable trapping ͑Sato and Akutsu, language of matrix-product states, ensures that the pref- 1996; Hieida et al., 1997, 2001; Okunishi et al., 1999a͒ ͉␴ ␴ ͘ ͓␴͔ actors of 1 ¯ L are given by products of A matri- and to the restricted solid-on-solid model ͑Akutsu and ces, hence this construction is a matrix-product state. Akutsu, 1998; Akutsu et al., 2001a, 2001b͒. While in Comparing this state to the representation of ͉␺͘ in Eq. DMRG the focus is to determine for each iteration ͑su- ͑56͒, one finds that the maps A are identical to the ͑pos- perblock size͒ the wave function numerically as precisely sibly transposed͒ basis transformation matrices A͓␴͔, as possible in order to derive the reduced basis transfor- with the exception of the position of the single sites: in mation, the product wave-function renormalization the S••E setup ͑bottom half of Fig. 6͒, there are no group operates directly on the A matrices itself: at each auxiliary state spaces between the two single sites, and ⌿͓␴ ␴ ͔ iteration, one starts with an approximation to the wave one map corresponds to ᐉ ᐉ+1 . In the S•E setup function and local ansatz matrices related to it by a ͑top half of Fig. 6͒, this anomaly disappears and cor- Schmidt decomposition. The wave function is then rectly normalized A͓␴ᐉ͔ can be formed from ⌿͓␴ᐉ͔. somewhat improved, by carrying out a few Lanczos The ͑finite-system͒ DMRG algorithm for the S•E steps at moderate effort, and again Schmidt decom- setup can now be reformulated in this picture as follows: posed. The transformation matrices thus obtained are sweeping forward and backward through the chain, one used to transform ͑improve͒ the local ansatz matrices, keeps for site ᐉ all A at other sites fixed and seeks the which in turn are combined with the decomposition ⌿ᐉ that minimizes the total energy. From this, one deter- ␺ weights to define a wave function for the next larger mines Aᐉ and moves to the next site, seeking ᐉ+1, and superblock. The final result is reached when both the so on, until all matrices have converged. In the final ansatz and transformation matrices become identical, as matrix-product state one evaluates correlators as in Eq. they should at the DMRG fixed point for reduced basis ͑48͒. It is important to note that in this setup there is no transformations. truncation error, as explained above in the language of

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Schmidt decomposition. Shifting the active site therefore grows exponentially with system size, restricting does not change the energy, and the next minimization DMRG to very small system sizes. ͑ ͒ can only decrease the energy or keep it constant . This All scenarios translate to classical systems of one ad- setup is therefore truly variational in the space of the ditional dimension, due to the standard quantum- states generated by the maps A and reaches a minimum classical mapping from d-to͑d+1͒-dimensional systems. of energy within that space ͑there is of course no guarantee of reaching the global minimum͒. By comparison, the setup S••E leads to a reduced basis transformation 1. DMRG applied to matrix-product states and always excludes two different auxiliary state spaces A state ͉␺˜ ͘ of the matrix-product form of Eq. ͑46͒ with from the minimization procedure. It is hence not strictly ˜ variational. dimension M can be written as In this setup the generalization to periodic boundary M˜ M˜ conditions is now easy. Additional auxiliary state spaces S E ͱ S E ͉␺˜ ͘ = ͚ ͉␺˜ ␣͉͘␺˜ ␣ ͘ → ͉␺͘ = ͚ w␣͉␺␣͉͘␺␣ ͘, ͑57͒ a1 and bL are introduced and maximally entangled as for ␣=1 ␣=1 all other bonds. There is now complete formal transla- ͑ ͒ where we have arbitrarily cut the chain into ͑left͒ system tional invariance of the ansatz see Fig. 7 . On this setup, ͑ ͒ one optimizes maps ͑matrices͒ A one by one, going for- and right environment with ward and backward. ͉␺˜ S͘ ͚ ͗␾ ͉͟ ͓␴ ͔͉␣͉͘␴S͘ ͑ ͒ ͑ ͒ ␣ = S A i , 58 Verstraete, Porras, and Cirac 2004 have shown that, ͕␴S͖ i෈S for a given M, they obtain roughly the same precision E S E for periodic boundary conditions as for open boundary and similarly ͉␺˜ ␣ ͘; ͉␺͘, ͉␺␣͘, and ͉␺␣ ͘ are the correspond- conditions. This compares extremely favorably with ing normalized states. An appropriate treatment of standard DMRG for periodic boundary conditions boundary sites is tacitly implied ͓cf. the discussion ͑ ͒ 2 ͑ ͔͒ ␳ where in the worst case up to M states are needed for before Eq. 55 . Then the density matrix ˆ S the same precision. ͚M˜ ͉␺S͗͘␺S͉ ˜ = ␣=1w␣ ␣ ␣ and has a finite spectrum of M nonvanishing eigenvalues. The truncated weight will thus be Ͼ ˜ B. Properties of DMRG density matrices zero if we choose M M for DMRG, as DMRG generates these states ͓see Eq. ͑55͔͒. In order to gain a theoretical understanding of In such cases, DMRG may be expected to become an DMRG performance, we now take a look at the prop- exact method up to small numerical inaccuracies. This ͑ erties of the reduced density matrices and their trunca- has been observed recurrently; Kaulke and Peschel in ͒ tion. Obviously, the ordered eigenvalue spectrum w␣ of Peschel, Hallberg, et al., 1999 provide an excellent ex- the reduced density matrix ␳ˆ should decay as quickly as ample, the non-Hermitian q-symmetric Heisenberg model with an additional boundary term. This Hamil- possible to minimize the truncated weight ⑀␳ =1 ͚M tonian is known to have matrix-product ground states of − ␣=1w␣ for optimal DMRG performance. This intu- itively clear statement can be quantified: there are four varying complexity ͑i.e., matrix sizes M˜ ͒ for particular major classes of density-matrix spectra, in descending choices of parameters ͑see also Alcaraz et al., 1994͒. order of DMRG performance. Monitoring the eigenvalue spectrum of the DMRG den- ͑ ͒ ϫ sity matrix for a sufficiently long system, Kaulke and i Density-matrix spectra for M M matrix-product Peschel found that indeed it collapses at these particular states as exact eigenstates of quantum systems, ˜ with a finite fixed number of nonvanishing eigenvalues: all eigenvalues but the M largest vanish at these ͑ ͒ values, leading to optimal DMRG performance. points Fig. 8 . Similarly, a class of two-dimensional quantum Hamiltonians with exact matrix-product ͑ ͒ ii Density-matrix spectra for non-matrix-product ground states has been studied using DMRG by Hieida states of one-dimensional quantum systems with et al. ͑1999͒. exponentially decaying correlations, and with leading exponential decay of w␣; spectra remain- 2. DMRG applied to generic gapped one-dimensional ing essentially unchanged for system sizes in ex- systems cess of the correlation length. It is quite easy to observe numerically that for gapped ͑iii͒ Density-matrix spectra for states of one- one-dimensional quantum systems the eigenvalue spec- dimensional quantum systems at criticality, with a trum of the density matrices decays essentially exponen- decay of w␣ that slows down with increasing sys- tially such that the truncated weight can be reduced ex- tem size, leading to DMRG failure to obtain ther- ponentially fast by increasing M, which is the hallmark modynamic limit behavior. of DMRG success. Moreover, the eigenvalue spectrum ͑iv͒ Density-matrix spectra for states of two- converges to some thermodynamic-limit form. dimensional quantum systems both at and away Peschel, Kaulke, and Legeza ͑1999͒ have confirmed from criticality, in which the number of eigenval- these numerical observations by studying exact density- ues to be retained to keep a fixed truncation error matrix spectra that may be calculated for the one-

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ground-state energies increasing by several orders of magnitude compared to gapped systems ͑Legeza and Fáth, 1996͒. Hence the double question arises of the decay of the eigenvalue spectrum for the density matrix of a given system size and the size dependency of this result. Chung and Peschel ͑2001͒ have shown for generic Hamiltonians quadratic in fermionic operators that the density-matrix spectrum is once again of the form of Eq. ͑ ͒ ⑀ 59 , but the energies j are now no longer given by a simple relationship linear in j. Instead, they show much ͑ ͒ FIG. 6. Pictorial representation of S•E above and S••E slower, curved growth, that slows down with system size. ͑ ͒ below DMRG schemes for open boundary conditions as in- Translating this into actual eigenvalues of the density troduced by Verstraete, Porras, and Cirac, 2004. Paired dots matrix, they show less than exponential decay, slowing represent auxiliary state spaces linked to a local state space. down with system size. This implies that for one- Straight lines symbolize maximal entanglement, ellipses and rectangles map to local state spaces as detailed in the text. dimensional quantum systems at criticality numerical Note the special role of the boundary sites. Adapted from Ver- convergence for a fixed system size will no longer be straete, Porras, and Cirac, 2004. exponentially fast in M. Maintaining a desired truncated weight in the thermodynamic limit implies a diverging number M of states to be kept. dimensional Ising model in a transverse field and the XXZ Heisenberg chain in its antiferromagnetic gapped regime, using a corner transfer-matrix method ͑Baxter, 4. DMRG in two-dimensional quantum systems ͒ 1982 . Due to the large interest in two-dimensional quantum ␳ In those cases, the eigenvalues of ˆ are given, up to a systems, we now turn to the question of DMRG conver- global normalization, as gence in gapped and critical systems. In the early days of ϱ DMRG, Liang and Pang ͑1994͒ observed numerically w ϰ expͩ− ͚ ⑀ n ͪ ͑59͒ that to maintain a given precision, an exponentially j j ϳ␣L ␣Ͼ j=0 growing number of states M , 1, had to be kept for system sizes LϫL. However, reliable information with fermionic occupation numbers nj =0,1 and an essen- from numerics is very difficult to obtain here, due to the ⑀ tially linear energy spectrum j: these are typically some very small system sizes in actual calculations. Chung and integer multiple of a fundamental scale ⑀. The density- Peschel ͑2000͒ have studied a ͑gapped͒ system of inter- matrix eigenvalue spectrum shows clear exponential de- acting harmonic oscillators, in which the density matrix cay only for large eigenvalues due to the increasing de- can be written as the bosonic equivalent of Eq. ͑59͒, with gree of degeneracy of the eigenvalue spectrum, as the eigenvalues, again up to a normalization, ⑀ ⑀ number of possible partitions of n into jnj grows. ϱ DMRG density matrices perfectly reproduce this behav- ϰ ͩ ͚ ⑀ † ͪ ͑ ͒ ͑ ͒ w exp − jbj bj . 61 ior Fig. 9 . Combining such exactly known corner j=0 transfer-matrix spectra with results from the theory of ϫ ഛ partitions, Okunishi et al. ͑1999b͒ derived the asymptotic Considering strip systems of size L N with N L, nu- form merical evaluations have shown that ⑀ ϳ ϫ ͑ ͒ 2 j const j/N, 62 w␣ ϳ exp͑− const ϫ ln ␣͒͑60͒ hence the eigenvalue decay slows down exponentially ␣ ͑ ͒ for the th eigenvalue see also Chan et al., 2002 . with inverse system size. This 1/N behavior can be un- These observations imply that for a desired truncated derstood by considering the spectrum of N chains with- weight the number of states to be kept remains finite out interchain interaction, which is given by the spec- even in the thermodynamic limit and that the truncated trum of the single chain with N-fold degeneracy weight decays exponentially fast in M, with some price introduced. Interaction will lift this degeneracy, but will to be paid due to the increasing degree of degeneracy not fundamentally change the slowdown this imposes on occurring for large M. the decay of the density-matrix spectrum ͓see also du Croo de Jongh and van Leeuwen ͑1998͒ for the generic argument͔. Taking bosonic combinatorics into account, 3. DMRG applied to one-dimensional systems at one finds criticality 2 w␣ ϳ exp͓− ͑const/N͒ln ␣͔, ͑63͒ Numerically, one observes at criticality that the eigenvalue spectrum decays dramatically slower and that for which is a consistent extension of Eq. ͑60͒. For a system increasing system size this phenomenon tends to aggra- of size 10ϫ10, typical truncation errors of 10−5 for M vate ͑Chung and Peschel, 2001͒, with errors in, e.g., =100, 10−7 for M=500, and 10−8 for M=1000 have been

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FIG. 7. Periodic boundary condition setup used in the algorithm of Verstraete, Porras, and Cirac ͑2004͒. Labeling as in previous figure. Adapted from Verstraete, Porras, and Cirac, 2004. reported ͑see Fig. 10͒, reflecting the very slow convergence of DMRG in this case. For a critical system, the noninteracting fermion model may be used once again as a model system ͑ ͒ FIG. 8. Largest density-matrix eigenvalues vs a tuning param- Chung and Peschel, 2001 . For M=2000 states, for this eter ␣ in a q=1/4-symmetric Heisenberg chain. Largest eigen- ϫ −2 simple model, the resulting truncation error is 5 10 value Ϸ1 invisible on logarithmic scale; eigenvalue collapse −1 −1 for systems of size 12ϫ12, 5ϫ10 for 16ϫ16, and 10 indicates pure matrix-product states of small finite dimension. for size 20ϫ20. Here, DMRG is clearly at its limits. From Kaulke and Peschel in Peschel, Hallberg, et al., 1999. Reprinted with permission. 5. DMRG precision for periodic boundary conditions theory as preserving the maximum entanglement be- While for periodic boundary conditions the overall tween system and environment as measured by the von properties of DMRG density matrices are the same as Neumann entropy of entanglement, those of their open-boundary-condition counterparts, their spectra have been observed numerically to decay ␳ ␳ ͑ ͒ much more slowly. Away from criticality, this is due to S =−Trˆ ln2 ˆ =−͚ w␣ln2w␣, 64 some ͑usually only approximate͒ additional factor-of-2 ␣ degeneracy of the eigenvalues. This can be explained by studying the amplitudes of density-matrix eigenstates. ͑ ͒ in an M-dimensional block state space. Chung and Peschel 2000 have demonstrated in their Latorre et al. ͑2004͒ have calculated the entropy of noncritical harmonic-oscillator model that the density- entanglement S for systems of length L embedded in matrix eigenstates associated with high eigenvalue L infinite ͑an͒isotropic XY chains and for systems embed- weight are strongly located close to the boundary be- ded in finite, periodic Heisenberg chains of length N, tween system and environment ͑Fig. 11͒. Hence, in a both with and without external field. Both models show periodic system, in which there are two boundary points, critical and noncritical behavior depending on aniso- there are for the high-weight eigenvalues two essentially tropy and field strength. In the limit N→ϱ, S ഛL, identical sets of eigenstates localized at the left and right L which is obtained by observing that entropy is maximal boundary, respectively, leading to the approximate if all 2L states are equally occupied with amplitude 2−L. double degeneracy of high-weight eigenvalues. At criti- →ϱ →ϱ cality, no such simple argument holds, but DMRG is They find that SL for L at criticality, but satu- → * Ϸ␰ similarly affected by a slower decay of spectra. rates as SL SL for L in the noncritical regime. At In a recent study, Verstraete, Porras, and Cirac ͑2004͒ criticality, however, SL grows far less than permitted by ഛ have shown that this strong deterioration of DMRG is SL L, but obeys logarithmic behavior, essentially due to its particular setup for simulating periodic boundary conditions, and have provided a new S = k log L + const. ͑65͒ formulation of the algorithm which produces results of L 2 the same quality as for open boundary conditions for the same number of states kept, at the cost of losing matrix The constant k is found to be given by k=1/6 for the sparseness ͑see Sec. III.A͒. anisotropic XY model at the critical field Hc =1 and by k=1/3 for the isotropic XY model at Hc =1 as well as ഛ ͑ the isotropic Heisenberg model for H Hc =1 for an C. DMRG and quantum information theory—A new Ͻ ͒ perspective isotropic XY model in field H Hc, see Fig. 12 . Away * from criticality, the saturation value SL decreases with As understood in the early phases of DMRG devel- decreasing correlation length ␰ ͑Fig. 13͒. opment ͑White, 1992; White and Noack, 1992͒, the rea- One-dimensional quantum spin chains at criticality son for the success of the method is that no system is are described by conformally invariant ͑1+1͒-dimen- considered in isolation, but embedded in a larger entity. sional field theories. In fact, Eq. ͑65͒ can be linked In fact, as discussed in Sec. II.B, DMRG truncation can ͑Gaite, 2003; Latorre et al., 2004͒ to the geometric en- be understood in the language of quantum information tropy ͑Callan and Wilczek, 1994͒ of such field theories,

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mance, which is in perfect agreement with the observations obtained by studying density-matrix spectra: ͑i͒ In one-dimensional quantum systems away from criticality, DMRG yields very precise results for the thermodynamic limit for some ﬁnite number * of states kept, Mϳ2SL, which grows with the correlation length. ͑ii͒ In one-dimensional quantum systems at criticality, the number of states that has to be kept will diverge as M͑L͒ϳLk, ͑68͒ with k from Eq. ͑65͒. This explains the failure of DMRG for critical one-dimensional systems as FIG. 9. Density-matrix eigenvalues wn vs eigenvalue number n →ϱ for a gapped XXY Heisenberg chain of L=98 and three values L .Ask is small, this statement has to be ⌬Ͼ⌬ qualified; DMRG still works for rather large finite of anisotropy c =1. Degeneracies are as predicted analytically. From Peschel, Kaulke, and Legeza, 1999. Reprinted with systems. permission. ͑iii͒ In higher-dimensional quantum systems, the number of states to be kept will diverge as

d−1 c + c¯ M͑L͒ϳ2L , ͑69͒ Sgeo = log L, ͑66͒ L 6 2 rendering the understanding of thermodynamic- where c ͑c¯͒ are the central charges, if one observes that limit behavior by conventional DMRG quite impossible; information is beyond retrieval just as in for the anisotropic XY model c=c¯ =1/2 and for the a black hole—whose entropy scales with its sur- ¯ Heisenberg model and isotropic XY model c=c=1. face, as the entanglement entropy would in a ͑ ͒ Geometric entropy arguments for d+1 -dimensional three-dimensional DMRG application. In any field theories use a bipartition of d-dimensional space by case, even for higher-dimensional systems, ͑ ͒ a d−1 -dimensional hypersurface, which is shared by DMRG may be a very useful method as long as system S and environment E. By the Schmidt decompo- system size is kept resolutely finite, as it is in sition, S and E share the same reduced density-matrix nuclear physics or quantum chemistry applica- spectrum, hence entanglement entropy, which is now ar- tions. Recent proposals ͑Verstraete and Cirac, gued to reside essentially on the shared hypersurface ͑cf. 2004͒ also show that it is possible to formulate the locus of highest-weight density-matrix eigenstates in generalized DMRG ansatz states in such a way Fig. 11; see also Gaite, 2001͒. Taking the thermodynamic that entropy shows correct size dependency in ͑infrared͒ limit, we find that entropy scales as the hyper- two-dimensional systems ͑see Sec. VI͒. surface area, Legeza et al. ͑2003a͒ have carried the analysis of d−1 DMRG state selection using entanglement entropy even ϰ ͩ Lͪ ͑ ͒ SL ␭ , 67 further, arguing that the acceptable truncated weight— not identical to, but closely related to the entropy of where ␭ is some ultraviolet cutoff which in condensed- entanglement, which emerges as the key quantity— matter physics we may fix at some lattice spacing. Intro- should be kept fixed during DMRG runs, determining ducing a gap ͑mass͒, an essentially infrared property, how many states M have to be retained at each iteration. This dynamical block state selection has already been into this field theory does not modify this behavior gen- ͑ erated on ultraviolet scales on the hypersurface. In d applied in various contexts Legeza and Sólyom, 2003; Legeza et al., 2003b͒. More recently, Legeza and Sólyom =1, a more careful argument shows that there is a ͑sub- ͑2004͒ have tightened the relationship between quantum leading͒ logarithmic correction as given above at critical- information theory and DMRG state selection by pro- ity or a correction saturating at some L away from criticality. posing a further refinement of state selection. M is now chosen variably to keep loss of quantum information be- S is the number of qubits corresponding to the en- L low some acceptable threshold. They argue that this loss tanglement information. To code this information in is given as DMRG, one needs a system Hilbert space of size M ജ SL SL ␹͑E͒ϵ ͑␳͒ ͑␳ ͒ ͑ ͒ ͑␳ ͒ ͑ ͒ 2 ; in fact, numerical results indicate that M and 2 S ˆ − ptypS ˆ typ − 1−ptyp S ˆ atyp . 70 are—to a good approximation—proportional. Taking ␳ ␳ ͑ ͒␳ ␳ the linear dimensions of total space and embedded sys- Here, ˆ =ptypˆ typ+ 1−ptyp ˆ atyp. For a given M, ˆ typ is ␳ ␳ tem to N,L→ϱ, quantum information theory hence formed from the M dominant eigenstates of ˆ , ˆ atyp provides us with a unified picture of DMRG perfor- from the remaining ones, with 1−ptyp being the trun-

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FIG. 10. Density-matrix eigenvalues wn for rectangular two- dimensional gapped systems of interacting harmonic oscilla- FIG. 11. Amplitudes of highest-weight single-particle eigen- tors. The leftmost curve corresponds to the one-dimensional states of the left-block density matrix for a one-dimensional case. From Chung and Peschel, 2000. Reprinted with permis- open chain of L=32 interacting harmonic oscillators. From sion. Chung and Peschel, 2000. Reprinted with permission.

␳ˆ cated weight and typ,atyp scaled to have trace one. 1 Legeza and Sólyom ͑2004͒ report a very clear linear re- ͑␻͒ ͑␻ ␩͒ ͑ ͒ CA = lim − ␲Im GA + i . 74 lationship between DMRG errors and ␹͑E͒. ␩→0+ ͑␻͒ ͑␻ ␩͒ In the following, I shall also use GA and CA +i where the limit ␩→0+ will be assumed to have been IV. ZERO-TEMPERATURE DYNAMICS taken in the former and omitted in the latter case. The role of ␩ in DMRG calculations is threefold: First, it As we have seen, DMRG is an excellent method for ensures causality in Eq. ͑72͒. Second, it introduces a fi- calculating ground states and selected excited eigen- nite lifetime ␶ϰ1/␩ to excitations, such that on finite states at almost machine precision. On the other hand, systems they can be prevented from traveling to the the targeting of specific states suggests that DMRG is open boundaries where their reflection would induce not suitable for calculating dynamical properties of spurious effects. Third, ␩ provides a Lorentzian broad- strongly correlated systems even at T=0, as the time ening of C ͑␻͒, evolution of general excited states will explore large A parts of the Hilbert space. Closer inspection has re- 1 ␩ C ͑␻ + i␩͒ = ͵ d␻ЈC ͑␻Ј͒ , ͑75͒ vealed, however, that the relevant parts of Hilbert space A ␲ A ͑␻ − ␻Ј͒2 + ␩2 can be properly addressed by DMRG. For some operator Aˆ , one may define a ͑time-dependent͒ Green’s func- which serves either to broaden the numerically obtained tion at T=0 in the Heisenberg picture by discrete spectrum of finite systems into some thermodynamic-limit behavior or to broaden analytical results for C for comparison to numerical spectra in ͑ Ј ͒ ͗ ͉ ˆ †͑ Ј͒ ˆ ͑ ͉͒ ͑͒͘ A iGA t − t = 0 A t A t 0 71 which ␩Ͼ0. Most DMRG approaches to dynamical correlations with tЈജt for a time-independent Hamiltonian Hˆ . Going center on the evaluation of Eq. ͑72͒. The first, which I to frequency range, the Green’s function reads refer to as Lanczos vector dynamics, was pioneered by Hallberg ͑1995͒ and calculates highly time-efficient, but 1 comparatively rough approximations to dynamical quan- ͑␻ ␩͒ ͗ ͉ ˆ † ˆ ͉ ͘ ͑ ͒ GA + i = 0 A A 0 , 72 tities, adopting the Balseiro-Gagliano method to E + ␻ + i␩ − Hˆ 0 DMRG. The second approach, which is referred to as the correction vector method ͑Ramasesha et al., 1997; where ␩ is some positive number to be taken to zero at Kühner and White, 1999͒, is yet another, older scheme the end. We may also use the spectral or Lehmann rep- adapted to DMRG, one much more precise, but numeri- resentation of correlations in the eigenbasis of Hˆ , cally much more expensive. A third approach, called dynamical DMRG, has been proposed by Jeckelmann ͑␻͒ ͦ͗ ͉ ˆ ͉ ͦ͘2␦͑␻ ͒ ͑ ͒ ͑ ͒ CA = ͚ n A 0 + E0 − En . 73 2002a ; while on the surface it is only a minor reformu- n lation of the correction vector method, it will be seen to be much more precise. This frequency-dependent correlation function is related Very recently, dynamical correlations as in Eq. ͑71͒ ͑␻ ␩͒ to GA +i as have also been calculated directly in real time using

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ϱ ͵ ␻␻2 ͑␻͒ ͗ ͉͓ ˆ † ˆ ͔͓ ˆ ˆ ͔͉ ͘ ͑ ͒ d CA = 0 A ,H H,A 0 , 78 −ϱ

where the first equation holds for ␩ജ0 and the latter two only as ␩→0. As DMRG is much more precise for equal-time than dynamical correlations, comparisons are made to DMRG results which for the purpose may be considered exact. Finite-size effects due to boundary conditions can be treated in various ways. They can be excluded completely by the use of periodic boundary conditions at the price of much lower DMRG precision FIG. 12. ͑Color in online edition͒ Diverging von Neumann ͑Hallberg, 1995͒. In cases where open boundary condi- entropy SL vs block length L for a critical isotropic XY chain tions are preferred, two situations should be distin- Ͻ in external fields H Hc. Increasing field strengths suppress guished. If Aˆ acts locally, such as in the calculation of an entropy. From Latorre et al., 2004. Reprinted with permission. optical conductivity, one may exploit the fact that finite ␩ exponentially suppresses excitations ͑Jeckelmann, ͒ time-dependent DMRG with adaptive Hilbert spaces 2002a . As they travel at some speed c through the sys- →ϱ ␩→ ␩ ͑Vidal, 2003, 2004; Daley et al., 2004; White and Feiguin, tem, a thermodynamic limit L , 0 with =c/L 2004͒, as will be discussed in Sec. IX.C. These may then may be taken consistently. For the calculation of dy- be Fourier transformed to a frequency representation. namical structure functions like those obtained in elastic As time-dependent DMRG is best suited for short times neutron scattering, Aˆ is a spatially delocalized Fourier ͑i.e., high frequencies͒, and the methods discussed in this transform, and another approach must be taken. The section are typically best for low frequencies, this alter- open boundaries introduce both genuine edge effects native approach may be very attractive to cover wider and a hard cut to the wave functions of excited states in frequency ranges. real space, leading to a large spread in momentum All approaches share the need for precision control space. To limit bandwidth in momentum space, filtering and the elimination of finite-system and/or boundary ef- is necessary. The filtering function should be narrow in fects. Beyond DMRG-specific checks of convergence, momentum space and broad in real space, while simul- precision may be checked by comparisons to indepen- taneously strictly excluding edge sites. Kühner and dently obtained equal-time correlations using the fol- White ͑1999͒ have introduced the so-called Parzen filter lowing sum rules: FP, 1−6͉x͉2 +6͉x͉3,0ഛ ͉x͉ ഛ 1/2 F ͑x͒ = ͭ ͮ ͑79͒ ϱ P 2͑1−͉x͉͒3, 1/2 ഛ ͉x͉ ഛ 1, ͵ ␻ ͑␻͒ ͗ ͉ ˆ † ˆ ͉ ͘ ͑ ͒ d CA = 0 A A 0 , 76 −ϱ ෈͓ ͔ where x=i/LP −1,1 , the relative site position in the filter for a total filter width 2LP. In momentum space FP ⌬ ͱ has a wave-vector uncertainty q=2 3/LP, which scales ϱ −1 as L if one scales LP with system size L. For finite-size ͵ ␻␻ ͑␻͒ ͗ ͉ ˆ †͓ ˆ ˆ ͔͉ ͘ ͑ ͒ d CA = 0 A H,A 0 , 77 extrapolations it has to be ensured that filter size does −ϱ not introduce a new size dependency. This can be ensured by introducing a Parzen-filter prefactor given by ͱ ␲ 140 /151LP.

A. Continued-fraction dynamics

The technique of continued-fraction dynamics was ﬁrst exploited by Gagliano and Balseiro ͑1987͒ in the framework of exact ground-state diagonalization. Obvi- ously, the calculation of Green’s functions as in Eq. ͑72͒ ˆ ͑ ␻ involves the inversion of H or more precisely, E0 + +i␩−Hˆ ͒, a typically very large sparse Hermitian matrix. This inversion is carried out in two, at least formally, FIG. 13. ͑Color in online edition͒ Saturating density-matrix exact steps. First, an iterative basis transformation tak- entropy SL vs block length L for an Ising chain in a transverse ഛ → ˆ ﬁeld H Hc. Saturation entropy grows with H Hc; diver- ing H to a tridiagonal form is carried out. Second, this gence is recovered at criticality ͑top curve͒. From Latorre et tridiagonal matrix is then inverted, allowing the evalua- al., 2004. Reprinted with permission. tion of Eq. ͑72͒.

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Let us call the diagonal elements of Hˆ in the tridiago- It should also be mentioned that, beyond the above 2 complications also arising for exact diagonalization with nal form an and the subdiagonal elements bn. The coef- 2 an exact Hˆ , additional approximations are introduced in ficients an ,bn are obtained as the Schmidt-Gram coefficients in the generation of a Krylov subspace of un- DMRG, as the Hamiltonian itself is of course not exact. normalized states starting from some arbitrary state, Instead of evaluating the continued fraction of Eq. ͑ ͒ ˆ ͉ ͘ 84 , one may also exploit the fact that, upon normaliza- which we take to be the excited state A 0 : ͉ ͘ tion of the Lanczos vectors fn and accompanying re- 2 ͉f ͘ = Hˆ ͉f ͘ − a ͉f ͘ − b2͉f ͘, ͑80͒ scaling of an and bn, the Hamiltonian is iteratively trans- n+1 n n n n n−1 formed into a tridiagonal form in a new approximate ͕͉ ͖͘ with orthonormal basis. When one transforms the basis fn by a diagonalization of the tridiagonal Hamiltonian ma- ͉ ͘ ˆ ͉ ͘ ͑ ͒ f0 = A 0 , 81 trix to the approximate energy eigenbasis of Hˆ ,͕͉n͖͘ with eigenenergies En, the Green’s function can be writ- ͗f ͉Hˆ ͉f ͘ ten within this approximation as a = n n , ͑82͒ n ͗f ͉f ͘ n n 1 ͑␻ ␩͒ ͚ ͗ ͉ ˆ †͉ ͗͘ ͉ ͉ ͘ GA + i = 0 A n n ␻ ␩ n n E0 + + i − En ͗f ͉Hˆ ͉f ͘ ͗f ͉f ͘ 2 n−1 n n n ͑ ͒ bn = ͗ ͉ ͘ = ͗ ͉ ͘ . 83 ϫ͗ ͉ ˆ ͉ ͘ ͑ ͒ fn−1 fn−1 fn−1 fn−1 n A 0 , 85 ͉ ͑͘ The global orthogonality of the states fn at least in where the sum runs over all approximate eigenstates. formal mathematics͒ and the tridiagonality of the new The dynamical correlation function is then given by representation ͑i.e., ͗f ͉Hˆ ͉f ͘=0 for ͉i−j͉Ͼ1͒ follow by in- i j ˆ 2 duction. It can then be shown quite easily by an expan- ␩ ͦ͗n͉A͉0ͦ͘ C ͑␻ + i␩͒ = ͚ , ͑86͒ ␻ ␩ A ␲ ͑ ␻ ͒2 ␩2 sion of determinants that the inversion of E0 + +i n E0 + − En + −Hˆ leads to a continued fraction such that the Green’s function G reads where the matrix elements in the numerator are simply A ͉ ͘ the f0 expansion coefficients of the approximate eigen- ͉ ͘ ͗0͉Aˆ †Aˆ ͉0͘ states n . ͑ ͒ ͑ ͒ GA z = , 84 For a given effective Hilbert-space dimension M, op- b2 1 timal precision within these constraints can be obtained z − a0 − 2 b2 by targeting not only the ground state, but also a se- z − a − 1 lected number of states of the Krylov sequence. While a z − ¯ first approach is to take arbitrarily the first n states gen- ␻ ␩ where z=E0 + +i . This expression can now be evalu- erated ͑say, five to ten͒ at equal weight, the approximate ated numerically, giving access to dynamical correla- eigenbasis formulation gives direct access to the relative tions. importance of the vectors. The importance of a Lanczos In practice, several limitations occur. The iterative ͉ ͘ ͉ ͘ ˆ ͉ ͘ 2 vector fn is given by, writing A =A 0 , generation of the coefficients an ,bn is equivalent to a Lanczos diagonalization of Hˆ with starting vector Aˆ ͉0͘. ␭ ͦ͗ ͉ ͦ͘2ͦ͗ ͉ ͦ͘2 ͑ ͒ n = ͚ A m m fn , 87 Typically, the convergence of the lowest eigenvalue of m the transformed tridiagonal Hamiltonian to the ground- which assesses the contribution the vector makes to an state eigenvalue of Hˆ will have happened after n approximative eigenstate ͉m͘, weighted by that eigen- ϳO͑102͒ iteration steps for standard model Hamilto- state’s contribution to the Green’s function. The density nians. Lanczos convergence is, however, accompanied matrix may then be constructed from the ground state by numerical loss of global orthogonality, which compu- and a number of Lanczos vectors weighted according to tationally is ensured only locally, invalidating the inver- Eq. ͑87͒. The weight distribution indicates the applica- sion procedure. The generation of coefficients has to be bility of the Lanczos vector approach: for the spin-1 stopped before that. Kühner and White ͑1999͒ have pro- Heisenberg chain at q=␲, where there is a one-magnon posed monitoring, for normalized vectors, ͗f ͉f ͘Ͼ⑀ as a 0 n band at ␻=⌬=0.41J, three target states carry 98.9% of termination criterion. The precision of this approach therefore depends on whether the continued fraction total weight, whereas for the spin-1/2 Heisenberg chain at q=␲ with a two-spinon continuum for 0ഛ␻ഛ␲J, the ˆ ͉ ͘ has sufficiently converged at termination. With A 0 as first three target states carry only 28.0%, indicating se- the starting vector, convergence will be fast if Aˆ ͉0͘ is a vere problems with precision ͑Kühner and White, 1999͒. long-lived excitation ͑close to an eigenstate͒ such as There is currently no precise rule for resolving the would be the case if the excitation were part of an exci- dilemma of targeting as many states as possible while tation band; this will typically not be the case if it is part retaining sufficient precision in the description of each of an excitation continuum. single one.

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As an example of the excellent performance of this equation for the imaginary part, and exploit the relation- method, one may consider the isotropic spin-1 Heisen- ship to the real part: berg chain, for which the single-magnon line is shown in ͓͑ ␻ ˆ ͒2 ␩2͔ ͉ ͑␻ ␩͒͘ ␩ ˆ ͉ ͑͒͘ Fig. 14. Exact diagonalization, quantum Monte Carlo, E0 + − H + Im c + i =− A 0 91 and DMRG are in excellent agreement, with the excep- → tion of the region q 0, where the single-magnon band Hˆ − E − ␻ ͉ ͑␻ ␩͒͘ 0 ͉ ͑␻ ␩͒͘ ͑ ͒ forms only the bottom of a magnon continuum. Here Re c + i = ␩ Im c + i . 92 Lanczos vector dynamics do not correctly reproduce the 2⌬ gap at q=0, which is much better resolved by quan- The standard method for solving a large sparse linear tum Monte Carlo. equation system is the conjugate-gradient method The intuition that excitation continua are badly ap- ͑Golub and van Loan, 1996͒, which effectively generates proximated by a sum over some O͑102͒ effective excited a Krylov space, as does the Lanczos algorithm. The main states is further corroborated by considering the spectral implementation work in this method is to provide +͑ ␲ ␻͓͒ + ͑ ͔͒ weight function S q= , use A=S in Eq. 74 for a Hˆ 2Im͉c͘. Two remarks are in order. The reduced basis spin-1/2 Heisenberg antiferromagnet. As shown in Fig. ˆ 2 15, Lanczos vector dynamics roughly capture the right representation of H is obtained by squaring the effective Hamiltonian generated by DMRG. This approxima- spectral weight, including the 1/␻ divergence, as can be tion is found to work extremely well as long as both real seen from the essentially exact correction vector curve, and imaginary parts of the correction vector are in- but no convergent behavior can be observed upon an cluded as target vectors. While the real part is not increase of the number of targeted vectors. The very fast ͑ ␻ Lanczos vector method is thus certainly useful for get- needed for the evaluation of spectral functions, E0 + ting a quick overview of spectra, but is not suited to −Hˆ ͒Im͉c͘ϳRe͉c͘ due to Eq. ͑92͒; and targeting Re͉c͘ detailed quantitative calculations of excitation continua, ensures minimal truncation errors in Hˆ Im͉c͘. The fun- only excitation bands. Nevertheless, this method has damental drawback of using a squared Hamiltonian is 1 been applied successfully to the S= 2 antiferromagnetic that, for all iterative eigenvalue or equation solvers, the Heisenberg chain ͑Hallberg, 1995͒, the spin-boson speed of convergence is determined by the matrix con- model ͑Nishiyama, 1999͒, the Holstein model ͑Zhang et dition number, which drastically deteriorates by the al., 1999͒, and spin-orbital chains in external fields ͑Yu squaring of a matrix. Many schemes are available to im- and Haas, 2000͒. Okunishi et al. ͑2001͒ have used it to prove the convergence of conjugate-gradient methods, extract spin chain dispersion relations. Garcia et al. usually based on providing the solution to some related ͑2004͒ have used continued-fraction techniques to pro- but trivial equation system, such as that formed from the vide a self-consistent impurity solver for dynamical diagonal elements of the large sparse matrix ͑Golub and mean-field calculations ͑Metzner and Vollhardt, 1989; van Loan, 1996͒. Georges et al., 1996͒. In the simplest form of the correction vector method, the density matrix is formed from targeting four states, ͉0͘, Aˆ ͉0͘,Im͉c͑␻+i␩͒͘, and Re͉c͑␻+i␩͒͘. B. Correction vector dynamics As has been shown by Kühner and White ͑1999͒,itis not necessary to calculate a very dense set of correction Even before the advent of DMRG, another way of vectors in ␻ space to obtain the spectral function for an obtaining more precise spectral functions had been pro- entire frequency interval. Assuming that the finite con- posed by Soos and Ramasesha ͑1989͒; it was first applied vergence factor ␩ ensures a good description of an entire using DMRG by Ramasesha et al. ͑1997͒ and Kühner range of energies of width Ϸ␩ by the correction vector, and White ͑1999͒. After preselection of a fixed fre- they applied the Lanczos vector method as detailed in quency ␻ one may introduce a correction vector: the last section to Aˆ ͉0͘, using the effective Hamiltonian 1 ͉c͑␻ + i␩͒͘ = Aˆ ͉0͘, ͑88͒ obtained by also targeting the correction vector, to ob- ␻ ␩ ˆ tain the spectral function in some interval around their E0 + + i − H anchor value for ␻. Comparing the results for some ␻ which, if known, allows for trivial calculation of the obtained by starting from neighboring anchoring fre- Green’s function and hence the spectral function at this quencies allows for excellent convergence checks ͑Fig. particular frequency: 16͒. In fact, they found that the best results were ob- ͑␻ ␩͒ ͗ ͉ ͑␻ ␩͒͘ ͑ ͒ tained for a two-correction-vector approach in which GA + i = A c + i . 89 two correction vectors were calculated and targeted for The correction vector itself is obtained by solving the ␻ ␻ ␻ ⌬␻ two frequencies 1, 2 = 1 + and the spectral func- large sparse linear equation system given by ͓␻ ␻ ͔ tion obtained for the interval 1 , 2 . This method was, for example, able to provide a high-precision result for ͑E + ␻ + i␩ − Hˆ ͉͒c͑␻ + i␩͒͘ = Aˆ ͉0͘. ͑90͒ 0 the spinon continuum in the S=1/2 Heisenberg chain To actually solve this non-Hermitian equation system, where standard Lanczos dynamics fails ͑Fig. 15͒. the current procedure is to split the correction vector Reducing the broadening factor ␩, it is also possible to into real and imaginary parts, to solve the Hermitian resolve finite-system peaks in the spectral function, ob-

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taining to some approximation both location and weight of the Green’s-function poles. The correction vector method has been applied to determine the nonlinear optical coefﬁcients of Hubbard chains and derived models by Pati et al. ͑1999͒; Kühner et al. ͑2000͒ have extracted the ac conductivity of the Bose-Hubbard model with nearest-neighbor interactions. Raas et al. ͑2004͒ have used it to study the dynamic correlations of single-impurity Anderson models and were able to resolve sharp dominant resonances at high energies, using optimized algorithms for the matrix inversion needed to obtain the correction vector.

FIG. 14. Single-magnon line of the S=1 Heisenberg antiferro- C. Dynamical DMRG magnet from exact diagonalization, quantum Monte Carlo, and DMRG for various system sizes and boundary conditions. A further important refinement of DMRG dynamics From Kühner and White, 1999. Reprinted with permission. is obtained by a reformulation of the correction vector method in terms of a minimization principle, which has ͑ ͒ ͑␻͒ ͑ ␻ ␩͒ ͑ ͒ been called “dynamical DMRG” Jeckelmann, 2002a . CA = lim lim CA L; + i , 96 While the fundamental approach remains unchanged, ␩→0 L→ϱ the large sparse equation system is replaced by a mini- where limits may not be interchanged and which is very mization of the functional hard to take numerically, may be taken as a single limit, ͑␻͒ ͓ ␻ ␩͑ ͔͒ ͑ ͒ ͑␻ ␺͒ ͗␺͉͑ ␻ ˆ ͒2 ␩2͉␺͘ ␩͗ ͉␺͘ CA = lim CA L; + i L , 97 WA,␩ , = E0 + − H + + A ␩͑L͒→0 + ␩͗␺͉A͘. ͑93͒ provided that ␩͑L͒→0asL→ϱ is chosen such that the finiteness of the system is not visible for the chosen ␩͑L͒ At the minimum, the minimizing state is and it thus seems to be in the thermodynamic limit from a level-spacing perspective. This implies that ␩͑L͒ ͉␺ ͘ =Im͉c͑␻ + i␩͒͘. ͑94͒ min Ͼ␦␻͑L͒, the maximum level spacing of the finite system ␻ Even more importantly, the value of the functional itself around energy . For a typical tight-binding Hamil- is tonian such as the Hubbard model, one finds c W ␩͑␻,␺͒ =−␲␩C ͑␻ + i␩͒, ͑95͒ ␩͑L͒ ജ , ͑98͒ A, A L such that for the calculation of the spectral function it is where c is the bandwidth ͑and, in such Hamiltonians, not necessary to explicitly use the correction vector. The also a measure of propagation velocity͒. The key argu- huge advantage is that if the correction vector is known ment is now that if ␩͑L͒ is chosen according to that ⑀ ͑ to some precision which will be essentially identical prescription, the scaling of numerical results broadened ͒ for the equation solver and the minimizer , the value of by ␩͑L͒ is the same as for some thermodynamic-limit the functional itself is, by general properties of varia- form known or conjectured analytically subject to ⑀2 tional methods, known to the much higher precision . Lorentzian broadening using the same ␩. From Lorent- Hence the DMRG algorithm is essentially implemented zian broadening of model spectra one can then show as in the correction vector method, with the same target that a local ␦ peak in an otherwise continuous spectrum vectors, until they converge under sweeping, but with with weight C0 scales as C0 /␲␩͑L͒, and that a power-law ͑␻ ␺͒ the minimization of WA,␩ , replacing the sparse ͑␻ ␻ ͒−␣ divergence − 0 at a band edge is signaled by a equation solver. Results that are obtained for a sequence ␩͑ ͒−␣ ␻ ␻ scaling as L . More model spectra are discussed by of i may then be extended to other by suitable inter- ͑ ͒ ͑ ͒ Jeckelmann 2002a . polation Jeckelmann, 2002a , also exploiting first de- Dynamical DMRG has been extensively used to study rivatives of spectral functions that are numerically acces- the spectrum and the optical conductivity of the ex- sible at the anchor points. tended Hubbard model ͑Jeckelmann et al., 2000; Essler For dynamical quantities, there are no strict state- et al., 2001; Jeckelmann, 2003a; Kim et al., 2004͒. The ments on convergence. However, convergence for large optical conductivity may be calculated as M seems to be monotonic, with CA typically underesti- 1 mated. ␴ ͑␻͒ ͑␻ ␩͒ ͑ ͒ 1 =− limIm GJ + i , 99 The high-quality numerical data obtained from dy- L␻ ␩→0 namical DMRG in fact allow for an extrapolation to the ˆ ͚ ͑ † † ͒ thermodynamic limit. As pointed out by Jeckelmann with the current operator J=−ie i␴ti c␴,ic␴,i+1−c␴,i+1c␴,i . ͑2002a͒, the double limit Another possibility is to use

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ϱ dom with Nsite= . Such bosonic degrees of freedom occur, for example, in the Bose-Hubbard model,

ˆ † † U ͑ ͒ ͑ ͒ HBH =−t͚ bi+1bi + bi bi+1 + ͚ ni ni −1 , 101 i 2 i which has come to the forefront of research due to the realization of a tunable quantum phase transition from a Mott insulating to a superfluid phase in ultracold bosonic atomic gases in optical lattices ͑Greiner et al., 2002͒. Another model of interest is the Holstein model, in which electrons ͑also spinless fermions or XY spins͒ couple to local ͑quantum͒ phonons that react dynami- FIG. 15. Spectral weight S+͑q=␲,␻͒ of the S=1/2 Heisenberg cally and are not a priori in some adiabatic limit, antiferromagnet from Lanczos vector and correction vector ˆ † † ␥ ͑ † ͒͑ ͒ DMRG. NL indicates the number of target states; M=256. HHol =−t͚ c␴,i+1c␴,i + c␴,ic␴,i+1 − ͚ bi + bi ni −1 Note that spectral weight times ␻ is shown. From Kühner and i i White, 1999. Reprinted with permission. ␻ ͑ † ͒ ͑ ͒ + ͚ bi bi + 1/2 . 102 i 1 ␴ ͑␻͒ ͓͑␻ ␩͒ ͑␻ ␩͔͒ ͑ ͒ It models electrons in a vibrating lattice and opens the 1 =− limIm + i GD + i , 100 L␻ ␩→0 way to polaron physics. Yet another model is the spin- boson model, which models the dissipative coupling of a ˆ ͚ ͑ ͒ with the dipole operator D=−e ii ni −1 . two-state system to a thermal reservoir given by bosonic In the noninteracting limit U=0, the optical conduc- oscillators: tivity is nonvanishing in a band 2͉⌬͉Ͻ␻Ͻ4t with square- root edge divergences. Dynamical DMRG quantita- ⌬ ␴z Hˆ =− ␴x + ͚ ␻ b†b + ͚ f ͑b† + b ͒. ͑103͒ tively reproduces all features of the exact solution, SB i i i i i i 2 i 2 i including the divergence. It is even possible to extract the degree of the singularity. Moving to the strongly interacting case Uӷt, one expects two continuous bands due to the dimerization at 2͉⌬͉ഛ͉␻−U͉ഛ4t and the A. Moderate number of degrees of freedom Hubbard resonance at ␻=U. Figure 17, which compares to analytical solutions, shows both that the bands are The simplest conceptual approach is to arbitrarily reproduced with excellent quality ͑up to broadening at truncate the local-state space to some Nmax for the the edges͒ and that the ␦ singularity is measured with bosonic degrees of freedom and to check for DMRG extremely high precision. Similarly, very precise spectral convergence both in M and in Nmax. This approach has functions for the Hubbard model away from half-filling been very successful in the context of the Bose-Hubbard have been obtained ͑Benthien et al., 2004͒. Nishimoto model in which the on-site Coulomb repulsion U sup- and Jeckelmann ͑2004͒ have shown that dynamical presses large occupation numbers. It has been used for DMRG may also be applied to impurity problems, such the standard Bose-Hubbard model ͑Kühner and Mon- as the flat-band single-impurity Anderson model, upon ien, 1998; Kühner et al., 2000͒, with a random potential suitable discretization of the band. Recently, Nishimoto ͑Rapsch et al., 1999͒, in a parabolic potential due to a et al. ͑2004͒ have extended this to a precise calculation of magnetic trap for cold atoms ͑Kollath et al., 2004͒, and the Green’s function of a single-impurity Anderson with non-Hermitian hopping and a pinning impurity to model with arbitrary band, thereby providing a high- model superconductor flux lines ͑Hofstetter et al., 2004͒. quality self-consistent impurity solver needed for dy- Typically, allowing for about three to five times the av- namical mean-field calculations ͑Metzner and Vollhardt, erage occupation is sufficient as Nmax. Similarly, the 1989; Georges et al., 1996͒. physics of the fluctuation of confined membranes ͑Nish- iyama, 2002a, 2002b, 2003͒ and quantum strings ͑Nish- iyama, 2001͒ necessitates the introduction of a larger V. BOSONS AND DMRG: MANY LOCAL DEGREES OF number of local vibrational states. Other applications FREEDOM that are more problematic have been to phonons, both with ͑Caron and Moukouri, 1996; Maurel and Lepetit, So far, our discussion of DMRG applications has been 2000; Maurel et al., 2001͒ and without ͑Caron and Mouk- largely restricted to quantum spin and electronic sys- ouri, 1997͒ coupling to magnetic or fermionic degrees of tems, which are characterized by a fixed, usually small freedom. While they are believed to be reliable in giving number of degrees of freedom per site. As algorithmic a generic impression of physical phenomena, for more ͓ performance relies heavily on Nsite small formally precise studies more advanced techniques are necessary ͑ 3 ͒ ͑ 2 ͔͒ ͑ O Nsite , in practice rather O Nsite , one may wonder compare the results of Caron and Moukouri, 1996 and whether DMRG is applicable to bosonic degrees of free- of Bursill, 1999͒.

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four-block approach. It was found that, in the neighbor- hood of the phase transition, roughly 30 phonon states were sufficient to obtain converged results. A similar scenario for the Kosterlitz-Thouless transition was obtained for spinless fermions in the Holstein model ͑Bur- sill et al., 1998͒. An approach more in the spirit of DMRG, the so- called local state reduction, was introduced by Zhang, Jeckelmann, and White ͑1998͒. While it can also be combined with exact diagonalization, I want to formulate it in a DMRG setup. Assuming a chain with fermionic and a small number of bosonic degrees of freedom on each site, one of the sites is chosen as the “big site” to which FIG. 16. Spectral weight of the S=1/2 Heisenberg antiferromagnet from correction vector DMRG. M=256 states kept. a further number of bare bosonic degrees of freedom is Spectral weights have been calculated for ␻ intervals starting added. Within a DMRG calculation, a density matrix is from various anchoring frequencies for the correction vector. formed for the big site to truncate the number of de- From Kühner and White, 1999. Reprinted with permission. grees of freedom down to some fixed number of optimal degrees of freedom. This procedure is repeated throughout the lattice in the finite-system algorithm, sweeping B. Large number of degrees of freedom for convergence and for the addition of further bosonic degrees of freedom. The standard prediction algorithm Essentially three approaches have been taken to re- makes the calculation quite fast. Physical quantities are duce the large, possibly divergent number of states per then extracted within the optimal phononic state space. site to a small number manageable by DMRG. As can be seen from Fig. 18, merely keeping two or Bursill ͑1999͒ has proposed a so-called four-block ap- three optimal states, in which high-lying bare states have proach that is particularly suited to periodic boundary non-negligible weight, may be as efficient as keeping of conditions and is a mixture of Wilson numerical renor- the order of a hundred bare states. This approach has malization group and DMRG ideas. Starting from four allowed the demonstration of the strong effect of quan- initial blocks of size 1 with M states ͑this may be a rela- tum lattice fluctuations in trans-polyacetylene ͑Barford, tively large number of electronic and phononic degrees Bursill, and Lavrentiev, 2002͒. Combined with Lanczos- of freedom͒ forming a ring, one solves for the ground vector dynamics, very precise dynamical susceptibilities state of that M4-state problem; density matrices are then have been extracted for spin-boson models ͑Nishiyama, formed to project out two blocks and form a new block 1999͒. Extensions of the method are offered by Fried- of double size with M2 states, which are truncated down man ͑2000͒ and Fehske ͑2000͒. to M using the density-matrix information. From four of Jeckelmann and White ͑1998͒ have devised a further these blocks, a new four-block ring is built, leading to a approach in which 2n bosonic degrees of freedom are doubling of system size at every step. Calculations may embodied by n fermionic pseudosites: writing the num- be simplified beyond the usual time-saving techniques ber of the bosonic degree of freedom as a binary num- by reducing the number of M4 product states to some ber, the degree of freedom is encoded by empty and full smaller number of states for which the product of their fermionic pseudosites as dictated by the binary number. weights in the density matrices is in excess of some very All operators on the bosonic degrees of freedom can small ⑀, of the order of 10−25 to 10−10. Translation opera- now be translated into ͑rather complicated͒ operators on tors applied to the resulting ground state of the four the fermionic pseudosites. Finite-system DMRG is then blocks on the ring allow the explicit construction of spe- applied to the resulting Hamiltonian. Jeckelmann and cific momentum eigenstates, in particular for k=0 and co-workers have been able to study polaronic self- k=␲. For both an XY-spin ͑Bursill, 1999͒ and isotropic- trapping of electrons in the Holstein model for up to 128 spin ͑Bursill et al., 1999͒ variety of the Holstein model, phonon states and have located very precisely the metal- ͑ this method allows one to trace out very precisely a insulator transition in this model Jeckelmann et al., ͒ Kosterlitz-Thouless phase transition from a quasi-long- 1999 . ranged antiferromagnet for small spin-phonon coupling to a dimerized, gapped phase. As Kosterlitz-Thouless VI. TWO-DIMENSIONAL QUANTUM SYSTEMS phase transitions exhibit an exponentially slow opening of the gap ͑Okamoto and Nomura, 1992͒, the exact lo- Since the spectacular discovery of high-Tc supercon- calization of the transition by analyzing the gap is prob- ductivity related to CuO2 planes, there has been a strong lematic. It is rather convenient to apply the level- focus on two-dimensional quantum systems. Early on, it spectroscopy technique ͑Nomura and Okamoto, 1994͒, was suggested that the Hubbard model or the t-J model which locates the phase transition by a small-system away from half-filling might be simple yet powerful * finite-size extrapolation of the crossing points g of the enough to capture essential features of high-Tc super- lowest-lying excitation of different symmetries, includ- conductivity. The analytical study of these quantum sys- ing k=0 and k=␲ states that are easily accessible in the tems is plagued by problems similar to the one-

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finite-system DMRG far from anything physically realistic. As finite-system DMRG provides only sequential local updates to that state, there is no guarantee that the system will not get trapped in some local minimum state. That such trappings do exist is seen from frequent observations that seemingly converged systems may, after many further sweeps, suddenly experience major “ground-state” energy drops into some new ͑possible͒ ground state. White and co-workers have followed the approach of exploiting local trappings by theorizing ahead of using FIG. 17. Optical conductivity for a Peierls-Hubbard model in ӷ ␩ ⍀ ␻ DMRG on possible ground-state types using other ana- the U t limit, L=128, =0.1, = −U: solid line, dynamical lytical or numerical techniques and of forcing these DMRG; dot-dashed line, broadened exact ␦ contribution; types of states onto the system by the application of dashed line, unbroadened thermodynamic-limit bands. Note suitable local magnetic fields and chemical potentials. log-linear scale. From Jeckelmann, 2002a. Reprinted with permission. These external fields are then switched off and the convergence behavior of the competing proposed states under finite-system DMRG observed. This procedure gen- dimensional case, as in many effective field theories d erates a set of states, each of which corresponds to a =2 is the lower critical dimension and few exact results local energy minimum. The associated physical proper- are available. Numerically, exact diagonalization meth- ties may be measured and compared ͑see White and ods are even more restricted in two dimensions, and Scalapino, 1998a and White and Scalapino, 2000 for a quantum Monte Carlo is difficult to use for fermionic discussion͒. In the multichain approach, the two- models away from half filling due to the negative-sign dimensional Heisenberg model ͑White, 1996b͒, the two- problem. Can DMRG help? dimensional t-J model ͑White and Scalapino, 1998a, The first step in any two-dimensional application of 1998b, 2000; Kampf et al., 2001; Chernychev et al., 2003͒ DMRG is to identify blocks and sites in order to apply with particular emphasis on stripe formation, and the the strategies devised in the one-dimensional case. As- six-leg ladder Hubbard model ͑White and Scalapino, suming nearest-neighbor interactions on a square lattice, 2003͒ have been studied. one might organize columns of sites into one supersite Among competing setups ͑du Croo de Jongh and van such that blocks are built from supersites, sweeping Leeuwen, 1998; Henelius, 1999; Farnell, 2003; Moukouri through the system horizontally. While this approach and Caron, 2003͒, Xiang et al. ͑2001͒ have set up a two- maintains short-ranged interactions, it must fail for any dimensional algorithm that uses a true DMRG calcula- two-dimensional strip of appreciable width L as the tion all along and builds LϫL systems from previously number of states per supersite grows exponentially as generated ͑L−1͒ϫ͑L−1͒ systems while keeping the lat- L tice structure intact. It can be applied to all lattices that Nsite and is only useful for narrow ladder systems. The standard approach ͓Noack, White, and Scalapino can be arranged to be square with suitable interactions. in Landau et al. ͑1994͒, Liang and Pang ͑1994͒, White For example, a triangular lattice is a square lattice with ͑1996b͔͒, known as the multichain approach, is to define additional next-nearest-neighbor interactions along one a suitable one-dimensional path over all sites of the two- diagonal in each square. dimensional lattice, such as the configuration shown in A one-dimensional path is organized as shown in Fig. Fig. 19. 20, where the pair of free sites may be zipped through One may now apply standard one-dimensional finite- the square along the path using the standard finite- system DMRG at the price of long-ranged interactions system algorithm, yielding arbitrary block sizes. In par- of range 2L both within and between blocks, as indi- ticular, one may obtain triangular upper or lower blocks cated in Fig. 19. An inherent difficulty is the preparation as shown in Fig. 21. Combining these blocks from a ͑L of blocks and operators for application of the finite- −1͒ϫ͑L−1͒ system and adding two free sites at the cor- system algorithm, as there is no precursor infinite- ners, one arrives at an LϫL system. Here, the upper left system DMRG run in some sequence of smaller systems free site can be zipped to be a neighbor of the lower- possible in this setup. Few compositions of smaller right free site, as it sits next to active block ends ͑i.e., the blocks in this scheme resemble the final system at all. ends where new sites are added͒. The pair of free sites While Liang and Pang ͑1994͒ have simply switched off can now be zipped through the system to yield the de- all non-nearest-neighbor interactions in the mapped sired triangular blocks for the step L→L+1. one-dimensional system and applied standard infinite- This approach systematically yields lower energies system DMRG in order to switch on all interactions in than the multichain approach for both two-dimensional finite-system runs, one can also grow blocks in a stan- square and triangular S=1/2 Heisenberg models with dard infinite-system DMRG run, where some very short, the exception of very small systems. Even for a rela- exactly solvable system is used as environment such that tively modest number of states kept ͑M=300͒, Xiang et the entire superblock remains treatable. This and similar al. ͑2001͒ report thermodynamic-limit extrapolations in warmup procedures will generate starting states for good agreement with large-scale quantum Monte Carlo

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 289 simulations: For the square lattice, they find Eϱ =−0.3346 versus a quantum Monte Carlo result Eϱ =−0.334 719͑3͒͑Sandvik, 1997͒. The potential of this approach seems far from exploited at the moment. So far, I have been tacitly assuming that interactions are of the same order along both spatial directions. Ex- perimentally, a relatively frequent situation is that one- dimensional quantum systems are weakly coupled along a second axis. At high enough temperatures this interac- FIG. 18. Optimal vs bare phonon states: ground-state energy convergence for a four-site Holstein model at half-filling vs tion will washed out, but at sufficiently low temperatures ͑ ͒ there will be a crossover from one- to two-dimensional number of optimal, bare phonon states kept. From Zhang, Jeckelmann, and White, 1998. behavior. Such systems may be studied by attempting a precise one-dimensional description and introducing the weak interchain interaction on the mean field or some one dimension͒ to a proliferation of indices on resulting other perturbative level. Moukouri and Caron ͑2003͒ tensors, a suitable truncation scheme is needed, as de- and Moukouri ͑2004͒ have used DMRG in a similar scribed by Verstraete and Cirac ͑2004͒. spirit by solving a one-dimensional system using stan- An appealing feature of this approach is that the en- dard DMRG with M states and determining the MЈ tropy of entanglement for a cut through a system of size lowest-lying states for the superblock. Chain lengths are LϫL will scale, for fixed M, linearly in system size, as it chosen such that the lowest-lying excitation of the finite should generically. The errors in energy seem to de- chain is down to the order of the interchain coupling. crease exponentially in M. M is currently very small ͑up These states are taken to be the states of a “site,” and to 5͒, as the algorithm scales badly in M; however, as M are combined to a new effective Hamiltonian, which is variational parameters are available on each bond, even once again treated using DMRG. Results for weakly a small M corresponds to large variational spaces. At the coupled spin chains are in good agreement with quan- time of writing, it is too early to assess the potential of tum Monte Carlo results; however, MЈ is currently se- this new approach; however, current results and the con- verely limited to several tens. ceptual clarity of this approach make it seem very prom- A severe limitation on two-dimensional DMRG is ising. provided by the exponential growth of M with L for a preselected truncated weight or ground-state precision ͑Sec. III.B͒. For frustrated and fermionic systems be- VII. BEYOND REAL-SPACE LATTICES: MOMENTUM- yond the very small exact diagonalization range ͑6ϫ6 SPACE DMRG, QUANTUM CHEMISTRY, SMALL GRAINS, for S=1/2 spins͒, DMRG may yet be the method of AND NUCLEAR PHYSICS choice as quantum Monte Carlo suffers from the In this section, I shall consider three groups of negative-sign problem and even medium-sized fermionic DMRG applications that seem to have very little in systems of size, say, 12ϫ12 sites would be most useful; in common at first sight: the study of translationally invari- models with non-Abelian symmetries, the implementa- ant low-dimensional models in momentum space, high- tion of the associated good quantum numbers has been precision quantum chemistry calculations for small mol- shown to reduce drastically the truncation error for a ͑ ecules, and studies of small grains and nuclei. However, given number of states McCulloch and Gulacsi, 2000, from a DMRG point of view, all share fundamental 2001, 2002; McCulloch et al., 2001͒. ͑ ͒ properties. Let me first discuss momentum-space Very recently, Verstraete and Cirac 2004 have pro- DMRG, move on to quantum chemistry, and finish by posed a new approach to two-dimensional quantum sys- considering small grains and nuclear physics. tems combining a generalization of their matrix-product formulation of DMRG ͑Verstraete, Porras, and Cirac, 2004͒ and imaginary-time evolution ͑Verstraete, Garcia- A. Momentum-space DMRG Ripoll, and Cirac, 2004͒, discussed in Sec. III.A and Sec. IX.C. One-dimensional matrix-product states are Real-space DMRG precision dramatically deterio- ϫ formed from matrix products of M M matrices rates when applied to long-ranged interactions or hop- ͑A͓␴͔͒␣␤, with M-dimensional auxiliary state spaces on pings. Moreover, momentum is not accessible as a good the bond to the left and right of each site. The two- quantum number in real-space DMRG. Momentum- dimensional generalization for a square lattice is given space DMRG, on the other hand, makes momentum a by the contraction of tensors ͑A͓␴͔͒␣␤␥␦ with good quantum number, works naturally with periodic M-dimensional auxiliary state spaces on the four adja- boundary conditions, and allows trivial manipulation of cent bonds. Finite-temperature and ground states are the single-particle dispersion relation. Momentum-space calculated by imaginary-time evolution of a totally DMRG has already yielded highly satisfying dispersion mixed state along the lines of Verstraete, Garcia-Ripoll, relations and momentum distributions in excellent and Cirac ͑2004͒. As tensorial contractions lead ͑in con- agreement with analytical results ͑Nishimoto, Jeckel- trast to the case of the ansatz matrices A͓␴͔ appearing in mann, et al., 2002͒.

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FIG. 19. Standard reorganization of a two-dimensional lattice as a zigzag one-dimensional chain with long-ranged interactions for DMRG treatment. Typical blocks during a finite- system DMRG application are shown. FIG. 20. Diagonal reorganization of a two-dimensional lattice as used by Xiang et al. ͑2001͒. Typical blocks during a finite- system DMRG application are shown. Consider for definiteness a Hubbard model with local interaction U, but long-ranged hopping tij, infinite-system stage of the algorithm. In momen- ˆ † ͑ ͒ tum space, another warmup procedure must be HLR = ͚ tijc␴,ic␴,j + U͚ ni↑ni↓ 104 ij␴ i found. ͑iii͒ On a one-dimensional lattice with short-ranged in arbitrary dimensions; site i is localized at ri and tij =t͑r −r ͒. With L lattice sites, Fourier transformations interactions it is natural to group sites as they are i j arranged in real space. In momentum space with 1 long-ranged interactions, there is no obvious site † ik·rj † ͑ ͒ c␴,k = ͱ ͚ e c␴,j 105 L j sequence, even in the one-dimensional case. Does the arrangement affect convergence properties? Is yield the band structure there an optimal arrangement? ⑀͑k͒ = ͚ e−ik·rt͑r͒͑106͒ Let us discuss these points in the following. r ͑i͒ DMRG operator representation for general two- and the Hamiltonian body-interaction Hamiltonians. In principle we should like to use DMRG to treat the generic U ˆ ⑀͑ ͒ † † † H = ͚ k c␴,kc␴,k + ͚ c↑k c↓k c↓k c↑k +k −k . Hamiltonian ␴ L 1 2 3 1 2 3 k, k1,k2,k3 Hˆ = ͚ T c†c + ͚ V c†c†c c , ͑108͒ ͑107͒ two-body ij i j ijkl i j k l ij ijkl The reciprocal lattice points k are now taken to be where Tij encodes kinetic energy and one-body “sites.” Note that L is not the linear size of the lattice, potentials and V a generic two-body ͑Coulomb͒ ͑ ijkl but is taken to be the total lattice size because of the interaction; for simplicity, we assume spin to be ͒ typical DMRG mapping to one dimension . contained in the orbital indices. In the worst-case The key idea ͑Xiang, 1996͒ is to arrange these sites scenario, all Vijkl are different, although symme- into a one-dimensional chain with long-ranged interac- tries and model properties may yield decisive sim- tions that is treated by the real-space finite-system plifications. In momentum space, for example, DMRG ͑Nishimoto, Jeckelmann, et al., 2002; Legeza et V Þ0 for one l only once ijk is fixed due to al., 2003a͒. In addition to the conventional good quan- ijkl momentum conservation. tum numbers, states will also be classified by total mo- As for operators living on the same block, mentum, which allows a further substantial segmenta- tion of Hilbert space ͑by a factor of order L͒. Hence for ͗ ͉ † ͉ ͘ Þ ͗ ͉ †͉ Ј͗͘ Ј͉ ͉ ͘ ͑ ͒ m ci cj m˜ ͚ m ci m m cj m˜ , 109 the same number of states kept, momentum-space Ј m· DMRG is much faster than real-space DMRG. To obtain an efficient implementation, several issues all operator pairings have to be stored separately. have to be addressed. Leaving aside the simpler case in which one or two of the four operators live on the single sites in ͑ ͒ i In momentum space there is a huge proliferation DMRG, and assuming that they are all either in ͑ 3͒ of interaction terms of O L that have to be gen- block S or E, this suggests that for L sites of the erated, stored, and applied to wave functions effi- order of O͑L4͒ operators have to be stored. As ciently. their form changes for each of the L possible ͑ii͒ For an application of the finite-system DMRG al- block sizes, the total memory consumption would gorithm, we need to provide blocks and operators be O͑L5M2͒ on disk for all blocks and O͑L4M2͒ in living on these blocks for all block sizes. In real RAM for the current block. At the same time, for space, they are naturally generated during the each of the L steps of a sweep, calculation time

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erator living on block S has now two matching operators in E. A further class of complementary operators

† † ͑ ͒ Okl = ͚ Vijklci cj 112 ij

allows the simplification

† † → ͑ ͒ ͚ Vijklci cj ckcl ͚ Oklckcl. 113 FIG. 21. Composition of blocks from a ͑L−1͒ϫ͑L−1͒ system ijkl kl and two free sites into an LϫL system as used by Xiang et al. ͑2001͒. The heavy solid line indicates the one-dimensional path Memory consumption for the second type of ͑ 3 2͒ through the lattice. The lattice subsets surrounded by the bro- complementary operator is O L M on disk and 2 2 ken lines are blocks A and B; the last added sites are indicated O͑L M ͒ in RAM. Taking all operator combina- by broken circles. Note that blocks overlap for the smaller tions together, global memory consumption is to lattice and are “pulled apart” for the big lattice. The ᭺ and b leading order O͑L3M2͒ on disk and O͑L2M2͒ in stand for the new sites added. RAM, which is a reduction by L2 compared to the naive estimate. In momentum space, due to momentum conservation, memory consumption is re- would be of order O͑L4M3͒,orO͑L5M3͒ for the duced by another factor of L to O͑L2M2͒ on disk entire calculation. and O͑LM2͒ in RAM. Memory consumption as well as the associated Using the complementary-operator technique, calculation time can, however, be reduced drasti- calculation times are dominated by the ͑2/2͒ terms. cally ͑Xiang, 1996͒. Let us consider the three pos- In analogy to the construction of ͑4/0͒ terms “on sible operator distributions on blocks S and E. the fly” by generating the new terms of the sum, ͑a͒ Four operators in one block ͑4/0͒: Terms transforming them, and adding them into the op- † † Vijklci cj ckcl are absorbed into a single-block erator, the Okl can be constructed at a computa- Hamiltonian operator during block growth. As- tional expense of O͑L3M2͒ for generating the L suming i,j,k are in the previous block and site l is new terms to be added to each of the L2 comple- added to form the current block, a representation mentary operators with M2 matrix elements each † † ͑ 2 3͒ 2 of ci cj ck in the previous block basis allows the for- and O L M for transforming the L operators ϫ † † ϫ mation of Vijkl ci cj ck cl in the block-plus-site into the current basis. Using the multiplication product basis, which is then transformed into the technique of Sec. II.I, multiplying the Hamiltonian basis of the current block and added into the to the state vector costs O͑L2M3͒ time at each step single-block Hamiltonian operator. For L blocks, or O͑L3M3͒ per sweep. Global calculation time per ͑ 3͒ † † ͑ 3 3͒ ͑ 4 2͒ O L representations each of ci cj ck are necessary. sweep is thus O L M +O L M , a reduction by These in turn can be compounded into complemen- L2 for the dominant first term ͑typically, MӷL for tary operators, the relevant DMRG applications͒.

† † ͑ ͒ ͑ ͒ Ol = ͚ Vijklci cj ck, 110 ii Setting up ﬁnite-system blocks. The standard prac- ijk tice currently adopted in momentum space ͑Xiang, 1996; Nishimoto, Jeckelmann, et al., 2002͒ so that is to use standard Wilsonian renormalization ͑ ͒ † † → ͑ ͒ Wilson, 1975, 1983 : for the chosen sequence of ͚ Vijklci cj ckcl ͚ Olcl. 111 ͑ ijkl l momenta, blocks are grown linearly as in infinite- system DMRG͒, but diagonalized without superblock embedding and the lowest-energy states re- The complementary operators can be constructed tained, with the important modification that it as discussed in Sec. II.G, assuming the knowledge should be ensured that at least one or two states † † ͑ 2͒ of two-operator terms ci cj . For L blocks, O L of are retained for each relevant Hilbert-space sec- those exist, leading to memory consumption tor, i.e., all sectors within a certain spread about O͑L3M2͒ on disk and O͑L2M2͒ in RAM. the average momentum, particle number, and magnetization. Otherwise DMRG may miss the ͑b͒ Three operators in a block ͑3/1͒: One applies the right ground state, as certain sectors that make an strategy of Eqs. ͑110͒ and ͑111͒, with O and c act- l l important contribution to the ground state may ing on different blocks. not be constructed during the finite-system sweep. ͑c͒ Two operators in a block ͑2/2͒: Again, the Left and right block sectors must be chosen such complementary-operator technique can be applied, that all of them find a partner in the other block with the modification that each complementary op- to combine to the desired total momentum, par-

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ticle number, and magnetization. Moreover, it is mental error, Legeza et al. ͑2003a͒ have pro- advantageous to choose corresponding states in posed a “dynamical block selection scheme” to different sectors such that obvious symmetries are vary M to maintain a fixed truncated weight. not broken, e.g., Sz ↔−Sz if the final state is They find that in this approach z known to be in the Stot=0 sector; it has also been ͑⑀ ͒ E ␳ − Eexact found that choosing initial block sizes too small = La⑀␳ ͑115͒ may result in much slower convergence or trap- Eexact ͑ ͒ ping in wrong minima Legeza et al., 2003a . is very well satisfied for aϷ1, where ⑀␳ is the trun- ͑iii͒ Determining the order of momentum sites. Nish- cated weight. A further advantage of this ap- imoto, Jeckelmann, et al. ͑2002͒ report that what proach is that the number of states needed for a works best for short-ranged Hubbard models is certain precision varies widely in momentum- ordering the levels according to increasing dis- space DMRG, so that for many DMRG steps tance to the Fermi energy of the noninteracting important savings in calculation time can be ͉⑀͑ ͒ ⑀ ͉ realized. limit, k − F , thus grouping levels that have strong scattering together, whereas for longer- All in all, momentum-space DMRG seems a promis- ranged Hubbard models a simple ordering ac- ing alternative to real-space DMRG, in particular for cording to ⑀͑k͒ works best. Similar results have longer-ranged interactions ͑for short-ranged interactions been reported by Legeza et al. ͑2003a͒, who could in one dimension, real-space DMRG remains the demonstrate trapping in wrong minima for inad- method of choice͒. However, momentum-space DMRG equate orderings; Legeza and Sólyom ͑2003͒ have presents additional complications ͑optimal ordering of provided a quantum-information-based method levels, efficient encoding͒ that may not yet have been to avoid such orderings. optimally solved. Nishimoto, Jeckelmann, et al. ͑2002͒ have carried out extensive convergence studies for Hub- bard models in one dimension and two dimen- B. Quantum chemistry sions with various hoppings. Convergence seems to be dominated by the ratio of interaction to A field in which DMRG will make increasingly impor- bandwidth, U/W rather than some U/t. The tant contributions in the next few years is quantum momentum-space DMRG algorithm becomes ex- chemistry. While the first quantum-chemical DMRG cal- ͑ ͒ act in the U/W→0 limit; convergence deterio- culations on cyclic polyene by Fano et al. 1998 and polyacetylene by Bendazzoli et al. ͑1999͒ were still very rates drastically with U/W, but is acceptable for much in the spirit of extended Hubbard models, more U/Wഛ1, which is Uഛ8t for the 2D Hubbard recent work has moved on to calculations in generic model with nearest-neighbor hopping. Generally bases with arbitrary interactions. Let me situate this lat- it seems that for momentum-space DMRG con- ter kind of DMRG application in the general context of vergence depends only weakly on the range of quantum chemistry. hopping as opposed to real-space DMRG. Within the framework of the Born-Oppenheimer ap- Momentum-space DMRG is worst at half-filling proximation, i.e., fixed nuclear positions, two major and deteriorates somewhat with dimension ͑if cal- ways of determining the electronic properties of mol- culations for the same physical scale U/W are ecules are given by Hartree-Fock ͑HF͒ and post-HF cal- compared͒. In two dimensions, for moderate U, culations and density-functional theory. Density- momentum-space DMRG is more efficient than functional theory is computationally rather inexpensive real-space DMRG. and well-suited to quick and quite reliable studies of As can be seen from the results of Nishimoto, ͑ ͒ medium-sized molecules, but is not overly precise, in Jeckelmann, et al. 2002 , even for values of M as particular for small molecules. Here, Hartree-Fock cal- large as several 1000, convergence is not achieved. culations, incorporating Fermi statistics exactly and Due to the complex growth scheme, it cannot be electron-electron interactions on a mean-field level, pro- taken for granted even after many sweeps that the vide good ͑initial͒ results and a starting point for further environmental error has been eliminated; as in refinement, so-called post-HF calculations. Quantum two-dimensional real-space DMRG, sudden chemistry DMRG is one such calculation. drops in energy are observed. In their application, ͑ ͒ For both the HF and post-HF calculations, one starts Nishimoto, Jeckelmann, et al. 2002 report that from a first-quantized Schrödinger equation ͑in atomic for fixed M a fit formula in 1/M, units͒ for the full molecule with N electrons; second 1 quantization is achieved by introducing some suitably ͩ ͪ a1 a2 ͑ ͒ Efit = Eϱ + + + , 114 chosen basis set ͕͉␸ ͖͘. In this way, a general two-body M M M2 ¯ i interaction Hamiltonian as in Eq. ͑108͒ can be derived. works very well and improves results by over an A HF calculation now solves this Hamiltonian at order of magnitude even for final M=2000. Build- mean-field level, and it can be reexpressed in terms of ing on the proportionality between truncated the HF orbitals. A closed-shell singlet ground state is weight and energy error for eliminated environ- then simply given by

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 293

͉ ͘ † † † † ͉ ͘ ͑ ͒ Quantum information techniques have been used by HF = c↑1c↓1 c↑N/2c↓N/2 0 . 116 ¯ Legeza and Sólyom ͑2003͒ both for momentum-space Within standard quantum chemistry post-HF calcula- and quantum chemistry DMRG to devise optimal order- tions, one improves now on this ground state by a ings. Studying various ad hoc orderings, they find fastest plethora of methods. Simple approaches consist in di- and most stable convergence under sweeping for those agonalizing in the subspace of one- or two-particle ex- orderings that maximize entanglement entropy ͓Eq. cited states; others choose a certain subset of electrons ͑22͔͒ between orbitals and the rest of the chain for or- and diagonalize the Hamiltonian fully in one subset of bitals at the chain center; orderings are hence chosen ͑ ͒ orbitals for those complete active space . Taking all such that strongly entangled orbitals are brought to the ͑ ͒ electrons excitations and all orbitals into account, one chain center ͑often those closest to the Fermi surface͒. obtains, within the originally chosen basis set, the exact This ordering can be constructed iteratively, starting many-body solution, known as the full configuration- from a first guess as obtained by the Cuthill-McKee or- interaction solution. While the approximate schemes dering. 5 7 typically scale as N to N in the number of orbitals, full Once the orbital ordering has been selected, current configuration-interaction solutions demand exponential applications follow a variety of warmup procedures to effort and are available only for some very small mol- build all sizes of blocks and to ensure that block states ecules. contain as many states as possible relevant for the final From the DMRG point of view, determining the result. This may be done by augmenting the block being ground state of the full second-quantized Hamiltonian is built by a toy environment block with some low-energy formally equivalent to an application of the momentum- states that provide an environment compatible with the space DMRG, with some additional complications. As block, i.e., leading to the desired total quantum numbers momentum conservation does not hold for Vijkl, there ͑Chan and Head-Gordon, 2002͒, doing infinite-system are O͑N͒ more operators. Moreover, the basis set must DMRG on left and right blocks while also targeting ex- be chosen carefully from a quantum chemist’s point of cited states ͑Mitrushenkov et al., 2001͒, and defining a view. As in momentum-space DMRG, a good sequence minimum number of states to be kept even if the non- of the strongly different orbitals has to be established for zero eigenvalues of the density matrix do not provide DMRG treatment so that convergence is optimized. enough states ͑Legeza et al., 2003a͒. Block-state choice Also, an initial setup of blocks must be provided, includ- can also be guided by entanglement entropy calculations ing operators. ͑Legeza and Sólyom, 2003͒. Orbital ordering in quantum chemistry turns out to be Various molecules have by now been studied using crucial to the performance of the method; a badly cho- DMRG, both for equilibrium and out-of-equilibrium sen ordering may lead to much slower convergence or configurations, ground states, and excitations. H2O has even trapping in some higher-energy local minimum. been serving as a benchmark in many quantum chemis- The simplest information available, the HF energy and try calculations and has been studied within various fi- occupation of the orbitals, has been exploited by White nite one-particle bases, more recently in the so-called and Martin ͑1999͒, Daul, Ciofini, et al. ͑2000͒, and double ␨ polarized ͑DZP͒ basis ͑Bauschlicher and Tay- Mitrushenkov et al. ͑2001, 2003͒ to enforce various or- lor, 1986͒ with 8 electrons in 25 orbitals ͑and 2 “frozen” derings of the orbitals. electrons in the 1s oxygen orbital͒ and the larger triple ␨ Chan and Head-Gordon ͑2002͒ have proposed the use plus double polarization ͑TZ2P͒ basis ͑Widmark et al., of the symmetric reverse Cuthill-McKee reordering 1990͒ of 10 electrons in 41 orbitals. While the larger ba- ͑Cuthill and McKee, 1969; Liu and Sherman, 1975͒, sis will generally yield lower energies, one can compare in which orbitals are reordered such that Tij matrix ele- how close various approximations come to an exact di- ments above a certain threshold are as band diagonal as agonalization ͑full configuration interaction if at all pos- possible. This reduces effective interaction range, lead- sible͒ solution within one particular basis set. In the ing to faster convergence in M and reduced sweep num- smaller DZP basis, the exact ground-state energy is at ber, as confirmed by Legeza et al. ͑2003b͒, who opti- −76.256 634 H ͑Hartree͒. While the Hartree-Fock solu- mized the position of the Fock term Tij tion is several hundred mH above, and the single and ͚ ͑ ͒ + k occupied 4Vikkj−2Vikjk and reported reductions in M double configuration-interaction solution about ten mH by a third or so. too high, and various coupled-cluster approximations Another approach exploits the short range of chemi- ͑Bartlett et al., 1990͒ are reaching chemical accuracy of 1 cal interactions which in the basis of typically delocal- mH with errors between 4.1 and 0.2 mH, DMRG results ized HF orbitals leads to a large effective range of inter- of various groups ͑White and Martin, 1999; Daul, actions detrimental to DMRG. This may be reduced by Ciofini, et al., 2000; Chan and Head-Gordon, 2002͒ are changing to more localized orbitals. Orbital-localization better than 0.2 mH at Mϳ400, reaching an error of techniques were first studied by Daul, Ciofini, et al. about 0.004 mH at Mϳ900 ͑Chan and Head-Gordon, ͑2000͒, who applied a localization procedure ͑Pipek and 2002; Legeza et al., 2003a͒. Moving to the larger TZ2P Mezey, 1989͒ to the unoccupied orbitals to avoid in- basis ͑Chan and Head-Gordon, 2003͒, in which there is creases in the starting configuration energy, but they did no exact solution, accuracies of all methods are ranking not find remarkable improvement over orderings of the similarly, with DMRG outperforming the best coupled- unmodified HF orbitals. cluster results with an error of 0.019 mH at Mϳ3000

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 294 U. Schollwöck: Density-matrix renormalization group and reaching an error of 0.005 mH at Mϳ5000, with the the exponentially large number of Hilbert-space states. extrapolated result being −76.314 715 H. The compari- White ͑2002͒ moves to the HF basis and carries out a son is even more favorable if the nuclei are arranged particle-hole transformation; the canonical transforma- away from their equilibrium configuration. tions are then carried out in the much smaller space of Excellent performance is also observed in the extrac- states not annihilated by one of the Vijkl and formed by ͑ tion of dissociation energy curves for N2 Mitrushenkov the minimum of creation operators from the HF vacuum ͒ ͑ ͒ ͑ ͒ † et al., 2001 and water Mitrushenkov et al., 2003 . ground state . For example, the term Vijkldi djdkdl, † Low-lying excited states have been studied for HHeH where the di ,di are particle-hole transformed fermionic ͑ ͒ ͑ ͒ Daul, Ciofini, et al., 2000 ,CH2 Legeza et al., 2003a , operators, implies the consideration of the “left” state ͑ ͒ †͉ ͘ † † †͉ ͘ and LiF Legeza et al., 2003b ; the latter case has pro- di 0 with HF energy El and the “right” state dj dkdl 0 vided a test system to study how efficiently methods map with HF energy Er. In this smaller state space, a se- out the ionic-neutral crossover of this alkali halogenide quence of canonical basis transformations may be imple- for increasing bond length. With relatively modest nu- mented as a differential equation, as originally proposed merical effort ͑M not in excess of 500 even at the com- by Wegner ͑1994͒ as the flow-equation method: and putationally most expensive avoided crossing͒ relative Glazek and Wilson ͑1994͒ as similarity renormalization. errors of 10−9,10−7, and 10−3 for the ground-state energy, With A some anti-Hermitian operator, a formal time de- first-excited-state energy, and the dipole moment com- pendence is introduced to the unitary transformation pared to the full configuration-interaction solution Hˆ ͑t͒=exp͑tA͒Hˆ ͑0͒exp͑−tA͒, where Hˆ ͑0͒ is the original ͑ ͒ Bauschlicher and Langhoff, 1988 have been obtained. Hamiltonian of Eq. ͑108͒ expressed in the HF basis. The The dipole-moment precision increases by orders of corresponding differential equation, magnitude away from the avoided crossing. To assess the further potential of quantum chemistry dHˆ ͑t͒ = ͓A,Hˆ ͑t͔͒, ͑117͒ DMRG for larger molecules and N orbital bases, it dt should be kept in mind that the generic N4 scaling of the algorithm—which does compare extremely favorably is modified by making A time dependent itself. One now with competing methods that scale as N5 to N7—is modi- expands fied by several factors. On the one hand, for a given ˆ ͑ ͒ ͑ ͒ ͑ ͒ desired error in energy, various authors report, as for H t = ͚ a␣ t h␣ 118 ␣ conventional DMRG, that ␦Eϳ⑀␳, the truncated weight, 2 which scales ͑Sec. III.B͒ as ⑀␳ =exp͑−k ln M͒. k, how- and ever, shrinks with size in a model-dependent fashion, the most favorable case being that orbitals have essentially A͑t͒ = ͚ s␣a␣͑t͒h␣, ͑119͒ chainlike interactions with only a few nearest-neighbor ␣ orbitals. It may be more realistic to think about molecu- where each h␣ is the product of creation and annihila- lar orbitals as arranged on a quasi-one-dimensional strip tion operators and each s␣ some constant yet to be cho- of some width L related to the number of locally inter- sen under the constraint of the anti-Hermiticity of A.To acting orbitals; in that case, for standard strip geom- avoid operator algebra, White introduces etries, the constant has been found to scale as L−1.SoN4 ␥ scaling is in my view overly optimistic. On the other ͓h␣,h␤͔ = ͚ B␣␤h␥, ͑120͒ hand, on larger molecules orbital-localization techniques ␥ may be more powerful than on the small molecules stud- where additional contributions to the commutator that ied so far, so that orbital interactions become much cannot be expressed by one of the operator terms in Eq. more sparse and scaling may actually improve. ͑118͒ are suppressed; this eliminates three-body interac- One other possible way of improving the scaling prop- tions generically generated by the transformation and erties of quantum chemistry DMRG might be the ca- not present in the original Hamiltonian. Then Eq. ͑117͒, nonical diagonalization approach proposed by White ͑ ͒ with A time dependent, reduces to a set of differential 2002 , which attempts to transform away numerically by equations, a sequence of canonical basis transformations as many as possible of the nondiagonal matrix elements of the da␥͑t͒ ␥ = ͚ B␣␤s␣a␣͑t͒a␤͑t͒, ͑121͒ second-quantized Hamiltonian Hˆ of Eq. ͑108͒. This is dt ␣␤ ˆ done so that entire orbitals can be removed from H, that can now be integrated numerically up to some time resulting in a new, smaller quantum chemistry problem t. It can now be shown that the goal of diminishing a␣͑t͒ in a different basis set, which may then be attacked by ˆ some DMRG technique like those outlined above. The for all nondiagonal contributions to H can be achieved removed orbitals, which are no longer identical with the efficiently by setting original ones, are typically strongly overlapping with the −1 s␣ = ͑E − E ͒ , ͑122͒ energetically very high-lying and low-lying orbitals of l r the original problem that are almost filled or empty. Of where El and Er are the HF energies of the left and right course, these transformations cannot be carried out for HF orbitals in h␣. Stopping after some finite time t, one

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 295 can now remove orbitals ͑e.g., with the lowest occu- observation and calculation of well-defined quasiparticle pancy, that is, almost full or empty before the particle- excitations in small interacting disordered systems with hole transformation͒ and use DMRG for the full many- high dimensionless conductance g ͑Gobert, Schechter, et body problem in the reduced orbital space. White ͑2002͒ al., 2004͒. finds this approach to yield very good results as com- Particle-hole DMRG has also been applied success- pared to full configuration-interaction calculations for a fully in nuclear physics ͑Dukelsky and Pittel, 2001; Dim- stretched water molecule, but as of now the full poten- itrova et al., 2002; Dukelsky et al., 2002͒ in the frame- tial of the method remains unexplored. work of the nuclear shell model, in which a nucleus is To conclude, in my opinion DMRG has been estab- modeled by core orbitals completely filled with neutrons lished as a competitive method with essentially the same and protons ͑a “doubly magic core”͒ and valence orbit- quality as the full configuration-interaction method in als partially filled with nucleons. The core is considered quantum chemistry and should be taken seriously. How- inert, and, starting from some Hartree-Fock-level orbital ever, efficient geometrical optimization techniques to configuration for the valence nucleons, a two-body determine lowest-energy molecular geometries are still Hamiltonian for these nucleons is solved using the lacking. particle-hole method. This is conceptually very similar to the post-HF approaches of quantum chemistry, with C. DMRG for small grains and nuclear physics model interactions for the nucleons as they are not so precisely known, such as pairing interactions, quadrupo- Let us reconsider the generic Hamiltonian of Eq. lar interactions, and the like. ͑108͒, where energy levels above and below the Fermi This approach has been very successful for nucleons energy are ordered in ascending fashion and where a in very large angular momentum shells interacting simple model interaction is chosen. This Hamiltonian through a pairing and a quadrupolar force in an oblate can then be treated in the following so-called particle- nucleus ͑Dukelsky and Pittel, 2001͒, with up to 20 par- hole reformulation of infinite-system real-space DMRG ticles of angular momentum j=55/2, obtaining energies ͑Dukelsky and Sierra, 1999, 2000͒: Starting from two ini- converged up to 10−6 for M=60; for 40 nucleons of j tial blocks from a small number of states, one ͑hole =99/2, four-digit precision was possible; in this case, block͒ from the lowest unoccupied one-particle levels 38% of energy was contained in the correlations ͑Dukel- above the Fermi energy and the other ͑particle block͒ sky et al., 2002͒. For more realistic calculations of the from the highest occupied one-particle levels below, we nucleus 24Mg, with four neutrons and four protons out- iteratively add further states to particle and hole blocks, side the doubly magic 16O core, convergence in M was moving away from the Fermi surface, determine the so slow that almost the complete Hilbert space had to be ground state, and in the usual DMRG procedure calcu- considered for good convergence ͑Dimitrova et al., late density matrices and the reduced basis transforma- 2002͒. The fundamental drawback of the particle-hole tions by tracing out particle and hole states, respectively. approach is that it does not allow for an easy implemen- This approach may work if there is a clear physical rea- tation of the finite-system algorithm ͑levels far away son that states far away from the Fermi surface will from the Fermi surface hardly couple to the system, giv- hardly influence low-energy physics and if interactions ing little relevant information via the density matrix͒ show translational invariance in energy space; otherwise and that angular momentum conservation is not ex- the infinite-system algorithm should fail. The advantage ploited. Pittel et al. ͑2003͒ are currently aiming at imple- of such simple models is that much larger systems can be menting an algorithm for nuclei using angular momen- treated than in quantum chemistry. tum, circumventing these difficulties ͑see also Dukelsky A model Hamiltonian which happens to combine both and Pittel, 2004͒. features is the reduced BCS Hamiltonian ˆ ͚ ͑⑀ ␮͒ † ␭ ͚ † † ͑ ͒ HBCS = j − cj␴cj␴ − d ci↑ci↓cj↓cj↑, 123 VIII. TRANSFER-MATRIX DMRG: CLASSICAL AND j␴ ij QUANTUM SYSTEMS where DMRG has been able to definitely settle long- standing questions ͑Dukelsky and Sierra, 1999, 2000; Conventional DMRG is essentially restricted to T=0 Dukelsky and Pittel, 2001͒ on the nature of the break- calculations, with some computationally expensive for- down of BCS superconductivity in small grains, ex- ays into the very-low-temperature regimes possible pected when the level spacing d in the finite system be- ͑Moukouri and Caron, 1996; Batista et al., 1998; Hall- comes of the order of the BCS gap ⌬ ͑Anderson, 1959͒. berg et al., 1999͒. Decisive progress was made by DMRG has conclusively shown that there is a smooth Nishino ͑1995͒, who realized that the DMRG idea could crossover between a normal and a superconducting re- also be used for the decimation of transfer matrices, gime in the pairing order parameter and other quanti- leading to the name of transfer-matrix renormalization ties, in contradiction to analytical approaches indicating group ͑TMRG͒. This opened the way to DMRG studies a sharp crossover. This approach to DMRG has been of classical statistical mechanics at finite temperature for extended to the study of the Josephson effect between systems on two-dimensional Lϫϱ strips. If one applies superconducting nanograins with discrete energy levels the generic mapping of d-dimensional quantum systems ͑Gobert, Schollwöck, and von Delft, 2004͒ and to the at finite temperature to ͑d+1͒-dimensional classical sys-

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L tems, TMRG also permits study of thermodynamic Nsite properties of one-dimensional quantum systems at finite ␭N ͑ ͒ Z = ͚ i , 126 temperature. i=1 ␭ Ͼ␭ ജ T ͑L͒ where 1 2 ¯ are the eigenvalues of ; the largest being positive and nondegenerate for positive Boltz- A. Classical transfer-matrix DMRG in two dimensions: mann weights, partition function and unnormalized den- TMRG sity matrix simplify in the thermodynamic limit N→ϱ to ␭N ͑ ͒ Consider the textbook transfer-matrix method for a Z = 1 127 one-dimensional classical system with Hamiltonian Hˆ and and N states ͉␴ ͘ on each site i, e.g., the Ising model. site i ␳ˆ = ␭N͉␭ ͗͘␭ ͉, ͑128͒ Assuming nearest-neighbor interactions, one performs a u 1 1 1 ˆ ͚ ˆ ͉␭ ͘ local decomposition of the Hamiltonian H= hi, where where 1 is the normalized eigenvector of the largest each hˆ contains the interaction between sites i and i+1. transfer-matrix eigenvalue. i Consider now Fig. 23. The unnormalized density ma- Taking the partition function of a system of length N trix has exactly the same form, up to the prefactor, of the with periodic boundary conditions, ͉␭ ͘ pure-state projector of standard DMRG, with 1 assuming the role of the target state there. Tracing out the ␤H ͚N ␤ ˆ − − i=1 hi ͑ ͒ ZL =Tre =Tre , 124 right “environment” half, the left “system” unnormalized reduced density matrix is given by ͚ ͉␴ ͗͘␴ ͉ and inserting identities ␴ i i , one obtains for a i NL/2 translationally invariant system Z =TrT N, where the site L ␳ ␭N͉␭ ͗͘␭ ͉ ␭N ͉ ͗͘ ͉ ͑ ͒ ϫ T ˆ uS =TrE 1 1 = ͚ w␣ w␣ w␣ , 129 Nsite Nsite transfer matrix reads 1 1 ␣=1 ␤ ͗␴ ͉T ͉␴ ͘ ͗␴ ͉ − hi͉␴ ͘ i i+1 = i e i+1 . where w␣ are the positive eigenvalues to the normalized eigenvectors ͉w␣͘ of the reduced system density matrix; ͚ From this, one deduces for N→ϱ the free energy per ␣w␣ =1. Completing the partial trace, one finds ␭ ͓T͔ ␭ site f=−kBT ln 1 , where 1 is the largest eigenvalue NL/2 T site of , from which all thermodynamic properties follow. N Z =Tr ␳ˆ = ␭ ͚ w␣, ͑130͒ Moving now to a two-dimensional geometry, on a strip S uS 1 ␣=1 of dimension LϫN with a short-ranged Hamiltonian, the transfer-matrix method may be applied in the limit so that indeed the best approximation to Z is obtained N→ϱ for some finite width LӶN, treating a row of L by retaining in a reduced basis the eigenvectors to the sites as one big site; transfer-matrix size then grows as largest eigenvalues w␣ as in conventional DMRG. L Let us explain the DMRG transfer-matrix renormal- Nsite, strongly limiting this approach in complete analogy to the exact diagonalization of quantum Hamiltonians. ization ͑Nishino, 1995͒ in more detail. Because the trans- Hence ͑Fig. 22͒ the first dimension, with finite system fer matrix factorizes into local transfer matrices, some length L, is attacked by DMRG applied to the transfer minor modifications of the standard DMRG approach instead of the Hamiltonian matrix, to keep transfer- are convenient. Assume as given an MϫM transfer ma- matrix size finite and sufficiently small by decimation, trix of a system of length L, while retaining the most important information. Results ͗ ␴͉T ͑L͉͒ ˜ ␴˜ ͘ ͑ ͒ are extrapolated to infinite size. The second dimension, m m , 131 with infinite system length, is accounted for by the one- where the sites at the active ͑growth͒ end are kept ex- dimensional transfer-matrix method. plicitly as opposed to standard DMRG, and ͉m͘,͉m˜ ͘ are To prove that this concept works, i.e., that an optimal block states ͑Fig. 24͒. Considering now the superblock of decimation principle can be set up for transfer matrices length 2L, we see that the transfer matrix reads in analogy to the case made for Hamiltonians, consider a short-ranged classical Hamiltonian at TϾ0onaLϫN ͗mS␴S␴EmE͉T ͑2L͉͒m˜ S␴˜ S␴˜ Em˜ E͘ →ϱ strip, where N . We now define an unnormalized S S ͑L͒ S S S E ͑2͒ S E ˆ = ͗m ␴ ͉T ͉m˜ ␴˜ ͗͘␴ ␴ ͉T ͉␴˜ ␴˜ ͘ ␳ −␤H ͑ ͒ density matrix ˆ u =e in reality an evolution operator ͑ ͒ by ϫ͗mE␴E͉T L ͉m˜ E␴˜ E͘. ͑132͒

͑ ͒ Lanczos diagonalization yields the eigenvector of ͗␴͉␳ ͉␴͘ ͗␴͉͓T L ͔N͉␴͘ ͑ ͒ ͑ ͒ ˜ ˆ u = ˜ , 125 T 2L ␭ to the maximum eigenvalue 1. Again in some analogy to conventional DMRG, the fact that the trans- where ␴˜ and ␴ label states of the L sites on top and fer matrix is a product can be used to speed up calcula- bottom of the strip. T ͑L͒ is the band transfer matrix of tions by decomposing ͉␾͘=T ͑2L͉͒␺͘ into three successive ␳ ͉␾͘ T ͑L͓͒T ͑2͒͑T ͑L͉͒␺͔͒͘ Fig. 22. The partition function Z=Tr ˆ u is then given as multiplications, = .

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 297

The obtained eigenvector now yields the reduced density matrices and the reduced basis transformations. The ϫ last step is to build the MNsite MNsite transfer matrix for a system of length L+1 ͑Fig. 25͒:

͗m␴͉T ͑L+1͉͒m˜ ␴˜ ͘ = ͚ ͗m͉n␶͗͘n␶͉T ͑L͉͒n˜ ␶˜͗͘␶␴͉T ͑2͉͒␶˜␴˜ ͘ nn˜ ␶␶˜ ϫ͗n˜ ␶˜͉m˜ ͘. ͑133͒ The TMRG procedure is repeated up to the desired ␭ final length, the free energy then obtained from 1; this allows the calculation of all thermodynamic quantities FIG. 22. Strategy for the DMRG treatment of two- through numerical differentiation of the free energy. To dimensional classical systems. avoid numerically unstable second derivatives for specific heat and susceptibility, it is convenient to consider the first-order derivative of the average energy extracted surface critical exponents that demonstrate the Q inde- ͗␴ ␴ ͘ ͗␭ ͉␴ ␴ ͉␭ ͘ pendence of the universality class of the surface transi- from expectation values such as i j = 1 i j 1 ob- ␤ ˆ ␤ ˆ tion ͑Igloi and Carlon, 1999͒. tained by replacements e− hi →␴ e− hi in the above cal- i TMRG has also been used to study critical properties culations. At the same time, correlation lengths may be ͑ extracted both from these expectation values or from of truly two-dimensional classical systems Honda and the two leading transfer-matrix eigenvalues, Horiguchi, 1997; Tsushima et al., 1997; Carlon, Chat- elain, and Berche, 1999; Sato and Sasaki, 2000͒, such as ␰ ͑␭ ␭ ͒ ͑ ͒ the spin-3/2 Ising model on a square lattice ͑Tsushima et = − 1/ln Re 2/ 1 . 134 al., 1997͒, the Ising model with line defects ͑Chung et al., To refine results, a finite-system calculation can be set up 2000͒, the three-state chiral clock model as example of a as in conventional DMRG. A main technical difference commensurate-incommensurate phase transition ͑Sato between conventional DMRG and classical transfer- and Sasaki, 2000͒, the 19-vertex model in two dimen- matrix DMRG is the absence of good quantum num- sions as a discrete version of the continuous XY model bers, which simplifies the algorithm but is hard on com- to study the Kosterlitz-Thouless phase transition putational resources. ͑Honda and Horiguchi, 1997͒, and the random A large body of work has emerged that strongly relies exchange-coupling classical Q-state Potts model which on the finite strip width provided by the TMRG, study- was demonstrated to have critical properties indepen- ing competing bulk and surface effects in two- dent of Q ͑Carlon, Chatelain, and Berche, 1999͒. Lay ϫϱ dimensional Ising models on a L strip. The confined and Rudnick ͑2002͒ have used the TMRG to study a Ising model has been used to model two coexisting continuous Ginzburg-Landau field-theory formulation phases in the presence of bulk and surface fields, as well of the two-dimensional Ising model by retaining the as gravity, in order to model wetting and coexistence largest eigenvalue eigenfunctions of bond transfer matri- phenomena and the competition of bulk and surface ef- ces as state space. fects in finite geometries. In the case of opposing surface Gendiar and Surda ͑2000͒ have extended the TMRG fields favoring phase coexistence at zero bulk field up to to periodic boundary conditions to obtain, at the ex- some temperature that ͑unintuitively͒ goes to the wet- pense of the expected lower DMRG precision much bet- ting temperature for L→ϱ ͑Parry and Evans, 1990͒,it ter thermodynamic-limit scaling behavior for L→ϱ. could be shown that gravity along the finite-width direc- Critical temperature and thermal and magnetic critical tion suppresses fluctuations such that it restores the bulk critical temperature as limiting temperature for coexist- exponents of the two-dimensional Ising model are all ence ͑Carlon and Drzewin´ ski, 1997, 1998͒. These studies extracted with modest computational effort at a stagger- were extended to the competition of surface and bulk ing six- to seven-digit precision compared to the On- fields ͑Drzewin´ ski et al., 1998͒. Further studies eluci- sager solution. dated finite-size corrections to the Kelvin equation at complete wetting for parallel surface and bulk fields ͑Carlon et al., 1998͒, the nature of coexisting excited states ͑Drzewin´ ski, 2000͒, the scaling of cumulant ratios B. Corner transfer-matrix DMRG: An alternative approach ͑Drzewin´ ski and Wojtkiewicz, 2000͒, and an analysis of the confined Ising model at and near criticality for com- Following Baxter ͑1982͒, one may conclude that the peting surface and bulk fields ͑Maciolek, Drzewin´ ski, essential aspect of the reduced density-matrix is that it and Ciach, 2001; Maciolek, Drzewin´ ski, and Evans, represents a half cut in the setup of Fig. 23 whose spatial 2001͒. In the case of the Potts model with QϾ4, where extension is to be taken to the thermodynamic limit. The there is a first-order phase transition in the bulk ͑Baxter, same setup can be obtained considering the geometry of 1982͒ but a second-order phase transition at the surface Fig. 26, where four corner transfer matrices are defined ͑Lipowsky, 1982͒, TMRG permitted the extraction of as

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 298 U. Schollwöck: Density-matrix renormalization group

FIG. 24. Transfer-matrix DMRG step.

most important of which define the reduced basis transformation. This reduced basis transformation is then car- FIG. 23. Pictorial representation of the partition function, the ried out on T ͑L+1͒ and C͑L+1͒, completing one corner ͑unnormalized͒ density matrix, and the reduced density matrix TMRG step. The crucial advantage of this algorithm is for a two-dimensional classical system on a strip geometry. The that it completely avoids the diagonalization of some black stripe represents a band transfer matrix. Adapted from large sparse matrix. Nishino in Peschel, Hallberg, et al., 1999. The corner TMRG allows us to calculate quantities similar to TMRG, all thermodynamic variables as some derivative of the free energy or using local expectation ͗␴Ј͉ ͑L͉͒␴͘ ͗␴ ␴ ͉T ͑2͉͒␴ ␴ ͘ ͑ ͒ C = ͚ ͟ i j k l , 135 values. For the two-dimensional Ising model, Nishino ␴ ͗ ͘ int ijkl and Okunishi ͑1997͒ obtained Ising critical exponents at where the product runs over all site plaquettes and the four-digit precision, using systems of up to L=20 000. sum over all internal site configurations, i.e., the states For the Q=5 two-dimensional Potts model, which has a on the sites linking to neighboring corner transfer matri- very weak first-order transition, they could determine ces are kept fixed. The corner-site state is invariant un- the latent heat LϷ0.027 compared to an exact L der application of the these matrices. The unnormalized =0.0265 ͑Baxter, 1982͒, which is hard to see in Monte reduced density matrix is then given by Carlo simulations due to metastability problems. Other applications have considered the spin-3/2 Ising model ͗␴͑5͉͒␳ˆ ͉␴͑1͒͘ = ͚ ͗␴͑5͉͒C͑L͉͒␴͑4͒͗͘␴͑4͉͒C͑L͉͒␴͑3͒͘ u ͑Tsushima and Horiguchi, 1998͒ and a vertex model with ␴͑4͒␴͑3͒␴͑2͒ seven vertex configurations, modeling an order-disorder ϫ͗␴͑3͉͒C͑L͉͒␴͑2͒͗͘␴͑2͉͒C͑L͉͒␴͑1͒͑͘136͒ transition ͑Takasaki et al., 2001͒. More recent extensions ͑ ͑L͒ study self-avoiding-walk models in two dimensions Fos- with the center site fixed. Diagonalizing C , and ob- ͒ ␭ ͉␭ ͘ ter and Pinettes, 2003a, 2003b . taining eigenvalues i and eigenvectors i , we see that One may also generalize the corner TMRG to three- ␳ ͑ ͒ the reduced density matrix ˆ u L then reads dimensional classical models ͑Nishino and Okunishi, ͒ ␳ ͑ ͒ ␭4͉␭ ͗͘␭ ͉͑͒ 1998 . While the implementation is cumbersome, the ˆ u L = ͚ i i i 137 i idea is simple: the semi-infinite cut line in the plane leading to four corner matrices gives way to a semi-infinite and the partition function, obtained by tracing out the cut plane in some volume leading to eight corner ten- reduced density matrix, is sors, two of which have an open side. Growing these ͑ ͒ ͚ ␭4 ͑ ͒ corner tensors involves concurrent growing of corner Z L i . 138 matrices, band transfer matrices, and plaquette transfer i Following the argument for classical TMRG, Nishino and Okunishi ͑1996͒ have introduced the corner transfer- matrix renormalization group. A sequence of increasingly large corner transfer matrices is built ͑Fig. 27͒ as

͗m␴␴C͉C͑L+1͉͒m˜ ␴˜ ␴C͘ = ͚ ͗␴␴C͉T ͑2͉͒␶C␴˜ ͘ nn˜ ␶C ϫ͗n␶C͉C͑L͉͒n˜ ␶C͘ ϫ͗m␴͉T ͑L͉͒n␶C͘ ϫ͗n˜ ␶C͉T ͑L͉͒m˜ ␴˜ ͘, ͑139͒ where previous reduced basis transformations are sup- posed to have taken place for C͑L͒ and T ͑L͒. Similarly, T ͑L+1͒ is built as in classical TMRG. Diagonalizing C͑L+1͒ ␭ yields eigenvalues i and associated eigenvectors, the M FIG. 25. Construction of the enlarged transfer matrix.

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 299 matrices. Numerical results, however, indicate clear practical limitations of the concept.

C. Quantum statistical mechanics in one dimension: Quantum TMRG

It is now straightforward—at least in principle—to ex- tend classical TMRG to the calculation of the thermodynamics of one-dimensional quantum systems due to the general relationship between d-dimensional quantum and ͑d+1͒-dimensional classical problems. This ap- ͑ ͒ FIG. 26. Corner transfer-matrix setup for the calculation of proach was first used by Bursill et al. 1996 and fully reduced density matrices and partition functions in two- ͑ ͒ developed by Wang and Xiang 1997 as well as by Shi- dimensional classical models. bata ͑1997͒. To appreciate the following, it is best to visualize quantum TMRG as an extension of the quantum transfer-matrix method ͑Betsuyaku, 1984, 1985͒ in N/2 L ␤ ͗␴2j+1␴2j+1͉ − h2i−1/L͉␴2j ␴2j͘ the same way conventional DMRG extends exact diago- ZL =Tr͟ ͟ 2i−1 2i e 2i−1 2i nalization. i=1 j=1 ␤ Consider the mapping of one-dimensional quantum ϫ͗␴2j␴2j ͉e− h2i/L͉␴2j−1␴2j−1͘. systems to two-dimensional classical systems, as used in 2i 2i+1 2i 2i+1 the quantum transfer-matrix and quantum Monte Carlo Figure 28 shows that as in a checkerboard only every ͑Suzuki, 1976͒ methods: Assume a Hamiltonian that second plaquette of the two-dimensional lattice is active ͑ ͒ contains nearest-neighbor interactions only and is in- i.e., black . To evaluate ZL the trace is taken over all variant under translations by an even number of sites. local states of the ͑2L+1͒ϫN sites, while noting that the We now introduce the well-known checkerboard decom- trace in Eq. ͑140͒ enforces periodic boundary conditions ͑ ͒ ˆ ˆ ˆ ˆ ͚N/2 ˆ ͉␴1͘ ͉␴2L+1͘ position Suzuki, 1976 H=H1 +H2. H1 = i=1 h2i−1 and along the imaginary-time direction, i = i . Noth- ˆ ͚N/2 ˆ ˆ ing specific needs to be assumed about boundary condi- H2 = i=1 h2i. Here hi is the local Hamiltonian linking sites i and i+1; neighboring local Hamiltonians will in tions along the real-space direction. Note that the orien- tation of the transfer matrix has changed compared to ͓ ˆ ˆ ͔Þ general not commute, hence H1 ,H2 0. However, all Fig. 22. ˆ ˆ terms in H1 or H2 commute. N is the system size in real Working backward from the partition function ZL,we space ͑which we shall take to infinity͒. can now identify transfer and density matrices. Introduc- ͑ ͒ ␶ ͑ ␤ ͒ Following Trotter 1959 , we consider a sequence of ing a local transfer matrix as ˆk =exp − hk /L , and ex- approximate partition functions ploiting the assumed restricted translational invariance, ␤ ␤ we find that the global transfer matrices ͑ − H1/L − H2/L͒L ͑ ͒ ZL ª Tr e e , 140 ͗␴1 ␴2L+1͉T ͑2L+1͉͒␯1 ␯2L+1͘ with Trotter number L. Then ¯ 1 ¯ L ͑ ͒ Z = lim ZL 141 ͗␴2j+1␯2j+1͉␶ ͉␴2j␯2j͘ L→ϱ = ͟ ˆ1 , j=1 holds: quantum effects are suppressed as neglected com- 2 mutators between local Hamiltonians scale as ͑␤/L͒ . ͗␯1 ␯2L+1͉T ͑2L+1͉͒␶1 ␶2L+1͘ ˆ ˆ ¯ 2 ¯ One now expands H1 and H2 in the exponentials of L ͑ ͒ ZL in Eq. 140 into the sums of local Hamiltonians and ͗␯2j␶2j͉␶ ͉␯2j−1␶2j−1͘ = ͟ ˆ2 inserts 2L+1 times the identity decomposition j=1 N ͑ ͒ ͉␴ ͗͘␴ ͉ ͑ ͒ can be defined see Fig. 28 , representing odd and even I = ͟ ͚ͩ i i ͪ, 142 i=1 ␴ bonds on the chain because of the alternating checker- i board decomposition. The local transfer matrices are sandwiching each exponential. Introducing an additional linking the states of two sites at different Trotter times label j for the identity decompositions, we find ͑2L+1͒ and are evaluated using the Trotter-time-independent ϫN sites, of which each site carries two labels, corre- ˆ ͉ ͘ ͉ ͘ eigenbasis of the local Hamiltonian, h i =Ei i : sponding to two dimensions, the real-space coordinate ͑subscript͒ i and the Trotter-space ͑imaginary-time͒ co- ␤ ˆ ␤ − h/L ͚ − Ei/L͉ ͗͘ ͉ ͑ ͒ ͑ ͒ e = e i i . 143 ordinate superscript j. Thinking of both coordinates as i spatial, one obtains a two-dimensional lattice of dimension ͑2L+1͒ϫN, for which the approximate partition Summing over all internal degrees of freedom ͉␯i͘,we function ZL reads obtain matrix elements

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T ͑3͒͑␴3␴3 ␴2␴2 ␴1␴1͒ 1 2; 1 3; 2 3 ͗␴3␴3͉␶ ͉␴2␴2͗͘␴2␴2͉␶ ͉␴1␴1͘ ͑ ͒ = ͚ 1 2 ˆ1 1 2 2 3 ˆ2 2 3 . 149 ␴2 2 The addition of plaquettes follows a zigzag pattern. Let us consider just the case in which the transfer matrix grows to comprise an even number of plaquettes. This number is L. Then the growth formula reads ͑Fig. 29͒ T ͑L+1͒͑␴L+1␴L+1 ␴L ␴L ␴1␴1͒ 1 2 ;m1 1 m3 3 ; 2 3 = ͚ ͗␴L+1␴L+1͉␶ˆ ͉␴L␴L͘T ͑L͒͑␴L␴L;m m ;␴1␴1͒. FIG. 27. Corner transfer-matrix growth using band and 1 2 1 1 2 2 3 1 3 2 3 ␴L plaquette transfer matrices. Solid states are summed over. 2 I distinguish between inner and outer states, the outer ͗␴1 ␴2L+1͉T T ͉␶1 ␶2L+1͘ ͑ ͒ states being those of the first and last row of the transfer ¯ 1 2 ¯ . 144 matrix. For the inner states, the compound notation m1 Using the global transfer matrices, the unnormalized and m3 indicates that they may have been subject to a density matrix reads reduced basis transformation beforehand. If the number of states m ␴L and m ␴L is in excess of the number of ␳ ͓T T ͔N/2 ͑ ͒ 1 1 3 3 ˆ uL = 1 2 145 states to be kept, a reduced basis transformation is car- ␴L → ␴L → and ried out, m1 1 m˜ 1 and m3 3 m˜ 3, using the reduced basis transformation as obtained in the previous super- ͓T T ͔N/2 ͑ ͒ ZL =Tr 1 2 , 146 block diagonalization. The superblock transfer matrix of 2L plaquettes is somewhat modified from the classical TMRG expres- now formed from the product of two such T ͑L+1͒, sum- sion. ␴1 ϵ␴2L+1 ␴L+1 ͑ ͒ ␭N/2 ␭N/2 ␭ ming over states 2 2 and 2 Fig. 30 . This su- As ZL = 1 + 2 +¯, where i are the eigenvalues of perblock transfer matrix is diagonalized using some T T ␭ T T 1 2, the largest eigenvalue 1 of 1 2 dominates in the large sparse eigensolver to get the maximum eigenvalue thermodynamic limit N→ϱ. The density matrix simpli- ␭ ͗␺ ͉ ͉␺ ͘ 1 and the right and left eigenvectors L and R , fies to ͗␺ ͉␺ ͘ which may be chosen mutually biorthonormal; L R ␳ˆ = ␭N/2͉␺ ͗͘␺ ͉, ͑147͒ =1. In view of the possible need for a reduced basis uL 1 R L transformation in the next step, nonsymmetric reduced ͗␺ ͉ ͉␺ ͘ where L and R are the left and right eigenvectors to density matrices are formed, ␭ eigenvalue 1. Due to the checkerboard decomposition, T T ␳ˆ =Tr͉␺ ͗͘␺ ͉, ͑150͒ the transfer matrix 1 2 is nonsymmetric, so that left and R L right eigenvectors are not identical. Normalization to where the trace is over all states in columns 1 and 3 ͗␺ ͉␺ ͘=1 is assumed. The free energy per site is given L R except m˜ 1 , m˜ 3 , ␴L+1, and ␴L+1, as they are the states as 1 3 subject to a subsequent reduced basis transformation. 1 Row and column matrices of the M left and right eigen- ␭ ͑ ͒ fL =− kBT ln 1. 148 vectors with highest eigenvalue weight yield the trans- 2 formation matrices for left and right basis states, i.e., in ͑ ␭ ͒ If ZL i.e., the eigenvalue 1 is calculated exactly, com- columns 1 and 3. This construction implies that basis ͗ ͉ ͉ ͘ putational effort scales exponentially with L as in exact vectors m1 and m3 will also be biorthonormal. The diagonalization. As the errors scale as ͑␤/L͒2, reliable procedure is repeated until the system has reached the calculations are restricted to small ␤, and the interesting desired final size of 2Lmax plaquettes. low-temperature limit is inaccessible. Instead, one can Several technical issues have to be discussed for quan- adopt the classical TMRG on the checkerboard transfer tum TMRG. An important reduction of the computa- matrix to access large L, hence low temperatures. tional load comes from the existence of modified good Conceptually, the main steps of the algorithm are un- quantum numbers derived from conservation laws of the changed from the classical case and the argument for the underlying Hamiltonian. Their existence was already ob- optimality of the decimation procedure is unchanged. served in the quantum transfer-matrix method ͑Nomura Explicit construction and decimation formulas are, how- and Yamada, 1991͒: The action of a local magnetization- ever, slightly changed from the classical case and are conserving Hamiltonian in Trotter direction between ͓ j+1͔z ͓ j+1͔z ͓ j ͔z much more complicated notationally because of the imaginary times j and j+1 obeys S1 + S2 = S1 ͑ ͓ j ͔z ͓ j+1͔z ͓ j ͔z ͓ j+1͔z ͓ j ͔z checkerboard decomposition see Wang and Xiang, 1997 + S2 or S1 − S1 =− S2 + S2 . This generalizes ͒ ͚ ͑ ͒j͓ j ͔z ͚ ͑ ͒j͓ j ͔z and Shibata, 1997, 2003 . to j −1 S1 =− j −1 S2 . Thus the staggered mag- At the start, the transfer matrix T ͑3͒ of two plaquettes netization, summed in the Trotter direction, is a con- is given by ͑Fig. 29͒ served quantity, i.e., constant along the real-space axis:

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1 ␳ˆ =Tr ͓͉␺ ͗͘␺ ͉ + ͉␺ ͗͘␺ ͉͔ ͑152͒ symm 2 R R L L for the density matrix is also conceivable, if one argues that this amounts to trying to optimally represent both the left and right eigenvectors at the same time. This choice is much more stable numerically, as no complex eigenvalue problem appears and diagonalization routines are much more stable, and it somehow also ac- counts for the information carried by both eigenvectors. FIG. 28. Checkerboard decomposition: active vs inactive A detailed study by Nishino and Shibata ͑1999͒ clearly plaquettes of the two-dimensional effective classical model. favors the asymmetric choice. They showed that if the asymmetric density matrix has real eigenvalues, it is for M retained states as precise as the symmetric choice for 2L 2M states; if they are complex, there is no such advan- ͑ ͒i+j͓ j͔z ͑ ͒ Q = ͚ −1 Si = cst. 151 tage. j=1 For the extraction of physical quantities it is now advantageous not to fix the inverse temperature ␤ and vary the Trotter number L, but to replace ␤/L by a fixed In fermionic systems, particle number conservation im- initial inverse temperature ␤ Ӷ1 in Eq. ͑140͒. Quantum ͚2L ͑ ͒i+j j 0 plies that P= j=1 −1 ni is an additional good quantum TMRG growth then reaches for 2L plaquettes all in- number ͑Shibata and Tsunetsugu, 1999a͒.AsinT=0 ␤ ␤ … ␤ verse temperatures 0 ,2 0 , ,L 0. At each step we ob- DMRG, the good quantum numbers are conserved by tain the free energy per site f͑T͒ and thus all thermody- the DMRG decimation process. Moreover, the eigen- namic quantities, such as the internal energy u, state with the maximum eigenvalue is always in the sub- magnetization m, specific heat at constant volume c , ͑ ͒ v space Q=0 or P=0, respectively : Thermodynamic and magnetic susceptibility ␹ by numerical differentia- quantities are independent of boundary conditions in tion. Temperatures less than a hundredth of the main high-temperature phases, which are found for all finite energy scale of the Hamiltonian can be reached. Conver- temperatures in one dimension. For open boundary con- →ϱ ␤ → gence must be checked both in M and in 0 0; the ditions in real space, the lack of an update at the open ␤2 latter convergence is in 0. For a fixed number of ends at every second Trotter time step implies the sub- plaquettes, finite-system DMRG runs can be carried out. traction of equal quantities when forming Q, leading to Severe numerical problems may occur for the second- Q=0. ␹ order derivatives cv and .Adirect calculation of u and An important technical complication occurs because m, which are expectation values, reduces the number of the transfer matrices to be diagonalized are asymmetric differentiations to one. due to the checkerboard decomposition. We therefore To do this, consider the internal energy ͗u͘ have to distinguish between left and right eigenvectors −1 ͓ ˆ −␤Hˆ ͔ T ͗␺ ͉ ͉␺ ͘ =Z Tr h1e . In the Trotter decomposition 1 L and R . Non-Hermitian diagonalization routines lack the stabilizing variational principle of the Hermitian changes to case and have to be monitored extremely carefully. The ͗␴1 ␴2L+1͉T ͑2L+1͉͒␯1 ␯2L+1͘ u simplest choice, very slow but stable, is the power ¯ ¯ L method applied to the transfer matrix and its transpose. ␤ ˆ ͗␴3␯3͉ ˆ − h1͉␴2␯2͘ ͗␴2j+1␯2j+1͉␶ ͉␴2j␯2j͑͘ ͒ Faster algorithms that have been used to obtain stable = h1e ͟ ˆ1 153 results are the Arnoldi and unsymmetric Lanczos meth- j=2 ͑ ͒ ͓͗␺ ͉ ͉␺ ͘ ͑ ͒ ods Golub and van Loan, 1996 , combined with rebior- and therefore L and R are the left right eigenvec- ␭ ͔ thogonalization. Once the eigenvectors have been ob- tors of 1 tained, the unsymmetric density matrix ͓Eq. ͑150͔͒ has ˆ −␤Hˆ ͗⌿ ͉T T ͉⌿ ͘ to be fully diagonalized. Here, two numerical problems Tr͓h e ͔ L u 2 R ͗u͘ = 1 = . ͑154͒ typically occur: the appearance of complex-conjugate ei- ␤ ˆ ␭ Tr͓e− H͔ 1 genvalue pairs with spurious imaginary parts and the loss of biorthonormality. The former is dealt with by dis- A direct calculation of cv from energy fluctuations is less carding the spurious imaginary parts and by taking the accurate. The susceptibility can be obtained from calcu- real and imaginary part of one of the eigenvectors as lating the magnetization m at two infinitesimally differ- two real-valued eigenvectors. The latter, less frequent ent fields. problem is dealt with by iterative reorthogonalization In the beginning, applications of the quantum TMRG ͑Ammon et al., 1999͒. focused on the thermodynamics of simple Heisenberg ͗␺ ͉ ͉␺ ͘ Both left and right eigenvectors L and R are spin chains ͑Shibata, 1997; Wang and Xiang, 1997; needed to construct the density matrix as in Eq. ͑150͒. Xiang, 1998͒; but modified ͑e.g., frustrated and dimer- Having both eigenvectors, a symmetric choice ͓with ized͒ spin chains are also easily accessible ͑Maisinger ͗␺ ͉ ͉͑␺ ͒͘†͔ R = R and Schollwöck, 1998; Klümper et al., 1999, 2000;

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multiple mobile impurities ͑Ammon and Imada, 2000a, 2000b, 2001͒. Dynamic properties at finite temperature are among the most frequently available experimental data. Quan- tum TMRG offers, in close analogy to quantum Monte Carlo calculations, a way to calculate these properties, albeit with far less than the usual precision. It has been applied to anisotropic spin chains ͑Naef et al., 1999͒ and the one-dimensional Kondo chain ͑Mutou et al., 1998b, 1999; Shibata and Tsunetsugu, 1999a͒ and has been used to calculate nuclear relaxation rates ͑Naef and Wang, 2000͒. One starts by calculating imaginary-time correlations G͑␶͒ and inserting two operators at different imaginary times and time distance ␶ into the transfer matrix analogous to Eq. ͑153͒. The spectral function ͑␻͒ T ͑3͒ A can now be extracted from the well-known rela- FIG. 29. Two-plaquette transfer matrix and transfer- ͑␶͒ matrix growth. b are summed over. tionship to G , ϱ 1 ͑␶͒ ͵ ␻ ͑␶ ␻͒ ͑␻͒ ͑ ͒ G = ␲ d K , A , 155 Johnston et al., 2000; Maeshima and Okunishi, 2000; 2 0 Wang et al., 2000; Karadamoglou et al., 2001; Shibata and Ueda, 2001͒; for frustrated chains, effective sites of where the kernel K͑␶,␻͒ is two original sites are formed to ensure nearest-neighbor ␶␻ ␤␻ ␶␻ interactions and applicability. Quantum TMRG studies K͑␶,␻͒ = e− + e− + ͑156͒ of ferrimagnetic spin chains have revealed an interesting combination of thermodynamics typical of ferromagnets and the extension to negative frequencies obtained ͑ ␻͒ −␤␻ ͑␻͒ and antiferromagnets ͑Maisinger et al., 1998; Yamamoto through A − =e A . This decomposition is not ͑ et al., 1998; Kolezhuk et al., 1999͒. In fact, the results are unique; other authors also use related expressions Naef et al., 1999͒, but the essential difficulty in using these remarkably similar to those obtained for spin ladders expressions is invariant: If we introduce a finite number ͑Hagiwara et al., 2000; Wang and Yu, 2000͒. of discrete imaginary times ␶ and frequencies ␻ , and a Electronic degrees of freedom have been studied for i j set K =K͑␶ ,␻ ͒, G =G͑␶ ͒, A =A͑␻ ͒, the spectral func- the t-J model by Ammon et al. ͑1999͒, and Sirker and ij i j i i j j tion A͑␻͒ is in principle obtained from inverting ͑by car- Klümper ͑2002a, 2002b͒ and for the Kondo lattice ͑Shi- rying out a singular-value decomposition of K ͒ bata et al., 1998; Shibata and Tsunetsugu, 1999b͒. Lately, ij quantum TMRG has also been applied to the algorith- ͑ ͒ Gi = ͚ KijAj, 157 mically very similar spin-orbit chain by Sirker and Kha- j liullin ͑2003͒. Rommer and Eggert ͑1999͒ studied one localized im- which, as has been known to practitioners of quantum purity linking two spin chains in the thermodynamic Monte Carlo for a long time, is a numerically unstable limit; there has also been strong interest in the case of procedure, because Kij is very badly conditioned. The ratio of its largest eigenvalues to its smallest eigenvalues is normally so big that it is not encoded properly in computer number representations. Naef et al. ͑1999͒ have argued, as is done in quantum Monte Carlo, that given the algorithm-based imprecisions in the imaginary-time data, one may merely try to find the spectral function that is ͑in a probabilistic sense͒ most compatible with the raw data. This maximum-entropy procedure leads to ͕ ͑␻ ͖͒ finding the set A j that maximizes

1 ␣S͓A͔ − ␹2͓A͔, ͑158͒ 2

where S͓A͔ is the relative entropy

͓ ͔ ͓ ͑ ͔͒͑ −␤␻͒͑͒ S A = ͚ Aj − mj − Ajln Aj/mj 1+e 159 j FIG. 30. Quantum transfer-matrix DMRG step. b are summed over; note the periodic boundary conditions. and ␹2͓A͔ accommodates the noisiness of the raw data,

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2 ͑ ͒ ␹2͓ ͔ ␴2 ͑ ͒ moval of particles at both ends Derrida et al., 1993 . A = ͚ ͩGi − ͚ KijAjͪ Ͳ i . 160 i j Excellent agreement with exact solutions for particular parameter sets ͑which correspond to matrix-product An- In the relative entropy, mj is some spectral function that sätze͒ and with other numerics was found. Kaulke and embodies previous knowledge relative to which infor- Peschel ͑1998͒ studied the q-symmetric Heisenberg ͑␻ ͒ mation of A j is measured. It may simply be assumed model out of equilibrium. to be flat. The factor 1+e−␤␻ ensures that contributions ␻Ͻ ␴2 for 0 are considered correctly. i measures the esti- mated error of the raw data and is in TMRG of the A. Transition matrices order of 10−6. It is found by looking for that order of ␴2 magnitude of i where results change least by varying it. In the field of nonequilibrium statistical mechanics ␣ is determined self-consistently such that, of all solu- there has been special interest in those one-dimensional tions that can be generated for various ␣, the one most systems with transitions from active to absorbing steady probable in the light of the input data is chosen ͑Jarrell states under a change of parameters. Active steady and Gubernatis, 1996͒. states show internal particle dynamics, whereas absorb- Another way of calculating A͑␻͒ is given by Fourier- ing steady states are truly frozen ͑e.g., a vacuum state͒. transforming the raw imaginary-time data to Matsubara These transitions are characterized by universal critical ␻ ͑ ͒ ␲ frequencies on the imaginary axis, n = 2n+1 / for fer- exponents, quite analogously to equilibrium phase tran- ␻ ␲ ␶ ␤ sitions ͑see Hinrichsen, 2000 for a review͒. Carlon, Hen- mions and n =2n/ for bosons, using i =i /L and kel, and Schollwöck ͑1999͒ have shown that DMRG is L ␤ ␻ ␶ able to provide reliable estimates of critical exponents ͑ ␻ ͒ i n i ͑␶ ͒ ͑ ͒ G i n = ͚ e G i . 161 for nonequilibrium phase transitions in one-dimensional L i=0 reaction-diffusion models. To this end they applied ͑ ␻ ͒ G i n is written in some Padé approximation and then DMRG to a ͑non-Hermitian͒ master equation of a ␻ analytically continued from i n to infinitesimally above pseudo-Schrödinger form. This approach has been at the the real frequency axis, ␻+i␩,␩=0+. basis of most later DMRG work on out-of-equilibrium The precision of quantum TMRG dynamics is far phenomena. Considering a chain of length L, in which lower than that of T=0 dynamics. Essential spectral fea- each site is either occupied by a particle ͑A͒ or empty tures are captured, but results are not very reliable ͑͒, the time evolution of the system is given in terms of quantitatively. A serious alternative is currently emerg- microscopic rules involving only neighboring sites. Typi- ing in time-dependent DMRG methods at finite tem- cal rules of evolution are site-to-site hopping ͑diffusion͒ perature ͑Sec. IX.C͒, but their potential in this field has A↔A or the annihilation of neighboring particles not yet been demonstrated. AA→, all with some fixed rates. Nonequilibrium phase transitions in the steady state may arise for com- → → IX. SYSTEMS OUT OF EQUILIBRIUM: NON-HERMITIAN peting reactions, such as AA and A , A AA AND TIME-DEPENDENT DMRG simultaneously present. The last model is referred to as the contact process. Its steady-state transition between The study of strongly correlated electronic systems, steady states of finite and vanishing density in the ther- from which DMRG originated, is dominated by at- modynamic limit is in the universality class of directed tempts to understand equilibrium properties of these percolation. systems. Hence the theoretical framework underlying all Once the reaction rates are given, the stochastic evo- previous considerations has been that of equilibrium sta- lution follows from a master equation, which can be tistical mechanics, which has been extremely well estab- written as lished for a long time. Much less understanding and in d͉P͑t͒͘ particular no unifying framework is available so far for =−Hˆ ͉P͑t͒͘, ͑162͒ nonequilibrium physical systems, the main difficulty be- dt ing the absence of a canonical ͑Gibbs-Boltzmann͒ en- where ͉P͑t͒͘ is the state vector containing the probabili- semble. Physical questions to be studied range from ties of all configurations, as opposed to the true quan- transport through quantum dots with strong voltage bias tum case, in which the coefficients are probability ampli- in the leads far from linear response, to reaction- tudes. The elements of the “Hamiltonian” Hˆ ͑which is diffusion processes in chemistry, to ion transport in bio- actually a transition matrix͒ are given by logical systems, to traffic flow on multilane systems. Quite generically these situations may be described by ͗␴͉Hˆ ͉␶͘ =−w͑␶ → ␴͒; ␴ Þ ␶, time-evolution rules that, if cast in some operator language, lead to non-Hermitian operators ͑Glauber, 1963͒. ͗␴͉Hˆ ͉␴͘ = ͚ w͑␴ → ␶͒, ͑163͒ The application of DMRG to such problems in one ␶Þ␴ spatial dimension was pioneered by Hieida ͑1998͒, who studied, using a transfer-matrix approach, the discrete- where ͉␴͘,͉␶͘ are the state vectors of two particle con- time asymmetric exclusion process, a biased hopping of figurations and w͑␶→␴͒ denotes the transition rates be- hard-core particles on a chain with injection and re- tween the two states and is constructed from the rates of

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 304 U. Schollwöck: Density-matrix renormalization group the elementary processes. Hˆ will in general be nontrivial left ground state must be targeted, although it ˆ contains no information, to get a good representation of Hermitian. Since H is a stochastic matrix, its columns the right eigenstates that are joined in a biorthonormal- ͗␺ ͉ add to zero. The left ground state L is hence given by ity relation ͑Carlon, Henkel, and Schollwöck, 1999͒. For ͗␺ ͉ ͗␴͉ ͑ ͒ non-Hermitian Hamiltonians, there is also a choice be- L = ͚ , 164 ␴ tween symmetric and nonsymmetric density matrices. It was found empirically ͑Carlon, Henkel, and Schollwöck, ͗␺ ͉ ˆ ͒ with ground-state energy E0 =0, since L H=0; the right 1999 that the symmetric choice ͉␺ ͘ ground state R is problem-dependent. All other eigen- 1 ˆ ജ ␳ˆ =Tr ͉͑␺ ͉͗␺ ͉ + ͉␺ ͗͘␺ ͉͒ ͑167͒ values Ei of H have a non-negative real part ReEi 0 E 2 R R L L ͑Alcaraz et al., 1994; van Kampen, 1997͒. Since the formal solution of Eq. ͑162͒ is was most efficient, as opposed to quantum TMRG, in which the nonsymmetric choice is to be preferred. This ˆ ͉P͑t͒͘ = e−Ht͉P͑0͒͘, ͑165͒ is probably due to numerical stability: a nonsymmetric ͉ ͑ϱ͒͘ density matrix makes this numerically subtle DMRG the system evolves towards its steady state P . Let variant even less stable. The same observation was made ⌫ Þ ⌫Ͼ ªinfiRe Ei for i 0. If 0, the approach towards the by Senthil et al. ͑1999͒ in the context of SUSY chains. steady state is characterized by a finite relaxation time The method has been applied to the asymmetric ex- ␶ ⌫ ⌫ =1/ , but if =0, that approach is algebraic. This situ- clusion model by Nagy et al. ͑2002͒. A series of papers ͑␶Þ ͒ ation is quite analogous to noncritical phases 0 and has been able to show impressive results on reptating critical points ͑␶=ϱ͒, respectively, which may arise in polymers exposed to the drag of an external field such as equilibrium quantum Hamiltonians. The big advantage in electrophoresis in the framework of the Rubinstein- of DMRG is that the steady-state behavior is addressed Duke model ͑Carlon, Drzewinski, and van Leeuwen, directly and that all initial configurations are inherently 2001, 2002; Paeßens and Schütz, 2002; Barkema and averaged over in this approach; the price to be paid is Carlon, 2003; Paeßens, 2003͒. DMRG has been able to restriction to relatively small system sizes, currently L show for the renewal time ͑longest relaxation time͒ ␶ a ϳ100, due to inherent numerical instabilities in non- crossover from an ͑experimentally observed͒ behavior Hermitian large sparse eigenvalue solvers. ␶ϳN3.3±0.1 for short polymers to the theoretically pre- Standard DMRG as in one-dimensional T=0 quan- dicted ␶ϳN3 for long polymers. tum problems is now used to calculate both left and Another field of application that is under active de- right lowest eigenstates. Steady-state expectation values bate is determination of the properties of the one- such as density profiles or steady-state two-point corre- dimensional pair-contact process with single-particle dif- lations are given by expressions fusion ͑PCPD͒, which is characterized by the reactions → → ͗n ͘ = ͗␺ ͉n ͉␺ ͘/͗␺ ͉␺ ͘, ͑166͒ AA , AA,AA AAA. While early field- i L i R L R theoretic studies ͑Howard and Täuber, 1997͒ showed where the subscripts L,R serve to remind us that two that its steady-state transition was not in the directed- distinct left and right ground states are employed in the percolation class, the first quantitative data came from a calculation. The gap allows the extraction of the relax- DMRG study ͑Carlon, Henkel, and Schollwöck, 2001͒ ation time on a finite lattice and the use of standard and suggested again a universality class different from finite-size scaling theory. The Bulirsch-Stoer- directed percolation. At present, despite further exten- transformation extrapolation scheme ͑Henkel and sive DMRG studies ͑Henkel and Schollwöck, 2001; Schütz, 1988͒ has been found very useful in extracting Barkema and Carlon, 2003͒ and additional simulational critical exponents from the finite-size scaling functions and analytical studies, no consensus on the critical be- for density profiles and gaps. In this way Carlon, Hen- havior of the one-dimensional PCPD has been reached kel, and Schollwöck ͑1999͒ determined both bulk and yet. It may be safely stated that DMRG has sparked a surface critical exponents of the contact process and major debate in this field. While the raw data as such are found their values to be compatible with the directed- not disputed, the extrapolations to the infinite-size limit percolation class, as expected. are highly controversial. The answer to this issue is be- In cases where the right ground state is also trivially yond the scope of this review. known ͑e.g., the empty state͒, DMRG calculations can be made more stable by shifting this trivially known eigenstate pair to some high energy in Hilbert space by B. Stochastic transfer matrices adding a term E ͉␺ ͗͘␺ ͉ to the Hamiltonian, turning shift R L ͑ ͒ the nontrivial first excitation into the more easily acces- Kemper et al. 2001 have made use of the pseudo- sible ground state. Hamiltonian formulation of stochastic systems as out- For a correct choice of the density matrix it is always lined above to apply the formalism of quantum TMRG. ͑ ͒ crucial to target the ground state, even if it has shifted The time evolution of Eq. 165 and the Boltzmann op- ˆ away; otherwise it will reappear due to numerical inac- erator e−␤H are formally identical, such that in both cases curacies in the representation of the shift operator ͑Car- the transfer-matrix construction outlined in Sec. VIII.C lon, Henkel, and Schollwöck, 2001͒. Moreover, the can be carried out. In the thermodynamic limit N→ϱ all

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 305 physical information is encoded in the left and right dependent response of quantum many-body systems to eigenvectors to the largest eigenvalue, which are ap- external time-dependent perturbations and to calculate proximately represented using the TMRG machinery. transport far from equilibrium. On the other hand, the In the stochastic TMRG the local transfer matrices recent mastery of storing ultracold bosonic atoms in a are direct evolutions in a time interval ⌬t obtained from magnetic trap superimposed by an optical lattice has al- the discretized time evolution, lowed us to drive, at will, by time-dependent variations of the optical lattice strength, quantum phase transitions ˆ ⌬ ͉P͑t + ⌬t͒͘ = e−H t͉P͑t͒͘. ͑168͒ from the superfluid ͑metallic͒ to the Mott insulating re- Otherwise, the formalism outlined in Sec. VIII.C carries gime, which is one of the key phase transitions in ͑ over identically, with some minor but important modifi- strongly correlated electron systems Greiner et al., ͒ cations. Boundary conditions in time are open, since pe- 2002 . riodic boundary conditions would set the future equal to The fundamental difficulty can be seen when consid- ͉␺͑ ͒͘ the past. Moreover, stochastic TMRG only works with a ering the time evolution of a quantum state t=0 unsymmetric density matrix constructed from left and right der the action of some time-independent Hamiltonian ͑ ͒ ˆ ͉␺ ͘ ͉␺ ͘ ͉␺ ͘ eigenvectors: Enss and Schollwöck 2001 have shown H n =En n . If the eigenstates n are known, ex- ͉␺͑ ͒͘ ͚ ͉␺ ͘ that the open boundary conditions imply that the un- panding t=0 = ncn n leads to the well-known time ␳ ͉␺ ͗͘␺ ͉ symmetric choice of the density matrix, ˆ =TrE R L , evolution which was superior in quantum TMRG, has no meaning- ͉␺͑ ͒͘ ͑ ͉͒␺ ͘ ͑ ͒ ful information because it has only one nonzero eigen- t = ͚ cnexp − iEnt n , 169 value. Moreover, for stochastic TMRG the use of the n ␳ ͑ ͒ ͓͉␺ ͗͘␺ ͉ ͉␺ ͗͘␺ ͉͔ ͉␺͑ ͒͘ symmetric choice, ˆ = 1/2 TrE L L + R R ,is where the modulus of the expansion coefficients of t mandatory. is time independent. A sensible Hilbert-space truncation The advantages of the stochastic TMRG are that it is then given by a projection onto the large-modulus works in the thermodynamic limit and that, although it eigenstates. In strongly correlated systems, however, we can only deal with comparatively short time scales of usually have no good knowledge of the eigenstates. In- some hundred time steps, it automatically averages over stead, one uses some orthonormal basis with unknown ͉ ͘ ͚ ͉␺ ͘ all initial conditions and is hence bias-free. It can be eigenbasis expansion, m = namn n . The time evolu- ͉␺͑ ͒͘ ͚ ͑ ͉͒ ͘ seen as complementary to the method of the last section. tion of the state t=0 = mdm 0 m then reads Unfortunately, it has also been shown by Enss and ͉␺͑ ͒͘ ͑ ͒ −iEnt ͉ ͘ϵ ͑ ͉͒ ͘ Schollwöck ͑2001͒ that there is a critical loss of precision t = ͚ ͚ͩ dm 0 amne ͪ m ͚ dm t m , strongly limiting the number L of possible time steps; it m n m is a property of probability-conserving stochastic trans- ͑170͒ fer matrices that the norm of the left and right eigenvec- where the modulus of the expansion coefficients is time tors to the largest eigenvalue diverges exponentially in dependent. For a general orthonormal basis, Hilbert- ͗␺ ͉␺ ͘ L, while biorthonormality L R =1 should hold ex- space truncation at one fixed time ͑i.e., t=0͒ will there- actly as well, which cannot be ensured by finite com- fore not ensure a reliable approximation of the time puter precision. evolution. Also, energy differences matter in time evo- Building on the observation by Enss and Schollwöck lution. The sometimes justified hope that DMRG yields ͑ ͒ 2001 that the local physics at some place and at time t a good approximation to the low-energy Hamiltonian is is determined by past events in a finite-sized light cone, hence of limited use. ͑ ͒ Kemper et al. 2003 have applied a variant of the corner All time-evolution schemes for DMRG so far follow ͑ ͒ transfer-matrix algorithm to the light cone Fig. 31 , re- one of two different strategies. Static Hilbert-space porting that the numerical loss-of-precision problem DMRG methods try to enlarge the truncated Hilbert now occurs at times several orders of magnitude larger, space optimally to approximate ͉␺͑t=0͒͘ so that it is big greatly enhancing the applicability of stochastic TMRG. ͑ ͑ ͒ enough to accommodate i.e., maintain large overlap This method has been applied by Enss et al. 2004 to with the exact result͉͒␺͑t͒͘ for a sufficiently long time to study scaling functions for aging phenomena in systems a very good approximation. More recently, adaptive without detailed balance. Hilbert-space DMRG methods keep the size of the truncated Hilbert space fixed, but try to change it as time evolves so that it also accommodates ͉␺͑t͒͘ to a very C. Time-dependent DMRG good approximation. So far, all physical properties discussed in this review and obtained via DMRG have been either true equilib- 1. Static time-dependent DMRG rium quantities, static or dynamic, or steady-state quan- ͑ ͒ tities. However, time-dependent phenomena in strongly Cazalilla and Marston 2002 were the first to exploit correlated systems are increasingly coming to the fore- DMRG to systematically calculate quantum many-body front of interest. On the one hand, in technological ap- effects out of equilibrium. After applying a standard plications such as are envisaged in nanoelectronics, it DMRG calculation to the Hamiltonian Hˆ ͑t=0͒, the will be of great interest to fully understand the time- time-dependent Schrödinger equation is numerically in-

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 306 U. Schollwöck: Density-matrix renormalization group

ˆ ͑ ͒ tegrated forward in time, building an effective Heff t ˆ ͑ ͒ ˆ ͑ ͒ ˆ ͑ ͒ =Heff 0 +Veff t , where Heff 0 is taken as the last super- ˆ ͑ ͒ ˆ ͑ ͒ block Hamiltonian approximating H 0 . Veff t as an approximation to Vˆ is built using the representations of FIG. 31. Transfer-matrix renormalization group ͑TMRG͒ ap- operators in the final block bases: plied to the causal light cone of a stochastic Hamiltonian evolving in real time: four corner transfer matrices make up ץ ͉␺͑ ͒͘ ͓ ˆ ˆ ͑ ͔͉͒␺͑ ͒͘ ͑ ͒ the light cone. From Kemper et al., 2003. Reprinted with per- t = Heff − E0 + Veff t t . 171 ץi t mission. ͉␺͑ ͒͘ ͉␺ ͘ The initial condition is obviously to take 0 = 0 obtained by the preliminary DMRG run. Forward integra- space, and then to move on to the next-larger DMRG tion can be carried out by step-size adaptive methods system where the procedure is repeated, which is very such as Runge-Kutta integration based on the infinitesi- time consuming. mal time-evolution operator It is important to note that the simple Runge-Kutta approach is not even unitary and should be improved by ͉␺͑t + ⌬t͒͘ = ͑1−iHˆ ͑t͒⌬t͉͒␺͑t͒͘. ͑172͒ using, e.g., the unitary Crank-Nicholson time evolution ͑Daley et al., 2004͒. As an application, Cazalilla and Marston have consid- Instead of considering time evolution as a differential ered a quantum dot weakly coupled to noninteracting equation, one may also consider the time-evolution op- leads of spinless fermions, where time dependency is in- ˆ troduced through a time-dependent chemical potential erator exp͑−iHt͒, which avoids the numerical delicacies coupling to the number of particles left and right of the of the Schrödinger equation. Schmitteckert ͑2004͒ com- dot, putes the transport through a small interacting nano- structure using this approach. To this end, he splits the ˆ ͑ ͒ ␦␮ ͑ ͒ ˆ ␦␮ ͑ ͒ ˆ ͑ ͒ problem into two parts: By obtaining a relatively large V t =− R t NR − L t NL, 173 number of low-lying eigenstates exactly ͑within time- ␦␮ which is switched on smoothly at t=0. Setting L independent DMRG precision͒, he can calculate their ␦␮ =− R, one may gauge-transform this chemical poten- time evolution exactly. For the subspace orthogonal to tial away into a time-dependent complex hopping from these eigenstates, he uses an efficient implementation of and to the dot, the matrix exponential ͉␺͑t+⌬t͒͘=exp͑−iHˆ ⌬t͉͒␺͑t͒͘ us- t ing a Krylov subspace approximation; the reduced Hil- t ͑t͒ = t expͫi͵ dtЈ␦␮ ͑tЈ͒ͬ. ͑174͒ bert space is determined by finite-system sweeps, con- q q L ϱ ͉␺͑ ͒͘ − currently targeting all ti at fixed times ti up to the time presently reached in the calculation. The limitation The current is then given by evaluating the imaginary to this algorithm comes from the fact that to target more part of the local hopping expectation value. Obviously, and more states as time evolves, the effective Hilbert in a finite system currents will not stay at some steady- space chosen has to be increasingly large. state value but go to zero on a time scale of the inverse system size when lead depletion has occurred. In this approach the hope is that an effective Hamil- 2. Adaptive time-dependent DMRG tonian obtained by targeting the ground state of the t Time-dependent DMRG using adaptive Hilbert =0 Hamiltonian will be capable of catching the states spaces was first proposed in essentially identical form by that will be visited by the time-dependent Hamiltonian Daley et al. ͑2004͒ and White and Feiguin ͑2004͒. Both during time evolution. Cazalilla and Marston argue that approaches are efficient implementations of an algo- on short time scales the perturbation can only access a rithm for the classical simulation of the time evolution of few excited states, which are well represented in the ef- weakly entangled quantum states invented by Vidal fective Hamiltonian. With one exception ͑Luo et al., ͑2003, 2004͓͒known as the time-evolving block- 2003͒ this seems borne out in their main application, decimation ͑TEBD͒ algorithm͔. This algorithm was transport through a quantum dot; in many other appli- originally formulated in the matrix-product-state lan- cations, this approach is not sufficient. guage. For simplicity, I shall explain the algorithm in its As one way out of this problem, it has been demon- DMRG context. A detailed discussion of the very strong strated ͑Luo et al., 2003͒ that using a density matrix that connection between adaptive time-dependent DMRG ͉␺͑ ͒͘ is given by a superposition of states ti at various and the original simulation algorithm is given by Daley ␳ ͚Nt ␣ ͉␺͑ ͒͗͘␺͑ ͉͒ ͚␣ ͑ ͒ times of the evolution, ˆ = i=0 i ti ti with i =1 et al. 2004 . It turns out that DMRG naturally attaches for the determination of the reduced Hilbert space is good quantum numbers to state spaces used by the much more precise, whereas simply increasing M is not TEBD algorithm, allowing for drastic increases in per- sufficient. Of course, these states are not known initially; formance due to the use of good quantum numbers. it was proposed to start from a small DMRG system, Time evolution in the adaptive time-dependent evolve it in time, take these state vectors, use them in DMRG is generated using a Trotter-Suzuki decomposi- the density matrix to determine the reduced Hilbert tion as discussed for quantum TMRG ͑Sec. VIII.C͒. As-

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 307 suming only nearest-neighbor interactions, the decom- after each bond update during an infinitesimal time step. position reads Hˆ =Hˆ +Hˆ . Hˆ =͚N/2hˆ and Hˆ Applications of adaptive time-dependent DMRG so far 1 2 1 i=1 2i−1 2 ͑ ͚N/2 ˆ ˆ for the time-dependent Bose-Hubbard model Daley et = i=1 h2i. Here hi is the local Hamiltonian linking sites i al., 2004͒ and spin-1 Heisenberg chains ͑White and and i+1; neighboring local Hamiltonians will in general Feiguin, 2004͒ have demonstrated that it allows access to ͓ ˆ ˆ ͔Þ not commute, hence H1 ,H2 0. However, all terms in large time scales very reliably. The errors mentioned can ˆ ˆ be well controlled by increasing M and can show signifi- H1 or H2 commute. The first-order Trotter decomposition of the infinitesimal time-evolution operator then cant accumulation only after relatively long times. reads

͑ ˆ ⌬ ͒ ͑ ˆ ⌬ ͒ ͑ ˆ ⌬ ͒ ͑⌬ 2͒ 3. Time-dependent correlations exp − iH t = exp − iH1 t exp − iH2 t + O t . Time-dependent DMRG also yields an alternative ap- ͑175͒ proach to time-dependent correlations which have been ˆ ˆ evaluated in frequency space in Sec. IV. Alternatively, Expanding H1 and H2 into the local Hamiltonians, one infinitesimal time step t→t+⌬t may be carried out by one may calculate expressions as performing the local time-evolution on ͑say͒ all even ˆ ˆ ͗␺͉c†͑t͒c ͑0͉͒␺͘ = ͗␺͉e+iHtc†e−iHtc ͉␺͑͘177͒ bonds first and then all odd bonds. In DMRG, a bond i j i j ͉␺͘ ͉␾͘ ͉␺͘ can be time evolved exactly if it is formed by the two constructing both and =cj using standard explicit sites typically occurring in DMRG states. We DMRG ͑targeting both͒, and calculating both ͉␺͑t͒͘ and may therefore carry out an exact time evolution by per- ͉␾͑t͒͘ using time-dependent DMRG. The desired cor- forming one finite-system sweep forward and backward relator is then simply given as through an entire one-dimensional chain, with time evo- ͗␺͑ ͉͒ †͉␾͑ ͒͘ ͑ ͒ lutions on all even bonds on the forward sweep and all t ci t 178 odd bonds on the backward sweep, at the price of the and can be calculated for all i and t as time proceeds. Trotter error of O͑⌬t2͒. This procedure necessitates that Frequency-momentum space is reached by Fourier ͉␺͑t͒͘ be available in the right block bases, which is en- transformation. Finite system sizes and edge effects im- sured by carrying out the reduced basis transformations pose physical constraints on the largest times and dis- on ͉␺͑t͒͘ that in standard DMRG form the basis of tances ͉i−j͉ or minimal frequencies and wave vectors ac- White’s prediction method ͑White, 1996b͒. The decisive cessible, but unlike dynamical DMRG, one run of the idea of Vidal ͑2003, 2004͒ was now to carry out a new time-dependent algorithm is sufficient to cover the en- Schmidt decomposition and make a new choice of the tire accessible frequency-momentum range. White and most relevant block basis states for ͉␺͑t͒͘ after each local Feiguin ͑2004͒ report a very successful calculation of the bond update. Therefore, as the quantum state changes one-magnon dispersion relation in an S=1 Heisenberg in time, so do the block bases such that an optimal rep- antiferromagnetic chain using adaptive time-dependent resentation of the time-evolving state is ensured. Choos- DMRG, but similar calculations are in principle feasible ing new block bases changes the effective Hilbert space, in all time-dependent methods. hence the name adaptive time-dependent DMRG. An appealing feature of this algorithm is that it can be very easily implemented in existing finite-system D. Finite temperature revisited: Time evolution and DMRG. One uses standard finite-system DMRG to gen- dissipation at TϾ0 erate a high-precision initial state ͉␺͑0͒͘ and continues to run finite-system sweeps, one for each infinitesimal time While our discussion so far has been restricted to time ͑ step, merely replacing the large sparse-matrix diagonal- evolution at T=0, it is possible to generalize to the pos- ͒ Ͼ ͑ ization at each step of the sweep by local bond updates sibly dissipative time evolution at T 0 Verstraete, ͒ for the odd and even bonds, respectively. Garcia-Ripoll, and Cirac, 2004; Zwolak and Vidal, 2004 . Errors are reduced by using the second-order Trotter Both proposals are in the spirit of adaptive Hilbert- decomposition space methods. One prepares a T=ϱ completely mixed state ␳ˆ ϱ, essentially the identity operator, from which a ˆ ⌬ ˆ ⌬ ˆ ⌬ ˆ ⌬ −iH t −iH1 t/2 −iH2 t −iH1 t/2 ͑⌬ 3͒ ͑ ͒ e = e e e + O t . 176 finite-temperature mixed state exp͑−␤Hˆ ͒ at ␤−1=TϾ0is As ⌬t−1 steps are needed to reach a fixed time t, the created by imaginary-time evolution, i.e., 2 overall error is then of order O͑⌬t ͒. One efficient way −␶Hˆ N −␶Hˆ N exp͑− ␤Hˆ ͒ = ͑e ͒ ␳ˆ ϱ͑e ͒ , ͑179͒ of implementing this is to group the half time steps of two subsequent infinitesimal evolution steps together, where ␤=2N␶ and ␶→0. The infinitesimal-time evolu- ˆ i.e., carry out the standard first-order procedure, but cal- tion operator e−␶H is Trotter decomposed into locally culate expectation values as averages of expectation val- acting bond evolution operators. The finite-temperature ͑ ˆ ⌬ ͒ ues measured before and after an exp −iH2 t step. state is then evolved in real time using Hamiltonian or Further errors are produced by the reduced basis Lindbladian dynamics. The differences reside essentially transformations and the sequential update of the repre- in the representation of mixed states and the truncation sentation of block states in the Schmidt decomposition procedure.

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 308 U. Schollwöck: Density-matrix renormalization group

Verstraete, Garcia-Ripoll, and Cirac ͑2004͒ consider −͉␺͑t+⌬t͒͘ʈ. If one uses the matrices composing the the T=0 matrix-product state of Eq. ͑47͒, where the state at t as initial guesses and keeps all but one A ma- ͓␴ ͔ ͑ ͒ Ai i are interpreted see Sec. III.A as maps from the trix fixed, one obtains a linear equation system for this tensor product of two auxiliary states ͑dimension M2͒ to A; sweeping through all A matrices several times is suf- a physical state space of dimension Nsite. This can be ficient to reach the fixed point that is the variational generalized to finite-temperature matrix-product density optimum. As temperature is lowered, M will be in- operators, creased to maintain a desired precision. Once the ther-

L mal state is constructed, real-time evolutions governed by Hamiltonians can be calculated similarly. In the case ␳ˆ = ͚ Trͫ͟ Mˆ ͓␴ ␴Ј͔͉ͬ␴͗͘␴Ј͉. ͑180͒ i i i of Lindbladian time evolutions, they are carried out di- ͕␴͖͕␴Ј͖ i=1 rectly on states of the form of Eq. ͑180͒, which is for- Here, the Mˆ ͓␴ ␴Ј͔ are now maps from the tensor prod- mally equivalent to a matrix-product state on a local i i 2 uct of four auxiliary state spaces ͑two left, two right of Nsite-dimensional state space. ͑ ͒ the physical site͒ of dimension M4 to the The approach of Zwolak and Vidal 2004 is based on ͑ ͒ N2 -dimensional local density-operator state space. The the observation that local density operators site ͚␳ ͉␴͗͘␴Ј͉ ϫ most general case allowed by quantum mechanics is for ␴␴Ј can be represented as Nsite Nsite Hermitian ˆ ˆ matrices. They now reinterpret the matrix coefficients as M to be a completely positive map. The dimension of M the coefficients of a local “superket” defined on a seems to be prohibitive, but it can be decomposed into a 2 Nsite-dimensional local state space. Any globally acting sum of d tensor products of A maps as ͑density͒ operator is now a global superket expanded as d a linear combination of tensor products of local super- ˆ ͓␴ ␴Ј͔ ͓␴ ͔ *͓␴Ј ͔ ͑ ͒ Mi i i = ͚ Ai iai Ai i ai . 181 kets, ai=1 N2 N2 ഛ 2 site site In general, d NsiteM , but it will be important to realize ␳ˆ = ͚ ͚ c␴¯ ␴¯ ͉␴¯ ␴¯ ͘. ͑184͒ that for thermal states d=N only. At T=0, one recov- ¯ 1¯ L 1 ¯ L site ␴¯ =1 ␴¯ =1 ers the standard matrix-product state, i.e., d=1. In order 1 L to actually simulate ␳ˆ, Verstraete, Garcia-Ripoll, and ͉␴ ͘ ͉␴ ͉͘␴Ј͘ Here, ¯ i = i i , the state space of the local superket. Cirac ͑2004͒ consider the purification We have now reached complete formal equivalence be- L tween the description of a pure state and a mixed state, ͉␺ ͘ ͚ ͫ͟ ˆ ͓␴ ͔͉ͬ␴ ͘ ͑ ͒ such that the time-evolving block decimation algorithm MPDO = Tr Ai iai a , 182 ͕␴͖͕a͖ i=1 ͑Vidal, 2003, 2004͒ or its DMRG incarnation ͑Daley et al., 2004; White and Feiguin, 2004͒ can be applied to the ␳ ͉␺ ͗͘␺ ͉ such that ˆ =Tra MPDO MPDO . Here, ancilla state time evolution of this state. ͕͉ ͖͘ spaces ai of dimension d have been introduced. Choosing the local totally mixed state as one basis In this form, a completely mixed state is obtained ␴ state of the local superket state space, say ¯ i =1,wefind from matrices A ͓␴ a ͔ϰ ·␦␴ , in which M may be 1 and i i i I i,ai that the global totally mixed state has only one nonzero normalization has been ignored. This shows d=Nsite. expansion coefficient, c =1 upon suitable normaliza- 1¯1 This state is now subjected to infinitesimal evolution in tion. This shows that the T=ϱ state, as in the previous ˆ imaginary time, e−␶H. As it acts on ␴ only, the dimension algorithm, takes a particularly simple form in this expan- of the ancilla state spaces need not be increased. Of sion; it can also be brought very easily to some matrix- course, for T=0 the state may be efficiently prepared product form corresponding to a DMRG system consist- using standard methods. ing of two blocks and two sites. The time-evolution The imaginary-time evolution is carried out after a operators, whether of Hamiltonian or Lindbladian ori- Trotter decomposition into infinitesimal time steps on gin, now map from operators to operators and are hence bonds. The local bond evolution operator Uˆ is con- referred to as superoperators. If one represents opera- i,i+1 tors as superkets, these superoperators are formally veniently decomposed into a sum of dU tensor products of on-site evolution operators, equivalent to standard operators. As in the T=0 case therefore they are now Trotter decomposed, applied in dU infinitesimal time steps to all bonds sequentially, with ˆ k k ͑ ͒ Ui,i+1 = ͚ uˆ i uˆ i+1. 183 Schmidt decompositions being carried out to determine k=1 block ͑super͒state spaces. The number of states kept is determined from the acceptable truncation error. dU is typically small, say 2 to 4. Applying now the time evolution at, say, all odd bonds exactly, the auxiliary Imaginary-time evolutions starting from the totally mixed state generate thermal states of a given tempera- state spaces are enlarged from dimension M to MdU. ͉␺˜ ͑ ture that may then be subjected to some evolution in One now has to find the optimal approximation t real time. It turns out again that the number of states to ⌬ ͒͘ ͉␺͑ ⌬ ͒͘ + t to t+ t using auxiliary state spaces of dimen- be kept grows with inverse temperature. sion M only. Hence the state spaces of dimension MdU For the simulation of dissipative systems, both ap- must be truncated optimally to minimize ʈ͉␺˜ ͑t+⌬t͒͘ proaches are effectively identical except for the optimal

Rev. Mod. Phys., Vol. 77, No. 1, January 2005 U. Schollwöck: Density-matrix renormalization group 309 approximation of Verstraete, Garcia-Ripoll, and Cirac Ammon, B., and M. Imada, 2001, J. Phys. Soc. Jpn. 70, 547. ͑2004͒ replacing the somewhat less precise truncation of Ammon, B., M. Troyer, T. M. Rice, and N. Shibata, 1999, Phys. the approach of Zwolak and Vidal ͑2004; see the discus- Rev. Lett. 82, 3855. sion of the precision achievable for various DMRG set- Anderson, P. W., 1959, J. Phys. Chem. Solids 11, 28. ups in Sec. III.A͒. Andersson, M., M. Boman, and S. Östlund, 1999, Phys. Rev. B 59, 10 493. Anusooya, Y., S. K. Pati, and S. Ramasesha, 1997, J. Chem. X. OUTLOOK Phys. 106, 10230. Anussoya, Y., S. K. Pati, and S. Ramasesha, 1999, J. Phys.: What lies ahead for DMRG? On the one hand, Condens. Matter 11, 2395. DMRG is quickly becoming a standard tool, to be rou- Aschauer, H., and U. Schollwöck, 1998, Phys. Rev. B 58, 359. tinely applied to the study of all low-energy properties Barford, W., and R. J. Bursill, 2001, Synth. Met. 119, 251. of strongly correlated one-dimensional quantum sys- Barford, W., R. J. Bursill, and M. Y. Lavrentiev, 1998, J. Phys.: tems, most often probably using black-box DMRG Condens. Matter 10, 6429. codes. On the other hand, there are several axes of Barford, W., R. J. Bursill, and M. Y. Lavrentiev, 2001, Phys. DMRG research that I expect to be quite fruitful in the Rev. B 63, 195108. future: DMRG might emerge as a very powerful tool in Barford, W., R. J. Bursill, and M. Y. Lavrentiev, 2002, Phys. quantum chemistry in the near future; its potential for Rev. B 65, 075107. time-dependent problems is far from being understood. Barford, W., R. J. Bursill, and R. W. Smith, 2002, Phys. Rev. B Last but not least I feel that two-dimensional model 66, 115205. Hamiltonians will be increasingly within reach of Barkema, G. T., and E. Carlon, 2003, Phys. Rev. E 68, 036113. DMRG methods, at least for moderate system sizes that Bartlett, R., J. Watts, S. Kucharski, and J. Noga, 1990, Chem. are still beyond the possibilities of exact diagonalization Phys. Lett. 165, 513. and quantum Monte Carlo. Batista, C. D., K. Hallberg, and A. A. Aligia, 1998, Phys. Rev. The strong link of DMRG to quantum information B 58, 9248. theory has only recently begun to be exploited—DMRG Batista, C. D., K. Hallberg, and A. A. Aligia, 1999, Phys. Rev. B 60, R12 553. variants to carry out complex calculations of interest in Bauschlicher, C. W., and S. Langhoff, 1988, J. Chem. Phys. 89, quantum information theory might emerge, while 4246. DMRG itself could be applied in a more focused man- Bauschlicher, C. W., and P. R. Taylor, 1986, J. Chem. Phys. 85, ner using the new insights about its intrinsic rationale. In 2779. that sense, DMRG would be at the forefront of a grow- Baxter, R. 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