DOCSLIB.ORG

Spectrum of Reduced Density Matrix: Bethe Ansatz Vs Valence Bond Solid States

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spectrum of Reduced Density Matrix: Bethe Ansatz Vs Valence Bond Solid States ENTROPY Spectrum of Reduced Density Matrix: Bethe Ansatz vs Valence Bond Solid states Vladimir Korepin C.N. Yang Institute for Theoretical Physics ⊲⊳ Vladimir Korepin Operator Structures in Quantum Information ENTROPY Reduced Density Matrix Was introduced by Dirac in 1930. Proceedings of Cambridge Philosophical Society, volume 26. Matrix elements of reduced density matrix are correlation functions. Miwas and Jimbo used reduced density matrix in the book. It is also useful in quantum information and entanglement. Brief introduction. Vladimir Korepin Operator Structures in Quantum Information ENTROPY Entropy of a subsystem of a pure system In classical case the following simple theorem is valid: If the entropy of a system is zero, then there is no entropy in any subsystem: In quantum case the entropy of a subsystem of a pure system can be positive. This means that the subsystem is entangled with the rest of the system. It is an important resource for quantum control. Vladimir Korepin Operator Structures in Quantum Information ENTROPY Example Two spins 1/2: ΨE,B = 1 ( ) | i √2 | ↑iE | ↓iB − | ↓iE | ↑iB Density matrix of one spin is proportional to identical matrix 1 ρ = tr ΨE,B ΨE,B = I B E | ih | 2 2 SE,B = 0 but SB = ln(2) > 0 Cannot happen for classical subsystems. This is a way to distinguish quantum system form classical . Vladimir Korepin Operator Structures in Quantum Information ENTROPY No factorization Two systems in general. One can choose the orthonormal basis in both spaces so that: d ΨE,B = a ψE ψB and a2 = 1 | i j | j i | j i | j | Xj=1 O Xj Theorem: If SB > 0 then d > 1. The subsystems E and B are entangled. Vladimir Korepin Operator Structures in Quantum Information ENTROPY Quantum Control Measurement of subsystem E can be organized to project on one of basis states ψE . The wave function will turn into | j0 i ΨE,B = ψE ψB | M i | j0 i | j0 i O with probability a2 . j0 This changes the| state| of another subsystem B. E and B entangled: one can control one system by another. This is quantum control useful for building quantum devices. Vladimir Korepin Operator Structures in Quantum Information ENTROPY Application to XX Model ∞ H = σx σx + σy σy + hσz , h 2 XX − n n+1 n n+1 n | |≤ n= X−∞ Solved by Lieb , Schultz & Mattis 1961 The gs is unique | i Entanglement of a block of L sequential spins with the rest of the ground state [environment]. Vladimir Korepin Operator Structures in Quantum Information ENTROPY Entanglement gs = B E | i | ∪ i density matrix of the ground state state ρ = gs gs | ih | Density matrix of the block is the reduced density matrix. ρB = TrE ρ Individual entries of ρB are correlation functions. Vladimir Korepin Operator Structures in Quantum Information ENTROPY Density matrix and correlations a In each lattice site j we have four ’Pauli’ matrices σj , here a = 0, x, y, z with σ0 = I. L L L aj aj ρ = 2− σ gs σ gs B j h | j | i aj j=1 j=1 j=X1,...,L O Y For gap-full models limit of large block exists. Vladimir Korepin Operator Structures in Quantum Information ENTROPY Quantum Entanglement Measures of entanglement for a pure state Von Neumann Entropy of subsystem [block] : S(ρ )= Tr (ρ ln ρ ) B − B B B Rényi Entropy of subsystem [block]: α ln TrBρ S(ρ , α)= B B 1 α − α is a parameter. Also known in literature as zeta function or replica trick. Vladimir Korepin Operator Structures in Quantum Information ENTROPY Calculation of Entropy Method of calcualtions of for correlation functions is based on Determinant Representation and Riemann-Hilbert Problem Fredholm integral operators of a special form: integrable integral operators. Developed by A.R. Its, A.G. Izergin, V.E. Korepin, N.M. Bogoliubov Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge 1993 Same method work for evaluation of the entropy Vladimir Korepin Operator Structures in Quantum Information ENTROPY Von Neumann Entropy Asymptotic L : →∞ 1 1 h 2 1 1 S(ρ ) ln L + ln 1 + ln 2 +Υ + O B → 3 6 − 2 3 1 L " # (ln L ) term was discovered Holzhey, Larsen, Wilczek 1994 Sub-leading terms: B.-Q. Jin, V.E. Korepin in 2004 ∞ e−t 1 cosh(t/2) Υ1 = dt + 2 3 0.495 − 0 3t t sinh t 2 − 2 sinh t 2 ≈ Z ( ( / ) ( / ) ) Vladimir Korepin Operator Structures in Quantum Information ENTROPY Rényi Entropy Asymptotic of Rényi entropy is 1 2 1 + α− h α S(ρB, α) ln 2L 1 +Υ{ } → 6 s − 2 B.-Q. Jin, V.E. Korepin 2003, see arXiv:quant-ph/0304108 Later reproduced by J. Cardy and P. Calabrese from conformal filed theory, arXiv:hep-th/0405152 Vladimir Korepin Operator Structures in Quantum Information ENTROPY XY model Gap-full models + ∞ H = (1 + γ)σx σx + (1 γ)σy σy + hσz XY − n n+1 − n n+1 n n= X−∞ Lieb , Schultz & Mattis 1961 Phases 1a. Moderate field: 2 1 γ2 < h < 2 − 1b. Weak field: 0 h < 2 1 γ2 p≤ − 2. Strong field: h > 2 p Vladimir Korepin Operator Structures in Quantum Information ENTROPY Phase diagram γ Case 1A 1 Case 1B Case 2 0 2 h Vladimir Korepin Operator Structures in Quantum Information ENTROPY Boundary between 1a and 1b Barouch-McCoy circle h = 2 1 γ2 − GS = GS + GS p | i | 1i | 2i GS = [cos(θ) n + sin(θ) n ] , | 1i | ↑ i | ↓ i n lattice ∈ Y GS = [cos(θ) n sin(θ) n ] | 2i | ↑ i− | ↓ i n lattice ∈ Y cos2(2θ) = (1 γ)/(1 + γ) − Partially ordered state. Vladimir Korepin Operator Structures in Quantum Information ENTROPY Von Neumann Entropy Entropy of a large block has a limit: ∞ τ τ 1 θ3 β(λ)+ 2 θ3 β(λ) 2 S(ρB)= ln − dλ 2 θ2 στ Z1 3 2 ! 1 λ + 1 I(k ) 1 (h/2)2 γ2 β(λ)= ln , τ = i ′ , k = − − 2πi λ 1 I(k) s 1 (h/2)2 − − 2 2 θ3 the Jacobi theta-function; (k ′) + k = 1 Entropy is constant on the ellipsis: k = const 1 dx Complete elliptic integral of the first kind I(k)= 0 √(1 x2)(1 x2k 2) − − R Vladimir Korepin Operator Structures in Quantum Information ENTROPY references First analytical expression for limiting entropy of the large block of spin in the ground state of XY spin chain on infinite lattece Its, Jin, Korepin: September 3 of 2004, see http://arxiv.org/abs/quant-ph/0409027 Shortly after on 16 Oct 2004 Ingo Peschel simplified our expression , see http://arxiv.org/abs/cond-mat/0410416 Vladimir Korepin Operator Structures in Quantum Information ENTROPY Rényi Entropy Limiting entropy in the region 0 < h < 2 is: ′ α α S(ρ , α)= 1 α ln k + 1 1 ln θ2(0,q )θ4(0,q ) + 1 ln 2 B 6 1 α k 2 3 1 α θ2(0,qα) 3 − − 3 I(k ) τ = i ′ iτ , q = eπiτ I(k) ≡ 0 F. Franchini, A. R. Its, B.-Q. Jin, V. E. Korepin 2006 Vladimir Korepin Operator Structures in Quantum Information ENTROPY Spectrum of the density matrix of very large block pm are eigenvalues of the limiting density matrix. They form exact geometric sequence: I k ′ −πτ0m ( ) pm = p0e , m = 0, 1, 2,... , τ0 = > 0 ∞ I(k) Here m is a label of eigenvalue. The maximum eigenvalue ′ 1 6 π p = (kk /4) / exp τ is unique 0 12 0 h i Degeneracy increases as eigenvalue diminishes: − − Degeneracy (192) 1/4 (m) 3/4 eπ√m/3, m → → ∞ A. R. Its, F.Franchini, V.E.Korepin and L.A. Takhtajan 2009 Vladimir Korepin Operator Structures in Quantum Information ENTROPY Affleck, Kennedy, Lieb and Tasaki Spin chain consists of N spin-1’s in the bulk and two spin-1/2 ~ on the boundary: Sk is spin-1 and ~sb is spin-1/2. N 1 − 1 2 H = S~ S~ + (S~ S~ )2 + + π + π . k k+1 3 k k+1 3 0,1 N,N+1 Xk=1 1 ~ ~ 1 ~ ~ 2 1 2 Sk Sk+1 + 6 (Sk Sk+1) + 3 is a projector on a state of spin 2. The π is a projector on a state with spin 3/2: 2 2 π = 1 + ~s S~ , π = 1 + ~s S~ . 0,1 3 0 1 N,N+1 3 N+1 N Vladimir Korepin Operator Structures in Quantum Information ENTROPY Ground state is VBS Kirillov-Korepin set up 1989. Ground state is unique: s ss ss ss ss ss ss s 1 a dot is spin- 2 ; circle means symmetrisation [makes spin 1]. A line is a anti-symmetrisation ( ) . Each projector annihilates the ground state (no| ↑↓i frustration). − | ↓↑i The ground state is called VBS. Exact formula for correlation function at any distance is: 3 1 x < S~ x S~ >= = p(x) 4 1 −3 Vladimir Korepin Operator Structures in Quantum Information ENTROPY Entropy of entanglement of VBS AKLT again. Fan, Korepin and Roychowdhury 2004 calculated exactly the entropy of the block of the size x on a finite lattice: 3(1 p(x)) 1 p(x) S(x)= − log − − 4 4 − 1 3p x 1 3p x + ( ) log + ( ) − 4 4 It does not depend on the size of the lattice.For large block S(x) ln 4 as x → →∞ Vladimir Korepin Operator Structures in Quantum Information ENTROPY Eigenvalues of the density matrix Fan, Korepin and Roychowdhury 2004 PRL: the density matrix of finite block of x spins on the finite lattice has only 4 non-zero eigenvalues.
Load more

Recommended publications
  • (Oxon) on the THEORY of MODERN QUANTUM ALGORITHMS
    SKOLKOVO INSTITUTE OF SCIENCE AND TECHNOLOGY draft version Jacob Daniel Biamonte, BS, DPhil (Oxon) ON THE THEORY OF MODERN QUANTUM ALGORITHMS Specialization 01.01.03 Mathematical Physics Dissertation submitted for the degree of Doctor of Physical and Mathematical Sciences arXiv:2009.10088v1 [quant-ph] 21 Sep 2020 Moscow | 2020 2 Contents Page Chapter 1. Introduction: Programming Ground States .......5 1.1 Computation and the Ising model . .9 1.2 Low-energy subspace embedding . 15 1.3 Two-body reductions . 19 1.4 P- vs. NP problems and physics . 23 1.5 Computational phase transitions . 24 1.5.1 Thermal states at the SAT phase transition . 29 Chapter 2. Quantum vs Probabilistic Computation ......... 32 2.1 Defining Mechanics . 32 2.1.1 Stochastic time evolution . 33 2.1.2 Quantum time evolution . 33 2.1.3 Properties in stochastic mechanics . 34 2.1.4 Hamiltonian properties in quantum mechanics . 35 2.1.5 Observables . 36 2.2 Walks on Graphs: quantum vs stochastic . 44 2.2.1 Normalized Laplacians . 47 2.2.2 Stochastic walk . 50 2.2.3 Quantum walk . 51 2.2.4 Perron's theorem . 54 2.3 Google page rank|a ground eigenvector problem . 59 2.4 Kitaev's quantum phase estimation algorithm . 61 2.5 Finding the ground state . 66 Chapter 3. Tensor Networks and Quantum Circuits ......... 69 3.1 Clifford gates . 69 3.2 Tensor network building blocks . 72 3.2.1 Reversible logic . 75 3.2.2 Heisenberg picture . 77 3.2.3 Stabilizer tensor theory . 79 3 Chapter 4. Variational Search and Optimization ..........
    [Show full text]
  • Algebraic Bethe Ansatz and Tensor Networks
    Algebraic Bethe Ansatz and Tensor Networks A Dissertation presented by You Quan Chong to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics Stony Brook University May 2015 Stony Brook University The Graduate School You Quan Chong We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation Vladimir E. Korepin - Dissertation Advisor Professor, C. N. Yang Institute for Theoretical Physics, Department of Physics and Astronomy Tzu-Chieh Wei - Chairperson of Defense Assistant Professor, C. N. Yang Institute for Theoretical Physics, Department of Physics and Astronomy Dominik A. Schneble - Committee Member Associate Professor, Department of Physics and Astronomy Scott Sutherland - Outside Member Associate Professor, Department of Mathematics This dissertation is accepted by the Graduate School Charles Taber Dean of the Graduate School ii Abstract of the Dissertation Algebraic Bethe Ansatz and Tensor Networks by You Quan Chong Doctor of Philosophy in Physics Stony Brook University 2015 We consider several exactly solvable models of strongly correlated elec- trons in one dimension, such as the Heisenberg XXX model, the supersym- metric t-J model and the Hubbard model. These models can be solved by using the method of graded algebraic Bethe ansatz. We use it to design graded tensor networks which can be contracted approximately to obtain a Matrix Product State. This overcomes a major shortcoming of current density matrix renormal- ization group (DMRG) methods which work well on the ground states, but have difficulty working with the excited states of such models. In addition, observables such as correlation functions are important as they are experimentally measurable, but have been analytically described in the double scaling limit only.
    [Show full text]
  • Entanglement and Quantum Spin Chains
    ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems ENTANGLEMENT AND QUANTUM SPIN CHAINS V. Korepin YITP of Stony Brook University New exactly solvable spin chain came out of quantum information V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Outline 1 ABitofHistoryofQuantumEntanglement 2 Quantum Information and Dynamical Systems V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems The dawn Aug 14, 1935 Shroedinger submitted his paper to Proceedings of Cambridge Philosophical Society. 1964 John Bell: entanglement distinguishes quantum mechanics from classical at fixed value of Plank constant. 1981 Richard Feynman: quantum computation. 1996 Bennett, Bernstein, Popescu and Schumacher introduced entropy of subsystem as a measure of entanglement discovery of entanglement entropy V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Application to physics: area Law 1986 L. Bombelli, R. Koul, J. Lee, R. Sorkin; 1993 M. Srednicki 1994 Holzhey, Larsen and Wilczek: ent entr of block of spins of c size x for 1D gapless models scales as S = 3 log x . 2003 Logarithmic scaling of Renyi entropy. B.-Q.Jin, Korepin, Journal Statistical Physics; arXiv.org April 15 of 2003. ln tr(ρα) −1 S (α)= 1+α ln x R 1−α → 6 c v πTx 2004 For finite temperature S = 3 log πT sinh v Korepin, PRL, vol 92, ei 096402, 05 March 2004,! ! "" arXiv:cond-mat/0311056 V. Ko r e p in F-H Formula ABitofHistoryofQuantumEntanglement Quantum Information and Dynamical Systems Environment and Error Correcting Codes Q-bits should be entangled with other qubits in quantum computer, but not entangled with the environment.
    [Show full text]
  • Entanglement Entropy & Spectrum • Holographic Spin Chain (VBS/CFT Correspondence) 3
    Entanglement spectra of two- dimensional solvable models Hosho Katsura (Gakushuin University) Collarborators: Anatol Kirillov (RIMS, Kyoto Univ.) Vladimir Korepin (YITP, Stony Brook) Naoki Kawashima (ISSP, Tokyo Univ.) Lou Jie (ISSP Fudan Univ.) Shu Tanaka (Tokyo Univ.) Ryo Tamura (NIMS) H.K., et al., J. Phys. A 43, 255303 (2010). J. Lou, S. Tanaka, H.K. & N. Kawashima, PRB 84, 245128 (2011). S. Tanaka, R. Tamura, & H.K., PRA 86, 032326 (2012). Outline 1. Introduction • AKLT model & valence-bond-solid (VBS) state • Schmidt decomposition & quantum entanglement 2. Entanglement spectra of 2d AKLT models • Gram matrix & Reflection symmetry • Entanglement entropy & spectrum • Holographic spin chain (VBS/CFT correspondence) 3. Entanglement spectra of quantum hard-square models • Tensor network states for interacting Rydberg atom systems • Entanglement spectra • Holographic minimal models (c<1) A brief history of AKLT and valence-bond-solid (VBS) state • Haldane gap problem Haldane’s conjecture (F. D. M. Haldane, Phys. Rev. Lett. 50 (‘83).) S=integer antiferromagnetic (AFM) Heisenberg chains are gapped. Experiment in S=1 spin chain: Ni(C2H8N2)2NO2(ClO4) (NENP) etc. • Affleck-Kennedy-Lieb-Tasaki (AKLT) model (PRL 59 (‘87), CMP 115 (‘87).) 1. Exact unique ground state valence-bond-solid (VBS) state 2. Rigorous proof of the ‘Haldane’ gap in this model 3. AFM correlation decays exponentially with distance. • Valence-bond-solid (VBS) state : spin singlet : symmetrization VBS state Scwinger boson(SB) rep. of spin operator Arovas, Auerbach a b & Haldane, PRL 60 (‘88). Constraint : S=1 state is spanned by . Matrix product state (MPS) Fannes et al.,(‘89), Kluemper et al.,(‘91). VBS on arbitrary graphs Square lattice Hexagonal lattice • Projected pair entangled state (PEPS), Tensor network state (TNS) Verstraete & Cirac (PRA 70 (R) (‘04)), Gross & Eisert (PRL 98 (‘07)).
    [Show full text]
  • SCIENTIFIC REPORT for the YEAR 2002 ESI, Boltzmanngasse 9, A-1090 Wien, Austria
    The Erwin SchrÄodinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria Scienti¯c Report for the Year 2002 Vienna, ESI-Report 2002 March 1, 2003 Supported by Federal Ministry of Education, Science, and Culture, Austria 2 ESI{Report 2002 ERWIN SCHRODINGERÄ INTERNATIONAL INSTITUTE OF MATHEMATICAL PHYSICS, SCIENTIFIC REPORT FOR THE YEAR 2002 ESI, Boltzmanngasse 9, A-1090 Wien, Austria, March 1, 2003 Honorary President: Walter Thirring, Tel. +43-1-4277-51516. President: Jakob Yngvason: +43-1-4277-51506. [email protected] Director: Peter W. Michor: +43-1-3172047-16. [email protected] Director: Klaus Schmidt: +43-1-3172047-14. [email protected] Administration: Maria Windhager, Eva Kissler, Ursula Sagmeister: +43-1-3172047-12, [email protected] Computer group: Andreas Cap, Gerald Teschl, Hermann Schichl. International Scienti¯c Advisory board: Jean-Pierre Bourguignon (IHES), Luis A. Ca®arelli (Austin, Texas), Giovanni Gallavotti (Roma), Krzysztof Gawedzki (IHES), Viktor Kac (MIT), Elliott Lieb (Princeton), Harald Grosse (Vienna), Harald Niederreiter (Vienna), Table of contents General remarks . 2 Report of the Review Panel . 3 Winter School in Geometry and Physics . 12 Arithmetic Groups and Automorphic Forms . 12 Stability Matters: A Symposium on Mathematical Physics . 12 PROGRAMS IN 2002 . 13 Developed Turbulence . 13 Arithmetic, automata, and asymptotics . 18 Quantum ¯eld theory on curved space time . 20 Aspects of foliation theory in geometry, topology and physics . 22 Noncommutative geometry and quantum ¯eld theory Feynman diagrams in mathematics and physics . 24 Mathematical population genetics and statistical physics . 25 CONTINUATION OF PROGRAMS FROM 2001 and earlier . 26 SENIOR FELLOWS and GUESTS via Director's shares .
    [Show full text]
  • Arxiv:Cond-Mat/0405155V3
    Universality of the one dimensional Bose gas with delta interaction Luigi Amico(1) and Vladimir Korepin(2) (1) MATIS-INFM & Dipartimento di Metodologie Fisiche e Chimiche (DMFCI), Universit´adi Catania, viale A. Doria 6, I-95125 Catania, Italy and (2) C.N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook, NY 11794-3840 We consider several models of interacting bosons in a one dimensional lattice. Some of them are not integrable like the Bose-Hubbard others are integrable. At low density all of these models can be described by the Bose gas with delta interaction. The lattice corrections corresponding to the different models are contrasted. I. INTRODUCTION duces quantum fluctuations that dominate the zero tem- perature phase diagram. At commensurate number of bosons per site the Bose-Hubbard model predicts a Mott The Bose gas with delta interaction is one of the insulator-superfluid quantum phase transition controlled most famous models for strongly interacting bosons [1, 2] by the ratio between the bosonic tunneling rate t and (the corresponding classical dynamics is described by the Coulomb repulsion. In optical applications the sys- the non-linear Schr¨odinger equation). In one dimension tem parameters are tuned by changing the laser intensity this model possesses infinitely many, mutually commut- [20]; the criticality of the superfluid-insulator quantum ing integrals of the motion and is therefore called inte- phase transition is substantially altered by the confining grable. The integrability of the Bose gas can be for- harmonic potential [21]. mulated within the Quantum Inverse Scattering Method (QISM) [3] that exploit a crucial property of the two- In this paper we focus on one dimension.
    [Show full text]
  • Arxiv:1806.04049V3 [Hep-Th] 20 Sep 2018 So Far
    R´enyi entropy of highly entangled spin chains Fumihiko Sugino∗ and Vladimir Korepiny ∗ Fields, Gravity & Strings Group, Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), 55, Expo-ro, Yuseong-gu, Daejeon 34126, Republic of Korea y C.N.Yang Institute for Theoretical Physics, Stony Brook University, NY 11794, USA [email protected], [email protected] Abstract Entanglement is one of the most intriguing features of quantum theory and a main resource in quantum information science. Ground states of quantum many-body systems with local interactions typically obey an \area law" meaning the entanglement entropy proportional to the boundary length. It is exceptional when the system is gapless, and the area law had been believed to be violated by at most a logarithm for over two decades. Recent discovery of Motzkin and Fredkin spin chain models is striking, since these models provide significant violation of the entanglement beyond the belief, growing as a square root of the volume in spite of local interactions. Although importance of intensive study of the models is undoubted to reveal novel features of quantum entanglement, it is still far from their complete understanding. In this article, we first analytically compute the R´enyi entropy of the Motzkin and Fredkin models by careful treatment of asymptotic analysis. The R´enyi entropy is an important quantity, since the whole spectrum of an entangled subsystem is reconstructed once the R´enyi entropy is known as a function of its parameter. We find non-analytic behavior of the R´enyi entropy with respect to the parameter, which is a novel phase transition never seen in any other spin chain studied arXiv:1806.04049v3 [hep-th] 20 Sep 2018 so far.
    [Show full text]
  • Measures of Entanglement in Dynamical Systems Density Matrix
    Measures of Entanglement in Dynamical Systems Density matrix of block of spins in the ground state of a Hamiltonian Vladimir Korepin Yang Institute for Theoretical Physics State University of New York at Stony Brook Abstract To build quantum computer or any other quan- tum device we need to maintain quantum con- trol. Entanglement is an important recourse for quantum control. Many-body physics is used to study measures of entanglement in ground state of different dynamical systems. 1 Binary system A&B , which is in a pure state: a { } unique wave function denoted by ΨA,B . it will be a | i unique ground state of a dynamical model: interacting spins, Bose gas or Hubbard model. d ΨA,B = ΨA ΨB , d> 1 | i | j i⊗| j i jX=1 Measure of entanglement. Von Neumann entropy of the subsystem S(ρ ) = T r (ρ ln ρ ) A − A A A ρ = T r ΨA,B ΨA,B A B | ih | John Preskill, Matthew Hastings, Jens Eisert, Frank Verstr aete, John Cardy 2 Another measure is R´enyi entropy of a subsystem: 1 α Sα(ρ ) = ln T r (ρ ) A 1 α A A where α is a parameter. − spectrum of density matrix is also in- teresting, even eigenvectors are impor- tant [measurement]. If whole binary system A&B is in a mixed states [thermodynamics]. Mutual entropy: I = S + S S {A,B} A B − A∪B Another measure of entanglement of mixed states was suggested by Asher Peres. We can describe the mixed binary system by a density matrix ρA,B.
    [Show full text]
  • Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases
    Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases Matthew Fahrbach∗ and Dana Randall† School of Computer Science, Georgia Institute of Technology December 23, 2020 Abstract The six-vertex model in statistical physics is a weighted generalization of the ice model on (i.e., Eulerian orientations) and the zero-temperature three-state Potts model (i.e., proper three-colorings). The phase diagram of the model depicts its physical properties and suggests where local Markov chains2 will be ecient. In this paper, we analyze the mixing time of Glauber dynamics for the six-vertexℤ model in the ordered phases. Specically, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require exponential time to converge to equilibrium. This is the rst rigorous result bounding the mixing time of Glauber dynamics in the ferroelectric phase. Our analysis demonstrates a fundamental connection between correlated random walks and the dynamics of intersecting lattice path models (or routings). We analyze the Glauber dynamics for the six-vertex model with free boundary conditions in the antiferroelectric phase and signicantly extend the region for which local Markov chains are known to be slow mixing. This result relies on a Peierls argument and novel properties of weighted non-backtracking walks. arXiv:1904.01495v2 [cs.DS] 22 Dec 2020 ∗Email: [email protected]. Supported in part by an NSF Graduate Research Fellowship under grant DGE-1650044. †Email: [email protected]. Supported in part by NSF grants CCF-1526900, CCF-1637031, and CCF-1733812.
    [Show full text]
  • Arxiv:1804.10133V3 [Cond-Mat.Quant-Gas] 17 Dec 2018
    Rise and fall of a bright soliton in an optical lattice Piero Naldesi1 ∗, Juan Polo Gomez1, Boris Malomed2;3, Maxim Olshanii4, Anna Minguzzi1 and Luigi Amico5;6;7;8;9 1 Universit´eGrenoble-Alpes, LPMMC, F-38000 Grenoble, France and CNRS, LPMMC, F-38000 Grenoble, France 2 Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, P.O.B. 39040, Ramat Aviv, Tel Aviv, Israel 3 Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Ramat Aviv, Tel Aviv, Israel 4 Department of Physics, University of Massachusetts Boston, Boston, MA 02125, USA 5 Dipartimento di Fisica e Astronomia, Via S. Sofia 64, 95127 Catania, Italy 6 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore 7 MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore 8 CNR-MATIS-IMM & INFN-Sezione di Catania, Via S. Sofia 64, 95127 Catania, Italy and 9 LANEF 'Chaire d'excellence', Universit`eGrenoble-Alpes & CNRS, F-38000 Grenoble, France (Dated: December 19, 2018) We study an ultracold atomic gas with attractive interactions in a one-dimensional optical lattice. We find that its excitation spectrum displays a quantum soliton band, corresponding to N-particle bound states, and a continuum band of other, mostly extended, states. For a system of a finite size, the two branches are degenerate in energy for weak interactions, while a gap opens above a threshold value of the interaction strength. We find that the interplay between degenerate extended and bound states has important consequences for both static and dynamical properties of the system.
    [Show full text]
  • The Hubbard Dimer: a Density Functional Case Study of a Many-Body Problem
    The Hubbard Dimer: A density functional case study of a many-body problem D. J. Carrascal1;2, J. Ferrer1;2, J. C. Smith3 and K. Burke3 1 Department of Physics, Universidad de Oviedo, 33007 Oviedo, Spain 2 Nanomaterials and Nanotechnology Research Center, Oviedo, Spain and 3 Departments of Chemistry and of Physics, University of California, Irvine, CA 92697, USA (Dated: September 3, 2015) This review explains the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site Hubbard model. The relationship to traditional quantum chemistry is included. Even in this elementary example, where the exact ground-state energy and site occupations can be found analytically, there is much to be explained in terms of the underlying logic and aims of Density Functional Theory. Although the usual solution is analytic, the density functional is given only implicitly. We overcome this difficulty using the Levy-Lieb construction to create a parametrization of the exact function with negligible errors. The symmetric case is most commonly studied, but we find a rich variation in behavior by including asymmetry, as strong correlation physics vies with charge-transfer effects. We explore the behavior of the gap and the many-body Green's function, demonstrating the `failure' of the Kohn-Sham method to reproduce the fundamental gap. We perform benchmark calculations of the occupation and components of the KS potentials, the correlation kinetic energies, and the adiabatic connection. We test several approximate functionals (restricted and unrestricted Hartree-Fock and Bethe Ansatz Local Density Approximation) to show their successes and limitations.
    [Show full text]
  • Deforming the Fredkin Spin Chain Away from Its Frustration-Free Point
    Deforming the Fredkin spin chain away from its frustration-free point Khagendra Adhikari∗ and K. S. D. Beachy Department of Physics and Astronomy, The University of Mississippi, University, Mississippi 38677, USA The Fredkin model describes a spin-half chain segment subject to three-body, correlated-exchange interactions and twisted boundary conditions. The model is frustration-free, and its ground state wave function is known exactly. Its low-energy physics is that of a strong xy ferromagnet with gapless excitations and an unusually large dynamical exponent. We study a generalized spin chain model that includes the Fredkin model as a special tuning point and otherwise interpolates between the conventional ferromagnetic and antiferromagnetic quantum Heisenberg models. We solve for the low-lying states, using exact diagonalization and density-matrix renormalization group calculations, in order to track the properties of the system as it is tuned away from the Fredkin point; we also present exact analytical results that hold right at the Fredkin point. We identify a zero-temperature phase diagram with multiple transitions and unexpected ordered phases. The Fredkin ground state turns out to be particularly brittle, unstable to even infinitesimal antiferromagnetic frustration. We remark on the existence of an “anti-Fredkin” point at which all the contributing spin configurations have a spin structure exactly opposite to those in the Fredkin ground state. I. INTRODUCTION z Stot = 0 The construction of a spin-1 quantum spin chain with an dimerization pattern z AFM exactly solvable ground state [1] and its later generalization to I = spin-S (for all integer S > 1) by Movassagh and Shor [2] have proved to be incredibly fruitful developments.
    [Show full text]