Energy Level Statistics of the Two-Dimensional Hubbard Model at Low ﬁlling

PHYSICAL REVIEW B VOLUME 55, NUMBER 14 1 APRIL 1997-II

Energy level statistics of the two-dimensional Hubbard model at low ﬁlling

Henrik Bruus* and Jean-Christian Angle`s d’Auriac† Centre de Recherches sur les Tre`s basses Tempe´ratures, CNRS, Boıˆte Postale 166, F-38042 Grenoble Ce´dex 9, France ͑Received 18 October 1996͒ The energy level statistics of the Hubbard model for LϫL square lattices (Lϭ3,4,5,6) at low filling ͑four electrons͒ is studied numerically for a wide range of coupling strength. All known symmetries of the model ͑space, spin, and pseudospin symmetry͒ have been taken into account explicitly from the beginning of the calculation by projecting into symmetry-invariant subspaces. The details of this group theoretical treatment are presented with special attention to the nongeneric case of Lϭ4, where a particular complicated space group appears. For all the lattices studied, a significant amount of levels within each symmetry invariant subspaces remains degenerated, but except for Lϭ4 the ground state is nondegenerate. We explain the remaining degeneracies, which occur only for very specific interaction-independent states, and we disregard these states in the statistical spectral analysis. The intricate structure of the Hubbard spectra necessitates a careful unfolding procedure, which is thoroughly discussed. Finally, we present our results for the level spacing distribution, the 2 number variance ⌺ , and the spectral rigidity ⌬3, which essentially all are close to the corresponding statistics for random matrices of the Gaussian ensemble independent of the lattice size and the coupling strength. Even very small coupling strengths approaching the integrable zero coupling limit lead to Gaussian ensemble statistics, stressing the nonperturbative nature of the Hubbard model. ͓S0163-1829͑97͒02613-1͔

I. INTRODUCTION study of low-lying excitations and the corresponding coherent part of the spectral densities, a topic we will study in The behavior of strongly correlated electronic systems re- forthcoming work. Here, we rather study the statistical prop- mains a central problem in contemporary condensed matter erties of the typical high-energy excitations which are related physics. Several years of intense studies have made it clear to the incoherent background of the typical spectral func- that the necessary theoretical skills and tools to deal with tions. More specifically, we study the statistical properties of strongly correlated fermion systems are lacking ͑see, e.g., the the spectra within the framework of random matrix theory recent reviews by Dagatto1 and Lieb2͒. Many exotic schemes ͑RMT͒. RMT was developed for the study of neutron scat- have been invented to accommodate a suitable theoretical tering resonances in nuclear physics in the 1950s and 1960s,6 framework, but the development of a predictive general but it has since been applied to a wide range of problems in theory does not seem to be in sight. In this state of affairs the many areas of physics7 ͑e.g., studies of conduction importance of performing numerical calculations of the fluctuations,8 microwave eigenmodes,9 and acoustical prop- ground-state properties and the energy spectrum of a given erties of solids10͒ and mathematics ͑e.g., studies of the dis- many-body Hamiltonian has grown. Computational results tribution of the zeros of the Riemann ␨ function11͒. More- can lead to the acceptance or rejection of the proposed ana- over, RMT has also been applied to various types of matrix lytical models, and they can guide the development of new ensembles like Hamiltonian matrices6 ͑as in our case͒ and analytical approaches. scattering matrices,8,12 as well as transfer matrices13 and In the Hubbard model and related models one important Glauber matrices14 of statistical mechanics models. Recently, parameter is the filling ␯. Much work has been devoted to RMT has been employed in the study of strongly correlated the high-density case near half filling, since it is believed to electronic systems. Examples are studies of the 2D t-J be relevant for high-temperature superconductivity,1,3 but model,15 2D tight-binding models,16 the 1D Bethe chain,17 also the low-filling regime is of interest; e.g., it plays an and the 1D Luttinger liquid.18 The work presented here with important role in theoretical studies of the breakdown of emphasis on mathematical and numerical methods is an ex- Fermi liquid theory in two dimensions ͑2D͒.4 In this paper tension of this line of research. Preliminary results of our we present a numerical study of the two-dimensional Hub- work have been published elsewhere.19 bard model at the low-filling regime ␯Ͻ0.25, a regime There are basically two ways of applying RMT. One way where the calculation is tractable. It is natural to choose four is to model a relevant matrix of the given physical system particles as a generic case close to the simple two-particle with a matrix drawn from a suitable random matrix ensemble case. The coupling strength is used as a perturbation param- and subsequently calculate average properties of the system eter, and we address the question of universality in the re- by averaging over the random matrix ensemble according to sponse of strongly correlated electron systems to this RMT. The other way, which we employ here, consists sim- perturbation.5 ply of characterizing the spectrum of a given physical system We describe an efficient method which allows for numeri- by comparing various statistical properties of the spectrum cal calculations of the exact energy spectrum. This method with the corresponding properties calculated within one of can relatively easily be extended to calculations of various the few universal statistical matrix ensembles of RMT. Green’s functions and spectral functions well suited for the Which of these ensembles describes properly a physical situ-

0163-1829/97/55͑14͒/9142͑18͒/$10.0055 9142 © 1997 The American Physical Society 55 ENERGY LEVEL STATISTICS OF THE TWO- . . . 9143 ation depends on the symmetries of the system. The given spectrum which one analyzes is of course deterministic, but statistical properties are given to it by considering quantities like, for example, the distribution of the energy spacings where all but one of the spectral variables have been integrated out. This is like pseudo-random-number generators which are perfectly deterministic and nevertheless have many properties in common with random sequences. The paper is organized as follows. In Sec. II we introduce the Hubbard model and the corresponding Hilbert space. In Sec. III we introduce the RMT quantities used in the charac- terization of the spectra, and we discuss in detail the special FIG. 1. Two four-electron states with Szϭ0 are shown: in ͑a͒ the zero-pair state 5,6;2,14 of the 4ϫ4 square lattice and in b spectral unfolding technique necessitated by the intricate na- ͉ ͘ ͑ ͒ the one-pair state ͉5,16;13,16͘ of the 5ϫ5 square lattice. The con- ture of the Hubbard spectra. In Sec. IV the entire symmetry vention of the ket notation is explained in the text. The numbering group consisting of space, spin, and pseudospin symmetry of of the lattice sites is the one employed in our computer calculations. the model is studied, and all the corresponding projection The signs of the 4ϫ4 lattice correspond to the bipartition of the operators are calculated. In Sec. V the model is diagonalized lattice, which is only possible for even site lattices. numerically and we study the raw spectrum, in particular the ground state and some unexpected remaining degeneracies encountered where xi and xiЈ are different. If during a calcu- higher in the spectrum. In Sec. VI we present the result of the lation a state is encountered with aϾb and/or cϾd, the nec- spectral statistics analysis of the model, and finally, in Sec. essary exchange operations including sign changes are per- VII we discuss the results and conclude. Appendixes A–D formed to restore it. contain mathematical details. Finally we note that for even L a bipartition of the lattice is possible. A given lattice site a can be identified by a set of II. HUBBARD MODEL Cartesian coordinates aϭ(a1 ,a2) simply counting the posi- Throughout this paper we study the simple one-band tion in the lattice ͓thus, e.g., site 0ϭ(0,0) and 9ϭ(1,2) in LϫL square lattice Hubbard model with periodic boundary Fig. 1͑a͔͒. Each site a can then be assigned with a sign (Ϫ1)a1ϩa2ϭexp(i a), where ϭ( , ). conditions containing a nearest-neighbor hopping term ϪtTˆ ␪aϵ ␲• ␲ ␲ ␲ and an on-site interaction term UUˆ : III. RANDOM MATRIX THEORY L2 L2 Within random matrix theory ͑RMT͒ one can study sev- ˆ ˆ ˆ ˆ† ˆ ˆ ˆ HϭϪtTϩUUϭϪt ͚ cj␴ci␴ϩU͚ ni ni , ͑1͒ eral statistical ensembles of matrices. Three important ex- ͗i,j͘,␴ i ↑ ↓ amples are the diagonal ensembles, the Gaussian ensembles ˆ † ˆ where ci␴ and ni␴ are the creation operator and the number for Hermitian matrices as, e.g., Hamiltonians, and the circu- operator, respectively, for an electron on site i with spin ␴. lar ensembles for unitary matrices as, e.g., scattering matri- No disorder is present in the model. Below half filling the ces. Since a main object of this work is to characterize the dimension NH of the Hilbert space grows rapidly as a func- spectrum of the Hubbard model within the framework of tion of L and the number Ne of electrons occupying the RMT, we are led to use the diagonal and the Gaussian en- lattice, and so we have confined ourselves to the low-filling sembles of square matrices. The first ensemble is the en- case of only four electrons, whereas we let L vary. More- semble of diagonal matrices D with statistically independent over, without loss of generality we always work in the sector diagonal elements Dii drawn from the same distribution where the z component Sz of the total spin is zero; the other P(Dii). This ensemble describes situations where the eigen- Sz sectors can be reached by use of the spin ladder operators values are essentially independent, which empirically has Sϩ and SϪ , which commute with the Hamiltonian. The been found to be the case for integrable models. One char- Szϭ0 sector is the largest of the spin sectors, and it has acteristic feature of such systems is a ‘‘soft’’ spectrum with 2 2 2 NHϭ͓L (L Ϫ1)/2͔ which for Lϭ3, 4, 5, and 6, the lattice large probabilities of having levels close together described sizes studied here, yields 1296, 14 400, 90 000, and 396 900, by the Poisson ͑exponential͒ distribution. The three others 2 respectively. The corresponding fillings ␯ϵNe/2L are 0.22, ensembles—denoted the Gaussian orthogonal ensemble 0.13, 0.08, and 0.06. ͑GOE͒, Gaussian unitary ensemble ͑GUE͒, and Gaussian In the occupation number basis we label the states as symplectic ensemble ͑GSE͒—are defined by requirering sta- follows,20 where a and b are two lattice sites occupied with tistical independence of the matrix elements of a matrix H spin-up electrons and c and d with spin-down electrons: and invariance of the probability distribution P(H) in matrix space under one of the three canonical similarity transforma- ˆ † ˆ † ˆ † ˆ † ͉a,b;c,d͘ϵca cb cc cd ͉vac͘. ͑2͒ tions: the orthogonal, the unitary, and the symplectic trans- ↑ ↑ ↓ ↓ formations, respectively.6 Which ensemble to choose de- Two explicit examples of such states as well as the lattice pends on the symmetry of the system. In fact, the ensembles site enumeration are shown in Fig. 1. In our work we make are universal in the sense that no details of the physical sys- sure that aϽb and cϽd, and we have ordered the basis tem play any role; only knowledge of the global symmetry is states such that the state ͉x1 ,x2 ;x3 ,x4͘ comes before the needed. The three Gaussian ensembles are found to describe state ͉x1Ј ,x2Ј ;x3Ј ,x4Ј͘ if xiϽxiЈ , where i is the first position situations of very complex or chaotic systems, and one char- 9144 HENRIK BRUUS AND JEAN-CHRISTIAN ANGLE` S D’AURIAC 55 acteristic feature of such systems is a ‘‘rigid’’ spectrum with finite random matrix one simply uses the limiting density of eigenvalue repulsion. In this work we need only to treat the states as the average density. In billiard systems one can GOE, which is found for systems with preserved time- make a Laurent series expansion in ͱE of N(E) and obtain reversal symmetry. the Weyl law for N¯(E) by truncating the series after a finite To perform a meaningful RMT analysis one has to sort number of terms, each of which has a physical interpretation. the spectrum in symmetry sectors corresponding to the sym- In our case, no such natural choices exist for N¯(E). We have metry group of the Hamiltonian since the symmetry- therefore used several methods to unfold the spectra. Each of invariant subspaces are orthogonal to one another. Each such these methods has a free parameter, but there is no unique symmetry-invariant subspace is characterized by a specific prescription of how to choose it. The best criterion is the set of quantum numbers, and the RMT analysis is performed insensitivity of the final result to the method employed and on sets of eigenlevels having the same quantum numbers. In to reasonable variations of the free parameter. Sec. IV we treat the complete symmetry group of the Hub- The first method is polynomial interpolation. It can be a bard model and calculate the corresponding projection opera- simple linear interpolation or running average where N¯(E ) tors of the symmetry-invariant subspaces. As illustrated in i is found as a linear fit of N(E) in an interval containing r Sec. VI significant errors are introduced in the analysis if levels on each side of the level E ; the free parameter is then some of the symmetries are neglected. i the parameter r. It can also be higher order polynomial in- Another caveat in the RMT analysis of finite spectra is the terpolation, e.g., in the form of interpolating between several notion of mixed phase space. As a function of some external linear interpolations—a method we used in our work on sta- parameter a given system can be driven from integrability tistical mechanics models.13,22 ͑the diagonal ensemble͒ to chaos ͑one of the Gaussian en- The second method7 defines ␧ ϭ␧ ϩd /d , where d sembles͒ or from one type of symmetry ͑say, the GOE͒ to i iϪ1 1 n k is the kth smallest spacing to E . Here n becomes the free another ͑say, the GUE͒. If the spectrum of such a system is i parameter. This method works also for complex eigenvalues studied in the middle of the transition, the spectrum is a of non-Hermitian matrices. mixture of two or more components, each described by one The third method is Fourier broadening of the step func- of the random ensembles.21 In the thermodynamic limit usu- tions (eϪE ) in Eq. 3 . The Fourier transforms from the ally only one component survives, but for finite spectra this ␪ i ͑ ͒ energy domain to the time domain of the step functions are mixing calls for a further sorting of the spectrum within each ¯ symmetry-invariant subspace. For the Hubbard model using found. In the following back transformation yielding N(E) the coupling strength U/t as the external parameter such a only the slow time components are kept. Choosing a cutoff situation does in fact arise as the analysis presented in Sec. ␶ beyond which all Fourier components are set to zero yields VB shows. ␪͑eϪEi)ϷSi͓͑eϩE*ϪEi)␶͔/␲ϪSi͓͑eϪE*ϩEi)␶͔/␲, where When the necessary sorting of the spectrum has been per- Si(x) is the sine integral and E* is an energy slightly larger formed the RMT analysis can begin. The first step is to un- than the largest energy in the spectrum to be unfolded. The fold the spectrum. free parameter is ␶, and by choosing 1/␶ to be of the order of the mean-level spacing a good N¯(E) is obtained. A. Unfolding the spectrum The fourth method is Gaussian broadening of the delta functions ␦(eϪEi) in Eq. ͑3͒, leading to the following ex- Naturally, it is necessary to make some kind of transfor- pression for N¯(E): mation or normalization of the spectrum of any given physical system to be able to make comparisons with the universal E 2 ¯ 1 ͑eϪEi͒ and dimensionless results of RMT. This operation is called N͑E͒ϭ ͚ exp Ϫ 2 de. ͑5͒ ͵Ϫϱ i 2 ͫ 2␴ ͬ the unfolding. By local rescaling of the spectrum with the ␴iͱ ␲ i local average level spacing the unfolding transforms the ac- The standard deviation or width ␴i of the Gaussians can tual energies Ei into dimensionless ‘‘unfolded energies’’ ␧i with a local density of 1. Thus, by unfolding one subtracts be taken as a constant for the entire spectrum; it is then the the regular slowly varying part of the spectrum and considers free parameter. However, due to the appearance of many only the fluctuations. It amounts to computing from the ac- minibands in the Hubbard spectrum for small values of U/t tual integrated density of states N(E), each having different densities, it is desirable to let ␴i adapt to local variations in the spectrum. We have developed the E following algorithm: Take ␣ levels to each side of level i, N͑E͒ϭ ͚ ␦͑eϪEi͒ deϭ͚ ␪͑eϪEi͒, ͑3͒ determine the local average level spacing ⌬iϭ(Eiϩ␣ ͵Ϫϱ i i ϪEiϪ␣)/(2␣), and set ␴iϭ0.608␣⌬i . By this assignment an averaged integrated density of state N¯(E). The unfolded 90% of the weight of the broadened peak falls in the interval energies ␧ are then given by ͓EiϪ␣⌬i ,Eiϩ␣⌬i͔ and ␣ becomes the free parameter. If a i gap ͑defined as a very atypical spacing͒ falls within the cho- ¯ sen range, we only take the states of the same side of the gap ␧iϭN͑Ei͒. ͑4͒ as Ei into account. The procedure is illustrated in Fig. 2. We The notions of ‘‘local density’’ and ‘‘averaged density’’ are discuss how to optimize the choice of ␣ in Sec. VI. Typi- not mathematically rigorous. For some systems there exists cally, we find ␣Ϸ4 to be a good choice. natural unfolding procedures. For example, it is a rigorous All four methods of unfolding yield essentially the same result that the density of states for a NϫN random matrix results. We decided to use the Gaussian broadening with approaches a semicircular form for N ϱ; thus for any given varying width, Eq. ͑5͒, since it was better suited to the study → 55 ENERGY LEVEL STATISTICS OF THE TWO- . . . 9145

clearly nothing else in common with GOE spectra. To test higher-order correlations one then looks at the two-point correlation function Y(x) and various weighted averages thereof.6 For the GOE in the large-N limit,

ds͑x͒ ϱ Y͑x͒ϭs͑x͒2ϩ s͑t͒ dt, dx ͵x with s(x)ϭsin(␲x)/(␲x). One average of Y(x) often studied is the number variance ⌺2(␭) deﬁned as the variance of the number of unfolded energy levels in intervals of length ␭ around the unfolded energy ␧0:

␭ ␭ 2 ⌺2͑␭͒ϭ N ␧ ϩ ϪN ␧ Ϫ Ϫ␭ , ͑7͒ ͳ ͫ uͩ 0 2 ͪ uͩ 0 2 ͪ ͬ ʹ FIG. 2. Unfolding of the spectrum using the Gauss broadening ␧0 with variable width, choosing ␣ϭ4. Shown is a part of the Hubbard where Nu(␧)ϵ͚i␪(␧Ϫ␧i) is the unfolded level staircase, spectrum of the invariant subspace (R,S)ϭ(6,0) around E/tϷ1.1 and where the brackets denote an averaging over ␧0. For the for Lϭ5 and U/tϭ1. The positions of the levels Ei are marked by Poissonian case ⌺2(␭)ϭ␭, while for the GOE case short vertical lines. For four levels marked by long vertical lines we ⌺2(␭)ϭ␭Ϫ2͐␭(␭Ϫx)Y(x) dx with a logarithmic show the actual broadened Gaussians. Note how the widths of the 0 Gaussians change as the local density of states changes, and note asymptotic behavior. how the width of the Gaussian centered near the gap ignores the Another average of the two-point correlation function is states beyond the gap. Also shown are the level staircase N(E) and the spectral rigidity ⌬3(␭) deﬁned as the least-squares de- the averaged integrated density of states N¯(E) ͑the smooth dotted viation of the unfolded level staircase Nu(␧) from the best- line͒. ﬁtting straight line in an interval of length ␭:

1 ␧0ϩ␭/2 of the Hubbard spectra with its many minibands at small ⌬ ͑␭͒ϭ min ͓N ͑␧͒ϪA␧ϪB͔2d␧ . ͑8͒ 3 ␭ ͵ u coupling strength U/t ͑see Sec. V A͒. ͳ ͑A,B͒ ␧0Ϫ␭/2 ʹ A final remark on the unfolded spectrum is that it is cus- ␧0 tomary to discard from the analysis the states closest to the For the Poissonian case the spectral rigidity is boundary of the spectrum or to the edges of the minibands. ⌬3(␭)ϭ␭/15, while for the GOE case ⌬3(␭)ϭ͓␭ The reason is that these levels in contrast to the levels in the ␭ 3 2 Ϫ͐0 f (x)Y(x) dx]/15, with f (x)ϭ(␭Ϫx) (2␭ Ϫ9␭x bulk of the spectrum do not interact with levels of both Ϫ3x2)/␭4 and again with a logarithmic asymptotic behavior. higher and lower energy. Hence such levels are nongeneric. We usually discarded a few percent of the total number of states on that account. This effect is a size effect that is IV. GROUP THEORY AND INVARIANT SUBSPACES expected to be negligible in the thermodynamic limit. The problem of diagonalizing the Hubbard Hamiltonian can be reduced considerably by group theoretical analysis. B. Quantities characterizing the spectrum Furthermore, as mentioned earlier and as illustrated by an example in Sec. VI A it is indispensable for the RMT analy- The simplest quantity one studies in RMT analysis is the sis. The symmetries are explicitly dealt with from the begin- probability distribution P(s) of unfolded energy spacings ning of the calculation by constructing the symmetry projec- sϭ Ϫ , where and are two consecutive un- ␧i ␧iϪ1 ␧i ␧iϪ1 tion operators corresponding to all known symmetries of the folded energies. One compares the actual P(s) with the same model and using them to project into symmetry-invariant quantity obtained for random matrices from one of the ma- subspaces of the full Hilbert space.23,24 The main object of trix ensembles introduced in Sec. III. For diagonal random this section is to construct the three projection operators matrices P(s) is the Poisson ͑exponential͒ distribution R , S , and J corresponding to the space symmetry, the P(s)ϭexp(Ϫs). For NϫN GOE matrices the distribution of Pspin symmetry,P P and the pseudospin symmetry, respectively. spacings is quite complicated for arbitrary N; however, it is always close to the exact spacing distribution of the 2ϫ2 GOE matrices known as the Wigner surmise, A. Space symmetry group The first symmetry we consider is the space group GL of ␲ ␲ the LϫL square lattice with periodic boundary conditions. It PGOE͑s͒ϭ sexp Ϫ s2 , ͑6͒ 2 ͩ 4 ͪ consists of all permutations g of the sites such that g(i) and g( j) are neighbors if and only if i and j are neighbors. In a which therefore is used in practice. straightforward manner an operator gˆ in Hilbert space can be The spacing distribution probes correlations between con- associated with each element g of G , gˆ a,b;c,d secutive states and is not sensitive to correlations of higher L ͉ ͘ ˆ order. For example, an artificial spectrum constructed by ϵ͉g(a),g(b);g(c),g(d)͘, thus forming a group GL of op- adding independent variables distributed according to Eq. ͑6͒ erators, which commutes with the Hubbard Hamiltonian Hˆ . will certainly show a Wignerian spacing distribution but has For general values of L the space group GL has been ana- 9146 HENRIK BRUUS AND JEAN-CHRISTIAN ANGLE` S D’AURIAC 55

of the elements ͕tx ,rx ,ty ,ry ,rd͖ is the hyperplane reflection rh ͑see Fig. 3͒ obtained in 4D by a reflection in the xz plane while keeping y and w fixed. G4 can be generated by replac- Ϫ1 Ϫ1 ing rd above with rh ͑we note that rdϭtx ty rhtytxrh), and a set of generators is ͕tx ,rx ,ty ,ry ,rh͖ though in fact as few 28 as two elements can be found that generate G4. Instead of 2 128 elements found by the formula NLϭ8L ,wefind 4 N4ϭ2ϫ4!ϭ384. We refer the reader to Appendix A and Appendix B for further details concerning G4 and GL , respectively. (RЈ) The complete table of characters ␹R (g)ofG4 is given in Table IV in Appendix A, while those of G , G , and G FIG. 3. The mapping of the 4ϫ4 square lattice with periodic 3 5 6 are found by combining Table V in Appendix B with the boundary conditions onto the unit hypercube in 4D. The four method of Ref. 20. Armed with the character tables the char- double arrows indicate the neighbor-preserving transformation r . h (RЈ) This transformation cannot be expressed by the ordinary transla- acter projection operator ˆ ͑where the subscript R is used PR tions and reflections in 2D; however, it can be interpreted as a to distinguish from S and J) can be constructed for each hyperplane reflection in 4D around the xz plane. lRЈ-dimensional irreducible representation RЈ:

lR lyzed in detail in Ref. 20. Here we will restrict ourselves to ˆ ͑RЈ͒ϵ Ј ␹͑RЈ͒*͑g͒gˆ . ͑9͒ R N ͚ R outlining this analysis and to correcting the particular cases P Lg෈GL of Lϭ2, which induces a simpler space group, and Lϭ4, Using ˆ (RЈ) for a given representation together with the which induces a much richer space group as briefly men- R Van VleckP basis-function generating algorithm23,24 in the tioned in Ref. 25. actual or a closely related Hilbert space it is straightfor- For LÞ2,4 the structure of G can simply be built up by L ward to construct an orthonormal l -dimensional basis forming direct26 and semi-direct27 products, denoted and R ͕␾(R) ,␾(R) ,...,␾(R)͖ and calculate an explicit irreducible ᭺s, respectively, of translation and reflection subgroups. Let 1 2 lR representation matrix ⌫(R)(g)ϵ͗␾(R)͉gˆ͉␾(R)͘. Finally the Tx (Ty) be the subgroups of order L of translations ͑isomor- ij i j phic with ) in the x (y) direction, and let r , r , and r row projection operator ˆ (RЈ)(g) can be constructed: L x y d PRk be the reflectionZ operations for the x axis, the y axis, and the l diagonal defined by rx(x)ϭϪx, ry(y)ϭϪy, and ˆ ͑RЈ͒ RЈ ͑RЈ͒ ˆ Rk ϵ ͚ ⌫kk *͑g͒g. ͑10͒ rd(x,y)ϭ(y,x), while e denotes the identity transformation. P NLg෈GL The two subgroups Gx ϭT ᭺s ͕e,r ͖ and Gy ϭT ᭺s ͕e,r ͖ of L x x L y y ˆ (RЈ) ˆ (RЈ) s Both projection operators R and Rk will be employed order 2L ͑isomorphic with CLvϭ L᭺ 2) are formed and P P Z Z xy x y in the diagonalization of the Hubbard Hamiltonian. combined into the direct product subgroup GL ϭGLGL . Finally GL is formed by the semidirect product xy s B. SU„2… spin symmetry GLϭGL ᭺͕e,rd͖. One can say that GL is generated by the elements ͕tx ,rx ,ty ,ry ,rd͖ although this is not the smallest It is easily verified that the Hubbard Hamiltonian, Eq. ͑1͒, possible set of generators. The order NL of commutes with the z component of the total spin operator s 2 ˆ Ne ˆ (i) GLϭ(CLv CLv)᭺ 2 is seen to be NLϭ8L . Szϭ͚iϭ1Sz as well as with the corresponding raising and Z s ˆ ˆ For Lϭ2 we find G2ϭ(͕e,rx͖ ͕e,ry͖)᭺͕e,rd͖ϭC4v lowering operators Sϩ and SϪ . The model therefore pos- with N2ϭ8. This result differs from that of the general case sesses a SU͑2͒ spin symmetry. We can therefore restrict our since txϭrx and tyϭry . diagonalization to the Szϭ0 sector, since the other sectors For the case Lϭ4 a richer group appears because the can be reached using the spin-raising and -lowering opera- 4ϫ4 lattice with periodic boundary conditions is isomorphic tors. The spin operators also commute with any space sym- with the group of transformations of the four-dimensional metry operator gˆ , and so the two symmetry groups form a unit hypercube. This isomorphism is easily seen by changing direct product group. The combination of two spin-up and the decimal enumeration of Fig. 1͑a͒ into a binary enumera- two spin-down electrons under the constraint Szϭ0 leads to tion such that the binary numbers of neighboring sites differ a total spin quantum number SЈϭ0, 1, or 2. The explicit by only 1 bit as shown in Fig. 3. They can immediately be construction in Appendix C with Oˆ ϭSˆ 2 yields the following interpreted as the coordinates of the corners of the hyper- ˆ (SЈ) three spin projection operators S : cube. Thus the group G4 of neighbor-preserving transforma- P tions is given by combining bit inversion operations 0 1 ↔ ˆ ͑0͉͒a,b;c,d͘ϭ 1 ͑ϩ2͉a,b;c,d͘ϩ͉a,c;b,d͘Ϫ͉a,d;b,c͘ with permutations of the 4 bits. In Appendix A we show that PS 6 s G4 is isomorphic with the group ( 2 2 2 2)᭺S4 ϩ2͉c,d;a,b͘ϩ͉b,d;a,c͘Ϫ͉b,c;a,d͘), corresponding to the semidirect productZ ofZ allZ combinationsZ of bit inversions at the four positions with the permutation ͑11͒ group S4 of the four positions. One neighbor-preserving transformation of the lattice which is not given by products ˆ ͑1͉͒a,b;c,d͘ϭ 1 ͑ϩ3͉a,b;c,d͘Ϫ3͉c,d;a,b͘), ͑12͒ PS 6 55 ENERGY LEVEL STATISTICS OF THE TWO- . . . 9147

͑2͒ 1 proved useful to introduce a special notation for one-pair and ˆ ͉a,b;c,d͘ϭ ͑ϩ1͉a,b;c,d͘Ϫ͉a,c;b,d͘ϩ͉a,d;b,c͘ PS 6 two-pair basis states as follows:

ϩ1͉c,d;a,b͘Ϫ͉b,d;a,c͘ϩ͉b,c;a,d͘). † † † † ͉P;b,d͘ϵ␪ Pc P c P cb cd ͉vac͘ϭ␪ P͉P,b;P,d͘, ͑13͒ ↓ ↑ ↑ ↓ ͉P;Q͘ϵ␪ ␪ c† c† c† c† ͉vac͘ϭϪ␪ ␪ ͉P,Q;P,Q͘. These expressions are generally valid; however, one should P Q P P Q Q P Q ↓ ↑ ↓ ↑ ͑15͒ note that the Pauli exclusion principle reduces the number of terms when electron pairs are present. For example, if aϭc For zero-pair states where a, b, c, and d are all different, (S ) (JЈϭJ ) and bϭd, one ﬁnds Ј ͉a,b;c,d͘ϭ␦ ͉a,b;c,d͘. Finally only ˆ 0 is nonzero, PS SЈ,0 PJ we note that the spin projectors of the state ͉a,b;c,d͘ in- volve all six states generated by permutations of the site ˆ ͑J0͒ J ͉a,b;c,d͘ϭ͉a,b;c,d͘, ͑16͒ indices. In Eqs. ͑11͒–͑13͒ only three orthogonal states P appear. The remaining three orthogonal states are the and for one-pair states where P, b, and d are all different, two Sϭ1 states ͉a,c;b,d͘Ϫ͉b,d;a,c͘ and ͉a,d;b,c͘ two projection operators are nonzero, Ϫ͉b,c;a,d͘ and the one Sϭ0 state ͉a,c;b,d͘ 2J ϩ1 ϩ͉b,d;a,c͘ϩ͉a,d;b,c͘ϩ͉b,c;a,d͘. ˆ ͑J0͒ 0 J ͉P;b,d͘ϭ ͉P;b,d͘ P 2͑J0ϩ1͒

C. SU„2… pseudospin symmetry Ϫ1 ϩ ͚ ͉Q;b,d͘, ͑17͒ The pseudospin symmetry of the Hubbard model has been 2͑J0ϩ1͒QÞP,b,d known for at least a quarter of century,29 but recently it was rediscovered and put in the more generalized context of the J ϩ1 1 30,31 ˆ ͑ 0 ͒ ␩-pairing mechanism. As discussed in Sec. II even L J ͉P;b,d͘ϭ ͉P;b,d͘ P 2͑J0ϩ1͒ lattices can be biparted using the site sign ␪i as an index, and three operators Jˆ , Jˆ , and Jˆ can be defined: 1 Ϫ ϩ z ϩ ͚ ͉Q;b,d͘, ͑18͒ 2͑J0ϩ1͒QÞP,b,d 1 L2 ˆ ˆ ˆ ˆ ˆ † ˆ † ˆ ˆ while for two-pair states where PÞQ, all three projection JϪϭ͚ ␪ici ci , Jϩϭ͚ ␪ici ci , Jzϭ ͚ ni␴Ϫ . i ↑ ↓ i ↓ ↑ 2 i␴ 2 operators are nonzero: ͑14͒ ͚RÞP,Q͚SÞP,Q,R͉R;S͘ ͑2J0ϩ1͉͒P;Q͘ ˆ ͑J0͒ P;Q ϭ ϩ It is seen that Jˆ creates a pair of electrons with phase on J ͉ ͘ ϩ ␪i P 2͑J0ϩ1͒͑2J0ϩ3͒ 2J0ϩ3 empty sites i. These three operators form the same algebra as ˆ ˆ ˆ ͑2J0ϩ1͚͒RÞP,Q͕͉R;Q͘ϩ͉R;P͖͘ SϪ , Sϩ , and Sz , hence the name pseudospin. Furthermore, Ϫ , ͑19͒ the Jˆ operators commute with both the space symmetry op- 2͑J0ϩ1͒͑2J0ϩ3͒ erators and the spin operators; and for the symmetrized Hub- ˆ 32 ˆ ͑J ϩ1͒ Ϫ͚RÞP,Q͚SÞP,Q,R͉R;S͘ ͉P;Q͘ bard model HЈ, trivially related to our model H, they even ˆ 0 ͉P;Q͘ϭ ϩ J 2 J ϩ1͒ J ϩ2͒ J ϩ2 commute with the Hamiltonian Hˆ Ј. Hence the ͑symmetrized͒ P ͑ 0 ͑ 0 0 model possesses an extra SU͑2͒ symmetry characterized J0͚RÞP,Q͕͉R;Q͘ϩ͉R;P͖͘ by the quantum numbers JЈ and Jz analogous to SЈ and ϩ , ͑20͒ 33 2͑J0ϩ1͒͑J0ϩ2͒ Sz for the spin. A detailed analysis shows that the combination of spin and pseudospin symmetries yields ͑J ϩ2͒ ͚RÞP,Q͚SÞP,Q,R͉R;S͘ ͉P;Q͘ ͓SU͑2͒ SU͑2͔͒/ 2ϭSO͑4͒ rather than the full SU͑2͒ ˆ 0 ͉P;Q͘ϭ ϩ 31 Z J SU͑2͒. The space symmetry group is completely inde- P 2͑J0ϩ2͒͑2J0ϩ3͒ ͑J0ϩ2͒͑2J0ϩ3͒ pendent of spin and pseudospin symmetries. Thus the full ͚RÞP,Q͕͉R;Q͘ϩ͉R;P͖͘ symmetry group for L even is ϭG SO(4) while for L ϩ . ͑21͒ L ͑J ϩ2͒͑2J ϩ3͒ odd it is ϭG SU(2). In ourG case the symmetry is low- 0 0 G L ered in a trivial way from spherical to cylindrical symmetry At this stage all the group theoretical ingredients are ready ˆ ˆ ˆ in pseudospin space since ͓U,JϮ͔ϭϮJϮ , but still we have for the diagonalization of the Hubbard Hamiltonian. ˆ ˆ 2 ˆ ˆ ͓H,J ͔ϭ͓H,Jz͔ϭ0 and JЈ and Jz both remain good quantum numbers. D. Symmetry-invariant subspaces For a given lattice with even L containing a fixed number 2 Using the projection operators the NH-dimensional Hil- Ne of electrons all states have Jzϭ(NeϪL )/2 ͑e.g., Ϫ6 for bert space can be broken down into smaller symmetry invari- Lϭ4 and Ϫ16 for Lϭ6). Defining J0ϵ͉Jz͉, the size J of the ant subspaces. Let be the group containing the symmetry pseudospin in this situation takes the values operations ␥, eachG being a product of a space symmetry ˆ (JЈ) J0 ,J0ϩ1,...,J0ϩNe/2. The projection operators J are transformation, a spin rotation, and ͑for L even͒ a pseudospin P found using the construction of Appendix C with rotation, ␥ϭg gS gJ . Here thus consists of one finite ˆ ˆ 2 ˆ ˆ ˆ 2 ˆ G OϭJ ϭJϩJϪϩJz ϪJz . Due to the explicit reference to group and one or two compact Lie groups. Let ␳ be a multi- pairs in the definition of the pseudospin operators, it has index describing an irreducible representation ⌫(␳) of . For G 9148 HENRIK BRUUS AND JEAN-CHRISTIAN ANGLE` S D’AURIAC 55

TABLE I. For Lϭ3, 4, 5, and 6 are shown the dimension NH of a␳-dimensional orthonormal basis is found by using the pro- the total unreduced Hilbert space and the dimension max͕a␳͖ of the jection operator largest symmetry-invariant subspace. Since matrix diagonalization is an n3 operation, the number ͚ a3/N3 provides an estimate of the ˆ ͑␳͒ ˆ ͑RЈ͒ ˆ ͑SЈ͒ ˆ ͑JЈ͒ ␳ ␳ H 0 ϵ R0 S J , ͑25͒ relative reduction in computer time by projecting into the invariant P P P P subspaces. which projects onto the ﬁrst row of the space group representation RЈ with the spin index SЈ and pseudospin index Lϭ3 Lϭ4 Lϭ5 Lϭ6 ˆ (␳) JЈ. The projection operator 0 is applied on one basis state after another while performingP an ongoing Gram-Schmidt N 1296 14 400 90 000 396 900 H orthonormalization procedure. This yields new basis states max ͕a ͖ 38 146 1794 5490 ␳ ␳ ͉w ͘, and when a such states are found, the procedure is ͚ a3/N3 14.1ϫ10Ϫ5 0.7ϫ10Ϫ5 2.3ϫ10Ϫ5 1.0ϫ10Ϫ5 i ␳ ␳ ␳ H terminated. The next step is to calculate the kinetic energy and poten- ˆ ˆ even ͓odd͔ L we have ␳ϭ(RЈ,SЈ,JЈ) ͓␳ϭ(RЈ,SЈ)͔. The tial energy matrix elements ͗wi͉T͉w j͘ and ͗wi͉U͉w j͘. These 23,24 matrix elements are stored in the computer and thereafter it is ‘‘celebrated’’ theorem states how many times a␳ each row of the irreducible representation ⌫(␳) with character a simple matter to pick any value of U/t and diagonalize the ␹(␳) appears in any given not necessarily irreducible repre- Hubbard Hamiltonian, Hˆ ϭϪtTˆϩUUˆ , using standard diago- sentation ⌫ with character ␹: nalization routines. The calculation of the symmetry invariant matrix blocks of Tˆ and Uˆ takes the amount of time of the 1 order of one diagonalization. ͑␳͒* a␳ϭ ͚ ͵ dgS ͵ dgJ␹ ͑␥͒␹͑␥͒. ͑22͒ NL GL SU͑2͒ SU͑2͒ A. Spectrum and ﬁrst-order perturbation theory We now choose ⌫ to be the following NH-dimensional re- of the ground state ducible representation: In this section we discuss some features of the raw spec-

NH trum of the Hubbard model. Only later in Sec. VI are we ˆ ˆ going to unfold the spectrum and look for universal features. ⌫ij͑␥͒ϵ͗i͉␥͉j͘, ␹͑␥͒ϭ͚ ͗i͉␥͉i͘. ͑23͒ iϭ1 As a function of the coupling strength U/t the spectrum clearly falls into three classes. In the weak-coupling limit Inserting into Eq. ͑22͒ these ␹(␥)’s which are directly linked (U/tӶW/t), to be studied in more detail below, the spec- to Hilbert space yield trum acquires a band width WϷ32t. It consists of a number M of well-separated minibands reminiscent of the huge de- N N H 1 H generacy at zero coupling due to size quantization. For ϭ ͑␳͒* ˆ ϭ ˆ ͑␳͒ a␳ ͵ d␥␹ ͑␥͚͒ ͗i͉␥͉i͘ ͚ ͗i͉ ͉i͘ intermediate-coupling strengths (U/tϷW/t) the spectrum iϭ1 lRiϭ1 P G becomes rather featureless. No apparent gaps or bands 1 NH emerge. In the strong-coupling limit (U/tӷW/t) the spec- ˆ ͑RЈ͒ ˆ ͑SЈ͒ ˆ ͑JЈ͒ ϭ ͗i͉ R S J ͉i͘, ͑24͒ trum splits up into three well-separated bands centered l i͚ϭ1 P P P R around the energies E/tϭ0,U/t,2U/t. These are the Hubbard bands corresponding to states containing zero, one, or two which is obtained by using ␹(␥)ϭ␹(g gS gJ) pairs of electrons. In Fig. 4 are shown one typical spectrum ϭ␹(g)␹(gS)␹(gJ) and the deﬁnitions of the character projection operators. Note that only the space group projection from each of the three coupling strength regimes. We now focus on the weak-coupling limit where it needs a normalizing factor 1/lR . The spin and pseudospin are both restricted to take only one value of their respective is natural to make Fourier transforms from real space to ˆ † ˆ † z component; hence their normalizing factor is 1. momentum space, ck␴ϵ(1/L)͚jexp(ik•rj)c j␴ , where The initial size of the Hubbard Hamiltonian to be diago- kϭ(kx,ky), with k␨ϭ2␲n/L and nϭ0,1, ...,LϪ1. The 2 nalized is ͓L(LϪ1)/2͔ . However, only states transforming eigenstates of the kinetic energy operator ϪtTˆ are according to the same row of the same irreducible represen- ˆ † ˆ † ˆ † ˆ † tation ␳ can have a nonzero matrix element. Hence the ͉k0 ,k1 ;k2 ,k3͘ϭck ck ck ck ͉vac͘, Hamiltonian matrix breaks up in blocks, one per representa- 1↑ 2↑ 3↓ 3↓ tion, and within each representation a further division into 3 ˆ lR equivalent blocks occurs. The relationship between NH , ϪtT͉k0 ,k1 ;k2 ,k3͘ϭϪ2t E͑kn͉͒k0 ,k1 ;k2 ,k3͘, ͑26͒ n͚ϭ0 a(RЈ,SЈ,JЈ) , and lRЈ is NHϭ͚(R,S,J)lRЈa(RЈ,SЈ,JЈ) . This reduction is considerable as shown in Table I. x y E͑kn͒ϭcos͑kn͒ϩcos͑kn͒.

V. NUMERICAL DIAGONALIZATION The Pauli principle prevents k0ϭk1 and k2ϭk3, and the (0) OF THE HUBBARD MODEL ground-state energy EL becomes

The numerical calculation of the exact spectra of the Hub- ͑0͒ EL ϭϪ12tϪ4tcos͑2␲/L͒. ͑27͒ bard model begins by determining the block size a␳ for all the irreducible representations ␳. Then for a given For large lattices this tends toward Ϫ16t and results in a ␳ϭ(RЈ,SЈ,JЈ), where JЈ is disregarded in case of odd L,an bandwidth Wϭ32t. For Lϭ3, 4, 5, and 6, respectively, the 55 ENERGY LEVEL STATISTICS OF THE TWO- . . . 9149

FIG. 4. The integrated density of states N(E) of the Hubbard model with four electrons on the 5ϫ5 lattice for the irreducible representation (R,S)ϭ(10,0) with U/tϭ0.1,10,1000. For U/tϭ0.1 the minibands are still clearly visible and N(E) is essentially featureless for U/tϭ10, while for U/tϭ1000 the three Hubbard bands at 0, 1000, and 2000 appear. The insets are magniﬁcations of N(E) where the arrows point. The smooth ¯N(E) is added.

actual bandwidths W/t are 20.0, 24.0, 26.2, and 28.0 while same form for Lϭ3,5,6: (0,0,1,1,Ϫ1), (0,0,1,Ϫ1,*), the number M of minibands for Uϭ0 are 7, 13, 42, and 29. (2␲/L,2/␲/L,*,*,Ϫ1), (4␲/L,0,*,1,*), (2␲/L,2␲/ The limitation of perturbation theory is demonstrated by a L,*,*,1), and ͑0,0,1,1,1͒, where the first set corresponds to first-order degenerate perturbation calculation for the ground the unique ground state. This result extends to the Lϭ4 lat- state. Let pL be the one-dimensional momentum component tice when the actual space group is replaced by the one generated by t ,r ,t ,r ,r . One state in the Lϭ4 triplet pLϭ2␲/L and construct the 5 two-dimensional momentum ͕ x x y y d͖ 0 1 2 ground state belongs to the (0,0,1,1,Ϫ1) representation and vectors 0ϭ(0,0), q ϭ(pL,0), q ϭ(0,pL), q ϭ(ϪpL,0), and q3ϭ(0,Ϫp ). The 16 states ͉␮,␭͘, defined as the other two states to (4␲/L,0,1,1,*). In all cases the L ground state is odd with respect to the diagonal reflection ␮ ␭ ͉␮,␭͘ϭ͉0 ,q ;0 ,q ͘, ␮,␭ϭ0,1,2,3, ͑28͒ rd , which is understandable since that symmetry suppresses ↑ ↑ ↓ ↓ the ability of having pairs along the diagonal. Finally we form the ground-state multiplet separated from the next mul- note that due to the aforementioned spin symmetry of the tiplet by an energy gap ⌬E ϭ2t͓1Ϫcos(p )͔. In momentum L L Hubbard model, the ground state of the Szϭ0 sector is in fact space the interaction operator UUˆ takes the form a global ground state, and furthermore since it has Sϭ0, the same energy level does not exist in any other S sector. Thus U z UUˆ ϭ cˆ† cˆ cˆ† cˆ , ͑29͒ we can conclude that the global ground state in the generic 2 ͚ k1ϩq k1 k2Ϫq k2 L k1 ,k2 ,q ↑ ↑ ↓ ↓ case (LÞ4) is nondegenerate.

and after some simple algebra the matrix elements of the B. Remaining degeneracies perturbation are found to be After taking all the known symmetries into account and U after projecting into the symmetry-invariant subspaces, it ͗m,l͉UUˆ ͉␮,␭͘ϭ ͑3␦ ␦ ϩ␦ ͒. ͑30͒ L2 m,␮ l,␭ mϩl,␮ϩ␭ The eigenvalues can be found analytically, and we end up with the following expression for the perturbed levels (1) EL,␤(U) with degeneracies d: U E͑1͒ ͑U͒ϭE͑0͒ϩ␤ , ␤ϭ3 ,4 ,5 ,7 . L,␤ L L2 ͑dϭ7͒ ͑dϭ4͒ ͑dϭ4͒ ͑dϭ1͒ ͑31͒ Thus in first-order perturbation theory the ground-state degeneracy is partly lifted, leaving a sevenfold-degenerate ground state for UϾ0. In Fig. 5 we compare the exact numerical calculation with Eq. ͑31͒ for Lϭ5 and 6. Note especially how in the exact calculation the degeneracy of the ground-state energy is lifted completely. This is also demonstrated in Table II, where it can be seen that also for Lϭ3 the ground state is FIG. 5. The evolution of the 16-fold degenerated ground state nondegenerate. However, the ground state of Lϭ4 remains multiplet as a function of U/L2. The exact numerical results for threefold degenerated even in the exact calculation. This re- Lϭ5 ͑dotted lines͒ and Lϭ6 ͑dashed lines͒ are contrasted with the flects the particular symmetry of the 4ϫ4 lattice, where the first-order perturbation calculation ͑solid straight lines͒. The inset is hyperplane reflection rh leads to the existence of threefold- a magnification of that portion of the plot marked by a rectangle, degenerate representations as shown in Appendix A. showing how the exact calculation leads to a splitting into three Using the quantum numbers (kx ,ky ,bx ,by ,c)of submultiplets with degeneracies 1, 2, and 4 of the sevenfold- Table V in Appendix B to interpret the representation degenerate perturbation theory ground state. The exact ground state label R in Table II we note that they are of the is nondegenerate. See also Table II. 9150 HENRIK BRUUS AND JEAN-CHRISTIAN ANGLE` S D’AURIAC 55

2 (1) TABLE II. For U/tL ϭ0.02 the sevenfold-degenerate perturbation theory ground state EL,␤ϭ3 given in row P is compared with the exact energies E of the 16 levels splitting off from the Uϭ0 ground state (0) EL The degeneracies of the submultiplets are given by the dimension lR of the corresponding irreducible representation R of the space group. Note that except for Lϭ4 the exact ground state is nondegenerate. Also listed are the quantum numbers (R,S,J) of the levels.

Lϭ3 Lϭ4 Lϭ5 Lϭ6

No. E/t (R,S)lR E/t (R,S,J)lR, E/t (R,S)lR E/t (R,S,J)lR P Ϫ9.94000 ͑Ϫ,Ϫ͒ 7 Ϫ11.94000 ͑Ϫ,Ϫ,Ϫ͒ 7 Ϫ13.17607͒͑Ϫ,Ϫ͒7Ϫ13.94000 ͑Ϫ,Ϫ,Ϫ͒ 7 0 Ϫ9.94122 ͑2,0͒ 1 Ϫ11.94268 ͑9,0,6͒ 3 Ϫ13.18059 ͑2,0͒ 1 Ϫ13.94676 ͑4,0,16͒ 1 1 Ϫ9.94110 ͑8,1͒ 4 Ϫ11.94232 ͑16,1,6͒ 6 Ϫ13.18012 ͑4,1͒ 2 Ϫ13.94617 ͑9,1,16͒ 2 2 Ϫ9.94097 ͑4,1͒ 2 Ϫ11.90425 ͑17,0,6͒ 6 Ϫ13.18005 ͑10,1͒ 4 Ϫ13.94606 ͑25,1,16͒ 4 3 Ϫ9.92188 ͑5,0͒ 4 Ϫ11.86560 ͑0,0,6͒ 1 Ϫ13.16204 ͑7,0͒ 4 Ϫ13.92870 ͑14,0,16͒ 4 4 Ϫ9.90229 ͑7,0͒ 4–Ϫ13.14302 ͑9,0͒ 4 Ϫ13.91032 ͑24,0,16͒ 4 5 Ϫ9.86204 ͑0,0͒ 1–Ϫ13.10590 ͑0,0͒ 1 Ϫ13.87494 ͑0,0,16͒ 1

turns out that within each subspace some further degenera- in Sec. VI of the spectral statistics we throw away the inte- cies remain. The reason is that the low filling of the lattice grable component and analyze only the generic noninte- allows for a kind of restricted permutation symmetry for the grable component. particle momentum components, resulting in energy eigen- The Tˆ /Uˆ states are formed by specific superpositions of ˆ ˆ states which are simultaneously eigenstates of T and U and eigenstates of Tˆ by permuting the eight momentum compo- therefore independent of U ͑though their eigenvalues might nents k␨ ␨ϭx,y and nϭ0, 1, 2, and 3, such that the sum of ˆ ˆ ␥ n depend on U). We denote such states T/U states or ͉␺S ͘, cosines in Eq. ͑26͒ remains unchanged. In the following we where S is the spin and ␥ the eigenvalue of Uˆ . With four mention some large classes of such states. The reader is re- electrons ␥ takes the values ␥ϭ0,1,2 corresponding to a su- ferred to Appendix D for details. perposition of states containing exactly ␥ pairs, i.e., doubly First we note that for even L proportionally many more occupied sites. A generic energy eigenstate is not a Tˆ /Uˆ state states remain degenerate as compared to L odd. This differ- ␨ and in that case we write ␥ϭ*. In Table III we show the ence is due to the momentum component knϭ␲, which only number of generic energy eigenstates and Tˆ /Uˆ states found exists for L even. Because cos(␲Ϫk)ϩcos(k)ϭ0 independent numerically. The Hubbard model is thus partly integrable. of k, such terms drop out when Tˆ is applied to a state con- The spectrum of each symmetry-invariant subspace is a mix- taining this combination and a certain degree of freedom is ture of an integrable component ͑the Tˆ /Uˆ states͒ and a non- left to form the Tˆ /Uˆ state superpositions. Starting with two- integrable component ͑the generic states͒, and in the analysis pair states ␥ϭ2 we find exactly one energy eigenstate

TABLE III. For Lϭ3, 4, 5, and 6 is shown the number of energy eigenstates found numerically in each of the four main groups indexed by ␥. The ﬁrst group, denoted ␥ϭ*, contains the generic energy eigenstates that are not eigenstates of Uˆ . The three other groups, denoted ␥ϭ0,1,2, respectively, contain the nongeneric Tˆ /Uˆ states that are simultaneous eigenstates of Uˆ ͑with eigenvalue ␥) and Tˆ . In parentheses are given the number of states that remain degenerated in the symmetry-invariant subspaces after taking the space, spin, and pseudospin symmetries into account. Note how the ␥ϭ* states exhibit no further degeneracy. In the last row denoted ‘‘%’’ are given the percentages of ␥ϭ* states out of the total number of states.

␥ SLϭ3 Lϭ4 Lϭ5 Lϭ6

* 0 540/͑0͒ 4169/͑0͒ 31977/͑0͒ 115896/͑0͒ * 1 621/͑0͒ 4472/͑0͒ 41662/͑0͒ 137199/͑0͒ * 20/͑0͒ 0/͑0͒ 0/͑0͒ 0/͑0͒ 000/͑0͒1176/͑1143͒ 523/͑316͒ 23555/͑23427͒ 019/͑0͒2548/͑2519͒ 3188/͑2823͒ 60306/͑60160͒ 0 2 126/͑80͒ 1820/͑1727͒ 12650/͑12362͒ 58905/͑58723͒ 100/͑0͒94/͑14͒ 0/͑0͒ 408/͑247͒ 110/͑0͒120/͑24͒ 0/͑0͒ 630/͑420͒ 120/͑0͒0/͑0͒ 0/͑0͒ 0/͑0͒ 200/͑0͒1/͑0͒ 0/͑0͒ 1/͑0͒ 210/͑0͒0/͑0͒ 0/͑0͒ 0/͑0͒ 220/͑0͒0/͑0͒ 0/͑0͒ 0/͑0͒ % 89.6% 60.0% 81.8% 63.8% 55 ENERGY LEVEL STATISTICS OF THE TWO- . . . 9151

2 FIG. 6. The spectral statistics P(s), ⌬3(␭), and ⌺ (␭) for the invariant subspace (R,S)ϭ(13,1) of the 5ϫ5 lattice as a function of the free parameter ␣ of the unfolding procedure using Gaussian broadening with variable width. In all three panels the values for ␣ are 2, 4, 5, 6, 8, 10, and 20. In the ﬁrst panel six of the seven curves ﬂuctuate around PGOE(s), hardly visible as a smooth dotted line. In the two other panels the Poisson case is given as a dotted straight line while the GOE case is given by a dotted smooth curve. For all data sets a few representative error bars are shown.

␥ϭ2 ͉␺Sϭ0 ͘. It depends on the momentum ␲, and hence it exits question now arises as to how to choose the free parameter only for even L. Similarly, for one-pair states only even L ␣ corresponding to how many energy levels each Gaussian ˆ ˆ L2 ␥ϭ1 essentially spreads out over to each side. The problem we leads to T/U states. A number (2 ) of states ͉␺Sϭ1 ͘ can easily be constructed. More care must be taken upon forming face is illustrated in Fig. 6 where it is seen that although the level spacing distribution P(s) is essentially independent of the corresponding Sϭ0 states. However, we have succeeded 2 ␥ϭ1 the choice of ␣, both the number variance ⌺ (␭) and the in constructing analytically the number of states ͉␺Sϭ0 ͘ re- spectral rigidity ⌬3(␭) vary with ␣. It is seen that ⌬3(␭)is quired by Table III. Finally, for zero-pair states the existence 2 ˆ ˆ less sensitive to changes in ␣ than ⌺ (␭). The former seems of the momentum ␲ is not required to form the T/U states, to saturate for large values of ␣, while the latter continues to but it certainly helps. Thus both even and odd values of L grow rapidly as ␣ enhances. We have chosen that value ␣0 lead to nongeneric energy eigenstates, but relatively more of ␣ which makes ⌺2(␭) ﬁt the corresponding GOE curve as such states are found for even L. It is easy to see that all well as it can. In general that leads to ␣0Ϸ4. We note that states with the maximal spin Sϭ2 are Tˆ /Uˆ states ␺␥ϭ0 . ͉ Sϭ2 ͘ for this choice of ␣, ⌬3(␭) is almost saturated, while P(s) This is a trivial consequence of choosing four different sites remains unchanged. Thus, two of the three statistics are es- ͑or momenta͒ and forming the superposition given in Eq. sentially independent of variations of ␣ around ␣0, while the L2 ͑13͒. There are (4 ) such states. The construction of states third is as close to the GOE behavior as it can be. ␥ϭ0 with S ϭ1 or 0 is more cumbersome, and so in To illustrate the importance of sorting the spectrum by ͉␺SϭSЈ͘ Ј Appendix D we have only given two examples of classes of group theory we show in Fig. 7 the level spacing distribution such states. for Lϭ4 with U/tϭ10 for the case where all symmetries are The main result of this section is that the degeneracies taken into account ͑‘‘full symm.’’͒ and for the case with that remain after space, spin, and pseudospin symmetry have lower symmetry ͑‘‘low symm.’’͒ where the spin and pseu- been taking into account is related to a restricted permutation dospin symmetries are being kept intact but where the space symmetry of the momentum components. We have not found group has been reduced by replacing among the generators the associated projection operators, but numerically and partly analytically we have established the fact that the degenerate states are simultaneously eigenstates of Tˆ and Uˆ with energies E(U)ϭE(0)ϩ␥U. By discarding these states we end up with nondegenerate Uˆ -dependent states. It is the spectral statistics of these states we analyze in the following section, and this analysis conﬁrms our claim that all symmetries indeed have been taken into account.

VI. SPECTRAL STATISTICS OF THE HUBBARD MODEL Having sorted the spectrum according to all symmetries including the restricted permutation symmetry of the momentum components the RMT analysis can be performed. As discussed in Sec. III the first step is the unfolding of the spectrum. There we mentioned how the unfolding procedure is not uniquely determined, and so we turn to that problem FIG. 7. P(s) for Lϭ4 with U/tϭ10 is calculated after sorting first. the spectra using either the full symmetry group of the Hamiltonian (छ) or an symmetry group artificially lowered (*) as described in the text. The full symmetry case compares well with the Wigner A. Optimization of the unfolding procedure surmise ͑smooth solid curve͒ whereas the lower-symmetry case We unfold the spectra by using the method of Gaussian compares well with the distribution of two GOE spectra mixed with broadening with variable width discussed in Sec. III A. The relative weights 0.72 and 0.28 ͑dotted smooth curve͒. 9152 HENRIK BRUUS AND JEAN-CHRISTIAN ANGLE` S D’AURIAC 55

FIG. 8. The probability distribution P(s) of the level spacings s averaged over the largest symmetry-invariant subspaces for the Hubbard model with four electrons on square LϫL lattices. To the left is shown P(s) as a function of lattice size L at medium-coupling strength. The data represent Lϭ4, 5, and 6 for U/tϭ10.0. To the right is shown P(s) as a function of coupling strength at ﬁxed low ﬁlling. The data represent U/tϭ0.1, 10, and 1000 for Lϭ5. The solid line is the Wigner distribution found for GOE random matrices.

the special hyperplane reflection rh with the ordinary diago- ration sets in. For all values of U/t we find the critical value nal reflection rd . For the full symmetry case a distribution ␭* where the departure from the GOE sets in to be roughly rather close to the Wigner surmise is found, whereas the 2. The precise origin of ␭* remains unclear. level repulsion is partially lost for the low-symmetry case, Finally, in Fig. 10 are shown the results for the spectral 2 and the data fit reasonably well the distribution found by rigidity ⌬3(␭). As ⌺ (␭) also ⌬3(␭) displays an excellent mixing two GOE spectra21 with relative weights 0.72 and agreement with the GOE for ␭Ͻ␭*Ϸ2, for all fillings and 0.28. This makes sense since by lowering the symmetry for all values of U/t. The deviations from the GOE beyond group artificially, spectra from the independent true ␭* are not so marked. The curves lie between the Poisson symmetry-invariant subspaces are being mixed. For the more line and the GOE curve, but rather close to the latter. severe symmetry reduction where the pseudospin is alto- It is remarkable how the results for the three statistics gether neglected and the space group is generated only by studied are fairly independent of the size of the lattice ͕tx ,ty͖, we find the level spacing distribution to be the Pois- ͑equivalent to the filling͒ and of the coupling strength. We son ͑exponential͒ distribution ͑not shown in the figure͒. find GOE-like behavior not only for all finite values of U/t including those close to the integrable Uϭ0 limit, but also

2 for filling factors as low as 0.06 close to the integrable B. Statistics P„S…, ⌺ „␭…, and ⌬3„␭… of the Hubbard model single-particle limit. However, as is evident from the behav- In this subsection we present the results of the spectral ior of especially ⌺2(␭) at large energy scales, the Hubbard statistical analysis of the Hubbard model at low filling. We model cannot be modeled exactly by a simple GOE random present only results for Lϭ4, 5, and 6 since Lϭ3 yields too matrix model. poor statistics due to its small invariant subspaces. Besides letting L vary we are also varying the coupling strength U/t and present results for weak-, intermediate-, and strong- VII. CONCLUSIONS AND DISCUSSION coupling regimes for Lϭ5. To improve on the statistics we In this paper the energy level statistics of the Hubbard have averaged over the largest invariant subspaces. For model for LϫL square lattices (Lϭ3,4,5,6) at low filling P(s) the sizes ␦P of the error bars shown in the figures are ͑four electrons͒ has been studied numerically for a wide estimated by ␦PiϭCͱni/hi , where ni is the number of range of the coupling strength. With great care all known points in bin i of the associated histogram and hi is the width symmetries of the model ͑space, spin, and pseudospin sym- of the bin, while C is the normalization factor rendering a metry͒ have been taken into account explicitly from the be- 2 total probability of 1. For ⌺ (␭) and ⌬3(␭) the error bars ginning of the calculation by projecting into symmetry- are estimated by ordinary variances obtained from the values invariant subspaces. The details of this group theoretical calculated in the many contiguous intervals of length ␭ treatment were presented with special attention to the nonge- throughout the unfolded spectrum. neric case of Lϭ4, where a particular complicated space The results for P(s) are shown in Fig. 8. It is seen that for group appears. The resulting reduction of the numerical di- all lattice sizes and for any value of the coupling strength the agonalization is significant, and the method presented can in level spacing distribution is fairly close to the Wigner distri- a straightforward manner be extended to larger lattices and bution of the GOE; it possesses a pronounced linear level higher fillings and thus form the basis of improved numerical repulsion for small s, a peak near sϭ0.8, signaling spectral studies of the Hubbard model and related models without rigidity, and a rapid falloff for sϾ2. disorder. In particular, this work can be used as a starting In Fig. 9 is shown ⌺2(␭) of the Hubbard model for the point for calculating various spectral functions, for which same parameters as for P(s) just mentioned. When ␭ is explicit forms of the eigenstates are required. This will be small the rigidity of the Hubbard spectrum is very close to dealt with in forthcoming work. that of the GOE random matrices, while for larger ␭ a satu- For all the lattices studied a significant amount of levels 55 ENERGY LEVEL STATISTICS OF THE TWO- . . . 9153

FIG. 9. The number variance ⌺2(␭) calculated for the same parameters as in Fig. 8. The data are compared to the results of the random diagonal matrix ensemble ͑Poisson͒ and the random full matrix ensemble ͑GOE͒, shown as the straight solid line and the curved solid line, respectively.

within each symmetry-invariant subspace remain degener- this sense the model seems always to be in the strong- ated, but except for Lϭ4 the ground state is nondegenerate. coupling limit. We explained the degenerate states as a consequence of a Largely, our results show GOE behavior of the spectral restricted permutation symmetry of the momentum compo- statistics of the typical high-lying excitations of the Hubbard nents. These states, all independent of U, form an integrable model at low ﬁlling. This indicates that at least the incoher- part of the spectrum, and after discarding them we end up ent part of the electronic spectral functions ͑related to the with nondegenerate spectra on which the level statistical coherent part describing the low-lying electronic excitations analysis could be performed. through sum rules͒ is out of reach by standard methods. On The intricate structure of the Hubbard spectra necessitated the other hand, it should be possible to model this part by a the development of a careful unfolding procedure as a pre- random matrix ansatz. This is in agreement with previous paratory step before the level statistical analysis. The proce- results of the t-J model near half ﬁlling.15 However, the dure we arrived at tested favorably in many cases of patho- cause of the deviations from the GOE we found in ⌺2(␭) logical spectra, and it seems to be very robust and applicable and ⌬3(␭) beyond ␭* remains an open question. A similar in general cases were no other natural unfolding procedure question has been answered in general for single-particle sys- exists. tems with mean-level spacing ⌬: In disordered ͑metallic͒ Finally, we have performed a level statistical analysis of systems ␭*⌬ϳប/␶D , where ␶D is the time it takes a particle the Hubbard spectra, and we presented results for the level to diffuse through the system,34 whereas for pure ͑ballistic͒ 2 spacing distribution P(s), the number variance ⌺ (␭), and systems ␭*⌬ϳប/␶0, where ␶0 is the period of the shortest 35 the spectral rigidity ⌬3(␭). The statistics for the different periodic orbit. Results are also beginning to emerge for lattice sizes and for a wide range of coupling strengths are disordered interacting lattice systems, where ␭* is related to essentially the same: P(s) shows good agreement with the the ratio U/W between some interaction strength U and the GOE. ⌺2(␭) agrees only with the GOE up to the disorder-induced single-particle bandwidth W.16,36 For these U/t-independent medium-sized energy scale ␭*Ϸ2 beyond systems the deviations from the GOE are due to the prefer- which a saturation sets in. ⌬3(␭) also agrees with the GOE ential basis supplied by the given disorder potential. For ex- for ␭Ͻ␭*. The deviation from the GOE beyond ␭* is not so ample, the system consisting of two interacting particles in a marked as that for ⌺2(␭). The curve falls between that of the disorder potential36 could be studied analytically by adding a GOE and the Poissonian case, but rather close to the former. random diagonal matrix modeling the disorder single- We stress that these results were also obtained for very small particle states to a random GOE matrix modeling the inter- coupling strengths approaching the integrable zero-coupling actions between these states, and it was found that ⌺2(␭) limit. This emphasizes the nonperturbative nature of the increased as a power law for ␭Ͼ␭*. This behavior is in model revealed by our analysis: Even the smallest deviation contrast to the saturation we found ͑see Fig. 9͒, which looks from the integrable limits leads to spectral statistics usually more like the result of the ballistic single-particle case.35 It is associated with nonintegrability and quantum chaos, and in perhaps not surprising that such a similarity exists between

FIG. 10. The spectral rigidity ⌬3(␭) calculated for the same parameters as in Fig. 8. The data are compared to the results of the random diagonal matrix ensemble ͑Poisson͒ and the random full matrix ensemble ͑GOE͒, shown as the straight solid line and the curved solid line, respectively. 9154 HENRIK BRUUS AND JEAN-CHRISTIAN ANGLE` S D’AURIAC 55 the disorder-free chaotic single-particle case and the The first step in calculating the irreducible representations q disorder-free Hubbard model rather than between two for G4 is to construct the character table ␹ (a)of . This is strongly correlated systems one with and the other without easily found as the product of the character table A disorder. However, exactly what physical mechanism pro- duces a preferential basis for the Hubbard model at low filling is not known, and neither is it known why a similar mechanism is suppressed for the t-J model near half filling, for 2 with itself 4 times. The 16 irreducible representations where much less pronounced deviations from the GOE are areZ identified by the index qϭ(q ,q ,q ,q ), with q ϭ0or 15 1 2 3 4 i found. These questions are topics for future work. 1, and the qth character ␹q(a) is given by the product of ai to the power qi : ACKNOWLEDGMENTS 4 q qi ␹ ͑a͒ϭ ai . ͑A2͒ It is a pleasure to thank Benoit Douçot, Jean-Louis Pi- i͟ϭ1 chard, Cle´ment Sire, and Dietmar Weinmann for stimulating discussions and Elliot H. Lieb for valuable comments. We Next step is to pick any qЈ and construct the associated little group (qЈ)ʕ defined as thank the Centre Grenoblois de Calcul Vectoriel for its kind B B help with the part of our numerical work that was carried out ͑qЈ͒ϵ͕b෈ ͉␹qЈ͑babϪ1͒ϭ␹qЈ͑a͒, for ᭙a෈ ͖. at its CRAY-94 computer. H.B. was supported by the Euro- B B A ͑A3͒ pean Commission under Grant No. ERBFMBICT 950414. The group is then written in a coset decomposition after (qЈ): B APPENDIX A: THE IRREDUCIBLE REPRESENTATIONS B OF G ϭ ͑qЈ͒b ͑qЈ͒b ͑qЈ͒b . ͑A4͒ 4 B B 1 B 2 ••• B MЈ For each of the MЈ coset representatives b an index qЈ is In Sec. IVA we showed how the group G4 of the j j neighbor-conserving transformation of the 4ϫ4 lattice is determined such that isomorphic with the point group of the four-dimensional hy- ␹qЈjϭ␹qЈ͑b abϪ1͒, for ᭙a෈ . ͑A5͒ percube. In this appendix we determine the structure of this j j A group and we sketch how the irreducible representations are Note that b1 is the identity and that q1Ј hence equals qЈ. The found analytically. The fundamental simplification is the ob- set orb(q )ϭ qЈ ,qЈ ,...,qЈ is called the orbit of q . Ј ͕ 1 2 MЈ͖ Ј servation that G4 has the structure of a semidirect product Now pick a qЉ outside orb(qЈ) and repeat the procedure. involving an invariant Abelian subgroup. The theorem of This is continued until each of the 16 q’s are associated with induced representations24 can then be used to find all irreduc- an orbit. The last thing to do before constructing the irreducible representations. The theorem is stated below. The coor- ible representations of G4 is to find the irreducible represen- dinate set of a corner in the unit hypercube in four dimen- tations ⌬qЈp of the little group (qЈ). This is usually a sions is given as a 4 bit binary number. Any transformation simple step due to the small size ofB (qЈ). of the hypercube can thus be written as a permutation of the The theorem of induced representationsB 24 states that all 4 bits followed by bit inversion (0 1) of all, some, or none qЈp ↔ irreducible representations ⌫ of the semidirect product of the bits. ᭺s , being an invariant Abelian subgroup, are found as In what follows any quadruple (x1 ,x2 ,x3 ,x4) is written in follows.A B A͑i͒ Pick one qЈ from each orbit. ͑ii͒ Construct the shorthand notation as (xi). The group of bit permutations is little group (qЈ) and its nЈ irreducible representations of course the permutation group S denoted in the follow- B p 4 ⌬qЈp, pϭ1,...,nЈ. ͑iii͒ Find the M coset representatives ing to be consistent with Ref. 24, on whichB the group theo- p Ј b ෈ , jϭ1,...,MЈ,of with respect to (qЈ). ͑iv͒ Then retical work in this appendix is based. Any element b෈ is j B B B B the matrix elements of ⌫qЈp for element ab are given by written as bϭ(bi)ϭ(b1 ,b2 ,b3 ,b4), listing the permutation of the numbers 1, 2, 3, and 4. The group of bit inversions is qЈp ⌫ ͑ab͒kt,jr denoted . It is easily seen that ϭ , since A A Z2 Z2 Z2 Z2 bit inversions do or do not take place on each of the 4 bit q qЈp Ϫ1 Ϫ1 ␹ Ј͑a͓͒⌬ ͑bkbbj ͔͒tr if bkbbj ෈ ͑qЈ͒, positions. Any element a෈ has the form aϭ(a ) ϭ B A i Ϫ1 ϭ(a ,a ,a ,a ), with a ϭϮ1, where ϩ1 means no bit in- ͭ0ifbkbbj ¹ ͑qЈ͒. 1 2 3 4 i B version and Ϫ1 means bit inversion. Any element of g ͑A6͒ ෈G4 can be written as gϭabϭ(aibi) and conversely all Here we will not give the explicit expressions of the irre- products ab෈G . contains 16 elements and contains 24 4 A B ducible representations of G4, but rather just briefly sketch so that G4 contains 384 elements. It is readily verified that the construction of them and calculate how many there are is an Abelian subgroup of G4. Furthermore, is an in- and what is the dimension of each of them. A A Ϫ1 variant subgroup since for ᭙a෈ ,᭙b෈ : bab First we chose qЈϭ(0000). From Eq. ͑A2͒ it is easily Ϫ1 Ϫ1 A B (0000) ϭb(ai)b ϭ(ab(i))bb ϭ(ab(i))෈ . Finally, the only seen that ␹ (a)ϭ1 for ᭙a෈ . Hence A (0000) Ϫ1 (0000) A common element of and is the identity. We can there- ␹ (bab )ϭ␹ (a) for ᭙a෈ , ᭙b෈ , and ac- A B A B fore conclude that G4 is a semidirect product of the invariant cording to Eq. ͑A3͒ we find (0000)ϭ . The coset repre- Abelian subgroup with : sentation of consists of onlyB one termB and the single coset A B B representative b1 is the identity. As a consequence we have G ϭ ᭺s ϭ͑ ͒᭺s S . ͑A1͒ orb(0000)ϭ͕(0000)͖. Finally, the irreducible representa- 4 A B Z2 Z2 Z2 Z2 4 55 ENERGY LEVEL STATISTICS OF THE TWO- . . . 9155

TABLE IV. The character table of G4 deﬁning the index R for each of the 20 irreducible representations and the index C for each of the 20 classes. Column Cϭ0 contains the dimension lR ranging from 1 to 8 of the representations.

C 012345678910111213141516171819گR

0 11111111111111111111 1 11111111111Ϫ1Ϫ1Ϫ1Ϫ1Ϫ1Ϫ1Ϫ1Ϫ1Ϫ1 2 1111111Ϫ1Ϫ1Ϫ1Ϫ11111Ϫ1Ϫ1Ϫ1Ϫ1Ϫ1 3 1111111Ϫ1Ϫ1Ϫ1Ϫ1Ϫ1Ϫ1Ϫ1Ϫ111111 4 22222Ϫ1Ϫ10000000011Ϫ2Ϫ2Ϫ2 5 22222Ϫ1Ϫ100000000Ϫ1Ϫ1222 633Ϫ13Ϫ10011Ϫ11Ϫ1Ϫ11Ϫ1001Ϫ3Ϫ3 733Ϫ13Ϫ100Ϫ1Ϫ11Ϫ111Ϫ11001Ϫ3Ϫ3 833Ϫ13Ϫ100Ϫ1Ϫ11Ϫ1Ϫ1Ϫ11Ϫ100Ϫ133 933Ϫ13Ϫ10011Ϫ1111Ϫ1100Ϫ133 10 4 Ϫ40001Ϫ1Ϫ22002Ϫ200Ϫ110Ϫ22 11 4 Ϫ40001Ϫ12Ϫ200Ϫ2200Ϫ110Ϫ22 12 4 Ϫ40001Ϫ12Ϫ2002Ϫ2001Ϫ102Ϫ2 13 4 Ϫ40001Ϫ1Ϫ2200Ϫ22001Ϫ102Ϫ2 14662Ϫ2Ϫ2000000Ϫ2Ϫ20200000 15 6 6 Ϫ2 Ϫ2200Ϫ2Ϫ202000000000 16 6 6 Ϫ2 Ϫ2200220Ϫ2000000000 17662Ϫ2Ϫ2000000220Ϫ200000 18 8 Ϫ8000Ϫ11000000001Ϫ10Ϫ44 19 8 Ϫ8000Ϫ1100000000Ϫ1104Ϫ4

0p tions ⌬ of (0000) are simply those of ϭS4 ͑or Td as the orb(0111)ϭ͕(0111),(1011),(1101),(1110)͖ and (1000) is B 24 B 0p B group also is called ͒; i.e., there are five different ⌬ ma- isomorphic with C3v . We end up with three more irreducible (0111)p trices with dimensions 1, 1, 2, 3, and 3, respectively. From representations ⌫ of G4 with dimensions 4, 4, and 8, (0111) 1p (1011) 1p (1101) 1p Eq. ͑A6͒ we find that j,kϭ1 and thus we have found five having entries of 0, ␹ ⌬ , ␹ ⌬ , ␹ ⌬ ,or ␹(1110)⌬1p. irreducible representations of G4 with dimensions 1, 1, 2, 3, and 3 of the form (0000)p(ab)ϭ (0000)(a) 0p(b). The last choice for qЈ turns out to be qЈϭ(0011), and we ⌫ ␹ ⌬ (0011) Ϫ1 (0011) Next we choose qЈϭ(1111). We find that, for ᭙a obtain ␹ (bab )ϭab(3)ab(4) , which equals ␹ (a) (1111) Ϫ1 1 (1111) for ᭙a if and only if b does not mix the pairs 1,2 with ෈ , ᭙b෈ , ␹ (bab )ϭ͟ a ϭ͟ a ϭ␹ (a). ෈ ͑ ͒ i b(i) j j ͑3,4͒. It isA easily seen that (0011) is a four-element group SoA in analogyB with qЈϭ(0000) we have (1111)ϭ , and isomorphic with , andB thus has 4 one-dimensional like before we find five irreducible representationsB B of G 2 2 4 irreducible representationsZ Z ⌬2p. The coset representation of with dimensions 1, 1, 2, 3, and 3 of the form (1111)p (1111) 0p 0p consists of six terms in this case, and ⌫ (ab)ϭ␹ (a)⌬ (b), with the same ⌬ matrices orb(0011)B ϭ͕(0011),(0101),(0110),(1001),(1010),(1100)͖. but different ␹ prefactors as for qЈϭ(0000). At this point we note that all 16 possible values of qЈ now We go on with qЈϭ(1000). This yields are a member of an orbit. The six coset representatives com- (1000) Ϫ1 (1000) ␹ (bab )ϭab(1) which equals ␹ (a) for ᭙a෈ if bined with the 4 one-dimensional irreducible representations A and only if b(1)ϭ1. Thus (1000) is the six-element sub- of (0011) yield through Eq. ͑A6͒ four new irreducible rep- B group of which leaves the first axis invariant. The coset resentationsB of G each being six dimensional and each hav- B 4 decomposition of contains four terms with coset represen- ing entries of the form 0, ␹(1100)⌬2p, ␹(1010)⌬2p, tatives that each leavesB one of the four axis invariant. Not ␹(1001)⌬2p, ␹(0110)⌬2p, ␹(0101)⌬2p,or␹(0011)⌬2p. In conclu- surprisingly we find orb(1000)ϭ͕(1000),(0100),(0010), sion we see that G4 has 20 irreducible representations with (0001)͖. The little group (1000) is isomorphic with C3v the dimensions listed in the first column of the character and has thus three irreducibleB representations ⌬1p with di- table shown in Table IV. This result is to be contrasted with mensions 1, 1, and 2, respectively. This combined with the the list of representation dimensions given in Ref. 20, where fact that index j,k in Eq. ͑A6͒ runs over the four coset rep- only translations and reflections are taken into account in the resentatives means that we have found three more irreducible analysis of the space group of the 4ϫ4 lattice. Note espe- (1000)p cially the three- and six-dimensional representations found representations ⌫ of G4 with dimensions 4, 4, and 8, and with entries of the form 0, ␹(1000)⌬1p, ␹(0100)⌬1p, here as opposed to dimensions equal powers of 2 in Ref. 20. ␹(0010)⌬1p,or␹(0001)⌬1p. APPENDIX B: THE IRREDUCIBLE REPRESENTATIONS Then we consider qЈϭ(0111). The character ␹qЈ now (0111) Ϫ1 OF G3, G5, AND G6 gives ␹ (bab )ϭab(2)ab(3)ab(4) which as above equals ␹(0111)(a) for ᭙a෈ if and only if b(1)ϭ1. The rest The irreducible representations R of the space groups A of the analysis is similar as the previous case: GL with Lϭ3,5,6 can be derived analytically following Ref. 9156 HENRIK BRUUS AND JEAN-CHRISTIAN ANGLE` S D’AURIAC 55

TABLE V. The irreducible representations R of GL for Lϭ3, 5, and 6, their corresponding quantum numbers (kx ,ky ,bx ,by ,c), and their dimensions lR . The symbol * refers to indeﬁnite reﬂection quantum numbers.

3ϫ3 lattice

R 012345 6 7 8 kx 000002␲2␲2␲2␲ 3 3 3 3 ky 000000 02␲2␲ 3 3 bx 1–11–11 * * * * by 1 –1 1 –1 –1 1 –1 * * c 1 1 –1 –1 * * * 1 –1 lR 111124 4 4 4 5ϫ5 lattice

R 012345 6 7 8 910111213 kx 000002␲2␲4␲4␲2␲2␲4␲4␲2␲ 5 5 5 5 5 5 5 5 5 ky 000000 0 0 02␲2␲4␲4␲4␲ 5 5 5 5 5 bx 1–11–11*** ** ** ** by 1 –1 1 –1 –1 1 –1 1 –1 * * * * * c 1 1 –1 –1 * * * * * 1 –1 1 –1 * lR 111124 4 4 4 4 4 4 4 8 6ϫ6 lattice

R 01234567891011121314151617181920212223242526 kx0␲0␲0␲0␲␲ 0␲␲␲␲2␲2␲␲␲␲␲2␲2␲2␲2␲␲␲␲ 3 3 333 3 3 3 3 3 3 33 ky0␲ 0␲ 0␲ 0␲ 000␲000 0␲␲ 00␲␲2␲2␲␲␲2␲ 3 3 333 bx1–1–11 1–1–11–11 1 1–11 * * * * * * * * * * * * * by 1–1–11 1–1–11 1–11–1–1–11 –1–11 1–1–11 * * * * * c 1111–1–1–1–1******* * ***** * 1–11–1* lR111111112222224 4 44444 4 4 4 448

20. In Table V each of them is specified by the lattice size ˆ If f 1ϭ0, we are done; if not, we continue by applying O to L and by its representation quantum numbers ͉␾1͘ and expand the result on ͉␾0͘, ͉␾1͘. The rest term is (k ,k ,b ,b ,c) related to the translation and reflection op- x y x y now denoted f 2͉␾2͘, where ͉␾2͘ is a unit vector perpendicu- erators t ,t ,r ,r ,r , respectively. Furthermore, the di- ͕ x y x y d͖ lar to both ͉␾0͘ and ͉␾2͘ and f 2 is a prefactor: mensions lR of the representations are listed. ˆ ˆ ˆ O͉␾1͘ϭ͗␾0͉O͉␾1͉͘␾0͘ϩ͗␾1͉O͉␾1͉͘␾1͘ϩ f 2͉␾2͘. APPENDIX C: PROJECTION OPERATORS ͑C2͒ OF CONTINUOUS SYMMETRIES Since we are working in a finite Hilbert space, this process is There exist several methods for calculating the projection guaranteed to yield a zero rest term after M steps, i.e., operators corresponding to a continuous symmetry given by f Mϭ0. The set SOˆ (͉␾0͘)ϭ͕͉␾0͘,͉␾1͘,...,͉␾MϪ1͖͘ thus a Hermitian operator Oˆ ; here we think of Oˆ as being either yields the smallest Oˆ -invariant subspace containing the start- the spin operator Sˆ 2 or the pseudospin operator Jˆ 2. In this ing vector ͉␾ ͘. The symmetry operator Oˆ is then diagonal- appendix we present an algorithm based on successive appli- 0 ized within SOˆ (͉␾0͘), yielding the eigenvalues ␻k and eigen- cations of Oˆ . Let ͉␾ ͘ be a given normalized state we want 0 vectors ͉␻k͘: to project into Oˆ -symmetry-invariant subspaces. When Oˆ MϪ1 acts on ͉␾0͘ a term proportional with ͉␾0͘ is generated to- ˆ gether with a rest term. The rest term is denoted f 1͉␾1͘, and O͉␻k͘ϭ␻k͉␻k͘, with ͉␻k͘ϭ cki͉␾i͘. ͑C3͒ i͚ϭ0 it defines a new unit vector ͉␾1͘ perpendicular to ͉␾0͘ while f 1 is a prefactor: ␻ The projection k of ␾ into the Oˆ -symmetry-invariant Oˆ ͉ 0͘ ˆ ˆ P O͉␾0͘ϭ͗␾0͉O͉␾0͉͘␾0͘ϩ f 1͉␾1͘. ͑C1͒ subspace corresponding to the eigenvalue ␻k is thus simply 55 ENERGY LEVEL STATISTICS OF THE TWO- . . . 9157

␻k ˆ ͉␾0͘ϭck*0͉␻k͘. ͑C4͒ i␲•a i[k•bϩ͑qϪk͒•d] PO ͉␺kq͘ϭ e e ͑1Ϫ␦b,d͉͒a,b;a,d͘ ͚a

Based on this equation we find the expressions for the pro- 1 ϭ͚ ͉p,k;␲Ϫp,qϪk͘Ϫ 2͚ ͉p,r;␲Ϫp,qϪr͘. jections in spin space, Eqs. ͑11͒–͑13͒, and in pseudospin p L pr space, Eqs. ͑16͒–͑21͒. ͑D2͒ Note how the double sum in Eq. ͑D2͒ involves all k vectors APPENDIX D: SIMULTANEOUS EIGENSTATES but that it only depends on q. To obtain Sϭ1 we simply form antisymmetric combinations of such states: OF Tˆ AND Uˆ Ϫ In this appendix we present the analytical construction of ͉␺kq͘ϭ͉␺kq͘Ϫ͉␺͑qϪk͒q͘ simultaneous eigenstates of Tˆ and Uˆ . The existence of these states explains the remaining degeneracies in the symmetry- ϭ ͉͑p,k;␲Ϫp,qϪk͘Ϫ͉p,qϪk;␲Ϫp,k͘). ͚p invariant subspaces, and they ͑found numerically͒ are listed in Table III grouped after the Uˆ eigenvalue ␥ ͑ϭ0,1,2 for ͑D3͒ Ϫ four-electron systems͒ and the spin S. We denote these states First, we note that ͉␺kq͘Þ0 if and only if qÞ2k. Second, ˆ ˆ ␥ ˆ since only permutations of the vectors k and (qϪk) are in- T/U states or ͉␺S ͘. A priori, eigenstates of U are most con- ˆ Ϫ Ϫ veniently described in real space whereas eigenstates of Tˆ volved, we find that H͉␺kq͘ϭE͉␺kq͘. And third, ˆ (1) Ϫ Ϫ L2 ˆ ˆ are naturally given in momentum space. To require a state to S ͉␺kq͘ϭϪ͉␺kq͘. Thus we have found (2 )T/U states P ␥ϭ1 be a Tˆ /Uˆ state imposes severe constraints. Below we find ͉␺Sϭ1 ͘. This yields exactly the number of states listed in the many of these states analytically. Since in the following we ␥ϭ1/Sϭ1 row of Table III: will be using both real space states and momentum space ␺␥ϭ1 ϭ ␺Ϫ . ͑D4͒ states, we will reserve the letters a, b, c, and d for sites in ͉ Sϭ1 ͘ ͉ kq͘ ␥ϭ1 real space and the letters k, q, p, and r for momenta. The states ͉␺Sϭ0 ͘ must be sought among linear combina- ϩ We begin from below in Table III by studying two-pair tions of ͉␺kq͘ defined as states, i.e., ␥ϭ2. In a naive sense, such a state must be a ϩ superposition of extremely localized states in real space or ͉␺kq͘ϭ͉␺kq͘ϩ͉␺͑qϪk͒q͘ correspondingly of very out-spread states in momentum space. This involves superpositions of many states with dif- ϭ͚ ͓͉p,k;␲Ϫp,qϪk͘ϩ͉p,qϪk;␲Ϫp,k͘] ferent wave vectors k and hence different energies given by p Eq. ͑26͒. It turns out that to obtain an energy eigenstate state 2 only states where k enters together with ␲Ϫk, where Ϫ ͉p,r;␲Ϫp,qϪr͘. ͑D5͒ L2͚ ␲ϭ(␲,␲), can be used since cos(k␨)ϩcos(␲Ϫk␨)ϭ0, and pr consequently these states only exist for even L. We construct The problem now, however, is that the double sum does not ␥ϭ2 the desired state ͉␺Sϭ0 ͘ by superposing two-pair states: vanish, hence preventing the state of being an energy eigenstate. Only upon forming differences ϩ Ϫ ϩ with k ͉␺kq͘ ͉␺kЈq͘ Ј Þk can we get rid of it. But the resulting state will only be an energy eigenstate if ͚ ͓cos(k␨)ϩcos(q␨Ϫk␨)͔ equals ␥ϭ2 i␲•͑aϩb͒ ␨ ͉␺Sϭ0 ͘ϵ e ͉a,b;a,b͘ϭ ͉k,p;␲Ϫk,␲Ϫp͘. ␨ ␨ ␨ ͚ab ͚kp ͚␨͓cos(kЈ )ϩcos(q ϪkЈ )͔. This is easily obtained if x x y ͑D1͒ kЈϭ(q Ϫk ,k ), since then we are only permuting the momentum components.

␥ϭ1 ϭ ϩ Ϫ ϩ . D6 From the real space representation it is seen immediately that ͉␺Sϭ0 ͘ ͉␺kq͘ ͉␺kЈq͘ ͑ ͒ Uˆ ͉␺␥ϭ2͘ϭ2͉␺␥ϭ2͘ and that ˆ (0)͉␺␥ϭ2͘ϭ͉␺␥ϭ2͘, while the Sϭ0 Sϭ0 PS Sϭ0 Sϭ0 By accident there can also exist other values of kЈ which momentum space representation directly yields fulfill the requirement, and besides combine pairs such as ˆ ␥ϭ2 ␥ϭ2 ϩ ϩ T͉␺Sϭ0 ͘ϭ0͉␺Sϭ0 ͘. In fact, as can be seen in the rows Ϫ we can also in some cases combine three ͉␺kq͘ ͉␺kЈq͘ ␥ϭ2 of Table III, this is the only Tˆ /Uˆ state with ␥ϭ2; e.g., ϩ ϩ ϩ ϩ ϩ states, 2͉␺kq͘Ϫ(͉␺pq͘ϩ͉␺p q͘), or four, (͉␺kq͘ϩ͉␺k q͘) ϩ Ј Ј it is easily seen from Eqs. ͑11͒–͑13͒ that no ␥ϭ2 state can Ϫ(͉␺ϩ ͘ϩ͉␺ ͘), or even more. By a straightforward com- ␥ϭ2 pq pЈq have Sϭ1 or 2. Note how ͉␺Sϭ0 ͘ is independent of the binatorial search we find the number of states listed in the coupling strength U, but that its energy is U dependent and ␥ϭ1/Sϭ0 row of Table III, and we have thus identified all of the form E(U)ϭE(0)ϩ2U. states in the ␥ϭ1 rows, and found them to be independent of Having explained the ␥ϭ2 rows of Table III we turn to U but with an energy dependence of the form the ␥ϭ1 rows. From Eqs. ͑11͒–͑13͒ we find that there exist E(U)ϭE(0)ϩU. no ␥ϭ1 states with Sϭ2. To find the ␥ϭ1 states with Finally, we turn to the ␥ϭ0 rows of Table III. First we Sϭ0,1 we construct states ͉␺kq͘ which manifestly contains note that all Sϭ2 states of the systems are found here. This ˆ exactly one pair, such that U͉␺kq͘ϭ͉␺kq͘: is easily proved by noting that the states with 9158 HENRIK BRUUS AND JEAN-CHRISTIAN ANGLE` S D’AURIAC 55

(S,Sz)ϭ(2,0) are formed by applying the spin-lowering op- ˆ ␴,␴ ⌸␦ ͉k1 ,k2 ;k3 ,k4͘ϭ͉k1 ,k2 ;k3 ,k4͘ erator SϪ ͑which commutes with H) twice to states with (S,Sz)ϭ(2,2). But the latter states cannot contain any dou- ϩ͉␲␦͑k1͒,␲␦͑k2͒;k3 ,k4͘ bly occupied sites since that would yield a lower than maxi- ϩ͉k1 ,k2 ;␲␦͑k3͒,␲␦͑k4͒͘ mal value of Sz . Clearly, these states as well as their energies are independent of U. Then we consider a large class of ϩ͉␲␦͑k1͒,␲␦͑k2͒;␲␦͑k3͒,␲␦͑k4͒͘, energy eigenstates with Sϭ0,1 and ␥ϭ0, which does not require the momentum component , and which therefore ˆ ¯␴,␴ ␲ ⌸␦ ͉k1 ,k2 ;k3 ,k4͘ϭ͉␲␦͑k1͒,k2 ;␲␦͑k3͒,k4͘ accounts for degeneracies for any value of L. Writing x y x y ͉(k ,k )͘ϭ͉k ͘ ͉k ͘ we construct ␥ϭ0 states ͉␾͘ obeying ϩ͉␲␦͑k1͒,k2 ;k3 ,␲␦͑k4͒͘ Uˆ ͉␾͘ϭ0 by symmetrizing one component, say, the x com- ϩ͉k1 ,␲␦͑k2͒;␲␦͑k3͒,k4͘ ˆ (2) ponent, and letting S act on the other: P ϩ͉k1 ,␲␦͑k2͒;k3 ,␲␦͑k4͒͘. ͉␾͘ϭ͉kx ,kx ;kx ,kx ͘ ˆ ͑2͉͒ky ,ky ;ky ,ky͘, 1 2 3 4 s PS 1 2 3 4 ͑D10͒ x x x x x x x x x x x x Direct inspection shows Uˆ (⌸ˆ ␴,␴Ϫ⌸ˆ ¯␴,␴) k ,k ;k ,k ϭ0, ͉k1 ,k2 ;k3 ,k4͘sϵ͉k1 ,k2 ;k3 ,k4͘ϩ͉k1 ,k2 ;k4 ,k3͘ ␦ ␦ ͉ 1 2 3 4͘ and by enforcing certain constraints on all eight momentum x x x x x x x x ϩ͉k2 ,k1 ;k3 ,k4͘ϩ͉k2 ,k1 ;k4 ,k3͘. components this state also becomes an energy eigenstate ˆ (SЈ) ͑D7͒ with energy Eϭ0, while applying the projector S renders the correct spin SϭSЈ: P Since only momentum permutations enter, ͉␾͘ is clearly an energy eigenstate. The proper spin states are found by the ␥ϭ0 ˆ ͑SЈ͒ ˆ ␴,␴ ˆ ¯␴,␴ ͉␺ ͘ϭ ͑⌸ Ϫ⌸ ͉͒k1 ,k2 ;k3 ,k4͘, standard projections: SϭSЈ PS ␦ ␦ with ␥ϭ0 ˆ ͑SЈ͒ ͉␺SϭS ͘ϭ S ͉␾͘. ͑D8͒ x y Ј P knϭ␲Ϫkn , nϭ1,2,3,4. ͑D11͒ Simple combinatorics yields 0, 30, 300, and 1680 Sϭ0 Finally, we note that for lattices containing the momentum states and 0, 90, 1050, and 6300 Sϭ1 states for Lϭ3, 4, 5, ␲/2 as is the case for Lϭ4, even more Tˆ /Uˆ states can be and 6, respectively. In analogy with the ␥ϭ1 case many constructed, in accordance with Table III. An example of this ˆ ˆ more T/U states can be constructed for Lϭ4 and 6 and can be obtained from Eq. ͑D11͒. If, for example, we let ␥ϭ0 when the momentum vector ␲ is taken into account. ␦ϭx, then it suffices to enforce the constraint kx ϭϮ␲/2 We give one example of a class of such states. For a given n x y while allowing any value for the y components. The result is momentum vector kϭ(k ,k ) we define for ␦ϭx,y,xy the y energy eigenstates with energy Eϭ͚ cos(k ). functions ␲ (k), n n ␦ ˆ ˆ ␥ We conclude that many of the T/U states ͉␺S ͘ found x y x y ␲x͑k͒ϭ͑k ϩ␲,k ͒, ␲y͑k͒ϭ͑k ,k ϩ␲͒, numerically have been constructed analytically, and in agreement with the numerical findings all these states are indepen- x y ␲xy͑k͒ϭ͑k ϩ␲,k ϩ␲͒, ͑D9͒ dent of U, while their energies are of the form E(U)ϭE(0)ϩ␥U. The analytic constructions reveal that ˆ ␴,␴ based on which we introduce two operators ⌸␦ and these states are due to a restricted permutation symmetry for ˆ ¯␴,␴ ⌸␦ : the momentum components of states in momentum space.

*Present address: Niels Bohr Institute, Blegdamsvej 17, DK-2100 Solids, edited by B. L. Altshuler, P. A. Lee, and R. A. Webb Copenhagen, Denmark. Electronic aaddress: [email protected] ͑North-Holland, Amsterdam, 1991͒, Chap. 9. †Electronic address: [email protected] 9 A. Kudrolli, V. Kidambi, and S. Sridhar, Phys. Rev. Lett. 75, 822 1 E. Dagatto, Rev. Mod. Phys. 66, 763 ͑1994͒. ͑1995͒. 2 E. H. Lieb, in Proceedings of the XIth International Congress of 10 C. Ellegaard et al., Phys. Rev. Lett. 77, 4918 ͑1996͒. Mathematical Physics, Paris, 1994, edited by D. Iagolnitzer ͑In- 11 M. V. Berry, Nonlinearity 1, 399 ͑1988͒. ternational Press, Paris, 1995͒, pp. 392–412. 12 R. Blu¨mel and U. Smilansky, Phys. Rev. Lett. 60, 477 ͑1988͒. 3 P. W. Anderson, Science 235, 1196 ͑1987͒. 13 H. Meyer, J.-C. Angle`s d’Auriac, and H. Bruus, J. Phys. A 29, 4 M. Fabrizio, A. Parola, and E. Tosatti, Phys. Rev. B 44, 1033 L483 ͑1996͒. ͑1991͒. 14 R. Me´lin, J. Phys. ͑France͒ I 6, 469 ͑1996͒. 5 M. Faas, B. D. Simons, X. Zotos, and B. L. Altshuler, Phys. Rev. 15 G. Montambaux, D. Poilblanc, J. Bellisard, and C. Sire, Phys. B 48, 5439 ͑1993͒. Rev. Lett. 70, 497 ͑1993͒. 6 M. L. Mehta, Random Matrices, 2nd ed. ͑Academic Press, San 16 R. Berkovits and Y. Avishai, J. Phys. Condens. Matter 8, 389 Diego, 1991͒. ͑1996͒. 7 F. Haake, Quantum Signatures of Chaos ͑Springer-Verlag, Ber- 17 T. Hsu and J.-C. Angle`s d’Auriac, Phys. Rev. B 47, 14 291 lin, 1991͒. ͑1993͒. 8 Y. Imry, Europhys. Lett. 1, 249 ͑1986͒; A. D. Stone, P. A. Mello, 18 R. Me´lin, B. Douc¸ot, and P. Butaud, J. Phys. ͑France͒ I 4, 737 K. A. Muttalib, and J.-L. Pichard, in Mesoscopic Phenomena in ͑1994͒. 55 ENERGY LEVEL STATISTICS OF THE TWO- . . . 9159

19 H. Bruus and J.-C. Angle`s d’Auriac, Europhys. Lett. 35, 321 28 Using standard cycle notation an example of two permu- ͑1996͒. tations generating all of G4 is ͑4,12͒͑5,13͒͑6,14͒͑7,15͒ and 20 G. Fano, F. Ortolani, and A. Parola, Phys. Rev. B 46, 1048 ͑0,4,8,9,10,14,2,3͒͑1,7,12,5,11,13,6,15͒. 29 ͑1992͒. O. J. Heilmann and E. H. Lieb, Trans. NY Acad. Sci. 33, 116 21 A. Pandey, Ann. Phys. ͑N.Y.͒ 119, 170 ͑1979͒. ͑1970͒. 30 22 H. Meyer, J.-C. Ange`s d’Auriac, and J.-M. Maillard, Phys. Rev. E C. N. Yang, Phys. Rev. Lett. 63, 2144 ͑1989͒. 31 ͑to be published͒. C. N. Yang and S. C. Zhang, Mod. Phys. Lett. B 4, 759 ͑1990͒. 32 23 M. Tinkham, Group Theory and Quantum Mechanics ͑McGraw- The symmetrized Hubbard model Hˆ Ј is obtained by the substitu- ˆ ˆ ˆ 1 ˆ 1 ˆ ˆ 2 Hill, New York, 1964͒. tion U UЈϭ͚i(ni Ϫ 2͒(ni Ϫ 2), leading to HЈϭHϩu(L /4 24 ˆ→ ↑ ↓ J. F. Cornwell, Group Theory in Physics, ͑Academic Press, Lon- Ϫ͚i␴ni␴/2). don, 1989͒, Vols. 1–3. 33 The pseudospin symmetry can be viewed as a consequence of a 25 Y. Hasagawa and D. Poilblanc, Phys. Rev. B 40, 9035 ͑1989͒. particle-hole symmetry which maps a Hubbard model H with 26 A group G is a direct product group if it possesses two subgroups coupling strength U and nu (nd) spin-up ͑spin-down͒ electrons ˜ ˜ G1 and G2 such that ͑i͒ ᭙g෈G ᭚g1෈G1 ᭚g2 onto a Hubbard model H with UϭϪU, ˜nuϭnu , and 2 ˜ ෈G2 :gϭg1g2, ͑ii͒ G1പG2ϭe (e is the identity͒, ͑iii͒ ᭙g1 ˜ndϭL Ϫnd . The pseudospin of H is the spin of H and vice ෈G1 ᭙g2෈G2 :g1g2ϭg2g1. One then writes GϭG1 G2. versa ͑Ref. 2͒. 27 34 A group G is a semidirect product group if it possesses two B. L. Al’tshuler and B. I. Shklovski, Zh. Eksp. Teor. Fiz. 91, 220 subgroups G1 and G2 such that ͑i͒ ᭙g෈G ᭚g1෈G1 ᭚g2 ͑1986͓͒Sov. Phys. JETP 64, 127 ͑1986͔͒. 35 ෈G2 :gϭg1g2, ͑ii͒ G1പG2ϭe, ͑iii͒ G1 is an invariant sub- M. V. Berry, Proc. R. Soc. 400, 229 ͑1991͒. Ϫ1 36 group, i.e., ᭙g1෈G1 ᭙g෈G:gg1g ෈G1. One then writes D. Weinmann and J.-L. Pichard, Phys. Rev. Lett. 77, 1556 s GϭG1᭺ G2. ͑1996͒.