# Exact ground states for the four-electron problem in a two-dimensional Hubbard square system

**ABSTRACT** We present exact explicit analytical results describing the exact ground

state of four electrons in a two-dimensional square Hubbard cluster

containing 16 sites taken with periodic boundary conditions. The

presented procedure, which works for arbitrary even particle number and

lattice sites, is based on explicitly given symmetry adapted base

vectors constructed in r space. The Hamiltonian acting on these states

generates a closed system of 85 linear equations, providing by its

minimum eigenvalue, the exact ground state of the system. The presented

results, described with the aim to generate further creative

developments, not only show how the ground state can be exactly obtained

and what kind of contributions enter in its construction, but also

emphasize further characteristics of the spectrum. On this line (i)

possible explications are found regarding why weak coupling expansions

often provide a good approximation for the Hubbard model at intermediate

couplings, or (ii) explicitly given low-lying energy states of the

kinetic energy, avoiding double occupancy, suggest new roots for pairing

mechanism attracting decrease in the kinetic energy, as emphasized by

kinetic energy driven superconductivity theories.

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**ABSTRACT:**An exact analytical diagonalization is used to solve the two dimensional Extended Hubbard Model for system with finite size. We have considered an Extended Hubbard Model (EHM) including on-site and off-site interactions with interaction energy U and V respectively, for square lattice containing 4*4 sites at one-eighth filling with periodic boundary conditions, recently treated by Kovacs et al [1]. Taking into account the symmetry properties of this square lattice and using a translation linear operator, we have constructed a r-space basis, only, with 85 state-vectors which describe all possible distributions for four electrons in the 4*4 square lattice. The diagonalization of the 85*85 matrix energy allows us to study the local properties of the above system as function of the on-site and off-site interactions energies, where, we have shown that the off-site interaction encourages the existence of the double occupancies at the first exited state and induces supplementary conductivity of the system.Physica Scripta 10/2011; 78(2). · 1.03 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**A technique based on the transformation in positive semidefinite form of the Hamiltonian (Ĥ) is developed to allow the study of the enlargement possibilities of the emergence domain of a given state. For this reason the calculation of exact ground states for the polyphenylene type of non-integrable chains is performed. Using different exact transcriptions of Ĥ in positive semidefinite form and different solutions of the matching equations, the same type of ground state is deduced in different regions of the parameter space, hence a more global view of its emergence possibilities is obtained. With this procedure an enlarged view of the appearance in the phase diagram of different studied phases can be achieved.Philosophical Magazine A 12/2012; 92(36):4657-4675.

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arXiv:cond-mat/0604319v1 [cond-mat.str-el] 12 Apr 2006

Exact ground states for the four electron problem in a

two-dimensional finite Hubbard square system.

Endre Kov´ acs and Zsolt Gul´ acsi

Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, Hungary

(Dated: February 6, 2008)

Abstract

We present exact explicit analytical results describing the exact ground state of four electrons

in a two dimensional square Hubbard cluster containing 16 sites taken with periodic boundary

conditions. The presented procedure, which works for arbitrary even particle number and lattice

sites, is based on explicitly given symmetry adapted base vectors constructed in r space. The

Hamiltonian acting on these states generates a closed system of 85 linear equations providing by

its minimum eigenvalue the exact ground state of the system. The presented results, described

with the aim to generate further creative developments, not only show how the ground state can be

exactly obtained and what kind of contributions enter in its construction, but emphasize further

characteristics of the spectrum. On this line i) possible explications are found regarding why weak

coupling expansions often provide a good approximation for the Hubbard model at intermediate

couplings, or ii) explicitly given low lying energy states of the kinetic energy, avoiding double

occupancy, suggest new roots for pairing mechanism attracting decrease in the kinetic energy, as

emphasized by kinetic energy driven superconductivity theories.

PACS numbers:

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I. INTRODUCTION

The growing interest in systems of highly correlated electrons has proposed many theoret-

ical descriptions where the dominant effects are treated in the frame of the Hubbard model

(Hubbard 1963). Since the exact solution of the model is known only in one dimension

(Lieb and Wu 1968), and the majority of driving forces in the field, as superconductivity

in cuprates, are two dimensional (2D) problems, a great variety of approximate treatments

have been proposed (Baeriswyl et al. 1995) in order to accommodate a suitable theoreti-

cal framework. Despite several years of intensive studies, it is apparent that the necessary

theoretical skills and tools to deal with this problem are still in fact relatively poor (Lieb

1994).

While most of the ongoing effort relating the Hubbard model is being concentrated

on the general case of large electron density, more than ten years ago it has been recog-

nized that the less analysed low-density limit not only that retain main aspects related to

the model behaviour (Falicov and Proetto 1993, Papavassiliou and Yartsev 1992), but can

provide key knowledge which could drive further at least nonperturbative developments

(Fabrizio, Parola and Tosatti 1991), firstly because is more hope to find non-approximated

solutions holding by their nature essential information. The history of this background has

started with the exact solution of the two-electron ground state problem on an arbitrary large

torus (Chen and Mei 1980, Mei and Chen 1988), solution of the two electron problem in 2D

(Parola et al. 1990), and continued with the three electron problem extended nonpertur-

batively in the low density limit (Fabrizio, Parola and Tosatti 1991). The numerical treat-

ment of the four electron problem on four sites follows by the test of the Bardeen-Cooper-

Schrieffer and resonant-valence-bond wave functions as approximated ground states of the

Hubbard model (Falicov and Proetto 1993), test of the unrestricted Hartree-Fock solution

(Louis et al. 1992), the calculation of Huckel-Hubbard correlation diagrams (Zhu and Jiang

1993), and deduction of absorption spectra of TCNQ particles (Papavassiliou and Yartsev

1992). The concurrently made numerical developments on 4 × 4 clusters around half filling

(Galan and Verges 1991, Parola et al. 1991) corroborated by group-theoretical studies of the

4×4 square system (Fano, Ortolani and Parola 1992) also lead to the development of the nu-

merical description of the 4 particle problem for energy level statistics (Bruus and d’Auriac

1996, 1997). In the last years the study of symmetry properties for Hubbard clusters in

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general has continued (Tjemberg 1998), extensions to Hubbard-Holstein model for clusters

holding 4 electrons has been given (Acquarone et al. 1999), and for spinless fermion case

numerical characterization of the 4 particle problem goes up to clusters of 20 × 20 exten-

sion (Zhang and Henley 2004). Despite the invested efforts, exact analytical results holding

essential information are not known at the moment in the problem.

The last period has provided strong new motivations for the further development of

the study of the low concentration limit of the Hubbard model.The reason for this

is that experimentally one started to encounter in condensed matter context rapidly in-

creasing situations containing small number of particles confined in a system or device, as

for example in the case of quantum dots (Maksym et al. 2000), quantum well structures

(Kochereshko et al. 2003), mesoscopic systems (Halfpap 2001), experimental entanglement

(Sackett et al. 2000) etc. Between the studied experiments, several are directly connected to

the Hubbard model, as in the case of charge transfer complexes (Saito et al. 2001), quantum

dots (Busser, Moreno and Dagotto 2004), or mesoscopic grains (Boyaci, Gedik and Kulik

2000). Furthermore, it has been observed that several measurements on strongly correlated

systems are acceptable reproduced on small Hubbard clusters, as in the case of the X-ray ab-

sorption for nickelates studied on 4-site cluster (Okada 2004). The same can be stated for the

4×4 cluster case used in describing manganites (Helberg 2001, Srinitiwarawong and Gehring

2002), photoemission intensities (Eroles, Batista and Aligia 1999), or thermodynamic prop-

erties provided by processes in the vicinity of the Fermi surface (Chiappe et al. 1999).

As can be observed, the accumulated knowledge to the present date regarding the ex-

treme low density limit described by few particles present in the system, excepting the

one and two particle cases, means only symmetry properties, and numerical results (often

approximated), which usually conceal explicit properties which could generate creative ad-

vancement at the level of the theoretical description and deep understanding. This is a

regrettable and unfortunate situation, since as is known from the accumulated experience in

solving exactly many-body systems (Mattis 1993), or as several times has been accentuately

stressed (Fabrizio, Parola and Tosatti 1991, Lieb 1994), key aspects of the unapproximated

descriptions are often hidden in the few particle cases.

On the presented background, with the aim to fill up at least partially this gap for

the two dimensional Hubbard case, and being driven by the intention to provide essential

information for the subjects mentioned above, we present the exact ground state of four

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interacting electrons in a 4 × 4 Hubbard cluster. Our main purpose is not to hide potential

essential characteristics behind numerical results, but to present explicit expressions, basic

resulting properties, and visible characteristics, which we strongly hope, are able to polarize

the creative thinking, and advancement in the field.

Our results are based on an r space description, which in our opinion is unjustly over-

shaded in the last period in treating such problems, in contradiction with its deep ability

to bring to light essential characteristics. We construct first a base vector set starting from

symmetry properties, and then show how a closed system of linear equations containing 85

components can be constructed characterizing the ground state manifold and providing by

its secular equation, through its minimum eigenvalue, the ground state wave function and

ground state energy. After this step the properties of the ground state are analyzed. The

concretely described situation is an L × L = N = 16 square lattice with periodic bound-

ary conditions in both directions, and Np= 4 particles. But the method itself works for

arbitrary even number of particles and arbitrary even L. The explicitly presented base vec-

tors and equations providing the ground state, not only show how the ground state can be

constructed and what kind of components build it up, but provides an insight into other

properties of the spectrum as well. Especially the characteristics regarding the kinetic energy

eigenstates should be mentioned on this line, important for understanding aspects related

to perturbative expansions, or kinetic energy driven superconductivity.

The remaining part of the paper is structurated as follows. Section II. presents the Hamil-

tonian, the deduction procedure and the ground state wave functions, Section III. describes

physical properties of the deduced eigenstates, Sect. IV. presents the conclusions of the

paper, while the Appendices A - B presenting mathematical details, close the presentation.

II. HAMILTONIAN AND GROUND STATE WAVE FUNCTIONS.

A. Presentation of the Hamiltonian

The Hamiltonian we use has the form of a standard Hubbard Hamiltonian

ˆH = −t

?

<i,j>,σ

(ˆ c†

i,σˆ cj,σ+ H.c.) + U

?

i

ˆ ni,↑ˆ ni,↓,(1)

where ˆ c†

i,σcreates an electron at site i with spin σ, t is the nearest-neighbour hopping

amplitude, U is the on-site Coulomb repulsion, and < i,j > represents nearest-neighbour

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sites, taken into account in the sum over sites only once. During our study we consider

an L × L cluster, N = L2, with periodic boundary conditions in both directions. All the

presented results will relateˆH/t, the only one microscopic parameter of the problem being

u = U/t. The case N = 16 will be presented in details, although the method is applicable

for arbitrary even N. Since Np= 4, the band filling for the presented case is 0.125. We

further mention that we consider during this paper u > 0.

B. The construction of the base wave vectors

1. The basic wave vector elements

In order to describe the Hilbert space region containing the ground state for the Np=

4 particle problem, we use an r-space representation for the wave vectors. In order to

characterize this, first the lattice sites of the considered system are numbered starting from

the down-left corner, the numbering being given along the lower lattice sites line, then going

upward with the notation, as shown in Fig.1.a. In the considered system (taking into account

that the ground state is a singlet state), three type of particle configurations may occur as

depicted in Fig.1.b,c,d.

1) We can have two double occupancies at sites i and j represented by two dots at sites

i and j (see Fig.1.b). 2) We may have a double occupancy at site i and two electrons with

opposite spins at sites j and k. In this case, the double occupancy at site i it is denoted as

before by a dot at i, while the two electrons with opposite spins placed on the different sites

(j, and k) are depicted by a dotted line connecting the sites j and k (see Fig.1.c). Finally, we

may have four single occupancies placed on different sites. From these, the electrons placed

at i and j have spin σ, while the electrons at sites k and l have spin −σ. This situation will

be represented by two continuous lines connecting the sites i and j, and k and l, respectively

(see Fig.1.d). The notations i,j,k,l are representing the numbering of the sites as specified

in Fig.1.a.

The mathematical expressions connected to the states represented in Fig.1.b,c,d are given

as follows. The two double occupancies present in Fig.1.b provide the state

|b? = (ˆ c†

i,↑ˆ c†

i,↓)(ˆ c†

j,↑ˆ c†

j,↓)|0?,(2)

where |0? represents the bare vacuum with no fermions present. The one double occupancy

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and the two electrons with opposite spins depicted in Fig.1.c are described as

|c? = (ˆ c†

i,↑ˆ c†

i,↓)[(ˆ c†

j,↑ˆ c†

k,↓) + (ˆ c†

k,↑ˆ c†

j,↓)]|0? .(3)

For the here presented situation we must have j ?= k and i ?= j,i ?= k respectively. Finally,

in describing mathematically the four single occupancies presented in Fig.1.d, we have

|d? = [(ˆ c†

j,↑ˆ c†

i,↑)(ˆ c†

l,↓ˆ c†

k,↓) + (ˆ c†

j,↓ˆ c†

i,↓)(ˆ c†

l,↑ˆ c†

k,↑)]|0?, (4)

where for the mathematical clarity, j > i and l > k is required (all sites being considered

different).

2.The translation of the basic elements

In order to construct the base vectors, the basic wave vector elements are translated by

the operation T. The T operator is considered a linear operator, hence the relation

T(A + B) = T(A) + T(B),(5)

holds, where A and B represent particle configurations as depicted in Fig.1.b,c,d. Further-

more T(A) means that the configuration of particles represented by A is translated to each

site of the system (for N = 16, the translation is made 16 times), and the obtained contribu-

tions are all added (see Fig.2). The argument of the T operator contains usually four terms

since the starting particle configuration is rotated by 180 degrees along the x,y, and z axes,

and then the obtained contributions are added. Less than four terms in the T argument

means that the mentioned rotations provide the same starting particle microconfiguration.

The translations are such effectuated that the relative inter-particle positions are all main-

tained, and arriving near the border of the system, the presence of the periodic boundary

conditions are explicitly taken into account. The translations must be given at the level of

the graphic plots, all the obtained 16 graphics must be added (see Fig.2), then each graphic

must be transformed in a mathematical form according to the rules presented in Eqs.(2-4).

At this point we must underline, that when the mathematical expression is written for a

given graph, a negative sign may emerge in some cases given by the restrictions prescribed

for Eqs.(2-4). For example, the last two terms in both rows of Fig.3 have mathematical ex-

pressions with negative sign. Similar reasons lead to sign changes obtained after rotations,

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reason which generates the sign changes observed in some T(...) arguments of base vectors

presented in Figs.4-18, for example in the case of the second contributions of |10?, or |11?,

etc. Furthermore, in some cases (see for example |12? or |15?), the rotations provide the

starting microconfiguration of particles, which leads to the decrease of the number of terms

present in the argument of the T operation. The sign changes also lead in some cases to

complete elimination of some possible particle microconfigurations.

3.The base wave vectors

Starting from the rules and operations described in details in Sec.II.B.1-2, the 85 base

wave vectors can be explicitly constructed as given in Fig.4-18. These are denoted by |n?,

n = 1,2,...,85, and are all orthogonal vectors. As can be seen, the base wave vectors are

obtained by subtracting two type of contributions. The first part of these is obtained from

a starting particle microconfiguration (presented as first terms in the argument of the first

operator T) which is rotated by 180 degrees along the x, y and z axes, and translated by the

operation T after this step. The second (subtracted) part of the wave vectors is obtained

from the first part by a rotation of the elements by 90 degree along the z axis.

C.The ground state wave function

First of all we are constructing a closed system of equations containing the base vectors

described above.

This system of 85 equations is obtained as follows. i) We start from the most condensed

and most interacting particle microconfiguration depicted by the base vector |1? (see Fig.4),

which cannot be eliminated on physical grounds from the ground state. ii) By applying the

Hamiltonian on the base vector |1? (see the first equation of Appendix A), we obtain as a

result new base vectors (|2? and |3?), which have the same symmetry properties as |1?. iii)

Since the Hamiltonian does not change the symmetry properties of the base vectors, one

must further applyˆH on the each resulting new base vector, up to the moment in which

the obtained system of equations closes. This happens for the studied system at the 85th

equation. The obtained closed system of equations is explicitly presented in Appendix A.

The minimum energy solution of the presented system of equations represents the ground

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state of the system.

III. PROPERTIES OF THE SOLUTIONS

A.The check of the ground state energies

First of all one can check that the system of equations presented in Appendix A indeed

provides the ground state of the problem. The obtained minimum eigenvalues provided by

Appendix A are

u = 0.0,E = −12.0000000000000000,

u = 0.5,E = −11.9126285145094126,

u = 1.0,E = −11.8364690235642218,

u = 1.5,E = −11.7695877912829268,

u = 2.0,E = −11.7104580743242632,

u = 2.5,E = −11.6578605150913841,

u = 3.0,E = −11.6108110503797608,

u = 3.5,E = −11.5685079573602838,

u = 4.0,E = −11.5302924026297138,(6)

which perfectly coincide with the numerically exact minimum eigenvalues obtained from ex-

act numerical diagonalization in the 14 400 dimensional full Hilbert space (Sorella and Hubsh

2000). We note that the eigenvalues are given in t units.

B.The U independent eigenstates

The study of the solutions of the system of equations presented in Eq.(A1) leads to an

extremely interesting observation, namely that almost half (exactly 40%) of the eigenstates

are U independent, consequently are eigenstates of the non-interacting system as well. For

example, the U independent 18 eigenvectors corresponding to zero eigenvalue are presented

in Appendix B. Besides these, one has also 16 eigenfunctions corresponding to eigenvalues

±4,±8, which are also U independent (see exemplification in Appendix B). The presence of

such eigenstates is important for following reasons.

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First of all, several authors observed (Galan and Verges 1991, Metzner and Vollhardt

1989) that weak coupling expansions often provide a good approximation for the Hubbard

model at intermediate coupling. A possible explication for this is the emergence of a huge

number of eigenstates in the spectrum of the Hubbard model with non-zero interaction,

which are present in the non-interacting case as well in the spectrum of the model.

A second aspect which must be mentioned here is that the eigenstates related to zero

eigenvalue presented in Appendix B are in fact many-body eigenstates of the kinetic energy

term with no double occupancy. Such states apparently totally avoid double occupancy at no

cost of energy, consequently are important in the study of the pairing mechanism in the Hub-

bard model (Cini, Perfetto and Stefanucci 2001, Cini and Stefanucci 2001, Perfetto and Cini

2004), and as seen from the presented results, easily emerge in the spectrum.

At this step we underline that contrary to the practice used in the theoretical studies up

to this moment (Balzarotti et al. 2004), double occupancy avoiding eigenstates emerge not

only at zero energy, but also at much lower energy values, closely situated to the ground

state energy. We exemplify this statement by the eigenstate |v2? presented in Eq.(B2),

corresponding to energy −8 (in t units). These eigenstates being U independent, their

energy remain unchanged if the strength of the on-site interaction is increased, while the

ground-state energy increases if U is increased. Consequently, starting from these states,

the contribution in the pairing mechanism of the kinetic energy eigenstates not containing

double occupancy could be much more efficiently taken into account than considered up

today. We note that this problem has interconnections to the problem of the kinetic en-

ergy driven superconductivity as well, strongly debated in the literature in the last period

(Anderson 1995, Feng 2003, Hirsch 2004, Yokoyama et al. 2004). Indeed, experimentally is

observed (Eckl, Hanke and Arrigoni 2003, van der Marel et al. 2003) that especially in Bi

based cuprates, a decrease in the kinetic energy is observed at the superconducting transi-

tion, which contradicts the usual (for example BCS) pairing theories (Hirsch 2004). If the

pairing process could be described on the states of the type |v2?, exemplified in our knowl-

edge in exact terms for the first time in this paper, a such a decrease could be much better

understood.

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C. The U dependent eigenstates

The major part of the spectrum (60%) is U dependent. From the eigenstates entering in

this category, there are states whose eigenvalue is exactly u, for example |w? = |8? − |9? +

|39?+|40?−|41?. Such states are interesting for two reasons. First, a detection possibility of

such states would lead to a direct experimental measurement of the U/t ratio. Second, since

all the contributions in |w? have double occupancy d = 1, this vector is also an eigenstate

of the interaction term with eigenvalue u, and in the same time, the eigenvector of the

kinetic energy term with eigenvalue zero. Consequently, the study of the zero kinetic energy

eigenvalue states must be given with care, since such type of states, as exemplified before,

not necessarily are avoiding the double occupancy.

The remaining part of the deduced U dependent states, are not eigenstates of the kinetic

energy, and the interacting part ofˆH separately, but are eigenvectors only for the sum

of both, e.g. the whole Hamiltonian. As seen from Eq.(6), the corresponding eigenvalues

have a smooth (less than linear) U dependence, and the ground states is obtained from this

category.

D. The system generating the ground state

In order to enhance further developments and enlighten further creative thinking, we pro-

vide explicitly in Appendix A the equations generating the ground state, together with the

explicit form of the base vectors providing the ground state (see Figs.4-18), hence describing

the Hilbert space region in which this is placed. We mention that if the base vectors are

defined only through the operation T, without subtracting two T components as shown in

Figs.4-18, a system containing 176 equations arises. This however can be cast in two block

diagonal components, one of which provides the equations presented in Appendix A.

IV.SUMMARY AND CONCLUSIONS

Driven by the aim to provide explicit expressions generating creative developments, we

present exact analytical results describing the ground state of the two dimensional Hubbard

model taken on an L × L = N square lattice at N = 16 (and periodic boundary conditions

in both directions), containing Np= 4 particles. The presented procedure allows the de-

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scription of an arbitrary even L and Np, and is based on symmetry adapted base vectors

constructed in r space. The Hamiltonian acting on the described base vectors provides a

closed system of linear equations (whose number is 85 for N = 16 and Np= 4), leading by

its secular equation, through its minimum eigenvalue, to the ground state wave function and

ground state energy of the system, deduced in a Hilbert subspace with almost three orders

of magnitude smaller in dimensions than the full Hilbert space of the problem. The deduced

explicit eigenstates characterize also other properties of the spectrum: i) The large number

of eigenstates which remain eigenstates of the non-interacting Hamiltonian as well shows

why weak coupling expansions often provide a good approximation for the Hubbard model

at intermediate coupling. ii) Zero energy eigenstates of the kinetic energy term (ˆHkin) which

are eigenstates of the Hamiltonian as well show how energy increasing double occupancies

can be avoided providing a possible support for the kinetic energy driven superconductivity.

iii) Low energy eigenstates of the kinetic energy term which completely avoid double occu-

pancy emphasize potentially new pairing possibilities in the low energy part of the spectrum

in the context of the kinetic energy driven superconductivity. iv) Zero energy eigenstates

of the kinetic energy term corresponding to double occupancy one underline that the zero

energyˆHkineigenstates must be handled with care, since as exemplified, these states not

necessarily represent double occupancy avoiding states.

Similarly obtained solutions for higher L or Np, corroborated with the study of the

emerging changes in the system of equations describing the ground state, remain a challenge

for future developments.

Acknowledgments

This work was supported by the Hungarian Scientific Research Fund through contract

OTKA-T-037212. The numerical calculations have been done at the Supercomputing Lab.

of the Faculty of Natural Sciences, Univ. of Debrecen, supported by OTKA-M-041537. Z.

G. kindly acknowledge test numerical results obtained from exact numerical diagonalization

on the full Hilbert space provided by Sandro Sorella and Arnd Hubsch.

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APPENDIX A: THE LINEAR SYSTEM OF EQUATIONS CONTAINING THE

GROUND STATE.

This Appendix presents the 85 linear equations describing the ground state of the studied

system.

ˆH|1? = 2u|1? + |2? − |3?,

ˆH|2? = |1? + u|2? + 8|4? − |6? − |7? + |10? + |11? + 4|12?,

ˆH|3? = −8|1? + u|3? − 8|5? + |6? + |7? − 2|8? − 2|9? − |10? − |11? − 2|13? − 2|14? − 4|15?,

ˆH|4? = |2? + 2u|4? − |17?,

ˆH|5? = −|3? + |23?,

ˆH|6? = −2|2? + |3? + u|6? − |16? + 2|17? − 2|18? − |19? + |20? + 2|21? − |23? + |24?

+|25? + |26? + |29? + |31?,

ˆH|7? = −2|2? + |3? + u|7? − |16? + 2|17? + |20? + 2|22? − |23? + |25? + |26? + |28? + |30?,

ˆH|8? = −|3? + u|8? − |16? − |21? − |22? + |23? − |27? − |33? − |34?,

ˆH|9? = −|3? + u|9? − |16? − |19? − |20? + |23? − |24? − |27? − |32?,

ˆH|10? = 2|2? − |3? + |16? − 2|17? − 2|18? − |19? + |23? + |24? − |25? − |26? − |28?

−|31? − |32? − 2|33?,

ˆH|11? = 2|2? − |3? + |16? − 2|17? + |23? − |25? − |26? − |29? − |30? − |32? − 2|34?,

ˆH|12? = |2? − |25?,

ˆH|13? = −|3? − |16? + |19? + |23? + |24? − |27? − |28? − |29?,

ˆH|14? = −|3? − |16? + |23? − |27? − |30? − |31?,

ˆH|15? = −|3? − |26? − |27?,

ˆH|16? = −|6? − |7? − 2|8? − 2|9? + |10? + |11? − 2|13? − 2|14? + u|16? − 8|36?

+4|46? − 4|54?,

ˆH|17? = −8|4? + |6? + |7? − |10? − |11? + u|17? + 4|36? − 2|40? − 2|41? − 4|44?

−4|47? + 2|63? + 2|64?,

ˆH|18? = −|6? − |10? + u|18? + |42? − |56? + |58?,

ˆH|19? = −|6? − 2|9? − |10? + 2|13? + u|19? − 2|35? − 4|39? − 2|42? − 2|55? + |57? + |58?,

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ˆH|20? = |6? + |7? − 2|9? + u|20? − 2|35? − 2|38? − 4|40? − 2|42? − |43? − |50? − |51?

+|57? + 2|61? + 2|65?,

ˆH|21? = |6? − |8? + u|21? − 2|41? + |42? − |43? − |50? − |56? − |59?,

ˆH|22? = |7? − |8? + u|22? − 2|38? − 2|41? − |51? − |59?,

ˆH|23? = 8|5? − |6? − |7? + 2|8? + 2|9? + |10? + |11? + 2|13? + 2|14? + 2|37?

+2|38? + 2|43? + 8|44? − 4|45? − 4|46?,

ˆH|24? = |6? − 2|9? + |10? + 2|13? − 2|35? − 4|39? − 2|55? − 2|56? + |57? − |58?,

ˆH|25? = |6? + |7? − |10? − |11? − 8|12? + 4|45? + 4|48? + 4|54? − 2|59? − 2|60?

+2|61? + 2|62?,

ˆH|26? = |6? + |7? − |10? − |11? − 4|15? − 8|47? + 4|48? − |49? − |50? − |51? − |52?,

ˆH|27? = −2|8? − 2|9? − 2|13? − 2|14? − 4|15? + 4|45? − |49? − |50? − |51? − |52?

−8|53? − 4|54?,

ˆH|28? = |7? − |10? − 2|13? + 2|35? − 2|38? + 2|42? − |43? − |51? − |52? − |57?

+2|62? + 4|63? − 2|65?,

ˆH|29? = |6? − |11? − 2|13? + 2|35? − 2|37? − |43? − |49? − |50? + 2|56? − |57?

+2|62? + 4|63? − 2|65?,

ˆH|30? = |7? − |11? − 2|14? − 2|37? − 2|38? − |49? − |51? − 2|60? + 4|64?,

ˆH|31? = |6? − |10? − 2|14? − 2|43? − |50? − |52? − 2|60? + 4|64?,

ˆH|32? = −2|9? − |10? − |11? − 2|35? − 2|37? − 4|40? − |43? − |49? − |52? − 2|56?

+|57? + 2|61? + 2|65?,

ˆH|33? = −|8? − |10? − 2|41? − |42? − |43? − |52? + |56? − |59?,

ˆH|34? = −|8? − |11? − 2|37? − 2|41? − |49? − |59?,

ˆH|35? = −|19? − |20? − |24? + |28? + |29? − |32? + |74? − |75?,

ˆH|36? = −|16? + |17? + 2u|36? + |66?,

ˆH|37? = |23? − |29? − |30? − |32? − 2|34? + 2|67? − |70? + |71?,

ˆH|38? = −|20? − 2|22? + |23? − |28? − |30? + 2|67? − |70? + |71?,

ˆH|39? = −|19? − |24? + u|39?,

ˆH|40? = −|17? − |20? − |32? + u|40? − |66? + |67? − |69?,

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ˆH|41? = −|17? − |21? − |22? − |33? − |34? + u|41? − |66? + |67? − |69?,

ˆH|42? = |18? − |19? − |20? + |21? + |28? − |33? + u|42? + |75? − |76?,

ˆH|43? = −|20? − 2|21? + 2|23? − |28? − |29? − 2|31? − |32? − 2|33? + 4|67? − 2|70? + 2|71?,

ˆH|44? = −|17? + |23? + |67?,

ˆH|45? = −|23? + |25? + |27? + |70?,

ˆH|46? = |16? − |23? − |71?,

ˆH|47? = −|17? − |26? − |69?,

ˆH|48? = |25? + |26? − |72?,

ˆH|49? = −|26? − |27? − |29? − |30? − |32? − 2|34? − 2|68? − 2|69? − |70? + |72? − |73?,

ˆH|50? = −|20? − 2|21? − |26? − |27? − |29? − |31? − 2|68? − 2|69? − |70? + |72?

−|73? + |74? − |75? − 2|76?,

ˆH|51? = −|20? − 2|22? − |26? − |27? − |28? − |30? − 2|68? − 2|69? − |70? + |72? − |73?,

ˆH|52? = −|26? − |27? − |28? − |31? − |32? − 2|33? − 2|68? − 2|69? − |70? + |72?

−|73? − |74? + |75? + 2|76?,

ˆH|53? = −|27? − |68?,

ˆH|54? = −|16? + |25? − |27? − |73?,

ˆH|55? = −|19? − |24? + |74? + |75?,

ˆH|56? = −|18? − |21? − |24? + |29? − |32? + |33? + |74? + |76?,

ˆH|57? = |19? + |20? + |24? − |28? − |29? + |32? − |74? − |75?,

ˆH|58? = 2|18? + |19? − |24? − |74? + |75? + 2|76?,

ˆH|59? = −|21? − |22? − |25? − |33? − |34? − |70? + |72? + |73?,

ˆH|60? = −|25? − |30? − |31? − |70? + |72? + |73?,

ˆH|61? = |20? + |25? + |32? + |70? − |72? − |73? − |74? − |75?,

ˆH|62? = |25? + |28? + |29? + |70? − |72? − |73? + |74? + |75?,

ˆH|63? = |17? + |28? + |29? + |66? − |67? + |69?,

ˆH|64? = |17? + |30? + |31? + |66? − |67? + |69?,

ˆH|65? = |20? − |28? − |29? + |32?,

ˆH|66? = 4|36? − 2|40? − 2|41? + 2|63? + 2|64? + u|66? + 4|78? − 4|85?,

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ˆH|67? = 2|37? + 2|38? + 2|40? + 2|41? + 2|43? + 4|44? − 2|63? − 2|64? + 16|77?

−4|78? + 4|79?,

ˆH|68? = −|49? − |50? − |51? − |52? − 4|53? − 8|80? + 4|81?,

ˆH|69? = −2|40? − 2|41? − 4|47? − |49? − |50? − |51? − |52? + 2|63? + 2|64? − 4|79?

−8|80? − 4|85?,

ˆH|70? = −2|37? − 2|38? − 2|43? + 4|45? − |49? − |50? − |51? − |52? − 2|59? − 2|60?

+2|61? + 2|62? − 8|79? + 8|82? − 4|84?,

ˆH|71? = 2|37? + 2|38? + 2|43? − 4|46? − 8|78? + 4|84?,

ˆH|72? = −4|48? + |49? + |50? + |51? + |52? + 2|59? + 2|60? − 2|61? − 2|62?

−8|81? − 8|82?,

ˆH|73? = −|49? − |50? − |51? − |52? − 4|54? + 2|59? + 2|60? − 2|61? − 2|62?

+4|84? − 8|85?,

ˆH|74? = 2|35? + |50? − |52? + 2|55? + 2|56? − |57? − |58? − 2|61? + 2|62? + 4|83?,

ˆH|75? = 2|35? + 2|42? − |50? + |52? + 2|55? − |57? + |58? − 2|61? + 2|62? + 4|83?,

ˆH|76? = −|42? − |50? + |52? + |56? + |58?,

ˆH|77? = |67?,

ˆH|78? = |66? − |67? − |71?,

ˆH|79? = |67? − |69? − |70?,

ˆH|80? = −|68? − |69?,

ˆH|81? = |68? − |72?,

ˆH|82? = |70? − |72?,

ˆH|83? = |74? + |75?,

ˆH|84? = −|70? + |71? + |73?,

ˆH|85? = −|66? − |69? − |73?.(A1)

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