Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1406.6472 · The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics

Exact ground state for the four-electron problem in a 2D finite honeycomb lattice

Réka Trencsényi, Konstantin E. Glukhov, Zsolt Gulácsi

Published Apr 23, 2014• 0 citations
arXivHTMLPDF

Abstract

Working in a subspace with dimensionality much smaller than the dimension of the full Hilbert space, we deduce exact four-particle ground states in 2D samples containing hexagonal repeat units and described by Hubbard type of models. The procedure identifies first a small subspace in which the ground state is placed, than deduces by exact diagonalization in . The small subspace is obtained by the repeated application of the Hamiltonian on a carefully chosen starting wave vector describing the most interacting particle configuration, and the wave vectors resulting from the application of , till the obtained system of equations closes in itself. The procedure which can be applied in principle at fixed but arbitrary system size and number of particles is interesting on its own since it provides exact information for the numerical approximation techniques which use a similar strategy, but apply non-complete basis for . The diagonalization inside provides an incomplete image of the low lying part of the excitation spectrum, but provides the exact . Once the exact ground state is obtained, its properties can be easily analysed. The is found always as a singlet state whose energy, interestingly, saturates in the limit. The unapproximated results show that the emergence probabilities of different particle configurations in the ground state presents ‘Zittern’ (trembling) characteristics which are absent in 2D square Hubbard systems. Consequently, the manifestation of the local Coulomb repulsion in 2D square and honeycomb types of systems presents differences, which can be a real source in the differences in the many-body behaviour.

DOI
10.1080/14786435.2014.904059

Related papers

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

E. Kovács, Zsolt Gulácsi

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.

arXiv:cond-mat/0604319•The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics•Apr 28, 2006•5 citations

Exact ground states for the four-electron problem in a Hubbard ladder

E. Kovács, Zsolt Gulácsi

The exact ground state of four electrons in an arbitrary large two-leg Hubbard ladder is deduced from nine analytic and explicit linear equations. The procedure used is described, and the properties of the ground state are analyzed. The method is based on the construction in r-space of the different type of orthogonal basis wave vectors which span the subspace of the Hilbert space containing the ground state. In order to do this, we start from the possible microconfigurations of the four particles within the system. These microconfigurations are then rotated, translated and spin-reversed in order to build up the basis vectors of the problem. A closed system of nine analytic linear equations is obtained whose secular equation, by its minimum energy solution, provides the ground-state energy and the ground-state wave function of the model.

arXiv:cond-mat/0604294•The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics•Mar 10, 2006•2 citations

Four electrons in a two-leg Hubbard ladder: exact ground states

Endre Kovács, Zsolt Gulácsi

In the case of a two-leg Hubbard ladder we present a procedure which allows the exact deduction of the ground state for the four particle problem in arbitrary large lattice system, in a tractable manner, which involves only a reduced Hilbert space region containing the ground state. In the presented case, the method leads to nine analytic, linear, and coupled equations providing the ground state. The procedure which is applicable to few particle problems and other systems as well is based on an r-space representation of the wave functions and construction of symmetry adapted orthogonal basis wave vectors describing the Hilbert space region containing the ground state. Once the ground state is deduced, a complete quantum mechanical characterization of the studied state can be given. Since the analytic structure of the ground state becomes visible during the use of the method, its importance is not reduced only to the understanding of theoretical aspects connected to exact descriptions or potential numerical approximation scheme developments, but is relevant as well for a large number of potential technological application possibilities placed between nano-devices and quantum calculations, where the few particle behavior and deep understanding are important key aspects to know.

arXiv:cond-mat/0602482•Journal of Physics A Mathematical and General•Nov 16, 2005•3 citations

An exact solution to the extended Hubbard model in 2D for finite size system

S Harir, Mohamed Bennai, Y. Boughaleb

An exact analytical diagonalization is used to solve the two-dimensional extended Hubbard model (EHM) for a system with finite size. We have considered an EHM including on-site and off-site interactions with interaction energies U and V , respectively, for a square lattice containing 4 × 4 sites at one-eighth filling with periodic boundary conditions, recently treated by Kovacs and Gulacsi (2006 Phil. Mag. 86 2073). Taking into account the symmetric 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 a 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 excited state and induces a supplementary conductivity of the system.

arXiv:1111.0076•Physica Scripta•Jul 25, 2008•4 citations

Nonperturbative results for few electrons in the two-dimensional Hubbard model

Michele Fabrizio, Alberto Parola, Erio Tosatti

Based on a generalization of the equation which solves exactly the two-electron problem, we have found a method which allows the nonperturbative, asymptotically exact solution for the ground state of any finite number n of electrons in the low-density limit (n/N\ensuremath{\rightarrow}0) of the N-site two-dimensional Hubbard model. The three-electron case is checked against the exact numerical diagonalization and the total energy is shown by finite-size scaling to agree well, to leading order of an expansion in powers of 1/lnN, with the asymptotic solution. We then indicate how to solve for an arbitrary electron number, including both closed and open shells. The case of four electrons is explicitly presented as an application. More generally, the closed-shell d=2 ground state is a nondegenerate singlet with zero total momentum. Similar to d=1, all observables differ from their U=0 value by a correction which is U independent. Unlike d=1 and similar to d=3, however, the ground state is a linear combination of the unperturbed U=0 Fermi sea with the set of two-particle excitations. Moreover, the momentum distribution n(k) can be evaluated explicitly, and shows a finite Fermi jump Z=1-4 ln2/${\mathrm{ln}}^{2}$(n/N). This result appears to argue in favor of a Fermi-liquid-like behavior for d=2 in the zero-density limit n/N\ensuremath{\rightarrow}0. The open-shell ground states display an interesting degeneracy between a nondegenerate singlet and high-spin states.

Physical review. B, Condensed matter•Jul 15, 1991•38 citations

The Ground- State Analytical Solution of Two-Interacting Electrons in a Two-Dimensional Square Lattice

Edison A. Enaibe, Daniel A. Babaiwa, Osafile E. Omosede

In this work, we developed the unit step model as an approximate solution to the single-band Hubbard Hamiltonian to solve variationally strongly correlated interacting elections on a two-dimensional (2D) square lattice. We also showed primarily how to derive possible electronic states available for several 2D N x N square lattices, although, with special emphasis on a 2D 5 x 5 square lattice. The results emerging from our present study was compared with the results of Gutzwiller variational approach (GVA) and correlated variational approach (CVA), at the large limit of the Coulomb interaction strength (U/4t).The approximation to the Hubbard Hamiltonian study is actually necessary because of the strong limitation and difficulty pose by the Hubbard Hamiltonian as we move away from finite - size lattices to larger N - dimensional lattices. Thus this work has provided a means of overcoming the finite - size lattice defects as we pass on to a higher dimension. We have shown in this study, that the repulsive Coulomb interaction which in part leads to the strong electronic correlations, would indicate that the two electron system prefer not to condense into s-wave superconducting singlet state (s = 0), at high positive values of the interaction strength. This study reveals that when the Coulomb interaction is zero, that is, for free electron system (non-interacting), the variational parameters which describe the probability distribution of lattice electron system is the same. The spectra intensity increases with increase in the interaction strength and it decreases to zero when the interaction strength is made negatively large. Keywords : unit step Hamiltonian, Hubbard Hamiltonian, 3D cubic lattice, interaction strength, total energy, lattice separation.

•Jan 1, 2013•0 citations
Working in a subspace with dimensionality much smaller than the dimension of the full Hilbert space, we deduce exact four-particle ground states in 2D samples containing hexagonal repeat units and described by Hubbard type of models. The procedure identifies first a small subspace in which the ground state is placed, than deduces by exact diagonalization in . The small subspace is obtained by the repeated application of the Hamiltonian on a carefully chosen starting wave vector describing the most interacting particle configuration, and the wave vectors resulting from the application of , till the obtained system of equations closes in itself. The procedure which can be applied in principle at fixed but arbitrary system size and number of particles is interesting on its own since it provides exact information for the numerical approximation techniques which use a similar strategy, but apply non-complete basis for . The diagonalization inside provides an incomplete image of the low lying part of the excitation spectrum, but provides the exact . Once the exact ground state is obtained, its properties can be easily analysed. The is found always as a singlet state whose energy, interestingly, saturates in the limit. The unapproximated results show that the emergence probabilities of different particle configurations in the ground state presents ‘Zittern’ (trembling) characteristics which are absent in 2D square Hubbard systems. Consequently, the manifestation of the local Coulomb repulsion in 2D square and honeycomb types of systems presents differences, which can be a real source in the differences in the many-body behaviour.
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.
The exact ground state of four electrons in an arbitrary large two-leg Hubbard ladder is deduced from nine analytic and explicit linear equations. The procedure used is described, and the properties of the ground state are analyzed. The method is based on the construction in r-space of the different type of orthogonal basis wave vectors which span the subspace of the Hilbert space containing the ground state. In order to do this, we start from the possible microconfigurations of the four particles within the system. These microconfigurations are then rotated, translated and spin-reversed in order to build up the basis vectors of the problem. A closed system of nine analytic linear equations is obtained whose secular equation, by its minimum energy solution, provides the ground-state energy and the ground-state wave function of the model.
In the case of a two-leg Hubbard ladder we present a procedure which allows the exact deduction of the ground state for the four particle problem in arbitrary large lattice system, in a tractable manner, which involves only a reduced Hilbert space region containing the ground state. In the presented case, the method leads to nine analytic, linear, and coupled equations providing the ground state. The procedure which is applicable to few particle problems and other systems as well is based on an r-space representation of the wave functions and construction of symmetry adapted orthogonal basis wave vectors describing the Hilbert space region containing the ground state. Once the ground state is deduced, a complete quantum mechanical characterization of the studied state can be given. Since the analytic structure of the ground state becomes visible during the use of the method, its importance is not reduced only to the understanding of theoretical aspects connected to exact descriptions or potential numerical approximation scheme developments, but is relevant as well for a large number of potential technological application possibilities placed between nano-devices and quantum calculations, where the few particle behavior and deep understanding are important key aspects to know.
An exact analytical diagonalization is used to solve the two-dimensional extended Hubbard model (EHM) for a system with finite size. We have considered an EHM including on-site and off-site interactions with interaction energies U and V , respectively, for a square lattice containing 4 × 4 sites at one-eighth filling with periodic boundary conditions, recently treated by Kovacs and Gulacsi (2006 Phil. Mag. 86 2073). Taking into account the symmetric 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 a 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 excited state and induces a supplementary conductivity of the system.
Based on a generalization of the equation which solves exactly the two-electron problem, we have found a method which allows the nonperturbative, asymptotically exact solution for the ground state of any finite number n of electrons in the low-density limit (n/N\ensuremath{\rightarrow}0) of the N-site two-dimensional Hubbard model. The three-electron case is checked against the exact numerical diagonalization and the total energy is shown by finite-size scaling to agree well, to leading order of an expansion in powers of 1/lnN, with the asymptotic solution. We then indicate how to solve for an arbitrary electron number, including both closed and open shells. The case of four electrons is explicitly presented as an application. More generally, the closed-shell d=2 ground state is a nondegenerate singlet with zero total momentum. Similar to d=1, all observables differ from their U=0 value by a correction which is U independent. Unlike d=1 and similar to d=3, however, the ground state is a linear combination of the unperturbed U=0 Fermi sea with the set of two-particle excitations. Moreover, the momentum distribution n(k) can be evaluated explicitly, and shows a finite Fermi jump Z=1-4 ln2/ln2{\mathrm{ln}}^{2}ln2(n/N). This result appears to argue in favor of a Fermi-liquid-like behavior for d=2 in the zero-density limit n/N\ensuremath{\rightarrow}0. The open-shell ground states display an interesting degeneracy between a nondegenerate singlet and high-spin states.
In this work, we developed the unit step model as an approximate solution to the single-band Hubbard Hamiltonian to solve variationally strongly correlated interacting elections on a two-dimensional (2D) square lattice. We also showed primarily how to derive possible electronic states available for several 2D N x N square lattices, although, with special emphasis on a 2D 5 x 5 square lattice. The results emerging from our present study was compared with the results of Gutzwiller variational approach (GVA) and correlated variational approach (CVA), at the large limit of the Coulomb interaction strength (U/4t).The approximation to the Hubbard Hamiltonian study is actually necessary because of the strong limitation and difficulty pose by the Hubbard Hamiltonian as we move away from finite - size lattices to larger N - dimensional lattices. Thus this work has provided a means of overcoming the finite - size lattice defects as we pass on to a higher dimension. We have shown in this study, that the repulsive Coulomb interaction which in part leads to the strong electronic correlations, would indicate that the two electron system prefer not to condense into s-wave superconducting singlet state (s = 0), at high positive values of the interaction strength. This study reveals that when the Coulomb interaction is zero, that is, for free electron system (non-interacting), the variational parameters which describe the probability distribution of lattice electron system is the same. The spectra intensity increases with increase in the interaction strength and it decreases to zero when the interaction strength is made negatively large. Keywords : unit step Hamiltonian, Hubbard Hamiltonian, 3D cubic lattice, interaction strength, total energy, lattice separation.