arXiv:cond-mat/0407756 · New Journal of Physics

Ground state of many-body lattice systems: an analytical probabilistic approach

Massimo Ostilli, Carlo Presilla

Published Aug 13, 2004• 5 citations

Abstract

On the grounds of a Feynman–Kac-type formula for Hamiltonian lattice systems, we derive analytical expressions for the matrix elements of the evolution operator. These expressions are valid at long times when a central limit theorem applies. As a remarkable result, we find that the ground-state energy as well as all the correlation functions in the ground state are determined semi-analytically by solving a simple scalar equation. Furthermore, explicit solutions of this equation are obtained in the noninteracting case.

DOI: 10.1088/1367-2630/6/1/107

Related papers

Ground state of many-body lattice systems: an analytical probabilistic approach

Massimo Ostilli, Carlo Presilla A

Jan 1, 20043 citations

An analytical probabilistic approach to the ground state of lattice quantum systems: exact results in terms of a cumulant expansion

Massimo Ostilli, Carlo Presilla A

arXiv:cond-mat/0412157Journal of Statistical Mechanics Theory and ExperimentApr 19, 20056 citations

Analytic solution for the ground-state energy of the extensive many-body problem

Lloyd C. L. Hollenberg, Nicholas S. Witte

arXiv:cond-mat/9602091Physical review. B, Condensed matterDec 15, 199622 citations

GROUND STATE OF MANY-BODY LATTICE SYSTEMS VIA A CENTRAL LIMIT THEOREM

Carlo Presilla A, MASSIMO OSTILLI

We review a novel approach to evaluate the ground-state properties of many-body lattice systems based on an exact probabilistic representation of the dynamics and its long time approximation via a central limit theorem. The choice of the asymptotic density probability used in the calculation is discussed in detail.

arXiv:cond-mat/0411225International Journal of Modern Physics BJul 30, 20062 citations

General nonperturbative estimate of the energy density of lattice Hamiltonians

Lloyd C. L. Hollenberg, Nicholas S. Witte

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsSep 1, 199424 citations

Analytic calculation of ground-state properties of correlated fermions with the Gutzwiller wave function

Walter Metzner, D. Vollhardt

Physical review. B, Condensed matterMay 1, 1988178 citations

arXiv:cond-mat/0407756 · New Journal of Physics

Ground state of many-body lattice systems: an analytical probabilistic approach

Massimo Ostilli, Carlo Presilla

Published Aug 13, 2004• 5 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1367-2630/6/1/107

Related papers

Ground state of many-body lattice systems: an analytical probabilistic approach

Massimo Ostilli, Carlo Presilla A

doi:10.1088/1367-2630/6/1/107 Abstract. On the grounds of a Feynman–Kac-type formula for Hamiltonian lattice systems, we derive analytical expressions for the matrix elements of the evolution operator. These expressions are valid at long times when a central limit theorem applies. As a remarkable result, we find that the ground-state energy as well as all the correlation functions in the ground state are determined semi-analytically by solving a simple scalar equation. Furthermore, explicit solutions of this equation are obtained in the noninteracting case. Contentsdoi:10.1088/1367-2630/6/1/107 Abstract. On the grounds of a Feynman–Kac-type formula for Hamiltonian lattice systems, we derive analytical expressions for the matrix elements of the evolution operator. These expressions are valid at long times when a central limit theorem applies. As a remarkable result, we find that the ground-state energy as well as all the correlation functions in the ground state are determined semi-analytically by solving a simple scalar equation. Furthermore, explicit solutions of this equation are obtained in the noninteracting case. Contents

Jan 1, 20043 citations

An analytical probabilistic approach to the ground state of lattice quantum systems: exact results in terms of a cumulant expansion

Massimo Ostilli, Carlo Presilla A

We present a large deviation analysis of a recently proposed probabilistic approach to the study of the ground-state properties of lattice quantum systems. The ground-state energy as well as the correlation functions in the ground state are exactly determined as series expansions in the cumulants of the multiplicities of the potential and hopping energies assumed by the system during its long-time evolution. Once these cumulants are known, even at a finite order, our approach provides the ground state analytically as a function of the Hamiltonian parameters. A scenario of possible applications of the analytic character of the present approach is discussed.We present a large deviation analysis of a recently proposed probabilistic approach to the study of the ground-state properties of lattice quantum systems. The ground-state energy as well as the correlation functions in the ground state are exactly determined as series expansions in the cumulants of the multiplicities of the potential and hopping energies assumed by the system during its long-time evolution. Once these cumulants are known, even at a finite order, our approach provides the ground state analytically as a function of the Hamiltonian parameters. A scenario of possible applications of the analytic character of the present approach is discussed.

arXiv:cond-mat/0412157Journal of Statistical Mechanics Theory and ExperimentApr 19, 20056 citations

Analytic solution for the ground-state energy of the extensive many-body problem

Lloyd C. L. Hollenberg, Nicholas S. Witte

A closed form expression for the ground-state energy density of the general extensive many-body problem is given in terms of the Lanczos tridiagonal form of the Hamiltonian. Given the general expressions of the diagonal and off-diagonal elements of the Hamiltonian Lanczos matrix, ${\mathrm{\ensuremath{\alpha}}}_{\mathit{n}}$(N) and ${\mathrm{\ensuremath{\beta}}}_{\mathit{n}}$(N), asymptotic forms \ensuremath{\alpha}(z) and \ensuremath{\beta}(z) can be defined in terms of a parameter z\ensuremath{\equiv}n/N (n is the Lanczos iteration and N is the size of the system). By application of theorems on the zeros of orthogonal polynomials we find the ground-state energy density in the bulk limit to be given, in general, by ${\mathit{E}}_{0}$ =inf[\ensuremath{\alpha}(z)-2\ensuremath{\beta}(z)]. \textcopyright{} 1996 The American Physical Society.A closed form expression for the ground-state energy density of the general extensive many-body problem is given in terms of the Lanczos tridiagonal form of the Hamiltonian. Given the general expressions of the diagonal and off-diagonal elements of the Hamiltonian Lanczos matrix, ${\mathrm{\ensuremath{\alpha}}}_{\mathit{n}}$(N) and ${\mathrm{\ensuremath{\beta}}}_{\mathit{n}}$(N), asymptotic forms \ensuremath{\alpha}(z) and \ensuremath{\beta}(z) can be defined in terms of a parameter z\ensuremath{\equiv}n/N (n is the Lanczos iteration and N is the size of the system). By application of theorems on the zeros of orthogonal polynomials we find the ground-state energy density in the bulk limit to be given, in general, by ${\mathit{E}}_{0}$=inf[\ensuremath{\alpha}(z)-2\ensuremath{\beta}(z)]. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9602091Physical review. B, Condensed matterDec 15, 199622 citations

GROUND STATE OF MANY-BODY LATTICE SYSTEMS VIA A CENTRAL LIMIT THEOREM

Carlo Presilla A, MASSIMO OSTILLI

arXiv:cond-mat/0411225International Journal of Modern Physics BJul 30, 20062 citations

General nonperturbative estimate of the energy density of lattice Hamiltonians

Lloyd C. L. Hollenberg, Nicholas S. Witte

Employing a theorem on lower bounds on the zeros of orthogonal polynomials, the plaquette expansion to order 1/ ${\mathit{N}}_{\mathit{p}}$ of the tridiagonal Lanczos matrix elements is solved for the ground-state energy density in the infinite lattice limit. The resulting nonperturbative expression for the estimate of the energy density in terms of the connected coefficients to order 〈 ${\mathit{H}}^{4}$ ${\mathrm{〉}}_{\mathit{c}}$ is completely general. This expression is applied to various Hamiltonian systems---the Heisenberg model in D dimensions and SU(2) and SU(3) lattice gauge theory in 3+1 dimensions. In all cases the analytic estimate to the energy density is not only a significant improvement on the trial state, but is typically accurate to a few percent. The energy density of the D-dimensional Heisenberg model is predicted to approach ${\mathit{scrE}}_{0}$ (N\'eel)-1/8 for large D. In the case of SU(2) and SU(3) the specific heat derived from the energy density peaks at the correct strong- to weak-coupling transition.Employing a theorem on lower bounds on the zeros of orthogonal polynomials, the plaquette expansion to order 1/${\mathit{N}}_{\mathit{p}}$ of the tridiagonal Lanczos matrix elements is solved for the ground-state energy density in the infinite lattice limit. The resulting nonperturbative expression for the estimate of the energy density in terms of the connected coefficients to order 〈${\mathit{H}}^{4}$${\mathrm{〉}}_{\mathit{c}}$ is completely general. This expression is applied to various Hamiltonian systems---the Heisenberg model in D dimensions and SU(2) and SU(3) lattice gauge theory in 3+1 dimensions. In all cases the analytic estimate to the energy density is not only a significant improvement on the trial state, but is typically accurate to a few percent. The energy density of the D-dimensional Heisenberg model is predicted to approach ${\mathit{scrE}}_{0}$(N\'eel)-1/8 for large D. In the case of SU(2) and SU(3) the specific heat derived from the energy density peaks at the correct strong- to weak-coupling transition.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsSep 1, 199424 citations

Analytic calculation of ground-state properties of correlated fermions with the Gutzwiller wave function

Walter Metzner, D. Vollhardt

A new analytic approach to the evaluation of ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function is presented. It is based on the observation that expectation values in terms of this wave function may be expressed by sums over different lattice sites. This makes the application of Wick's theorem and the resulting contractions extremely simple, since the latter involve only anticommuting numbers as in a Grassmann algebra. Expressions for the momentum distribution ${n}_{\mathrm{k}\ensuremath{\sigma}}$ and the Hubbard interaction in terms of a power series in a particular correlation parameter are derived which are valid for all dimensions. An explicit diagrammatic evaluation of the coefficients is described. In one dimension these coefficients may be determined to all orders which yields an approximation-free calculation of ${n}_{\mathrm{k}}$ and ground-state energy E for arbitrary density n and interaction strength U. In the case of a half-filled band and large U, hitherto unexpected nonanalyticities are found. The results allow for the first approximation-free assessment of the properties of the Gutzwiller wave function. It is shown that the well-known Gutzwiller approximation may be derived diagrammatically, too. The approximation is seen to yield the exact results for expectation values in terms of this wave function in the limit of infinite dimensionality.A new analytic approach to the evaluation of ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function is presented. It is based on the observation that expectation values in terms of this wave function may be expressed by sums over different lattice sites. This makes the application of Wick's theorem and the resulting contractions extremely simple, since the latter involve only anticommuting numbers as in a Grassmann algebra. Expressions for the momentum distribution ${n}_{\mathrm{k}\ensuremath{\sigma}}$ and the Hubbard interaction in terms of a power series in a particular correlation parameter are derived which are valid for all dimensions. An explicit diagrammatic evaluation of the coefficients is described. In one dimension these coefficients may be determined to all orders which yields an approximation-free calculation of ${n}_{\mathrm{k}}$ and ground-state energy E for arbitrary density n and interaction strength U. In the case of a half-filled band and large U, hitherto unexpected nonanalyticities are found. The results allow for the first approximation-free assessment of the properties of the Gutzwiller wave function. It is shown that the well-known Gutzwiller approximation may be derived diagrammatically, too. The approximation is seen to yield the exact results for expectation values in terms of this wave function in the limit of infinite dimensionality.

Physical review. B, Condensed matterMay 1, 1988178 citations

{\mathit{E}}_{0}

=inf[\ensuremath{\alpha}(z)-2\ensuremath{\beta}(z)]. \textcopyright{} 1996 The American Physical Society.

Employing a theorem on lower bounds on the zeros of orthogonal polynomials, the plaquette expansion to order 1/

{\mathit{N}}_{\mathit{p}}

of the tridiagonal Lanczos matrix elements is solved for the ground-state energy density in the infinite lattice limit. The resulting nonperturbative expression for the estimate of the energy density in terms of the connected coefficients to order 〈

{\mathit{H}}^{4}

{\mathrm{〉}}_{\mathit{c}}

is completely general. This expression is applied to various Hamiltonian systems---the Heisenberg model in D dimensions and SU(2) and SU(3) lattice gauge theory in 3+1 dimensions. In all cases the analytic estimate to the energy density is not only a significant improvement on the trial state, but is typically accurate to a few percent. The energy density of the D-dimensional Heisenberg model is predicted to approach

{\mathit{scrE}}_{0}

(N\'eel)-1/8 for large D. In the case of SU(2) and SU(3) the specific heat derived from the energy density peaks at the correct strong- to weak-coupling transition.

{n}_{\mathrm{k}}

and ground-state energy E for arbitrary density n and interaction strength U. In the case of a half-filled band and large U, hitherto unexpected nonanalyticities are found. The results allow for the first approximation-free assessment of the properties of the Gutzwiller wave function. It is shown that the well-known Gutzwiller approximation may be derived diagrammatically, too. The approximation is seen to yield the exact results for expectation values in terms of this wave function in the limit of infinite dimensionality.