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

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Improved Hamiltonian variational technique for lattice models

Elbio R. Dagotto, Adriana Moreo

Published Feb 15, 1985• 116 citations
Publisher

Abstract

In the context of the Hamiltonian variational approach we propose a systematic method for the improvement of any given trial wave function. The approach has many similarities with the Lanczos scheme and it may be used for quantum-mechanical problems as well as field-theory and statistical-mechanics systems. We apply the method to the Mathieu equation and to the Ising one-dimensional model in a finite lattice. The agreement between our results and the exact ones is excellent in the whole range of parameters. We also briefly discuss the application of these ideas to lattice gauge theories and the similarities with other recently proposed methods.

DOI
10.1103/physrevd.31.865

Related papers

Effective hamiltonian approach for strongly correlated lattice models

Sandro Sorella

We review a recent approach for the simulation of many-body interacting systems based on an efficient generalization of the Lanczos method for Quantum Monte Carlo simulations. This technique allows to perform systematic corrections to a given variational wavefunction, that allow to estimate exact energies and correlation functions, whenever the starting variational wavefunction is a qualitatively correct description of the ground state. The stability of the variational wavefunction against possible phases, not described at the variational level can be tested by using the ''effective Hamiltonian'' approach. In fact Monte Carlo methods, such as the ''fixed node approximation'' and the ''generalized Lanczos technique'' (Phys. Rev. B 64,024512, 2001) allow to obtain exact ground state properties of an effective Hamiltonian, chosen to be as close as possible to the exact Hamiltonian, thus yielding the most reasonable estimates of correlation functions. We also describe a simplified one-parameter scheme that improve substantially the efficiency of the generalized Lanczos method. This is tested on the t-J model, with a special effort to obtain accurate pairing correlations, and provide a possible non-phonon mechanism for High temperature superconductivity.

arXiv:cond-mat/0201388•Jan 22, 2002•0 citations

Simple numerical approach to quantum lattice models

Eduardo R. Gagliano, Silvia B. Bacci

The purpose of this paper is to present some simple variants of a recently developed iterative algorithm for improvement of any given trial wave function. These modifications improve the convergence rate of the iterative process and allow us to obtain a good variational state for larger lattices. We apply the method to the one-dimensional Gross-Neveu model with one flavor in a finite-size lattice. The ground-state energy, the density-density correlation function, and the single-particle Green's function are calculated. They are in good agreement with existing results.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jul 15, 1987•15 citations

A perturbative and variational approach to quantum lattice Hamiltonians

José G. Esteve, Germán Sierra

We propose a method to construct the ground state \ensuremath{\psi}(\ensuremath{\lambda}) of local lattice Hamiltonians with the generic form ${\mathit{H}}_{0}$+\ensuremath{\lambda}${\mathit{H}}_{1}$, where \ensuremath{\lambda} is a coupling constant and ${\mathit{H}}_{0}$ is a Hamiltonian with a nondegenerate ground state ${\mathrm{\ensuremath{\psi}}}_{0}$. The method is based on the choice of an exponential ansatz \ensuremath{\psi}(\ensuremath{\lambda})=exp[U(\ensuremath{\lambda})]${\mathrm{\ensuremath{\psi}}}_{0}$, which is a sort of generalized lattice version of a Jastrow wave function. We combine perturbative and variational techniques to get successive approximations of the operator U(\ensuremath{\lambda}). Perturbation theory is used to set up a variational method which in turn produces nonperturbative results. The computation with this kind of ansatz leads us to associate to the original quantum-mechanical problem a statistical-mechanical system defined in the same spatial dimension. In some cases these statistical-mechanical systems turn out to be integrable, which allows us to obtain exact upper bounds to the energy. The general ideas of our method are illustrated in the example of the Ising model in a transverse field.

arXiv:cond-mat/9409098•Physical review. B, Condensed matter•Apr 1, 1995•1 citation

Variational approach to Hamiltonian lattice theories

D. Horn

A variational calculation of the vacuum energy of a Hamiltonian lattice theory is formulated in terms of a finite box Hamiltonian for a cluster of points. The box Hamiltonian contains surface terms which are proportional to order parameters of the system. It is tested on the Ising model and applied to $Z(N)$ spin models in 1 + 1 and 2 + 1 dimensions. $Z(3)$ is found to have an exceptional phase-transition structure. The application of the method to local gauge theories is discussed.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 15, 1981•9 citations

Demonstration of systematic im provements in application of the variational method to weakly bound potentials

W. N. Mei

In applying the variational method, we demonstrate that, with careful study of the general features of the trial wave functions, we can achieve systematic improvements in calculating the eigenvalues of quantum mechanical systems. We apply five different sets of trial wave functions to calculate the ground state and first excited state energies of weakly bound potentials, such as V(x) = ∣x∣, √∣x∣ and ln (∣x∣). We demonstrate that accurate results can be obtained through a thorough understanding of the asymptotic behaviours. With proper adjustments, we show how to improve trial wave functions. Finally, by calculating to the fifth excited states, we compare the exact and variational results with those obtained from the semi-classical method, and discuss the merits of the different approaches.

International Journal of Mathematical Education in Science and Technology•Jul 1, 1999•0 citations

Eigenvalue approximation methods for quantum lattice Hamiltonians

Peter Hornby

The thesis consists of five major parts or chapters, the first of which is an introduction and review relating the field of investigation to problems in quantum field theory and statistical mechanics.Although the thesis generally deals with eigenvalue bounding techniques, these methods are applied with a view to calculating the critical properties of infinite systems [7].The approach advocated in the thesis intends the use of approximate methods to calculate the thermodynamics of large, but finite lattices.Finite size scaling [3] can then be used to estimate the bulk system thermodynamics, especially critical phenomena, from the approximate large lattice data.The approximations used include projected mean fields [5], for both ground and excited states, correlated product wave functions [6], which are extended to kink states, and blocking approximations [2].Part of this work is motivated by the promise these wave functions have in their application to variationally guided Monte Carlo methods for finite lattices [4], Also, it is shown that the critical indices predicted by the finite size rescaling formalism, are left invariant by the application of an approximate (blocking) renormalisation group to the finite lattices involved.Moreover, this invariance of the finite size rescaling equations is used to define a sequence transformation suitable for improving

Bulletin of the Australian Mathematical Society•Jun 1, 1985•0 citations
In the context of the Hamiltonian variational approach we propose a systematic method for the improvement of any given trial wave function. The approach has many similarities with the Lanczos scheme and it may be used for quantum-mechanical problems as well as field-theory and statistical-mechanics systems. We apply the method to the Mathieu equation and to the Ising one-dimensional model in a finite lattice. The agreement between our results and the exact ones is excellent in the whole range of parameters. We also briefly discuss the application of these ideas to lattice gauge theories and the similarities with other recently proposed methods.
We review a recent approach for the simulation of many-body interacting systems based on an efficient generalization of the Lanczos method for Quantum Monte Carlo simulations. This technique allows to perform systematic corrections to a given variational wavefunction, that allow to estimate exact energies and correlation functions, whenever the starting variational wavefunction is a qualitatively correct description of the ground state. The stability of the variational wavefunction against possible phases, not described at the variational level can be tested by using the ''effective Hamiltonian'' approach. In fact Monte Carlo methods, such as the ''fixed node approximation'' and the ''generalized Lanczos technique'' (Phys. Rev. B 64,024512, 2001) allow to obtain exact ground state properties of an effective Hamiltonian, chosen to be as close as possible to the exact Hamiltonian, thus yielding the most reasonable estimates of correlation functions. We also describe a simplified one-parameter scheme that improve substantially the efficiency of the generalized Lanczos method. This is tested on the t-J model, with a special effort to obtain accurate pairing correlations, and provide a possible non-phonon mechanism for High temperature superconductivity.
The purpose of this paper is to present some simple variants of a recently developed iterative algorithm for improvement of any given trial wave function. These modifications improve the convergence rate of the iterative process and allow us to obtain a good variational state for larger lattices. We apply the method to the one-dimensional Gross-Neveu model with one flavor in a finite-size lattice. The ground-state energy, the density-density correlation function, and the single-particle Green's function are calculated. They are in good agreement with existing results.
We propose a method to construct the ground state \ensuremath{\psi}(\ensuremath{\lambda}) of local lattice Hamiltonians with the generic form H0{\mathit{H}}_{0}H0​+\ensuremath{\lambda}H1{\mathit{H}}_{1}H1​, where \ensuremath{\lambda} is a coupling constant and H0{\mathit{H}}_{0}H0​ is a Hamiltonian with a nondegenerate ground state ${\mathrm{\ensuremath{\psi}}}_{0}$. The method is based on the choice of an exponential ansatz \ensuremath{\psi}(\ensuremath{\lambda})=exp[U(\ensuremath{\lambda})]${\mathrm{\ensuremath{\psi}}}_{0}$, which is a sort of generalized lattice version of a Jastrow wave function. We combine perturbative and variational techniques to get successive approximations of the operator U(\ensuremath{\lambda}). Perturbation theory is used to set up a variational method which in turn produces nonperturbative results. The computation with this kind of ansatz leads us to associate to the original quantum-mechanical problem a statistical-mechanical system defined in the same spatial dimension. In some cases these statistical-mechanical systems turn out to be integrable, which allows us to obtain exact upper bounds to the energy. The general ideas of our method are illustrated in the example of the Ising model in a transverse field.
A variational calculation of the vacuum energy of a Hamiltonian lattice theory is formulated in terms of a finite box Hamiltonian for a cluster of points. The box Hamiltonian contains surface terms which are proportional to order parameters of the system. It is tested on the Ising model and applied to Z(N)Z(N)Z(N) spin models in 1 + 1 and 2 + 1 dimensions. Z(3)Z(3)Z(3) is found to have an exceptional phase-transition structure. The application of the method to local gauge theories is discussed.
In applying the variational method, we demonstrate that, with careful study of the general features of the trial wave functions, we can achieve systematic improvements in calculating the eigenvalues of quantum mechanical systems. We apply five different sets of trial wave functions to calculate the ground state and first excited state energies of weakly bound potentials, such as V(x) = ∣x∣, √∣x∣ and ln (∣x∣). We demonstrate that accurate results can be obtained through a thorough understanding of the asymptotic behaviours. With proper adjustments, we show how to improve trial wave functions. Finally, by calculating to the fifth excited states, we compare the exact and variational results with those obtained from the semi-classical method, and discuss the merits of the different approaches.
The thesis consists of five major parts or chapters, the first of which is an introduction and review relating the field of investigation to problems in quantum field theory and statistical mechanics.Although the thesis generally deals with eigenvalue bounding techniques, these methods are applied with a view to calculating the critical properties of infinite systems [7].The approach advocated in the thesis intends the use of approximate methods to calculate the thermodynamics of large, but finite lattices.Finite size scaling [3] can then be used to estimate the bulk system thermodynamics, especially critical phenomena, from the approximate large lattice data.The approximations used include projected mean fields [5], for both ground and excited states, correlated product wave functions [6], which are extended to kink states, and blocking approximations [2].Part of this work is motivated by the promise these wave functions have in their application to variationally guided Monte Carlo methods for finite lattices [4], Also, it is shown that the critical indices predicted by the finite size rescaling formalism, are left invariant by the application of an approximate (blocking) renormalisation group to the finite lattices involved.Moreover, this invariance of the finite size rescaling equations is used to define a sequence transformation suitable for improving