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:cond-mat/9803107 · Physical Review Letters

Green Function Monte Carlo with Stochastic Reconfiguration

Sandro Sorella

Published May 18, 1998• 267 citations
arXivHTMLPDF

Abstract

A new method for the stabilization of the sign problem in the Green function Monte Carlo technique is proposed. The method is devised for real lattice Hamiltonians and is based on an iterative ``stochastic reconfiguration'' scheme which introduces some bias but allows a stable simulation with constant sign. The systematic reduction of this bias is possible in principle. The method is applied to the frustrated ${J}_{1}{\ensuremath{-}J}_{2}$ Heisenberg model, and tested against exact diagonalization data. Evidence of a finite spin gap for ${J}_{2}{/J}_{1}>\ensuremath{\sim}0.4$ is found in the thermodynamic limit.

DOI
10.1103/physrevlett.80.4558

Related papers

Green function Monte Carlo with stochastic reconfiguration: An effective remedy for the sign problem

Sandro Sorella, Luca Capriotti

A recent technique, proposed to alleviate the ``sign problem disease,'' is discussed in detail. As is well known, the ground state of a given Hamiltonian H can be obtained by applying the propagator ${e}^{\ensuremath{-}H\ensuremath{\tau}}$ to a trial wave function ${\ensuremath{\psi}}_{T}$ and sampling statistically the state ${\ensuremath{\psi}}_{\ensuremath{\tau}}{=e}^{\ensuremath{-}H\ensuremath{\tau}}{\ensuremath{\psi}}_{T}$ for large imaginary time $\ensuremath{\tau}.$ However, the sign problem may appear in the simulation and such statistical propagation would be practically impossible without employing some approximation such as the ``fixed node'' (FN) one. The present method allows the improvement of the FN dynamics with a systematic correction scheme. This is possible by the simple requirement that, after a short imaginary time propagation via the FN Hamiltonian, a number p of correlation functions can be further constrained to be exact by small perturbations of the FN state, which is free from the sign problem. By iterating this procedure, the Monte Carlo average sign, which is almost zero when there is a sign problem, remains stable and finite even for large $\ensuremath{\tau}.$ The proposed algorithm is tested against exact diagonalization data available on finite lattices. It is also shown, in some test cases, that the dependence of the results upon the few parameters entering the stochastic technique can be very easily controlled, unless for exceptional cases.

arXiv:cond-mat/9902211•Physical review. B, Condensed matter•Jan 15, 2000•118 citations

Quantum Monte Carlo scheme for frustrated Heisenberg antiferromagnets

Jacek Wojtkiewicz

When one tries to simulate quantum spin systems by the Monte Carlo method, often the ``minus-sign problem'' is encountered. In such a case, an application of probabilistic methods is not possible. In this paper, the method has been proposed which allows to avoid the minus-sign problem for certain class of frustrated Heisenberg models. The systems where this method is applicable are, for instance, the pyrochlore lattice and the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. The method works in singlet sector. It relies on the expression of wave functions in dimer (pseudo)basis and writing down the Hamiltonian as a sum over plaquettes. In such a formulation, matrix elements of the exponent of Hamiltonian are positive.

arXiv:cond-mat/0611550•Physical Review B•May 15, 2007•2 citations

Generalized Lanczos algorithm for variational quantum Monte Carlo

Sandro Sorella

We show that the standard Lanczos algorithm can be efficiently implemented statistically and self-consistently improved, using the stochastic reconfiguration method, which has been recently introduced to stabilize the Monte Carlo sign problem instability. With this scheme a few Lanczos steps over a given variational wave function are possible even for large size as a particular case of a more general and more accurate technique that allows to obtain lower variational energies. This method has been tested extensively for a strongly correlated model like the $t\ensuremath{-}J$ model. With the standard Lanczos technique it is possible to compute any kind of correlation functions, with no particular computational effort. By using the fact that the variance $〈{H}^{2}〉\ensuremath{-}〈H{〉}^{2}$ is zero for an exact eigenstate, we show that the approach to the exact solution with few Lanczos iterations is indeed possible even for $\ensuremath{\sim}100$ electrons for reasonably good initial wave functions. The variational stochastic reconfiguration technique presented here allows in general a many-parameter energy optimization of any computable many-body wave function, including for instance generic long-range Jastrow factors and arbitrary site-dependent orbital determinants. This scheme improves further the accuracy of the calculation, especially for long-distance correlation functions.

arXiv:cond-mat/0009149•Physical review. B, Condensed matter•Jun 19, 2001•368 citations

Numerical study of the two-dimensional Heisenberg model using a Green function Monte Carlo technique with a fixed number of walkers

Matteo Calandra, Sandro Sorella

We describe in detail a simple and efficient Green function Monte Carlo technique for computing both the ground state energy and the ground state properties by the ``forward walking'' scheme. The simplicity of our reconfiguration process, used to maintain the walker population constant, allows us to control any source of systematic error in a rigorous and systematic way. We apply this method to the Heisenberg model and obtain accurate and reliable estimates of the ground state energy, the order parameter, and the static spin structure factor $S(q)$ for several momenta. For the latter quantity we also find very good agreement with available experimental data on the ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$ antiferromagnet.

arXiv:cond-mat/9802020•Physical review. B, Condensed matter•May 1, 1998•199 citations

Reduction of the sign problem near T=0 in quantum Monte Carlo simulations

Jonathan D’Emidio, Stefan Weßel, Frédéric Mila

Building on a recent investigation of the Shastry-Sutherland model [S. Wessel et al., Phys. Rev. B 98, 174432 (2018)], we develop a general strategy to eliminate the Monte Carlo sign problem near the zero-temperature limit in frustrated quantum spin models. If the Hamiltonian of interest and the sign-problem-free Hamiltonian, obtained by making all off-diagonal elements negative in a given basis, have the same ground state and this state is a member of the computational basis, then the average sign returns to one as the temperature goes to zero. We illustrate this technique by studying the triangular and kagome lattice Heisenberg antiferrromagnet in a magnetic field above saturation, as well as the Heisenberg antiferromagnet on a modified Husimi cactus in the dimer basis. We also provide detailed Appendices on using linear programming techniques to automatically generate efficient directed loop updates in quantum Monte Carlo simulations.

arXiv:2003.13356•Physical review. B./Physical review. B•Aug 20, 2020•8 citations

Incorporation of density-matrix wave functions in Monte Carlo simulations: Application to the frustrated Heisenberg model

M. S. L. du Croo de Jongh, J. M. J. van Leeuwen, Wim van Saarloos

We combine the density-matrix renormalization-group (DMRG) technique with Green function Monte Carlo (GFMC) simulations using a special representation of the DMRG wave function. As a test case we apply the method to the two-dimensional frustrated Heisenberg antiferromagnet. By supplementing the branching in GFMC simulations with stochastic reconfiguration we get a stable simulation with a small variance also in the region where the fluctuations due to the minus sign problem are maximal. The sensitivity of the results to the choice of the guiding wave function is extensively investigated. In agreement with earlier calculations it is found from the DMRG wave function that for small ratios of the next-nearest-to-nearest neighbor coupling strength the system orders as a N\'eel-type antiferromagnet and for large ratios as a columnar antiferromagnet. The spin stiffness suggests an intermediate regime without magnetic long-range order. The energy curve indicates that the columnar phase is separated from the intermediate phase by a first-order transition. The combination of the DMRG and GFMC techniques allows us to substantiate this picture by calculating also the spin correlations in the system. We observe a pattern of spin correlations in the intermediate regime which is in between dimerlike and plaquette-type ordering, states that have recently been suggested. It is a state with strong dimerization in one direction and weaker dimerization in the perpendicular direction and thus it lacks the square symmetry of the plaquette state.

arXiv:cond-mat/0002116•Physical review. B, Condensed matter•Dec 1, 2000•45 citations
A new method for the stabilization of the sign problem in the Green function Monte Carlo technique is proposed. The method is devised for real lattice Hamiltonians and is based on an iterative ``stochastic reconfiguration'' scheme which introduces some bias but allows a stable simulation with constant sign. The systematic reduction of this bias is possible in principle. The method is applied to the frustrated J1\ensuremath−J2{J}_{1}{\ensuremath{-}J}_{2}J1​\ensuremath−J2​ Heisenberg model, and tested against exact diagonalization data. Evidence of a finite spin gap for J2/J1>\ensuremath∼0.4{J}_{2}{/J}_{1}>\ensuremath{\sim}0.4J2​/J1​>\ensuremath∼0.4 is found in the thermodynamic limit.
A recent technique, proposed to alleviate the ``sign problem disease,'' is discussed in detail. As is well known, the ground state of a given Hamiltonian H can be obtained by applying the propagator ${e}^{\ensuremath{-}H\ensuremath{\tau}}$ to a trial wave function ${\ensuremath{\psi}}_{T}$ and sampling statistically the state ${\ensuremath{\psi}}_{\ensuremath{\tau}}{=e}^{\ensuremath{-}H\ensuremath{\tau}}{\ensuremath{\psi}}_{T}$ for large imaginary time $\ensuremath{\tau}.$ However, the sign problem may appear in the simulation and such statistical propagation would be practically impossible without employing some approximation such as the ``fixed node'' (FN) one. The present method allows the improvement of the FN dynamics with a systematic correction scheme. This is possible by the simple requirement that, after a short imaginary time propagation via the FN Hamiltonian, a number p of correlation functions can be further constrained to be exact by small perturbations of the FN state, which is free from the sign problem. By iterating this procedure, the Monte Carlo average sign, which is almost zero when there is a sign problem, remains stable and finite even for large $\ensuremath{\tau}.$ The proposed algorithm is tested against exact diagonalization data available on finite lattices. It is also shown, in some test cases, that the dependence of the results upon the few parameters entering the stochastic technique can be very easily controlled, unless for exceptional cases.
When one tries to simulate quantum spin systems by the Monte Carlo method, often the ``minus-sign problem'' is encountered. In such a case, an application of probabilistic methods is not possible. In this paper, the method has been proposed which allows to avoid the minus-sign problem for certain class of frustrated Heisenberg models. The systems where this method is applicable are, for instance, the pyrochlore lattice and the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. The method works in singlet sector. It relies on the expression of wave functions in dimer (pseudo)basis and writing down the Hamiltonian as a sum over plaquettes. In such a formulation, matrix elements of the exponent of Hamiltonian are positive.
We show that the standard Lanczos algorithm can be efficiently implemented statistically and self-consistently improved, using the stochastic reconfiguration method, which has been recently introduced to stabilize the Monte Carlo sign problem instability. With this scheme a few Lanczos steps over a given variational wave function are possible even for large size as a particular case of a more general and more accurate technique that allows to obtain lower variational energies. This method has been tested extensively for a strongly correlated model like the $t\ensuremath{-}J$ model. With the standard Lanczos technique it is possible to compute any kind of correlation functions, with no particular computational effort. By using the fact that the variance $〈{H}^{2}〉\ensuremath{-}〈H{〉}^{2}$ is zero for an exact eigenstate, we show that the approach to the exact solution with few Lanczos iterations is indeed possible even for $\ensuremath{\sim}100$ electrons for reasonably good initial wave functions. The variational stochastic reconfiguration technique presented here allows in general a many-parameter energy optimization of any computable many-body wave function, including for instance generic long-range Jastrow factors and arbitrary site-dependent orbital determinants. This scheme improves further the accuracy of the calculation, especially for long-distance correlation functions.
We describe in detail a simple and efficient Green function Monte Carlo technique for computing both the ground state energy and the ground state properties by the ``forward walking'' scheme. The simplicity of our reconfiguration process, used to maintain the walker population constant, allows us to control any source of systematic error in a rigorous and systematic way. We apply this method to the Heisenberg model and obtain accurate and reliable estimates of the ground state energy, the order parameter, and the static spin structure factor S(q)S(q)S(q) for several momenta. For the latter quantity we also find very good agreement with available experimental data on the La2CuO4{\mathrm{La}}_{2}{\mathrm{CuO}}_{4}La2​CuO4​ antiferromagnet.
Building on a recent investigation of the Shastry-Sutherland model [S. Wessel et al., Phys. Rev. B 98, 174432 (2018)], we develop a general strategy to eliminate the Monte Carlo sign problem near the zero-temperature limit in frustrated quantum spin models. If the Hamiltonian of interest and the sign-problem-free Hamiltonian, obtained by making all off-diagonal elements negative in a given basis, have the same ground state and this state is a member of the computational basis, then the average sign returns to one as the temperature goes to zero. We illustrate this technique by studying the triangular and kagome lattice Heisenberg antiferrromagnet in a magnetic field above saturation, as well as the Heisenberg antiferromagnet on a modified Husimi cactus in the dimer basis. We also provide detailed Appendices on using linear programming techniques to automatically generate efficient directed loop updates in quantum Monte Carlo simulations.
We combine the density-matrix renormalization-group (DMRG) technique with Green function Monte Carlo (GFMC) simulations using a special representation of the DMRG wave function. As a test case we apply the method to the two-dimensional frustrated Heisenberg antiferromagnet. By supplementing the branching in GFMC simulations with stochastic reconfiguration we get a stable simulation with a small variance also in the region where the fluctuations due to the minus sign problem are maximal. The sensitivity of the results to the choice of the guiding wave function is extensively investigated. In agreement with earlier calculations it is found from the DMRG wave function that for small ratios of the next-nearest-to-nearest neighbor coupling strength the system orders as a N\'eel-type antiferromagnet and for large ratios as a columnar antiferromagnet. The spin stiffness suggests an intermediate regime without magnetic long-range order. The energy curve indicates that the columnar phase is separated from the intermediate phase by a first-order transition. The combination of the DMRG and GFMC techniques allows us to substantiate this picture by calculating also the spin correlations in the system. We observe a pattern of spin correlations in the intermediate regime which is in between dimerlike and plaquette-type ordering, states that have recently been suggested. It is a state with strong dimerization in one direction and weaker dimerization in the perpendicular direction and thus it lacks the square symmetry of the plaquette state.