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/0002116 · Physical review. B, Condensed matter

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

Published Dec 1, 2000• 45 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.62.14844

Related papers

Density Matrix Renormalisation Group Variants for Spin Systems

De Jongh, M. S. L. du Croo de Jongh

This thesis gives an extension for the Density Matrix Renormalisation Group (DMRG) to two dimensions and described a newly developed combination of the DMRG and a Green Function Monte Carlo simulation (GFMC). The first two chapters focus on the DMRG method. The properties are reviewed and it is shown that good quality results can be obtained for two-dimensional systems through finite-size scaling. In the third and fourth chapter the properties of the two-dimensional frustrated Heisenberg model are analysed using a similar combination of DMRG and finite-size scaling. The final chapter presents the combination of the DMRG and GFMC with Stochastic Reconfiguration (Sorella, Capriotti cond-mat/9902211) We find an intermediate phase between Neel and collinear long-range order. This phase shows spatial inhomogeneities with a tendency towards plaquette formation.

•Aug 13, 1999•0 citations

Density Matrix Renormalisation Group Variants for Spin Systems

M. S. L. du Croo de Jongh

This thesis gives an extension for the Density Matrix Renormalisation Group (DMRG) to two dimensions and described a newly developed combination of the DMRG and a Green Function Monte Carlo simulation (GFMC). The first two chapters focus on the DMRG method. The properties are reviewed and it is shown that good quality results can be obtained for two-dimensional systems through finite-size scaling. In the third and fourth chapter the properties of the two-dimensional frustrated Heisenberg model are analysed using a similar combination of DMRG and finite-size scaling. The final chapter presents the combination of the DMRG and GFMC with Stochastic Reconfiguration (Sorella, Capriotti cond-mat/9902211) We find an intermediate phase between Neel and collinear long-range order. This phase shows spatial inhomogeneities with a tendency towards plaquette formation.

arXiv:cond-mat/9908200•Aug 13, 1999•0 citations

Green's function Monte Carlo combined with projected entangled pair state approach to the frustrated J 1 − J 2 Heisenberg model

He-Yu Lin, Yibin Guo, Rong-Qiang He +2 more

The tensor network algorithm, a family of prevalent numerical methods for quantum many-body problems, aptly captures the entanglement properties intrinsic to quantum systems, enabling precise representation of quantum states. However, its computational cost is notably high, particularly in calculating physical observables like correlation functions. To surmount the computational challenge and enhance efficiency, we propose integrating the Green's function Monte Carlo (GFMC) method with the projected entangled pair state (PEPS) ansatz. This approach combines the high-efficiency characteristics of Monte Carlo with the sign-free nature of tensor network states and proves effective in addressing the computational bottleneck. To showcase its prowess, we apply this hybrid approach to investigate the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice, a model notorious for its sign problem in quantum Monte Carlo simulations. Our results reveal a substantial improvement in the accuracy of ground-state energy when utilizing a preliminary PEPS as the guiding wave function for GFMC. By calculating the structure factor and spin-spin correlation functions, we further characterize the phase diagram, identifying a possible columnar valence-bond state phase within the intermediate parameter range of $0.52<{J}_{2}/{J}_{1}<0.58$. This comprehensive study underscores the efficacy of our combined approach, demonstrating its ability to accurately simulate frustrated quantum spin systems while ensuring computational efficiency.

arXiv:2406.12207•Physical review. B./Physical review. B•Jun 17, 2024•6 citations

Ground-state correlations of quantum antiferromagnets: A Green-function Monte Carlo study

Nandini Trivedi, David M. Ceperley

We have studied via a Green-function Monte Carlo (GFMC) method the S=(1/2 Heisenberg quantum antiferromagnet in two dimensions. We use a well-known transformation to map the spin problem onto a system of hard-core bosons that allows us to exploit interesting analogies between magnetism and superfluidity. The GFMC method is a zero-temperature stochastic method that projects out the component of the true ground state in a given variational wave function. This method is complementary to previously used finite-temperature Monte Carlo methods and is well suited to studying the ground state and low-lying excited states. Starting with even a simple wave function, e.g., the classical N\'eel state, the GFMC method can obtain the short-range correlations very accurately, and we find the ground-state energy per site ${\mathit{E}}_{0}$/J=-0.6692(2). We show that it is important to include the zero-point motion of the elementary excitations in the ground state and by a spin-wave analysis find that it produces long-range correlations in the wave function. Upon inclusion of such long-range correlations, we obtain a staggered magnetization ${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$=0.31(2) and the structure factor scrS(q)\ensuremath{\sim}q at long wavelengths. Using the Feynman-Bijl relation, from the slope we deduce the renormalization of the spin-wave velocity by quantum fluctuations to be ${\mathit{Z}}_{\mathit{c}}$==c/${\mathit{c}}_{\mathit{s}}$=1.14(5).

Physical review. B, Condensed matter•Mar 1, 1990•238 citations

Phase diagrams of two-dimensional frustrated spin systems

Ansgar Kalz

In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.

•Jan 1, 2012•0 citations

Green's function Monte Carlo combined with projected entangled pair state approach to the frustrated J1-J2 Heisenberg model

Lin, He-Yu, Yibin Guo, Rong-Qiang He +2 more

The tensor network algorithm, a family of prevalent numerical methods for quantum many-body problems, aptly captures the entanglement properties intrinsic to quantum systems, enabling precise representation of quantum states. However, its computational cost is notably high, particularly in calculating physical observables like correlation functions. To surmount the computational challenge and enhance efficiency, we propose integrating the Green's function Monte Carlo (GFMC) method with the projected entangled pair state (PEPS) ansatz. This approach combines the high-efficiency characteristics of Monte Carlo with the sign-free nature of tensor network states and proves effective in addressing the computational bottleneck. To showcase its prowess, we apply this hybrid approach to investigate the antiferromagnetic J1-J2 Heisenberg model on the square lattice, a model notorious for its sign problem in quantum Monte Carlo simulations. Our results reveal a substantial improvement in the accuracy of ground-state energy when utilizing a preliminary PEPS as the guiding wave function for GFMC. By calculating the structure factor and spin-spin correlation functions, we further characterize the phase diagram, identifying a possible columnar valence-bond state phase within the intermediate parameter range of 0.52<J2/J1<0.58. This comprehensive study underscores the efficacy of our combined approach, demonstrating its ability to accurately simulate frustrated quantum spin systems while ensuring computational efficiency.

•Jan 1, 2024•1 citation
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.
This thesis gives an extension for the Density Matrix Renormalisation Group (DMRG) to two dimensions and described a newly developed combination of the DMRG and a Green Function Monte Carlo simulation (GFMC). The first two chapters focus on the DMRG method. The properties are reviewed and it is shown that good quality results can be obtained for two-dimensional systems through finite-size scaling. In the third and fourth chapter the properties of the two-dimensional frustrated Heisenberg model are analysed using a similar combination of DMRG and finite-size scaling. The final chapter presents the combination of the DMRG and GFMC with Stochastic Reconfiguration (Sorella, Capriotti cond-mat/9902211) We find an intermediate phase between Neel and collinear long-range order. This phase shows spatial inhomogeneities with a tendency towards plaquette formation.
This thesis gives an extension for the Density Matrix Renormalisation Group (DMRG) to two dimensions and described a newly developed combination of the DMRG and a Green Function Monte Carlo simulation (GFMC). The first two chapters focus on the DMRG method. The properties are reviewed and it is shown that good quality results can be obtained for two-dimensional systems through finite-size scaling. In the third and fourth chapter the properties of the two-dimensional frustrated Heisenberg model are analysed using a similar combination of DMRG and finite-size scaling. The final chapter presents the combination of the DMRG and GFMC with Stochastic Reconfiguration (Sorella, Capriotti cond-mat/9902211) We find an intermediate phase between Neel and collinear long-range order. This phase shows spatial inhomogeneities with a tendency towards plaquette formation.
The tensor network algorithm, a family of prevalent numerical methods for quantum many-body problems, aptly captures the entanglement properties intrinsic to quantum systems, enabling precise representation of quantum states. However, its computational cost is notably high, particularly in calculating physical observables like correlation functions. To surmount the computational challenge and enhance efficiency, we propose integrating the Green's function Monte Carlo (GFMC) method with the projected entangled pair state (PEPS) ansatz. This approach combines the high-efficiency characteristics of Monte Carlo with the sign-free nature of tensor network states and proves effective in addressing the computational bottleneck. To showcase its prowess, we apply this hybrid approach to investigate the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice, a model notorious for its sign problem in quantum Monte Carlo simulations. Our results reveal a substantial improvement in the accuracy of ground-state energy when utilizing a preliminary PEPS as the guiding wave function for GFMC. By calculating the structure factor and spin-spin correlation functions, we further characterize the phase diagram, identifying a possible columnar valence-bond state phase within the intermediate parameter range of 0.52<J2/J1<0.580.52<{J}_{2}/{J}_{1}<0.580.52<J2​/J1​<0.58. This comprehensive study underscores the efficacy of our combined approach, demonstrating its ability to accurately simulate frustrated quantum spin systems while ensuring computational efficiency.
We have studied via a Green-function Monte Carlo (GFMC) method the S=(1/2 Heisenberg quantum antiferromagnet in two dimensions. We use a well-known transformation to map the spin problem onto a system of hard-core bosons that allows us to exploit interesting analogies between magnetism and superfluidity. The GFMC method is a zero-temperature stochastic method that projects out the component of the true ground state in a given variational wave function. This method is complementary to previously used finite-temperature Monte Carlo methods and is well suited to studying the ground state and low-lying excited states. Starting with even a simple wave function, e.g., the classical N\'eel state, the GFMC method can obtain the short-range correlations very accurately, and we find the ground-state energy per site E0{\mathit{E}}_{0}E0​/J=-0.6692(2). We show that it is important to include the zero-point motion of the elementary excitations in the ground state and by a spin-wave analysis find that it produces long-range correlations in the wave function. Upon inclusion of such long-range correlations, we obtain a staggered magnetization ${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$=0.31(2) and the structure factor scrS(q)\ensuremath{\sim}q at long wavelengths. Using the Feynman-Bijl relation, from the slope we deduce the renormalization of the spin-wave velocity by quantum fluctuations to be Zc{\mathit{Z}}_{\mathit{c}}Zc​==c/cs{\mathit{c}}_{\mathit{s}}cs​=1.14(5).
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
The tensor network algorithm, a family of prevalent numerical methods for quantum many-body problems, aptly captures the entanglement properties intrinsic to quantum systems, enabling precise representation of quantum states. However, its computational cost is notably high, particularly in calculating physical observables like correlation functions. To surmount the computational challenge and enhance efficiency, we propose integrating the Green's function Monte Carlo (GFMC) method with the projected entangled pair state (PEPS) ansatz. This approach combines the high-efficiency characteristics of Monte Carlo with the sign-free nature of tensor network states and proves effective in addressing the computational bottleneck. To showcase its prowess, we apply this hybrid approach to investigate the antiferromagnetic J1-J2 Heisenberg model on the square lattice, a model notorious for its sign problem in quantum Monte Carlo simulations. Our results reveal a substantial improvement in the accuracy of ground-state energy when utilizing a preliminary PEPS as the guiding wave function for GFMC. By calculating the structure factor and spin-spin correlation functions, we further characterize the phase diagram, identifying a possible columnar valence-bond state phase within the intermediate parameter range of 0.52<J2/J1<0.58. This comprehensive study underscores the efficacy of our combined approach, demonstrating its ability to accurately simulate frustrated quantum spin systems while ensuring computational efficiency.