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/9601070

Variational Wavefunction for Quantum Antiferromagnets

Franjo Franjic, Sandro Sorella

Published Jan 18, 1996• 0 citations
arXivHTMLPDF

Abstract

We present here a new approach to determine an accurate variational wavefunction for general quantum antiferromagnets, completely defined by the requirement to reproduce the simple and well known spin-wave expansion. By this wavefunction, it is possible to obtain the correct behavior of the long distance correlation functions for the one dimensional $S=1/2$ antiferromagnet, i.e. {\em a system without long range order}. The qualitative difference between the integer and half integer case is also easily understood with this variational approach. Finally we present numerical results for the 2D XY model, showing that the present wavefunction has an overlap $> 0.99 $ to the exact ground state of the $S=1/2$ model for finite system up to $6 \times 6$ clusters.

DOI
10.48550/arxiv.cond-mat/9601070

Related papers

Spin-Wave Wave Function for Quantum Spin Models

Franjo Franjic, Sandro Sorella

We present a new approach to determine an accurate variational wave function for general quantum spin models, completely defined by a consistency requirement with the simple and well-known linear spin-wave expansion. With this wave function, it is also possible to obtain the correct behavior of the long distance correlation functions for the 1D S = 1/2 antiferromagnet. In 2D the proposed spin-wave wave function represents an excellent approximation to the exact ground state of the S = 1/2 XY model. We obtained accurate values for the correlation functions and discuss their physical relevance.

Progress of Theoretical Physics•Mar 1, 1997•36 citations

Generalized dimer states

M W Long, C A Hayward

We present a new class of variational wavefunctions that can be used to investigate quantum antiferromagnetic systems. The states are a natural generalization of the nearest-neighbour dimer phases which have been thoroughly studied in the literature. We describe a one-dimensional class of total-spin-singlet wavefunctions, which include as special cases the two nearest-neighbour dimer phases and the exact solution to the total-spin-two projector Hamiltonian. These different solutions are 'smoothly' interpolated by a range of singlets with short-range correlations. Although the states have only short-range order, we can obtain over 98% of the ground-state energy of the nearest-neighbour Heisenberg model. We apply our basis to the J 1 -J 2 Heisenberg model and the single hole in the t 1 -t 2 infinite-U Hubbard model, successfully predicting the behaviour observed in finite-size scaling calculations.

Journal of Physics Condensed Matter•Apr 4, 1994•2 citations

Spin-Wave Theory on Finite Lattices: Application to the J 1 - J 2 Heisenberg Model

Q. F. Zhong, Sandro Sorella

We present a new method for a systematic spin-wave expansion for the quantum fluctuations of a generic spin Hamiltonian in a finite lattice, where the inverse spin magnitude 1/ S is a well-defined expansion parameter. The first two leading contributions of the spin-spin correlation function are evaluated for the J 1 - J 2 Heisenberg model. Very good agreement between our finite-size predictions and the exact diagonalization and Monte Carlo results is found for J 2 / J 1 < 0.2 and S = 1/2, thus confirming the existence of antiferromagnetic long-range order in this J region. For J 2 / J 1 > 0.3 the expansion is poorly converging, suggesting a possible breakdown of the spin-wave approximation. Here our calculation seems consistent with a possible spin liquid ground state.

Europhysics Letters (EPL)•Feb 10, 1993•48 citations

Variational ground states of two-dimensional antiferromagnets in the valence bond basis

Jie Lou, Anders W. Sandvik

We study a variational wave function for the ground state of the two-dimensional $S=1∕2$ Heisenberg antiferromagnet in the valence bond basis. The expansion coefficients are products of amplitudes $h(x,y)$ for valence bonds connecting spins separated by $(x,y)$ lattice spacings. In contrast to previous studies, in which a functional form for $h(x,y)$ was assumed, we here optimize all the amplitudes for lattices with up to $32\ifmmode\times\else\texttimes\fi{}32$ spins. We use two different schemes for optimizing the amplitudes; a Newton conjugate-gradient method and a stochastic method which requires only the signs of the first derivatives of the energy. The latter method performs significantly better. The energy for large systems deviates by only $\ensuremath{\approx}0.06%$ from its exact value (calculated using unbiased quantum Monte Carlo simulations). The spin correlations are also well reproduced, falling $\ensuremath{\approx}2%$ below the exact ones at long distances (corresponding to an $\ensuremath{\approx}1%$ underestimation of the sublattice magnetization). The amplitudes $h(r)$ for valence bonds of long length $r$ decay as ${r}^{\ensuremath{-}3}$. We also discuss some results for small frustrated lattices.

arXiv:cond-mat/0605034•Physical Review B•Sep 25, 2007•39 citations

Variational cluster approach to spontaneous symmetry breaking: The itinerant antiferromagnet in two dimensions

Christopher Dahnken, Markus Aichhorn, Werner R. Hanke +2 more

Based on the self-energy-functional approach proposed recently [M. Potthoff, Eur. Phys. J. B 32, 429 (2003)], we present an extension of the cluster-perturbation theory to systems with spontaneously broken symmetry. Our method applies to models with local interactions and accounts for both short-range correlations and long-range order. Short-range correlations are accurately taken into account via exact diagonalization of finite clusters. Long-range order is described by variational optimization of a ficticious symmetry-breaking field. In comparison with related cluster methods, our approach is more flexible and, for a given cluster size, less demanding numerically, especially at zero temperature. An application of the method to the antiferromagnetic phase of the Hubbard model at half-filling shows good agreement with results from quantum Monte Carlo calculations. We demonstrate that the variational extension of the cluster-perturbation theory is crucial to reproduce salient features of the single-particle spectrum.

arXiv:cond-mat/0309407•Physical Review B•Dec 10, 2004•170 citations

Gutzwiller correlated wave functions in finite dimensionsd: A systematic expansion in 1/d

Florian Gebhard

We present a systematic formalism for the variational evaluation of ground-state properties of Hubbard-type models in finite dimensions d. The formalism starts from generalized Gutzwiller correlated wave functions, which are then studied in a systematic (1/d) expansion around the limit of high dimensions (d=\ensuremath{\infty}). The limit of d\ensuremath{\rightarrow}\ensuremath{\infty} has recently been introduced by Metzner and Vollhardt (MV) for itinerant lattice fermion systems. The approach, presented in this paper, is particularly efficient since results in d=\ensuremath{\infty} are obtained without having to calculate a single graph. Our results confirm the finding of MV that counting approximations in the spirit of the Gutzwiller approximation become exact in d=\ensuremath{\infty} for translationally invariant wave functions. This type of approximation is no longer exact for more complicated (e.g., antiferromagnetic) wave functions. In addition, we completely reproduce the results of the Kotliar-Ruckenstein path-integral approach to the Hubbard model. Performing a (1/d) expansion for the Gutzwiller wave function, we show that the lowest orders in (1/d) are sufficient to reproduce all numerical findings in d=2,3 quantitatively. We therefore conclude that the limit of d=\ensuremath{\infty} is a very fruitful starting point for the study of finite-dimensional systems. On the basis of our study we propose new variational wave functions for the numerical investigation of antiferromagnetism in the Hubbard, the t-J, and the spin-1/2 Heisenberg model. For the first two models we calculate in d=\ensuremath{\infty} only, for the Heisenberg model we also derive corrections up to order (1/d). To this order we obtain complete agreement with linear spin-wave theory. Since our trial state is based on a fermionic description of the Heisenberg model, we interpret this analytically determined wave function as Fermi sea of spin-1/2 quasiparticles (``spinons'').

Physical review. B, Condensed matter•May 1, 1990•157 citations
We present here a new approach to determine an accurate variational wavefunction for general quantum antiferromagnets, completely defined by the requirement to reproduce the simple and well known spin-wave expansion. By this wavefunction, it is possible to obtain the correct behavior of the long distance correlation functions for the one dimensional S=1/2S=1/2S=1/2 antiferromagnet, i.e. {\em a system without long range order}. The qualitative difference between the integer and half integer case is also easily understood with this variational approach. Finally we present numerical results for the 2D XY model, showing that the present wavefunction has an overlap >0.99> 0.99 >0.99 to the exact ground state of the S=1/2S=1/2S=1/2 model for finite system up to 6×66 \times 66×6 clusters.
We present a new approach to determine an accurate variational wave function for general quantum spin models, completely defined by a consistency requirement with the simple and well-known linear spin-wave expansion. With this wave function, it is also possible to obtain the correct behavior of the long distance correlation functions for the 1D S = 1/2 antiferromagnet. In 2D the proposed spin-wave wave function represents an excellent approximation to the exact ground state of the S = 1/2 XY model. We obtained accurate values for the correlation functions and discuss their physical relevance.
We present a new class of variational wavefunctions that can be used to investigate quantum antiferromagnetic systems. The states are a natural generalization of the nearest-neighbour dimer phases which have been thoroughly studied in the literature. We describe a one-dimensional class of total-spin-singlet wavefunctions, which include as special cases the two nearest-neighbour dimer phases and the exact solution to the total-spin-two projector Hamiltonian. These different solutions are 'smoothly' interpolated by a range of singlets with short-range correlations. Although the states have only short-range order, we can obtain over 98% of the ground-state energy of the nearest-neighbour Heisenberg model. We apply our basis to the J 1 -J 2 Heisenberg model and the single hole in the t 1 -t 2 infinite-U Hubbard model, successfully predicting the behaviour observed in finite-size scaling calculations.
We present a new method for a systematic spin-wave expansion for the quantum fluctuations of a generic spin Hamiltonian in a finite lattice, where the inverse spin magnitude 1/ S is a well-defined expansion parameter. The first two leading contributions of the spin-spin correlation function are evaluated for the J 1 - J 2 Heisenberg model. Very good agreement between our finite-size predictions and the exact diagonalization and Monte Carlo results is found for J 2 / J 1 < 0.2 and S = 1/2, thus confirming the existence of antiferromagnetic long-range order in this J region. For J 2 / J 1 > 0.3 the expansion is poorly converging, suggesting a possible breakdown of the spin-wave approximation. Here our calculation seems consistent with a possible spin liquid ground state.
We study a variational wave function for the ground state of the two-dimensional S=1∕2S=1∕2S=1∕2 Heisenberg antiferromagnet in the valence bond basis. The expansion coefficients are products of amplitudes h(x,y)h(x,y)h(x,y) for valence bonds connecting spins separated by (x,y)(x,y)(x,y) lattice spacings. In contrast to previous studies, in which a functional form for h(x,y)h(x,y)h(x,y) was assumed, we here optimize all the amplitudes for lattices with up to $32\ifmmode\times\else\texttimes\fi{}32$ spins. We use two different schemes for optimizing the amplitudes; a Newton conjugate-gradient method and a stochastic method which requires only the signs of the first derivatives of the energy. The latter method performs significantly better. The energy for large systems deviates by only $\ensuremath{\approx}0.06%$ from its exact value (calculated using unbiased quantum Monte Carlo simulations). The spin correlations are also well reproduced, falling $\ensuremath{\approx}2%$ below the exact ones at long distances (corresponding to an $\ensuremath{\approx}1%$ underestimation of the sublattice magnetization). The amplitudes h(r)h(r)h(r) for valence bonds of long length rrr decay as ${r}^{\ensuremath{-}3}$. We also discuss some results for small frustrated lattices.
Based on the self-energy-functional approach proposed recently [M. Potthoff, Eur. Phys. J. B 32, 429 (2003)], we present an extension of the cluster-perturbation theory to systems with spontaneously broken symmetry. Our method applies to models with local interactions and accounts for both short-range correlations and long-range order. Short-range correlations are accurately taken into account via exact diagonalization of finite clusters. Long-range order is described by variational optimization of a ficticious symmetry-breaking field. In comparison with related cluster methods, our approach is more flexible and, for a given cluster size, less demanding numerically, especially at zero temperature. An application of the method to the antiferromagnetic phase of the Hubbard model at half-filling shows good agreement with results from quantum Monte Carlo calculations. We demonstrate that the variational extension of the cluster-perturbation theory is crucial to reproduce salient features of the single-particle spectrum.
We present a systematic formalism for the variational evaluation of ground-state properties of Hubbard-type models in finite dimensions d. The formalism starts from generalized Gutzwiller correlated wave functions, which are then studied in a systematic (1/d) expansion around the limit of high dimensions (d=\ensuremath{\infty}). The limit of d\ensuremath{\rightarrow}\ensuremath{\infty} has recently been introduced by Metzner and Vollhardt (MV) for itinerant lattice fermion systems. The approach, presented in this paper, is particularly efficient since results in d=\ensuremath{\infty} are obtained without having to calculate a single graph. Our results confirm the finding of MV that counting approximations in the spirit of the Gutzwiller approximation become exact in d=\ensuremath{\infty} for translationally invariant wave functions. This type of approximation is no longer exact for more complicated (e.g., antiferromagnetic) wave functions. In addition, we completely reproduce the results of the Kotliar-Ruckenstein path-integral approach to the Hubbard model. Performing a (1/d) expansion for the Gutzwiller wave function, we show that the lowest orders in (1/d) are sufficient to reproduce all numerical findings in d=2,3 quantitatively. We therefore conclude that the limit of d=\ensuremath{\infty} is a very fruitful starting point for the study of finite-dimensional systems. On the basis of our study we propose new variational wave functions for the numerical investigation of antiferromagnetism in the Hubbard, the t-J, and the spin-1/2 Heisenberg model. For the first two models we calculate in d=\ensuremath{\infty} only, for the Heisenberg model we also derive corrections up to order (1/d). To this order we obtain complete agreement with linear spin-wave theory. Since our trial state is based on a fermionic description of the Heisenberg model, we interpret this analytically determined wave function as Fermi sea of spin-1/2 quasiparticles (``spinons'').