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arXiv:cond-mat/0009149 · Physical review. B, Condensed matter

Generalized Lanczos algorithm for variational quantum Monte Carlo

Sandro Sorella

Published Jun 19, 2001• 368 citations
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Abstract

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.

DOI
10.1103/physrevb.64.024512

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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−Jt\ensuremath{-}Jt\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 〈H2〉\ensuremath−〈H〉2〈{H}^{2}〉\ensuremath{-}〈H{〉}^{2}〈H2〉\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∼100\ensuremath{\sim}100\ensuremath∼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 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.
Variational wave functions used in the variational Monte Carlo (VMC) method are extensively improved to overcome the biases coming from the assumed variational form of the wave functions. We construct a highly generalized variational form by introducing a large number of variational parameters to the Gutzwiller-Jastrow factor as well as to the one-body part. Moreover, the projection operator to restore the symmetry of the wave function is introduced. These improvements enable to treat fluctuations with long-ranged as well as short-ranged correlations. A highly generalized wave function is implemented by the Pfaffians introduced by Bouchaud et al., together with the stochastic reconfiguration method introduced by Sorella for the parameter optimization. Our framework offers much higher accuracy for strongly correlated electron systems than the conventional variational Monte Carlo methods.
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.
We present a technique for optimizing hundreds of thousands of variational parameters in variational quantum Monte Carlo. By introducing iterative Krylov subspace solvers and by multiplying by the Hamiltonian and overlap matrices as they are sampled, we remove the need to construct and store these matrices and thus bypass the most expensive steps of the stochastic reconfiguration and linear method optimization techniques. We demonstrate the effectiveness of this approach by using stochastic reconfiguration to optimize a correlator product state wave function with a Pfaffian reference for four example systems. In two examples on the two dimensional Fermionic Hubbard model, we study 16 and 64 site lattices, recovering energies accurate to 1% in the smaller lattice and predicting particle-hole phase separation in the larger. In two examples involving an ab initio Hamiltonian, we investigate the potential energy curve of a symmetrically dissociated $4\ifmmode\times\else\texttimes\fi{}4$ hydrogen lattice as well as the singlet-triplet gap in free base porphin. In the hydrogen system we recover 98% or more of the correlation energy at all geometries, while for porphin we compute the gap in a 24 orbital active space to within 0.02 eV of the exact result. The number of variational parameters in these examples ranges from $4\ifmmode\times\else\texttimes\fi{}{10}^{3}$ to $5\ifmmode\times\else\texttimes\fi{}{10}^{5}$.
Gutzwiller-projected fermionic states can be efficiently implemented within quantum Monte Carlo calculations to define extremely accurate variational wave functions for Heisenberg models on frustrated two-dimensional lattices, not only for the ground state but also for low-energy excitations. The application of few Lanczos steps on top of these states further improves their accuracy, allowing calculations on large clusters. In addition, by computing both the energy and its variance, it is possible to obtain reliable estimations of exact results. Here, we report the cases of the frustrated Heisenberg models on square and Kagome lattices.
A scheme of optimization of a trial wave function in the variational Monte Carlo calculation is developed. It provides a more precise description of the ground state for strongly correlated electronic systems with highly improved trial wave function. A general type of variational wave function constructed from a linear combination of Slater determinants multiplied by Jastrow correlation factors is proposed. They can explicitly contain low-energy fluctuations and long-range correlations. Using this optimization method, we examine the properties of these wave functions by applying them to a one-dimensional Hubbard model for which the exact solution is available for comparison. It is found that these wave functions prove to be substantial improvements compared to the conventional Jastrow-type wave functions.