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arXiv:1411.5198 · Physical Review B

Variational Monte Carlo method in the presence of spin-orbit interaction and its application to Kitaev and Kitaev-Heisenberg models

Moyuru Kurita, Youhei Yamaji, Satoshi Morita, Masatoshi Imada

Published Jul 10, 2015• 14 citations
arXivHTMLPDF

Abstract

We propose an accurate variational Monte Carlo method applicable in the presence of the strong spin-orbit interactions. The algorithm is applicable even in a wider class of Hamiltonians that do not have the spin-rotational symmetry. Our variational wave functions consist of generalized Pfaffian-Slater wave functions that involve mixtures of singlet and triplet Cooper pairs, Jastrow-Gutzwiller-type projections, and quantum number projections. The generalized wave functions allow describing states including a wide class of symmetry-broken states, ranging from magnetic and/or charge ordered states to superconducting states and their fluctuations, on equal footing without any ad hoc ansatz for variational wave functions. We detail our optimization scheme for the generalized Pfaffian-Slater wave functions with complex-number variational parameters. Generalized quantum number projections are also introduced, which imposes the conservation of not only the momentum quantum number but also Wilson loops. As a demonstration of the capability of the present variational Monte Carlo method, the accuracy and efficiency is tested for the Kitaev and Kitaev-Heisenberg models, which lack the SU(2) spin-rotational symmetry except at the Heisenberg limit. The Kitaev model serves as a critical benchmark of the present method: The exact ground state of the model is a typical gapless quantum spin liquid far beyond the reach of simple mean-field wave functions. The newly introduced quantum number projections precisely reproduce the ground state degeneracy of the Kitaev spin liquids, in addition to their ground state energy. An application to a closely related itinerant model described by a multiorbital Hubbard model with the spin-orbit interaction also shows promising benchmark results. The strong-coupling limit of the multiorbital Hubbard model is indeed described by the Kitaev model. Our framework offers accurate solutions for the systems where strong electron correlation and spin-orbit interaction coexist.

DOI
10.1103/physrevb.92.035122

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We propose an accurate variational Monte Carlo method applicable in the presence of the strong spin-orbit interactions. The algorithm is applicable even in a wider class of Hamiltonians that do not have the spin-rotational symmetry. Our variational wave functions consist of generalized Pfaffian-Slater wave functions that involve mixtures of singlet and triplet Cooper pairs, Jastrow-Gutzwiller-type projections, and quantum number projections. The generalized wave functions allow describing states including a wide class of symmetry-broken states, ranging from magnetic and/or charge ordered states to superconducting states and their fluctuations, on equal footing without any ad hoc ansatz for variational wave functions. We detail our optimization scheme for the generalized Pfaffian-Slater wave functions with complex-number variational parameters. Generalized quantum number projections are also introduced, which imposes the conservation of not only the momentum quantum number but also Wilson loops. As a demonstration of the capability of the present variational Monte Carlo method, the accuracy and efficiency is tested for the Kitaev and Kitaev-Heisenberg models, which lack the SU(2) spin-rotational symmetry except at the Heisenberg limit. The Kitaev model serves as a critical benchmark of the present method: The exact ground state of the model is a typical gapless quantum spin liquid far beyond the reach of simple mean-field wave functions. The newly introduced quantum number projections precisely reproduce the ground state degeneracy of the Kitaev spin liquids, in addition to their ground state energy. An application to a closely related itinerant model described by a multiorbital Hubbard model with the spin-orbit interaction also shows promising benchmark results. The strong-coupling limit of the multiorbital Hubbard model is indeed described by the Kitaev model. Our framework offers accurate solutions for the systems where strong electron correlation and spin-orbit interaction coexist.
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.
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We extend our low-scaling variational Monte Carlo (VMC) algorithm to optimize the symmetry-projected Jastrow mean-field (SJMF) wave functions. These wave functions consist of a symmetry-projected product of a Jastrow and a general broken-symmetry mean-field reference. Examples include Jastrow antisymmetrized geminal power, Jastrow-Pfaffians, and resonating valence bond states, among others, all of which can be treated with our algorithm. We will demonstrate using benchmark systems including the nitrogen molecule, a chain of hydrogen atoms, and 2-D Hubbard model that a significant amount of correlation can be obtained by optimizing the energy of the SJMF wave function. This can be achieved at a relatively small cost when multiple symmetries including spin, particle number, and complex conjugation are simultaneously broken and projected. We also show that reduced density matrices can be calculated using the optimized wave functions, which allows us to calculate other observables such as correlation functions and will enable us to embed the VMC algorithm in a complete active-space self-consistent field calculation.
Variational quantum Monte Carlo (VMC) method applied to interacting electron systems has played a pivotal role for the studies of electronic states of atoms, molecules and condensed matters. First, the many-body wave functions optimized by the VMC method provide good guiding functions in more accurate, diffusion quantum Monte Carlo (DMC) calculations for identical systems. Second, the VMC calculations themselves can give a good upper bound for the exact ground-state energy of many-electron systems when an appropriate form for the variational wave function is employed. In addition to a conventional form for the variational wave function that is expressed by the product of the Jastrow factor and the single Slater determinant, recent investigations have found that the expression in terms of many Slater determinants can provide a very accurate description of electronic states, especially for intrinsically multi-reference systems, which corresponds to the configuration interaction (CI) expansion method in conventional quantum chemistry. Although the CI method needs an extremely large number of expansion coefficients as variational parameters, the number of parameters can be reduced efficiently by using an idea of (molecular) orbital pair correlations. The purpose of the present paper is to examine the feasibility and usefulness of this idea within the framework of VMC method with the Jastrow–Slater wave function, where the ground state of a water molecule is investigated as a test case. In this study we rely on a variational wave function composed of a product of the Jastrow and Slater parts as 1⁄4 J S; ð1Þ
The conventional tensor-network states employ real-space product states as reference wave functions. Here, we propose a many-variable variational Monte Carlo (mVMC) method combined with tensor networks by taking advantages of both to study fermionic models. The variational wave function is composed of a pair product wave function operated by real-space correlation factors and tensor networks. Moreover, we can apply quantum number projections, such as spin, momentum, and lattice symmetry projections, to recover the symmetry of the wave function to further improve the accuracy. We benchmark our method for one- and two-dimensional Hubbard models, which show significant improvement over the results obtained individually either by mVMC or by tensor network. We have applied the present method to a hole-doped Hubbard model on the square lattice, which indicates the stripe charge/spin order coexisting with a weak ddd-wave superconducting order in the ground state for the doping concentration of less than 0.3, where the stripe oscillation period gets longer with increasing hole concentration. The charge homogeneous and highly superconducting state also exists as a metastable excited state for the doping concentration less than 0.25.
We develop a time-dependent variational Monte Carlo (t-VMC) method for quantum dynamics of strongly correlated electrons. The t-VMC method has been recently applied to bosonic systems and quantum spin systems. Here we propose a time-dependent trial wave function with many variational parameters, which is suitable for nonequilibrium strongly correlated electron systems. As the trial state, we adopt the generalized pair-product wave function with correlation factors and quantum-number projections. This trial wave function has been proven to accurately describe ground states of strongly correlated electron systems. To show the accuracy and efficiency of our trial wave function in nonequilibrium states as well, we present our benchmark results for relaxation dynamics during and after interaction quench protocols of fermionic Hubbard models. We find that our trial wave function well reproduces the exact results for the time evolution of physical quantities such as energy, momentum distribution, spin structure factor, and superconducting correlations. These results show that the t-VMC with our trial wave function offers an efficient and accurate way to study challenging problems of nonequilibrium dynamics in strongly correlated electron systems.