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arXiv:1403.4150 · Physical Review B

Variational Monte Carlo method for electron-phonon coupled systems

Takahiro Ohgoe, Masatoshi Imada

Published May 29, 2014• 28 citations
arXivHTMLPDF

Abstract

We develop a variational Monte Carlo (VMC) method for electron-phonon coupled systems. The VMC method has been extensively used for investigating strongly correlated electrons over the last decades. However, its applications to electron-phonon coupled systems have been severely restricted because of its large Hilbert space. Here, we propose a variational wave function with a large number of variational parameters, which is suitable and tractable for systems with electron-phonon coupling. In the proposed wave function, we implement an unexplored electron-phonon correlation factor, which takes into account the effect of the entanglement between electrons and phonons. The method is applied to systems with diagonal electron-phonon interactions, i.e., interactions between charge densities and lattice displacements (phonons). As benchmarks, we compare VMC results with previous results obtained by the exact diagonalization, the Green function Monte Carlo method and the density matrix renormalization group for the Holstein and Holstein-Hubbard model. From these benchmarks, we show that the present method offers an efficient way to treat strongly coupled electron-phonon systems.

DOI
10.1103/physrevb.89.195139

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We develop a variational Monte Carlo (VMC) method for electron-phonon coupled systems. The VMC method has been extensively used for investigating strongly correlated electrons over the last decades. However, its applications to electron-phonon coupled systems have been severely restricted because of its large Hilbert space. Here, we propose a variational wave function with a large number of variational parameters, which is suitable and tractable for systems with electron-phonon coupling. In the proposed wave function, we implement an unexplored electron-phonon correlation factor, which takes into account the effect of the entanglement between electrons and phonons. The method is applied to systems with diagonal electron-phonon interactions, i.e., interactions between charge densities and lattice displacements (phonons). As benchmarks, we compare VMC results with previous results obtained by the exact diagonalization, the Green function Monte Carlo method and the density matrix renormalization group for the Holstein and Holstein-Hubbard model. From these benchmarks, we show that the present method offers an efficient way to treat strongly coupled electron-phonon systems.
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.
Abstract Interactions between electrons and phonons play a crucial role in quantum materials. Yet, there is no universal method that would simultaneously accurately account for strong electron-phonon interactions and electronic correlations. By combining methods of the variational quantum eigensolver and the variational non-Gaussian solver, we develop a hybrid quantum-classical algorithm suitable for this type of correlated systems. This hybrid method tackles systems with arbitrarily strong electron-phonon coupling without increasing the number of required qubits and quantum gates, as compared to purely electronic models. We benchmark our method by applying it to the paradigmatic Hubbard-Holstein model at half filling, and show that it correctly captures the competition between charge density wave and antiferromagnetic phases, quantitatively consistent with exact diagonalization.
Interactions between electrons and phonons play a crucial role in quantum materials. Yet, there is no universal method that would simultaneously accurately account for strong electron-phonon interactions and electronic correlations. By combining methods of the variational quantum eigensolver and the variational non-Gaussian solver, we develop a hybrid quantum-classical algorithm suitable for this type of correlated systems. This hybrid method tackles systems with arbitrarily strong electron-phonon coupling without increasing the number of required qubits and quantum gates, as compared to purely electronic models. We benchmark the new method by applying it to the paradigmatic Hubbard-Holstein model at half filling, and show that it correctly captures the competition between charge density wave and antiferromagnetic phases, quantitatively consistent with exact diagonalization.
This thesis investigates the interplay between electron-electron and electron-phonon interactions in low-dimensional lattice models, specifically within the Hubbard model incorporating Su-Schrieffer-Heeger (SSH) coupling with phonons. SSH interactions, which modulate the electron hopping amplitude through lattice displacements, have recently attracted renewed interest due to their potential relevance in strongly correlated materials. The main objective of this work is to determine whether the inclusion of SSH couplings in the Hubbard model can stabilize superconducting or spin-gapped metallic states, as well as other exotic phases of matter. While the half-filled SSH-Hubbard model has been extensively studied in one dimension and only partially examined in two dimensions, its phase diagram in the doped regime remains largely unexplored. This thesis aims to fill this gap, focusing on the emergence of correlated phases away from half-filling in both one and two spatial dimensions. We use the variational Monte Carlo method with a trial wave function defined in a hybrid electron-phonon Hilbert space. In addition to Jastrow factors that account for correlation effects, the wave function includes an antisymmetric component that captures backflow-inspired electron-phonon correlations. This approach enables us to accurately model both electron-electron and electron-phonon interactions. Moreover, since the variational method avoids the sign problem, we can simulate the system in the doped regime, an area typically inaccessible to exact quantum Monte Carlo methods (QMC), which are limited to half-filling. In one dimension, we observe the stabilization of a Luther-Emery liquid, a metallic phase with gapless charge excitations and a finite spin gap, upon doping a Peierls insulating phase. This finding resolves a long-standing debate regarding its realization in SSH chains. Benchmarking with density-matrix renormalization group results confirms the reliability of our approach. In two dimensions, we investigate the SSH-Hubbard model on the square lattice, obtaining results that are consistent with previous exact QMC studies at half-filling. Upon doping, we observe that SSH phonons can induce s-wave superconductivity in the weakly correlated regime. However, we suggest that the enhancement of d-wave superconductivity by SSH coupling in the strongly correlated regime may significantly depend on the specific implementation of the SSH interaction. In one particular case, we find that d-wave superconductivity is slightly suppressed by SSH coupling. These results are further confirmed by a Hartree-Fock analysis performed in the anti-adiabatic limit. Overall, this thesis highlights the crucial role of lattice effects in correlated electron systems, shedding light on the distinct contributions of SSH-type couplings in one and two dimensions. Our findings provide new insights into the microscopic mechanisms behind exotic metallic phases and set the stage for further studies of phonon-mediated phenomena in correlated systems.
An optimized phonon approach for the numerical diagonalization of interacting electron-phonon systems is proposed. The variational method is based on an expansion in coherent states that leads to a dramatic truncation in the phonon space. The reliability of the approach is demonstrated for the extended Holstein model showing that different types of lattice distortions are present at intermediate electron-phonon couplings as observed in strongly correlated systems. The connection with the density matrix renormalization group is discussed.
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.