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arXiv:2101.07045

The phase diagram of the Hubbard model by Variational Auxiliary Field quantum Monte Carlo

Sandro Sorella

Published Jan 18, 2021• 27 citations
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Abstract

A systematically improvable wave function is proposed for the numerical solution of strongly correlated systems. With a stochastic optimization method, based on the auxiliary field quantum Monte Carlo technique, an effective temperature Teff is defined, probing the distance of the ground state properties of the model in the thermodynamic limit from the ones of the proposed correlated mean-field ansatz. In this way their uncertainties from the unbiased zero temperature limit may be estimated by simple and stable extrapolations well before the so called sign problem gets prohibitive. At finite Teff the convergence of the energy to the thermodynamic limit is indeed shown to be possible in the Hubbard model already for relatively small square lattices with linear dimension L ~10, thanks to appropriate averages over several twisted boundary conditions. Within the estimated energy accuracy of the proposed variational ansatz, two clear phases are identified, as the energy is lowered by spontaneously breaking some symmetries satisfied by the Hubbard Hamiltonian: a) a stripe phase where both spin and translation symmetries are broken, and b) a strong coupling d-wave superconducting phase when the particle number is not conserved and global U(1) symmetry is broken. On the other hand the symmetric phase is stable in a wide region at large doping and small coupling.

DOI
10.48550/arxiv.2101.07045

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A systematically improvable wave function is proposed for the numerical solution of strongly correlated systems. With a stochastic optimization method, based on the auxiliary field quantum Monte Carlo technique, an effective temperature Teff is defined, probing the distance of the ground state properties of the model in the thermodynamic limit from the ones of the proposed correlated mean-field ansatz. In this way their uncertainties from the unbiased zero temperature limit may be estimated by simple and stable extrapolations well before the so called sign problem gets prohibitive. At finite Teff the convergence of the energy to the thermodynamic limit is indeed shown to be possible in the Hubbard model already for relatively small square lattices with linear dimension L ~10, thanks to appropriate averages over several twisted boundary conditions. Within the estimated energy accuracy of the proposed variational ansatz, two clear phases are identified, as the energy is lowered by spontaneously breaking some symmetries satisfied by the Hubbard Hamiltonian: a) a stripe phase where both spin and translation symmetries are broken, and b) a strong coupling d-wave superconducting phase when the particle number is not conserved and global U(1) symmetry is broken. On the other hand the symmetric phase is stable in a wide region at large doping and small coupling.
A systematically improvable wave function is proposed for the numerical solution of strongly correlated systems. With a stochastic optimization method, based on the auxiliary field quantum Monte Carlo technique, an effective temperature Teff{T}_{\mathrm{eff}}Teff​ is defined, probing the distance of the ground-state properties of the model in the thermodynamic limit from the ones of the proposed correlated mean-field ansatz. In this way, their uncertainties from the unbiased zero temperature limit may be estimated by simple and stable extrapolations well before the so-called sign problem gets prohibitive. At finite Teff{T}_{\mathrm{eff}}Teff​, the convergence of the energy to the thermodynamic limit is indeed shown to already be possible in the Hubbard model for relatively small square lattices with linear dimension $L\ensuremath{\simeq}10$, thanks to appropriate averages over several twisted boundary conditions. Within the estimated energy accuracy of the proposed variational ansatz, two clear phases are identified, as the energy is lowered by spontaneously breaking some symmetries satisfied by the Hubbard Hamiltonian: (a) a stripe phase where both spin and translation symmetries are broken and (b) a strong coupling ddd-wave superconducting phase when the particle number is not conserved and global U(1)U(1)U(1) symmetry is broken. On the other hand, the symmetric phase is stable in a wide region at large doping and small coupling.
Understanding the properties of strongly correlated materials is a tough work since most of the times analytical tools cannot be applied due to the complicated nature of the problem. Here we apply quantum Monte Carlo techniques in order to tackle with one of the most important and unknown phenomena of many materials: the mechanism of superconductivity. We study the Hubbard-Holstein model which is one of the most simple theoretical models including strong correlation and electron-phonon coupling. By using twist-averaging boundary conditions we eliminate finite-size errors in the Monte Carlo simulations. With this useful tool we first solve the Hubbard-Holstein model by variational Monte Carlo by reporting the phase diagram of the model for different phonon frequencies at half filling. Then we investigate the phase separation away from half filling. Finally we try to attack the problem with an essentially unbiased method based on the auxiliary field quantum Monte Carlo technique and accelerated Langevin dynamics. By curing the sign problem via Cauchy integration in the complex plane where the auxiliary fields have been defined, we report the effect of the electron-phonon coupling on some observables such as magnetization, spin and charge structure factors. We show preliminary results that are already meaningful to understand the nature of the transition between magnetism and charge order in the model at half filling.
Ground-state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization, and momentum distribution are calculated for interaction strengths of U/tU/tU/t from 2 to 8, for a range of densities including half-filling and n=0.3,0.5,0.6,0.75n=0.3,0.5,0.6,\phantom{\rule{0.16em}{0ex}}0.75n=0.3,0.5,0.6,0.75, and 0.8750.8750.875. At half-filling, the results are numerically exact. Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem. Our results are obtained with several advances in the computational algorithm, which are described in detail. We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions, as well as the interplay with different forms of Hubbard-Stratonovich decompositions. We study the use of different twist angle sets when applying the twist averaged boundary conditions. We propose the use of quasirandom sequences, which improves the convergence to the thermodynamic limit over pseudorandom and other sequences. With it and a careful finite size scaling analysis, we are able to obtain accurate values of ground-state properties in the thermodynamic limit. Detailed results for finite-sized systems up to $16\ifmmode\times\else\texttimes\fi{}16$ are also provided for benchmark purposes.
Superconductivity in the ttt-t′t't′ Hubbard Model from Symmetry-Preserving Neural-Network Quantum States
Despite its fundamental importance in the theory of strongly correlated electrons, the nature of the ground state of the two-dimensional doped Hubbard model remains intensely debated. Variational approaches provide a powerful route to this problem, but their conclusions can depend sensitively on the chosen wave-function parameterization, the mean-field initialization, or the pinning fields used to guide the optimization, as well as on boundary conditions. This can favor one type of symmetry breaking over another, making it difficult to distinguish the genuine interplay of intertwined or competing orders from biases induced by the variational parameterization. Here, we introduce the Symmetry-Preserving Backflow Pairing (SBP) ansatz, a neural-network wave function that respects translational symmetry by construction and thereby avoids these broken-symmetry minima. The SBP ansatz reaches state-of-the-art variational energies for the ttt-t′t't′ Hubbard model on lattices up to 24×2424\times2424×24 with 504504504 electrons, below those of competing pure stripe solutions. By extrapolating to the thermodynamic limit, we find robust evidence for ddd-wave superconducting order, resolving a long-standing question about the 1/81/81/8-doped model at t′/t=−0.2t'/t=-0.2t′/t=−0.2 and U/t=8.0U/t=8.0U/t=8.0. Built on general principles of symmetry and locality, the SBP wave function provides a broadly applicable variational representation for challenging interacting fermionic systems.
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