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Physical review. B./Physical review. B

Systematically improvable mean-field variational ansatz for strongly correlated systems: Application to the Hubbard model

Sandro Sorella

Published Mar 15, 2023• 13 citations
Publisher

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 ${T}_{\mathrm{eff}}$ 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 ${T}_{\mathrm{eff}}$, 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 $d$-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
10.1103/physrevb.107.115133

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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{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≃10L\ensuremath{\simeq}10L\ensuremath≃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.
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.
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.
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.
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.
The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.
We investigate the ground-state properties of two strongly-correlated systems: the two-dimensional t−Jt-Jt−J model that can be used to study the high-temperature superconducting phase in the doped antiferromagnet, and the frustrated N\'eel antiferromagnet described by the J1−J2J_{1}-J_{2}J1​−J2​ Heisenberg model on the square lattice, which is widely considered as the prototype model for spin frustration. In this thesis, we apply to these two systems state-of-the-art quantum Monte Carlo techniques, including the variational and the Green's function Monte Carlo with the fixed-node approximation. Few Lanczos steps are used to systematically improve the accuracy of the trial wave functions. By introducing a suitable regularization scheme for the variational calculations with few Lanczos steps, stable and controllable simulations can be performed up to very large cluster sizes with very good accuracy. In the two-dimensional t−Jt{-}Jt−J model at J/t=0.4J/t=0.4J/t=0.4, we show that the accuracy of the Gutzwiller-projected variational state (containing dx2−y2d_{x^2-y^2}dx2−y2​ pairing) can be improved much by few Lanczos steps; in addition, the fixed-node Monte Carlo with these systematically improvable trial wave functions gives results that are comparable with the best accurate DMRG ones. Our main outcome is that the ground state is homogeneous and no evidence of stripes is detected around the doping δ=1/8\delta=1/8δ=1/8. Indeed, our best approximation to the ground state does not show any tendency towards charge inhomogeneity. Furthermore, our results show that a uniform state containing superconductivity and antiferromagnetism is stabilized at low hole doping, i.e., δ≲0.1\delta \lesssim 0.1δ≲0.1. In the J1−J2J_{1}-J_{2}J1​−J2​ Heisenberg model on the square lattice, we use the projected mean-field state that is built from Abrikosov fermions having a Z2Z_2Z2​ gauge structure and four Dirac points in the spinon spectrum. No spin or dimer order is found in the strongly frustrated regime and our calculations imply that a spin liquid phase may faithfully represent the exact ground state around J2/J1=0.5J_{2}/J_{1}=0.5J2​/J1​=0.5. The few Lanczos step technique is used to systematically improve the accuracy of the variational states both for the ground state and for few relevant low-energy excitations. This procedure allows us to estimate, in a valuable and systemic way, the spin gaps within thermodynamical limit and to show a solid evidence of an unconventional gapless excitation spectrum in the strongly frustrated regime, i.e., J2/J1≃0.5J_{2}/J_{1}\simeq 0.5J2​/J1​≃0.5. In particular, we found gapless triplet excitations at momenta (π,0)(\pi,0)(π,0) and (0,π)(0,\pi)(0,π), which are compatible with the presence of four Dirac points at momenta (±π2,±π2)(\pm \frac{\pi}{2},\pm \frac{\pi}{2})(±2π​,±2π​) in the spinon spectrum.