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arXiv:1902.01463 · Physical review. B./Physical review. B

Auxiliary field quantum Monte Carlo for multiband Hubbard models: Controlling the sign and phase problems to capture Hund's physics

Hongxia Hao, Brenda M. Rubenstein, Hao Shi

Published Jun 20, 2019• 20 citations
arXivHTMLPDF

Abstract

In the study of strongly correlated, many-electron systems, the Hubbard-Kanamori (HK) model has emerged as one of the prototypes for transition metal oxide physics. The model is multiband in nature and contains Hund's coupling terms, which have pronounced effects on metal-insulator transitions, high-temperature superconductivity, and other physical properties. In the following, we present a complete theoretical framework for treating the HK model using the ground-state auxiliary field quantum Monte Carlo (AFQMC) method and analyze its performance on few-band models whose parameters approximate those observed in ruthenates, rhodates, and other materials exhibiting Hund's physics. Unlike previous studies, the constrained path and phaseless approximations are used to respectively control the sign and phase problems, which enables high-accuracy modeling of the HK model's ground-state properties within parameter regimes of experimental interest. We demonstrate that, after careful consideration of the Hubbard-Stratonovich transformations and trial wave functions employed, relative errors in the energy of less than $1%$ can routinely be achieved for moderate to large values of the Hund's coupling constant. Crucially, our methodology also accurately predicts magnetic ordering and phase transitions. The results presented open the door to more predictive modeling of Hund's physics within a wide range of strongly correlated materials using AFQMC.

DOI
10.1103/physrevb.99.235142

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In the study of strongly correlated, many-electron systems, the Hubbard-Kanamori (HK) model has emerged as one of the prototypes for transition metal oxide physics. The model is multiband in nature and contains Hund's coupling terms, which have pronounced effects on metal-insulator transitions, high-temperature superconductivity, and other physical properties. In the following, we present a complete theoretical framework for treating the HK model using the ground-state auxiliary field quantum Monte Carlo (AFQMC) method and analyze its performance on few-band models whose parameters approximate those observed in ruthenates, rhodates, and other materials exhibiting Hund's physics. Unlike previous studies, the constrained path and phaseless approximations are used to respectively control the sign and phase problems, which enables high-accuracy modeling of the HK model's ground-state properties within parameter regimes of experimental interest. We demonstrate that, after careful consideration of the Hubbard-Stratonovich transformations and trial wave functions employed, relative errors in the energy of less than 11%1 can routinely be achieved for moderate to large values of the Hund's coupling constant. Crucially, our methodology also accurately predicts magnetic ordering and phase transitions. The results presented open the door to more predictive modeling of Hund's physics within a wide range of strongly correlated materials using AFQMC.
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.
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.
Spurred by recent technological advances, there is a growing demand for computational methods that can accurately predict the dynamics of correlated electrons. Such methods can provide much-needed theoretical insights into the electron dynamics probed via time-resolved spectroscopy experiments and observed in non-equilibrium ultracold atom experiments. In this article, we develop and benchmark a numerically exact Auxiliary Field Quantum Monte Carlo (AFQMC) method for modeling the dynamics of correlated electrons in real time. AFQMC has become a powerful method for predicting the ground state and finite temperature properties of strongly correlated systems mostly by employing constraints to control the sign problem. Our initial goal in this work is to determine how well AFQMC generalizes to real-time electron dynamics problems without constraints. By modeling the repulsive Hubbard model on different lattices and with differing initial electronic configurations, we show that real-time AFQMC is capable of accurately capturing long-lived electronic coherences beyond the reach of mean field techniques. While the times to which we can meaningfully model decrease with increasing correlation strength and system size as a result of the exponential growth of the dynamical phase problem, we show that our technique can model the short-time behavior of strongly correlated systems to very high accuracy. Crucially, we find that importance sampling, combined with a novel adaptive active space sampling technique, can substantially lengthen the times to which we can simulate. These results establish real-time AFQMC as a viable technique for modeling the dynamics of correlated electron systems and serve as a basis for future sampling advances that will further mitigate the dynamical phase problem.
to develop quantum Monte Carlo (QMC) approaches for excited states and for more systematic and accurate treatment of ground states of strongly correlated sytems and heavier elements. We use the auxiliary-field quantum Monte Carlo (AFQMC) framework, which is a many-body total energy method that looks like many coupled mean-field calculations. I will discuss this method’s relations and dierences with traditional density-functional and quantum chemistry methods, and with other forms of QMC. The AFQMC method uses importance-sampled random walks in the space of nonorthogonal Slater determinants to project out the many-body ground state, or excited state. The random walks are constrained by a trial wave function (TWF) to remove the so-called sign problem, giving approximate (but non-perturbative) results. With singledeterminant TWFs from Hartree-Fock or density-functional theory (DFT), this approach can be thought of as a post-DFT method with no parameter. In this form, the accuracy is typically comparable to the quantum chemistry coupled-cluster CCSD(T) method near equilibrium geometries, and better than CCSD(T) when bonds are stretched. The computational cost of AFQMC scales as N 3 with characteristic size of the system. Its accuracy can be further improved at the cost of increased computing (releasing the constraint) or better TWF. With AFQMC, we can systematically target selected orbitals to correlate. This allows many-body calculations in solids with a frozen core, with down-folded Hamiltonians, or possibly embeded in a DFT calculations to extend scales. Results from a few on-going applications will be presented, including band structures in simple solids, magnetic phases in MnO, and Cobalt adsoption on graphene.
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.
We present an extension of constrained-path auxiliary-field quantum Monte Carlo (CP-AFQMC) for the treatment of correlated electronic systems coupled to phonons. The algorithm follows the standard CP-AFQMC approach for description of the electronic degrees of freedom while phonons are described in first quantization and propagated via a diffusion Monte Carlo approach. Our method is tested on the one- and two-dimensional Holstein and Hubbard-Holstein models. With a simple semiclassical trial wave function, our approach is remarkably accurate for $\ensuremath{\omega}/(2dt\ensuremath{\lambda})<1$ for all parameters in the Holstein model considered in this study where ddd is the dimensionality, $\ensuremath{\omega}$ is the phonon frequency, ttt is the electronic hopping strength, and $\ensuremath{\lambda}$ is the dimensionless electron-phonon coupling strength. In addition, we empirically show that the autocorrelation timescales as $1/\ensuremath{\omega}$ for $\ensuremath{\omega}/t\ensuremath{\lesssim}1$, which is an improvement over the $1/{\ensuremath{\omega}}^{2}$ scaling of the conventional determinant quantum Monte Carlo algorithm. In the Hubbard-Holstein model, the accuracy of our algorithm is found to be consistent with that of standard CP-AFQMC for the Hubbard model when the Hubbard UUU term dominates the physics of the model, and is nearly exact when the ground state is dominated by the electron-phonon coupling scale $\ensuremath{\lambda}$. The ap- proach developed in this work should be valuable for understanding the complex physics arising from the interplay between electrons and phonons in both model lattice problems and ab initio systems.