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The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics

Recent advances in determinant quantum Monte Carlo

Chia‐Chen Chang, Sergiy Gogolenko, Jeffrey Perez, Zhaojun Bai, Richard T. Scalettar

Published Oct 10, 2013• 29 citations
Publisher

Abstract

Determinant quantum Monte Carlo is a method for studying magnetic, transport and thermodynamic properties of interacting fermions on a lattice. It is widely used to explore the physics of strongly correlated quantum systems, from cuprate superconductors to ultracold atoms trapped on optical lattices. This paper contains a description of recent algorithmic advances in the determinant quantum Monte Carlo technique. Focus will be on algorithms developed for hybrid multicore processor and GPU platforms. The resulting speed-up of the simulations will be quantified. Simulations’ results will also be presented, with an emphasis on physical quantities that can now be computed for large numbers of sites.

DOI
10.1080/14786435.2013.845314

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Determinant quantum Monte Carlo is a method for studying magnetic, transport and thermodynamic properties of interacting fermions on a lattice. It is widely used to explore the physics of strongly correlated quantum systems, from cuprate superconductors to ultracold atoms trapped on optical lattices. This paper contains a description of recent algorithmic advances in the determinant quantum Monte Carlo technique. Focus will be on algorithms developed for hybrid multicore processor and GPU platforms. The resulting speed-up of the simulations will be quantified. Simulations’ results will also be presented, with an emphasis on physical quantities that can now be computed for large numbers of sites.
Analog quantum simulation based on ultracold atoms in optical lattices has catalyzed significant breakthroughs in the study of quantum many-body systems. These simulations rely on the statistical sampling of electronic Fock states, which are not easily accessible in classical algorithms. In this work, we modify the determinant quantum Monte Carlo by integrating a Fock-state update mechanism alongside the auxiliary field. This method enables efficient sampling of Fock-state configurations. The Fock-state restrictive sampling scheme further enables the pre-selection of multiple ensembles at no additional computational cost, thereby broadening the scope of simulation to more general systems and models. Employing this method, we analyze static correlations of the Hubbard model up to the fourth order and achieve quantitative agreement with cold-atom experiments. The simulations of dynamical spectroscopies of the Hubbard and Kondo-lattice models further demonstrate the reliability and advantage of this method. Ultracold atoms in optical lattices have advanced the study of quantum many-body systems but are challenging to replicate with classical algorithms. The authors propose a modified quantum Monte Carlo method which integrates Fock-state updates, enabling efficient simulations that match experimental results and extend to broader systems and models.
Many strongly correlated quantum systems are difficult to study numerically because of the exponential growth of the Hilbert space combined with the failure of numerical methods in important parameter regimes. This includes the long-standing problems of high-temperature superconductivity and frustrated magnetism. A promising solution to this dilemma are quantum simulators, well-controlled (synthetic) quantum systems, emulating model Hamiltonians not only of condensed matter systems, where parameters can be tuned at will and model complexity can be added step by step. This dissertation presents numerical simulation studies that are motivated by quantum simulators. Using large-scale quantum Monte Carlo computer simulations and mean-field theory the semiquantitative ground state phase diagram of a Penning ion trap quantum simulator, realizing a two-dimensional model of quantum magnetism with long-range interactions, is obtained. We study also thermal and quantum phase transitions in one-dimensional transverse-field Ising models with long-range interactions, which exhibit a Kosterlitz-Thouless transition and universality classes with continuously varying critical exponents. The second part of this thesis is concerned with the Fermi-Hubbard model, which has been realized recently with ultracold atoms in optical lattices, both for repulsive and attractive interactions. Equipped with a quantum gas microscope, which allows single-site single-atom detection, these experimental setups can collect histograms of the particle number from repeated projective measurements. This dissertation presents a method to compute such full counting statistics for quadratic operators such as particle number or magnetization in the framework of determinantal quantum Monte Carlo. From an analysis of the full counting statistics of the particle number for different subsystem sizes the size of preformed pairs or Cooper pairs in the attractive Hubbard model is inferred. Furthermore, excellent agreement of the numerically computed probability distribution of the staggered magnetization with a recent quantum gas experiment demonstrates that current experiments are capable of resolving the differences between the Hubbard model and its limiting Heisenberg model. An extensive method section reviews the stochastic series expansion quantum Monte Carlo method for Ising models in a transverse field, which is free of the sign problem for arbitrary interactions, with special emphasis on the efficient simulation of long-range interactions. Also a self-contained introduction to the determinantal quantum Monte Carlo method is provided.
Over the past several decades, computational approaches to studying strongly-interacting systems have become increasingly varied and sophisticated. This book provides a comprehensive introduction to state-of-the-art quantum Monte Carlo techniques relevant for applications in correlated systems. Providing a clear overview of variational wave functions, and featuring a detailed presentation of stochastic samplings including Markov chains and Langevin dynamics, which are developed into a discussion of Monte Carlo methods. The variational technique is described, from foundations to a detailed description of its algorithms. Further topics discussed include optimisation techniques, real-time dynamics and projection methods, including Green's function, reptation and auxiliary-field Monte Carlo, from basic definitions to advanced algorithms for efficient codes, and the book concludes with recent developments on the continuum space. Quantum Monte Carlo Approaches for Correlated Systems provides an extensive reference for students and researchers working in condensed matter theory or those interested in advanced numerical methods for electronic simulation.
Determinant Quantum Monte Carlo (DQMC) provides numerically exact solutions for strongly correlated fermionic systems but faces significant computational challenges with increasing system size. While submatrix updates were originally developed for Hirsch-Fye QMC with onsite interactions at finite temperatures, their comprehensive application in DQMC has remained unexplored despite noted algorithmic similarities. We present the first comprehensive application of submatrix updates in DQMC, significantly extending beyond the original scope by enabling simulations with extended interactions and at zero temperature. Building upon conventional fast updates and delay updates, our generalized implementation achieves an order-of-magnitude improvement in computational efficiency, enabling simulations of the half-filled Hubbard model on lattices up to 8,000 sites - a scale previously challenging with standard DQMC implementations. This enhanced computational capability allows us to accurately determine the finite-temperature phase diagram of the 3D Hubbard model at half-filling. Our findings not only shed light on the phase transitions within these complex systems but also pave the way for more effective simulations of strongly correlated electrons, potentially guiding experimental efforts in cold atom simulations of the 3D Hubbard model.
The self-learning Monte Carlo method is a powerful general-purpose numerical method recently introduced to simulate many-body systems. In this work, we extend it to an interacting fermion quantum system in the framework of the widely used determinant quantum Monte Carlo. This method can generally reduce the computational complexity and moreover can greatly suppress the autocorrelation time near a critical point. This enables us to simulate an interacting fermion system on a $100\ifmmode\times\else\texttimes\fi{}100$ lattice even at the critical point and obtain critical exponents with high precision.
Monte Carlo simulations of strongly interacting fermionic systems are plagued by the fermion sign problem, making the nonperturbative study of many interesting regimes of dense quantum matter, or of theories of odd numbers of fermion flavors, challenging. Moreover, typical fermion algorithms require the computation (or sampling) of the fermion determinant. We focus instead on the meron cluster algorithm, which can solve the fermion sign problem in a class of models without involving the determinant. We develop and benchmark new meron algorithms to simulate fermions coupled to Z2 and U(1) gauge fields in the presence of appropriate four-fermi interactions. Such algorithms can be used to uncover potential exotic properties of matter, particularly relevant for quantum simulator experiments. We demonstrate the emergence of the Gauss’ law at low temperatures for a U(1) model in (1+1)D . Published by the American Physical Society 2024