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arXiv:2305.08917 · Physical review. D/Physical review. D.

Quantum Monte Carlo for gauge fields and matter without the fermion determinant

Debasish Banerjee, Emilie Huffman

Published Feb 27, 2024• 5 citations
arXivHTMLPDF

Abstract

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

DOI
10.1103/physrevd.109.l031506

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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
Typical fermion algorithms require the computation (or sampling) of the fermion determinant. We focus instead on cluster algorithms which do not involve the determinant and involve a more physically relevant sampling of the configuration space. We develop new cluster algorithms and design classes of models for fermions coupled to Z2\mathbb{Z}_2Z2​ and U(1)U(1)U(1) gauge fields that are amenable to being simulated by these cluster algorithms in a sign-problem free way. Such simulations should contain rich phase diagrams and are particularly relevant for quantum simulator experiments.
State-of-the-art algorithms for simulating fermions coupled to gauge fields often rely on integrating fermion degrees of freedom. While successful in simulating QCD at zero chemical potential, at finite density these approaches are hindered by the sign problem. We discuss the simulation of lattice gauge theories in the Hamiltonian formalism and present a generalized meron-cluster algorithm for the simulation of the U(1)U\left(1\right)U(1) Quantum Link Model for spin 1/21/21/2. This enables the study of models directly relevant to current quantum simulators and is a promising first step toward constructing new efficient algorithms for more complicated gauge theories.
Reliable simulations of correlated quantum systems, including high-temperature superconductors and frustrated magnets, are increasingly desired nowadays to further our understanding of essential features in such systems. Quantum Monte Carlo (QMC) is a unique numerically exact and intrinsically unbiased method to simulate interacting quantum many-body systems. More importantly, when QMC simulations are free from the notorious fermion sign problem, they can reliably simulate interacting quantum models with large system size and low temperature to reveal low-energy physics such as spontaneously broken symmetries and universal quantum critical behaviors. Here, we concisely review recent progress made in developing new sign-problem-free QMC algorithms, including those employing Majorana representation and those utilizing hot-spot physics. We also discuss applications of these novel sign-problem-free QMC algorithms in simulations of various interesting quantum many-body models. Finally, we discuss possible future directions of designing sign-problem-free QMC methods.
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.
Lattice gauge theories coupled to fermionic matter account for many interesting phenomena in both high-energy physics and condensed-matter physics. Certain regimes, e.g., at finite fermion density, are difficult to simulate with traditional Monte Carlo algorithms due to the so-called sign problem. We present a variational, sign-problem-free Monte Carlo method for lattice gauge theories with continuous gauge groups and apply it to (2+1)(2+1)(2+1)-dimensional compact QED with dynamical fermions at finite density. The variational ansatz is formulated in the full gauge-field basis, i.e., without having to resort to truncation schemes for the U(1)\text{U}(1)U(1) gauge-field Hilbert space. The ansatz consists of two parts: first, a pure gauge part based on Jastrow-type ansatz states (which can be connected to certain neural-network ansatz states) and, second, a fermionic part based on gauge-field-dependent fermionic Gaussian states. These are designed in such a way that the gauge-field integral over all fermionic Gaussian states is gauge-invariant and at the same time still efficiently tractable. To ensure the validity of the method we benchmark the pure gauge part of the ansatz against another variational method and the full ansatz against an existing Monte Carlo simulation where the sign problem is absent. Moreover, in limiting cases where the exact ground state is known we show that our ansatz is able to capture this behavior. Finally, we study a sign-problem affected regime by probing density-induced phase transitions.
Quantum Monte Carlo (QMC) simulations involving fermions have a notorious sign problem. Some well-known exceptions to the auxiliary field QMC algorithm rely on the factorizibility of the fermion determinant. Recently, a fermionic QMC algorithm [C. Wu, J. Hu, and S. Zhang, Phys. Rev. Lett. 91, 186402 (2003)] has been found in which the fermion determinant may not necessarily be factorizable, but can instead be expressed as a product of complex conjugate pairs of eigenvalues, thus eliminating the sign problem for a much wider class of models. In this paper, we present the general conditions for the applicability of this algorithm and point out that it is deeply related to the time-reversal symmetry of the fermion matrix. We apply this method to various models of strongly correlated systems at all doping levels and lattice geometries, and show that many phases can be simulated without the sign problem.