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Quantum-Atom Optics Downunder (QAO)

Stochastic Phase-Space Methods for Fermions

Joel Frederick Corney, P. D. Drummond

Published Jan 1, 2007• 0 citations
Publisher

Abstract

We give an overview of the Gaussian phase-space methods for fermions, focusing on the example of the Hubbard model, which describes ultracold fermions in a lattice. The phase-space methods do not suffer the sort of sign problem encountered by other quantum Monte Carlo methods, and provide an efficient means of calculating exact zero-temperature Hubbard results. However at finite temperature, there are known issues to do with the boundedness of the underlying distribution. We show how ‘stochastic gauges’ can be used to control the distribution tails and thereby improve the accuracy of the simulations at low temperatures.

DOI
10.1364/qao.2007.qwe5

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We give an overview of the Gaussian phase-space methods for fermions, focusing on the example of the Hubbard model, which describes ultracold fermions in a lattice. The phase-space methods do not suffer the sort of sign problem encountered by other quantum Monte Carlo methods, and provide an efficient means of calculating exact zero-temperature Hubbard results. However at finite temperature, there are known issues to do with the boundedness of the underlying distribution. We show how ‘stochastic gauges’ can be used to control the distribution tails and thereby improve the accuracy of the simulations at low temperatures.
We introduce a new class of quantum Monte Carlo methods, based on a Gaussian quantum operator representation of fermionic states. The methods enable first-principles dynamical or equilibrium calculations in many-body Fermi systems, and, combined with the existing Gaussian representation for bosons, provide a unified method of simulating Bose-Fermi systems. As an application relevant to the Fermi sign problem, we calculate finite-temperature properties of the two dimensional Hubbard model and the dynamics in a simple model of coherent molecular dissociation.
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.
All known Quantum-Monte-Carlo algorithms for fermions suffer from the so-called “minus-sign-problem” which is detrimental to the application of these simulation methods to fermion systems at very low temperatures and/or of very many lattice sites. We identify the origin of this fundamental problem, demonstrate that it is a very general feature, not necessarily related to the presence of fermionic degrees of freedom. We describe an novel algorithm which does not suffer from the minus-sign problem. Illustrative results for the two-dimensional Hubbard model are presented
In general, Quantum Monte Carlo methods suffer from a sign problem when simulating fermionic systems. This causes the efficiency of a simulation to decrease exponentially with the number of particles and inverse temperature. To circumvent this issue, a nodal constraint is often implemented, restricting the Monte Carlo procedure from sampling paths that cause the many-body density matrix to change sign. Unfortunately, this high-dimensional nodal surface is not a priori known unless the system is exactly solvable, resulting in uncontrolled errors. We will discuss two possible routes to extend the applicability of finite-temperatue path integral Monte Carlo. First we extend the regime where signful simulations are possible through a novel permutation sampling scheme. Afterwards, we discuss a method to variationally improve the nodal surface by minimizing a free energy during simulation. Applications of these methods will include both free and interacting electron gases, concluding with discussion concerning extension to inhomogeneous systems.
The quantum dynamics and grand canonical thermodynamics of many-mode (one-, two-, and three-dimensional) interacting Bose gases are simulated from first principles. The model uses a lattice Hamiltonian based on a continuum second-quantized model with two-particle interactions, external potential, and interactions with an environment, with no further approximations. The interparticle potential can be either an (effective) delta function as in Bose-Hubbard models, or extended with a shape resolved by the lattice. Simulations are of a set of stochastic equations that in the limit of many realizations correspond exactly to the full quantum evolution of the many-body systems. These equations describe the evolution of samples of the gauge P distribution of the quantum state, details of which are developed. Conditions under which general quantum phase-space representations can be used to derive stochastic simulation methods are investigated in detail, given the criteria: 1) The simulation corresponds exactly to quantum mechanics in the limit of many trajectories. 2) The number of equations scales linearly with system size, to allow the possibility of efficient first-principles quantum mesoscopic simulations. 3) All observables can be calculated from one simulation. 4) Each stochastic realization is independent to allow straightforward use of parallel algorithms. Special emphasis is placed on allowing for simulation of open systems. In contrast to typical Monte Carlo techniques based on path integrals, the phase-space representation approach can also be used for dynamical calculations. Two major (and related) known technical stumbling blocks with such stochastic simulations are instabilities in the stochastic equations, and pathological trajectory distributions as the boundaries of phase space are approached. These can (and often do) lead to systematic biases in the calculated observables. The nature of these problems are investigated in detail. Many phase-space distributions have, however, more phase-space freedoms than the minimum required for exact correspondence to quantum mechanics, and these freedoms can in many cases be exploited to overcome the instability and boundary term problems, recovering an unbiased simulation. The stochastic gauge technique, which achieves this in a systematic way, is derived and heuristic guidelines for its use are developed. The gauge P representation is an extension of the positive P distribution, which uses coherent basis states, but allows a variety of useful stochastic gauges that are used to overcome the stability problems. Its properties are investigated, and the resulting equations to be simulated for the open interacting Bose gas system are derived. The dynamics of the following many-mode systems are simulated as examples: 1) Uniform one-dimensional and two-dimensional Bose gases after the rapid appearance of significant two-body collisions (e.g. after entering a Feshbach resonance). 2) Trapped bosons, where the size of the trap is of the same order as the range of the interparticle potential. 3) Stimulated Bose enhancement of scattered atom modes during the collision of two Bose-Einstein condensates. The grand canonical thermodynamics of uniform one-dimensional Bose gases is also calculated for a variety of temperatures and collision strengths. Observables calculated include first to third order spatial correlation functions (including at finite interparticle separation) and momentum distributions. The predicted phenomena are discussed. Improvements over the positive P distribution and other methods are discussed, and simulation times are analyzed for Bose-Hubbard lattice models from a general perspective. To understand the behavior of the equations, and subsequently optimize the gauges for the interacting Bose gas, single- and coupled two-mode dynamical and thermodynamical models of interacting Bose gases are investigated in detail. Directions in which future progress can be expected are considered. Lastly, safeguards are necessary to avoid biased averages when exponentials of Gaussian-like trajectory distributions are used (as here), and these are investigated.