Abstract
This paper presents a method for alleviating sign problems in lattice path integrals, including those associated with finite fermion density in relativistic systems. The method makes use of information gained from some systematic expansion -- such as perturbation theory -- in order to accelerate the Monte Carlo. The method is exact, in the sense that no approximation to the lattice path integral is introduced. Thanks to the underlying systematic expansion, the method is systematically improvable, so that an arbitrary reduction in the sign problem can in principle be obtained. The Thirring model (in 0 + 1 and 1 + 1 dimensions) is used to demonstrate the ability of this method to reduce the finite-density sign problem.
Highlights
Lattice Monte Carlo methods are able to provide nonperturbative access to observables in quantum field theories
They are unique in this respect for many strongly coupled theories. Under certain circumstances, such as at finite density of relativistic fermions and the Hubbard model away from half-filling, lattice methods are made dramatically less efficient by the so-called sign problem. This sign problem is a central obstacle to first-principles calculations in many regimes of strongly coupled theories, including ab initio studies of the nuclear equation of state
In this paper we will examine a new method, inspired by two observations: first, that the partition function is unchanged if a function that integrates to zero is added to the Boltzmann factor, and second, that lattice methods can encounter a fatal sign problem even in regimes under good control by perturbation theory
Summary
Lattice Monte Carlo methods are able to provide nonperturbative access to observables in quantum field theories. They are unique in this respect for many strongly coupled theories Under certain circumstances, such as at finite density of relativistic fermions and the Hubbard model away from half-filling, lattice methods are made dramatically less efficient by the so-called sign problem. The average of the exponential of the imaginary part of the action, often termed the “average phase,” is equal to the ratio of the physical to quenched partition functions Z=ZQ, and characteristically scales like e−βV Resolving this exponentially small quantity, by averaging many quantities of unit magnitude, requires ∼e2βV samples; the reweighting procedure incurs an exponential cost in the volume. In this paper we will examine a new method, inspired by two observations: first, that the partition function is unchanged if a function that integrates to zero is added to the Boltzmann factor, and second, that lattice methods can encounter a fatal sign problem even in regimes under good control by perturbation theory (or any other systematic expansion). VI, discussing in particular a relation between this method and the method of field complexification
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