Numerical estimation of fermionic observables often requires introduction of auxiliary bosonic fieldsthat have no direct physical relevance through a Hubbard-Stratonovich transformation. The varianceof some fermionic observables (for example, appropriately constructed local four-fermion operatorsin 2d Gross-Neveu model) when they are estimated using such auxiliary fields may not correspondto any physical observable and in particular may diverge. In such cases it is not clear how onecan reliably estimate the corresponding observable. An important example may be found in thecontext of exceptional configurations in quenched QCD. We demonstrate simpler examples in toymodels by constructing particular four-fermion operators. We propose two methods to overcome thisissue. The first method is a discrete Hubbard-Stratonovich transformation that does not suffer fromsuch divergences for these classes of observables. While this method works in principle, it is notpractically feasible. The second method we propose is a reweighting method.
Numerical estimation of fermionic observables often requires introduction of auxiliary bosonic fieldsthat have no direct physical relevance through a Hubbard-Stratonovich transformation. The varianceof some fermionic observables (for example, appropriately constructed local four-fermion operatorsin 2d Gross-Neveu model) when they are estimated using such auxiliary fields may not correspondto any physical observable and in particular may diverge. In such cases it is not clear how onecan reliably estimate the corresponding observable. An important example may be found in thecontext of exceptional configurations in quenched QCD. We demonstrate simpler examples in toymodels by constructing particular four-fermion operators. We propose two methods to overcome thisissue. The first method is a discrete Hubbard-Stratonovich transformation that does not suffer fromsuch divergences for these classes of observables. While this method works in principle, it is notpractically feasible. The second method we propose is a reweighting method.
Monte Carlo calculations of fermionic systems with continuous auxiliary fields frequently suffer from a diverging variance for fermionic observables. If the simulations have an infinite variance problem, one cannot estimate observables reliably even with an arbitrarily large number of samples. In this paper, we explore a method to solve this problem using sampling based on the distribution of a system with an extra time slice. The necessary reweighting factor is computed both perturbatively and through a secondary Monte Carlo. We show that the Monte Carlo reweighting coupled to the use of an unbiased estimator of the reweighting factor leads to a method that eliminates the infinite variance problem at a very small extra cost. We compute the double occupancy in the Hubbard model at half-filling to demonstrate the method and compare the results to well-established results obtained by other methods.
Fermionic Monte Carlo calculations with continuous auxiliary fields often encounter infinite variance problem from fermionic observables. This issue renders the estimation of observables unreliable, even with an infinite number of samples. In this work, we show that the infinite variance problem stems from the fermionic determinant. Also, we propose an approach to address this problem by employing a reweighting method that utilizes the distribution from an extra time-slice. Two strategies to compute the reweighting factor are explored: one involves truncating and analytically calculating the reweighting factor, while the other employs a secondary Monte Carlo estimation. With Hubbard model as a testbed, we demonstrate that utilizing the sub-Monte Carlo estimation, coupled with an unbiased estimator, offers a solution that effectively mitigates the infinite variance problem at a minimal additional cost.
For important classes of many-fermion problems, quantum Monte Carlo (QMC) methods allow exact calculations of ground-state and finite-temperature properties without the sign problem. The list spans condensed matter, nuclear physics, and high-energy physics, including the half-filled repulsive Hubbard model, the spin-balanced atomic Fermi gas, and lattice quantum chromodynamics calculations at zero density with Wilson Fermions, and is growing rapidly as a number of problems have been discovered recently to be free of the sign problem. In these situations, QMC calculations are relied on to provide definitive answers. Their results are instrumental to our ability to understand and compute properties in fundamental models important to multiple subareas in quantum physics. It is shown, however, that the most commonly employed algorithms in such situations have an infinite variance problem. A diverging variance causes the estimated Monte Carlo statistical error bar to be incorrect, which can render the results of the calculation unreliable or meaningless. We discuss how to identify the infinite variance problem. An approach is then proposed to solve the problem. The solution does not require major modifications to standard algorithms, adding a "bridge link" to the imaginary-time path integral. The general idea is applicable to a variety of situations where the infinite variance problem may be present. Illustrative results are presented for the ground state of the Hubbard model at half-filling.
The ground state of a many-electron system can be projected from a trial state using an integral of one-electron projectors over a set of Hubbard-Stratonovich auxiliary fields. We have explored several alternative formulations of this method and find that the convergence of the ground-state energy with the number of field variables can be dramatically improved by choosing the appropriate one. This is illustrated using the two-dimensional Hubbard model as an example. These results are also used to examine the error associated with neglecting the so-called ``sign problem'' common to quantum Monte Carlo many-fermion calculations.
We tutorially review the determinantal Quantum Monte Carlo method for fermionic systems, using the Hubbard model as a case study. Starting with the basic ingredients of Monte Carlo simulations for classical systems, we introduce aspects such as importance sampling, sources of errors, and finite-size scaling analyses. We then set up the preliminary steps to prepare for the simulations, showing that they are actually carried out by sampling discrete Hubbard-Stratonovich auxiliary fields. In this process the Green's function emerges as a fundamental tool, since it is used in the updating process, and, at the same time, it is directly related to the quantities probing magnetic, charge, metallic, and superconducting behaviours. We also discuss the as yet unresolved "minus-sign problem", and two ways to stabilize the algorithm at low temperatures.
We tutorially review the determinantal Quantum Monte Carlo method for fermionic systems, using the Hubbard model as a case study. Starting with the basic ingredients of Monte Carlo simulations for classical systems, we introduce aspects such as importance sampling, sources of errors, and finite-size scaling analyses. We then set up the preliminary steps to prepare for the simulations, showing that they are actually carried out by sampling discrete Hubbard-Stratonovich auxiliary fields. In this process the Green's function emerges as a fundamental tool, since it is used in the updating process, and, at the same time, it is directly related to the quantities probing magnetic, charge, metallic, and superconducting behaviours. We also discuss the as yet unresolved ‘minus-sign problem', and two ways to stabilize the algorithm at low temperatures.