On the thermodynamic properties of fictitious identical particles and the application to fermion sign problem

Abstract

By generalizing the recently developed path integral molecular dynamics for identical bosons and fermions, we consider the finite-temperature <p>thermodynamic properties of fictitious identical particles with a real parameter ξ interpolating continuously between bosons (ξ=1) and fermions (ξ=-1). Through general analysis and numerical experiments we find that the average energy may have good analytical property as a function of this real parameter ξ, which provides the chance to calculate the thermodynamical properties of identical fermions by an extrapolation with a simple polynomial function after accurately calculating the thermodynamic properties of the fictitious particles for ξ{greater than or equal to}0. Using several examples, it is shown that our method can efficiently give accurate energy values for finite-temperature fermionic systems. Our work provides a chance to circumvent the fermion sign problem for some quantum systems.

On the Thermodynamics of Fermions at any Temperature based on Parametrized Partition Function

“…For 10 particles and λ = 0.5, with the method in Ref. 28,29 , for different temperatures, we obtain energies for ξ = 0 and ξ = 1 with PIMD shown in the inset of Fig. 4, which enables us to obtain two functions f 0 (T ) and f 1 (T ).…”

“…For N identical particles, we consider the following parametrized partition function 28,29 with a real parameter ξ,…”

“…Using e −β Ĥ = e −∆β Ĥ • • • e −∆β Ĥ with ∆β = β/P and the technique of path integral, the partition function Z(ξ, β) with a general parameter ξ can be also mapped as a classical system of interacting ring polymers 28,29 , based on the idea of recursion formula for identical particles 7,25 . The so called exact numerical simulation of the thermodynamics for a quantum system is through this path integral formalism so that Z(ξ, β) can be written as the high dimensional integral of all the coordinates of N P beads.…”

“…In principle, path integral Monte Carlo/molecular dynamics takes all quantum effects into account but when we try to apply this methodology to fermions, we encounter an insurmountable difficulty known as fermion sign problem [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] , where the probabilities used for sampling become negative. In this work, we consider the recently developed parametrized path integral formulation 28,29 and propose a scheme to overcome the difficulties associated with the numerical simulation of Fermi systems, hopefully obtaining a method to numerically study the ab initio properties of Fermi systems.…”

“…The recently developed parametrized partition function 28 provides a scheme to smoothly interpolate the thermodynamics of bosons, distinguishable particles and fermions; in particular, the energy as a function of the interpolation parameter ξ can be shown to be a smooth monotonic function. In a previous work 29 , an attempt was made to infer the thermodynamics of fermions by numerically simulating the parametrized partition function for ξ ≥ 0 through path integral molecular dynamics (PIMD), and then extrapolate the results to ξ = −1 corresponding to fermions; of course, direct simulation for ξ < 0 is infeasible due to fermion sign problem where the probability distribution in importance sampling becomes negative, rendering any sampling methods inapplicable.…”