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arXiv:2606.05442

Newton's Identity in Finite-Bead Fermionic Partition Function

A. Chaudhary, J. Valenzuela

Published Jun 3, 2026• 0 citations
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Abstract

For non-interacting fermions in a harmonic trap, the partition function at any discrete number of imaginary time slices (or beads) and for any choice of short-time propagator admits an exact recursion relation derived directly from the contracted determinant form of the path integral. This finite-bead recursion is distinct from earlier continuum-limit recursions, which do not apply to the discrete time partition functions. By identifying a direct correspondence between this recursion and Newton's identity, application of a closed-form result from the theory of partitions provides an exact expression for the one-dimensional $n$-fermion finite-bead partition function. From this, the Thermodynamic and Hamiltonian energies and specific heats are analytically calculated for any $n$, $N$, $τ$, and propagator choice.

DOI
10.48550/arxiv.2606.05442

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For non-interacting fermions in a harmonic trap, the partition function at any discrete number of imaginary time slices (or beads) and for any choice of short-time propagator admits an exact recursion relation derived directly from the contracted determinant form of the path integral. This finite-bead recursion is distinct from earlier continuum-limit recursions, which do not apply to the discrete time partition functions. By identifying a direct correspondence between this recursion and Newton's identity, application of a closed-form result from the theory of partitions provides an exact expression for the one-dimensional nnn-fermion finite-bead partition function. From this, the Thermodynamic and Hamiltonian energies and specific heats are analytically calculated for any nnn, NNN, τττ, and propagator choice.
In the Path Integral Monte Carlo (PIMC) simulation of fermions in a harmonic trap, with and without pairwise harmonic interactions, the partition functions for any discrete number of imaginary time slices (or beads) and for any choice of the short-time propagator can be analytically obtained from the contracted determinant form of the propagator. This work shows that the resulting recursion relation can be reformulated in the λλλ-ring language, yielding a closed-form finite-bead partition function in two dimensions in terms of permutation statistics. This closed-form partition function can be evaluated by a special algorithm in O(n2)O(n^2)O(n2) time, providing an exact and numerically stable scheme for reproducing the energies of the original (undoable) fermion PIMC simulation for n=104n=10^4n=104 or more fermions. This result provides a concrete framework in which the numerical instability of the sign problem is completely bypassed, and serves as a counterexample to the prevailing view that all truly fermionic PIMC sign problems are NP-hard.
While the thermodynamics of a system of non-interacting fermions can be straightforwardly determined in the grand canonical ensemble, results for a specific number of N particles are more difficult to obtain. In this talk we will present a general recursion scheme for the canonical partition function of free fermions with a quadratic energy level spacing as might be present for ultracold fermionic atoms confined inside a box trap. Exact results for the entropy, specific heat and one and two particle occupation probabilities are numerically obtained to arbitrary precision and compared with their corresponding grand canonical values. The numerical stability of the recursion relation allows us to quantify deviations from Wicks theorem in the canonical ensemble for a variety of temperatures and densities.
This work shows that the recently discovered operator contraction identity for solving the discrete path integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the path integral Monte Carlo energy at any time step for any number of fermions is known analytically or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interactions can shift the sign problem to larger/smaller imaginary time but do not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in D dimension, with n = D + 1 fermions, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three dimensions. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and are compared to results obtained by modern neural networks.
This work shows that the recently discovered operator contraction identity for solving the discreet Path Integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the Path Integral Monte Carlo energy at any time step for any number of fermions is known analytically, or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interaction can shift the sign problem to larger/smaller imaginary time but does not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in DDD dimension, with n=D+1n=D+1n=D+1 fermion, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three-dimension. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and compared to results obtained by modern neural networks.
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For a system of bosons that interact through a class of general memory kernels, a recurrence relation for the partition function is derived within the path-integral formalism. This approach provides a generalization to previously known treatments in the literature of harmonically coupled systems of identical particles. As an example the result is applied to the specific heat of a simplified model of an open quantum system of bosons, harmonically coupled to a reservoir of distinguishable fictitious masses.