Newton's Identity in Finite-Bead Fermionic Partition Function

Published 3 Jun 2026 in physics.comp-ph and cond-mat.stat-mech | (2606.05442v1)

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.

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Summary

The paper presents a closed-form expression for the finite-bead fermionic partition function using Newton’s identity mapping, valid for arbitrary n, N, and propagator choice.
It details a recursive formulation that connects the determinant structure of the path integral to elementary symmetric polynomials, enabling exact evaluations.
The analytic framework facilitates precise thermodynamic calculations and benchmarking of discretization schemes in path integral Monte Carlo simulations.

Newton’s Identity and Finite-Bead Fermionic Partition Functions

Overview

The paper "Newton’s Identity in Finite-Bead Fermionic Partition Function" (2606.05442) presents a detailed derivation and analysis of the partition function for non-interacting $n$ -fermion systems confined to a one-dimensional harmonic potential using path integral formulations with a finite number of imaginary time slices (beads). Distinctly, the work demonstrates that the partition function admits a closed-form, finite-bead recursion—valid for arbitrary bead number $N$ , imaginary time $\tau$ , particle number $n$ , and choice of short-time propagator—by directly mapping the path integrals’ determinant structure to Newton’s identities for elementary symmetric polynomials. The equivalence enables the direct application of closed-form combinatorial results (Macdonald's theorem) for exact evaluation, and supports analytic calculation of internal energies and specific heats at finite discretization, without recourse to the continuum limit.

Finite-Bead Path Integral Structure and Recursion

The authors analyze the path-integral formulation for non-interacting fermions in a 1D harmonic trap, starting from the Slater determinant representation of the propagator over $N$ beads. For any short-time propagator (e.g., primitive approximation and higher-order alternatives), the contraction of Gaussian integrals over beads leads to an exact determinant structure for the $n$ -particle imaginary-time propagator.

By systematically expanding the determinant and performing the intermediate Gaussian integrals, the $n$ -fermion, finite-bead partition function $Z_n^N$ can be recursively expressed in terms of single-particle cycle (trace) terms $z_i^N$ and lower-order partition functions. This recursion has the explicit form:

$Z_n^N = \frac{1}{n} \sum_{i=1}^n (-1)^{i-1} z_i^N Z_{n-i}^N,\quad Z_0^N=1,$

and holds identically for all $N$ 0, $N$ 1, $N$ 2, and short-time propagator choice. This is in contrast to classical recursions that are only valid in the $N$ 3 (continuum) limit [Ford71, 10.1063/1.464180, PhysRevE.55.227, Schmidt_2002]. The explicit evaluation of cycle traces $N$ 4 for the harmonic oscillator yields

$N$ 5

where $N$ 6, with $N$ 7 parameterizing the contracted propagator (dependent on the specific propagator and bead number).

Newton’s Identity Correspondence and Closed Form Expression

The recursion derived for $N$ 8 is formally identical to Newton’s identity relating elementary symmetric polynomials (ESPs) and power sums:

$N$ 9

establishing a direct mapping between ESPs and $\tau$ 0, and between power sums and $\tau$ 1. Utilizing Macdonald’s closed-form results for ESPs with infinite roots, the finite-bead $\tau$ 2-fermion partition function is obtained in closed form:

$\tau$ 3

In the continuum limit, this closed-form reduces to established results previously derived only via continuum recursions [PhysRevE.55.227]. The result emphasizes that all finite-bead effects are analytically encapsulated, and the method is robust for arbitrary $\tau$ 4, $\tau$ 5, and propagator.

Analytic Thermodynamics at Finite Bead Number

Using the closed-form partition function, the paper provides analytic expressions for thermodynamic observables at fixed bead number:

Thermodynamic internal energy:

$\tau$ 6

Specific heat:

$\tau$ 7

Hamiltonian energy and corresponding specific heat are constructed analogously, leveraging parameters of the contracted propagator.

Notably, the thermodynamic and Hamiltonian energies both converge for large $\tau$ 8 to the expected result for the 1D fermion harmonic oscillator, $\tau$ 9, for all $n$ 0. The validity for arbitrary propagator choice allows for analytic benchmarking and convergence analysis of different propagation schemes (e.g., primitive vs. symplectic 4A propagators) with respect to bead number and system size, without requiring full numerical PIMC evaluation at each parameter choice.

Implications and Extensions

The analytic tractability and closed-form results at finite bead number enable:

Systematic benchmarking of PIMC discretization schemes, quantification of bead-number errors, and propagation selection in practical simulations.
Direct extension to higher dimensions (with technical complexity) and prospects for addressing the fermionic sign problem when dimensionality induces antisymmetry constraints, as in $n$ 1 [PhysRevE.107.035305].
Transferability of the recursive structure to interacting systems, as shown formally for the integrand structure (Appendix B), suggesting a program for generalized recursion and integration strategies in the presence of interactions.
Potential application of group-theoretic and combinatorial methods (via symmetric polynomials) for more general classes of quantum partition functions.

Conclusion

The paper rigorously establishes that the finite-bead $n$ 2-fermion partition function for non-interacting particles in a harmonic trap satisfies an exact recursion directly equivalent to Newton's identity, and is amenable to a closed-form analytic solution using symmetric function theory. This framework provides a robust analytic foundation for understanding discretization in PIMC approaches and paves the way for both practical algorithmic advances and further theoretical investigation into the structure of quantum canonical partition functions and the fermion sign problem at finite discretization. Future directions include the higher-dimensional generalization and systematic study of interacting systems informed by the recursive/combinatorial methodologies developed here.

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