Diagrammatic Monte Carlo for Fermionic R\'enyi Entanglement Entropy

TL;DR

This paper introduces a diagrammatic Monte Carlo method for calculating the Renyi entanglement entropy in interacting lattice fermions, enabling high-order, large-system computations without a traditional sign problem.

Contribution

It develops a novel diagrammatic Monte Carlo framework based on the fermionic graded-swap representation for entanglement entropy, suitable for large systems and high-order corrections.

Findings

Benchmark against exact diagonalization for a 3x3 Hubbard cluster.

Successful large lattice calculations with dominant memory limitations.

Provides a pathway for entanglement calculations in sign-problem-limited regimes.

Abstract

We develop a direct diagrammatic Monte Carlo framework for the Renyi entanglement entropy of interacting lattice fermions. The method starts from the fermionic graded-swap representation of Z_n[A]=Tr_A\rho_A^n, which converts the entropy problem into a replicated path integral with mixed temporal boundary conditions on the entangling region. In this representation the replica momenta are half-shifted, q_m=(2m+1)\pi/n, and the interaction expansion has a determinant form suitable for connected-determinant summation. We combine this expansion with a many-configuration Markov-chain Monte Carlo sampler to obtain order-by-order corrections for very large systems to very high orders. As a benchmark, we compare the order-by-order coefficients of a 3*3 Hubbard cluster with exact diagonalization. We then report a production calculation for a large periodic lattice with a square subregions. The dominant system-size limitation is therefore memory rather than a conventional auxiliary-field sign problem. The results provide a step toward diagrammatic calculations of fermionic entanglement observables in regimes where direct quantum Monte Carlo sampling is costly or sign-problem limited.