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arXiv:2605.21632

Diagrammatic Monte Carlo for Fermionic Rényi Entanglement Entropy

Boyuan Shi

Published May 20, 2026• 0 citations
arXivHTMLPDF

Abstract

We develop a direct diagrammatic Monte Carlo method for the second Rényi entropy of inter- acting lattice fermions. Fermionic replica sewing yields a connected-determinant (CDet) expansion, which we sample with many-configuration Markov-chain Monte Carlo. On a 3 \times 3 Hubbard clus- ter, deterministic coefficients through third order agree with exact-diagonalization derivatives for three reference shifts, while sampling extends the series to tenth order. For a reference shift with a continuous stability region, conformal-Borel resummation also gives good agreement with the exact entropy. We further compute coefficients through ninth order on 40 \times 40 and 50 \times 50 periodic lat- tices, with subsystems of up to 676 sites. The resummed results from different reference shifts show good agreement at a common truncation order. Their shift and order dependence provide checks on the stability of the continuation. These calculations extend connected-diagram sampling to spatial Rényi observables on large finite fermionic systems.

DOI
10.48550/arxiv.2605.21632

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We develop a direct diagrammatic Monte Carlo method for the second Rényi entropy of inter- acting lattice fermions. Fermionic replica sewing yields a connected-determinant (CDet) expansion, which we sample with many-configuration Markov-chain Monte Carlo. On a 3 \times 3 Hubbard clus- ter, deterministic coefficients through third order agree with exact-diagonalization derivatives for three reference shifts, while sampling extends the series to tenth order. For a reference shift with a continuous stability region, conformal-Borel resummation also gives good agreement with the exact entropy. We further compute coefficients through ninth order on 40 \times 40 and 50 \times 50 periodic lat- tices, with subsystems of up to 676 sites. The resummed results from different reference shifts show good agreement at a common truncation order. Their shift and order dependence provide checks on the stability of the continuation. These calculations extend connected-diagram sampling to spatial Rényi observables on large finite fermionic systems.
This thesis develops diagrammatic methods for thermodynamic and subsystem quantities of interacting lattice fermions, providing reference calculations for quantum simulators and many-body approximations. I formulate an inverse-flavour expansion about a self-consistent Hartree saddle, enumerate non-Gaussian diagrams, and use a discrete Lehmann representation to integrate selected internal vertices analytically. Density results cover 31 state points of the six-flavour Hubbard model on a 70×7070\times7070×70 lattice at T/t=0.15T/t=0.15T/t=0.15. I generalize combinatorial summation to flavour-dependent propagators and expansion shifts. CoS++ combines partial contractions to obtain connected sums for interaction subsets. A determinant-polynomial sieve evaluates the same sums through local coefficient filters; at ninth order it is 24--25 times faster than the specified CoS baselines in three-flavour dense-random-matrix benchmarks. Thermodynamic applications include a 2048-site honeycomb lattice and a three-flavour square lattice with 1024 sites. For the second Renyi entropy, I derive a fermionic replica path integral and a stable evaluation of its two-time Green functions. The connected-determinant calculation reaches tenth order on a 3×33\times33×3 cluster and ninth order on larger lattices. On 50×5050\times5050×50 sites at (U/t,μ/t,βt)=(3,1,8)(U/t,μ/t,βt)=(3,1,8)(U/t,μ/t,βt)=(3,1,8), the ninth-order continuation at shift α=1.3α=1.3α=1.3 gives S2/Ns=0.06695S_2/N_s=0.06695S2​/Ns​=0.06695 with propagated Monte Carlo uncertainty 3.0×10−53.0\times10^{-5}3.0×10−5. Comparing two expansion shifts at their respective retained orders gives an endpoint difference of 2.6×10−42.6\times10^{-4}2.6×10−4.
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