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arXiv:2507.10503 · Journal of High Energy Physics

Modular Hamiltonian and entanglement entropy in the BMS free fermion theory

Peng-Xiang Hao, Wen-Xin Lai, Wei Song, Zehua Xiao

Published Feb 9, 2026• 2 citations
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Abstract

A bstract We study the modular Hamiltonian and the entanglement entropy of the BMS-invariant free fermion model. Starting from the modular Hamiltonian on a half-line interval, we calculate the modular Hamiltonian for a region consisting of two disjoint intervals using the uniformization map and a background field method. The resulting modular Hamiltonian contains a local term which generates a geometrical flow, and a bi-local term which mixes operators between the two intervals. We further derive a relation between Wightman functions and the modular Hamiltonian using both diagonalization and a coherent state method. This enables us to compute the entanglement entropy for multi-disjoint intervals. Our explicit results in the BMS free fermion model are compatible with earlier ones based on symmetry and holography.

DOI
10.1007/jhep02(2026)099

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Abstract We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv: 1104.1004 , for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m} ∪ {2m + 1, 2m + 2, …, 3m}, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.
We calculate the analytic form of the vacuum modular Hamiltonian for a two interval region and the algebra of a current $j(x)=\ensuremath{\partial}\ensuremath{\phi}(x)$ corresponding to a chiral free scalar $\ensuremath{\phi}$ in d=2d=2d=2. We also compute explicitly the mutual information between the intervals. This model shows a failure of Haag duality for two intervals that translates into a loss of a symmetry property for the mutual information usually associated with modular invariance. Contrary to the case of a free massless fermion, the modular Hamiltonian turns out to be completely nonlocal. The calculation is done diagonalizing the density matrix by computing the eigensystem of a correlator kernel operator. These eigenvectors are obtained by a novel method that involves solving an equivalent problem for a holomorphic function in the complex plane where multiplicative boundary conditions are imposed on the intervals. Using the same technique we also rederive the free fermion modular Hamiltonian in a more transparent way.
Abstract In this work, we perturbatively calculate the modular Hamiltonian to obtain the entanglement entropy in a free fermion theory on a torus with three typical deformations, e.g., TT‾ T\overline{T} TT T T ¯ deformation, local bilinear operator deformation, and mass deformation. For TT‾ T\overline{T} TT T T ¯ deformation, we find that the leading order correction of entanglement entropy is proportional to the expectation value of the undeformed modular Hamiltonian. As a check, in the high/low-temperature limit, the entanglement entropy coincides with that obtained by the replica trick in the literature. Following the same perturbative strategy, we obtain the entanglement entropy of the free fermion vacuum state up to second-order by inserting a local bilinear operator deformation in a moving mirror setting. In the uniformly accelerated mirror, the first-order and second-order correction of entanglement entropy vanishes in the late time limit. For mass deformation, we derive the entanglement entropy up to first-order deformation and comment on the second-order correction.