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arXiv:2211.12545 · Journal of High Energy Physics

Modular parallel transport of multiple intervals in 1+1-dimensional free fermion theory

Bowen Chen, Bartłomiej Czech, Ling-Yan Hung, Gabriel Wong

Published Mar 20, 2023• 7 citations
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Abstract

Abstract Modular parallel transport is a generalization of Berry phases, applied to modular (entanglement) Hamiltonians. Here we initiate the study of modular parallel transport for disjoint field theory regions. We study modular parallel transport in the kinematic space of multi-interval regions in the vacuum of 1+1-dimensional free fermion theory — one of the few theories for which modular Hamiltonians on disjoint regions are known. We compute explicitly the generators of modular parallel transport, and explain why their relatively simple form follows from a half-sided modular inclusion. We also compute explicitly the curvature two-form of modular parallel transport. We contrast all calculations with the expected behavior of modular parallel transport in holographic theories, emphasizing the role of non-local terms that couple distinct intervals.

DOI
10.1007/jhep03(2023)147

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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.
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A bstract We study the modular Hamiltonians of an interval for the massless Dirac fermion on the half-line. The most general boundary conditions ensuring the global energy conservation lead to consider two phases, where either the vector or the axial symmetry is preserved. In these two phases we derive the corresponding modular Hamiltonian in explicit form. Its density involves a bi-local term localised in two points of the interval, one conjugate to the other. The associated modular flows are also established. Depending on the phase, they mix fields with different chirality or charge that follow different modular trajectories. Accordingly, the modular flow preserves either the vector or the axial symmetry. We compute the two-point correlation functions along the modular flow and show that they satisfy the Kubo-Martin-Schwinger condition in both phases. The entanglement entropies are also derived.
We revisit the calculation of multi-interval modular Hamiltonians for free fermions using a Euclidean path integral approach. We show how the multi-interval modular flow is obtained by gluing together the single interval modular flows. Using this relation, we obtain an exact expression for the multi-interval modular Hamiltonian and entanglement entropy in agreement with existing results. An essential ingredient in our derivation is the introduction of the modular action. This determines the non-local field theory describing the free fermion reduced density matrix, and makes manifest it’s non-local conformal symmetry and U(1) Kacs-Moody symmetry.