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arXiv:1706.07110 · Journal of High Energy Physics

Entanglement entropy with background gauge fields

Bom Soo Kim

Published Aug 1, 2017• 6 citations
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Abstract

We study the entanglement entropy, the Rényi entropy, and the mutual (Rényi) information of Dirac fermions on a 2 dimensional torus in the presence of constant gauge fields. We derive their general formulas using the equivalence between twisted boundary conditions and the background gauge fields. Novel and interesting physical consequences have been presented in arXiv:1705.01859 . Here we provide detailed computations of the entropies and mutual information in a low temperature limit, a large radius limit, and a high temperature limit. The high temperature limit reveals rather different physical properties compared to those of the low temperature one: there exist two non-trivial limits that depend on a modulus parameter and are not smoothly connected.

DOI
10.1007/jhep08(2017)041

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We construct and analytically compute entanglement and the Renyi entropies of Dirac fermions on a 2 dimensional torus in the presence of background chemical potential, current source and Wilson loop, by employing correlation functions of the electromagnetic vertex operators of Zn\mathbb{Z}_nZn​ orbifold theory. The entropies reveal numerous novelties. They probe the energy levels of compact fermions through the chemical potential, demonstrate interference phenomena with the current source being `beat frequency,' and experience topological phase transitions by dialing the Wilson loop, in the low temperature limit. In the large radius limit, they depend only on the Wilson loop via topological transitions, which are tightly linked to conformal dimension of the electromagnetic operators.
We construct and analytically compute entanglement and the Rényi entropies of Dirac fermions on a 2 dimensional torus in the presence of background chemical potential, current source and Wilson loop, by employing correlation functions of the electromagnetic vertex operators of Zn\mathbb{Z}_nZn​ orbifold theory. The entropies reveal numerous novelties. They probe the energy levels of compact fermions through the chemical potential, demonstrate interference phenomena with the current source being `beat frequency,' and experience topological phase transitions by dialing the Wilson loop, in the low temperature limit. In the large radius limit, they depend only on the Wilson loop via topological transitions, which are tightly linked to conformal dimension of the electromagnetic operators.
Entanglement and the Rényi entropies for Dirac fermions on 2 dimensional torus in the presence of chemical potential, current source, and topological Wilson loop are unified in a single framework by exhausting all the ingredients of the electromagnetic vertex operators of Zn orbifold conformal field theory. We employ different normalizations for different topological sectors to organize various topological phase transitions in the context of entanglement entropy. Pictorial representations for the topological transitions are given for the n=2 Rényi entropy. Our analytic computations reveal numerous novelties and provide resolutions for existing issues. We have settled to provide non-singular entanglement entropies that are also continuous across the topological sectors. Surprisingly, in infinite space, these entropies become exact and depend only on the Wilson loop. On a circle, we resolve to find the entropies subtly depend on the chemical potential at zero temperature, which is useful for probing the ground state energy levels of quantum systems.
We compute the Entanglement Entropy (EE) of a bipartition in 2D pure non-abelian U(N) gauge theory. We obtain a general expression for EE on an arbitrary Riemann surface. We find that due to area-preserving diffeomorphism symmetry EE does not depend on the size of the subsystem, but only on the number of disjoint intervals defining the bipartition. In the strong coupling limit on a torus we find that the scaling of the EE at small temperature is given by S(T)−S(0)=O(mgapTe−mgapT), which is similar to the scaling for the matter fields recently derived in literature. In the large N limit we compute all of the Renyi entropies and identify the Douglas–Kazakov phase transition.
We compute the leading contribution to the ground-state R\'enyi entropy ${S}_{\ensuremath{\alpha}}$ for a region of linear size LLL in a Fermi liquid in ddd dimensions. The result contains a universal boundary law violating term simply related to the more familiar entanglement entropy. We also obtain a universal crossover function that smoothly interpolates between the zero-temperature result and the ordinary thermal R\'enyi entropy of a Fermi liquid. Formulas for the entanglement entropy of more complicated regions, including nonconvex and disconnected regions, are obtained from the conformal field theory formulation of Fermi surface dynamics. These results permit an evaluation of the quantum mutual information between different regions in a Fermi liquid. We also study the number fluctuations in a Fermi liquid. Taken together, these results give a reasonably complete characterization of the low-energy quantum information content of Fermi liquids.
We study both entanglement and the Rényi entropies for the 2 dimensional massless Dirac fermions in the presence of topological Wilson loops, which are qualitatively different from those with a chemical potential and a current source. In the language of Zn orbifold theories, the Wilson loop is interpreted as an electric operator while the orbifold twist operator as a magnetic operator. Topological transitions for the generalized entropies are driven by both electric and magnetic parameters via the restriction on the operator's conformal weight. By adapting different normalizations for different topological sectors, we achieve two goals: entanglement entropy can be obtained with a smooth limit from the generalized Rényi entropy, and the entropies are continuous across the different topological sectors that include general Wilson loops winding sectors. We provide exact results for the entropies in infinite space, which depend only on the topological Wilson loops, independent of the chemical potential and the current source.