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Entanglement Entropy, Current, and Chemical Potential

Bom Soo Kim

Published May 4, 2017• 1 citation
OpenAlex

Abstract

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 $\mathbb{Z}_n$ 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.

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Entanglement Entropy, Chemical Potential, Current Source, and Wilson Loop

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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 $\mathbb{Z}_n$ 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.

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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 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.
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.
In this work we consider the time evolution of charged R\'enyi entanglement entropies after exciting the vacuum with local fermionic operators. In order to explore the information contained in charged R\'enyi entropies, we perform computations of their excess due to the operator excitation in two-dimensional conformal field theory, free fermionic field theories in various dimensions as well as holography. In the analysis we focus on the dependence on the entanglement charge, the chemical potential and the spacetime dimension. We find that excesses of charged (R\'enyi) entanglement entropy can be interpreted in terms of charged quasiparticles. Moreover, we show that by appropriately tuning the chemical potential, charged R\'enyi entropies can be used to extract entanglement in a certain charge sector of the excited state.
It is generally believed that the semiclassical AdS3 higher spin gravity could be described by a two dimensional conformal field theory with W \mathcal{W} W -algebra symmetry in the large central charge limit. In this paper, we study the single interval entanglement entropy on the torus in the CFT with a W3 {\mathcal{W}}_{\mathcal{3}} W3​ deformation. More generally we develop the monodromy analysis to compute the two-point function of the light operators under a thermal density matrix with a W3 {\mathcal{W}}_{\mathcal{3}} W3​ chemical potential to the leading order. Holographically we compute the probe action of the Wilson line in the background of the spin-3 black hole with a chemical potential. We find exact agreement.