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The figure is a torus with a cut that represents the replica boundary condition (red dashed lines with an arrow at the end of the cut) and Wilson loop contribution (blue dashed line with arrows).

Entanglement and its extension, Renyi entropy, are useful in measuring the quantum information encoded in a quantum state and are at the heart of the quantum theories, encompassing the quantum mechanics, quantum field theories, quantum gravity and quantum information science. These entropies have been extensively studied recently in the literature.

Quantum fields can be manipulated by the background gauge fields, such as electric and magnetic fields. In the quantum world, gauge potentials are more useful. The time and space components of the 1+1 dimensional gauge potential are called a chemical potential and current source, respectively. Our analysis also include a topological contribution called Wilson loop.

We construct the most general formula of entanglement and the Renyi entropies for Dirac fermions on a 2 dimensional torus in the presence of chemical potential, current source, and/or topological Wilson loop. Then we perform analytic and exact computations of the the entropies.

In the zero temperature limit, the Renyi and entanglement entropies depend non-trivially on all of the three, chemical potential, current source, and Wilson loops. They pick up finite contributions whenever chemical potential coincides with one of the energy levels of the Dirac fermions. This demonstrates their usefulness to probe the energy spectra of quantum systems. This turns out to be contrary to earlier results.

The entropies are periodic in current source, which plays the role of ‘beat frequency’ when one of the modulus parameters is dialed. The entropies of the periodic fermions are finite, while those of the anti-periodic fermions vanish. This novel feature can be achieved by changing the modulus parameter.

This picture shows a typical behavior of entanglement entropy as a function of the Wilson loop parameter w in the low temperature limit. It reveals the phase transition as w is increased. The entropy is continuous, while its derivative experiences a discontinuity.

Wilson loops parameter also reveals the characteristics of the entropies in the zero temperature limit. Entanglement entropy experiences phase transitions when the Wilson loop parameter coincides with an odd integer. As the parameter is increased, the entropies show increasing tendencies along with non-monotonicity within each transition ranges.

In the large radius limit, the entropies' dependences on chemical potential and current source vanish as fast as the square of the sub-system size over that of the total system size. This supports a recent claim that the entropies of an interval in infinite space are independent of chemical potential. We further generalize the claim for multiple intervals and in the presence of current source. On the other hand, the entropies depend on the Wilson loops in the large radius limit, for the conformal dimension of vertex operators takes part in the spin structure independent entropies.

This graph represent the spin structure independent part of entanglement entropy as a function of Wilson loop parameter w. This contribution survives in the large radius limit as well as in the zero temperature limit. Thus entanglement entropy depends on the Wilson loop. We see the discontinuity of the entropy for the different topological sectors.

- B. S. Kim, "Entanglement Entropy, Chemical Potential, Current Source, and Wilson Loop" arXiv:1705.01859 [hep-th].
- B. S. Kim, "Entanglement Entropy with Background Gauge Fields," JHEP 1708, 041 (2017). arXiv:1706.07110 [hep-th].
- B. S. Kim, "Entanglement Entropy and Wilson Loop," arXiv:1808.09976 [hep-th].

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