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 representing the magnetic parameter of the electromagnetic twist operators of Zn orbifold theories) and Wilson loop contribution (blue dashed line with arrows representing the electric parameter of the same operator). The torus has two different cycles (represented by the black dashed lines), and Chemical potential and currents source can be understood as parts of the twisted boundary conditions on these two cycles. Here we exploit all the possible ingredients of the electromagnetic twist operators of Zn orbifold conformal field theories, and thus the resulting entropy formulas are comprehensive and exhaustive.
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 controlled and 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. They are encoded into the electromagnetic twist vertex operator of Zn orbifold conformal field theories as depicted in the figure above. In the presence of Wilson loops, we propose a general scheme to parametrize the order of topological transitions. Here we adapt the second order phase transitions, Entanglement and Renyi entropies are continuous, while their first derivatives are discontinuous at each transition point. With the general scheme, we perform analytic and exact computations of the the entropies. The salient features of entanglement entropy for topological Wilson loops are clearly visualized in the infinite space limit.
A. Infinite space entanglement entropy: The spin structure independent Renyi and entanglement entropies have non-trivial dependences only on the Wilson loops parameter. This result turns out to be exact in infinite space. It is consistent with the previous claim that entanglement entropy is independent of the chemical potential in infinite space, where topological sectors were not considered.
Using the replica trick, we compute the Renyi entropy in the presence of topological Wilson loops w. The topological transitions happens due to the restriction of the conformal dimension of the vertex operator and depends on the number of replica copies n. The following figure captures five different Renyi entropies as a function of Wilson loop parameter.
We can view this as a limiting process of getting entanglement entropy from the Renyi entropies. As we take entanglement entropy limit, each topological transition range for Wilson loops increase, while the slope does not change. This is exciting news because we have succeeded to have entanglement entropy in the presence of Wilson loops for the first time. Previous attempts found singular limits for entanglement entropy. Here is the plot for the entanglement entropy. There are topological transitions when the Wilson loop parameter is odd integer multiple of Pi.
B. Infinite space corrections for entanglement entropy: 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. In particular, the entropies 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 novel feature 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 can be achieved by changing the modulus parameter.
We further study the role of Wilson loops for the entropies at finite space. We compute the Renyi entropies in the low temperature limit for different replica numbers. This can be considered as a limiting process of getting entanglement entropy, while the Renyi entropies are also physically relevant and have their own physical significances.
The following picture shows a typical behavior of entanglement entropy (for three different sub-system sizes) as a function of the Wilson loop parameter w in the low temperature limit. The entropies are continuous and show oscillatory features. The oscillation period in Wilson loop parameter depends on the sub-system size.
- B. S. Kim, "Entanglement Entropy and Wilson Loop," Nucl. Phys. B 948, 114771 (2019). arXiv:1808.09976 [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, Chemical Potential, Current Source, and Wilson Loop" arXiv:1705.01859 [hep-th].