Flavored N $$ \mathcal{N} $$ = 4 SYM — a highly entangled quantum liquid

Han-Chih Chang,Andreas Karch,Christoph F Uhlemann

doi:10.1007/jhep09(2014)110

Han-Chih Chang, Andreas Karch + Show 1 more

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https://doi.org/10.1007/jhep09(2014)110

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Abstract

We study N=4 SYM theory coupled to fundamental N=2 hypermultiplets in a state of finite charge density. The setup can be described holographically as a configuration of D3 and D7 branes with a non-trivial worldvolume gauge field on the D7. The phase has been identified as a new form of quantum liquid, where certain properties are those of a Fermi liquid while others are clearly distinct. We focus on the entanglement among the flavors, as quantified by the entanglement entropy. The expectation for a Fermi liquid would be a logarithmic enhancement of the area law, but we find a more drastic enhancement instead. The leading contributions are volume terms with a non-trivial shape dependence, signaling extensive entanglement among the flavors. At finite temperature these correlations are confined to a region of size given by the inverse temperature.

Highlights

General interest, as many of the investigations of entanglement entropy have focused on the vacuum and analytical results for more general states are still rather scarce
We study N = 4 SYM theory coupled to fundamental N = 2 hypermultiplets in a state of finite charge density
To obtain the flavor contribution to the entanglement entropy, one would first have to compute the backreaction of the D7 branes with the worldvolume gauge field on the background geometry created by the D3 branes

Summary

Entanglement entropy at zero temperature

In the following we compute the entanglement entropy at zero temperature. To give the results in a clear form, we expand it as SEE = SE(0E) + SE(1E) + O t20 , SE(1E) = SE(1E), q=0 + ∆SE(1E). Following [22], the O(t0) change in the entanglement entropy due to the backreaction of the flavor branes can be expressed as an integral over the minimal surface in the unperturbed geometry. In the last equality we have introduced the regularized volume of the hyperbolic space which is the original minimal surface This nicely reproduces the zero-density results of [22, 26]. The first term in square brackets yields the zero-density result, and the second one the renormalized entropy. We can turn to the finite-density contribution, given by the second term in the square brackets in (3.14). Due to the factor Vd−2, the entanglement entropy for the strip is not a function of q d−1 alone, but we note that the rescaled entropy d−2∆SE(1E) is

Entanglement thermodynamics and shape-dependent volume terms

Transition to thermal entropy at finite temperature

Discussion