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arXiv:1303.3472

Holographic Entanglement Entropy and Hidden Fermi surfaces

Zhong-Ying Fan

Published Mar 14, 2013• 1 citation
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Abstract

We prove that the purely classical gravity dual of Fermi and non-Fermi liquids exist by employing the logarithmic behavior of entanglement entropy to probe Fermi surfaces. For isotropic systems, the logarithmic behavior originates only from UV-IR intermediate regions of the minimal area surface. For anisotropic systems, the surfaces' configuration becomes largely modified by spatial anisotropy and series of solutions exist. By imposing the null energy condition we show that the specific heat behaves as $C\propto T^α$ where $α\leq1$, in both systems. In the end, we also present an effective gravity modle for anisotropic background.

DOI
10.48550/arxiv.1303.3472

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We prove that the purely classical gravity dual of Fermi and non-Fermi liquids exist by employing the logarithmic behavior of entanglement entropy to probe Fermi surfaces. For isotropic systems, the logarithmic behavior originates only from UV-IR intermediate regions of the minimal area surface. For anisotropic systems, the surfaces' configuration becomes largely modified by spatial anisotropy and series of solutions exist. By imposing the null energy condition we show that the specific heat behaves as C∝TαC\propto T^αC∝Tα where α≤1α\leq1α≤1, in both systems. In the end, we also present an effective gravity modle for anisotropic background.
General scaling arguments, and the behavior of the thermal entropy density, are shown to lead to an infrared metric holographically representing a compressible state with hidden Fermi surfaces. This metric is characterized by a general dynamic critical exponent, zzz, and a specific hyperscaling violation exponent, $\ensuremath{\theta}$. The same metric exhibits a logarithmic violation of the area law of entanglement entropy, as shown recently by Ogawa et al. [e-print arXiv:1111.1023 (unpublished)]. We study the dependence of the entanglement entropy on the shape of the entangling region(s), on the total charge density, on temperature, and on the presence of additional visible Fermi surfaces of gauge-neutral fermions; for the latter computations, we realize the needed metric in an Einstein-Maxwell-dilaton theory. All our results support the proposal that the holographic theory describes a metallic state with hidden Fermi surfaces of fermions carrying gauge charges of deconfined gauge fields.
We consider the holographic entanglement entropy of (d + 2)-dimensional semi-local quantum liquids, for which the dual gravity background in the deep interior is AdS 2 × $ \mathbb{R} $ d multiplied by a warp factor which depends on the radial coordinate. The entropy density of this geometry goes to zero in the extremal limit. The thermodynamics associated with this semi-local background is discussed via dimensional analysis and scaling arguments. For the case of an asymptotically AdS UV completion of this geometry, we show that the entanglement entropy of a strip and an annulus exhibits a phase transition as a typical length of the different shapes is varied, while there is no sign of such a transition for the entanglement entropy of a sphere. Moreover, for the spherical entangling region, the leading order contribution to the entanglement entropy in the IR is calculated analytically. It exhibits an area law behaviour and agrees with the numerical result.
In this paper, we investigate a holographic model of superconductor with hidden Fermi surfaces, which was defined by the logarithmic violation of area law of entanglement entropy. We works in fully back-reacted background using standard Einstein-Maxwell-Dilaton action with additional complex scalar filed which was charged under the Maxwell field. Particularly, we analyze the behavior of entanglement entropy during the phase transition. At the critical point, the finite part of the entanglement entropy has a discontinuity of slope and tends to a lower value in the superconducting phase all the way down to the zero temperature limit, indicating the reorganization of degrees of freedom of the system across the phase transition.
A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area. This area law inspired the holographic principle, which was later realized concretely in gauge-gravity duality. In this context, entanglement entropy is given by the area of a minimal surface in a dual spacetime. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires Rényi entropies. Here we show that all Rényi entropies satisfy a similar area law in holographic theories and are given by the areas of dual cosmic branes. This geometric prescription is a one-parameter generalization of the minimal surface prescription for entanglement entropy. Applying this we provide the first holographic calculation of mutual Rényi information between two disks of arbitrary dimension. Our results provide a framework for efficiently studying Rényi entropies and understanding entanglement structures in strongly coupled systems and quantum gravity.
We study the holographic entanglement entropy of nonlocal field theories at both extremality and finite temperature. The gravity duals, constructed by Nozaki et al. [J. High Energy Phys. 10 (2012) 193], are characterized by a parameter www. Both the zero-temperature backgrounds and the finite-temperature counterparts are exact solutions of Einstein-Maxwell-dilaton theory. For the extremal case we consider the examples where the entangling regions are a strip and a sphere. We find that the leading-order behavior of the entanglement entropy always exhibits a volume law when the size of the entangling region is sufficiently small. We also clarify the condition under which the next-to-leading-order result is universal. For the finite-temperature case we obtain analytic expressions in both the high-temperature and low-temperature limits. In the former case the leading-order result approaches the thermal entropy, while the finite contribution to the entanglement entropy at extremality can be extracted by taking the zero-temperature limit in the latter case. Moreover, we observe some peculiar properties of the holographic entanglement entropy when w=1w=1w=1.