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arXiv:1603.02159 · Nuclear Physics B

Entanglement rules for holographic Fermi surfaces

Dibakar Roychowdhury B

Published May 26, 2016• 1 citation
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Abstract

In this paper, based on the notion of Gauge/Gravity duality, we explore the laws of entanglement thermodynamics for most generic classes of Quantum Field Theories with hyperscaling violation. In our analysis, we note that for Quantum Field Theories with compressible quark like excitation, the first law of entanglement thermodynamics gets modified due to the presence of an additional term that could be identified as the entanglement chemical potential associated with hidden Fermi surfaces of the boundary theory. Most notably, we find that the so called entanglement chemical potential does not depend on the size of the entangling region and is purely determined by the quark d.o.f. encoded within the entangling region.

DOI
10.1016/j.nuclphysb.2016.05.019

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In this paper, based on the notion of Gauge/Gravity duality, we explore the laws of entanglement thermodynamics for most generic classes of Quantum Field Theories with hyperscaling violation. In our analysis, we note that for Quantum Field Theories with compressible quark like excitation, the first law of entanglement thermodynamics gets modified due to the presence of an additional term that could be identified as the entanglement chemical potential associated with hidden Fermi surfaces of the boundary theory. Most notably, we find that the so called entanglement chemical potential does not depend on the size of the entangling region and is purely determined by the quark d.o.f. encoded within the entangling region.
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 [e-print arXiv:1111.1023 (unpublished)].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.
Entanglement entropy in conformal field theories is known to satisfy a first law. For spherical entangling surfaces, this has been shown to follow via the AdS/CFT correspondence and the holographic prescription for entanglement entropy from the bulk first law for Killing horizons. The bulk first law can be extended to include variations in the cosmological constant Λ, which we established in earlier work. Here we show that this implies an extension of the boundary first law to include varying the number of degrees of freedom of the boundary CFT. The thermodynamic potential conjugate to Λ in the bulk is called the thermodynamic volume and has a simple geometric formula. In the boundary first law it plays the role of a chemical potential. For the bulk minimal surface Σ corresponding to a boundary sphere, the thermodynamic volume is found to be proportional to the area of Σ, in agreement with the variation of the known result for entanglement entropy of spheres. The dependence of the CFT chemical potential on the entanglement entropy and number of degrees of freedom is similar to how the thermodynamic chemical potential of an ideal gas depends on entropy and particle number.
We use holographic duality to study the entanglement entropy (EE) of Conformal Field Theories (CFTs) in various spacetime dimensions d, in the presence of various deformations: a relevant Lorentz scalar operator with constant source, a temperature T , a chemical potential μ, a marginal Lorentz scalar operator with source linear in a spatial coordinate, and a circle-compactified spatial direction. We consider EE between a strip or sphere sub-region and the rest of the system, and define the “entanglement density” (ED) as the change in EE due to the deformation, divided by the sub-region’s volume. Using the deformed CFTs above, we show how the ED’s dependence on the strip width or sphere radius, L, is useful for characterizing states of matter. For example, the ED’s small-L behavior is determined either by the dimension of the perturbing operator or by the first law of EE. For Lorentz-invariant renormalization group (RG) flows between CFTs, the “area theorem” states that the coefficient of the EE’s area law term must be larger in the UV than in the IR. In these cases the ED must therefore approach zero from below as L→∞. However, when Lorentz symmetry is broken and the IR fixed point has different scaling from the UV, we find that the ED often approaches the thermal entropy density from above, indicating area theorem violation.
We give a unified overview of the zero temperature phases of compressible quantum matter: i.e., phases in which the expectation value of a globally conserved U(1) density, Q\mathcal{Q}Q, varies smoothly as a function of parameters. Provided the global U(1) and translational symmetries are unbroken, such phases are expected to have Fermi surfaces, and the Luttinger theorem relates the volumes enclosed by these Fermi surfaces to ⟨Q⟩⟨\mathcal{Q}⟩⟨Q⟩. We survey models of interacting bosons and/or fermions and/or gauge fields which realize such phases. Some phases have Fermi surfaces with the singularities of Landau's Fermi liquid theory, while other Fermi surfaces have non-Fermi liquid singularities. Compressible phases found in models applicable to condensed-matter systems are argued to also be present in models obtained by applying chemical potentials (and other deformations allowed by the residual symmetry at nonzero chemical potential) to the paradigmatic supersymmetric gauge theories underlying gauge-gravity duality: the Aharony-Bergman-Jafferis-Maldacena model in spatial dimension d=2d=2d=2, and the N=4\mathcal{N}=4N=4 super Yang-Mills theory in d=3d=3d=3.
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.
This thesis collects three works about holographic computations of entanglement entropy. In the first one it is shown how to compute numerically, following the Ryu-Takayanagi prescription, the entanglement entropy for arbitrarily shaped entangling regions in three dimensional conformal field theories. The other two focus on holographic theories with hyperscaling violating exponents: the time dependence after a holographic quench is analyzed and the arising of new universal terms due to the presence of non smooth boundaries is shown.