Hidden Fermi surfaces in compressible states of gauge-gravity duality

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Hidden Fermi surfaces in compressible states of gauge-gravity duality

Liza Huijse, Subir Sachdev, and Brian Swingle

Phys. Rev. B 85, 035121 – Published 25 January 2012

Abstract

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, $z$ , and a specific hyperscaling violation exponent, $θ$ . 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 [e-print arXiv:1111.1023 (unpublished)].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.

Received 11 December 2011

DOI:https://doi.org/10.1103/PhysRevB.85.035121

Authors & Affiliations

Liza Huijse^1,2, Subir Sachdev^1,2, and Brian Swingle¹

¹Department of Physics, Harvard University, Cambridge MA 02138, USA
²Kavli Institute for Theoretical Physics, University of California, Santa Barbara CA 93106-4030, USA

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Vol. 85, Iss. 3 — 15 January 2012

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Images

Figure 1
Geometry of holographic entanglement. The $d$ spatial coordinates are $x_{i} \equiv (x_{j}, x_{d})$ , with $j = 1, ..., d - 1$ , and the emergent holographic direction is $r$ . The entangling region of the compressible quantum state is shown shaded; its boundary is the surface described in coordinate patches by $x_{d} = w (x_{j})$ , $j = 1, ..., d - 1$ , which has surface area $Σ$ . This boundary is extended along the holographic direction into the surface written locally as $x_{d} = W (r, x_{j})$ ; the holographic entanglement[55] is proportional to the surface area, $A$ , of this extended boundary. The area $A$ has to be minimized while keeping the surface $x_{d} = w (x_{j})$ fixed.Reuse & Permissions
Figure 2
Geometry of mutual entanglement bewteen regions $A$ and $B$ .Reuse & Permissions
Figure 3
Schematic illustration of the holographic geometries of various compressible phases. The phases are labeled non-Fermi liquid (NFL), fractionalized Fermi liquid (FL*), and Fermi liquid (FL), following the notation of Ref. 29. The boundary theory at $r = 0$ has total charge density $Q$ , which sources the bulk electric field $E_{r}$ ; we define $E_{r}$ to equal the left-hand side of Eq. (2.8). The NFL phase is described by the theory in Sec. 2 and has the IR metric (1.1). The shading in the bulk region represents the density of mesinos. The FL* phase is described in Sec. 4a using the Thomas-Fermi approximation. The FL* phase has mesinos that are in the bulk shaded region at $r \sim \sqrt{Q}$ , which corresponds to the shaded region in Fig. 5; the region $r ≫ \sqrt{Q}$ , with $E_{r} = Q - Q_{mesino}$ , determines the entanglement entropy. The FL phase is not described in the present paper: its geometry is confining and terminates at a finite $r$ where $E_{r} = 0$ , as discussed in Ref. 36.Reuse & Permissions
Figure 4
Ratio of fractionalized charge to total charge ${\hat{Q}}_{quark} / \hat{Q}$ as a function of the relevant coupling $φ_{0}$ for ${\hat{m}, \hat{β}} = {3 / 16, 10}$ . The black dot denotes the location of the transition between the FL* and NFL phases. In the FL* phase indicated by the blue drawn line, ${\hat{Q}}_{quark} / \hat{Q} < 1$ , whereas in the NFL phase indicated by the dashed red line, we have ${\hat{Q}}_{quark} / \hat{Q} = 1$ .Reuse & Permissions
Figure 5
The plot shows the minimal and maximal radii for which the fluid is present as a function of the relevant coupling $φ_{0}$ . The factors of $\hat{μ}$ ensure that all quantities are given in dimensionless units. Remember that the IR corresponds to $r \to \infty$ . The black dot denotes the location of the transition between the FL* and NFL phases. The fluid is present at all radii for which ${\hat{μ}}_{loc} (r) > \hat{m}$ , this is indicated by the shaded region; this corresponds to the shaded region in Fig. 3. It follows from scaling arguments that the fluid is present at $r \sim \sqrt{Q_{quark}}$ .Reuse & Permissions

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