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arXiv:1205.5291 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Holographic order parameter for charge fractionalization

Sean A. Hartnoll, Đorđe Radičević

Published Sep 5, 2012• 23 citations
arXivHTMLPDF

Abstract

Nonlocal order parameters for deconfinement, such as the entanglement entropy and Wilson loops, depend on spatial surfaces $\ensuremath{\Sigma}$. These observables are given holographically by the area of a certain bulk spatial surface $\ensuremath{\Gamma}$ ending on $\ensuremath{\Sigma}$. At finite charge density it is natural to consider the electric flux through the bulk surface $\ensuremath{\Gamma}$ in addition to its area. We show that this flux provides a refined order parameter that can distinguish ``fractionalized'' phases, with charged horizons, from what we term ``cohesive'' phases, with charged matter in the bulk. Fractionalization leads to a volume law for the flux through the surface, the flux for deconfined but cohesive phases is between a boundary and a volume law, while finite density confined phases have vanishing flux through the surface. We suggest two possible field theoretical interpretations for this order parameter. The first is as information extracted from the large $N$ reduced density matrix associated with $\ensuremath{\Sigma}$. The second is as surface operators dual to polarized bulk ``D-branes,'' carrying an electric dipole moment.

DOI
10.1103/physrevd.86.066001

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Nonlocal order parameters for deconfinement, such as the entanglement entropy and Wilson loops, depend on spatial surfaces \ensuremathΣ\ensuremath{\Sigma}\ensuremathΣ. These observables are given holographically by the area of a certain bulk spatial surface \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ ending on \ensuremathΣ\ensuremath{\Sigma}\ensuremathΣ. At finite charge density it is natural to consider the electric flux through the bulk surface \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ in addition to its area. We show that this flux provides a refined order parameter that can distinguish fractionalized'' phases, with charged horizons, from what we term cohesive'' phases, with charged matter in the bulk. Fractionalization leads to a volume law for the flux through the surface, the flux for deconfined but cohesive phases is between a boundary and a volume law, while finite density confined phases have vanishing flux through the surface. We suggest two possible field theoretical interpretations for this order parameter. The first is as information extracted from the large NNN reduced density matrix associated with \ensuremathΣ\ensuremath{\Sigma}\ensuremathΣ. The second is as surface operators dual to polarized bulk ``D-branes,'' carrying an electric dipole moment.
We study in detail various information theoretic quantities with the intent of distinguishing between different charged sectors in fractionalized states of gauge theories. For concreteness, we focus on a simple holographic (2+1)(2+1)(2+1)-dimensional strongly coupled electron fluid whose charged states organize themselves into a fractionalized and coherent patterns at sufficiently low temperatures. However, we expect that our results are quite generic and applicable to a wide range of systems, including non-holographic. The probes we consider include the entanglement entropy, mutual information, entanglement of purification and the butterfly velocity. The latter turns out to be particularly useful, given the universal connection between momentum and charge diffusion in the vicinity of a black hole horizon. The RT surfaces used to compute the above quantities, though, are largely insensitive to the electric flux in the bulk. To overcome this challenge, we propose a generalized entanglement functional that is motivated through the Iyer-Wald formalism, applied to a gravity theory coupled to a U(1)U(1)U(1) gauge field. We argue that this functional gives rise to a coarse grained measure of entanglement in the boundary theory which is obtained by tracing over (part) of the fractionalized and cohesive charge degrees of freedom. Based on the above, we construct a candidate for an entropic ccc-function that accounts for the existence of bulk charges. We explore some of its general properties and their significance, and discuss how it can be used to efficiently account for charged degrees of freedom across different energy scales.
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We compute holographic entanglement entropy in two strongly coupled nonlocal field theories: the dipole and the noncommutative deformations of SYM theory. We find that entanglement entropy in the dipole theory follows a volume law for regions smaller than the length scale of nonlocality and has a smooth cross-over to an area law for larger regions. In contrast, in the noncommutative theory the entanglement entropy follows a volume law for up to a critical length scale at which a phase transition to an area law occurs. The critical length scale increases as the UV cutoff is raised, which is indicative of UV/IR mixing and implies that entanglement entropy in the noncommutative theory follows a volume law for arbitrary large regions when the size of the region is fixed as the UV cutoff is removed to infinity. Comparison of behaviour between these two theories allows us to explain the origin of the volume law. Since our holographic duals are not asymptotically AdS, minimal area surfaces used to compute holographic entanglement entropy have novel behaviours near the boundary of the dual spacetime. We discuss implications of our results on the scrambling (thermalization) behaviour of these nonlocal field theories. ar X iv
We identify conditions for the entanglement entropy as a function of spatial region to be compatible with causality in an arbitrary relativistic quantum field theory. We then prove that the covariant holographic entanglement entropy prescription (which relates entanglement entropy of a given spatial region on the boundary to the area of a certain extremal surface in the bulk) obeys these conditions, as long as the bulk obeys the null energy condition. While necessary for the validity of the prescription, this consistency requirement is quite nontrivial from the bulk standpoint, and therefore provides important additional evidence for the prescription. In the process, we introduce a codimension-zero bulk region, named the entanglement wedge, naturally associated with the given boundary spatial region. We propose that the entanglement wedge is the most natural bulk region corresponding to the boundary reduced density matrix.
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.
It was proposed by Ryu and Takayanagi that the entanglement entropy in conformal field theory (CFT) is related through the AdS/CFT correspondence to the area of a minimal surface in the bulk. We apply this holographic geometrical method of calculating the entanglement entropy to study the vacuum case of a CFT which is holographically dual to empty anti-de Sitter (AdS) spacetime. We present all possible minimal surfaces spanned on one or two spherical boundaries at AdS infinity. We give exact analytical expressions for the regularized areas of these surfaces and identify finite renormalized quantities. In the case of two disjoint boundaries the existence of two different phases of the entanglement entropy is confirmed. A trivial phase corresponds to two disconnected minimal surfaces, while the other one corresponds to a tube connecting the spherical boundaries. A transition between these phases is reminiscent of the finite temperature deconfinement transition in the CFT on the boundary. The exact analytical results are thus consistent with previous numerical and approximate computations. We also briefly discuss the character of a spacetime extension of the minimal surface spanned on two uniformly accelerated boundaries.