Brian Swingle

Hidden Fermi surfaces in compressible states of gauge-gravity duality

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 specic hyperscaling violation exponent, . The same metric exhibits a logarithmic violation of the area law of entanglement entropy, as shown recently by Ogawa et al. (arXiv:1111.1023). 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 deconned gauge elds.

Entanglement Renormalization and Holography

We show how recent progress in real space renormalization group methods can be used to define a generalized notion of holography inspired by holographic dualities in quantum gravity. The generalization is based upon organizing information in a quantum state in terms of scale and defining a higher-dimensional geometry from this structure. While states with a finite correlation length typically give simple geometries, the state at a quantum critical point gives a discrete version of anti-de Sitter space. Some finite temperature quantum states include black hole-like objects. The gross features of equal time correlation functions are also reproduced.

Holographic Complexity Equals Bulk Action

We conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in anti-de Sitter spacetime, as well as black holes perturbed with static shells and with shock waves. This conjecture evolved from a previous conjecture that complexity is dual to spatial volume, but appears to be a major improvement over the original. In light of our results, we discuss the hypothesis that black holes are the fastest computers in nature.We conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in AdS, as well as black holes perturbed with static shells and with shock waves. This conjecture evolved from a previous conjecture that complexity is dual to spatial volume, but appears to be a major improvement over the original. In light of our results, we discuss the hypothesis that black holes are the fastest computers in nature. Published in PRL 116, 191301 as “Holographic Complexity Equals Bulk Action?” ar X iv :1 50 9. 07 87 6v 3 [ he pth ] 1 0 M ay 2 01 6 The interior of a black hole is the purest form of emergent space: once the black hole has formed, the interior grows linearly for an exponentially long time. One of the few holographic ideas about the black hole interior is that its growth is dual to the growth of quantum complexity [1, 2]. This duality is a conjecture but it has passed a number of tests. In the context of AdS/CFT duality, the conjecture has taken a fairly concrete form: the volume of a certain maximal spacelike slice, which extends into the black hole interior, is proportional to the computational complexity of the instantaneous boundary conformal field theory (CFT) state [3]. The conjecture is an example of the proposed connection between tensor networks and geometry—the geometry being defined by the smallest tensor network that prepares the state. (See also [1, 2, 4–8].) For the case of the two-sided AdS black hole the conjecture is schematically described by Complexity ∼ V G`AdS , (1) where V is the volume of the Einstein-Rosen bridge (ERB), `AdS is the radius of curvature of the AdS spacetime, and G is Newton’s constant. Multiplying and dividing Eq. 1 by `AdS suggests a new perspective on the identification of complexity and geometry, Complexity ∼ W G`AdS , (2) whereW ≡ `AdSV now has units of spacetime volume and represents the world volume of the ERB. Further noting that 1/`AdS is proportional to the cosmological constant of the AdS space inspires a new conjecture which we suspect may have deep implications for the connection between quantum information and gravitational dynamics. We propose: CA-conjecture: Complexity = Action π~ . (3) (The detailed calculations are presented in [9].) The systems we will consider are those whose low-energy bulk physics is described by the Einstein-Maxwell action

Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories

As experiments are increasingly able to probe the quantum dynamics of systems with many degrees of freedom, it is interesting to probe fundamental bounds on the dynamics of quantum information. We elaborate on the relationship between one such bound—the Lieb-Robinson bound—and the butterfly effect in strongly-coupled quantum systems. The butterfly effect implies the ballistic growth of local operators in time, which can be quantified with the “butterfly” velocity vB . Similarly, the Lieb-Robinson velocity places a state independent ballistic upper bound on the size of time evolved operators in non-relativistic lattice models. Here, we argue that vB is a state-dependent effective Lieb-Robinson velocity. We study the butterfly velocity in a wide variety of quantum field theories using holography and compare with free particle computations to understand the role of strong coupling. We find that, depending on the way length and time scale, vB acquires a temperature dependence and decreases towards the IR. We also comment on experimental prospects and on the relationship between the butterfly velocity and signaling.

Measuring the scrambling of quantum information

We provide a general protocol to measure out-of-time-order correlation functions. These correlation functions are of broad theoretical interest for diagnosing the scrambling of quantum information in interacting quantum systems and have recently received particular attention in the study of chaos and black holes within holographic duality. Measuring them requires an echo-type sequence in which the sign of a many-body Hamiltonian is reversed. We illustrate our protocol by detailing an implementation employing cold atoms and cavity quantum electrodynamics to probe spin models with nonlocal interactions. To verify the feasibility of the scheme with current technology, we analyze the effects of dissipation in a chaotic kicked-top model. Finally, we propose a number of other experimental platforms where similar out-of-time-order correlation functions can be measured.

Slow scrambling in disordered quantum systems

Recent work has studied the growth of commutators as a probe of chaos and information scrambling in quantum many-body systems. In this work we study the effect of static disorder on the growth of commutators in a variety of contexts. We find generically that disorder slows the onset of scrambling, and, in the case of a many-body localized state, partially halts it. We access the many-body localized state using a standard fixed point Hamiltonian, and we show that operators exhibit slow logarithmic growth under time evolution. We compare the result with the expected growth of commutators in both localized and delocalized non-interacting disordered models. Finally, based on a scaling argument, we state a conjecture about the effect of weak interactions on the growth of commutators in an interacting diffusive metal.

Entanglement Entropy and the Fermi Surface

Free fermions with a finite Fermi surface are known to exhibit an anomalously large entanglement entropy. The leading contribution to the entanglement entropy of a region of linear size L in d spatial dimensions is S∼L(d-1)logL, a result that should be contrasted with the usual boundary law S∼L(d-1). This term depends only on the geometry of the Fermi surface and on the boundary of the region in question. I give an intuitive account of this anomalous scaling based on a low energy description of the Fermi surface as a collection of one-dimensional gapless modes. Using this picture, I predict a violation of the boundary law in a number of other strongly correlated systems.

Quantum Butterfly Effect in Weakly Interacting Diffusive Metals

We study scrambling, an avatar of chaos, in a weakly interacting metal in the presence of random potential disorder. It is well known that charge and heat spread via diffusion in such an interacting disordered metal. In contrast, we show within perturbation theory that chaos spreads in a ballistic fashion. The squared anticommutator of the electron field operators inherits a light-cone like growth, arising from an interplay of a growth (Lyapunov) exponent that scales as the inelastic electron scattering rate and a diffusive piece due to the presence of disorder. In two spatial dimensions, the Lyapunov exponent is universally related at weak coupling to the sheet resistivity. We are able to define an effective temperature-dependent butterfly velocity, a speed limit for the propagation of quantum information, that is much slower than microscopic velocities such as the Fermi velocity and that is qualitatively similar to that of a quantum critical system with a dynamical critical exponent

Linearity of holographic entanglement entropy

z > 1

Geometric proof of the equality between entanglement and edge spectra

.