arXiv subjects

Christopher P. Herzog

Publications and source records attributed to Christopher P. Herzog.

At least 37 records · Page 2Linked to original sources

arXiv2020

Interface Conformal Anomalies

We consider two $d \geq 2$ conformal field theories (CFTs) glued together along a codimension one conformal interface. The conformal anomaly of such a system contains both bulk and interface contributions. In a curved-space setup, we compute the heat kernel coefficients and interface central charges in free theories. The results are consistent with the known boundary CFT data via the folding trick. In $d=4$, two interface invariants generally allowed as anomalies turn out to have vanishing interface charges. These missing invariants are constructed from components with odd parity with respect to flipping the orientation of the defect. We conjecture that all invariants constructed from components with odd parity may have vanishing coefficient for symmetric interfaces, even in the case of interacting interface CFT.

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arXiv2020

The $O(N)$ model with $ϕ^6$ potential in ${\mathbb R}^2 \times {\mathbb R}^+$

We study the large $N$ limit of $O(N)$ scalar field theory with classically marginal $ϕ^6$ interaction in three dimensions in the presence of a planar boundary. This theory has an approximate conformal invariance at large $N$. We find different phases of the theory corresponding to different boundary conditions for the scalar field. Computing a one loop effective potential, we examine the stability of these different phases. The potential also allows us to determine a boundary anomaly coefficient in the trace of the stress tensor. We further compute the current and stress-tensor two point functions for the Dirichlet case and decompose them into boundary and bulk conformal blocks. The boundary limit of the stress tensor two point function allows us to compute the other boundary anomaly coefficient. Both anomaly coefficients depend on the approximately marginal $ϕ^6$ coupling.

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arXiv2020

The Large D Limit of Einstein's Equations

We review recent progress in taking the large dimension limit of Einstein's equations. Most of our analysis is classical in nature and concerns situations where there is a black hole horizon although we briefly discuss various extensions that include quantum gravitational effects. The review consists of two main parts: the first a discussion of general aspects of black holes and effective membrane theories in this large dimension limit, and the second a series of applications of this limit to interesting physical problems. The first part includes a discussion of quasinormal modes which leads naturally into a description of effective hydrodynamic-like equations that describe the near horizon geometry. There are two main approaches to these effective theories -- a fully covariant approach and a partially gauge-fixed one -- which we discuss in relation to each other. In the second part we divide the applications up into three main categories: the Gregory-Laflamme instability, black hole collisions and mergers, and the anti-de Sitter/conformal field theory correspondence (AdS/CFT). AdS/CFT posits an equivalence between a gravitational theory and a strongly interacting field theory, allowing us to extend our spectrum of applications to problems in hydrodynamics, condensed matter physics, and nuclear physics. A final, shorter part of the review describes further promising directions where there have been, as yet, few published research articles.

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arXiv2020

On Marginal Operators in Boundary Conformal Field Theory

The presence of a boundary (or defect) in a conformal field theory allows one to generalize the notion of an exactly marginal deformation. Without a boundary, one must find an operator of protected scaling dimension $Δ$ equal to the space-time dimension $d$ of the conformal field theory, while with a boundary, as long as the operator dimension is protected, one can make up for the difference $d-Δ$ by including a factor $z^{Δ-d}$ in the deformation where $z$ is the distance from the boundary. This coordinate dependence does not lead to a reduction in the underlying $SO(d,1)$ global conformal symmetry group of the boundary conformal field theory. We show that such terms can arise from boundary flows in interacting field theories. Ultimately, we would like to be able to characterize what types of boundary conformal field theories live on the orbits of such deformations. As a first step, we consider a free scalar with a conformally invariant mass term $z^{-2} ϕ^2$, and a fermion with a similar mass. We find a connection to double trace deformations in the AdS/CFT literature.

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arXiv2020

How a-type anomalies can depend on marginal couplings

Even dimensional defects and boundaries in conformal field theory support type $a$ anomalies on their world-volume. We show that the one-point functions of marginal operators, in the presence of defects and boundaries, are anomalous, and that the Wess-Zumino consistency condition relates them to the derivative of the $a$-anomaly with respect to the marginal coupling. We also argue that the constant term $F$ for odd dimensional surfaces can depend on marginal parameters.

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arXiv2020

Duality and Transport for Supersymmetric Graphene from the Hemisphere Partition Function

We use localization to compute the partition function of a four dimensional, supersymmetric, abelian gauge theory on a hemisphere coupled to charged matter on the boundary. Our theory has eight real supercharges in the bulk of which four are broken by the presence of the boundary. The main result is that the partition function is identical to that of ${\mathcal N}=2$ abelian Chern-Simons theory on a three-sphere coupled to chiral multiplets, but where the quantized Chern-Simons level is replaced by an arbitrary complexified gauge coupling $τ$. The localization reduces the path integral to a single ordinary integral over a real variable. This integral in turn allows us to calculate the scaling dimensions of certain protected operators and two-point functions of abelian symmetry currents at arbitrary values of $τ$. Because the underlying theory has conformal symmetry, the current two-point functions tell us the zero temperature conductivity of the Lorentzian versions of these theories at any value of the coupling. We comment on S-dualities which relate different theories of supersymmetric graphene. We identify a couple of self-dual theories for which the complexified conductivity associated to the U(1) gauge symmetry is $τ/2$.

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arXiv2018

Holographic Bjorken Flow at Large-D

We use gauge/gravity duality to study the dynamics of strongly coupled gauge theories undergoing boost invariant expansion in an arbitrary number of space-time dimensions (D). By keeping the scale of the late-time energy density fixed, we explore the infinite-D limit and study the first few corrections to this expansion. In agreement with other studies, we find that the large-D dynamics are controlled by hydrodynamics and we use our computation to constrain the leading large-D dependence of a certain combination of transport coefficients up to 6-th order in gradients. Going beyond late time physics, we discuss how non-hydrodynamic modes appear in the large-D expansion in the form of a trans-series in D, identical to the non-perturbative contributions to the gradient expansion. We discuss the consequence of this trans-series in the non-convergence of the large-D expansion.

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arXiv2018

Superconformal Models for Graphene and Boundary Central Charges

In the context of boundary conformal field theory, we investigate whether the boundary trace anomaly can depend on marginal directions in the presence of supersymmetry. Recently, it was found that a graphene-like non-supersymmetric conformal field theory with a four-dimensional bulk photon and a three-dimensional boundary electron has two boundary central charges that depend on an exactly marginal direction, namely the gauge coupling. In this work, we supersymmetrize this theory, paying special attention to the boundary terms required by supersymmetry. We study models with 4, 8, and 16 Poincaré supercharges in the bulk, half of which are broken by the boundary. In all cases, we find that at all orders in perturbation theory, the gauge coupling is not renormalized, providing strong evidence that these theories are boundary conformal field theories. Moreover, the boundary central charges depend on the coupling. One possible exception to this dependence on marginal directions is that the difference between the two charges is coupling independent at one-loop in the maximally supersymmetric case. In our analysis, a possible boundary Chern-Simons term is incorporated by a bulk $θ$-term.

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arXiv2017

The Large Dimension Limit of a Small Black Hole Instability in Anti-de Sitter Space

We study the dynamics of a black hole in an asymptotically $AdS_d \times S^d$ space-time in the limit of a large number of dimensions, $d \to \infty$. Such a black hole is known to become dynamically unstable below a critical radius. We derive the dispersion relation for the quasinormal mode that governs this instability in an expansion in $1/d$. We also provide a full nonlinear analysis of the instability at leading order in $1/d$. We find solutions that resemble the lumpy black spots and black belts previously constructed numerically for small $d$, breaking the $SO(d+1)$ rotational symmetry of the sphere down to $SO(d)$. We are also able to follow the time evolution of the instability. Due possibly to limitations in our analysis, our time dependent simulations do not settle down to stationary solutions. This work has relevance for strongly interacting gauge theories; through the AdS/CFT correspondence, the special case $d=5$ corresponds to maximally supersymmetric Yang-Mills theory on a spatial $S^3$ in the microcanonical ensemble and in a strong coupling and large number of colors limit.

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arXiv2017

Boundary Conformal Field Theory and a Boundary Central Charge

We consider the structure of current and stress tensor two-point functions in conformal field theory with a boundary. The main result of this paper is a relation between a boundary central charge and the coefficient of a displacement operator correlation function in the boundary limit. The boundary central charge under consideration is the coefficient of the product of the extrinsic curvature and the Weyl curvature in the conformal anomaly. Along the way, we describe several auxiliary results. Three of the more notable are as follows: (1) we give the bulk and boundary conformal blocks for the current two-point function; (2) we show that the structure of these current and stress tensor two-point functions is essentially universal for all free theories; (3) we introduce a class of interacting conformal field theories with boundary degrees of freedom, where the interactions are confined to the boundary. The most interesting example we consider can be thought of as the infrared fixed point of graphene. This particular interacting conformal model in four dimensions provides a counterexample of a previously conjectured relation between a boundary central charge and a bulk central charge. The model also demonstrates that the boundary central charge can change in response to marginal deformations.

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arXiv2016

The Edge of Entanglement: Getting the Boundary Right for Non-Minimally Coupled Scalar Fields

In entanglement computations for a free scalar field with coupling to background curvature, there is a boundary term in the modular Hamiltonian which must be correctly specified in order to get sensible results. We focus here on the entanglement in flat space across a planar interface and (in the case of conformal coupling) other geometries related to this one by Weyl rescaling of the metric. For these "half-space entanglement" computations, we give a new derivation of the boundary term and revisit how it clears up a number of puzzles in the literature, including mass corrections and twist operator dimensions. We also discuss how related boundary terms may show up in other field theories.

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arXiv2016

The holographic dual of a Riemann problem in a large number of dimensions

We study properties of a non equilibrium steady state generated when two heat baths are initially in contact with one another. The dynamics of the system we study are governed by holographic duality in a large number of dimensions. We discuss the "phase diagram" associated with the steady state; the dual, dynamical, black hole description of this problem; and its relation to the fluid/gravity correspondence.

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arXiv2016

Estimation for Entanglement Negativity of Free Fermions

In this letter we study the negativity of one dimensional free fermions. We derive the general form of the $\mathbb{Z}_{N}$ symmetric term in moments of the partial transposed (reduced) density matrix, which is an algebraic function of the end points of the system. Such a path integral turns out to be a convenient tool for making estimations for the negativity.

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arXiv2015

Relativistic Hydrodynamics and Non-Equilibrium Steady States

We review recent interest in the relativistic Riemann problem as a method for generating a non-equilibrium steady state. In the version of the problem under con- sideration, the initial conditions consist of a planar interface between two halves of a system held at different temperatures in a hydrodynamic regime. The new double shock solutions are in contrast with older solutions that involve one shock and one rarefaction wave. We use numerical simulations to show that the older solutions are preferred. Briefly we discuss the effects of a conserved charge. Finally, we discuss deforming the relativistic equations with a nonlinear term and how that deformation affects the temperature and velocity in the region connecting the asymptotic fluids.

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arXiv2015

Universal Entanglement and Boundary Geometry in Conformal Field Theory

Employing a conformal map to hyperbolic space cross a circle, we compute the universal contribution to the vacuum entanglement entropy (EE) across a sphere in even-dimensional conformal field theory. Previous attempts to derive the EE in this way were hindered by a lack of knowledge of the appropriate boundary terms in the trace anomaly. In this paper we show that the universal part of the EE can be treated as a purely boundary effect. As a byproduct of our computation, we derive an explicit form for the A-type anomaly contribution to the Wess-Zumino term for the trace anomaly, now including boundary terms. In d=4 and 6, these boundary terms generalize earlier bulk actions derived in the literature.

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arXiv2015

Thermal Corrections to Rényi entropies for Free Fermions

We calculate thermal corrections to Rényi entropies for free massless fermions on a sphere. More specifically, we take a free fermion on $\mathbb{R}\times\mathbb{S}^{d-1}$ and calculate the leading thermal correction to the Rényi entropies for a cap like region with opening angle $2θ$. By expanding the density matrix in a Boltzmann sum, the problem of finding the Rényi entropies can be mapped to the problem of calculating a two point function on an $n$ sheeted cover of the sphere. We follow previous work for conformal field theories to map the problem on the sphere to a conical region in Euclidean space. By using the method of images, we calculate the two point function and recover the Rényi entropies.

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arXiv2015

Thermal Corrections to Renyi Entropies for Conformal Field Theories

We compute thermal corrections to Rényi entropies of $d$ dimensional conformal field theories on spheres. Consider the $n$th Rényi entropy for a cap of opening angle $2 θ$ on $S^{d-1}$. From a Boltzmann sum decomposition and the operator-state correspondence, the leading correction is related to a certain two-point correlation function of the operator (not equal to the identity) with smallest scaling dimension. More specifically, via a conformal map, the correction can be expressed in terms of the two-point function on a certain conical space with opening angle $2πn$. In the case of free conformal field theories, this two-point function can be computed explicitly using the method of images. We perform the computation for the conformally coupled scalar. From the $n \to 1$ limit of our results, we extract the leading thermal correction to the entanglement entropy, reproducing results of arXiv:1407.1358.

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arXiv2014

Universal Thermal Corrections to Entanglement Entropy for Conformal Field Theories on Spheres

We consider entanglement entropy of a cap-like region for a conformal field theory living on a sphere times a circle in d space-time dimensions. Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading correction to the entanglement entropy in a low temperature expansion. The correction has a universal form for any conformal field theory that depends only on the size of the mass gap, its degeneracy, and the angular size of the cap. We confirm our result by calculating the entanglement entropy of a conformally coupled scalar numerically. We argue that an apparent discrepancy for the scalar can be explained away through a careful treatment of boundary terms. In an appendix, to confirm the accuracy of the numerics, we study the mutual information of two cap-like regions at zero temperature.

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