Entanglement in Quantum Critical Phenomena, Holography and Gravity Dmitri V. Fursaev Joint Institute for Nuclear Research Dubna, RUSSIA Banff, July 31, 2006 hep-th/ hep-th/

gravity - quantum information - condensed matter finding entanglement entropy in spin chains near a critical point finding a minimal surface in a curved space one dimension higher

plan of the talk ● quantum entanglement in 2D critical phenomena and CFT’s ● geometrical structure of entanglement entropy ● gravitational coupling in quantum gravity and entanglement entropy ● new gravity analogs in condensed matter systems (applications) ● “holographic formula” for entanglement entropy (in QFT’s dual to AdS gravity)

Quantum Entanglement Quantum state of particle «1» cannot be described independently from particle «2» (even for spatial separation at long distances)

measure of entanglement - entropy of entanglement density matrix of particle «2» under integration over the states of «1» «2» is in a mixed state when information about «1» is not available S – measures the loss of information about “1” (or “2”)

Ising spin chains off-critical regime at large N critical regime

RG-evolution of the entropy entropy does not increase under RG-flow (as a result of integration of high energy modes) IR UV is UV fixed point

Explanation Near the critical point the Ising model is equivalent to a 2D quantum field theory with mass m proportional to At the critical point it is equivalent to a 2D CFT with 2 massless fermions each having the central charge 1/2

Entanglement in 2D models: analytical results a is a UV cutoff Calabrese, Cardy hep-th/ ground state entanglement on an interval massive case: massless case: is the length of

analytical results (continued) ground state entanglement for a system on a circle system at a finite temperature is the length of

effective action and geometrical structure of entanglement entropy -effective action is defined on manifolds with cone-like singularities - “inverse temperature” - “partition function”

example: 2D theory at a finite temperature T case conical singularity is located at the separating point - standard partition function

effective action on a manifold with conical singularities is the gravity action (even if the manifold is locally flat) curvature at the singularity is non-trivial: derivation of entanglement entropy in a flat space has to do with gravity effects!

many-body systems in higher dimensions spin lattice continuum limit A – area of a flat separation surface B which divides the system into two parts (pure quantum states!) entropy per unit area in a QFT is determined by a UV cutoff!

geometrical structure of the entropy edge ( L = number of edges) separating surface (of area A ) sharp corner ( C = number of corners) (method of derivation: spectral geometry) (Fursaev, hep-th/ ) for ground state a is a cutoff C – topological term (first pointed out in D=3 by Preskill and Kitaev)

gravitational coupling - gravitational force between two bodies is determined by the microscopical properties of a fundamental theory

● gravitational constant as a measure of quantum entanglement in the fundamental theory - entanglement entropy per unit area for degrees of freedom of the fundamental theory in a flat space CONJECTURE (Fursaev, hep-th/ )

arguments: ● entropy density is determined by UV-cutoff ● the conjecture is valid for area density of the entropy of black holes ● entanglement entropy can be derived form the effective gravity action ● entropy in QFT’s which admit AdS duals

BLACK HOLE THERMODYNAMICS Bekenstein-Hawking entropy - area of the horizon - measure of the loss of information about states under the horizon

some references: ● black hole entropy as the entropy of entanglement (Srednicki 93, Sorkin et al 86) ● iduced gravity (Sakharov 68) as a condition (Jacobson 94, Frolov, Fursaev, Zelnikov 96) ● application to de Sitter horizon (Hawking, Maldacena, Strominger 00) ● entropy of certain type black holes in string theory as the entanglement entropy in 2- and 3- qubit systems (Duff 06, Kallosh & Linde 06) ● yields the value for the fundamental entropy in flat space in terms of gravity coupling ● horizon entropy is a particular case our conjecture :

● applications: new gravity analogs in condensed matter systems In condensed matter systems one can define an effective gravity constant whereis the ground state entanglement entropy per unit area Requirements: ● lattice models (cutoff) ● second order phase transition ● description in terms of a massive QFT near the critical point Advantage: one does not need to introduce effective metric in the system

theories with extra dimensions the conjecture should hold in higher dimensions: fundamental entanglement entropy per unit area of the separating surface is is the higher-dimensional gravitational coupling What is the separating surface in higher dimensions? ● Kaluza-Klein-like theories: ● brane-world models (only gravity is higher dimensional): - space of extra dimensions extension of the separating surface to higher dimensions has to be determined by the dynamical gravity equations in the bulk is

Holographic Formula for the Entropy Ryu and Takayanagi, hep-th/ , CFT which admit a dual description in terms of the Anti-de Sitter (AdS) gravity one dimension higher Let be the extension of the separating surface in d-dim. CFT 1) is a minimal surface in (d+1) dimensional AdS space 2) “holographic formula” holds: is the area of is the gravity coupling in AdS

the holographic formula enables one to compute entanglement entropy in strongly coupled theories by using geometrical methods

example in d=2: CFT on a circle - AdS radius A is the length of the geodesic in AdS - UV cutoff -holographic formula reproduces the entropy for a ground state entanglement - central charge in d=2 CFT

Sketch of the proof of the holographic formula Fursaev, hep-th/ AdS/CFT representation for CFT partition function (with specific boundary conditions) is (a conformal) boundary of (3D AdS / 2D CFT)

the proof (continued) in semiclassical approximation extremality of the action requires be a minimal surface there are conical singularities in the bulk located on

consequences possibility to consider entropy in stationary but not static theories (Riemannian sections) choice of the minimal surface in case of several options theories with different phases and phase transitions higher-curvature corrections in the bulk entropy in brane-world models (Randall and Sundrum)

choice of the minimal surface in finite-temperature cases and topology Euclidean BTZ black hole slice of the torus The bulk manifold is obtained by cutting and gluing along n copies of the torus

Summary - Entanglement and critical phenomena in condensed matter systems (d=2,...) - Entanglement in quantum gravity: relation to gravity coupling in a fundamental theory -New gravity analogs in condensed matter (lattice models) - “Holographic” representation of entanglement entropy: geometrical way of computation + new ideas