Irreversibility of RG flows from relative entropy

Horacio Casini Instituto Balseiro, Centro Atómico Bariloche

In collaboration with Ignacio Salazar, Eduardo Teste, Gonzalo Torroba

Relative entropy

Two states in the same Hilbert space or algebra

is the modular Hamiltonian or entanglement Hamiltonian

Relative entropy measures distinguishability of two states (roughly inverse of

number of experiments needed to distinguish them)

It is positive and increasing with the size of the region: States become more

distinguishable for larger regions.

Boundary entropy. g-theorem Gonzalo Torroba, Ignacio Salazar, H.C.

CFT in 1+1 with boundary conditions or an impurity at the origin. Local interaction at x=0 (local Hamiltonian

description =bulk locality)

Thermal entropy. Boundary entropy Affleck, Ludwig (1991)

g-theorem Friedan, Konechny (2004)

Entanglement entropy of an interval containing the origin

At the fix points conformal boundary condition. Classified Cardy (1989)

But in general g(T) and g(R) not easily related to each other

Calabrese, Cardy (2004)

A simple example: a free «Kondo» model

The model interpolates between the conformal boundary conditions

IR

UV

Momentum dependent reflection coefficient

Log(g) goes from log(2) in the UV to 0 in the IR

Thermal boundary entropy

Entanglement entropy. Boundary contribution

Is there a g-theorem in terms of entanglement entropy? Holographic version based on NEC in the bulk by Fujita, Takayanagi, Tonni (2011)

has the right sign: Relative entropy is increasing with size, g(r) would be decreasing

Stress tensor expectation values vanish in the bulk

Whatever the contribution in the origin, it is increasing linearly with r, ruining proof of monotonicity of g(r)

Numerics shows relative entropy increases linearly.

Modular Hamiltonian

This relative entropy distinguishes too much!

relevant boundary perturbation

The modular Hamiltonian on the impurity region vanishes as we move to the null horizon. Less distinguishability because «high temperature» at the impurity position.

Let us compare the states on a different Cauchy surfaces

The entropy is not modified because the evolution is unitary inside the diamond (local Hamiltonian evolution for full system)

It is «Rindler like» near the boundaries in null coordinates, and forgets about the value of r near the boundary

However the two states evolve with different unitaries. The expectation value of modular Hamiltonian will change with Cauchy surface

Relative entropy is increasing on the null Cauchy surface

In the free fermion model

C-theorem and «area theorem» from relative entropy Eduardo Testé, Gonzalo Torroba, HC

Relative entropy between two vacuum states of different theories.

We will use a CFT as reference state the other theory is obtained by perturbing with a relevant operator

We will use spheres since we know the modular Hamiltonian for spheres

For two states evolved with the same time independent local Hamiltonian we do not care about the choice of Cauchy surface, both DS and DK will be the same

«Interaction representation» in which the CFT state does not evolve and the other does

Or Cauchy surface dependent mapping between Hilbert spaces

Has zero trace but depends on the surface

It vanishes for the null surface. But can be justified only for

Converges for

for curvature scale a of the order of the cutoff

Modular Hamiltonian

is constant since the evolution is unitary inside the diamond

Converges for

Relative entropy on spatial surfaces:

Converges for

Dominated by modular Hamiltonian,

Relative entropy on null surface:

Converges for

Some consequences

d=2)

UV)

IR)

d>2)

The area term decreases towards the IR

Or Newton’s constant increases toward the IR

Conclusion: In terms of relative entropy these monotonicity theorems are a common QM phenomenon: as the algebra of operators get bigger we have more operators available to distinguish states and the relative entropy increases But relativity enters in two ways: First the modular Hamitonian of the CFT on spheres is local and universal in terms of Second, surprisingly, to get usefull information, we must not distinguish the states too much using the full power of the relative entropy. Moving the surface to the null horizon distinguishability decreases: Correlators along null directions are UV correlators. In other words: sometimes relative entropy is very large, corresponding to very different states. Still we can obtain tight constraints on the entropy difference using relative entropy if there is a unitary in the algebra that brings these two states to be much closer. Other meassures of distinguishability are monotonic, fidelity, etc. Higher dimensions?