We consider relative entropy in Field Theory as a well defined (non-divergent) quantity of interest. We establish a monotonicity property with respect to the couplings in the theory. As a consequence, the relative entropy in a field theory with a hierarchy of renormalization group fixed points ranks the fixed points in decreasing order of criticality. We argue from a generalized $H$ theorem that Wilsonian RG flows induce an increase in entropy and propose the relative entropy as the natural quantity which increases from one fixed point to another in more than two dimensions.We consider relative entropy in Field Theory as a well defined (non-divergent) quantity of interest. We establish a monotonicity property with respect to the couplings in the theory. As a consequence, the relative entropy in a field theory with a hierarchy of renormalization group fixed points ranks the fixed points in decreasing order of criticality. We argue from a generalized $H$ theorem that Wilsonian RG flows induce an increase in entropy and propose the relative entropy as the natural quantity which increases from one fixed point to another in more than two dimensions.

arXiv:hep-th/9610040Oct 7, 19962 citations

Non-equilibrium entropy

S. V. Peletminsky, A.J. Sokolovsky

Physics Letters AMar 1, 19751 citation

Entanglement Entropy at Infinite-Randomness Fixed Points in Higher Dimensions

Yu‐Cheng Lin, Ferenc Iglói, Heiko Rieger

The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong-disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite-randomness fixed point. Here we identify a double-logarithmic multiplicative correction to the area law for the entanglement entropy. This contrasts with the pure area law valid at the infinite-randomness fixed point in the diluted transverse Ising model in higher dimensions.The entanglement entropy of the two-dimensional random transverse Ising model is studied with a numerical implementation of the strong-disorder renormalization group. The asymptotic behavior of the entropy per surface area diverges at, and only at, the quantum phase transition that is governed by an infinite-randomness fixed point. Here we identify a double-logarithmic multiplicative correction to the area law for the entanglement entropy. This contrasts with the pure area law valid at the infinite-randomness fixed point in the diluted transverse Ising model in higher dimensions.

arXiv:0704.0418Physical Review LettersOct 2, 200778 citations

A note on relative entropy

Rodney W. Benoist, Jean-Paul Marchand, Walter Wyss

Letters in Mathematical PhysicsMay 1, 19798 citations

Entanglement entropy in the O(N) model

Max A. Metlitski, Carlos A Fuertes, Subir Sachdev A

It is generally believed that in spatial dimension $d>1$ , the leading contribution to the entanglement entropy $S=\ensuremath{-}\text{tr}\text{ }{\ensuremath{\rho}}_{A}\text{ }\text{log}\text{ }{\ensuremath{\rho}}_{A}$ scales as the area of the boundary of subsystem $A$ . The coefficient of this ``area law'' is nonuniversal. However, in the neighborhood of a quantum critical point $S$ is believed to possess subleading universal corrections. In the present work, we study the entanglement entropy in the quantum $O(N)$ model in $1<d<3$ . We use an expansion in $ϵ=3\ensuremath{-}d$ to evaluate (i) the universal geometric correction to $S$ for an infinite cylinder divided along a circular boundary; (ii) the universal correction to $S$ due to a finite correlation length. Both corrections are different at the Wilson-Fisher and Gaussian fixed points, and the $ϵ\ensuremath{\rightarrow}0$ limit of the Wilson-Fisher fixed point is distinct from the Gaussian fixed point. In addition, we compute the correlation length correction to the Renyi entropy ${S}_{n}=\frac{1}{1\ensuremath{-}n}\text{log}\text{ }\text{tr}\text{ }{\ensuremath{\rho}}_{A}^{n}$ in $ϵ$ and large- $N$ expansions. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, this correction generally scales as ${N}^{2}$ rather than the naively expected $N$ . Moreover, the Renyi entropy has a phase transition as a function of $n$ for $d$ close to 3.It is generally believed that in spatial dimension $d>1$, the leading contribution to the entanglement entropy $S=\ensuremath{-}\text{tr}\text{ }{\ensuremath{\rho}}_{A}\text{ }\text{log}\text{ }{\ensuremath{\rho}}_{A}$ scales as the area of the boundary of subsystem $A$. The coefficient of this ``area law'' is nonuniversal. However, in the neighborhood of a quantum critical point $S$ is believed to possess subleading universal corrections. In the present work, we study the entanglement entropy in the quantum $O(N)$ model in $1<d<3$. We use an expansion in $ϵ=3\ensuremath{-}d$ to evaluate (i) the universal geometric correction to $S$ for an infinite cylinder divided along a circular boundary; (ii) the universal correction to $S$ due to a finite correlation length. Both corrections are different at the Wilson-Fisher and Gaussian fixed points, and the $ϵ\ensuremath{\rightarrow}0$ limit of the Wilson-Fisher fixed point is distinct from the Gaussian fixed point. In addition, we compute the correlation length correction to the Renyi entropy ${S}_{n}=\frac{1}{1\ensuremath{-}n}\text{log}\text{ }\text{tr}\text{ }{\ensuremath{\rho}}_{A}^{n}$ in $ϵ$ and large-$N$ expansions. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, this correction generally scales as ${N}^{2}$ rather than the naively expected $N$. Moreover, the Renyi entropy has a phase transition as a function of $n$ for $d$ close to 3.

arXiv:0904.4477Physical Review BSep 23, 2009182 citations

Weighted entropy

Silviu Guiaşu

Reports on Mathematical PhysicsSep 1, 1971156 citations

H

theorem that Wilsonian RG flows induce an increase in entropy and propose the relative entropy as the natural quantity which increases from one fixed point to another in more than two dimensions.

It is generally believed that in spatial dimension

d>1

, the leading contribution to the entanglement entropy $S=\ensuremath{-}\text{tr}\text{ }{\ensuremath{\rho}}_{A}\text{ }\text{log}\text{ }{\ensuremath{\rho}}_{A}$ scales as the area of the boundary of subsystem

A

. The coefficient of this ``area law'' is nonuniversal. However, in the neighborhood of a quantum critical point

S

is believed to possess subleading universal corrections. In the present work, we study the entanglement entropy in the quantum

O(N)

model in

1<d<3

. We use an expansion in $ϵ=3\ensuremath{-}d$ to evaluate (i) the universal geometric correction to

S

for an infinite cylinder divided along a circular boundary; (ii) the universal correction to

S

due to a finite correlation length. Both corrections are different at the Wilson-Fisher and Gaussian fixed points, and the $ϵ\ensuremath{\rightarrow}0$ limit of the Wilson-Fisher fixed point is distinct from the Gaussian fixed point. In addition, we compute the correlation length correction to the Renyi entropy ${S}_{n}=\frac{1}{1\ensuremath{-}n}\text{log}\text{ }\text{tr}\text{ }{\ensuremath{\rho}}_{A}^{n}$ in

ϵ

and large-

N

expansions. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, this correction generally scales as

{N}^{2}

rather than the naively expected

N

. Moreover, the Renyi entropy has a phase transition as a function of

n

for

d

close to 3.