Entanglement entropy and boundary renormalization group flow: Exact results in the Ising universality class

Abstract

The entanglement entropy in one dimensional critical systems with boundaries has been associated with the noninteger ground state degeneracy. This quantity, being a characteristic of boundary fixed points, decreases under renormalization group flow, as predicted by the g-theorem. Here, using conformal field theory methods, we exactly calculate the entanglement entropy in the boundary Ising universality class. Our expression can be separated into the well known bulk term and a boundary entanglement term, displaying a universal flow between two boundary conditions, in accordance with the g-theorem. These results are obtained within the replica trick approach, where we show that the associated twist field, a central object generating the geometry of an n-sheeted Riemann surface, can be bosonized, giving simple analytic access to multiple quantities of interest. We argue that our result applies to other models falling into the same universality class. This includes the vicinity of the quantum critical point of the two-channel Kondo model, allowing to track in real space the presence of a region containing one half of a qubit with entropy 1 2 log(2), associated with a free local Majorana fermion.

Universal finite-size scaling around tricriticality between topologically ordered, symmetry-protected topological, and trivial phases

“…It is depicted in Fig.(14). This result is consistent with that reported in the literature on the universal flow of boundary entropy 33,38,39 .…”

“…. Operatorsψ L/R correspond to two localized Majorana fermion operators, satisfying 2ψ 2 L/R = 1, and the mass term, m(x), is given by 54) is consistent with the action proposed in the literature 33,34 , which is used to describe the boundary Ising chain. In the model of boundary Ising chain , boundary magnetic field drives the RG flow from free boundary condition 35,36 to fixed boundary condition.…”

Measuring fermionic entanglement: Entropy, negativity, and spin structure

“…One may unify the even/odd expressions by absorbing the fermionic minus sign in a relabelling of the FT index k as to run over half integers for even n, such thatÛ fer A = (n−1)/2 k=−(n−1)/2 e 2πik nN A k . This was noticed by the conformal field theory community and used to solve for the entropies in critical fermionic systems [90][91][92][93][94]; we herein showed this to hold in fact for any (non-critical) charge conserving system.…”

Entanglement and symmetry resolution in two dimensional free quantum field theories

“…We can write the complex fermionic field as ψ k ∼ e iϕ k and the twist field as T n,k,α (z) = e i( k n + α 2πn )ϕk . By introducing the vertex operators V β (z) = e iβϕ(z) , the twist fields take the form T n,k,α (z) = [46,75]. Let us observe that at first sight this result could be misleading since the outcome for bosons in eq.…”