Bipartite fidelity for models with periodic boundary conditions

TL;DR

This paper investigates the bipartite fidelity in critical lattice models with periodic boundary conditions, deriving conformal field theory predictions and confirming them through exact and numerical lattice calculations for free-fermionic models.

Contribution

It provides the first derivation of conformal field theory predictions for bipartite fidelity with periodic boundaries and validates these predictions with lattice model computations.

Findings

Conformal field theory accurately predicts the leading terms of bipartite fidelity.

Lattice calculations for the XX spin chain and dense polymers agree with CFT predictions.

Asymptotic analysis confirms the theoretical framework for models with periodic boundary conditions.

Abstract

For a given statistical model, the bipartite fidelity $\mathcal F$ is computed from the overlap between the groundstate of a system of size $N$ and the tensor product of the groundstates of the same model defined on two subsystems $A$ and $B$, of respective sizes $N_A$ and $N_B$ with $N = N_A + N_B$. In this paper, we study $\mathcal F$ for critical lattice models in the case where the full system has periodic boundary conditions. We consider two possible choices of boundary conditions for the subsystems $A$ and $B$, namely periodic and open. For these two cases, we derive the conformal field theory prediction for the leading terms in the $1/N$ expansion of $\mathcal F$, in a most general case that corresponds to the insertion of four and five fields, respectively. We provide lattice calculations of $\mathcal F$, both exact and numerical, for two free-fermionic lattice models: the XX spin chain and the model of critical dense polymers. We study the asymptotic behaviour of the lattice results for these two models and find an agreement with the predictions of conformal field theory.