Boundary Symmetry Breaking in CFT and the String Order Parameter

TL;DR

This paper investigates how boundary conditions in one-dimensional critical quantum systems break global symmetries, using string-order parameters within Boundary Conformal Field Theory to characterize and predict the symmetry breaking patterns and their scaling behaviors.

Contribution

It provides a detailed Boundary CFT analysis of boundary-induced symmetry breaking and introduces a logarithmic scaling law for the string-order parameter, supported by numerical tests.

Findings

Logarithmic scaling of the string-order parameter with system size.

Boundary conditions can induce symmetry breaking detectable by string-order parameters.

Numerical results agree with theoretical predictions for free fermionic systems.

Abstract

We consider the ground state of a one-dimensional critical quantum system carrying a global symmetry in the bulk, which is explicitly broken by its boundary conditions. We probe the system via a string-order parameter, showing how it detects the symmetry breaking pattern. We give a precise characterization of the mechanism depicted above in Boundary CFT, and we find a general logarithmic scaling for the order parameter. As a first example we analyze the breaking of a $U(1)$ symmetry for complex free theories induced by a boundary pairing term. Moreover, we give predictions for the breaking of $U(N)$ in free theories, arising from a boundary mixing. We test our predictions with numerical calculations for some lattice realizations of free fermionic system with boundary symmetry breaking, finding a good agreement.