The quantum Ising chain with a generalized defect

TL;DR

This paper investigates how generalized defects affect the finite-size scaling and integrability of the quantum Ising chain, revealing exact solutions for certain defects and numerical insights for others, with implications for conformal field theory.

Contribution

It provides an exact solution for the scaling spectrum with a universal defect preserving Z2 symmetry and explores the non-continuous effects of other defects on scaling dimensions.

Findings

Exact scaling spectrum for Z2-symmetric defect

Continuum limit matches ordinary defect results

Non-continuous dependence of scaling dimensions on defect parameters

Abstract

The finite-size scaling properties of the quantum Ising chain with different types of generalized defects are studied. These not only mean an alteration of the coupling constant as previously examined, but an additional arbitrary transformation in the algebra of observables at one site of the chain. One can distinguish between two classes of generalized defects: those which do not affect the finite-size integrability of the Ising chain, and on the other hand those that destroy this property. In this context, finite-size integrability is always understood as a synonym for the possibility to write the Hamiltonian of the finite chain as a bilinear expression in fermionic operators by means of a Jordan-Wigner transformation. Concerning the first type of defect, an exact solution for the scaling spectrum is obtained for the most universal defect that preserves the global Z_2 symmetry of the chain. It is shown that in the continuum limit this yields the same result as for one properly chosen `ordinary' defect, that is changing the coupling constant only, and thus the finite-size scaling spectra can be described by irreps of a shifted u(1) Kac-Moody algebra. The other type of defect is examined by means of numerical finite-size calculations. In contrast to the first case, these suggest a non-continuous dependence of the scaling dimensions on the defect parameters. A conjecture for the operator content involving only one primary field of a Virasoro algebra with central charge c=1/2 is given.