Sine-square deformation of solvable spin chains and conformal field theories

TL;DR

This paper investigates sine-square deformation (SSD) in solvable spin chains, Dirac fermions, and conformal field theories, showing that SSD preserves ground states and suppresses boundary effects in critical systems.

Contribution

It provides an exact analytical understanding of SSD's effects on ground states and boundary suppression in critical quantum systems, connecting to Virasoro and Kac-Moody algebra structures.

Findings

Ground state of open SSD systems matches that of periodic uniform chains.

Vacuum state is an exact eigenstate and, for some CFTs, the ground state of the SSD Hamiltonian.

SSD effectively suppresses boundary effects in one-dimensional critical systems.

Abstract

We study solvable spin chains, one-dimensional massless Dirac fermions, and conformal field theories (CFTs) with sine-square deformation (SSD), in which the Hamiltonian density is modulated by the function $f(x)=\sin^2 (\pi x/\ell)$, where $x$ is the position and $\ell$ is the length of the system. For the XY chain and the transverse field Ising chain at criticality, it is shown that the ground state of an open system with SSD is identical to that of a uniform chain with periodic boundary conditions. The same holds for the massless Dirac fermions with SSD, corresponding to the continuum limit of the gapless XY chain. For general CFTs, we find that the Hamiltonian of a system with SSD has an expression in terms of the generators of the Virasoro algebra. This allows us to show that the vacuum state is an exact eigenstate of the sine-square deformed Hamiltonian. Furthermore, for a restricted class of CFTs associated with affine Lie (Kac-Moody) algebras, including $c=1$ Gaussian CFT, we prove that the vacuum is an exact ground state of the deformed Hamiltonian. This explains why the SSD has succeeded in suppressing boundary effects in one-dimensional critical systems, as observed in previous numerical studies.