Zero-energy states in conformal field theory with sine-square deformation

TL;DR

This paper investigates the spectral properties of sine-square deformed conformal field theories, revealing the uniqueness of the zero-energy vacuum state and introducing a regularization method to analyze zero-energy states systematically.

Contribution

It demonstrates the absence of finite norm eigenstates in SSD CFTs except the vacuum and introduces a regularized Hamiltonian related by a unitary transformation for systematic analysis.

Findings

No finite norm eigenstates other than the vacuum in SSD CFTs.

A regularized SSD Hamiltonian is unitarily equivalent to the undeformed Hamiltonian.

The regularization enables computation of expectation values in non-normalizable zero-energy states.

Abstract

We study the properties of two-dimensional conformal field theories (CFTs) with sine-square deformation (SSD). We show that there are no eigenstates of finite norm for the Hamiltonian of a unitary CFT with SSD, except for the zero-energy vacuum state $|0\rangle$. We then introduce a regularized version of the SSD Hamiltonian which is related to the undeformed Hamiltonian via a unitary transformation corresponding to the Mobius quantization. The unitary equivalence of the two Hamiltonians allows us to obtain zero-energy states of the deformed Hamiltonian in a systematic way. The regularization also provides a way to compute the expectation values of observables in zero-energy states that are not necessarily normalizable.