The BCS wave function, matrix product states, and the Ising conformal field theory

TL;DR

This paper characterizes many-body lattice wave functions derived from Ising conformal field theory, showing they can be expressed as BCS states, and connects these to topological phases like the p+ip superconductor.

Contribution

It provides a novel matrix product state formalism for Ising CFT wave functions, proves their BCS form, and links 2D configurations to topological superconductor phases.

Findings

States can be written as BCS states

Ground state of critical Ising Hamiltonian obtained from this construction

2D configurations relate to p+ip superconductor phases

Abstract

We present a characterization of the many-body lattice wave functions obtained from the conformal blocks (CBs) of the Ising conformal field theory (CFT). The formalism is interpreted as a matrix product state using continuous ancillary degrees of freedom. We provide analytic and numerical evidence that the resulting states can be written as BCS states. We give a complete proof that the translationally invariant 1D configurations have a BCS form and we find suitable parent Hamiltonians. In particular, we prove that the ground state of the finite-size critical Ising transverse field (ITF) Hamiltonian can be obtained with this construction. Finally, we study 2D configurations using an operator product expansion (OPE) approximation. We associate these states to the weak pairing phase of the $p+ip$ superconductor via the scaling of the pairing function and the entanglement spectrum.