Conformal data and renormalization group flow in critical quantum spin chains using periodic uniform matrix product states

TL;DR

This paper demonstrates that periodic uniform Matrix Product States can accurately approximate low-energy states in critical quantum spin chains, enabling precise extraction of conformal data and nonperturbative investigation of renormalization group flows between conformal field theories.

Contribution

It shows that puMPS Bloch states can be used to identify low-energy eigenstates with their CFT counterparts and analyze RG flows nonperturbatively.

Findings

Accurate extraction of conformal data from low-energy spectra.

Nonperturbative analysis of RG flow between CFTs.

Excellent numerical agreement with analytical RG flow predictions.

Abstract

We establish that a Bloch-state ansatz based on periodic uniform Matrix Product States (puMPS), originally designed to capture single-quasiparticle excitations in gapped systems, is in fact capable of accurately approximating all low-energy eigenstates of critical quantum spin chains on the circle. When combined with the methods of [Milsted, Vidal, Phys. Rev. B 96 245105] based on the Koo-Saleur formula, puMPS Bloch states can then be used to identify each low-energy eigenstate of a chain made of up to hundreds of spins with its corresponding scaling operator in the emergent conformal field theory (CFT). This enables the following two tasks, that we demonstrate using the quantum Ising model and a recently proposed generalization thereof due to O'Brien and Fendley [Phys. Rev. Lett. 120, 206403]. (i) From the spectrum of low energies and momenta we extract conformal data (specifying the emergent CFT) with unprecedented numerical accuracy. (ii) By changing the lattice size, we investigate nonperturbatively the RG flow of the low-energy spectrum between two CFTs. In our example, where the flow is from the Tri-Critical Ising CFT to the Ising CFT, we obtain excellent agreement with an analytical result [Klassen and Melzer, Nucl. Phys. B 370 511] conjectured to describe the flow of the first spectral gap directly in the continuum.