Extracting the number of type-B Goldstone modes and the dynamical critical exponent for a type of scale-invariant states

TL;DR

This paper introduces a finite-entanglement scaling method to analyze scale-invariant states with degenerate ground states, enabling extraction of Goldstone modes and critical exponents from tensor network simulations.

Contribution

It presents a novel scheme combining finite-entanglement and block-size scaling to determine the number of type-B Goldstone modes and the dynamical critical exponent.

Findings

Number of type-B Goldstone modes equals the fractal dimension.

Method successfully applied to various quantum spin models.

Reveals fractal structure underlying ground state subspace.

Abstract

A generic scheme is proposed to perform a finite-entanglement scaling analysis for scale-invariant states, which appear to be highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes. This allows us to extract the number of type-B Goldstone modes and the dynamical critical exponent, in combination with a finite block-size scaling analysis, from numerical simulations of quantum many-body systems in the context of tensor network representations. The number of type-B Goldstone modes is identical to the fractal dimension, thus reflecting an abstract fractal underlying the ground state subspace. As illustrative examples, we investigate the spin-$s$ Heisenberg ferromagnetic model, the $\rm{SU}(3)$ ferromagnetic model and the $\rm{SO}(4)$ spin-orbital model.