Fractal dimension and the counting rule of the Goldstone modes

TL;DR

This paper links the fractal dimension of certain orthonormal basis states in quantum many-body systems to the number of type-B Goldstone modes, revealing a scale-invariant structure with logarithmic entanglement entropy scaling.

Contribution

It introduces a novel connection between fractal dimensions and Goldstone modes, supported by an exact Schmidt decomposition and field-theoretic predictions.

Findings

Entanglement entropy scales logarithmically with block size.

Fractal dimension equals the number of type-B Goldstone modes.

Orthonormal basis states exhibit self-similarity and scale invariance.

Abstract

It is argued that there are a set of orthonormal basis states, which appear as highly degenerate ground states arising from spontaneous symmetry breaking with a type-B Goldstone mode, and they are scale-invariant, with a salient feature that the entanglement entropy $S(n)$ scales logarithmically with the block size $n$ in the thermodynamic limit. As it turns out, the prefactor is half the number of type-B Goldstone modes $N_B$. This is achieved by performing an exact Schmidt decomposition of the orthonormal basis states, thus unveiling their self-similarities in the real space--the essence of a fractal. Combining with a field-theoretic prediction [O. A. Castro-Alvaredo and B. Doyon, Phys. Rev. Lett. \textbf{108}, 120401 (2012)], we are led to the identification of the fractal dimension $d_f$ with the number of type-B Goldstone modes $N_B$ for the orthonormal basis states in quantum many-body systems undergoing spontaneous symmetry breaking.