Fractal dimension and the counting rule of the Goldstone modes

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.

15 pages, 3 figures; minor changes and references added

References in corpus (4)

• Counting rule for Nambu-Goldstone modes in nonrelativistic systems
• A relativistic non-relativistic Goldstone theorem: gapped Goldstones at finite charge density
• Counting rule of Nambu-Goldstone modes for internal and spacetime symmetries: Bogoliubov theory approach
• Dispersion relations of Nambu-Goldstone modes at finite temperature and density

Cited by in corpus (2)

• Fractals and spontaneous symmetry breaking with type-B Goldstone modes: a perspective from entanglement
• Goldstone modes and the golden spiral in the ferromagnetic spin-1 biquadratic model