Spontaneous symmetry breaking with type-B Goldstone modes in the SO($2s+1$) ferromagnetic model: an entanglement perspective

Abstract

Spontaneous symmetry breaking with type-B Goldstone modes is investigated in the SO($2s+1$) ferromagnetic model. A set of orthonormal basis states in the ground state subspace are constructed, which admit an exact Schmidt decomposition, exposing self-similarities in real space of an abstract fractal underlying the ground state subspace. Focusing on the SO(5) and the SO(6) ferromagnetic spin chains as illustrative examples, finite system-size scaling analysis of the entanglement entropy for this set of orthonormal basis states confirms that the entanglement entropy scales logarithmically with block size in the thermodynamic limit. The prefactor in front of the logarithm is half the number of type-B Goldstone modes $N_B$, which is identified as the fractal dimension $d_f$ for these orthonormal basis states. For the SO($2s+1$) ferromagnetic model $N_B = d_f =s$ for integer $s$ and $N_B = d_f =s+1/2$ for half-odd-integer $s$.