Emergent continuous symmetry and ground-state factorization induced by long-range interactions

Abstract

The spontaneous breaking of a $Z_2$ symmetry typically gives rise to emergent excitations possessing the same symmetry with a renormalized mass. Contrary to this conventional wisdom, we present a theory in which the low-lying excitation in the broken-symmetry phase acquires a continuous symmetry, even when the underlying symmetry of the system is discrete. In the presence of anisotropic long-range interactions, the order parameter renormalizes the relative strength of the particle-conserving and particle-nonconserving interactions. When one of the two renormalized interactions vanishes, a conservation law absent in the original Hamiltonian emerges, giving rise to a continuous symmetry. A striking consequence of the emergent continuous symmetry and conservation law is that it constrains quantum correlations in the ground-state to be zero, leading to the ground-state factorization in the presence of strong interactions. Our finding is a universal feature of quantum phase transitions in fully-connected systems and in their lattice generalizations; therefore, it can be observed in a wide range of physical systems.

Figures (3)

• Figure 1: The mean-field energy landscape as a function of the real and imaginary part of a dimensionless complex order parameter $\bar{\alpha}=\bar{X}+i \bar{P}$. (a) The normal phase with $\bar{X} = \bar{P} = 0$ as the sole minimum. (b) A specific instance and (c) a generic instance of the broken discrete symmetry phase, where the principal curvatures at the local minima (blue) are equal and different, respectively. (d) The broken continuous symmetry leads to a circle of degenerate minima, leading to the Goldstone mode.
• Figure 2: The ground-state entanglement diagram for the anisotropic Dicke model in $g_{+}$-$g_{-}$ plane. In addition to the well-known factorized ground state along the line of $g_+=g_-$ in the normal phase, the ground-state factorization occurs also in the $Z_2$-symmetry broken superradiant phase along the emergent Tavis-Cummings line ($g_+g_-=1$) where the conversation law for the total number of particles, absent in the bare Hamiltonian, appears. The emergent anti-TC line ($g_+g_-=-1$, dashed line), on the other hand, does not feature the ground-state factorization.
• Figure 3: (a) The ground-state entanglement diagram for the LMG model. The emergent continuous symmetry appears along the dashed magenta line. (b) The entanglement along the line (cyan) $\gamma_y = 1.05\gamma_x$ indicated in (a) shows the ground-state factorization as it crosses the emergent continuous symmetry line.