Continuous Symmetry Breaking in 1D Long-Range Interacting Quantum Systems

Abstract

Continuous symmetry breaking (CSB) in low-dimensional systems, forbidden by the Mermin-Wagner theorem for short-range interactions, may take place in the presence of slowly decaying long-range interactions. Nevertheless, there is no stringent bound on how slowly interactions should decay to give rise to CSB in 1D quantum systems at zero temperature. Here, we study a long-range interacting spin chain with U(1) symmetry and power-law interactions V(r)∼1/r^{α}. Using a number of analytical and numerical techniques, we find CSB for α smaller than a critical exponent α_{c}(≤3) that depends on the microscopic parameters of the model. Furthermore, the transition from the gapless XY phase to the gapless CSB phase is mediated by the breaking of conformal and Lorentz symmetries due to long-range interactions, and is described by a universality class akin to, but distinct from, the Berezinskii-Kosterlitz-Thouless transition. Signatures of the CSB phase should be accessible in existing trapped-ion experiments.

FIG. 1.

Phase diagram for the Hamiltonian (1) based on a finite-size DMRG calculation of the effective central charge c_eff = 6[S(N₁) − S(N₂)]/[log(N₁) − log(N₂)] [37]. Here S(N) is the ground-state entanglement entropy for a chain of size N split in two equal halves. We choose N₁ = 100 and N₂ = 110 in our calculation. The XY phase has conformal symmetry and is identified by c_eff = 1. The XY-to-CSB phase boundary is numerically obtained by finding the place where c_eff starts to increase appreciably (4%) above 1 (the black squares fitted by the black line). The dotted (purple) line is the XY-to-CSB transition line obtained from perturbative field theory calculation in the Supplemental Material [34]. The XY-to-AFM phase boundary is obtained by finding the place where c_eff starts to decrease appreciably (1%) below its value at J_z = 1 and α = ∞ (the white squares fitted by the white line).

FIG. 2.

The RG flow in the vicinity of the phase transition, denoted by the thick (red) line, between the CSB phase and the XY phase. We have defined δ = 3 − α − 1/(2K). The RG flow is given by g_LR ~ δ² + a(1 + δ), where the parameter a quantifies the distance from the critical point. For δ < 0, the a = 0 contour (denoted by the red line) describes the critical line. The flows with a > 0 and those with a < 0 and δ > 0 proceed to infinity characterizing the CSB phase. The trajectories with a < 0 and δ < 0 flow to the wavy line characterizing the XY phase.