Phases and phase transitions in a dimerized spin-$\mathbf{\frac{1}{2}}$ XXZ chain

Abstract

We revisit the phase diagram of the dimerized XXZ spin-$\frac{1}{2}$ chain with nearest-neighbor couplings which was studied numerically in Phys. Rev. B 106, L201106 (2022). The model has isotropic $XY$ couplings which have a uniform value and $ZZ$ couplings which have a dimerized form, with strengths $J_a$ and $J_b$ on alternate bonds. We find a rich phase diagram in the region of positive $J_a, ~J_b$. We provide a detailed understanding of the different phases and associated quantum phase transitions using a combination of mean-field theory, low-energy effective Hamiltonians, renormalization group calculations employing the technique of bosonization, and numerical calculations using the density-matrix renormalization group (DMRG) method. The phase diagram consists of two Ising paramagnetic phases called IPM$_0$ and IPM$_\pi$, and a phase with Ising Neel order called IN; all these phases are gapped. The phases IPM$_0$ and IPM$_\pi$ are separated by a gapless phase transition line given by $0 \le J_a = J_b \le 1$ which is described by a conformal field theory with central charge $c=1$. There are two gapless phase transition lines separating IPM$_0$ from IN and IPM$_\pi$ from IN; these are described by conformal field theories with $c=\frac{1}{2}$ corresponding to quantum Ising transitions. The $c=1$ line bifurcates into the two $c=\frac{1}{2}$ lines at the point $J_a = J_b = 1$; the shape of the bifurcation is found analytically using RG calculations. A symmetry analysis shows that IPM$_0$ is a topologically trivial phase while IPM$_\pi$ is a time-reversal symmetry-protected topological phase (SPT) with spin-$\frac{1}{2}$ states at the two ends of an open system. The numerical results obtained by the DMRG method are in good agreement with the analytical results. Finally we propose experimental platforms for testing our results.