Phases and phase transitions in a dimerized spin- 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 by Mondal et al. [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 ${J}_{a},\phantom{\rule{0.28em}{0ex}}{J}_{b}$. 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 ${\mathrm{IPM}}_{0}$ and ${\mathrm{IPM}}_{\ensuremath{\pi}}$, and a phase with Ising Neel (IN) order; all these phases are gapped. The phases ${\mathrm{IPM}}_{0}$ and ${\mathrm{IPM}}_{\ensuremath{\pi}}$ are separated by a gapless phase transition line $0\ensuremath{\le}{J}_{a}={J}_{b}\ensuremath{\le}1$---the well-known Tomonaga Luttinger liquid captured by a conformal field theory (CFT) with central charge $c=1$. In contrast, there are two phase transition lines separating ${\mathrm{IPM}}_{0}$ or the ${\mathrm{IPM}}_{\ensuremath{\pi}}$ from IN that are described by $c=\frac{1}{2}$ CFTs and correspond to quantum Ising transitions. In fact, 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 ${\mathrm{IPM}}_{0}$ is a topologically trivial phase while ${\mathrm{IPM}}_{\ensuremath{\pi}}$ is a time-reversal symmetry-protected topological phase (SPT) with spin-$\frac{1}{2}$ states at the two ends of an open system. A gap opens in the low-energy spectrum whenever one moves away from one of the phase transition lines; the scaling of the gap with the distance from a transition line is found analytically using the RG method. The numerical results obtained by the DMRG method are in good agreement with the analytical results. Further, we show that useful insights into the nature of the phase diagram of the above Hamiltonian, particularly the SPT as well as the phase transitions, is obtained via Kramers-Wannier duality to a deformed quantum Ashkin-Teller model. Finally, we propose experimental platforms for testing our results.