Gapless and ordered phases in spin-1/2 Kitaev-XX-Gamma chain

TL;DR

The authors analyze a 1D spin-1/2 Kitaev-XX-Gamma chain, solving it exactly via the Jordan–Wigner transformation and validating the results with DMRG. They reveal six distinct phases: four gapped phases with ferromagnetic or antiferromagnetic order along the $(1,1,0)$ or $(1,-1,0)$ directions, and two gapless phases hosting two helical Majorana Majorana branches. Critical lines separate these phases with dynamical exponents $z=1$ (gapless lines and deconfined transitions) and $z=2$ (special lines at $J=0$ and $J=-1$), while the gapped regions exhibit two-fold ground-state degeneracy due to symmetry considerations in a nonsymmorphic group. The study highlights the rich interplay of symmetry, magnetic order, and quantum criticality in a realizable 1D compass-Gamma model, providing a bridge to 2D Kitaev materials and highlighting deconfined criticality in one dimension.

Abstract

In this work, we study the spin-1/2 Kitaev chain with additional XX and symmetric off-diagonal Gamma interactions. By a combination of Jordan-Wigner transformation and density matrix renormalization group (DMRG) numerical simulations, we obtain the exact solution of the model and map out the phase diagram containing six distinct phases. The four gapped phases display ferromagnetic and antiferromagnetic magnetic orders along the (1, 1, 0)- and (1, -1, 0)-spin directions, whereas in the gapless phases, the low energy spectrum consists of two branches of helical Majorana fermions with unequal velocities. Transition lines separating different phases include deconfined quantum critical lines with dynamical critical exponent z = 1 and quadratic critical lines with z = 2. Our work reveals the rich interplay among symmetry, magnetic order, and quantum criticality in the Kitaev-XX-Gamma chain

Figures (9)

• Figure 1: Phase diagram of the 1D spin-1/2 Kitaev–XX–$\Gamma$ model with $K=1$. AFM-I and FM-I denote antiferromagnetic and ferromagnetic phases polarized along the spin direction $(1,1,0)$, whereas AFM-II and FM-II denote the corresponding phases along $(1,-1,0)$. At the parameter points $(0,0)$ and $(-1,0)$ the model reduces to the pure Kitaev chain.
• Figure 2: Bond pattern of the Kitaev-XX-Gamma chain.
• Figure 3: Energy dispersions in the gapless phase for (a) $J=0,\Gamma=0$, (b) $J=1,\Gamma=0$, (c) $J=-0.5,\Gamma=1$, (d) $J=-1,\Gamma=1$.
• Figure 4: (a) $E_1-E_0$ vs. $N$ at $(J=1,\Gamma=1)$, (b) $E_2-E_0$ vs. $N$ at $(J=1,\Gamma=1)$, (c) $E_1-E_0$ vs. $1/N$ at $(J=1,\Gamma=0)$, (d) $E_1-E_0$ vs. $1/N^2$ at $(J=-1,\Gamma=1)$, where $E_0$, $E_1$, $E_2$ represent the lowest, next lowest, and third lowest energies, respectively. The chain length $N$ is varied from $2$ to $60$ in (a,b), and from $10$ to $28$ in (c,d). The maximum bond dimension $m$ and truncation error $\epsilon$ in DMRG calculations are taken as $m= 1500$, $\epsilon=10^{-12}$.
• Figure 5: Correlation functions $C^{\pm}(r)$ as a function of $r$ for (a) $(J=1,\Gamma=1)$, (b) $(J=1,\Gamma=-1)$, (c) $(J=-2,\Gamma=1)$, and (d) $(J=-2,\Gamma=-1)$. DMRG numerics are performed on systems of $N=100$ sites using open boundary conditions. The maximum bond dimension $m$ and truncation error $\epsilon$ in DMRG calculations are taken as $m= 1500$, $\epsilon=10^{-12}$.