Electric field effects in one-dimensional spin-1/2 $K_1J_1Γ_1Γ_1^\prime K_2J_2$ model with ferromagnetic Kitaev coupling

TL;DR

The paper probes how static electric fields affect the one-dimensional spin-1/2 $K_1J_1\Gamma_1\Gamma_1'K_2J_2$ chain in the FM Kitaev regime, combining SU(2)$_1$ WZW field theory with large-scale DMRG. It shows that a field along $(1,1,1)$ preserves the Luttinger-liquid phase and can modestly tune the Luttinger parameter $\kappa$, while fields in other directions generically drive the system into a dimerized phase via the leading $\epsilon$ operator. The analysis relies on a six-sublattice rotation to simplify the Gamma terms, a careful symmetry classification of electric-field induced couplings, and numerical verification of LL versus dimerized behavior for multiple field directions. The findings provide a systematic framework for understanding electric-field effects in 1D generalized Kitaev models and lay groundwork for exploring related phenomena in quasi-1D and 2D Kitaev systems.

Abstract

We perform a systematic study on the effects of electric fields in the Luttinger liquid phase of the one-dimensional spin-$1/2$ $K_1J_1Γ_1Γ_1^\prime K_2J_2$ model in the region of ferromagnetic nearest-neighboring Kitaev coupling. We find that while electric fields along $(1,1,1)$-direction maintain the Luttinger liquid behavior, fields along other directions drive the system to a dimerized state. An estimation is made on how effective a $(1,1,1)$-field is for tuning the Luttinger parameter in real materials. Our work is useful for understanding the effects of electric fields in one-dimensional generalized Kitaev spin models, and provides a starting point for exploring the electric-field-related physics in two dimensions based on a quasi-one-dimensional approach.

Figures (6)

• Figure 1: Bond patterns of (a) the $K_1J_1\Gamma_1\Gamma^\prime_1K_2J_2$ chain without sublattice rotation, (b) the $K_1J_1\Gamma_1\Gamma^\prime_1K_2J_2$ chain after the six-sublattice rotation, (c) couplings induced by electric fields after the six-sublattice rotation. Black squares represent unit cells for the bond patterns.
• Figure 2: Schematic plot of the Luttinger liquid phase of spin-1/2 $K_1J_1\Gamma_1$ model in the $(K_1<0,\Gamma_1>0,J_1>0)$ region, in which $K_1,J_1,\Gamma_1$ are parametrized by $\theta,\phi$ in accordance with Eq. (\ref{['eq:parametrize_KJG']}). The line marked with green color corresponds to Kitaev-Gamma model, which has an emergent SU(2)$_1$ conformal symmetry at low energies where the transition point $\phi_1$ is $0.33\pi$, as discussed in Ref. Yang2020. The system remains in the Luttinger liquid phase in the $K_1J_1\Gamma_1\Gamma^\prime_1K_2J_2$ model for small enough $\Gamma_1^\prime,K_2,J_2$.
• Figure 3: (a) Staggered energy density $E_A(r)$ as a function of $r_L$ on a log-log scale and (b) entanglement entropy $S_L(x)$ as a function of $\frac{1}{6}\sin(\frac{\pi x}{L})$ in the absence of electric fields, where $r_L=\frac{L}{\pi} \sin(\frac{\pi x}{L})$. In (a,b), DMRG numerics are performed on a system of $L=96$ sites using open boundary conditions, with bond dimension $m$ and truncation error $\epsilon$ chosen as $m=1000$, $\epsilon=10^{-10}$. The parameters of $K_1,J_1,\Gamma_1,\Gamma_1^\prime, K_2,J_2$ in DMRG numerics are taken in accordance with Eq. (\ref{['eq:numerics_KJG']}) and Eq. (\ref{['eq:numerics_coefficients']}).
• Figure 4: (a) Staggered energy density $E_A(r)$ as a function of $r_L$ on a log-log scale and (b) entanglement entropy $S_L(x)$ as a function of $\frac{1}{6}\sin(\frac{\pi x}{L})$ with electric field $\vec{E}=\frac{4}{\sqrt{4}}(1,1,1)$, where $r_L=\frac{L}{\pi} \sin(\frac{\pi x}{L})$. In (a,b), DMRG numerics are performed on a system of $L=96$ sites using open boundary conditions, with bond dimension $m$ and truncation error $\epsilon$ chosen as $m=1000$, $\epsilon=10^{-10}$. The parameters of $K_1,J_1,\Gamma_1,\Gamma_1^\prime, K_2,J_2$ in DMRG numerics are taken in accordance with Eq. (\ref{['eq:numerics_KJG']}) and Eq. (\ref{['eq:numerics_coefficients']}).
• Figure 5: (a) Staggered energy density $E_A(r)$ as a function of $r_L$ on a log-log scale and (b) entanglement entropy $S_L(x)$ as a function of $\frac{1}{6}\sin(\frac{\pi x}{L})$ for $\vec{E}=(0,0,4)$ and $\lambda_{D-}^{\text{(0)}}=0$, (c) staggered energy density $E_A(r)$ as a function of $r_L$ on a log-log scale and (d) dimerization $O^\prime_{\text{dim},i}$ as a function of site $i$ for electric field $\vec{E}=(0,0,4)$ and $\lambda_{D-}^{\text{(0)}}\neq 0$, where $r_L=\frac{L}{\pi} \sin(\frac{\pi x}{L})$. In (a,b,c), DMRG numerics are performed on a system of $L=96$ sites using open boundary conditions, and in (d), DMRG numerics are performed on a system of $L=96$ sites using periodic boundary condition. Bond dimension $m$ and truncation error $\epsilon$ are chosen as $m=1000$, $\epsilon=10^{-10}$. The parameters of $K_1,J_1,\Gamma_1,\Gamma_1^\prime, K_2,J_2$ in DMRG numerics are taken in accordance with Eq. (\ref{['eq:numerics_KJG']}) and Eq. (\ref{['eq:numerics_coefficients']}).