Quantum Spin Liquids in Weak Mott Insulators with a Spin-Orbit Coupling

TL;DR

The paper addresses stabilizing quantum spin liquids in weak Mott insulators with SOC on the triangular lattice. By deriving an effective spin model that includes DM and SOC-mediated ring exchange terms and analyzing it with iDMRG, the authors show that CSL and VBS phases can persist in a weak SOC regime due to compensation between the DM interaction and SOC-mediated ring exchange. They provide qualitative and quantitative arguments, including a renormalized DM picture and a XXZ-type heuristic, to explain the stabilization and the extent of the CSL region. The work suggests a viable route toward experimental realization of spin liquids in SOC-active materials and motivates exploration across different lattices and multi-band systems.

Abstract

The weak Mott insulating regime of the triangular lattice Hubbard model exhibits a rich magnetic phase diagram as a result of the ring exchange interaction in the spin Hamiltonian. These phases include the Kalmeyer-Laughlin type chiral spin liquid (CSL) and a valence bond solid (VBS). A natural question arises regarding the robustness of these phases in the presence of a weak spin-orbit coupling (SOC). In this study, we derive the effective spin model for the spin-orbit coupled triangular lattice Hubbard model in the weak Mott insulting regime, including all SOC-mediated spin-bilinears and ring-exchange interactions. We then construct a simplified spin model keeping only the most relevant SOC-mediated spin interactions. Using infinite density matrix renormalization group (iDMRG) we show that the CSL and VBS phases of the triangular lattice Hubbard model can be stabilized in the presence of a weak SOC. The stabilization results from a compensation between the Dzyaloshinskii-Moriya interaction and a SOC-mediated ring exchange interaction. We also provide additional qualitative arguments to intuitively understand the compensation mechanism in the iDMRG quantum phase diagrams. This mechanism for stabilization can potentially be useful for the experimental realization of quantum spin liquids.

Figures (14)

• Figure 1: (a) The triangular lattice spanned by the primitive vectors $\mathbf{a}_1=(0,1)$, $\mathbf{a}_2=(\sqrt{3}/2,1/2)$. The naming convention for the different types of bonds is shown. The three types of rhombuses (rings) are labeled by $(ijkl)$ counted anti-clockwise where $i$ is a sharp vertex of the rhombus. (b) Brillouin zone of the triangular lattice and some of the high-symmetry points. (c) Sign of the $\mathbf{v}_{ij}$ vector on moving along different directions in the triangular lattice, green is positive, red is negative. $\mathbf{\delta}_n$ are the three positive directions of $\mathbf{v}_{ij}$.
• Figure 2: (a) Quantum Phase diagram obtained from iDMRG by starting from a CSL phase at $J_1=1$, $J_2=J_3=0.01$, $J_r=0.2$ ($t/U=0.097$). The SOC generates the nearest neighbour DM interaction $D_z$ and a SOC-mediated ring exchange interaction $J_{r_1}$. The two types of $120^{\circ}$ orders in (a) are distinguished by their handedness $\omega$. The $D_z$ term prefers the $120^{\circ}_{-}$ phase with $\omega<0$ and the $J_{r_1}$ term prefers $120^{\circ}_{+}$ phase with $\omega>0$. The compensation between these opposing tendencies leads to the stability of the CSL phase in a narrow diagonal region, where their combined effect is minimized. Along the vertical axis $J_{r_1}/J_r=0$, the CSL phase persists until $D_z/J_1=0.0125$, as in inset. Observables along the vertical dashed line at $J_{r_1}/J_r=0.4$ are shown in (b)-(f). The CSL has a nonzero scalar chirality $\chi$, a high entanglement entropy $S_E$, and a weak intensity at $S(K)$, compared to the $120^{\circ}_{\pm}$ phases. Location of the peaks in the second derivative of the energy in (d), can also be used to identify the location of the phase boundaries.
• Figure 3: (a) Quantum Phase diagram obtained from iDMRG by starting from a VBS phase at $J_1=1$, $J_2=J_3=0.012$,$J_r=0.24$ ($t/U=0.105$). Similar to Fig. \ref{['fig:Fig-2_CSL_phase_diagram']}, the compensation between the DM interaction $D_z$ and SOC mediated-ring exchange $J_{r_1}$ leads to the VBS phase being stabilized in the diagonal region. In addition to the $120^{\circ}_{\pm}$ orders, there are narrow regions of the CSL accompanying the VBS. Additionally, a magnetic ordered (MO) phase with long-range correlations, distinct from the $120^{\circ}_{-}$ is identified. Observables along the vertical dashed line $J_{r_1}/J_r=0.3$ are shown in (b)-(f). The VBS is characterized by a weak intensity at $S(K)$ and strong peaks in the dimer structure factor $D_3(M")$. To clearly distinguish the CSL around $D_z/J_1 \sim 0.055$ from its neighbouring states, a zoomed-in plot is provided in Fig. \ref{['fig:zoom']}.
• Figure 4: The two equal-weight dominant dimer coverings in the VBS phase. The singlet bonding axes in the dimer coverings are along $\mathbf{a}_2$ and $\mathbf{a}_3$ respectively.
• Figure 5: The expectation values of $\langle H_{D_z}\rangle$, $\langle H_{J_{r_1}}\rangle$ and $\langle H_{D_z}+ H_{J_{r_1}}\rangle$ in the ground state calculated along: (a) $J_{r_1}/J_r=1.2$ in Fig. \ref{['fig:Fig-2_CSL_phase_diagram']}, and (b) $J_{r_1}/J_r=0.3$ in Fig. \ref{['fig:Fig-3_VBS_phase_diagram']}. The background colors denote the various phases in the corresponding figures. $\langle H_{D_z}+ H_{J_{r_1}}\rangle$ is close to zero (and its absolute value is minimized) in the region where the CSL (as in (a)) and the VBS (as in (b)) are stabilized, indicating that in this region $H_{D_z}$ and $H_{J_{r_1}}$ compensate for each other.