Kitaev-Ising-$J_1$-$J_2$ model: a density matrix renormalization group study

TL;DR

The work investigates the Kitaev honeycomb model augmented by XX Ising interactions on NN and NNN bonds, mapped to a square-qubit lattice and studied with density matrix renormalization group on a $10\times10$ setup. Parameterized by $K=\cos\varphi$, $J_1=\sin\varphi\cos\alpha$, and $J_2=\sin\varphi\sin\alpha$, the study maps a rich phase diagram including gapped/gapless Kitaev spin liquids (KSL), commensurate magnetic orders, and incommensurate quantum paramagnetic phases (QPM1, QPM2), as diagnosed by $W_p$, magnetic order parameters, and the spin structure factor $S(\boldsymbol{k})$. The authors show that anisotropic Ising couplings can stabilize QPM phases even without a magnetic field, and they report entanglement signatures with $S\sim\sqrt{N}$ and $D\sim e^{\sqrt{N}}$ in highly entangled phases, alongside a DMRG optimization strategy to avoid local minima in low-entanglement regimes. These results have implications for quantum simulations of Kitaev physics on superconducting qubit platforms where parasitic couplings may destabilize the KSL, and they provide practical methodology to improve 2D DMRG performance in highly entangled regimes.

Abstract

We numerically study the Kitaev honeycomb model with the additional XX Ising interaction between the nearest and the next nearest neighbors (Kitaev-Ising-$J_1$-$J_2$ model), by using the density matrix renormalization group (DMRG) method. Such additional interaction correspond to the nearest and diagonal interactions on the square lattice. Phase diagram of the bare Kitaev model consist of low entangled commensurate magnetic phases and entangled Kitaev spin liquid. Anisotropic Ising interaction allows the entangled quantum paramagnetic phases in the phase diagram, which in the absence of the magnetic field previously was predicted for more complex type of interaction. We study the scaling law of the entanglement entropy and the bond dimension of the matrix product state with the size of the system. In addition, we propose an optimization algorithm to prevent DMRG from getting stuck in the low-entangled phases.

Figures (6)

• Figure 1: The Kitaev-Ising-$J_1$-$J_2$ model on the hexagonal (a) and square (b) lattices. Black lines represent the pure Kitaev model coupling $K$, blue dashed lines with dots indicate to the NN XX coupling $J_1$, and red dashed lines denote to the NNN XX coupling $J_2$. Shaded sectors denote the sites involved in the calculation of magnetic order parameters \ref{['eq:sigma_order_1']},\ref{['eq:sigma_order_2']}.
• Figure 2: Phase diagram of the Kitaev-Ising-$J_1$-$J_2$ model, along with schematic spin configurations for commensurate phases. Red (blue) points represent spin-up (spin-down) states with $\left<S^x\right>=+1/2$ ($\left<S^x\right>=-1/2$).
• Figure 3: Spin-structure factor $S(\mathbf{k})$ of the Kitaev-Ising-$J_1$-$J_2$ model for representative momenta in different phases. The larger black and white dashed hexagone denotes the extended Brillouin zone, while the inner hexagone represents the first Brilloin zone. (a) AFM KSL phase, plotted at $\alpha=0,\,\,\varphi=0$. (b) FM KSL phase at $\alpha=0,\,\,\varphi=\pi$. (c) QPM1 phase at $\alpha=0,\,\,\varphi=0.85\pi$. (d) QPM2 phase at $\alpha=0,\,\,\varphi=1.85\pi$. (e) FM phase at $\alpha=0,\,\,\varphi=1.5\pi$. (f) Neel phase $\alpha=0,\,\,\varphi=0.5\pi$. (g) $x$-stripy phase at $\alpha=0.5\pi,\,\,\varphi=0.75\pi$. (h) $y$-stripy phase at $\alpha=0.5\pi,\,\,\varphi=0.25\pi$.
• Figure 4: The spin-spin correlation function is plotted as a function of the distance $\left|i-j\right|$ between spins. The calculations for different phases were performed at the same values of $\alpha$ and $\varphi$ as in the Fig. \ref{['fig:ss_factor']}.
• Figure 5: Scaling of the entanglement entropy (top) and the maximum bond dimension (bottom) with system size. The calculations for different phases were performed at the same values of $\alpha$ and $\varphi$ as in the Fig. \ref{['fig:ss_factor']}. The lattice size $N$ varies from $3\times3$ to $10\times10$.