Disentangling Kitaev Quantum Spin Liquid

Abstract

In this work, we investigate the Kitaev honeycomb model employing the recently developed Clifford Circuits Augmented Matrix Product States (CAMPS) method. While the model in the gapped phase is known to reduce to the toric code model - whose ground state is entirely constructible from Clifford circuits - we demonstrate that the very different gapless quantum spin liquid (QSL) phase can also be significantly disentangled with Clifford circuits. Specifically, CAMPS simulations reveal that approximately two-thirds of the entanglement entropy in the isotropic point arises from Clifford-circuit contributions, enabling dramatically more efficient computations compared to conventional matrix product state (MPS) methods. Crucially, this finding implies that the Kitaev QSL state retains significant Clifford-simulatable structure, even in the gapless phase with non-abelian anyon excitations when time reversal symmetry is broken. This property not only enhances classical simulation efficiency significantly but also suggests substantial resource reduction for preparing such states on quantum devices. As an application, we leverage CAMPS to study the Kitaev-Heisenberg model and determine the most accurate phase boundary between the anti-ferromagnetic phase and the Kitaev QSL phase in the model. Our results highlight how Clifford circuits can effectively disentangle the intricate entanglement of Kitaev QSLs, opening avenues for efficiently simulating related and similar strongly correlated models.

Figures (4)

• Figure 1: Schematic of the honeycomb lattice geometry and boundary conditions used in this work. Periodic boundary conditions are applied along the vertical direction, while the horizontal direction is open. The three types of Kitaev bonds ($x$ (green), $y$ (red), and $z$ (blue)) are indicated by differently colored lines. The mapping of the two-dimensional lattice ($6 \times 6$ in the plot) to a one-dimensional chain for MPS simulations is illustrated by the numbering.
• Figure 2: (a) Convergence of the ground-state energy of the Kitaev honeycomb model as a function of the bond dimension $D$ for both MPS (blue) and CAMPS (orange). The system size is $16\times16$. While the MPS energy decreases slowly and requires $D \gtrsim 2000$ to approach convergence, the CAMPS reaches a significantly lower energy already at very small bond dimensions $D \lesssim 100$. Notably, the CAMPS result at a very small bond dimension $D=20$ already achieves a lower variational energy than the MPS calculation with $D=1200$. (b) entanglement entropy (EE) as a function of system size $L$ for MPS and CAMPS. Simple linear fits (dashed lines) show that the slope of CAMPS ($0.09$) is substantially smaller than that of MPS ($0.17$).(c) Ratio of entanglement entropies, $\mathrm{EE}_{\mathrm{MPS}}/\mathrm{EE}_{\mathrm{CAMPS}}$, as a function of $L$. Across all sizes, the ratio basically remains above 2.9, confirming that MPS consistently requires more entanglement to describe the same physical state.
• Figure 3: Ground-state energy of the Kitaev--Heisenberg model on the $16\times16$ lattice as a function of the parameter $\varphi$ (in the unit of $\pi$). The MPS data are indicated by triangles, whereas the CAMPS data are indicated by circles. The CAMPS data display significantly improvement over MPS. The dashed magenta lines represent a linear extrapolation based on the CAMPS energies. The intersection between the two lines identifies the phase-transition point, marked by the red dot. The transition point of the $16\times16$ lattice is $\varphi_c=0.48500(7)$
• Figure 4: Finite-size estimates of the phase-transition point $\varphi_c$ (in the unite of $\pi$) obtained from CAMPS calculations for system sizes $L = 8, 10, 12, 14,$ and $16$ ($L=16$ results are shown in Fig. \ref{['Kitaev_Heisenberg_result']} and other results can be found in the supplementary materials) The extracted transition points exhibit weak size dependence and converge rapidly as $L$ increases. The shaded band represents the final uncertainty range of the transition point, considering both the spread of the finite-size data and the extrapolation uncertainty.