Field-induced phase transitions in the Kitaev-Heisenberg model: A sign-problem-free quantum Monte Carlo study and possible application to $α$-RuCl3

TL;DR

The work addresses whether a field-induced quantum spin liquid can arise in the honeycomb Kitaev–Heisenberg model with $K=-2J$ under a magnetic field along $[001]$, a question relevant to $\alpha$-RuCl$_3$. The authors perform sign-free, numerically exact quantum Monte Carlo (SSE with directed loops) on a transformed, sign-problem-free version of the model to access zero and finite temperatures. They find a direct transition from zigzag (or stripe) order to a spin-polarized state with $3D$ XY universality, $h_c \approx 0.6970(1)$ for $K=2,J=-1$ (and $h_c \approx 1.1531(1)$ for $K=-2,J=1$), and a terminating BKT line described by ${\rho_s}(T_{\rm BKT})=2T_{\rm BKT}/\pi$ at finite $T$, with no intermediate quantum spin liquid phase. These results constrain the realization of a field-induced QSL in $\alpha$-RuCl$_3$ for $[001]$ fields and imply that in-plane fields or additional interactions (e.g., a $\Gamma$ term) may be required to stabilize a QSL, guiding future experiments and theory.

Abstract

The frustrated magnet $α$-RuCl3 is one of the prime candidates for realizing a Kitaev quantum spin liquid (QSL). However, the existence of a field-induced intermediate QSL phase in this material remains under debate. Here, we employ sign-free numerically exact quantum Monte Carlo simulations to investigate the Kitaev-Heisenberg (KH) model on the honeycomb lattice with $K=-2J$ under an applied magnetic field along the z-direction. Our findings reveal that the system undergoes a direct quantum phase transition from a zigzag magnetically ordered phase to a spin-polarized phase at zero temperature, which belongs to the 3D XY universality class. At finite temperatures, a Berezinskii-Kosterlitz-Thouless transition line separates the spin-polarized phase from a quasi-long-range ordered state, eventually terminating at the quantum critical point. Our results convincingly show that there is no intermediate QSL phase in the KH model with a z-direction magnetic field, which we believe will shed important light on understanding experimental observations in $α$-RuCl3.

Figures (11)

• Figure 1: Phase diagram of the KH model at ($K=2, J=-1$) on the honeycomb lattice under a uniform magnetic field along the $[001]$ direction. $h$ denotes the strength of the magnetic field, and $T$ is the temperature. The thermal phase transition line separating a quasi-long-range ordered state from the spin-polarized state is classified as a BKT transition. At zero temperature, a quantum phase transition in the 3D XY universality class occurs at $h_c \approx 0.697$, where the BKT line terminates.
• Figure 2: Transformation of the KH model under a magnetic field. (a) Visualization of the honeycomb lattice structure in $\alpha$-${\rm RuCl_3}$. The Kitaev term in the Hamiltonian is illustrated by three distinct bond types, each represented by a different color. Lattice sites are marked with four unique shapes, corresponding to the four spin transformation types. (b) The spin transformation rules illustrating how the spin components $(S^x, S^y, S^z)$ are transformed for each lattice site type. (c) In the original model, the magnetic field $h$ is uniformly applied along the [001] direction. (d) In the transformed model, the effective magnetic field $h$ exhibits a stripe-like pattern, with the black dot representing the [001] direction and the light gray dot indicating the [00$\bar{1}$] direction.
• Figure 3: Structure factor $M_{2}^{x,y,z}(\vec{Q})$ in the first Brillouin zone: comparing zigzag state ($h=0.5$) and polarized state ($h=1.5$) for system size $L=18$ at inverse temperature $\beta=L$. (a-c) The zigzag state at $h=0.5$, with $M_{2}^{x,y,z}(\vec{Q})$ showing peak values at $M_{x}$, $M_y$, and $\Gamma$ points, respectively. The peak of $M_{2}^{z}$ at $\Gamma$ is induced by the $z$-direction field, while the zigzag order is an in-plane ordered state. (d-f) The spin-polarized state at $h=1.5$, where $M_{2}^{z}(\vec{Q})$ exhibits a peak at the $\Gamma$ point, and the in-plane zigzag order vanishes.
• Figure 4: Phase transition at zero temperature in the KH model with a uniform magnetic field. (a) The RG invariant ratio of zigzag order shows crossing points, indicating the transition at critical magnetic field $h_c=0.6970(1)$. Inset: an FSS analysis of $h_c$. (b) The same transition point, $h_c=0.6969(1)$, is obtained from $L\rho_s$, consistent with the RG invariant ratio. Inset: an FSS analysis of $h_c$. (c) Magnetic susceptibility $\chi={\rm d} M^z/{\rm d}h$ exhibits a single peak around the quantum phase transition point, suggesting the absence of an intermediate spin liquid phase. The dashed line marks $h_c=0.697$.
• Figure 5: Data collapse of (a) $R(h, L)$ and (b) $M_2^x(\vec{Q}^{*},L)$ using $h_c=0.6970$ and 3D XY critical exponents $\eta=0.03810$ and $\nu=0.67169$Hasenbusch3dxy.