Classical Kitaev model in a magnetic field

Abstract

Motivated by experiments on spin-orbit coupled magnets with Kitaev exchange in magnetic fields, we present an analysis of the classical Kitaev honeycomb model in the presence of a magnetic field. We show that there is a spin liquid regime that exists within a finite window of fields from zero up to a finite threshold before transitioning into the polarized paramagnet. We uncover constraints that spins need to satisfy in the ground state and show that they determine the exact limiting zero temperature behavior of the heat capacity and magnetic susceptibility within the spin liquid as a function of field. When the field is finite, both the two-point spin and the quadrupolar correlations are short-ranged, in contrast to the zero-field case. We rationalize an effective mass for the quadrupolar correlations in terms of a coarse-grained theory with fluctuating effective charge degrees of freedom. Finally, we show that weak site-dilution does not change the magnetization within the spin liquid -- a kind of "perfect" compensation of the site dilution.

Figures (12)

• Figure 1: (a) Phase diagram of the classical anti-ferromagnetic Kitaev model in a magnetic field at zero temperature. For arbitrary field direction there is a field-induced spin liquid phase for field $|\boldsymbol{B}|\leq 2K$. (b) Magnetization as a function of field. The magnetization is linear in the spin liquid phase, with $\boldsymbol{M} = \boldsymbol{B}/(2K)$, until saturating at $|\boldsymbol{B}|=2K$. (c) Average susceptibility $\chi \equiv \sum_{\mu}\chi_{\mu\mu}/3$ as a function of field is constant in the spin liquid phase and $\propto 1/|\boldsymbol{B}|$ in the polarized phase.
• Figure 2: A hexagon surrounded by a canted Néel state. Dark gray sites belong to the A sublattice, light gray to the B sublattice. The triangles inside the circles denote the corresponding (color-coded) components of the $\boldsymbol{S}_A$ and $\boldsymbol{S}_B$ Néel vectors. We allow for the small circles inside the big circles to take different values to find a weathervane mode.
• Figure 3: Thermodynamics of the classical anti-ferromagnetic Kitaev model in an $[111]$ field. Monte Carlo simulations for a $L=24$ system with periodic boundary conditions taking approximately $10^6$ samples using heat-bath, over-relaxation and parallel tempering updates. (a) Heat capacity shows a smooth function of temperature from the classical spin liquid regime into the polarized paramagnetic regime. (b) At low temperatures the heat capacity approaches $3/4$ in the spin liquid phase ($|\boldsymbol{B}|<2K$) before switching to $1$ in the polarized phase ($|\boldsymbol{B}|>2K$). (c) The average susceptibility $\chi \equiv \sum_{\mu} \chi_{\mu\mu}/3$ is also featureless as a function of temperature, approaching a constant in the spin liquid phase at $T=0$. (d) At low temperature there is a discontinuity in the susceptibility at the critical field.
• Figure 4: Appearance of a pinch point at $\boldsymbol{k}=0$ in the spin-quadrupole correlator, $S^{zz}_{Q}(\boldsymbol{k})$ for the anti-ferromagnetic Kitaev model at zero field as temperature $T$ is lowered. Monte Carlo simulations for a $L=120$ system with periodic boundary conditions over $10^6$ sweeps using heat-bath, over-relaxation and parallel tempering updates, taking samples of the quadrupolar structure factor every $10$ sweeps.
• Figure 5: Scaling of the pinch point width in the spin-quadrupole structure factor in the Kitaev model at zero field. System size is $L=120$ and was run for $10^6$ sweeps. For each temperature the wave-vector dependence of $S^{zz}_Q(\boldsymbol{k})$ is fit over a range of wave-vectors near $\boldsymbol{k}=0$ using Eq. (\ref{['eq:finite-temp-pp']}) (see upper plots). The temperature dependence of $1/\xi(T)$ is then fit to $1/\xi^2 = {1/\xi(0)^2 + aT}$ yielding $1/\xi(0) \approx 10^{-6}$. The temperature dependent stiffness $\kappa$ is fit to $\kappa = \kappa(0) + a T^b$, yielding $\kappa(0) \approx 3.1$