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Physical review. B, Condensed matter

Phase transition of theS=1/2 quantum anisotropic Heisenberg antiferromagnet on the triangular lattice

Naoki Suzuki, Fumitaka Matsubara

Published Mar 1, 1995• 8 citations
Publisher

Abstract

We study the S=1/2 anisotropic Heisenberg antiferromagnet on finite triangular lattices with N\ensuremath{\le}24 sites: H=2J${\mathcal{J}}_{\mathrm{〈}\mathit{i},}$${\mathit{j}}_{\mathrm{〉}}$(${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$${\mathit{S}}_{\mathit{j}}^{\mathit{x}}$ +${\mathit{S}}_{\mathit{i}}^{\mathit{y}}$${\mathit{S}}_{\mathit{j}}^{\mathit{y}}$ +\ensuremath{\Delta}${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${\mathit{S}}_{\mathit{j}}^{\mathit{z}}$) with 0\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}1. The specific heat C and the chiral-order parameter 〈${\mathrm{\ensuremath{\chi}}}^{2}$〉 are calculated using a quantum transfer Monte Carlo method. Remarkable differences in the size dependences of C and 〈${\mathrm{\ensuremath{\chi}}}^{2}$〉 are found between the cases of \ensuremath{\Delta}\ensuremath{\le}0.4 and of \ensuremath{\Delta}\ensuremath{\ge}0.6. For \ensuremath{\Delta}\ensuremath{\le}0.4, the peak height of C increases with increasing N and an extrapolation of 〈${\mathrm{\ensuremath{\chi}}}^{2}$〉 to the thermodynamic limit gives a finite, nonzero value at low temperatures. In contrast with these, for \ensuremath{\Delta}\ensuremath{\ge}0.6, the peak height does not increase and the extrapolation of 〈${\mathrm{\ensuremath{\chi}}}^{2}$〉 gives a smaller value even at very low temperatures indicating the absence of a long-range chiral order. From the results, we suggest that the chiral-ordered phase transition occurs at a finite, nonzero temperature when \ensuremath{\Delta}\ensuremath{\le}${\mathrm{\ensuremath{\Delta}}}_{\mathit{c}}$ with ${\mathrm{\ensuremath{\Delta}}}_{\mathit{c}}$\ensuremath{\gtrsim}0.4. The phase diagram of the model is predicted.

DOI
10.1103/physrevb.51.6402

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We study the S=1/2 anisotropic Heisenberg antiferromagnet on finite triangular lattices with N\ensuremath{\le}24 sites: H=2J{\mathcal{J}}_{\mathrm{〈}\mathit{i},}$${\mathit{j}}_{\mathrm{〉}}({\mathit{S}}_{\mathit{i}}^{\mathit{x}}$${\mathit{S}}_{\mathit{j}}^{\mathit{x}} +{\mathit{S}}_{\mathit{i}}^{\mathit{y}}$${\mathit{S}}_{\mathit{j}}^{\mathit{y}} +\ensuremath{\Delta}{\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${\mathit{S}}_{\mathit{j}}^{\mathit{z}}) with 0\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}1. The specific heat C and the chiral-order parameter 〈\ensuremathχ2{\mathrm{\ensuremath{\chi}}}^{2}\ensuremathχ2〉 are calculated using a quantum transfer Monte Carlo method. Remarkable differences in the size dependences of C and 〈\ensuremathχ2{\mathrm{\ensuremath{\chi}}}^{2}\ensuremathχ2〉 are found between the cases of \ensuremath{\Delta}\ensuremath{\le}0.4 and of \ensuremath{\Delta}\ensuremath{\ge}0.6. For \ensuremath{\Delta}\ensuremath{\le}0.4, the peak height of C increases with increasing N and an extrapolation of 〈\ensuremathχ2{\mathrm{\ensuremath{\chi}}}^{2}\ensuremathχ2〉 to the thermodynamic limit gives a finite, nonzero value at low temperatures. In contrast with these, for \ensuremath{\Delta}\ensuremath{\ge}0.6, the peak height does not increase and the extrapolation of 〈\ensuremathχ2{\mathrm{\ensuremath{\chi}}}^{2}\ensuremathχ2〉 gives a smaller value even at very low temperatures indicating the absence of a long-range chiral order. From the results, we suggest that the chiral-ordered phase transition occurs at a finite, nonzero temperature when \ensuremath{\Delta}\ensuremath{\le}\ensuremathΔc{\mathrm{\ensuremath{\Delta}}}_{\mathit{c}}\ensuremathΔc​ with \ensuremathΔc{\mathrm{\ensuremath{\Delta}}}_{\mathit{c}}\ensuremathΔc​\ensuremath{\gtrsim}0.4. The phase diagram of the model is predicted.
We study the quantum phase transition of an S=12S=\frac{1}{2}S=21​ anisotropic $\ensuremath{\alpha}$ $(\ensuremath{\equiv}{J}_{z}/{J}_{xy})$ Heisenberg antiferromagnet on a triangular lattice. We calculate the sublattice magnetization and the long-range helical order parameter and their Binder ratios on finite systems with $N\ensuremath{\le}36$ sites. The NNN dependence of the Binder ratios reveals that the classical ${120}^{\ensuremath{\circ}}$ N\'eel state occurs for $\ensuremath{\alpha}\ensuremath{\lesssim}0.55$, whereas a critical collinear state occurs for $1/\ensuremath{\alpha}\ensuremath{\lesssim}0.6$. This result is at odds with a widely held belief that the ground state of a Heisenberg antiferromagnet is the ${120}^{\ensuremath{\circ}}$ N\'eel state, but it also provides a possible mechanism explaining experimentally observed spin liquids.
We study the S= quantum antiferromagnetic XY model on finite triangular lattices with N sites in both longitudinal and transverse magnetic fields. We calculate physical quantities in the ground state using a diagonalization for spins N\ensuremath{\leqslant}27, and those at finite temperatures using a quantum transfer Monte Carlo method for N\ensuremath{\leqslant}24. In the longitudinal magnetic field, the long-range chiral order parameter seems to have a finite, nonzero value at low temperatures suggesting the occurrence of a classical umbrella-type phase. In the transverse magnetic field, the 1/3 plateau of the magnetization curve appears even at low temperatures, in contrast with the classical model. The magnetic-field dependences of the order parameters suggest that the chiral-ordered, the ferrimagnetic, and the spin-flop phases appear successively as the magnetic field is increased. The transition temperatures are estimated from the peak position of the specific heat, and the phase diagrams are predicted in both longitudinal and transverse magnetic fields.
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An antiferromagnetic Ising system on a triangular lattice is studied by means of a transfer matrix and the phenomenological scaling method. System consists of a semi-infinite strip and helical boundary conditions are imposed at the end of each row. At external magnetic fields h =0 and 6 J , where J is a nearest neighbor coupling constant, a unite residual entropy is obtained at the zero temperature. With a small field, specific heat is found to have two peaks; a high-temperature peak corresponding to the enhanced short-range order, and a low-temperature peak to the appearance of a long-range order. A phase diagram in h - T space is obtained by means of the phenomenological scaling, and agrees fairly well with the one obtained by Monte Carlo simulation. The main deficiency is the finite transition temperature at h =0, which may be due to a crossover to the XY-like phase transition.