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Journal of the Physical Society of Japan

Ground-State Properties and Phase Diagram of the QuantumXXZAntiferromagnet on a Triangular Lattice

Tsutomu Momoi, Masuo Suzuki

Published Oct 15, 1992• 24 citations
Publisher

Abstract

Long-range order, especially chiral order, in the two-dimensional spin \(\frac{1}{2} XXZ\) antiferromagnet H = J ∑( S i x S j x + S i y S j y +Δ S i z S j z ) on a triangular lattice is studied, partially by showing exact ground states and partially using the spin-wave theory. Exact ground states in the region Δ≤-0.5 are found. These states are not disordered by quantum fluctuation. Most of the ground states at Δ=-0.5 have a so-called 120° structure. Ground-state properties in the region -0.5≤Δ≤1 are studied using the spin-wave theory. It is shown that quantum fluctuations are enhanced in the ground states as Δ is increased. The phase diagram of the relevant system at low temperatures is also discussed using the spin-wave theory.

DOI
10.1143/jpsj.61.3732

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Long-range order, especially chiral order, in the two-dimensional spin (\frac{1}{2} XXZ) antiferromagnet H = J ∑( S i x S j x + S i y S j y +Δ S i z S j z ) on a triangular lattice is studied, partially by showing exact ground states and partially using the spin-wave theory. Exact ground states in the region Δ≤-0.5 are found. These states are not disordered by quantum fluctuation. Most of the ground states at Δ=-0.5 have a so-called 120° structure. Ground-state properties in the region -0.5≤Δ≤1 are studied using the spin-wave theory. It is shown that quantum fluctuations are enhanced in the ground states as Δ is increased. The phase diagram of the relevant system at low temperatures is also discussed using the spin-wave theory.
In this work we use the linear spin-wave approximation to study the quantum two-dimensional Heisenberg model on triangular lattice in the presence of nonmagnetic impurities. In particular we discuss how site disorder changes the ground-state properties of this important frustrated antiferromagnetic spin system. Our main focus is on the role of the nonmagnetic impurities which generates an influence of the quantum order-disorder transition and the effect of the ground-state spin structure in many spin systems.
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.
Spin liquids are important for a fundamental understanding of quantum magnetism and may find applications in quantum computing, but it is still not clear under which general circumstances these exotic, quantum-disordered phases occur. To gather deeper insights, we analyze a generalization of the spatially anisotropic triangular lattice (SATL), the spatially completely anisotropic triangular lattice (SCATL), using Takahashi's modified spin-wave theory, complemented by exact diagonalizations. For Heisenberg interactions, calculations on the SATL predict quantum disorder for several materials in which experiments have found magnetic long-range order. In the SCATL, the corresponding parameter values yield ordered ground states and this discrepancy disappears. We also study the model with XYXYXY interactions, which can be implemented with current technology in ultracold atoms in optical lattices. This allows to estimate the stability of quantum-disordered phases towards imperfect implementations of the coupling parameters. For both kinds of interactions, we find indications for extended quantum-disordered phases. In part of the parameter space of the XYXYXY model, exact diagonalization hints at a gapped state with chiral long-range order. In both models, our results suggest that two gapped nonmagnetic regions, identified as distinct in the SATL, could actually be continuously connected via the additional anisotropy of the SCATL. Further, we find that different kinds of order are always separated by disordered phases. We propose that this is a quite universal feature of two-dimensional frustrated antiferromagnets with continuous symmetry in the couplings. The SCATL may therefore---besides being relevant for experiments---yield fundamental insight into quantum-disordered phases.
We study the S=1/2 anisotropic Heisenberg antiferromagnet on finite triangular lattices with N\ensuremath{\le}24 sites: H=2JJ〈i,{\mathcal{J}}_{\mathrm{〈}\mathit{i},}J〈i,​j〉{\mathit{j}}_{\mathrm{〉}}j〉​(Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​Sjx{\mathit{S}}_{\mathit{j}}^{\mathit{x}}Sjx​ +Siy{\mathit{S}}_{\mathit{i}}^{\mathit{y}}Siy​Sjy{\mathit{S}}_{\mathit{j}}^{\mathit{y}}Sjy​ +\ensuremath{\Delta}Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​Sjz{\mathit{S}}_{\mathit{j}}^{\mathit{z}}Sjz​) 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.
We have studied the pseudoscalar chiral long-range order, based on the local operator ${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\psi}}}$=S1{\mathbf{S}}_{1}S1​\ensuremath{\cdot}(S2{\mathbf{S}}_{2}S2​\ifmmode\times\else\texttimes\fi{}S3{\mathbf{S}}_{3}S3​), of the S=1/2 XY antiferromagnet on the triangular lattice. At finite temperature the chirality has been studied via a seventh-order high temperature series expansion of the square of the fluctuation in the pseudoscalar chiral long-range order. An exact diagonalization study on small finite lattices has been used to investigate the chirality long-range order parameter at T=0. While Pad\'e approximant analysis of the series suggests the possibility of a finite-temperature phase transition to a low temperature phase with pseudoscalar chiral long-range order, exact diagonalization results indicate that no such order exists at zero temperature.
We study the low-field ground-state (GS) properties of the antiferromagnetic transverse-field Ising model with long-range interactions (afLRTFIM) on the triangular lattice. We use the method of perturbative continuous unitary transformations (pCUT) to derive an effective model for the degenerate GS space of the antiferromagnetic nearest-neighbour (NN) Ising model on a finite system, by treating the transverse-field (TF) and the long-range interactions (LRI) as a perturbation. We determine a level-crossing between the gapped plain stripe phase at small TF and the gapped clock-ordered phase at intermediate TF at hc≅0.129 for α=6, N=36 spins in order three perturbation theory on the finite system. We discuss the qualitative layout of the quantum phase diagram of the afLRTFIM on the triangular lattice.