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Journal of the Physical Society of Japan

Thermodynamic Properties of Spin 1/2 Triangular Antiferromagnet

Masatoshi Imada

Published Jan 15, 1987• 33 citations
Publisher

Abstract

Finite size calculation up to 18 spins combined with the extrapolation to the thermodynamic limit is performed to obtain thermodynamic properties of spin 1/2 triangular antiferromagnet. It shows that both of isotropic and Ising-like Heisenberg systems are in quantum spin fluid state at low temperatures with the singlet ground state. In contrast with the classical system, the result is consistent with the picture that the sublattice ordering has no relevance and no phase transition occurs.

DOI
10.1143/jpsj.56.311

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Finite size calculation up to 18 spins combined with the extrapolation to the thermodynamic limit is performed to obtain thermodynamic properties of spin 1/2 triangular antiferromagnet. It shows that both of isotropic and Ising-like Heisenberg systems are in quantum spin fluid state at low temperatures with the singlet ground state. In contrast with the classical system, the result is consistent with the picture that the sublattice ordering has no relevance and no phase transition occurs.
After decades of debate, now there is a rough consensus that at zero temperature the spin-1/21/21/2 Heisenberg antiferromagnet on the triangular lattice is three-sublattice ${120}^{\ensuremath{\circ}}$ magnetically ordered, in contrast to a quantum spin liquid as originally proposed. However, there remains considerable discrepancy in the magnetization reported among various methods. To resolve this issue, in this work we revisit this model by the tensor-network state algorithm. The ground-state energy per bond Eb{E}_{b}Eb​ and magnetization per spin M0{M}_{0}M0​ in the thermodynamic limit are obtained with high precision. The former is estimated to be ${E}_{b}=\ensuremath{-}0.18334(10)$. This value agrees well with that from the series expansion. The three-sublattice magnetic order is firmly confirmed and the magnetization is determined as M0=0.161(5){M}_{0}=0.161(5)M0​=0.161(5). It is about 3232%32 of its classical value and slightly below the lower bound from the series expansion. In comparison with the best estimated value by Monte Carlo and density-matrix renormalization group, our result is about 2020%20 smaller. This magnetic order is consistent with further analysis of the three-body correlation. Our work thus provides benchmark results for this prototypical model.
Monte Carlo simulations of the Ising model on the triangular lattice with the antiferromagnetic nn interaction J and the ferromagnetic nnn interaction J ' are carried out. Several physical quantities (magnetizations, energies, susceptibilities, specific heats of sublattices and of the total system) are calculated. A phase which resembles the partially disordered phase proposed by Mekata is observed in the finite temperature region. But it is shown in this region that the sublattice magnetizations vanish at the thermodynamic limit and the Kosterlitz-Thouless phase transition is predicted by the finite size analysis.
We study, on the basis of the general entangled-plaquette variational ansatz, the ground-state (GS) properties of the spin- antiferromagnetic Heisenberg model on the triangular lattice. Our numerical estimates are in good agreement with available exact results and comparable, for large system sizes, to those computed via the best alternative numerical approaches, or by means of variational schemes based on specific (i.e. incorporating problem-dependent terms) trial wave functions (WFs). The extrapolation to the thermodynamic limit of our results for lattices comprising up to N =324 spins yields an upper bound of the GS energy per site (in units of the exchange coupling) of − 0.5458(2) [−0.4074(1) for the XX model], while the estimated infinite-lattice order parameter is 0.3178(5) (i.e. ∼64% of the classical value).
We study the ground-state and finite-temperature properties of the one-dimensional isotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions which compete with each other. Extrapolating the exact results for finite-size systems of up to 20 spins, we estimate the ground-state energy and the ground-state two-spin correlation function in the infinite-size limit. Using the cluster transfer matrix method, we also calculate the temperature dependence of the internal energy and the specific heat and that of the inverse correlation length and the wave number, by both of which the asymptotic behavior of the two-spin correlation function in the long-distance limit is characterized. We compare the results of these calculations with those obtained previously for the case where the nearest-neighbor interaction is antiferromagnetic.
Thermodynamic properties of antiferromagnetic quantum X Y -model ( s =1/2) on a triangular lattice are studied by the Decoupled Cell Monte Carlo Method (DCM). The decoupled cell (DC) used in this study consists of 7 spins. The size N of systems are 15 * 15, 30 * 30, 45 * 45, respectively. Detailed studies of spin pair correlation functions for both x - and z -components are performed at various temperatures. The results suggest that the existence of the shortrange three sublattice structure. The mean square magnetization of a sublattice for the x -component does not depend on N , whereas that of the z -component shows a strong dependence on N below a certain temperature.
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.