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Physical Review Letters

Critical Behavior of Spin S Antiferromagnetic Ising Model on Triangular Lattice

Adam Lipowski, Tsuyoshi Horiguchi, Dorota Lipowska

Published May 8, 1995• 25 citations
Publisher

Abstract

We have shown that the ground-state problem of the spin $S$ antiferromagnetic Ising model on the triangular lattice maps to a certain solid-on-solid (SOS) model. For $S<{S}_{c}\ensuremath{\sim}3$ this SOS model is in a rough phase while for $S>{S}_{c}$ it enters a flat phase. The rough phase, whose stability is examined, corresponds to the critical ground state of the Ising model. We suggest that the flat phase corresponds to the ground state with hidden long-range order.

DOI
10.1103/physrevlett.74.3888

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Journal of the Physical Society of Japan•Apr 25, 2011•1 citation
We have shown that the ground-state problem of the spin SSS antiferromagnetic Ising model on the triangular lattice maps to a certain solid-on-solid (SOS) model. For S<Sc\ensuremath∼3S<{S}_{c}\ensuremath{\sim}3S<Sc​\ensuremath∼3 this SOS model is in a rough phase while for S>ScS>{S}_{c}S>Sc​ it enters a flat phase. The rough phase, whose stability is examined, corresponds to the critical ground state of the Ising model. We suggest that the flat phase corresponds to the ground state with hidden long-range order.
The zero-temperature triangular Ising antiferromagnet is mapped onto a solid-on-solid (SOS) model. The system undergoes a roughening transition characterised by a critical exponent alpha = 1 / 2 , by the absence of excitations in the smooth phase, and by domain wall excitations (stripes) in the rough phase. At infinite SOS temperature the height-height correlation function is explicitly calculated with the aid of known four-point Ising correlations. The authors point out that a certain six-vertex model with a comparable SOS interpretation has an identical critical temperature, critical exponent and critical amplitude. This is in support of existing ideas on university in systems with striped phases.
The critical coarsening dynamics of the spin S =1/2, 3/2 antiferromagnetic Ising model on a triangular lattice is studied by the dynamic Monte Carlo simulation using a heat bath algorithm. The triangular antiferromagnetic Ising (TAI) model possesses an intrinsic geometrical frustration and a large degeneracy of ground state which may affect the equilibrium and non-equilibrium critical behaviors. The S =1/2 TAI has no phase transition at a finite temperature, but it was suggested that the S =3/2 TAI has the Kosterlitz–Thouless (KT)-type phase transition at a finite temperature. We employ a finite size scaling approach for the correlation function from the short-time dynamics of the S =1/2, 3/2 TAI to calculate the values of the static critical exponent η and the dynamic exponent z. For the S =1/2 TAI, our dynamic scaling analysis provides η =0.498±0.006 and z =2.278±0.020 at T =0, agreeing with the previous results. For the S =3/2 TAI, after identifying a KT-transition temperature TKT =0.51±0.01, we find the temperature-dependent η ranging from 0.301±0.008 at T =0.51 to 0.224±0.016 at T =0 along the KT-line whereas the value of z =2.20±0.06 is constant for T≤TKT. It is shown that the spin S =3/2 TAI model and the two-dimensional XY model, sharing the KT-type phase transition, exhibit similar static critical and coarsening dynamics behavior. For both the S =1/2, 3/2 TAI, the value of z (η) is compatible with (larger than) that of the Ising model at Tc and the XY model for T≤TKT in two-dimension. Our results imply that although the quasi-long-range order disappears with ηXY =0 in the two-dimensional XY model at T =0, the S =3/2 TAI still maintains it with η TAI =0.224 due to the effect of a frustration and a high degeneracy of ground state.
We map the ground-state ensemble of antiferromagnetic Ising model of spin-S on a triangular lattice to an interface model whose entropic fluctuations are proposed to be described by an effective Gaussian free energy, which enables us to calculate the critical exponents of various operators in terms of the stiffness constant of the interface. Monte Carlo simulations for the ground-state ensemble utilizing this interfacial representation are performed to study both the dynamical and the static properties of the model. This method yields more accurate numerical results for the critical exponents. By varying the spin magnitude in the model, we find that the model exhibits three phases with a Kosterlitz-Thouless phase transition at ${\mathrm{S}}_{\mathrm{KT}}$2 and a locking phase transition at SL{\mathrm{S}}_{\mathrm{L}}SL​\ensuremath{\leqslant}3. The phase diagram at finite temperatures is also discussed.
The critical ground state of the antiferromagnetic triangular Ising model is studied with finite-size scaling, with the use of exact techniques and numerical transfer-matrix methods. Exact solution of the model in zero field yields a conformal anomaly equal to 1, and values of critical exponents. For nonzero fields, dominant eigenvalues of the transfer matrix are computed for systems with linear dimensions up to 27 with the use of an efficient coding of spin states as a system of nonintersecting strings. The results are in perfect agreement with the Gaussian-model--Coulomb-gas scenario proposed by Nienhuis et al. for this model. Quantitative agreement is found with the predictions for exponents of spin-wave and vortex operators. Furthermore, we locate the field-induced Kosterlitz-Thouless transition to a long-range-ordered state at a field Hc{\mathit{H}}_{\mathit{c}}Hc​=0.266\ifmmode\pm\else\textpm\fi{}0.010.
A two-dimensional Ising model on a particular topological structure, a ``triangles-in-triangles'' Kagom\'e structure, which is called a triangular-Kagom\'e lattice, is investigated. In this model, we consider two interaction parameters Jaa{J}_{aa}Jaa​ and Jab{J}_{ab}Jab​, corresponding to the interactions among spins on the inner sites and between spins on the inner and vertex sites of the triangular in the Kagom\'e lattice, respectively. By summing over all spins at inner sites in the partition function, we arrive at the partition function of the Kagom\'e lattice. The effective interaction of the corresponding Kagom\'e lattice is always ferromagnetic, even for antiferromagnetic Jaa{J}_{aa}Jaa​. The critical properties of the system depend only on ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}$. When ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}>\ensuremath{-}1$, the system has long range order at low temperature. However, when ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}<\ensuremath{-}1$, the partition function of the triangular-Kagom\'e lattice can be related to that of the Kagom\'e lattice with effective interaction at a temperature higher than its critical temperature. Therefore, the system will be in paramagnetic phase at all temperatures for ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}<\ensuremath{-}1$. The phase diagram for ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}$ is given exactly.
We propose a non-disordered antiferromagnetic Heisenberg spin model on a triangular lattice, which exhibits a critical state and a glassy state at low temperatures. The system has three phases, the paramagnetic, the intermediate critical, and the glassy phase. The intermediate phase is characterized by a quasi-long-range order, whereas only short-range order exists in the glassy phase. Using Monte Carlo simulation, the phase transitions and properties of the two phases at low temperatures are examined. Our spin model shows that the variable-length spin and the competition between bilinear and biquadratic interactions are essential for forming the critical and glassy phases.