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arXiv:2109.03178

Melting of three-sublattice order in triangular lattice Ising antiferromagnets: Power-law order, $Z_6$ parafermionic multicriticality, and weakly first order transitions

Geet Rakala, Nisheeta Desai, Saumya Shivam, Kedar Damle

Published Sep 7, 2021• 1 citation
arXivHTMLPDF

Abstract

The nature of the thermal melting process by which triangular-lattice Ising antiferromagnets lose their low-temperature ferrimagnetic three-sublattice order depends on the range of the interactions: It changes character when second and third neighbour ferromagnetic interactions become comparable to the nearest-neighbour antiferromagnetic coupling. We present a detailed numerical characterization of the corresponding threshold at which two-step melting of three-sublattice order gives way to a direct first-order transition at which this order is lost. The multicritical behaviour at this threshold is argued to be in the universality class of the $Z_6$ parafermion conformal field theory with central charge $c=5/4$. The presence of this multicritical threshold influences the melting behaviour and long-wavelength properties over a fairly large range of parameters, and at temperatures that are of the same order as the exchange interactions. It is therefore of potential experimental relevance in the context of easy-axis triangular lattice antiferromagnets that display such low temperature ordering.

DOI
10.48550/arxiv.2109.03178

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Melting of three-sublattice order in triangular lattice Ising antiferromagnets: Power-law order, Z6Z_6Z6​ parafermionic multicriticality, and weakly first order transitions
The nature of the thermal melting process by which triangular-lattice Ising antiferromagnets lose their low-temperature ferrimagnetic three-sublattice order depends on the range of the interactions: It changes character when second and third neighbour ferromagnetic interactions become comparable to the nearest-neighbour antiferromagnetic coupling. We present a detailed numerical characterization of the corresponding threshold at which two-step melting of three-sublattice order gives way to a direct first-order transition at which this order is lost. The multicritical behaviour at this threshold is argued to be in the universality class of the Z6Z_6Z6​ parafermion conformal field theory with central charge c=5/4c=5/4c=5/4. The presence of this multicritical threshold influences the melting behaviour and long-wavelength properties over a fairly large range of parameters, and at temperatures that are of the same order as the exchange interactions. It is therefore of potential experimental relevance in the context of easy-axis triangular lattice antiferromagnets that display such low temperature ordering.
A transverse magnetic field $\mathrm{\ensuremath{\Gamma}}$ is known to induce antiferromagnetic three-sublattice order of the Ising spins ${\ensuremath{\sigma}}^{z}$ in the triangular lattice Ising antiferromagnet at low enough temperature. This low-temperature order is known to melt on heating in a two-step manner, with a power-law ordered intermediate temperature phase characterized by power-law correlations at the three-sublattice wave vector Q\mathbf{Q}Q: $\ensuremath{\langle}{\ensuremath{\sigma}}^{z}(\stackrel{P\vec}{R}){\ensuremath{\sigma}}^{z}(0)\ensuremath{\rangle}\ensuremath{\sim}cos(\mathbf{Q}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{R})/{|\stackrel{P\vec}{R}|}^{\ensuremath{\eta}(T)}$ with the temperature-dependent power-law exponent $\ensuremath{\eta}(T)\ensuremath{\in}(1/9,1/4)$. Here, we use a quantum cluster algorithm to study the ferromagnetic easy-axis susceptibility ${\ensuremath{\chi}}_{u}(L)$ of an $L\ifmmode\times\else\texttimes\fi{}L$ sample in this power-law ordered phase. Our numerical results are consistent with a recent prediction of a singular LLL dependence ${\ensuremath{\chi}}_{u}(L)\ensuremath{\sim}{L}^{2\ensuremath{-}9\ensuremath{\eta}}$ when $\ensuremath{\eta}(T)$ is in the range (1/9,2/9)(1/9,2/9)(1/9,2/9). This finite-size result implies, via standard scaling arguments, that the ferromagnetic susceptibility ${\ensuremath{\chi}}_{u}(B)$ to a uniform field BBB along the easy axis is singular at intermediate temperatures in the small BBB limit, ${\ensuremath{\chi}}_{u}(B)\ensuremath{\sim}{|B|}^{\ensuremath{-}\frac{4\ensuremath{-}18\ensuremath{\eta}}{4\ensuremath{-}9\ensuremath{\eta}}}$ for $\ensuremath{\eta}(T)\ensuremath{\in}(1/9,2/9)$, although there is no ferromagnetic long-range order in the low temperature state. Additionally we establish similar two-step melting behavior (via a study of the order parameter susceptibility ${\ensuremath{\chi}}_{\mathbf{Q}}$) in the case of the ferrimagnetic three-sublattice ordered phase which is stabilized by ferromagnetic next-neighbor couplings (J2{J}_{2}J2​) and confirm that the ferromagnetic susceptibility obeys the predicted singular form in the associated power-law ordered phase.
Critical properties of an Ising bilayer system, consisting of two triangular planes with antiferromagnetic (AF) JAJ_{\rm A}JA​ and ferromagnetic (FM) JBJ_{\rm B}JB​ exchange interactions, coupled by the interlayer exchange JABJ_{\rm AB}JAB​, are studied by Monte Carlo simulation. The FM plane is found to induce in the AF one some effects similar to those resulting from the joint presence of a next-nearest-neighbor FM coupling and an external magnetic field in the single triangular lattice Ising antiferromagnet. The similarities include the appearance of the quasi-long-range-order BKT phase with algebraically decaying correlations for smaller JABJ_{\rm AB}JAB​, as well as the phase transitions belonging to the three-state Potts universality class for larger JABJ_{\rm AB}JAB​. On the other hand, no first-order phase transitions are observed. A new phenomenon is a cross-over from the BKT to the Wannier state at higher temperatures, that can be stabilized over a relatively wide temperature range. In the strong JABJ_{\rm AB}JAB​ limit, the critical behavior of the entire bilayer is governed by the behavior of the plane with a dominant intraplane interaction. If JAJ_{\rm A}JA​ is dominant, then the strong JABJ_{\rm AB}JAB​ coupling makes the system kinetically freeze at lower temperatures.
The three-state Potts antiferromagnetic model on the triangular lattice (3PAFT) has a weak first-order transition in the pure case. The first-order nature follows from analytic arguments and while it has been observed in some simulations, the precision of (and internal disagreement between) these observations has left much to be desired. The 3PAFT is the only two-dimensional nearest-neighbour Potts model with an ordered phase that does not have an infinite degeneracy, that remains without an exact value for its critical point. This model is expected to exhibit interesting glassy behaviour and a possible crossover to a second-order transition upon dilution. A careful characterization of the nature of the pure transition is needed in order to explore the physics of the dilute model. We find clear indication of a first-order transition at T t =0.62731+or-0.00006, using a cluster method that can be extended to the dilute case and histogram reweighting analysis. Our estimate is two orders of magnitude more precise than previous simulations and falls within the error bounds of a reanalysis of existing series expansion (T t =0.628+or-0.004) reported on herein.
We study a triangular Ising model subject to nearest-neighbor pair interactions KKK, an external magnetic field HHH, and triangular three-spin interactions PPP. K>0(<0)K>0 (<0)K>0(<0) corresponds to ferromagnetic (antiferromagnetic) coupling. The phase diagram and thermodynamics are invariant under $(K,H,P)\ensuremath{\leftrightarrow}(K,\ensuremath{-}H,\ensuremath{-}P)$. A simple position-space renormalization group is formulated, based on three interpenetrating three-spin cells, which preserves the three-sublattice symmetry of the antiferromagnetic phase. The complete (K,H,PK,H,PK,H,P) phase diagram consists of five interconnecting transition surfaces (two second-order and three first-order), which bound the four single-phase regions (\ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\uparrow}, \ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\downarrow}, \ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\downarrow}, and \ensuremath{\downarrow}\ensuremath{\downarrow}\ensuremath{\downarrow}). The edges of these surfaces form loci of second-order, critical-endpoint, and Baxter-Wu transitions. When P=0P=0P=0, this system is known to provide a useful model of the order-disorder transition in monolayer films adsorbed on a hexagonal substrate. Calculation of the magnetization M(K,H)M(K,H)M(K,H) and heat capacity cM(K,H){c}_{M}(K,H)cM​(K,H) of the Ising model allows quantitative comparison with experimental data for helium on graphite. The observed dependence Tc(n){T}_{c}(n)Tc​(n) of the transition temperature on coverage is well fit with a single effective antiferromagnetic pair interaction. The corresponding calculated heat capacities (no additional adjustable parameters) are in remarkable agreement with experimental data near and below Tc{T}_{c}Tc​.
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.
The Ising model with nearest and next nearest neighbor antiferromagnetic interactions on the triangular lattice displays, for Jnnn/Jnn = 0.1, three phase transitions in different universality classes as the magnetic field is increased. We have studied this model using Monte Carlo and renormalization group techniques. The transition from the paramagnetic to the 2 × 1 phase (universality class of the Heisenberg model with cubic anisotropy) is found to be first order; the transition from the paramagnetic phase to the [Formula: see text] phase (universality class of the three state Potts model) is continuous; and the transition from the paramagnetic to the 2 × 2 phase (universality class of the four state Potts model) is found to change from first order to continuous as the field is increased. We have mapped out the phase diagram and determined the critical exponents for the continuous transitions. A novel technique, using a Landau-like free energy functional determined from Monte Carlo calculations, to distinguish between first order and continuous transitions, is described.