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Physical review. B, Solid state

Phase diagram of the triangular Ising model: Renormalization-group calculation with application to adsorbed monolayers

Michael Schick C, James S. Walker, Michael Wortis

Published Sep 1, 1977• 199 citations
Publisher

Abstract

We study a triangular Ising model subject to nearest-neighbor pair interactions $K$, an external magnetic field $H$, and triangular three-spin interactions $P$. $K>0 (<0)$ 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 ($K,H,P$) 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 $P=0$, 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 $M(K,H)$ and heat capacity ${c}_{M}(K,H)$ of the Ising model allows quantitative comparison with experimental data for helium on graphite. The observed dependence ${T}_{c}(n)$ 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 ${T}_{c}$.

DOI
10.1103/physrevb.16.2205

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arXiv:2412.10808•Physical review. E•Jun 3, 2025•1 citation
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↔(K,\ensuremath−H,\ensuremath−P)(K,H,P)\ensuremath{\leftrightarrow}(K,\ensuremath{-}H,\ensuremath{-}P)(K,H,P)\ensuremath↔(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​.
The lattice-gas reformulation of the spin-12\frac{1}{2}21​, triangular Ising antiferromagnet in a field is expected to mimic the behavior of He atoms adsorbed on grafoil (exfoliated graphite). These adsorbed helium systems have been observed to have a second-order phase transition for surface densities near one-third coverage. We have derived and analyzed seventh-order staggered susceptibility series for the triangular Ising antiferromagnet to locate the second-order phase boundary. Our results are in good quantitative agreement with results from real-space renormalization-group calculations for the model and with results from experiments on the helium layers.
The phase diagrams of Ising antiferromagnets in a magnetic field HHH are investigated for various values of the ratio RRR between nearest- and next-nearest-neighbor interaction. While meanfield approximations and the existing real-space renormalization-group treatments yield phase diagrams which are sometimes even qualitatively incorrect, accurate results are obtained from Monte Carlo calculations. For R<0R<0R<0 only an antiferromagnetically ordered phase exists. Its transition to the disordered phase is first order for temperatures below the tricritical point (Tt{T}_{t}Tt​,Ht{H}_{t}Ht​). For $R\ensuremath{\rightarrow}0$ also ${T}_{t}\ensuremath{\rightarrow}0$. For R=0R=0R=0 we find very good agreement with the results of M\"uller-Hartmann and Zittartz. For R>0R>0R>0 and H1<H<H2{H}_{1}<H<{H}_{2}H1​<H<H2​ a new phase with anomalous high ground-state degeneracy is found (two sublattices have only one-dimensional order). These sublattices undergo order-disorder transitions at T=0T=0T=0, such that for T>0T>0T>0 one is left with a "superantiferromagnetic" phase. At low temperatures in this phase a pronounced tendency is observed to form a simpler (2 \ifmmode\times\else\texttimes\fi{} 2) superstructure but with many antiphase domain boundaries. For $R\ensuremath{\rightarrow}\frac{1}{2}$ and H<H1H<{H}_{1}H<H1​ the regime of the antiferromagnetic phases goes to zero temperature, while for R>12R>\frac{1}{2}R>21​ the superantiferromagnetic phase exists also for H<H1H<{H}_{1}H<H1​. The order-disorder transition associated with this phase seems to have non-Ising critical exponents which vary as a function of RRR and HHH. Estimates for the exponents lead us to suggest that Suzuki's "weak universality" is valid. The behavior of the model at T=0T=0T=0 is related to known results on hard-core lattice gases. It is shown that it is useful to interpret the transitions at T=0T=0T=0 as generalized percolation transitions. Since the model may have applications to adsorbate phases in registered structures at (100) surfaces of cubic crystals, the transcription of our results to temperature-coverage phase diagrams and adsorption isotherms is discussed in detail, and possible experimental applications are mentioned.
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.
We use simple position-space renormalization-group procedures involving only nearest-neighbor interactions to discuss the critical behavior of the ferromagnetically coupled semi-infinite Ising model with a free surface in two and three dimensions. In three dimensions for suitably enhanced surface couplings there are surface transitions and simultaneous surface-bulk transitions as well as the bulk transition. A phase diagram, critical exponents for the various transitions, and some representative magnetization profiles are calculated. Despite their simplicity, the renormalization transformations exhibit complete sets of fixed points for describing the various phase transitions and yield phase diagrams which are a significant improvement over mean-field theory. However, the calculated magnetization profiles show some unphysical features due to the finite cluster size on which the transformations are based.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.
The model of localized fermions on the triangular lattice is analyzed in means of the Monte Carlo simulations in the grand canonical ensemble. The Hamiltonian of the system has a form of the extended Hubbard model (at the atomic limit) with nearest-neighbor Ising-like magnetic J interactions and onsite Coulomb U interactions. The model is investigated for both signs of J, arbitrary U interaction, and arbitrary chemical potential μ (or, equivalently, arbitrary particle concentration n). Based on the specific heat capacity and sublattice magnetization analyses, the phase diagrams of the model are determined. For ferromagnetic case (J<0), the transition from the ordered phase (which is a standard ferromagnet and can be stable up to k_{B}T/|J|≈0.61) is found to be second order (for sufficiently large temperatures k_{B}T/|J|≳0.2) or first order (for -1<U/|J|<-0.65 at the half-filling, i.e., n=1). In the case of J>0, the ordered phase occurs in a range of -1/2<U/|J|<0 (for n=1), while for larger U the state with short-range order is also found (also for n≠1). The ordered phase is characterized by an antiferromagnetic arrangement of magnetic moments in two sublattices forming the hexagonal lattice. The transition from this ordered phase, which is found also for μ≠0 (n≠1) and U/|J|>-1/2 is always second order for any model parameters. The ordered phase for J>0 can be stable up to k_{B}T/|J|≈0.06.