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arXiv:1512.09059 · Physical review. E

Phase diagram of a truncated tetrahedral model

Roman Krčmár, Andrej Gendiar, Tomotoshi Nishino

Published Aug 23, 2016• 18 citations
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Abstract

Phase diagram of a discrete counterpart of the classical Heisenberg model, the truncated tetrahedral model, is analyzed on the square lattice, when the interaction is ferromagnetic. Each spin is represented by a unit vector that can point to one of the 12 vertices of the truncated tetrahedron, which is a continuous interpolation between the tetrahedron and the octahedron. Phase diagram of the model is determined by means of the statistical analog of the entanglement entropy, which is numerically calculated by the corner transfer matrix renormalization group method. The obtained phase diagram consists of four different phases, which are separated by five transition lines. In the parameter region, where the octahedral anisotropy is dominant, a weak first-order phase transition is observed.

DOI
10.1103/physreve.94.022134

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Phase diagram of a discrete counterpart of the classical Heisenberg model, the truncated tetrahedral model, is analyzed on the square lattice, when the interaction is ferromagnetic. Each spin is represented by a unit vector that can point to one of the 12 vertices of the truncated tetrahedron, which is a continuous interpolation between the tetrahedron and the octahedron. Phase diagram of the model is determined by means of the statistical analog of the entanglement entropy, which is numerically calculated by the corner transfer matrix renormalization group method. The obtained phase diagram consists of four different phases, which are separated by five transition lines. In the parameter region, where the octahedral anisotropy is dominant, a weak first-order phase transition is observed.
In the context of a discrete analog of the classical Heisenberg model, we investigate the critical behavior of the icosahedron model, where the interaction energy is defined as the inner product of neighboring vector spins of unit length pointing to the vertices of the icosahedron. The effective correlation length and magnetization of the model are calculated by means of the corner-transfer-matrix renormalization group (CTMRG) method. A scaling analysis with respect to the cutoff dimension m in CTMRG reveals a second-order phase transition characterized by the exponents ν=1.62±0.02 and β=0.12±0.01. We also extract the central charge from the classical analog of entanglement entropy as c=1.90±0.02, which cannot be explained by the minimal series of conformal field theory.
We study the Kitaev--Heisenberg model on a triangular lattice by using the two-dimensional density-matrix renormalization group method. Calculating the ground-state energy and spin structure factors, we obtain a ground-state phase diagram of the Kitaev--Heisenberg model. As suggested by previous studies, we find a 120∘^\circ∘ antiferromagnetic (AFM) phase, a Z2\mathbb{Z}_2Z2​-vortex crystal phase, a nematic phase, a dual Z2\mathbb{Z}_2Z2​-vortex crystal phase (the dual counterpart of the Z2\mathbb{Z}_2Z2​-vortex crystal phase), a Z6\mathbb{Z}_6Z6​ ferromagnetic phase, and a dual ferromagnetic phase (the dual counterpart of the $\mathbb{Z}_6 $ ferromagnetic phase). Spin correlations discontinuously change at phase boundaries because of first-order phase transitions. We also study the relation among the von Neumann entanglement entropy, entanglement spectrum, and phase transitions of the model. We find that the Schmidt gap closes at phase boundaries and thus the entanglement entropy clearly changes as well. This is different from the Kitaev--Heisenberg model on a honeycomb lattice, where the Schmidt gap and entanglement entropy are not necessarily a good measure of phase transitions.
We obtain phase diagrams of a spin-3/2 Ising model on a square lattice by using a corner transfer matrix renormalization group method. We find that three phases appear in the phase diagrams; they are the ferromagnetic(FM) phase, the paramagnetic(PM) phase and a ferrimagnetic(FI) phase. It turns out that a tricritical point appears in a region of space spanned by parameters of coupling constants and the temperature. We estimate values of critical exponent ν and α at a transition temperature between the FM phase and the PM phase, at that between the FI phase and the FM phase, and also at that between the FI phase and the PM phase, by using the finite size scaling analysis.
The asymmetric tetrahedron is composed of all edges of a tetrahedron represented by Ising interactions except for one, which has a Heisenberg-type interaction. This asymmetric tetrahedron is arranged by connecting a vertex whose edges are only Ising-type interactions to another vertex with the same structure for another tetrahedron. The process is replicated and this kind of lattice we call the asymmetric Ising–Heisenberg chain. We have studied the ground-state phase diagram for this kind of model. In particular, we consider two situations in the Heisenberg-type interaction: (i) the asymmetric tetrahedral spin-(1/2, 1/2) Ising-XYZ chain and (ii) the asymmetric tetrahedral spin-(1/2, 1) Ising-XXZ chain, where we have found a rich phase diagram and a number of multicritical points. Additionally we have also studied their thermodynamical properties and the correlation function, using the decorated transformation. We have mapped the asymmetric tetrahedral Ising–Heisenberg chain in an effective Ising chain, and we have also concluded that it is possible to evaluate the partition function including a longitudinal external magnetic field.
Anisotropy is important for the existence of true long range order in two dimensional (2D) systems. This is firmly exemplified by the qqq-state clock models in which discreteness drives the quasi-long range order into a true long range order at low temperature for q>4q> 4q>4. Previously we studied 2D edge-cubic spin model, which is one of the discrete counterpart of the continuous Heisenberg model, and observed two finite temperature phase transitions, each corresponds to the breakdown of octahedral (OhO_hOh​) symmetry and C3hC_{3h}C3h​ symmetry, which finally freezes into ground state configuration. The present study investigates discret models with polyhedral symmetry, obtained by e equally partioning the 4π4π4π of the solid angle of a sphere. There are five types of models if spins are only allowed to point to the vertices of the polyhedral structures such as Tetrahedron, Octahedron, Hexahedron, Icosahedron and Dodecahedron. By using Monte Carlo simulation with cluster algorithm we calculate order parameters and estimate the critical temperatures exponents of each model. We found a systematic decrease in critical temperatures as the number of spin states increases (from the Tetrahedral to Dodecahedral spin model).
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.