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arXiv:2412.10808 · Physical review. E

Localized fermions on the triangular lattice with Ising-like interactions

Lubomíra Regeciová, Konrad Jerzy Kapcia

Published Jun 3, 2025• 1 citation
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Abstract

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.

DOI
10.1103/4lmb-6qqv

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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.
In this paper the two-dimensional extended Hubbard model with intersite magnetic Ising-like interaction in the atomic limit is analyzed by means of the classical Monte Carlo method in the grand canonical ensemble.Such an eective simple model could describe behavior of insulating (anti)ferromagnets.In the model considered the Coulomb interaction (U ) is on-site and the magnetic interactions in z-direction (J > 0, antiferromagnetic) are restricted to nearest-neighbors.Simulations of the model have been performed on a square lattice consisting of N = L × L = 400 sites (L = 20) in order to obtain the full phase diagram for U/(4J) = 1.Results obtained for on-site repulsion (U > 0) show that, apart from homogeneous non-ordered (NO) and ordered magnetic (antiferromagnetic, AF) phases, there is also a region of phase separation (PS: AF/NO) occurrence.We present a phase diagram as well as some thermodynamic properties of the model for the case of U/(4J) = 1 (and arbitrary chemical potential and arbitrary electron concentration).The AFNO transition can be second-order as well as rst-order and the tricritical point occurs on the diagram.
The extended Hubbard model in the atomic limit, which is equivalent to lattice S=1/2 fermionic gas, is considered on the triangular lattice. The model includes onsite Hubbard U interaction and both nearest-neighbor (W1) and next-nearest-neighbor (W2) density–density intersite interactions. The variational approach treating the U term exactly and the Wl terms in the mean-field approximation is used to investigate thermodynamics of the model and to find its finite temperature (T>0) phase diagrams (as a function of particle concentration) for W1>0 and W2<0. Two different types of charge-order (i.e., DCO and TCO phases) within 3×3 unit cells as well as the nonordered (NO) phase occur on the diagram. Moreover, several kinds of phase-separated (PS) states (NO/DCO, DCO/DCO, DCO/TCO, and TCO/TCO) are found to be stable for fixed concentration. Attractive W2<0 stabilizes PS states at T=0 and it extends the regions of their occurrence at T>0. The evolution of the diagrams with increasing of |W2|/W1 is investigated. It is found that some of the PS states are stable only at T>0. Two different critical values of |W2|/W1 are determined for the PS states, in which two ordered phases of the same type (i.e., two domains of the DCO or TCO phase) coexist.
The adsorbed atoms exhibit tendency to occupy a triangular lattice formed by periodic potential of the underlying crystal surface. Such a lattice is formed by, e.g., a single layer of graphane or the graphite surfaces as well as (111) surface of face-cubic center crystals. In the present work, an extension of the lattice gas model to S=1/2 fermionic particles on the two-dimensional triangular (hexagonal) lattice is analyzed. In such a model, each lattice site can be occupied not by only one particle, but by two particles, which interact with each other by onsite U and intersite W1 and W2 (nearest and next-nearest-neighbor, respectively) density-density interaction. The investigated hamiltonian has a form of the extended Hubbard model in the atomic limit (i.e., the zero-bandwidth limit). In the analysis of the phase diagrams and thermodynamic properties of this model with repulsive W1>0, the variational approach is used, which treats the onsite interaction term exactly and the intersite interactions within the mean-field approximation. The ground state (T=0) diagram for W2≤0 as well as finite temperature (T>0) phase diagrams for W2=0 are presented. Two different types of charge order within 3×3 unit cell can occur. At T=0, for W2=0 phase separated states are degenerated with homogeneous phases (but T>0 removes this degeneration), whereas attractive W2<0 stabilizes phase separation at incommensurate fillings. For U/W1<0 and U/W1>1/2 only the phase with two different concentrations occurs (together with two different phase separated states occurring), whereas for small repulsive 0<U/W1<1/2 the other ordered phase also appears (with tree different concentrations in sublattices). The qualitative differences with the model considered on hypercubic lattices are also discussed.
Motivated by recent experiments with a Penning ion trap quantum simulator, we perform numerically exact Stochastic Series Expansion quantum Monte Carlo simulations of long-range transverse-field Ising models on a triangular lattice for different decay powers $\ensuremath{\alpha}$ of the interactions. The phase boundary for the ferromagnet is obtained as a function of $\ensuremath{\alpha}$. For antiferromagnetic interactions, there is strong indication that the transverse field stabilizes a clock ordered phase with sublattice magnetization $(M,\ensuremath{-}\frac{M}{2},\ensuremath{-}\frac{M}{2})$ with unsaturated M<1M<1M<1 in a process known as ``order by disorder'' similar to the nearest-neighbor antiferromagnet on the triangular lattice. Connecting the known limiting cases of nearest-neighbor and infinite-range interactions, a semiquantitative phase diagram is obtained. Magnetization curves for the ferromagnet for experimentally relevant system sizes and with open boundary conditions are presented.
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