Localized fermions on the triangular lattice with Ising-like interactions

TL;DR

This work investigates localized fermions on a triangular lattice described by an extended Hubbard model with onsite interaction $U$ and nearest-neighbor Ising-like coupling $J$, using grand-canonical Monte Carlo to map phase diagrams in $(\mu,U,T)$. By analyzing specific heat, sublattice magnetizations, particle number, and energy distributions, it reveals distinct ordering patterns for $J<0$ (ferromagnetic) and $J>0$ (antiferromagnetic) cases, including first- and second-order transitions and a tricritical region for the ferromagnetic case. The ferromagnetic phase exhibits a NO–F transition that is first-order at low $T$ for certain $U$ ranges and second-order otherwise, with a maximal $k_B T/|J|$ near $0.605$ as $U\to\infty$; the antiferromagnetic case is strongly frustrated, supporting a honeycomb-like AF arrangement with only second-order transitions and a maximal $k_B T/|J|$ around $0.06$. Overall, the results illuminate how frustration and correlated electrons shape finite-temperature order on the triangular lattice and offer benchmarks for experiments with frustrated magnets or ultracold atomic systems.

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 $k_{B}T/|J| \approx 0.61$) is found to be second-order (for sufficiently large temperatures $k_{B}T/|J| \gtrsim 0.2$) 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 $n \neq 1$). 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 $μ\neq 0$ ($n \neq 1$) and $U/|J|>-1/2$ is always second-order for any model parameters. The ordered phase for $J>0$ can be stable up to $k_{B}T/|J| \approx 0.06$.

Figures (7)

• Figure 1: Schematic three dimensional phase diagrams (in $\mu$-$U$-$k_{B}T$ space) for ferromagnetic case ($J<0$, the left panel) and antiferromagnetic case ($J>0$, the right panel). The blue and red surfaces denote first order (discontinuous) and second order (continuous) transitions.
• Figure 2: Phase diagrams for the model with ferromagnetic ($J=-1$) interactions, shown for three different lattice sizes ($L = 12^{2},24^{2}, 36^{2}$). Triangle markers indicate first-order phase transitions, while circular markers represent second-order phase transitions. In the bottom-left corner, Monte Carlo averaged spin configurations for the identified ferromagnetic (F) phase at $U=-0.6$, $\mu=0.0$, and $k_BT=0.15$ and non-ordered (NO) phase at $U=-0.6$, $\mu=0.0$, and $k_BT=0.4$, are shown for a lattice of size $L = 6 \times 6$.
• Figure 3: Specific heat capacities (top row), sublattice magnetizations (second row), and the energy distribution functions near the phase transition on a $24 \times 24$ lattice (third line) for ferromagnetic interaction ($J=-1$) and representative values of $U$ ($U = -1.0, -0.7, -0.6, 5.0$, from the left) at $\mu=0$.
• Figure 4: Specific heat capacities (top row), sublattice magnetizations (second row), number of particles (third row), and the energy distribution functions near the phase transition on a $24 \times 24$ lattice (fourth line) for ferromagnetic interaction ($J=-1$) and representative values of $U$ and $\mu$ (from the left: $U = -0.7$ and $\mu = -0.15, -0.10$; $U = -0.6$ and $\mu = -0.18, -0.12$).
• Figure 5: Phase diagrams for the model with antiferromagnetic ($J=1$) interactions, shown for three different lattice sizes ($L = 12^{2},24^{2}, 36^{2}$). Circular markers represent second-order phase transitions and diamonds illustrates the crossover from short-range order (SO) to non-ordered phase (NO). In the bottom-left corner, Monte Carlo averaged spin configurations for the identified antiferromagnetic phase (AF) at $U=1$, $\mu=-0.6$, and $k_BT=0.02$, and for the short-ordered (SO) phase at $U=1$, $\mu=-0.3$, and $k_BT=0.02$, are shown for a lattice of size $L = 6 \times 6$.