Ground states of the Ising model at fixed magnetization on a triangular ladder with three-spin interactions

TL;DR

This paper addresses the zero-temperature ground-state problem of an Ising model with three-spin interactions on a two-leg triangular ladder under fixed magnetization. It introduces a linear-programming framework that encodes realizability constraints through a parametrization of local triangle configurations, enabling exact enumeration of all ground states. The authors identify three ground-state classes—periodic, phase-separated, and ordered but aperiodic—and construct phase diagrams at the critical magnetizations $m=0$ and $m=1/3$, detailing how these phases evolve with intermediate $m$; when magnetization is treated as a free parameter, only periodic configurations appear, with allowed values $m\in\{0,\pm1/3,\pm1\}$ away from phase boundaries. The work provides an exact, scalable methodology for constrained ground-state enumeration of frustrated Ising models, with potential relevance to ultracold-atom simulations and the study of complex ordering in low-dimensional systems, and resolves the issue of inconstructible vertices in this chain setting.

Abstract

We study the Ising model at fixed magnetization on a triangular ladder with three-spin interactions. By recasting the ground-state determination as a linear programming (LP) problem, we solve it exactly using standard LP techniques. We construct the phase diagram for arbitrary fixed magnetization and identify three types of ground states: periodic, phase-separated, and ordered but aperiodic. When magnetization is treated as a free parameter, the ground state adopts only periodic configurations with the average magnetization per site $0$, $\pm 1/3$ or $\pm 1$, except for the phase boundaries.

Figures (4)

• Figure 1: Sketch of the lattice for the model \ref{['eq:H']}. The sites are enumerated along the zig-zag chain. Periodic boundary conditions are imposed, and each leg contains $L$ sites, giving $2L$ sites in total.
• Figure 2: All possible spin configurations on triangular plaquettes. Solid and open circles represent ${\sigma = +1}$ and ${\sigma = -1}$, respectively. The symbols $u_i$ and $v_i$ label the configurations and also denote their occurrence frequencies ${(0 \leqslant u_i, v_i \leqslant 1)}$ in a given state.
• Figure 3: Phase diagrams at $m=0$, $m=1/3$ for $J>0$ and $J<0$. Top row: $m=0$. Bottom row: $m=1/3$. Left column: $J<0$. Right column: $J>0$. In each region, only one representative state is depicted for clarity. For an arbitrary magnetization value $m$, the phase diagram consists of phases with the same structure as those at ${m=0}$ and ${m=1/3}$, but with different number of particles. The boundaries between phases are accordingly shifted.
• Figure 4: Phase diagram for unrestricted magnetization. Only periodic ground states are realized, with $m$ taking one of the critical values, except for the phase boundaries.