Classical Heisenberg and XY models on zigzag ladder lattices with nearest-neighbor bilinear-biquadratic exchange: Exact solution for the ground-state problem

TL;DR

This work provides an exact, complete ground-state solution for classical Heisenberg and XY models with nearest-neighbor bilinear-biquadratic exchange on lattices built from isosceles triangles, using a cluster method focused on a triangular plaquette. The analysis reduces the problem to minimization of the triangle energy $E = K \cos \alpha + L(\cos \beta + \cos \gamma) - A \cos^2 \alpha - B(\cos^2 \beta + \cos^2 \gamma)$ over the tetrahedral angle space, yielding explicit ground-state structures and a global phase diagram that extends to 2D and 3D zigzag ladders. The main findings include two spiral phases, Sp$_1$ and Sp$_2$, with both continuous and discontinuous transitions and tricritical points, plus highly degenerate canted CF states and degeneracy lifting when moving from 2D to 3D lattices. These results advance the understanding of frustration-induced order in bilinear-biquadratic spin systems and highlight how lattice geometry governs ground-state degeneracies and phase transitions with potential relevance to real magnetic materials. The methodology and exact solutions provide a framework for analyzing similar frustrated spin models on complex lattices.

Abstract

An exact and complete solution of the ground-state problem for the classical Heisenberg and XY models with nearest-neighbor bilinear-biquadratic exchange on two- and three-dimensional lattices composed of isosceles triangles is determined with the use of a cluster method. It is shown how the geometric frustration due to the presence of triangles as structural units leads to the emergence of a rich phase diagram with incommensurate spiral orderings and their collinear limits, as well as canted and noncoplanar (conical) structures. Surprisingly, there are two different spiral phases with both continuous and discontinuous phase transitions between them. One of these phases is degenerate on two-dimensional partially anisotropic triangular lattice. This degeneracy is lifted on three-dimensional lattices. Canted phase is highly degenerate and this degeneracy persists on three-dimensional lattices.

Figures (12)

• Figure 1: (a) An elementary triangular plaquette for zigzag ladder lattices with spins $\textbf{S}_1$, $\textbf{S}_2$, and $\textbf{S}_3$ (unit 3-vectors) at its vertices. There are linear ($K$ and $L$) and biquadratic ($A$ and $B$) pairwise interactions (per one plaquette) between the spins. (b) Tetrahedron of values for angles $\alpha$, $\beta$, and $\gamma$ between the spins. Vertices of the tetrahedron correspond to three collinear spins; edges and faces to three coplanar ones; for edges two of three spins are collinear. (c) Cross section of the tetrahedron at a fixed value of $\alpha$.
• Figure 2: Ground-state phase diagrams for the bilinear-biquadratic Heisenberg and XY models on one isosceles triangle. (a) $A = -0.2|L|$ (Heisenberg model), (b) $A = -0.2|L|$ (XY model), (c) $A = 0$ (both models), (d) $A = 0.2|L|$ (both models). Black and red lines correspond to continuous and discontinuous transitions, respectively. Dotted lines correspond to $A \rightarrow +\infty$. The diagram of type (a) for the Heisenberg model occurs if $-\frac{4}{9}\sqrt6 L \approx -1.08866 L < A < 0$ and the diagram of type (b) for XY model occurs if $-\frac{1}{2} < A < 0$.
• Figure 3: Ground-state phase diagram for the bilinear-biquadratic (a), (c) Heisenberg and (b), (d) XY models on one isosceles triangle at $A = -3.0$ and $A = -2.0$ ($L = 1.0$). Diagrams of these types occur when $A < -\frac{4}{9}\sqrt6 L \approx -1.08866 L~~(L > 0)$. Black and red lines correspond to continuous and discontinuous transitions, respectively. The tricritical points and the points of minimum $\alpha$ value for Sp$_2$ structure are indicated with magenta color. At $A = -3.0$ and $L = 1.0$, $\alpha \approx 0.68673486445\pi$ [see Figs. (a) and (b)].
• Figure 4: Ground-state phase diagram for the XY model on one isosceles triangle at $A = -1.0|L|$. Diagrams of these types arise when $-\frac{4}{9}\sqrt6 |L| < A < -\frac{1}{2}|L|$.
• Figure 5: Ground-state phase diagram for the bilinear-biquadratic (a), (c) Heisenberg and (b), (c) XY models on one isosceles triangle at $L = 0$.