The Finite-Temperature Behavior of a Triangular Heisenberg Antiferromagnet

TL;DR

This work studies the finite-temperature behavior of the classical triangular-lattice Heisenberg antiferromagnet with up to second- and third-nearest-neighbor couplings using nematic bond theory to compute the temperature-dependent free energy and the static structure factor. By mapping the $J_1$-$J_2$-$J_3$ phase diagram, it identifies discontinuous lattice-symmetry-breaking transitions, stripe and other ordered phases, and spiral-spin-liquid regimes characterized by a ring of degenerate minima in momentum space when $J_3=J_2/2$. The results show how the specific-heat hump and transition temperatures depend on $J_2$ and $J_3$, and demonstrate entropy-driven order-by-disorder selecting among ring states. These findings provide quantitative benchmarks for neutron-scattering and specific-heat measurements in triangular-lattice magnets and help to infer exchange parameters in real materials.

Abstract

We investigate the classical antiferromagnetic Heisenberg model on the triangular lattice with up to third-nearest neighbor couplings using nematic bond theory. This approach allows us to compute the free energy and the neutron scattering static structure factor at finite temperatures. We map out the phase diagram with a particular emphasis on finite-temperature phase transitions that break lattice rotational symmetries, spiral spin liquids and the broad specific heat hump that is ubiquitous in the antiferromagnetic 120 degree phase.

Figures (9)

• Figure 1: Triangular lattice with first- ($J_1$), second- ($J_2$) and third-nearest ($J_3$) neighbor interactions. $\vec{a}_1=(1,0)$, $\vec{a}_2 = (-1/2, \sqrt{3}/2)$ and $\vec{a}_3 = (-1/2, -\sqrt{3}/2)$ are the triangular lattice vectors.
• Figure 2: Zero-temperature phase diagram with $J_1=1$Rastelli1980. Solid(dashed) lines indicate discontinuous(continuous) phase transitions. On the lower right, we show the first Brilloin zone of the triangular lattice with dashed lines indicating the high-symmetry lines $\Gamma$K and $\Gamma$M.
• Figure 3: Specific heat vs. temperature for different values of $J_2$ (indicated by the legends). $J_3=0$. $L=300$.
• Figure 4: $T_{\rm hump}\xspace$ (green) and $T_c$ (orange) vs. $J_2$ for $J_3=0$. The pink symbols show the lower $T_c$ associated with the phase transition between the ordered M and K states. Both orange and pink points represent discontinuous phase transitions. The orange dashed line indicates a crossover region where the ordering wave vector goes smoothly from M to $\Gamma$M as the temperature is lowered. $L=300$.
• Figure 5: $T_{\rm hump}\xspace$ (green) and $T_c$ (orange) vs. $J_2$ for different values of $J_3$ indicated by the numbers (right). Each curve is lifted by an amount $(J_3-0.05) \times 10$ to avoid overlaps. The vertical bars indicate the positions of the $T=0$ phase boundaries from Fig. \ref{['fig:phasediagram_zeroT']}. All phase transitions are discontinuous. $L=300$.