Adjoint ferromagnets

TL;DR

The paper studies ferromagnets whose elementary magnets carry the adjoint representation of SU(N) and interact via two-body quadratic couplings, revealing a phase diagram with a paramagnetic singlet and two ferromagnetic phases (A and B) that can coexist as stable or metastable states. Using a mean-field thermodynamic limit, it derives saddle-point equilibrium equations and a stability analysis, uncovering a discrete conjugation symmetry that can be spontaneously broken in phase A. The analysis shows that the stable phases correspond to specific Young tableau patterns and SU(N) breaking channels, with the phase structure strongly dependent on the group rank N and featuring multiple critical temperatures and metastability, which persist into the large-N limit. The work generalizes previous results for other representations, highlights the role of conjugation symmetry in ferromagnets with self-conjugate representations, and outlines potential experimental and theoretical extensions, including external fields and higher-order interactions.

Abstract

We derive the phase structure and thermodynamics of ferromagnets consisting of elementary magnets carrying the adjoint representation of $SU(N)$ and coupled through two-body quadratic interactions. Such systems have a continuous $SU(N)$ symmetry as well as a discrete conjugation symmetry. We uncover a rich spectrum of phases and transitions, involving a paramagnetic and two distinct ferromagnetic phases that can coexist as stable and metastable states in different combinations over a range of temperatures. The ferromagnetic phases break $SU(N)$ invariance in various channels, leading to spontaneous magnetization. Interestingly, the conjugation symmetry also breaks over a range of temperatures and group ranks $N$, providing a realization of a spontaneously broken discrete symmetry.

Figures (8)

• Figure 1: Plots of the left hand side of (\ref{['eqq1']}) for $T<T_0(\rho)$ (blue) and for $T>T_0(\rho)$ (yellow).
• Figure 2: Eigenvalues of $T\mathbb{1}-c\Lambda$ for a solution of type A and $N=5$, for $p_3=1$ (left) and $p_3=2$ (right). In the former case, all eigenvalues are positive for temperatures $T_{\rm A}^{\rm (m)} < T < 1$, whereas in the latter one several eigenvalues are negative in the full temperature range in which the solutions exist. Temperatures are in units of $T_0$, and multi-colored lines represent degenerate eigenvalues.
• Figure 3: Eigenvalues of $T\mathbb{1}-c\Lambda$ for a solution of type B and $N=5$, for $p_1=1$ (left) and $p_1=2$ (right). Temperatures are in units of $T_0$. For $p_1=1$, the solution is stable for temperatures below $T_{\rm B}^{\rm (un)}$, whereas for $p_1 = 2$ the solution is unstable for all temperatures. Multi-colored lines represent degenerate eigenvalues.
• Figure 4: Plot of the left hand side of (\ref{['u13w1']}) for $T<T_0$.
• Figure 5: Left plot: the left hand side of (\ref{['u13w1']}) for $T>T_0$, for $N=3,4,5$. Right plot: the left hand side of (\ref{['u13w1']}) for $N \geqslant 6$ for $T_0 < T^{(c)}_A < T$ (in blue) and for $T_0 < T<T^{(c)}_A$ (in yellow).