Adjoint ferromagnets

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.

💡 Research Summary

The paper investigates a ferromagnetic model in which each “atom” carries the adjoint representation of the SU(N) group and interacts with all other atoms via a two‑body quadratic (Heisenberg‑type) coupling. Using a mean‑field (large‑n) approach, the authors derive the free‑energy per site as a function of magnetization parameters x_i associated with the Cartan subalgebra. Because the adjoint representation is self‑conjugate, the system possesses not only a continuous SU(N) symmetry but also an additional discrete conjugation symmetry (z → z⁻¹). The character of the adjoint representation is χ(z)=∑_i z_i ∑_j z_j⁻¹ − 1, which has zero total U(1) charge; consequently the magnetization variables satisfy the constraint Σ_i x_i = 0.

Introducing logarithmic variables w_i = ln z_i and λ = ln χ, the saddle‑point equations become x_i = ∂λ/∂w_i and T w_i = c x_i (with c the ferromagnetic coupling). By shifting w_i → α_i = w_i + μ, the equations reduce to a universal nonlinear relation α_i – (T₀(ρ)/T) sinh α_i = μ, where ρ = Σ_i e^{α_i} = Σ_i e^{-α_i} ≥ N and T₀(ρ)=2ρ/(ρ²–1)·c. The ratio T₀(ρ)/T determines the number of solutions: for T > T₀(ρ) only the trivial solution α_i = 0 (the singlet, i.e., paramagnetic phase) exists; for T < T₀(ρ) three real solutions appear (α₁>0, α₂, α₃<0). The multiplicities (p₁, p₂, p₃) of each solution must satisfy p₁+p₂+p₃ = N and the constraints Σ_i p_i sinh α_i = 0 and ρ = Σ_i p_i e^{α_i} = Σ_i p_i e^{-α_i}. These conditions uniquely determine a Young tableau (YT) that encodes the distribution of “boxes” (positive magnetization) and “antiboxes” (negative magnetization) among the rows.

The free energy at equilibrium can be written as F(T) = (T₀ ρ²/(N²–1)) Σ_i sinh²α_i – T ln(ρ²–1), which, together with the stability criteria derived from the Hessian matrix Λ_{ij}=∂²λ/∂w_i∂w_j, allows the authors to classify stable, metastable, and unstable configurations. The first stability condition Λ ≥ 0 is automatically satisfied; the second condition T 1 – c Λ ≥ 0 imposes non‑trivial restrictions.

A systematic numerical scan over all possible (p₁,p₂,p₃) shows that almost all configurations are unstable, except for two families of Young tableaux:

• Phase A – Two rows of equal length ℓ₁ and the remaining N‑2 rows empty (e.g., for N=4 the tableau looks like ••··). This phase appears in a pair related by conjugation (A and A*), thereby breaking the discrete conjugation symmetry spontaneously. The continuous symmetry breaking pattern is SU(N) → SU(N‑1) × U(1).

• Phase B – One long row of length ℓ₁ and N‑2 rows of length ℓ₂ = ℓ₁/2 (e.g., ••···). This configuration is self‑conjugate, so the conjugation symmetry remains intact. The continuous symmetry breaking pattern is SU(N) → SU(N‑2) × U(1) × U(1).

The singlet (all α_i = 0) corresponds to the high‑temperature paramagnetic state. Each of the three phases (singlet, A, B) possesses its own critical temperature T_c(N). For small N (N=2,3) the phase diagram is simple: a single Curie‑like transition separates the singlet from either A or B. As N increases, the critical temperatures rearrange, metastable regions broaden, and multiple transitions can occur. Notably, for N ≥ 13 the A and B phases can coexist over a sizable temperature interval, and in the large‑N limit (N → ∞) a triple critical point emerges where all three phases become simultaneously marginally stable. The scaling parameter T₀ approaches a constant T₀ = 2N c/(N²–1) in this limit.

The authors also examine a reducible representation consisting of a fundamental and an anti‑fundamental (fundamental‑antifundamental) plus a singlet. Although the representation differs from the adjoint by the presence of an extra singlet state, the overall phase structure is qualitatively identical: the same A and B ferromagnetic phases appear, with the singlet simply shifting the free‑energy baseline.

In conclusion, the adjoint‑representation SU(N) ferromagnet exhibits a surprisingly rich landscape despite the presence of both continuous and discrete symmetries. Two distinct ferromagnetic phases (A and B) break SU(N) in different channels and either preserve or break the conjugation symmetry, while a paramagnetic singlet dominates at high temperature. The dependence of critical temperatures and metastability on the group rank N, together with the emergence of a triple critical point at large N, provides a novel playground for exploring symmetry breaking, metastability, and phase coexistence in systems with high‑dimensional internal symmetries. The results are relevant for experimental platforms such as ultracold atoms with engineered SU(N) interactions, multi‑orbital solid‑state systems, and lattice gauge‑theory analogues, where the adjoint degrees of freedom can be realized and the predicted phase diagram potentially observed. Future work may incorporate external SU(N) fields, higher‑body interactions, and quantum fluctuations beyond mean‑field to refine the picture and connect more directly with realistic experimental conditions.