Ferromagnetic phase transitions in $SU(N)$

We study the thermodynamics of a non-abelian ferromagnet consisting of “atoms” each carrying a fundamental representation of <math altimg="si1.svg"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math>, coupled with long-range two-body quadratic interactions. We uncover a rich structure of phase transitions from non-magnetized, global <math altimg="si1.svg"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math>-invariant states to magnetized ones breaking global invariance to <math altimg="si2.svg"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo linebreak="badbreak" linebreakstyle="after">−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math>. Phases can coexist, one being stable and the other metastable, and the transition between states involves latent heat exchange, unlike in usual <math altimg="si3.svg"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math> ferromagnets. Coupling the system to an external non-abelian magnetic field further enriches the phase structure, leading to additional phases. The system manifests hysteresis phenomena both in the magnetic field, as in usual ferromagnets, and in the temperature, in analogy to supercooled water. Potential applications are in fundamental situations or as a phenomenological model.