We derive the phase structure and thermodynamics of ferromagnets consisting of elementary magnets carrying the adjoint representation of <italic>SU</italic>( <italic>N</italic>) and coupled through two-body quadratic interactions, using a mean field approach. Such systems have a continuous <italic>SU</italic>( <italic>N</italic>) symmetry as well as a discrete conjugation symmetry. We uncover a rich spectrum of phase 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 <italic>SU</italic>( <italic>N</italic>) invariance in various channels, leading to spontaneous magnetization. Interestingly, the conjugation symmetry also breaks over a range of temperatures and group ranks <italic>N</italic>, providing a realization of a spontaneously broken discrete symmetry.

Greece

Joaquín López-Suárez

United States

Alexios P. Polychronakos

Konstantinos Sfetsos

Adjoint ferromagnets

We study the thermodynamics of nonabelian ferromagnets consisting of atoms in the 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> and interacting with two-body and three-body interactions. Using a mean field approach, we uncover an intricate phase structure, depending on the relative strength and sign of the two-body and three-body coupling constants. In the case where two-body interactions are ferromagnetic and three-body ones are antiferromagnetic, we uncover a rich cascade of phase transitions, the appearance of phases with two distinct polarization directions being the most striking novel feature. Our results are relevant to magnetic systems where higher-body interactions cannot be neglected.

Nonabelian ferromagnets with three-body interactions

We present a general formalism for deriving the thermodynamics of ferromagnets consisting of “atoms” carrying an arbitrary irreducible representation of <math altimg='si1.svg'><mi>S</mi><mi>U</mi><mo stretchy='false'>(</mo><mi>N</mi><mo stretchy='false'>)</mo></math> and coupled through long-range two-body quadratic interactions. Using this formalism, we derive the thermodynamics and phase structure of ferromagnets with atoms in the doubly symmetric or doubly antisymmetric irreducible representations. The symmetric representation leads to a paramagnetic and a ferromagnetic phase with transitions similar to the ones for the fundamental representation studied before. The antisymmetric representation presents qualitatively new features, leading to a paramagnetic and two distinct ferromagnetic phases that can coexist over a range of temperatures, two of them becoming metastable. Our results are relevant to magnetic systems of atoms with reduced symmetry in their interactions compared to the fundamental case.

Ferromagnets from higher <italic>SU</italic>( <italic>N</italic>) representations

The non-Abelian ferromagnet recently introduced by the authors, consisting of atoms in the fundamental representation of <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math>, is studied in the limit where N becomes large and scales as the square root of the number of atoms n. This model exhibits additional phases, as well as two different temperature scales related by a factor <math><mi>N</mi><mo>/</mo><mi>ln</mi><mo>⁡</mo><mi>N</mi></math>. The paramagnetic phase splits into a “dense” and a “dilute” phase, separated by a third-order transition and leading to a triple critical point in the scale parameter <math><mi>n</mi><mo>/</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup></math> and the temperature, while the ferromagnetic phase exhibits additional structure, and a new paramagnetic-ferromagnetic metastable phase appears at the larger temperature scale. These phases can coexist, becoming stable or metastable as temperature varies. A generalized model in which the number of <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math>-equivalent states enters the partition function with a nontrivial weight, relevant, e.g., when there is gauge invariance in the system, is also studied and shown to manifest similar phases, the dense-dilute phase transition becoming second-order in the fully gauge invariant case.

Triple critical point and emerging temperature scales in ( ) Ferromagnetism at Large

Particle production in integrable field theories may exist depending on the vacuum around which excitations are defined. To tackle this and analogous issues with conventional field theoretical tools, we consider the integrable λ-deformed model for <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> together with a timelike coordinate. We construct the corresponding four-dimensional plane wave background (a deformation of the Nappi–Witten solution) keeping also post-plane wave corrections, as well as all the non-trivial λ-dependence. After imposing the classical light-cone gauge and the Virasoro constraints, we obtain an interacting field theory for the transverse physical modes which are massive. We explicitly demonstrate the absence of particle production to leading order in the large k-expansion. This is based crucially on the form of the interaction vertices and their dependence on the λ-deformation parameter. In addition, we compute the S-matrix for the two-particle elastic scattering exactly in λ and to leading order in the large k-expansion. Our method can be applied to any integrable theory with at least one isometry in order to check the classical integrability of the parent theory.

George Georgiou

CERN

Scattering in integrable pp-wave backgrounds: -matrix and Absence of Particle Production

We study the multiplicity of irreducible representations in the decomposition of n fundamentals of <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> weighted by a power of their dimension in the large n and large N double scaling limit. A nontrivial scaling is obtained by keeping <math><mi>n</mi><mo>/</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup></math> fixed, which plays the role of an order parameter. We find that the system generically undergoes a fourth order phase transition in this parameter, from a dense phase to a dilute phase. The transition is enhanced to third order for the unweighted multiplicity, and disappears altogether when weighting with the first power of the dimension. This corresponds to the infinite temperature partition function of non-Abelian ferromagnets, and the results should be relevant to the thermodynamic limit of such ferromagnets at high temperatures.

Phase transitions in the decomposition of ( ) Representations

We provide the first supersymmetric embedding of an integrable λ-deformation to type-II supergravity. Specifically, that of the near horizon of the NS1-NS5 brane intersection, geometrically corresponding to AdS 3 × S 3 × T 4. We show that the deformed background preserves 1/4 of the maximal supersymmetry. In the Penrose limit we show that it preserves no-more than one half of the maximal supersymmetry.

Germany

Georgios Itsios

Konstantinos Siampos

Supersymmetric backgrounds from λ-deformations

We study the thermodynamics of a non-abelian ferromagnet consisting of “atoms” each carrying a fundamental representation of <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math>, coupled with long-range two-body quadratic interactions. We uncover a rich structure of phase transitions from non-magnetized, global <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math>-invariant states to magnetized ones breaking global invariance to <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</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><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</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.

Ferromagnetic phase transitions in ( )

We compute the multiplicity of the irreducible representations in the decomposition of the tensor product of an arbitrary number n of fundamental representations of <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math>, and we identify a duality in the representation content of this decomposition. Our method utilizes the mapping of the representations of <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> to the states of free fermions on the circle, and can be viewed as a random walk on a multidimensional lattice. We also derive the large-n limit and the response of the system to an external non-abelian magnetic field. These results can be used to study the phase properties of non-abelian ferromagnets and to take various scaling limits.

Composing arbitrarily many ( ) Fundamentals

The simplest example of the λ-deformation connects the <math><mrow><mi>SU</mi><mi>(</mi><mn>2</mn><mi>)</mi></mrow></math> Wess-Zumino-Witten model with the non-Abelian T-dual (NATD) of the <math><mrow><mi>SU</mi><mi>(</mi><mn>2</mn><mi>)</mi></mrow></math> principal chiral model. We analyze spinning strings with one spin propagating through the truncation of an S-dual embedding of the λ-deformation of the target space into type IIB supergravity. We show that the situation apart from the NATD limit parallels the undeformed case. We demonstrate that regular spinning strings are either folded or circular, and that nearly degenerate spinning strings are either nearly point-like, fast, or slow. The effects of the λ-deformation are both the overall increment of the energy of spinning strings and the enlargement of the gap between the energies of folded and circular strings. In the NATD limit, we prove that circular strings disappear and that fast strings realize the dispersion relation of Gubser-Klebanov-Polyakov strings.