The de Almeida–Thouless Line in Hierarchical Quantum Spin Glasses

We determine explicitly and discuss in detail the effects of the joint presence of a longitudinal and a transversal (random) magnetic field on the phases of the Random Energy Model and its hierarchical generalization, the GREM. Our results extent known results both in the classical case of vanishing transversal field and in the quantum case for vanishing longitudinal field. Following Derrida and Gardner, we argue that the longitudinal field has to be implemented hierarchically also in the Quantum GREM. We show that this ensures the shrinking of the spin glass phase in the presence of the magnetic fields as is also expected for the Quantum Sherrington–Kirkpatrick model.

The quantum random energy model as a limit of p-spin interactions

We consider the free energy of a mean-field quantum spin glass described by a [Formula: see text]-spin interaction and a transversal magnetic field. Recent rigorous results for the case [Formula: see text], i.e. the quantum random energy model (QREM), are reviewed. We show that the free energy of the [Formula: see text]-spin model converges in a joint thermodynamic and [Formula: see text] limit to the free energy of the QREM.

Disorder-free spin glass transitions and jamming in exactly solvable mean-field models

We construct and analyze a family of MM-component vectorial spin systems which exhibit glass transitions and jamming within supercooled paramagnetic states without quenched disorder. Our system is defined on lattices with connectivity c=\alpha Mc=αM and becomes exactly solvable in the limit of large number of components M \to \inftyM→∞. We consider generic pp-body interactions between the vectorial Ising/continuous spins with linear/non-linear potentials. The existence of self-generated randomness is demonstrated by showing that the random energy model is recovered from a MM-component ferromagnetic pp-spin Ising model in M \to \inftyM→∞ and p \to \inftyp→∞ limit. In our systems the quenched disorder, if present, and the self-generated disorder act additively. Our theory provides a unified mean-field theoretical framework for glass transitions of rotational degree of freedoms such as orientation of molecules in glass forming liquids, color angles in continuous coloring of graphs and vector spins of geometrically frustrated magnets. The rotational glass transitions accompany various types of replica symmetry breaking. In the case of repulsive hardcore interactions in the spin space, the criticality of the jamming or SAT/UNSTAT transition becomes the same as that of hardspheres.