Trajectory phase transitions in non-interacting systems: all-to-all dynamics and the random energy model.

We study the fluctuations of time-additive random observables in the stochastic dynamics of a system of [Formula: see text] non-interacting Ising spins. We mainly consider the case of all-to-all dynamics where transitions are possible between any two spin configurations with uniform rates. We show that the cumulant generating function of the time-integral of a normally distributed quenched random function of configurations, i.

Existence of Replica-Symmetry Breaking in Quantum Glasses.

By controlling quantum fluctuations via the Falk-Bruch inequality we give the first rigorous argument for the existence of a spin-glass phase in the quantum Sherrington-Kirkpatrick model with a "transverse" magnetic field if the temperature and the field are sufficiently low. The argument also applies to the generalization of the model with multispin interactions, sometimes dubbed as the transverse p-spin model.

Dimerization and Néel Order in Different Quantum Spin Chains Through a Shared Loop Representation.

The ground-states of the spin- antiferromagnetic chain with a projection-based interaction and the spin-1/2 XXZ-chain  at anisotropy parameter share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planar -state Potts model at . The multifaceted relation is used here to directly relate the distinct forms of translation symmetry breaking which are manifested in the ground-states of these two models: dimerization for at all , and Néel order for at . The results presented include: (i) a translation to the above quantum spin systems of the results which were recently proven by Duminil-Copin-Li-Manolescu for a broad class of two-dimensional random-cluster models, and (ii) a short proof of the symmetry breaking in a manner similar to the recent structural proof by Ray-Spinka of the discontinuity of the phase transition for .

Extended states in a Lifshitz tail regime for random Schrödinger operators on trees.

We resolve an existing question concerning the location of the mobility edge for operators with a hopping term and a random potential on the Bethe lattice. The model has been among the earliest studied for Anderson localization, and it continues to attract attention because of analogies which have been suggested with localization issues for many particle systems. We find that extended states appear through disorder enabled resonances well beyond the energy band of the operator's hopping term.

Quantum-classical transitions in Lifshitz tails with magnetic fields.

We consider Lifshitz's model of a quantum particle subject to a repulsive Poissonian random potential and address various issues related to the influence of a constant magnetic field on the leading low-energy tail of the integrated density of states. In particular, we propose the magnetic analog of a 40-year-old landmark result of Lifshitz for short-ranged single-impurity potentials U. The Lifshitz tail is shown to change its character from purely quantum, through quantum classical, to purely classical with an increasing range of U.