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.

Rounding of first order transitions in low-dimensional quantum systems with quenched disorder.

We prove that the addition of an arbitrarily small random perturbation to a quantum spin system rounds a first-order phase transition in the conjugate order parameter in d < or = 2 dimensions, or for cases involving the breaking of a continuous symmetry in d < or = 4. This establishes rigorously for quantum systems the existence of the Imry-Ma phenomenon which for classical systems was proven by Aizenman and Wehr.