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.H. Lesc

In this paper we study the many-body localization (MBL) transition and relate it to the eigenstate structure in the Fock space. Besides the standard entanglement and multifractal probes, we introduce the radial probability distribution of eigenstate coefficients with respect to the Hamming distance in the Fock space and relate the cumulants of this distribution to the properties of the quasilocal integrals of motion in the MBL phase. We demonstrate nonself-averaging property of the many-body fractal dimension Dq and directly relate it to the jump of Dq as well as of the localization length of the integrals of motion at the MBL transition. We provide an example of the continuous many-body transition confirming the above relation via the self-averaging of Dq in the whole range of parameters. Introducing a simple toy model, which hosts ergodic thermal bubbles, we give analytical evidences both in standard probes and in terms of newly introduced radial probability distribution that the MBL transition in the Fock space is consistent with the avalanche mechanism for delocalization, i.e., the Kosterlitz-Thouless scenario. Thus, we show that the MBL transition can been seen as a transition between ergodic states to nonergodic extended states and put the upper bound for the disorder scaling for the genuine Anderson localization transition with respect to the noninteracting case.

We study an effective Hamiltonian for the standard ν=1/3 fractional quantum Hall system in the thin cylinder regime. We give a complete description of its ground state space in terms of what we call Fragmented Matrix Product States, which are labeled by a certain family of tilings of the one-dimensional lattice. We then prove that the model has a spectral gap above the ground states for a range of coupling constants that includes physical values. As a consequence of the gap we establish the incompressibility of the fractional quantum Hall states. We also show that all the ground states labeled by a tiling have a finite correlation length, for which we give an upper bound. We demonstrate by example, however, that not all superpositions of tiling states have exponential decay of correlations.

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

We identify the ground-state of a truncated version of Haldane's pseudo-potential Hamiltonian in the thin cylinder geometry as being composed of exponentially many fragmented matrix product states. These states are constructed by lattice tilings and their properties are discussed. We also report on a proof of a spectral gap, which implies the incompressibility of the underlying fractional quantum Hall liquid at maximal filling ν = 1/3. Low-energy excitations and an extensive number of many-body scars at positive energy density, but nevertheless low complexity, are also identified using the concept of tilings.

We prove Goldschmidt's formula (Goldschmidt in Phys Rev B 47:4858-4861, 1990) for the free energy of the quantum random energy model. In particular, we verify the location of the first order and the freezing transition in the phase diagram. The proof is based on a combination of variational methods on the one hand, and bounds on the size of percolation clusters of large-deviation configurations in combination with simple spectral bounds on the hypercube's adjacency matrix on the other hand.

The ground-states of the spin-S antiferromagnetic chain HAF with a projection-based interaction and the spin-1/2 XXZ-chain HXXZ at anisotropy parameter Δ=cosh(λ) share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planar Q-state Potts model at Q−−√=2S+1=2cosh(λ). 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 HAF at all S>1/2, and Néel order for HXXZ at λ>0. 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 Q>4. Altogether, the quantum manifestation of the change between Q=4 and Q>4 is a transition from a gapless ground-state to a pair of gapped and extensively distinct ground-states.

We extend the random characteristics approach to Wigner matrices whose entries are not required to have a normal distribution. As an application, we give a simple and fully dynamical proof of the weak local semicircle law in the bulk.

We give a direct alternative proof of an area law for the entanglement entropy of the ground state of disordered oscillator systems—a result due to Nachtergaele et al (2013 J. Math. Phys. 54 042110). Instead of studying the logarithmic negativity, we invoke the explicit formula for the entanglement entropy of Gaussian states to derive the upper bound. We also contrast this area law in the disordered case with divergent lower bounds on the entanglement entropy of the ground state of one-dimensional ordered oscillator chains.