##### Existence of Replica-Symmetry Breaking in Quantum Glasses

H. Leschke, C. Manai, R. Ruder, S. Warzel

Physical Review Letters 127, 207204 (2021).

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

##### Rare thermal bubbles at the many-body localization transition from the Fock space point of view

G. De Tomasi, I.M. Khaymovich, F. Pollmann, S. Warzel

Physical Review B 104, 024202 (2021).

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.

##### Spectral Gaps and Incompressibility in a 𝜈 = 1/3 Fractional Quantum Hall System

B. Nachtergaele, S. Warzel, A. Young

Communications in Mathematical Physics 383, 1093–1149 (2021).

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.

##### Bounds on the bipartite entanglement entropy for oscillator systems with or without disorder

V. Beaud, J. Sieber and S. Warzel.

Journal of Physics A: Mathematical and Theoretical 52, 235202 (2019).

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.