arXiv subjects

Simone Warzel

Publications and source records attributed to Simone Warzel.

At least 55 records · Page 3Linked to original sources

arXiv2018

Kac-Ward formula and its extension to order-disorder correlators through a graph zeta function

A streamlined derivation of the Kac-Ward formula for the planar Ising model's partition function is presented and applied in relating the kernel of the Kac-Ward matrices' inverse with the correlation functions of the Ising model's order-disorder correlation functions. A shortcut for both is facilitated by the Bowen-Lanford graph zeta function relation. The Kac-Ward relation is also extended here to produce a family of non planar interactions on $\mathbb{Z}^2$ for which the partition function and the order-disorder correlators are solvable at special values of the coupling parameters/temperature.

math-ph↗

arXiv2018

The Phase Transition in the Ultrametric Ensemble and Local Stability of Dyson Brownian Motion

We study the ultrametric random matrix ensemble, whose independent entries have variances decaying exponentially in the metric induced by the tree topology on $\mathbb{N}$, and map out the entire localization regime in terms of eigenfunction localization and Poisson statistics. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model. In the simpler case of the Rosenzweig-Porter model, the analysis yields a complete characterization of the transition in the local statistics. The proofs are based on the flow of the resolvents of matrices with a random diagonal component under Dyson Brownian motion, for which we establish submicroscopic stability results for short times. These results go beyond norm-based continuity arguments for Dyson Brownian motion and complement the existing analysis after the local equilibration time.

math.PR↗

arXiv2018

Non-Ergodic Delocalization in the Rosenzweig-Porter Model

We consider the Rosenzweig-Porter model $H = V + \sqrt{T}\, Φ$, where $V$ is a $N \times N$ diagonal matrix, $Φ$ is drawn from the $N \times N$ Gaussian Orthogonal Ensemble, and $N^{-1} \ll T \ll 1$. We prove that the eigenfunctions of $H$ are typically supported in a set of approximately $NT$ sites, thereby confirming the existence of a previously conjectured non-ergodic delocalized phase. Our proof is based on martingale estimates along the characteristic curves of the stochastic advection equation satisfied by the local resolvent of the Brownian motion representation of $H$.

math-ph↗

arXiv2018

Emergent Planarity in two-dimensional Ising Models with finite-range Interactions

The known Pfaffian structure of the boundary spin correlations, and more generally order-disorder correlation functions, is given a new explanation through simple topological considerations within the model's random current representation. This perspective is then employed in the proof that the Pfaffian structure of boundary correlations emerges asymptotically at criticality in Ising models on $\mathbb Z^2$ with finite-range interactions. The analysis is enabled by new results on the stochastic geometry of the corresponding random currents. The proven statement establishes an aspect of universality, seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases.

math-ph↗

arXiv2018

The Localization Transition in the Ultrametric Ensemble

We study the hierarchical analogue of power-law random band matrices, a symmetric ensemble of random matrices with independent entries whose variances decay exponentially in the metric induced by the tree topology on $\mathbb{N}$. We map out the entirety of the localization regime by proving the localization of eigenfunctions and Poisson statistics of the suitably scaled eigenvalues. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model.

math-ph↗

arXiv2018

Bounds on the entanglement entropy of droplet states in the XXZ spin chain

We consider a class of one-dimensional quantum spin systems on the finite lattice $Λ\subset\mathbb{Z}$, related to the XXZ spin chain in its Ising phase. It includes in particular the so-called droplet Hamiltonian. The entanglement entropy of energetically low-lying states over a bipartition $Λ= B \cup B^c$ is investigated and proven to satisfy a logarithmic bound in terms of $\min\lbrace n,\vert B\vert,\vert B^c\vert\rbrace$, where $n$ denotes the maximal number of down spins in the considered state. Upon addition of any (positive) random potential the bound becomes uniformly constant on average, thereby establishing an area law. The proof is based on spectral methods: a deterministic bound on the local (many-body integrated) density of states is derived from an energetically motivated Combes--Thomas estimate.

math-ph↗

arXiv2017

Edge switching transformations of quantum graphs

Discussed here are the effects of basics graph transformations on the spectra of associated quantum graphs. In particular it is shown that under an edge switch the spectrum of the transformed Schrödinger operator is interlaced with that of the original one. By implication, under edge swap the spectra before and after the transformation, denoted by $\{ E_n\}_{n=1}^{\infty}$ and $\{\widetilde E_n\}_{n=1}^{\infty}$ correspondingly, are level-2 interlaced, so that $E_{n-2}\le \widetilde E_n\le E_{n+2}$. The proofs are guided by considerations of the quantum graphs' discrete analogs.

math-ph↗

arXiv2017

Low-energy Fock-space localization for attractive hard-core particles in disorder

We study a one-dimensional quantum system with an arbitrary number of hard-core particles on the lattice, which are subject to a deterministic attractive interaction as well as a random potential. Our choice of interaction is suggested by the spectral analysis of the XXZ quantum spin chain. The main result concerns a version of high-disorder Fock-space localization expressed here in the configuration space of hard-core particles. The proof relies on an energetically motivated Combes-Thomas estimate and an effective one-particle analysis. As an application, we show the exponential decay of the two-point function in the infinite system uniformly in the particle number.

math-ph↗

arXiv2016

Renormalization Group Analysis of the Hierarchical Anderson Model

We apply Feshbach-Krein-Schur renormalization techniques in the hierarchical Anderson model to establish a criterion on the single-site distribution which ensures exponential dynamical localization as well as positive inverse participation ratios and Poisson statistics of eigenvalues. Our criterion applies to all cases of exponentially decaying hierarchical hopping strengths and holds even for spectral dimension $d > 2$, which corresponds to the regime of transience of the underlying hierarchical random walk. This challenges recent numerical findings that the spectral dimension is significant as far as the Anderson transition is concerned.

math-ph↗

arXiv2016

Pfaffian Correlation Functions of Planar Dimer Covers

The Pfaffian structure of the boundary monomer correlation functions in the dimer-covering planar graph models is rederived through a combinatorial / topological argument. These functions are then extended into a larger family of order-disorder correlation functions which are shown to exhibit Pfaffian structure throughout the bulk. Key tools involve combinatorial switching symmetries which are identified through the loop-gas representation of the double dimer model, and topological implications of planarity.

math-ph↗

arXiv2016

Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas

We consider the Tonks-Girardeau gas subject to a random external potential. If the disorder is such that the underlying one-particle Hamiltonian displays localization (which is known to be generically the case), we show that there is exponential decay of correlations in the many-body eigenstates. Moreover, there is no Bose-Einstein condensation and no superfluidity, even at zero temperature.

math-ph↗

arXiv2015

Boosted Simon-Wolff Spectral Criterion and Resonant Delocalization

Discussed here are criteria for the existence of continuous components in the spectra of operators with random potential. First, the essential condition for the Simon-Wolff criterion is shown to be measurable at infinity. By implication, for the iid case and more generally potentials with the K-property the criterion is boosted by a zero-one law. The boosted criterion, combined with tunneling estimates, is then applied for sufficiency conditions for the presence of continuous spectrum for random Schrödinger operators. The general proof strategy which this yields is modeled on the resonant delocalization arguments by which continuous spectrum in the presence of disorder was previously established for random operators on tree graphs. In another application of the Simon-Wolff rank-one analysis we prove the almost sure simplicity of the pure point spectrum for operators with random potentials of conditionally continuous distribution.

math-ph↗

arXiv2015

Decay of Determinantal and Pfaffian Correlation Functionals in One-dimensional Lattices

We establish bounds on the decay of time-dependent multipoint correlation functionals of one-dimensional quasi-free fermions in terms of the decay properties of their two-point function. At a technical level, this is done with the help of bounds on certain bordered determinants and pfaffians. These bounds, which we prove, go beyond the well-known Hadamard estimates. Our main application of these results is a proof of strong (exponential) dynamical localization of spin-correlation functions in disordered $XY$-spin chains.

math-ph↗

arXiv2015

Exponential vanishing of the ground-state gap of the QREM via adiabatic quantum computing

In this note we use ideas of Farhi, Goldstone, Gosset, Gutmann, Nagaj and Shor who link a lower bound on the run time of their quantum adiabatic search algorithm to an upper bound on the energy gap above the ground-state of the generators of this algorithm. We apply these ideas to the quantum random energy model (QREM). Our main result is a simple proof of the conjectured exponential vanishing of the energy gap of the QREM.

math-ph↗

arXiv2015

Multiparticle localization for disordered systems on continuous space via the fractional moment method

We investigate spectral and dynamical localization of a quantum system of $ n $ particles on $ \mathbb{R}^d $ which are subject to a random potential and interact through a pair potential which may have infinite range. We establish two conditions which ensure spectral and dynamical localization near the bottom of the spectrum of the $ n $-particle system: i)localization is established in the regime of weak interactions supposing one-particle localization, and ii)localization is also established under a Lifshitz-tail type condition on the sparsity of the spectrum. In case of polynomially decaying interactions, we provide an upper bound on the number of particles up to which these conditions apply.

math-ph↗

arXiv2014

Resonances and Partial Delocalization on the Complete Graph

Random operators may acquire extended states formed from a multitude of mutually resonating local quasi-modes. This mechanics is explored here in the context of the random Schrödinger operator on the complete graph. The operators exhibits local quasi modes mixed through a single channel. While most of its spectrum consists of localized eigenfunctions, under appropriate conditions it includes also bands of states which are delocalized in the $\ell^1$-though not in $\ell^2$-sense, where the eigenvalues have the statistics of Šeba spectra. The analysis proceeds through some general observations on the scaling limits of random functions in the Herglotz-Pick class. The results are in agreement with a heuristic condition for the emergence of resonant delocalization, which is stated in terms of the tunneling amplitude among quasi-modes.

math-ph↗

arXiv2014

On the ubiquity of the Cauchy distribution in spectral problems

We consider the distribution of the values at real points of random functions which belong to the Herglotz-Pick (HP) class of analytic mappings of the upper half plane into itself. It is shown that under mild stationarity assumptions the individual values of HP functions with singular spectra have a Cauchy type distribution. The statement applies to the diagonal matrix elements of random operators, and holds regardless of the presence or not of level repulsion, i.e. applies to both random matrix and Poisson-type spectra.

math-ph↗

arXiv2014

An invitation to trees of finite cone type: random and deterministic operators

Trees of finite cone type have appeared in various contexts. In particular, they come up as simplified models of regular tessellations of the hyperbolic plane. The spectral theory of the associated Laplacians can thus be seen as induced by geometry. Here we give an introduction focusing on background and then turn to recent results for (random) perturbations of trees of finite cone type and their spectral theory.

math.SP↗