Dynamical delocalization for discrete Schrödinger operators

Speaker: 

Simon Becker

Institution: 

Cambridge University

Time: 

Tuesday, March 17, 2020 - 2:00pm

We study discrete magnetic random Schrödinger operators on the square and honeycomb lattice, both with single-site potentials in weak magnetic fields under weak disorder. We show that there is, in the case of the honeycomb lattice, both strong dynamical localization and delocalization close to the conical point. We obtain similar results for the discrete random Schrödinger operator on the Z2-lattice close to the bottom and top of its spectrum. As part of this analysis, we give a rigorous derivation of the quantum hall effect for both models derived from the density of states for which we obtain an asymptotic expansion in the disorder parameter. The expansion implies (leading order in the disorder parameter) universality of the integrated density of states. Finally, we show that on the hexagonal lattice the Dirac cones occur for any rational magnetic flux. We use this observation to study the self-similarity of the Hall conductivity and transport properties of the random operator close to any rational magnetic flux.

Joint work with Rui Han.

 Irreducibility of Fermi variety for periodic Schr\"odinger operators in higher dimensions

Speaker: 

Wencai Liu

Institution: 

TAMU

Time: 

Thursday, March 12, 2020 - 2:00pm to 3:00pm

Location: 

RH 340

This is an ongoing project.  Let $H_0$ be a discrete periodic  Schr\"odinger operator on $\Z^d$:

$$H_0=-\Delta+v_0,$$

where $-\Delta$ is the discrete Laplacian and $v_0$ is periodic in the sense that it is well defined on  $\Z^d/q_1\Z\oplus q_2 \Z\oplus\cdots\oplus q_d\Z$. For $d=2$, we tentatively proved that the Fermi variety $F_{\lambda}(v_0)/\Z^2$ is irreducible except for one value of  $\lambda$. We also construct a non-constant periodic function $v_0$ such that its Fermi variety is reducible for  some $\lambda$, which disproves a conjecture by  Gieseker, Kn\"orrer and Trubowitz.

Under some assumptions of irreducibility of Fermi variety $F_{\lambda}(v_0)/\Z^d$, we show that $H=-\Delta +v_0+v$ does not have any embedded eigenvalues provided that $v$ decays  exponentially. The assumptions are conjectured to be true for any periodic function $v_0$. As an application, we show that when $d=2$, $H=-\Delta +v_0+v$ does not have any embedded eigenvalues provided that $v$ decays exponentially. 

 

On the relation between strong ballistic transport and exponential dynamical localization

Speaker: 

Ilya Kachkovskiy

Institution: 

MSU

Time: 

Thursday, March 5, 2020 - 2:00am to 3:00am

Host: 

Location: 

RH 340

Abstract: We establish strong ballistic transport for a family of discrete quasiperiodic Schrodinger operators as a consequence of exponential dynamical localization for the dual family. The latter has been, essentially, shown by Jitomirskaya and Kruger in the one-frequency setting and by Ge--You--Zhou in the multi-frequency case. In both regimes, we obtain strong convergence of $\frac{1}{T}X(T)$ to the asymptotic velocity operator $Q$, which improves recent perturbative results by Zhao and provides the strongest known form of ballistic motion. In the one-frequency setting, this approach allows to treat Diophantine frequencies non-perturbatively and also consider the weakly Liouville case.

The proof is based on the duality method. Originally, localization for the dual model allows to obtain ballistic transport in expectation. Combined with dynamical localization bounds, the improved convergence allows to replace ``in expectation’’ by ``almost surely’'.

Logarithmic lower bounds for the entanglement entropy of droplet states for the XXZ model on the ring

Speaker: 

Ruth Schulte

Institution: 

Ludwig-Maximilians-Universitat Munich

Time: 

Thursday, February 27, 2020 - 2:00pm to 3:00pm

Location: 

RH 340P

We study the free XXZ quantum spin model defined on a ring of size L and show that the bipartite entanglement entropy of eigenstates belonging to the first energy band above the vacuum ground state satisfy a logarithmically corrected area law. 

 

Speculative and hedging interaction model in oil and U.S. dollar markets—Long-term investor dynamics and phases

Speaker: 

Michael Campbell

Institution: 

Eureka (SAP)

Time: 

Thursday, January 9, 2020 - 2:00pm to 3:00pm

Location: 

RH 340 P

We develop the rational dynamics for the long-term investor among boundedly rational speculators in the Carfì–Musolino speculative and hedging model. Numerical evidence is given that indicates there are various phases determined by the degree of nonrational behavior of speculators. The dynamics are shown to be influenced by speculator “noise”. This model has two types of operators: a real economic subject (Air, a long-term trader) and one or more investment banks (Bank, short-term speculators). It also has two markets: oil spot market and U.S. dollar futures. Bank agents react to Air and equilibrate much more quickly than Air, thus we consider rational, best-local-response dynamics for Air based on averaged values of equilibrated Bank variables. The averaged Bank variables are effectively parameters for Air dynamics that depend on deviations-from-rationality (temperature) and Air investment (external field). At zero field, below a critical temperature, there is a phase transition in the speculator system which creates two equilibriums for bank variables, hence in this regime the parameters for the dynamics of the long-term investor Air can undergo a rapid change, which is exactly what happens in the study of quenched dynamics for physical systems. It is also shown that large changes in strategy by the long-term Air investor are always preceded by diverging spatial volatility of Bank speculators. The phases resemble those for unemployment in the “Mark 0” macroeconomic model.

 

Eigensystem multiscale analysis for the Anderson model via the Wegner estimate

Speaker: 

Abel Klein

Institution: 

UCI

Time: 

Thursday, October 3, 2019 - 2:00pm

Host: 

Location: 

RH 340P

We present a new approach to the  eigensystem multiscale analysis (EMSA) for the Anderson model that relies on the Wegner estimate.  The EMSA  treats all energies  of the finite volume operator in an energy interval at the same time,  simultaneously establishing localization of  all eigenfunctions with eigenvalues in the energy interval  with high probability.  It implies all the usual manifestations of localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization)   for the Anderson model. The new method removes the restrictive level spacing hypothesis used in the previous versions of the EMSA, allowing  for single site probability distributions that are  H\"older continuous of order  $\alpha \in (0,1]$. (Joint work with Alex Elgart.)

Continuity of spectra and  spectral measure of quasi-periodic Schr\"odinger operators 

Speaker: 

Xin Zhao

Institution: 

Nanjing Universuty

Time: 

Thursday, October 10, 2019 - 2:00pm to 3:00pm

Location: 

rh 340p

In this talk, we first consider quasi-periodic Schr\"odinger operators with finitely differentiable potentials. If the potential is analytic, there are numerous results. But not every result holds if one replaces the analyticity with a smoothness condition. We will give some positive results in this aspect,  generalizing some interesting results in the analytic case to the finitely smooth case. This includes the global reducibility results, generalized Chamber's formula and their applications to the study of continuity of the spectra. Finally we will give a recent result on the continuity of spectral measure of multi frequency quasi-periodic Schr\"odinger operators with small analytic quasi-periodic potentials.

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