# Projected Green’s Function Methods Applied to Quasi-Periodic Systems and the Dry Ten Martini Problem, 1

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Projected Green’s Functions (pGf), Gxx(⍵), have long been used to describe the localization of quantum systems. More recently, pGf zeros have been used to determine physical observables of topological invariants in free-fermion systems, including topological obstructions to bulk localization and bulk-boundary correspondence. In this talk, I will discuss how these pGfs appear in transfer matrices and what their zeros can tell us about the solutions to transfer matrix equations – linking the localization and topological perspectives. Using these methods, we re-examine the almost-Matthieu operator and notice new guarantees on analytic regions of its resolvent.

# Projected Green’s Function Methods Applied to Quasi-Periodic Systems and the Dry Ten Martini Problem

## Speaker:

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## Time:

## Location:

Projected Green’s Functions (pGf), Gxx(⍵), have long been used to describe the localization of quantum systems. More recently, pGf zeros have been used to determine physical observables of topological invariants in free-fermion systems, including topological obstructions to bulk localization and bulk-boundary correspondence. In this talk, I will discuss how these pGfs appear in transfer matrices and what their zeros can tell us about the solutions to transfer matrix equations – linking the localization and topological perspectives. Using these methods, we re-examine the almost-Matthieu operator and notice new guarantees on analytic regions of its resolvent.

# An Elementary Humanomics Approach to Boundedly Rational Quadratic Models

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**We take a refreshing new look at boundedly rational quadratic models in economics using some elementary modeling of the principles put forward in the book Humanomics by Vernon L. Smith and Bart J. Wilson. A simple model is introduced built on the fundamental Humanomics principles of gratitude/resentment felt and the corresponding action responses of reward /punishment in the form of higher/lower payoff transfers. There are two timescales: one for strictly self-interested action, as in economic equilibrium, and another governed by feelings of gratitude/resentment. One of three timescale scenarios is investigated: one where gratitude /resentment changes much more slowly than economic equilibrium (“quenched model”). Another model, in which economic equilibrium occurs over a much slower time than gratitude /resentment evolution (“annealed” model) is set up, but not investigated. The quenched model with homogeneous interactions turns out to be a non-frustrated spin-glass model. For this particular model, the Nash equilibrium has no predictive power of Humanomics properties since the rewards are the same for self-interested behavior, resentful behavior, and gratitude behavior. Accordingly, we see that the boundedly rational Gibbs equilibrium does indeed lead to richer properties.**

# Local Noether theorem for quantum lattice systems and topological invariants of gapped states

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We study generalizations of the Berry phase for quantum lattice systems in arbitrary dimensions. For a smooth family of gapped ground states in d dimensions, we define a closed (d+2)-form on the parameter space which generalizes the curvature of the Berry connection. Its cohomology class is a topological invariant of the family. When the family is equivariant under the action of a compact Lie group G, topological invariants take values in the equivariant cohomology of the parameter space. These invariants unify and generalize the Hall conductance and the Thouless pump. A key role in these constructions is played by a certain differential graded Frechet-Lie algebra attached to any quantum lattice system. As a by-product, we describe ambiguities in charge densities and conserved currents for arbitrary lattice systems with rapidly decaying interactions.

# Approximating the ground state eigenvalue via the effective potential

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we study 1-d random Schrödinger operators on a finite interval with Dirichlet boundary conditions. We are interested in the approximation of the ground state energy using the minimum of the effective potential. For the 1-d continuous Anderson Bernoulli model, we show that the ratio of the ground state energy and the minimum of the effective potential approaches

*π*^2/8

# Spectral and Dynamical contrast on highly correlated Anderson-type models

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# Massive particle interferometry with lattice solitons: robustness against ionization

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We revisit the proposal of Castin and Weiss [Phys. Rev. Lett. vol. 102, 010403 (2009)] for using the scattering of a quantum matter-wave soliton on a barrier in order to create a coherent superposition state of the soliton being entirely to the left of the barrier and being entirely to the right of the barrier. In that proposal, is was assumed that the scattering is perfectly elastic, i.e. that the center-of-mass kinetic energy of the soliton is lower than the chemical potential of the soliton. Here we relax this assumption. Also, we introduce an interferometric scheme, which uses interference of soltions, that can be used to detect the degree of coherence between the reflected and transmitted part of the soliton. Using exact diagonalization, we numerically simulate a complete interferometric cycle for a soliton consisting of six atoms. We find that the interferometric fringes persist even when the center-of-mass kinetic energy of the soliton is above the energy needed for complete dissociation of the soliton into constituent atoms.

# Spectral and Dynamical contrast on highly correlated Anderson-type models

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We present simple, physically motivated, examples where small geometric changes on a two-dimensional graph , combined with high disorder, have a significant impact on the spectral and dynamical properties of the random Schr\"odinger operator obtained by adding a random potential to the graph's adjacency operator. Differently from the standard Anderson model, the random potential will be constant along any vertical line, hence the models exhibit long range correlations. Moreover, one of the models presented here is a natural example where the transient and recurrent components of the absolutely continuous spectrum, introduced by Avron and Simon, coexist and allow us to capture a sharp phase transition present in the model. Joint work with Matos and Schenker

# Oscillations in the nucleation preexponential

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Oscillations in the preexponential of the nucleation rate are due to the discrete nature of small nuclei. An accurate elementary expression to describe such oscillations is derived in the limit of a high nucleation barrier. The result is applied to the standard Becker-Döring equation in two and three dimensions, and to the lowest-energy nucleation path in a cold lattice gas with Glauber and Metropolis dynamics (equivalent to an Ising model on a square lattice) where oscillation effects can be more pronounced.