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Name | Title | Time | ⇈ |
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Name | Title | Time | ⇈ |

Plenary | |||

Plenary --- Saturday October 8th --- CULC 152 --- Chair: Robert Seiringer | |||

Michael Aizenman | Emergent Pfaffian Relations in Quasi-Planar Models | Sat 8th 8:30am | |

Abstract: | |||

Alessandro Giuliani | Universality of transport coefficients in the Haldane-Hubbard model. | Sat 8th 11:00am | |

Abstract: In this talk I will review some selected aspects of the theory of interacting electrons on the honeycomb lattice, with special emphasis on the mathematics of the Haldane-Hubbard model: this is a model for interacting electrons on the hexagonal lattice, in the presence of nearest and next-to-nearest neighbor hopping, as well as of a transverse dipolar magnetic field. I will discuss the key properties of its phase diagram, most notably the phase transition from a standard insulating phase to a Chern insulator, across a critical line, where the system exhibits semi-metallic behavior. I will also review the universality of its transport coefficients, including the quantization of the transverse conductivity within the gapped phases, and that of the longitudinal conductivity on the critical line. The methods of proof combine constructive Renormalization Group methods with the use of Ward Identities and the Schwinger-Dyson equation. Based on joint works with Vieri Mastropietro, Marcello Porta, Ian Jauslin. | |||

Plenary --- Sunday October 9th --- CULC 152 --- Chair: Jan Philip Solovej | |||

Yoshiko Ogata | A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization. | Sun 9th 8:30am | |

Abstract: Recently, the classification problem of gapped Hamiltonians attracts a lot of attentions. We consider this problem for a class of Hamiltonians on quantum spin chains. This class is characterized by five qualitative properties. On the other hand, the Hamiltonians are MPS(Matrix product state)-Hamiltonian with some structure. This structure enable us to classify them. | |||

Michael Weinstein | Honeycomb Schroedinger Operators in the Strong Binding Regime | Sun 9th 11:00am | |

Abstract: We discuss the Schroedinger operator for a large class of periodic potentials with the symmetry of a hexagonal tiling of the plane. The potentials we consider are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure, corresponding to the single electron model of graphene and its artificial analogues.We consider the regime of strong binding, where the depth of the potential wells is large. Our main result is that for sufficiently deep potentials, the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, converge uniformly to those of the two-band tight-binding model, introduced by PR Wallace (1947) in his pioneering study of graphite. We then discuss corollaries, in the strong binding regime, on (a) the existence of spectral gaps for honeycomb potentials with PT symmetry-breaking perturbations, and (b) the existence of topologically protected edge states for honeycomb structures with "rational edges". This is joint work with CL Fefferman and JP Lee-Thorp. | |||

Plenary --- Monday October 10th --- CULC 152 --- Chair: Abel Klein | |||

Svetlana Jitomirskaya | Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions and universal hierarchical structure of eigenfunction | Mon 10th 8:30am | |

Abstract: We will review recent results on sharp arithmetic spectral transitions in some popular models: Harper's, extended Harper's, Maryland, as well as the general class of analytic potentials (papers joint with A. Avila, R. Han, H. Kruger, W. Liu, C. Marx, F. Yang, S. Zhang, and Q. Zhou) and then focus on a recently discovered universal hierarchical structure in the behavior of quasiperiodic eigenfunctions (joint work with W. Liu). The structure is governed by the continued fraction expansion of the frequency and explains some predictions in physics literature. | |||

Maciej Zworski | Microlocal methods in dynamical systems | Mon 10th 11:00am | |

Abstract: Microlocal analysis exploits mathematical manifestations of the classical/quantum (particle/wave) correspondence and has been a successful tool in spectral theory and partial differential equations. We can say that these last two fields lie on the quantum/wave side.Recently, microlocal methods have been applied to the study of classical dynamical problems, in particular of chaotic (Anosov) flows. I will illustrate this by proving that the order of vanishing of the dynamical zeta function at zero for negatively curved surfaces is given by the absolute value of the Euler characteristic (joint work with S Dyatlov). | |||

Plenary --- Tuesday October 11th --- CULC 152 --- Chair: Pavel Exner | |||

Peter Kuchment | Analytic properties of dispersion relations and spectra of periodic operators | Tue 11th 8:30am | |

Abstract: The talk will survey some known results and unresolved problems concerning analytic properties of dispersion relations and their role in various spectral theory problems for periodic operators of mathematical physics, such as spectral structure, embedded impurity eigenvalues, Greens function asymptotics, Liouville theorems, etc. | |||

Fernando Brandao | Quantum Approximate Markov Chains and the Locality of Entanglement Spectrum | Tue 11th 11:00am | |

Abstract: In this talk I will show that quantum many-body states satisfying an area law for entanglement have a local entanglement spectrum, i.e. the entanglement spectrum can be approximated by the spectrum of a local model acting on the boundary of the region. The result follows from a version of the Hammersley-Clifford Theorem (which states that classical Gibbs states are equivalent to Markov networks) for quantum approximate Markov chains. In particular I'll argue that those are in one-to-one correspondence to 1D quantum Gibbs states. | |||

Short | |||

Short --- Saturday October 8th --- CULC 152 --- Chair: Robert Seiringer | |||

Nelson Javier Buitrago Aza | Large Deviation Principles for Weakly Interacting Fermions | Sat 8th 10:00am | |

Abstract: We show that the Gärtner–Ellis scaled cumulant generating function of fluctuation measures associated to KMS states of weakly interacting fermions on the lattice can be written as the limit of a sequence of logarithms of Gaussian Grassmann–Berezin integrals. Moreover, the covariances of the Gaussian integrals have a uniform determinant bound. As a consequence, the Grassmann integral representation may be used to obtain convergent expansions of the generating function in terms of powers of its parameter. The derivation and analysis of these expansions are studied via Brydges–Kennedy tree expansions. The proof of uniformity of the determinant bound given here uses Hölder inequalities for Schatten norms as a key argument. | |||

Søren Fournais | The semi-classical limit of large fermionic systems | Sat 8th 10:15am | |

Abstract: We study a system of $N$ fermions in the regime where the intensity of the interaction scales as $1/N$ and with an effective semi-classical parameter $\hbar=N^{-1/d}$ where $d$ is the space dimension. For a large class of interaction potentials and of external electromagnetic fields, we prove the convergence to the Thomas-Fermi minimizers in the limit $N\to\infty$. The limit is expressed using many-particle coherent states and Wigner functions. The method of proof is based on a fermionic de Finetti-Hewitt-Savage theorem in phase space and on a careful analysis of the possible lack of compactness at infinity.This is bases on joint work with Mathieu Lewin and Jan Philip Solovej | |||

David Müller | Lieb-Thirring and Cwickel-Lieb-Rozenblum inequalities for perturbed graphene with a Coulomb impurity | Sat 8th 10:30am | |

Abstract: We study the two dimensional massless Coulomb-Dirac operator restricted to its positive spectral subspace and prove estimates on the negative eigenvalues created by electromagnetic perturbations. | |||

Takuya Mine | Spectral shift function for the magnetic Schroedinger operators | Sat 8th 10:45am | |

Abstract: The spectral shift function (SSF) for the Schroedinger operator is usually defined only when the scalar potential decays sufficiently fast. In the case of the magnetic Schroedinger operator in the Euclidean plane, the vector potential has long-range decay if the total magnetic flux is non-zero, and then SSF cannot be defined in the ordinary sense. In this talk, we show that SSF for the magnetic Schroedinger operator in the Euclidean plane can be defined in some weak sense, even if the total magnetic flux is non-zero. In particular, we give some explicit formula for the SSF for the Aharonov-Bohm magnetic field. | |||

Short --- Monday October 10th --- CULC 152 --- Chair: Abel Klein | |||

Matthew Cha | The complete set of infinite volume ground states for Kitaev's abelian quantum double models | Mon 10th 10:00am | |

Abstract: We study the set of infinite volume ground states of Kitaev's quantum double model on $\mathbb{Z}^2$ for an arbitrary finite abelian group $G$. In the finite volume, the ground state space is frustration-free and the low-lying excitations correspond to abelian anyons. The ribbon operators act on the ground state space to create pairs of single excitations at their endpoints. It is known that in the infinite volume these models have a unique frustration-free ground state. We show that the complete set of ground states decomposes into $|G|^2$ different charged sectors, corresponding to the different types of abelian anyons (or superselection sectors). In particular, all pure ground states are equivalent to the single excitation states. Our proof proceeds by showing that each ground state can be obtained as the weak$*$-limit of the finite volume ground states of the quantum double model with a suitable boundary term. The boundary terms allow for states which represent an excitation pair with one excitation in the bulk and one pinned to the boundary to be included in the ground state space. This is joint work with P. Naaijkens and B. Nachtergaele. | |||

Christoph Fischbacher | The proper dissipative extensions of a dual pair | Mon 10th 10:15am | |

Abstract: Let A and (−B) be dissipative operators on a Hilbert space H and let (A,B) form a dual pair, i.e. A⊂B*, resp. B⊂A*. We present a method of determining the proper dissipative extensions A' of this dual pair, i.e. A⊂A'⊂B* provided that D(A)∩D(B) is dense in H. Applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed. | |||

Eugene Dumitrescu | Discrimination of correlated and entangling quantum channels with selective process tomography | Mon 10th 10:30am | |

Abstract: The accurate and reliable characterization of quantum dynamical processes underlies efforts to validate quantum technologies, where discrimination between competing models of observed behaviors inform efforts to fabricate and operate qubit devices. We present a novel protocol for quantum channel discrimination that leverages advances in direct characterization of quantum dynamics (DCQD) codes. We demonstrate that DCQD codes enable selective process tomography to improve discrimination between entangling and correlated quantum dynamics. Numerical simulations show selective process tomography requires only a few measurement configurations to achieve a low false alarm rate and that the DCQD encoding improves the resilience of the protocol to hidden sources of noise. Our results show that selective process tomography with DCQD codes is useful for efficiently distinguishing sources of correlated crosstalk from uncorrelated noise in current and future experimental platforms. | |||

Yese J. Felipe | Quantum Music: Applying Quantum Theory to Music Theory and Composition | Mon 10th 10:45am | |

Abstract: Classical and popular music is written so that the melody, harmony, rhythm is independent of the listener and the instance when it is played, providing the same experience to all listeners. By applying concepts from Quantum Theory to Music Theory, a linear combination of melodies and harmonies can be composed, establishing a unique experience for different listeners. The application of some concepts of Quantum Theory and their effects on the outcome of a quantum musical composition will be discussed. | |||

Short --- Tuesday October 11th --- CULC 152 --- Chair: Pavel Exner | |||

Thomas Norman Dam | The Spin-Boson model in the strong interaction limit | Tue 11th 10:00am | |

Abstract: The Spin-Boson model is a model from QFT which describes a two level system coupled to a scalar field. In this talk, I will present new results about the strong interaction limit of the massive Spin-Boson model. As the interaction approaches infinity, one can under very general assumptions describe the asymptotics of the resolvent (in a suitable sense), the ground state energy and the ground state eigenvector.One application of these results is to show the existence of a non degenerate exited state in the strong interaction limit and prove that the energy of the excited state converges to the energy of the ground state. If time allows, I will also talk about the strategy to prove the above mentioned results. This is joint work with Jacob Schach Møller. | |||

Atsuhide Ishida | Non-existence of the wave operators for the repulsive Hamiltonians | Tue 11th 10:15am | |

Abstract: We consider the quantum systems described by the Schroedinger equation equipped with so-called repulsive part. In this quantum system, we can see the characteristic property in which the free dynamics of the particle disperse in an exponential order in time. I will report in this talk that we can find a counter example of the slow decaying interaction potential such that the wave operators do not exist and we come to conclusion of the borderline between the short-range and long-range. | |||

Shanshan Li | Continuous Time Quantum Walks in finite Dimensions | Tue 11th 10:30am | |

Abstract: We consider the quantum search problem with a continuous time quantum walk for networks of finite spectral dimension of the network Laplacian. For general networks of fractal (integer or non-integer) dimension, for which in general the fractal dimension is not equal to the spectral dimension, it suggests that the spectral dimension is the scaling exponent that determines the computational complexity of the search. Our results are consistent with those of Childs and Goldstone [Phys. Rev. A 70 (2004), 022314] for lattices of integer dimension. For general fractals, we find that the Grover limit of quantum search can be obtained whenever the spectral dimension is larger than four. This complements the recent discussion of mean-field networks by Chakraborty et al. [Phys. Rev. Lett. 116 (2016), 100501] showing that for all those networks spatial search by quantum walk is optimal. | |||

Poster | |||

Poster --- Sunday October 9th --- CULC --- Chair: | |||

Kimmy Cushman | Lie Algebras in Quantum Field Theories | Sun 9th 10:00am | |

Abstract: We discuss the progress made in the study of Lie Algebras through the lens of quantum field theory. We compare the physical and mathematical interpretations of Clifford Algebras. We plan to investigate the effects of choice of bases and new bases representations for this algebra on Dirac Spinor theories. | |||

Shingo Kukita | non-Markovian dynamics from singular perturbation method | Sun 9th 10:00am | |

Abstract: We derived a complete positive map representing non-Markovian dynamics for a finite dimensional system by using a singular perturbation method. A mixing property of the environment coupled with the target system plays an important role. In this presentation, we will explain our derivation of the complete positive map and compere its dynamics with that of other non-Markovian master equations. | |||

Josiah Park | Asymptotics for Steklov Eigenvalues on Non-Smooth Domains | Sun 9th 10:00am | |

Abstract: We study eigenfunctions and eigenvalues of the Dirichlet-to-Neumann operator on boxes in Euclidean spaces. We consider bounds on the counting function for the Steklov spectrum on such domains. | |||

Diane Pelejo | Maximum Fidelity under Mixed Unitary or Unital Quantum Channels | Sun 9th 10:00am | |

Abstract: Let $\rho_1$ and $\rho_2$ be fixed quantum states. We describe a simple algorithm to determine the maximum value for the fidelity $F(\rho_1,\Phi(\rho))$ between $\rho_1$ and an image $\Phi(\rho_2)$ of $\rho_2$ under any mixed unitary channel $\Phi$ or under any unital channel $\Phi$. | |||

Itaru Sasaki | Embedded Eigenvalues and Neumann-Wigner Potentials for Relativistic Schrodinger Operators | Sun 9th 10:00am | |

Abstract: We construct Neumann-Wigner type potentials for the massive relativistic Schrodinger operator in one and three dimensions for which a strictly positive eigenvalue embedded in the continuous spectrum exists. We show that in the non-relativistic limit these potentials converge to the classical Neumann-Wigner potentials. Thus, the potentials constructed this talk can be considered as a relativistic generalization of the Neumann-Wigner potentials. | |||

Cem Yuce | Self-accelerating Parabolic Cylinder Waves in 1-D | Sun 9th 10:00am | |

Abstract: We introduce a new self-accelerating wave packet solution of the Schrodinger equation in one dimension. We obtain an exact analytical parabolic cylinder wave for the inverted harmonic potential. We show that truncated parabolic cylinder waves exhibits their accelerating feature. | |||

Quantum mechanics on graphs and similar structures | |||

Graphs --- Saturday October 8th --- Skiles 006 --- Chair: Delio Mugnolo | |||

Pavel Exner | Singular Schrödinger operators with interactions supported by sets of codimension one | Sat 8th 2:00pm | |

Abstract: In this talk we discuss Schr\"odinger operators with a singular `potentials' supported by a subsets $\Gamma$ of a configuration space having codimension one. Some of them can be formally written as $-\Delta-\alpha \delta(x-\Gamma)$ with $\alpha>0$, where $\Gamma$ is a manifold in~$\mathbb{R}^d$, but we introduce also more singular interactions like $\delta'$ as well as the most general ones parametrized by a family of four function on $\Gamma$. We discuss relations between the spectra of these operators and the geometry of $\Gamma$ using, in particular, inequalities between operators corresponding to different `potentials'. | |||

Cesar de Oliveira | Approximations of Neumann nonuniformly collapsing strips | Sat 8th 2:30pm | |

Abstract: Consider the Neumann Laplacian in the region below the graph of $\varepsilon g(x)$ for smooth $g: [a,\infty) \to (0,\infty)$ and diverging $\lim_{x\to\infty}g(x)=\infty$. The effective operator as $\varepsilon \to 0$ is found to have Robin boundary conditions at $a$. Then we recover such effective operator through suitable uniformly collapsing regions as~$\varepsilon \to 0$; in such approach, we have (roughly) got norm resolvent convergence for~$g$ diverging less than exponential and strong resolvent convergence otherwise. | |||

Claudio Cacciapuoti | Existence of Ground State for the NLS on Star-like Graphs | Sat 8th 3:00pm | |

Abstract: We consider a nonlinear Schrödinger equation (NLS) on a Star-like graph (a graph composed by a compact core to which a finite number of half-lines are attached). At the vertices of the graph interactions of delta-type can be present and an overall external potential is admitted. Our goal is to show that the NLS dynamics on a star-like graph admits a ground state of prescribed mass $m$ under mild and natural hypotheses. By ground state of mass $m$ we mean a minimizer of the NLS energy functional constrained to the manifold of mass ($L^2$-norm) equal to $m$. When existing, the ground state is an orbitally stable standing wave for the NLS evolution. We prove that a ground state exists whenever the quadratic part of the energy admits a simple isolated eigenvalue at the bottom of the spectrum (the linear ground state) and $m$ is sufficiently small. This is a major generalization of a result previously obtained for a graph with a single vertex (a star graph) with a delta interaction in the vertex and without potential terms. The main tools of the proof are concentration-compactness and bifurcation techniques. This is a joint work in collaboration with Domenico Finco and Diego Noja. | |||

Zhiqin Lu | Ground State of Quantum Layers | Sat 8th 4:00pm | |

Abstract: I will give a survey of the existence of ground state of quantum layers in this talk, and I will also present some new results and discuss the relation of this spectrum problem with differential geometry. Some of the results are joint with Julie Rowlett and David Krejcirik. | |||

Hiroaki Niikuni | Schrödinger operators on a zigzag supergraphene-based carbon nanotube | Sat 8th 4:30pm | |

Abstract: In this talk, we study the spectrum of a periodic Schrödinger operator on a zigzag super carbon nanotube, which is a generalization of thezigzag carbon nanotube. We prove that its absolutely continuous spectrum has the band structure.Moreover, we show that its eigenvalues with infinite multiplicities consisting ofthe Dirichlet eigenvalues and points embedded in the spectral band for some corresponding Hill operator. We also give the asymptotics for the spectral band edges. | |||

Petr Siegl | Non-self-adjoint graphs | Sat 8th 5:00pm | |

Abstract: On finite metric graphs, we consider Laplace operators subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity transforms to self-adjoint Laplacians. Among other things, we describe a simple way to relate the similarity transforms between Laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions.The talk is based on: [1] A. Hussein, D. Krejcirik and P. Siegl: Non-self-adjoint graphs, Transactions of the AMS, 367, (2015) 2921-2957. | |||

Graphs --- Sunday October 9th --- Skiles 006 --- Chair: Peter Kuchment | |||

James Kennedy | Eigenvalue estimates for quantum graphs | Sun 9th 2:00pm | |

Abstract: A classical problem in the analysis of (partial) differential operators such as the Laplacian on domains or manifolds is to understand how their eigenvalues depend on the underlying geometry of the object on which they are defined. This dependence can take various forms, such as asymptotics or trace formulae, but we will be interested in bounds on the (low) eigenvalues of the operator. A basic example of this is the Faber--Krahn inequality, which states that the first eigenvalue of the Dirichlet Laplacian is smallest among all domains with given volume when the domain is a ball.Interest in problems of this nature on metric graphs, which in the prototype case simply concerns estimating the eigenvalues the Laplacian with Kirchhoff conditions at the vertices, seems only to have developed in the last couple of years (with a few notable exceptions, such as works of Nicaise and Friedlander). This is also at odds with the relatively well-developed parallel body of literature on the eigenvalues of discrete and normalised Laplacians. This will be a first attempt to provide a natural framework for such eigenvalue estimates in the easiest case of the spectral gap of the Kirchhoff Laplacian: which geometric and algebraic quantities of a graph, such as total length, diameter, number of edges or vertices, connectivity, Betti number etc. enable one to control the eigenvalue(s), and how? Which bounds are possible? We shall attempt to demonstrate that on the one hand such questions can be surprisingly subtle. But on the other, one can come a long way armed with little more than elementary variational principles, a workhorse of the PDE theory which becomes very powerful on graphs, but seems to have been largely overlooked by much of the graph theory community until recently. This talk is based on joint, ongoing work with Gregory Berkolaiko, Pavel Kurasov, Gabriela Malenova and Delio Mugnolo. | |||

Ram Band | Quantum graphs which optimize the spectral gap | Sun 9th 2:30pm | |

Abstract: A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of the spectral gap (the first positive Laplacian eigenvalue) on the choice of edge lengths. In particular, starting from a certain discrete graph, we seek the quantum graph for which an optimal (either maximal or minimal) spectral gap is obtained. We fully solve the minimization problem for all graphs. We develop tools for investigating the maximization problem and solve it for some families of graphs.The talk is based on a joint work with Guillaume Levy. | |||

Boris Gutkin | Spectral statistics of nearly unidirectional quantum graphs | Sun 9th 3:00pm | |

Abstract: Quantum Hamiltonian systems with unidirectional classical dynamics posses a number of intriguing spectral properties. In particular, their energy levels are quasi-degenerate and have anomalous spectral statistics. We look at the unidirectional quantum graphs as a toy model for this phenomenon. Their spectrum is doubly degenerate with the same statistics as in the Gaussian Unitary Ensembles of random matrices. However, adding a backscattering at one of the graph's bonds lifts the degeneracies. Based on a random matrix model we derive an analytic expression for the anomalous nearest neighbor distribution between energy levels. As we show the result agrees excellently with the actual statistics in most of the cases. Yet, it exhibits quite substantial deviations for classes of graphs with strong localization of eigenfunctions. The talk is based on the joint work (arXiv:1503.01342) with M. Akila. | |||

Evans Harrell | Pointwise control of eigenfunctions on quantum graphs | Sun 9th 4:00pm | |

Abstract: Pointwise bounds on eigenfunctions are useful for establishing localization of quantum states, and they have implications for the distribution of eigenvalues and for physical properties such as conductivity. In the low-energy regime, localization is associated with exponential decrease through potential barriers. We adapt the Agmon method to control this tunneling effect for quantum graphs with Sobolev and pointwise estimates. It turns out that as a generic matter, the rate of decay is controlled by an Agmon metric related to the classical Liouville-Geen approximation for the line, but more rapid decay is typical, arising from the geometry of the graph. In the high-energy regime one expects states to oscillate but to be dominated by a 'landscape function' in terms of the potential and features of the graph. We discuss the construction of useful landscape functions for quantum graphs. | |||

Gueorgui Raykov | Local Eigenvalue Asymptotics of the Perturbed Krein Laplacian | Sun 9th 4:30pm | |

Abstract: I will consider the Krein Laplacian on a regular bounded domain, perturbed by a real-valued multiplier V vanishing on the boundary. Assuming that V has a definite sign, I will discuss the asymptotics of the eigenvalue sequence which converges to the origin. In particular, I will show that the effective Hamiltonian that governs the main asymptotic term of this sequence, is the harmonic Toeplitz operator with symbol V, unitarily equivalent to a pseudodifferential operator on the boundary. This is a joint work with Vincent Bruneau (Bordeaux, France). The partial support of the Chilean Science Foundation Fondecyt under Grant 1130591 is gratefully acknowledged. | |||

Kenichi Ito | Branching form of the resolvent at threshold for discrete Laplacians | Sun 9th 5:00pm | |

Abstract: We compute an explicit expression of the resolvent around the threshold zero for an ultra-hyperbolic operator of signature $(p,q)$, which includes the Laplacian as a special case. In particular, we classify a branching form of the resolvent; The resolvent has a square-root singularity if $(p,q)$ is odd-even or even-odd, a logarithm singularity if $(p,q)$ is even-even, and a dilogarithm singularity if $(p,q)$ is odd-odd. We apply the same computation scheme to the discrete Laplacian around thresholds embedded in continuous spectrum as well as those at end points, and obtain similar results, presenting a practical procedure to expand the resolvent around these thresholds. This talk is based on a recent joint work with Arne Jensen (Aalborg University). | |||

Graphs --- Monday October 10th --- Skiles 006 --- Chair: Maciej Zworski | |||

Françoise Truc | Topological Resonances on Quantum Graphs | Mon 10th 2:00pm | |

Abstract: In this paper, we try to put the results of Smilansky and al. on "Topological resonances" on a mathematical basis.A key role in the asymptotic of resonances near the real axis for Quantum Graphs is played by the set of metrics for which there exists compactly supported eigenfunctions. We give several estimates of the dimension of this semi-algebraic set, in particular in terms of the girth of the graph. The case of trees is also discussed. | |||

Jens Bolte | Spectra of interacting particles on quantum graphs | Mon 10th 2:30pm | |

Abstract: One reason for the success of one-particle quantum graph models is that their spectra are determined by secular equations involving finite-dimensional determinants. In general, one cannot expect this to extend to interacting many-particle models. In this talk I will introduce two-particle quantum graph models with interactions that allow one to express eigenfunctions in terms of a Bethe ansatz. From this a secular equation will be determined, and eigenvalues can be calculated numerically. The talk is based on joint work with George Garforth. | |||

Jon Harrison | n-particle quantum statistics on graphs | Mon 10th 3:00pm | |

Abstract: For particles in three or more dimensions the forms of quantum statistics of indistinguishable particles are either Bose-Einstein or Fermi-Dirac corresponding to the two abelian representations of the first homology group of the configuration space. Restricting particles to the plane the fundamental group of the configuration space is the braid group and a new form of particle statistics corresponding to its abelian representations appears, anyon statistics. Restricting the dimension of the space further to a quasi-one-dimensional quantum graph opens new forms of statistics determined by the connectivity of the graph. We develop a full characterization of abelian quantum statistics on graphs which leads to an alternative proof of the structure theorem for the first homology group of the n-particle configuration space. For two connected graphs the statistics are independent of the particle number. On three connected non-planar graphs particles are either bosons or fermions while in three connected planar graphs they are anyons. Graphs with more general connectivity exhibit interesting mixtures of these behaviors which we illustrate. For example, a graph can be constructed where particles behave as bosons, fermions and anyons depending on the region of the graph that they inhabit. An advantage of this direct approach to analysis of the first homology group is that it makes the physical origin of these new forms of statistics clear. This is work with Jon Keating, Jonathan Robbins and Adam Sawicki at Bristol. | |||

Tracy Weyand | Zeta Functions of the Dirac Operator on Quantum Graphs | Mon 10th 4:00pm | |

Abstract: The spectral zeta function generalizes the Riemann zeta function by replacing the sum over integers with a sum over a spectrum. Here we consider the spectrum of the Dirac operator acting on a metric graph. Since all eigenvalues are roots of a secular equation, we can calculate the spectral zeta function by applying the argument principle to a particular contour integral. This will be done first for a rose graph, and then for general graphs with self-adjoint vertex matching conditions. We will also discuss how this function can then be used to compute the spectral determinant. | |||

Jiri Lipovsky | How to find the effective size of a non-Weyl graph | Mon 10th 4:30pm | |

Abstract: We study the asymptotics of the number of resolvent resonances in a quantum graph with attached halflines. It has been proven that in some cases the constant by the leading term of the asymptotics (the effective size of the graph) is smaller than one expects by the Weyl law since some resonances escape to infinity. We show how to find this effective size by the method of pseudo-orbit expansion. Furthermore, we prove two theorems on the effective size of certain type of graphs with standard (Kirchhoff) coupling. | |||

Vladimir Rabinovich | Essential spectrum of Schrödinger operators with no periodic potentials on periodic graphs | Mon 10th 5:00pm | |

Abstract: We consider Schrödinger operators $H$ with bounded uniformly continuous electric potentials on periodic graphs $\Gamma$ provided by the standard Kirchhoff-Neumann conditions at every vertex. Following to [1-4] we define for $H$ a family of limit operators and we show that the essential spectrum of $H$ is the union of the spectra of all limit operators.We give applications of this result to calculations of the essential spectra of Schr\"{o}dinger operators on periodic graphs with periodic electric potentials perturbed by a slowly oscillating at infinity terms. Bibliography 1: V.S.Rabinovich, S. Roch, B.Silbermann, Limit Operators and its Applications in the Operator Theory, In ser. Operator Theory: Advances and Applications, vol 150, ISBN 3-7643-7081-5, Birkhäuser Velag, 2004, 392 pp. 2: V. Rabinovich, Essential spectrum of perturbed pseudodifferential operators. Applications to the Schrödinger, Klein-Gordon, and Dirac operators, Russian Journal of Math. Physics, Vol.12, No.1, 2005, p. 62-80 3: V.S. Rabinovich, S. Roch, The essential spectrum of Schrödinger operators on lattice, Journal of Physics A, Math. Theor. 39 (2006) 8377-8394 4: V.S. Rabinovich, S. Roch, Essential spectra of difference operators on $\mathbb{Z}^n$-periodic graphs, J. of Physics A: Math. Theor. ISSN 1751-8113, 40 (2007) 10109-10128 | |||

New mathematical topics arising in current theoretical physics | |||

New topics --- Saturday October 8th --- Skiles 202 --- Chair: Robert Seiringer | |||

Martin Fraas | Perturbation Theory of Non-Demolition Measurements | Sat 8th 2:00pm | |

Abstract: In a non-demolition measurement an observable on a quantum system is measured through a direct measurement on a sequence of probes subsequently interacting with the system. Recent interest in developing a theory of this process originates in the photon counting experiments of Haroche. Mathematically the problem is equivalent to the study of statistics of long products of completely positive maps -- which all commute in the non-demolition case. I will describe a mathematical theory of non-demolition measurements for observables with arbitrary spectra, and a theory describing the process for small Hamiltonian perturbations of the non-demolition case. The talk is based on joint works with M. Ballesteros, N. Crawford, J. Fröhlich and B. Schubnel. | |||

Emil Prodan | A geometric identity for index theory | Sat 8th 2:30pm | |

Abstract: The index theorem for the Hall conductivity in 2-dimensions given by Bellissard at al [J. Math. Phys. 1994] relies on a remarkable geometric identity discovered by Alain Connes just a few years before. Relatively recently, this geometric identity was extended to higher dimensions, enabling index theorems for certain non-linear transport coefficients. This in turn confirmed the stability agains strong disorder of various invariants for topological insulators. In this talk I will describe the geometrical principles behind these generalizations. | |||

Rainer Dick | Dressing up for length gauge: Mathematical aspects of a debate in quantum optics | Sat 8th 3:00pm | |

Abstract: A debate about the correct form of the interaction Hamiltonian in quantum optics has been going on since Lamb’s investigation of optical line shapes in 1952. Surprisingly, the debate has never been settled, but rather intensified in recent years with the observation of phenomena on atomic time scales in attosecond spectroscopy. In short, the debate concerns the description of matter-photon interactions through vector potentials (“velocity gauge”) or electric fields (“length gauge”) in the Schrödinger equation. Observational evidence is inconclusive, since the observationally preferred interaction terms depend on observed systems and parameters. Indeed, more experimental observations seem to favor the length gauge, which is surprising from a fundamental theory perspective. I will review the problem both from a theoretical and an experimental perspective, and then point out that the underlying transformation between velocity gauge and length gauge is actually an incomplete gauge transformation which should rather be addressed as a basic dressing operation for the Schrödinger field. This observation and a study of the coupled Schrödinger-Maxwell system will help us to understand why predictions in velocity gauge and length gauge differ, and why length gauge may be preferred in quantum optical systems. | |||

Vit Jakubsky | On dispersion of wave packets in Dirac materials | Sat 8th 4:00pm | |

Abstract: We show that a wide class of quantum systems with translational invariance can host dispersionless, soliton-like, wave packets. We focus on the settings where the effective, two-dimensional Hamiltonian acquires the form of Dirac operator. The proposed framework for construction of the dispersionless wave packets is illustrated on systems with topologically nontrivial effective mass. Our analytical predictions are accompanied by a numerical analysis and possible experimental realizations are discussed. | |||

Brian Swingle | Tensor networks, entanglement, and geometry | Sat 8th 4:30pm | |

Abstract: Tensor networks are entanglement-based tools which are useful for representing quantum many-body states, especially thermal states of local Hamiltonians. I will discuss some recent results constructing tensor networks for a wide variety of states of quantum matter. I will also briefly describe recent conjectures relating tensor networks and entanglement to the emergence of quantum gravity via the AdS/CFT correspondence. Based in part on 1607.05753, 1602.02805, and 1407.8203 with John McGreevy and Shenglong Xu. | |||

Volkher Scholz | Matrix product approximations to multipoint functions in two-dimensional conformal field theory | Sat 8th 5:00pm | |

Abstract: Matrix product states (MPS) illustrate the suitability of tensor networks for the description of interacting many-body systems: ground states of gapped 1-D systems are approximable by MPS as shown by Hastings [J. Stat. Mech. Theor. Exp., P08024 (2007)]. In contrast, whether MPS and more general tensor networks can accurately reproduce correlations in critical quantum systems, respectively quantum field theories, has not been established rigorously. Ample evidence exists: entropic considerations provide restrictions on the form of suitable Ansatz states, and numerical studies show that certain tensor networks can indeed approximate the associated correlation functions. Here we provide a complete positive answer to this question in the case of MPS and 2D conformal field theory: we give quantitative estimates for the approximation error when approximating correlation functions by MPS. Our work is constructive and yields an explicit MPS, thus providing both suitable initial values as well as a rigorous justification of variational methods. | |||

Boris Gutkin | Quantum chaos in many-particle systems | Sat 8th 5:30pm | |

Abstract: Upon quantisation, systems with classically chaotic dynamics exhibit universal spectral and transport properties effectively described by Random Matrix Theory. Semiclassically this remarkable phenomenon can be attributed to the existence of pairs of classical orbits with small action differences. So far, however, the scope of the theory has, by and large, been restricted to single-particle systems. I will discuss an extension of this program to chaotic systems with a large number of particles. The crucial step is introducing a two-dimensional symbolic dynamics which allows an effective representation of periodic orbits in many-particle chaotic systems with local interactions. By using it we show that for a large number of particles the dominant correlation mechanism among periodic orbits essentially differs from the one of the single-particle theory. Its implications on spectral properties of many-particle quantum systems will be discussed as well. | |||

New topics --- Sunday October 9th --- Skiles 202 --- Chair: Robert Seiringer | |||

Anushya Chandran | Heating in periodically driven Floquet systems | Sun 9th 2:00pm | |

Abstract: Periodically driven quantum systems (Floquet systems) do not have a conserved energy. Thus, statistical mechanical lore holds that if they thermalize, it must be to infinite temperature. I will first show this holds in undriven systems that satisfy the eigenstate thermalization hypothesis. I will then present two counter-examples to infinite temperature heating. The first is the bosonic O(N) model at infinite N, in which the steady states are paramagnetic and have non-trivial correlations. The second is the Clifford circuit model, which can fail to heat depending on the choice of circuit elements. The resulting steady states can then be localized or delocalized but not ergodic. Such models shed light on the nature of interacting Floquet localization. | |||

Pieter Naaijkens | Operator algebras and data hiding in topologically ordered systems | Sun 9th 2:30pm | |

Abstract: The total quantum dimension is an invariant of topological phases, related to the anyonic excitations a topologically ordered state supports. In this talk I will discuss the total quantum dimension in the thermodynamic limit of topologically ordered quantum spin systems. In particular, I will discuss how the anyons can be used to hide data in the state. While not a practical way of data hiding, it sheds new light on the total quantum dimension: in particular, I will outline how deep results from operator algebra (and subfactors in particular) can be used to quantify how much information can be hidden, and how this is related to the quantum dimension. Joint work with Leander Fiedler and Tobias Osborne. | |||

Subir Sachdev | The Sachdev-Ye-Kitaev models of non-Fermi liquids and black holes | Sun 9th 3:00pm | |

Abstract: The SYK models are simple Hamiltonians of fermions with random all-to-all interactions. Their ground states largely self-average over disorder, and have a gapless excitation spectrum with no quasiparticle structure. They provide a model of non-Fermi liquids, and also, remarkably, of black holes in two-dimensional anti-de Sitter space | |||

Marco Merkli | Evolution of a two-level system strongly coupled to a thermal bath | Sun 9th 4:00pm | |

Abstract: We consider a quantum process where electric charge, or excitation energy, is exchanged between two agents, and in the presence of a thermal environment. In some chemical processes in biology (photosynthesis), the agent-reservoir interaction energy is large, at least of the same size as the agents' energy difference. We present a rigorous analysis of the effective dynamics of the agents in this coupling regime, valid for all times. In particular, we derive a generalization of the Marcus formula from quantum chemistry, predicting the reaction rate. Our generalization shows that by coupling one agent more strongly to the environment than the other one, a significant speedup of the process can be achieved. Our analytic method is based on a resonance expansion of the reduced agent dynamics, cast in the framework of the strongly coupled spin-boson system. | |||

Hal Tasaki | What is thermal equilibrium and how do we get there? | Sun 9th 4:30pm | |

Abstract: We discuss the foundation of equilibrium statistical mechanics in terms of isolated macroscopic quantum systems. We shall characterize thermal equilibrium based on "typicality" picture and a large-deviation type consideration. We then present a simple (and hopefully realistic) condition based on the notion of effective dimension which guarantees that a nonequilibrium initial state evolves into the thermal equilibrium. | |||

Paul Goldbart | Universality in transitionless quantum driving | Sun 9th 5:00pm | |

Abstract: A time-dependent quantum system, if prepared in some instantaneous eigenstate of its Hamiltonian, typically exhibits nonadiabaticity: it develops quantum amplitudes to be found in orthogonal instantaneous eigenstates. When the time dependence is slow, these amplitudes are small, as seen explicitly, e.g., in the Landau-Majorana-Zener model. Berry (2009) has shown how to construct Hamiltonian terms that stifle nonadiabaticity, regardless of the pace of the time dependence of the original Hamiltonian: this is transitionless quantum driving. We discuss the extension of transitionless quantum driving to systems possessing exact degeneracies amongst their instantaneous energy eigenvalues and, as a result, exhibit the Wilczek-Zee (1984) nonabelian extension of Berry's connection (1984). We also discuss how a particular stifling term serves to protect adiabaticity for a surprisingly large family of systems. We conclude by mentioning some settings in which transitionless quantum driving should be realizable, experimentally. This talk is based on work done with Rafael Hipolito.F. Wilczek and A. Zee (1984) Appearance of gauge structure in simple dynamical systems, Physical Review Letters 52, 2111-2114. M. V. Berry (2009) Transitionless quantum driving, Journal of Physics A: Mathematical and Theoretical 42, 365303 [9 pages]. M. V. Berry (1984) Quantal phase factors accompanying adiabatic changes, Proceedings of the Royal Society of London Series A 392, 45-57. | |||

New topics --- Monday October 10th --- Skiles 202 --- Chair: Bruno Nachtergaele | |||

Takahiro Morimoto | Classification theory of topological insulators with Clifford algebras and its application to interacting fermions | Mon 10th 2:00pm | |

Abstract: The topological classification of noninteracting fermionic ground states is established as the tenfold way. Systems of non-interacting fermions are divided into ten symmetry classes. For each dimension, five out of ten symmetry classes contain nontrivial topological insulators (TIs) or superconductors (TSCs) characterized by Z or Z_2 topological numbers. Later, it was revealed that the noninteracting topological classification Z is unstable to interactions and reduces to Z_8 (Z_16) in the case of 1D (3D) time-reversal symmetric TSCs.In this talk, first, we review the classification theory of noninteracting topological insulators in terms of an extension problem of associated Clifford algebras. This enables us to concisely derive the tenfold way classification and also to classify topological crystalline insulators [1]. Then we apply the Clifford algebra approach to the breakdown of the ten-fold way in the presence of quartic fermion-fermion interactions for any dimension of space [2]. Specifically, we study the effects of interactions on the boundary gapless modes of TIs in terms of boundary dynamical masses. Breakdown of the noninteracting topological classification occurs when the quantum non-linear sigma models for the boundary dynamical masses favor quantum disordered phases. For the ten-fold way, we find that (i) Z_2 is always stable, (ii) Z in even dimensions is always stable, (iii) Z in odd dimensions is unstable and reduces to Z_N that can be identified explicitly for any dimension and any defining symmetries. We also apply our method to the topological crystalline insulator (SnTe) and find the reduction of the noninteracting topological classification Z to Z_8. [1] T. Morimoto, and A. Furusaki, Phys. Rev. B 88, 125129 (2013). . [2] T. Morimoto, A. Furusaki, and C. Mudry, Phys. Rev. B 92, 125104 (2015 | |||

Carlos sa de Melo | Effects of spin-orbit coupling on the Berezinskii-Kosterlitz-Thouless transition. | Mon 10th 2:30pm | |

Abstract: We investigate the Berezinskii-Kosterlitz-Thouless (BKT) transition in a two-dimensional (2D) neutral Fermi system with spin-orbit coupling (SOC), as a function of the two-body binding energy and a perpendicular Zeeman field [1,2]. By including a generic form of the SOC, as a function of Rashba and Dresselhaus terms, we study the evolution between the equal Rashba-Dresselhaus (ERD) and the Rashba-only (RO) cases. We show that in the ERD case, at fixed non-zero Zeeman field, the BKT transition temperature TBKT is increased by the effect of the SOC for all values of the binding energy. We also find a significant increase in the value of the Clogston limit compared to the case without SOC. Furthermore, we demonstrate that the superfluid density tensor becomes anisotropic (except in the RO case), leading to an anisotropic phase-fluctuation action that describes elliptic vortices and anti-vortices, which become circular in the RO limit. This deformation constitutes an important experimental signature for superfluidity in a 2D Fermi system with ERD SOC. Finally, we show that the anisotropic sound velocity exhibit anomalies at low temperatures in the vicinity of quantum phase transitions between topologically distinct uniform superfluid phases.[1] Jeroen P. A. Devreese, Jacques Tempere, and Carlos A. R. Sá de Melo, Phys. Rev. Lett. 113, 165304 (2014). [2] Jeroen P. A. Devreese, Jacques Tempere, and Carlos A. R. Sá de Melo, Physical Review A 92, 043618 (2015). | |||

Maksym Serbyn | Properties of many-body localized phase: entanglement spec | Mon 10th 3:00pm | |

Abstract: Many body localization allows quantum systems to escape thermalization via emergence of extensive number of conserved quantities. I will demonstrate how the existence of these local conserved quantities is manifested in various properties of many-body localized phase. I will demonstrate the power-law form of the entanglement spectrum in the MBL phase, which follows from existence of local conserved quantities. I will discuss general implications of this result for variational studies of highly excited eigenstates in many-body localized systems, and show an implementation of a matrix-product state algorithm which allows us to access the eigenstates of large systems close to the delocalization transition. In addition, I will discuss statistics of matrix elements of local operators and use it to probe delocalization transition. | |||

Po-Yao Chang | Entanglement negativity in many-body physics | Mon 10th 4:00pm | |

Abstract: Entanglement measures are powerful techniques of extracting quantum information in a many-body state. However, most of the entanglement measures focus on a bipartite system in a pure state. To characterize quantum entanglement of a tripartite system in a mixed state, entanglement negativity is proposed. This talk will present the current developments of computing entanglement negativity and their applications. Three methods will be demonstrated: an overlap matrix approach for free-fermion systems[1], the conformal field theory approach for a local quantum quench[2], and a surgery method for Chern-Simons theories[3].[1] P.-Y. Chang and X. Wen, Phys. Rev. B 93, 195140 (2016). [2] X. Wen, P.-Y. Chang and S. Ryu, Phys. Rev. B 92, 075109 (2015). [3] X. Wen, P.-Y. Chang and S. Ryu, arXiv:1606.04118. | |||

Israel Klich | Novel quantum phase transition from bounded to extensive entanglement entropy. | Mon 10th 4:30pm | |

Abstract: I will describe a continuous family of frustration-free Hamiltonians with exactly solvable ground states. We prove that the ground state of our model is non-degenerate and exhibits a novel quantum phase transition from bounded entanglement entropy to a massively entangled state with volume entropy scaling. The ground state may be interpreted as a deformation away from the uniform superposition of colored Motzkin paths, showed by Movassagh and Shor to have a large (square-root) but sub-extensive scaling of entanglement into a state with an extensive entropy. | |||

Shina Tan | Exact relations for two-component Fermi gases with contact interactions | Mon 10th 5:00pm | |

Abstract: Ultracold atomic gases created in experiments are so dilute that the average inter-atomic distance is much larger than the characteristic range of the atomic interaction, and so cold that the thermal de Broglie wave length is much larger than that range. Normally they are weakly interacting. By tuning them near a Feshbach resonance, near which the two-body scattering length can be made arbitrarily large, however, people can easily make them strongly interacting. When the scattering length is much larger than the range, we can consider an idealized model in which the range of the interaction is taken to be zero. Within such a model, the scattering length becomes the only parameter for the atomic interactions, if the atoms are fermionic and there are no more than two spin states involved. In such a model, the momentum distribution of the atoms behaves as C/k^4+O(1/k^6) when the wave number k goes to infinity. The coefficient C is known as the contact. There are some exact relations relating the energy, pressure, and the two-body short-range correlation functions, etc. All of them involve the contact C. In particular, the energy of such a gas is a linear functional of the momentum distribution, for both the ground state and all excited states. This is true even if the scattering length is comparable to or larger than the average interatomic distance, such that the gas is strongly interacting. | |||

Nicholas Read | Compactly-supported Wannier functions, algebraic K-theory, and tensor network states | Mon 10th 5:30pm | |

Abstract: | |||

Quantum information | |||

QI --- Saturday October 8th --- Skiles 268 --- Chair: Christopher King | |||

Mario Berta | Multivariate Trace Inequalities | Sat 8th 2:00pm | |

Abstract: We prove several trace inequalities that extend the Golden-Thompson and the Araki-Lieb-Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb’s triple matrix inequality. As an example application of our four matrix extension of the Golden-Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong subadditivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities. | |||

Debbie Leung | Embezzlement of entanglement, conservation laws, and nonlocal games | Sat 8th 2:30pm | |

Abstract: Consider two remote parties Alice and Bob, who share quantum correlations in the form of a pure entangled state. Without further interaction, the 'Schmidt coefficients' of the entangled state are invariant; in particular, the amount of entanglement is conserved. van Dam and Hayden found that reordering these coefficients (corresponding to allowed local operations) can effect an apparent violation of the conservation law nearly perfectly, a phenomenon called 'embezzlement'. We discuss how the same mathematics can explain coherent manipulation of spins in NMR and other approximate violation of conservation laws. We show how this phenomenon gives rise to a quantum generalization of nonlocal games that cannot be won with finite amount of entanglement. (Joint work with Ben Toner, John Watrous and Jesse Wang.) | |||

Beth Ruskai | Extreme Points of Unital Quantum Channels | Sat 8th 3:00pm | |

Abstract: Several new classes of extreme points of unital and trace-preserving completely positive (CP) maps are analyzed. One class is not extreme in either the convex set of unital CP maps or the set of trace-preserving CP maps and is factorizablle. Another class is extreme for both the set of unital CP maps and the set of trace-preserving CP maps, except for certain critical parameters. For those parameters the linear dependence of the matrices in the Choi product condition are associated with representations of the symmetric group. | |||

Mark Wilde | Universal Recoverability in Quantum Information | Sat 8th 4:00pm | |

Abstract: The quantum relative entropy is well known to obey a monotonicity property (i.e., it does not increase under the action of a quantum channel). Here we present several refinements of this entropy inequality, some of which have a physical interpretation in terms of recovery from the action of the channel. The recovery channel given here is explicit and universal, depending only on the channel and one of the arguments to the relative entropy. Time permitted, we discuss several application to the 2nd law of thermodynamics, uncertainty relations, and Gaussian quantum information. | |||

Jeongwan Haah | Local Approximate Quantum Error Correction | Sat 8th 4:30pm | |

Abstract: We study the fundamental limits on reliably storing quantum information in lattices of qubits by deriving tradeoﬀ bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing error correction criteria in exact settings. Our tradeoﬀ bounds relate the spatial dimension of the lattice, the number of physical qubits, the number of encoded qubits, the code distance, the accuracy parameter that quantiﬁes how well erasure can be recovered, and the locality parameter that speciﬁes the length scale at which the recovery operates. Connection to the topological order will be discussed. Joint work with S. Flammia, M. Kastoryano, and I. Kim. | |||

QI --- Sunday October 9th --- Skiles 268 --- Chair: Beth Ruskai | |||

Isaac Kim | Markovian marginals | Sun 9th 2:00pm | |

Abstract: We introduce the notion of so called Markovian marginals, which is a natural framework for constructing solutions to the quantum marginal problem. We show that a set of reduced density matrices on overlapping supports necessarily has a global state that is compatible with all the given reduced density matrices, provided that they satisfy certain (nonlinear) local constraints. | |||

Ke Li | Discriminating quantum states: the multiple Chernoff distance | Sun 9th 2:30pm | |

Abstract: Suppose we are given n copies of one of the quantum states {rho_1,..., rho_r}, with an arbitrary prior distribution that is independent of n. The multiple hypothesis Chernoff bound problem concerns the minimal average error probability P_e in detecting the true state. It is known that P_e~exp(-En)decays exponentially to zero. However, this error exponent E is generally unknown, except for the case r=2.In this talk, I will give a solution to the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szkola's conjecture that E=min_{i eq j} C(rho_i, rho_j). The right-hand side of this equality is called the multiple quantum Chernoff distance, and C(rho_i,rho_j):=max_{0 <= s <= 1} {-log Tr rho_i^s rho_j^(1-s)} has been previously identified as the optimal error exponent for testing two hypotheses, rho_i versus rho_j. The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and Szkola's lower bound. Specialized to the case r=2, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al. | |||

Graeme Smith | Uniformly additive entropic formulas | Sun 9th 3:00pm | |

Abstract: Information theory establishes the fundamental limits on data transmission, storage, and processing. Quantum information theory unites information theoretic ideas with an accurate quantum-mechanical description of reality to give a more accurate and complete theory with new and more powerful possibilities for information processing. The goal of both classical and quantum information theory is to quantify the optimal rates ofinterconversion of different resources. These rates are usually characterized in terms of entropies. However, nonadditivity of many entropic formulas often makes finding answers to information theoretic questions intractable. In a few auspicious cases, such as the classical capacity of a classical channel, the capacity region of a multiple access channel and the entanglement assisted capacity of a quantum channel, additivity allows a full characterization of optimal rates. Here we present a new mathematical property of entropic formulas, uniform additivity, that is both easily evaluated and rich enough to capture all known quantum additive formulas. We give a complete characterization of uniformly additive functions using the linear programming approach to entropy inequalities. In addition to all known quantum formulas, we find a new and intriguing additive quantity: the completely coherent information. We also uncover a remarkable coincidence---the classical and quantum uniformly additive functions are identical; the tractable answers in classical and quantum information theory are formally equivalent. | |||

Stefan Boettcher | The Renormalization Group Solution of Quantum Walks on Complex Networks | Sun 9th 4:00pm | |

Abstract: Replacing the stochastic evolution operator in the master equation of the classical random walk with a unitary operator leads to a spectrum of new phenomena. Such a quantum walk has gain considerable interest in quantum information sciences as it is the "engine" that drives Grover's quantum search to gain a quadratic speed-up over classical randomized algorithms. The spreading dynamics on regular lattices already leads to numerous fascinating features, such as localization and violation of Polya's theorem, however, the motion is universally ballistic in all dimensions and reveals little insight about the intricate nature of the quantum dynamics. We use the renormalization group to produce non-trivial, exact results for the asymptotic scaling of the probability density function for quantum walks on various complex networks (Sierpinski, Migdal-Kadanoff, Hanoi). These elucidate the subtle interplay of quantum effects and internal ("coin") degrees of freedom with the geometry of the network and the spectral properties of the evolution operator by which one can control the behavior. | |||

Michael Walter | Entanglement in Random Tensor Networks | Sun 9th 4:30pm | |

Abstract: Motivated by recent research in quantum information and gravity, we study tensor networks with large bond dimension, obtained by contracting random stabilizer states. We find that their bipartite and multipartite entanglement properties are dictated by the geometry of the network and explain how this relates to non-standard entropy inequalities. We further consider 'holographic' bulk-boundary mappings defined by such tensor networks and discuss their properties as quantum subsystem codes. Techniques used include spin models for random tensor averages and a new formula for the third moment of a random stabilizer state. | |||

QI --- Monday October 10th --- Skiles 268 --- Chair: Renato Renner | |||

Nilanjana Datta | Contractivity properties of a quantum diffusion semigroup | Mon 10th 2:00pm | |

Abstract: We consider a quantum generalization of the classical heat equation, and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality for Gaussian states. The former leads to an ultracontractivity result. This in turn implies that the largest eigenvalue and the purity of any state, evolving under the action of the semigroup, decrease inverse polynomially in time, while its entropy increases logarithmically in time. This is joint work with Cambyse Rouze' and Yan Pautrat. | |||

William Slofstra | Tsirelson's problem and linear system games | Mon 10th 2:30pm | |

Abstract: In quantum information, we frequently consider (for instance, whenever we talk about entanglement) a composite system consisting of two separated subsystems. A standard axiom of quantum mechanics states that a composite system can be modeled as the tensor product of the two subsystems. However, there is another less restrictive way to model a composite system, which is used in quantum field theory: we can require only that the algebras of observables for each subsystem commute within some larger subalgebra. Tsirelson's question (which comes in several variants) asks whether the correlations arising from commuting-operator models can always be represented by tensor-product models. I will give examples of linear system non-local games which cannot be played perfectly with tensor-product strategies, but can be played perfectly with commuting-operator strategies, resolving (one version of) Tsirelson's question in the negative. From these examples, we can also derive other consequences for the theory of non-local games, such as the undecidability of determining whether a non-local game has a perfect commuting-operator strategy. | |||

Carlos Ortiz-Marrero | Categories and Topological Quantum Computing | Mon 10th 4:00pm | |

Abstract: Quantum computation is defined to be any computational model based upon the theoretical ability to manufacture, manipulate, and measure quantum states. (2+1)-dimensional topological phases of matter (TPM) promise a route to quantum computation where quantum information is topologically protected against decoherence. In this talk, we will explore the underling mathematical theory that is driving the classification of these TPM. We will mainly concentrate on the algebraic/categorical structure behind such phases and explain where this structure fits in describing TPM. Finally, we will discuss some recent developments in the mathematical classification pertinent to TPM, namely the classificationof (pre-)modular categories. | |||

Vern Paulsen | Perfect embezzlement of a Bell State | Mon 10th 4:30pm | |

Abstract: Van Dam and Hayden showed that if Alice and Bob each have finite dimensional state spaces, then using local unitary operations and a shared entangled state on some bipartite resource space, with vanishingly small error, they can "appear" to produce an entangled state. Hence, the term "embezzlement". We prove that perfect embezzlement is impossible in this framework even when the shared resource space is allowed to be infinite dimensional. But if one allows the commuting operator model, then one can embezzle perfectly. We then relate this to recent work on the conjectures of Tsirelson and Connes. Finally, we show that this implies a perfect commuting strategy for a game of Regev and Vidick which has no perfect bipartite strategy. | |||

Anna Vershynina | Quantum analogues of geometric inequalities for Information Theory | Mon 10th 5:00pm | |

Abstract: Geometric inequalities, such as entropy power inequality or the isoperimetric inequality, relate geometric quantities, such as volumes and surface areas. Entropy power inequality describes how the entropy power of a sum of random variables behaves to the sum of entropy powers. The isoperimetric inequality for entropies relates the entropy power and the Fisher information, and implies that Gaussians have minimal entropy power among random variables with a fixed Fisher information. Classically, these inequalities have useful applications for obtaining bounds on channel capacities, and deriving Log-Sobolev inequalities.In my talk I provide quantum analogues of certain well-known inequalities from classical information theory, with the most notable being the isoperimetric inequality for entropies. The latter inequality is useful for the study of convergence of certain semigroups to fixed points. In the talk I demonstrate how to apply the isoperimetric inequality for entropies to show exponentially fast convergence of quantum Ornstein-Uhlenbeck (qOU) semigroup to a fixed point of the process. The inequality representing the fast convergence can be viewed as a quantum analogue of a classical Log-Sobolev inequality. As a separate result, necessary for the fast convergence of qOU semigroup, I argue that gaussian thermal states minimize output entropy for the attenuator semigroup among all states with a given mean-photon number. (based on a joint work with S. Huber and R. Koenig) | |||

Quantum mechanics with random features | |||

Random --- Saturday October 8th --- Skiles 005 --- Chair: Simone Warzel | |||

Houssam Abdul Rahman | Entanglement and Transport in the disordered Quatum XY chain. | Sat 8th 2:00pm | |

Abstract: For a class of disordered Quantum XY chains, we prove that the dynamical entanglement of a broad class of product states satisfies a constant bound. Corollaries include area laws for eigenstates and thermal states. These results correspond to the absence of information transport. We also present and discuss some new results about the particle number transport and the energy transport in the disordered XY chain. We will draw the relation between these results and the notion of Many-body localization. | |||

Dhriti Dolai | Spectral Statistics of Random Schroedinger Operators with Non-Ergodic Random Potential | Sat 8th 2:30pm | |

Abstract: It is known from earlier result of Gordon-Jaksic-Molchanov-Simon [1], that the spectrum of the random Schrodinger operators with unbounded potentials (non stationary) is pure point. Recently we obtain the eigenvalue statistics for this model and it is turn out that the statistics is Poisson. It is an analogous of Minami’s work on stationary potential [2].This is a joint work with Anish Mallick. References [1] Gordon, Y. A; Jaksic, V; Molchanov, S; Simon, B: Spectral properties of random Schrodinger operators with unbounded potentials, Comm. Math. Phys. 157(1), 23-50, 1993. [2] Minami, Nariyuki: Local Fluctuation of the Spectrum of a Multidimensional Anderson Tight Binding Model, Commun. Math. Phys. 177(3), 709-725, 1996. [3] Dolai, Dhriti; Mallick, Anish: Spectral Statistics of Random Schrdinger Operators with Unbounded Potentials, arXiv:1506.07132 [math.SP]. [4] Combes, Jean-Michel; Germinet, Francois; Klein, Abel: Generalized Eigenvalue-Counting Estimates for the Anderson Model, J Stat Physics. 135(2), 201-216, 2009. | |||

Milivoje Lukic | KdV equation with almost periodic initial data | Sat 8th 3:00pm | |

Abstract: The KdV equation is known to be integrable for some classes of initial data, such as decaying, periodic, and finite-gap quasiperiodic. In this talk, we will describe recent progress for almost periodic initial data, centered around a conjecture of Percy Deift that the solution is almost periodic in time. We will discuss the proof of existence, uniqueness, and almost periodicity in time, in the regime of absolutely continuous and sufficiently 'thick' spectrum. In particular, this result proves Deift's conjecture for small analytic quasiperiodic initial data with Diophantine frequency. The talk is based on joint work with Ilia Binder, David Damanik, and Michael Goldstein. | |||

Tatyana Shcherbyna | Local regime of 1d random band matrices | Sat 8th 4:00pm | |

Abstract: Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss an application of the supersymmetric method (SUSY) to the analysis of the bulk local regime of some specific types of RBM.We present rigorous SUSY results about the crossover for 1d RBM on the level of characteristic polynomials, as well as some progress in studying of the density of states and usual second correlation function. | |||

Joe Chen | Spectral decimation and its application to spectral analysis on infinite fractal lattices | Sat 8th 4:30pm | |

Abstract: The method of spectral decimation originated from Rammal and Toulouse in the 80s, and has since been developed to tackle spectral problems on self-similar fractals by Bellissard, Fukushima, Shima, Malozemov, Teplyaev, etc.In this talk we present two concrete spectral problems on infinite fractal lattices which are inspired by the study of quasi-periodic and random Schrodinger operators. In both problems, we use spectral decimation in an essential way, and reduce the problem to the analysis of a certain 1-dimensional complex dynamical system. We hope that these models can help enlighten the mechanisms behind the spectral properties of more complicated Schrodinger operators. 1) On the integer half-line ($\mathbb{Z}_+$) endowed with a fractal self-similar Laplacian parametrized by a single parameter $p\in (0,1)$, we prove that the Laplacian spectrum is purely singularly continuous whenever $p\neq \frac{1}{2}$. (If $p=\frac{1}{2}$ one recovers the usual Laplacian on $\mathbb{Z}_+$, whose spectrum is absolutely continuous.) To our knowledge this may be the simplest toy model for exhibiting purely singularly continuous spectrum. 2) On the infinite Sierpinski gasket lattice (SGL), we establish an exponential decay of the resolvent associated with the Laplace or Schrodinger operator, based on spectral decimation and a heat kernel upper estimate. This leads to a proof of Anderson localization on SGL by the methods of Simon-Wolff and Aizenman-Molchanov. This is based on joint works with S. Molchanov (UNC-Charlotte) and A. Teplyaev (UConn). | |||

Random --- Sunday October 9th --- Skiles 005 --- Chair: Gunter Stolz | |||

Vojkan Jaksic | Adiabatic theorems and Landauer's principle in quantum statistical mechanics | Sun 9th 2:00pm | |

Abstract: The Landauer principle asserts that the energy cost of erasure of one bit of information by the action of a thermal reservoir in equilibrium at temperature T is never less than k_BT log 2. We discuss Landauer's principle for quantum statistical models describing a finite level quantum system S coupled to an infinitely extended thermal reservoir R and link the saturation of Landauer's bound to adiabatic theorems in quantum statistical mechanics (for states and relative entropy). Furthermore, by extending the adiabatic theorem to Renyi's relative entropy, we extend the Landauer principle to the level the Full Counting Statistics (FCS) of energy transfer between S and R. This allows to elucidate the nature of Landauer's principle FCS fluctuations.This talk is based on joint works with Tristan Benoist, Martin Fraas, and Claude-Alain Pillet. | |||

Abel Klein | Eigensystem multiscale analysis for Anderson localization in energy intervals I | Sun 9th 2:30pm | |

Abstract: We perform an eigensystem multiscale analysis for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model in an energy interval. In particular, it yields localization for the Anderson model in a nonempty interval at the bottom of the spectrum. This eigensystem multiscale analysis in an energy interval treats all energies of the finite volume operator at the same time, establishing level spacing and localization of eigenfunctions with eigenvalues in the energy interval in a fixed box with high probability. In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems (eigenvalues and eigenfunctions). In any given scale we only have decay for eigenfunctions with eigenvalues in the energy interval, and no information about the other eigenfunctions. For this reason, going to a larger scale requires new arguments that were not necessary in our previous eigensystem multiscale analysis for the Anderson model at high disorder, where in a given scale we have decay for all eigenfunctions. | |||

Alexander Elgart | Eigensystem multiscale analysis for Anderson localization in energy intervals II | Sun 9th 3:00pm | |

Abstract: We perform an eigensystem multiscale analysis for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model in an energy interval. In particular, it yields localization for the Anderson model in a nonempty interval at the bottom of the spectrum. This eigensystem multiscale analysis in an energy interval treats all energies of the finite volume operator at the same time, establishing level spacing and localization of eigenfunctions with eigenvalues in the energy interval in a fixed box with high probability.In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems (eigenvalues and eigenfunctions). In any given scale we only have decay for eigenfunctions with eigenvalues in the energy interval, and no information about the other eigenfunctions. For this reason, going to a larger scale requires new arguments that were not necessary in our previous eigensystem multiscale analysis for the Anderson model at high disorder, where in a given scale we have decay for all eigenfunctions. | |||

Jeffrey Schenker | Localization in the disordered Holstein model | Sun 9th 4:00pm | |

Abstract: The Holstein model (in the one particle sector) describes a lattice particle interacting with independent Harmonic oscillators at each site of the lattice. We consider this model with on site disorder in the particle potential. This is proposed a simple model in which it may be possible to test some ideas regarding multi/many-body localization. Provided the oscillator frequency is not too small and the hopping is weak, we are able to prove localization for the eigenfunctions, in particle position and in oscillator Fock space. Some open problems regarding the character of high energy eigenstates will be discussed. (Joint work with Rajinder Mavi.) | |||

Per von Soosten | Localizationiin the Hierarchical Anderson Model | Sun 9th 4:30pm | |

Abstract: We will consider a hierarchical version of the classical Anderson model on the lattice and present results to the extent that the hierarchical model remains localized throughout its range of parameters. Our argument is based on renormalization ideas that transform the Hamiltonian into a regime of high disorder. This talk is based on joint work with Simone Warzel. | |||

Many-body systems and statistical mechanics | |||

Many-body --- Saturday October 8th --- Skiles 249 --- Chair: Vieri Mastropietro | |||

Christian Hainzl | Spectral theoretic aspects of the BCS theory of superconductivity | Sat 8th 2:00pm | |

Abstract: The critical temperature in the BCS theory of superconductivity, in the presence of external fields, is determined by a linear two-body operator. I present the corresponding operator and its properties in the case of bounded potentials as well as in the case of a constant external magnetic field. | |||

Marius Lemm | Condensation of fermion pairs in a domain | Sat 8th 2:30pm | |

Abstract: We consider a gas of fermions at zero temperature and low density, interacting via a microscopic two body potential which admits a bound state. The particles are confined to a domain with Dirichlet (i.e. zero) boundary conditions. Starting from the microscopic BCS theory, we derive an effective macroscopic Gross-Pitaevskii (GP) theory describing the condensate of fermion pairs. The GP theory also has Dirichlet boundary conditions. Along the way, we prove that the GP energy, defined with Dirichlet boundary conditions on a bounded Lipschitz domain, is continuous under interior and exterior approximations of that domain. This is joint work with Rupert L. Frank and Barry Simon. | |||

Marcello Porta | Mean field evolution of fermionic systems | Sat 8th 3:00pm | |

Abstract: In this talk I will discuss the dynamics of interacting fermionic systems in the mean field regime. Compared to the bosonic case, fermionic mean field scaling is naturally coupled with a semiclassical scaling, making the analysis more involved. As the number of particles grows, the quantum evolution of the system is expected to be effectively described by Hartree-Fock theory. The next degree of approximation is provided by a classical effective dynamics, corresponding to the Vlasov equation.I will consider initial data which are close to quasi-free states, at zero (pure states) or at positive temperature (mixed states), with an appropriate semiclassical structure. Under mild regularity assumptions on the interaction potential, I will show that the time evolution of such initial data stays close to a quasi-free state, with reduced one-particle density matrix given by the solution of the time-dependent Hartree-Fock equation. The result can be extended to Coulomb interactions, under the assumption that the solution of the time-dependent Hartree-Fock equation preserves the semiclassical structure of the initial data. If time permits, the convergence from the time-dependent Hartree-Fock equation to the Vlasov equation will also be discussed. The results hold for all semiclassical times, and give effective bounds on the rate of convergence towards the effective dynamics as the number of particles goes to infinity. | |||

Michele Correggi | Local Density Approximation for the Almost-bosonic Anyon Gas | Sat 8th 4:00pm | |

Abstract: We study a one-parameter one-body energy functional with a self-consistent magnetic field, which describes a quantum gas of almost-bosonic anyons in the average-field approximation. For the homogeneous gas we prove the existence of the thermodynamic limit of the energy at fixed effective statistics parameter and the independence of such a limit from the shape of the domain.This result is then used in a local density approximation to derive an effective Thomas-Fermi like model for the trapped anyon gas in the limit of a large effective statistics parameter (i.e., "less-bosonic" anyons). Joint work with D. Lundholm, N. Rougerie | |||

Rupert Frank | Derivation of an effective evolution equation for a strongly coupled polaron | Sat 8th 4:30pm | |

Abstract: Fröhlich's polaron Hamiltonian describes an electron coupled to the quantized phonon field of an ionic crystal. We show that in the strong coupling limit the dynamics of the polaron is approximated by an effective non-linear partial differential equation due to Landau and Pekar, in which the phonon field is treated as a classical field.
The talk is based on joint works with B. Schlein and with Z. Gang. | |||

Peter Pickl | Derivation of the Maxwell-Schrödinger Equations from the Pauli-Fierz Hamiltonian | Sat 8th 5:00pm | |

Abstract: We consider the spinless Pauli-Fierz Hamiltonian which describes a quantum system of non-relativistic identical particles coupled to the quantized electromagnetic field. We study the time evolution in a mean-field limit where the number N of charged particles gets large while the coupling to the radiation field is rescaled by 1/√N. At time zero we assume that almost all charged particles are in the same one-body state (a Bose-Einstein condensate) and we assume also the photons to be close to a coherent state. We show that at later times and in the limit N -> ∞ the charged particles as well as the photons exhibit condensation, with the time evolution approximately described by the Maxwell-Schrödinger system, which models the coupling of a non-relativistic particle to the classical electromagnetic field. | |||

Nicolas Rougerie | Rigidity of the Laughlin liquid | Sat 8th 5:30pm | |

Abstract: The Laughlin state is a well-educated ansatz for the ground state of 2D particles subjected to large magnetic fields and strong interactions. It is of importance to understand the rigidity of its response to perturbations. Indeed, this is a crucial ingredient in the Fractional Quantum Hall Effect, where the Laughlin state is the cornerstone of our current theoretical understanding.In this talk we shall consider general N-particle wave functions that have the form of a product of the Laughlin state and an analytic function of the N variables. This is the most general form of a wave function that can arise through a perturbation of the Laughlin state by external potentials or impurities, while staying in the lowest Landau level and maintaining the strong correlations of the original state. We show that the perturbation can only shift or lower the 1-particle density but nowhere increase it above a maximum value. Regardless of the analytic prefactor, the density satisfies the same bound as the Laughlin function itself in the limit of large particle number. Consequences of this incompressibility bound for the response of the Laughlin state to external fields will be discussed. joint work with Elliott H. Lieb and Jakob Yngvason | |||

Many-body --- Sunday October 9th --- Skiles 249 --- Chair: Benjamin Schlein | |||

Ian Jauslin | Ground state construction of bilayer graphene | Sun 9th 2:00pm | |

Abstract: We consider a model of weakly-interacting electrons in bilayer graphene. Bilayer graphene is a 2-dimensional crystal consisting of two layers of carbon atoms in a hexagonal lattice. Our main result is an expression of the free energy and two-point Schwinger function as convergent power series in the interaction strength. In this talk, I discuss the properties of the non-interacting model, and exhibit three energy regimes in which the energy bands are qualitatively different. I then sketch how this decomposition may be used to carry out the renormalization group analysis used to prove our main result. This is joint work with Alessandro Giuliani. | |||

Phan Thanh Nam | Stability of 2D focusing many-boson systems | Sun 9th 2:30pm | |

Abstract: We consider a 2D quantum system of N bosons, interacting via a pair potential of the form $N^{2\beta-1}w(N^\beta (x-y))$. In the focusing case $w<0$, the stability of the second kind of the system is not obvious. We will show that if the system is trapped by an external potential $|x|^s$ and $\beta<(s+1)/(s+2)$, then the leading order behavior of ground states in the large N limit is described by the corresponding cubic nonlinear Schr\"odinger energy functional. In particular, our result covers the dilute regime $\beta>1/2$, where the range of the interaction is much smaller than the average distance between particles. This is joint work with Mathieu Lewin and Nicolas Rougerie. | |||

Shannon Starr | Robust Bounds for Emptiness Formation Probability for Dimers | Sun 9th 3:00pm | |

Abstract: Emptiness formation probability is a measurable quantity associated to a ground state or equilibrium state of a quantum spin system. It was originally promoted by V Korepin. For the XXZ chain, a relation with the 6 vertex model discovered by Lieb allows for robust bounds using the reflection positivity technique. For dimers, emptiness formation probability for a lattice rotated by 45 degrees is more natural. The basic technique applies but there are extra mathematical issues, including discovering a quantum spin system associated to the lattice model. This is joint work with Scott Williams, a student at UAB. | |||

Bruno Nachtergaele | Stability of Frustration-Free Ground States of Quantum Lattice Systems | Sun 9th 4:00pm | |

Abstract: We study frustration-free quantum lattice systems with a non-vanishing spectral gap above one or more (infinite-volume) ground states. The ground states are called stable if arbitrary perturbations of the Hamiltonian that are uniformly small throughout the lattice have only a perturbative effect. In the past several years such stability results have been obtained in increasing generality aimed at applications to topological phases. We discuss the works by Bravyi-Hastings-Michalakis and Michalakis-Zwolak, and some recent extensions of these results to systems with spontaneous symmetry breaking in joint work with Robert Sims and Amanda Young. | |||

Jan Philip Solovej | Zero modes for Dirac operators with magnetic links | Sun 9th 4:30pm | |

Abstract: The occurence of zero modes for Dirac operators with magnetic fields is the cause of break down of stability of matter for charged systems. All known examples of magnetic fields leading to zero modes are geometrically very complex. In order to better understand this geometry I will discuss singular magnetic fields supported on a finite number of possibly interlinking field lines (magnetic links). I will show that the occurence of zero modes is intimately connected to the twisting and interlinking of the field lines. The result will rely on explicitly calculating appropriate spectral flows for the Dirac operators. This is joint work with Fabian Portmann and Jeremy Sok. | |||

Quantum mechanics of large systems with random features | |||

Random/Many-body --- Monday October 10th --- Skiles 249 --- Chair: Alessandro Giuliani/Jeffrey Schenker | |||

Rafael Ducatez | Anderson localization for infinitely many interacting particules under Hartree Fock theory | Mon 10th 2:00pm | |

Abstract: We prove the occurrence of Anderson localisation for a system of infinitely many particles interacting with a short range potential, within the ground state Hartree-Fock approximation. We assume that the particles hop on a discrete lattice and that they are submitted to an external periodic potential which creates a gap in the non-interacting one particle Hamiltonian. We also assume that the interaction is weak enough to preserve a gap. We prove that the mean-field operator has exponentially localised eigenvectors, either on its whole spectrum or at the edges of its bands, depending on the strength of the disorder. | |||

Francois Huveneers | A random matrix approach to Many-Body Localization | Mon 10th 2:30pm | |

Abstract: The localized phase in interacting systems is usually understood in a perturbative sense, as a robustness of Anderson localization when perturbing away from the non-interacting limit. In this talk, I will present a new approach, relying as much as possible on random matrix theory, which is generally used to describe ergodic systems (cf. ETH). The localized phase emerges then as an instability of the random matrix theory when adding disordered spins. This new view point is especially useful to analyze the influence of ergodic spots on the localized phase: It yields a detailed description of the boundary region near the spot, and naturally leads to the discussion of the stability of the localized phase upon bringing it in contact with a piece of ergodic material. I will also describe how the theory can be tested, and I will show some (preliminary) numerical results. From a joint work with Wojciech De Roeck (arXiv:1608.01815) | |||

John Imbrie | Constructive Methods for Localization and Eigenvalue Statistics | Mon 10th 3:00pm | |

Abstract: Convergent expansions for eigenvalues and eigenvectors lead to new insights in many-body and single-body quantum systems with disorder. I will review recent work elucidating the way randomness localizes eigenfunctions, smooths out eigenvalue distributions, and produces eigenvalue separation. | |||

Alain Joye | Representations of CCR describing infinite coherent states | Mon 10th 4:00pm | |

Abstract: We investigate the infinite volume limit of quantized photon fields in multimode coherent states. We show that for states containing a continuum of coherent modes, it is natural to consider their phases to be random and identically distributed. The infinite volume states give rise to Hilbert space representations of the canonical commutation relations which are random as well and can be expressed with the help of Itô stochastic integrals. We analyze the dynamics of the infinite coherent state alone and that of open systems consisting of small quantum systems coupled to the infinite coherent state. Under the free field dynamics, the initial phase distribution is shown to be driven to the uniform distribution, and coherences in small quantum systems interacting with the infinite coherent state, are shown to exhibit Gaussian time decay, instead of the exponential decay caused by infinite thermal states.Joint work with Marco Merkli. | |||

Chris Laumann | Many-body localization in mean-field quantum glasses | Mon 10th 4:30pm | |

Abstract: The central assumption of statistical mechanics is that interactions between particles establish local equilibrium. Isolated quantum systems, however, need not equilibrate; for example, this happens when sufficient quenched disorder causes localization. Unfortunately there are few tractable models to study this phenomenon.In this talk, I will briefly review the basic phenomenology of many-body localization and then introduce a family of mean-field spin glass models known to be tractable: the quantum p-spin models. I will argue that the quantum dynamics in these models exhibits a localized phase that cannot be detected in the canonical thermodynamic analysis. The properties of the phase and the mobility edge which separates it from the ergodic regime can be analytically estimated using several techniques. The localized eigenstates concentrate on clusters within Hilbert space which exhibit distinct magnetization patterns as characterized by an eigenstate variant of the Edwards-Anderson order parameter. Based on joint work with: C. L. Baldwin, A. Pal, A. Scardicchio | |||

Vieri Mastropietro | Localization of Interacting Fermions in the Aubry-André Model | Mon 10th 5:00pm | |

Abstract: We establish exponential decay of the zero temperature correlations of a fermionic system with a quasi-periodic Aubry-André potential and a many body short range interaction, for weak hopping and interactions and almost everywhere in the frequency and phase. Such decay indicates localization of the ground state.The proof is based on rigorous Renormalization Group methods and it is inspired by techniques developed to deal with KAM Lindstedt series. New problems are posed by the simultaneous presence of loops and small divisors. | |||

Robert Seiringer | Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas | Mon 10th 5:30pm | |

Abstract: 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. (Joint work with Simone Warzel.) |