Titles and Abstracts

Dmitry Abanin, University of Geneva

Title:
Ergodicity, entanglement, and many-body localization

Abstract:
We will give an introduction into dynamics of isolated, quantum many-body systems, often using tools and insights from the quantum information theory. We will describe two generic possibilities: ergodic systems, which thermalize, and many-body localized (MBL) systems, which avoid thermalization and break ergodicity. The mechanism of quantum thermalization -- the eigenstate thermalization hypothesis, as well as its implications, will be discussed. We will further explain the basic properties of the many-body localized phase, including the emergent robust integrability, area-law entanglement of MBL eigenstates, and dynamics. Finally, we will consider periodically driven (Floquet) systems, focusing on many-body effects and ways to realize new robust phases of matter in those systems. We will close by giving an overview of recent experiments. 

Sven Bachmann, LMU Munich

Title:
Adiabatic dynamics and quasi-adiabatic flow within gapped phases

Abstract:
These lectures will focus on the fundamental understanding of gapped phases of quantum lattice systems — some of which may be topologically ordered. I will first review an essential tool in their description, the so-called quasi-adiabatic flow relating ground state manifolds of systems in the same phase in a local way. I will then turn to a parallel dynamical question and prove an adiabatic theorem for driven many-body systems in a gapped phase. If time allows, I may briefly describe how this can be used to prove the validity of linear response theory for macroscopic, interacting systems. 

Mark Rudner, NBI, University of Copenhagen

Title:
Topology in periodically-driven systems

Abstract:
Recent work on topological materials has revealed a wide variety of intriguing phenomena that may arise when particles move in "non-trivial" Bloch bands.  At the same time, modern advances in experimental capabilities for controlling electronic, atomic, and optical systems open new possibilities for dynamically controlling the behaviors of a range of quantum systems. In this talk I will review the basic ideas behind topological band theory, and then explain how periodic driving can be used to gain dynamical control over the topological properties of quantum matter.  In the driven case, intriguing new types of robust non-equilibrium topological phenomena emerge. I will show how this occurs, and discuss recent and proposed experiments aimed at harnessing these exciting possibilities.

Gunter Stolz, University of Alabama at Birmingham

Title:
Rigorous results on many-body localization in quantum spin chains

Abstract:
Disordered quantum spin systems provide interesting models to study many-body localization (MBL), a phenomenon which has received strong attention in physics and quantum information theory over the last decade.

In these lectures we will mostly focus on two explicit spin systems which have led to some of the first rigorous results on MBL, the XY chain (isotropic as well as anisotropic) and the XXZ chain, with disorder introduced in form of a random exterior magnetic field.

The XY chain, the simplest illustrative example, maps to a quasi-free Fermion system via the Jordan-Wigner transform. In the presence of a random field, the effective Hamiltonian governing the Fermion system is the one-dimensional Anderson model. We will use this to conclude that the disordered XY chain satisfies several physically expected forms of MBL, including (i) a zero-velocity Lieb-Robinson bound on many-body transport, (ii) exponential decay of spatial correlations of eigenstates and thermal states (exponential clustering), and (iii) an area law for the entanglement entropy of eigenstates.

The XXZ chain is more challenging. Its main feature is particle number conservation (with particles corresponding to, say, down-spins). This leads to interacting Fermion systems, which need to be controlled for arbitrarily large particle number. We will describe the Ising phase of the XXZ chain, which exhibits a so-called droplet regime at low energy, characterized by states in which spins form a single cluster of down-spins. We will then discuss recent joint work with A. Elgart and A. Klein, which proves that the addition of a random field leads to many-body localization of the droplet spectrum in the form of dynamical exponential clustering of all eigenstates.

Frank Verstraete, University of Vienna and Ghent University

Title:
The mathematics of tensor networks

Abstract:
TBA

Simone Warzel, TU Munich

Title:
Localization in random media: from one-particle to many-particle models

Abstract:
This minicourse starts with a brief introduction and overview over results and proof ideas of Anderson localisation in one-particle lattice models. In particular, we will learn how to establish localisation estimates in the high disorder regime using the fractional moment method.
The second part of the course concerns more recent results for special many-particle models in disorder. I will give an overview of models for which ideas from the one-particle case extend to proofs of localisation in the many-particle set-up.

Albert Werner, QMATH, University of Copenhagen

Title:
Topological classification and bulk-edge-correspondence for symmetric quantum walks

Abstract:
We outline a theory of symmetry protected topological phases of one-dimensional quantum walks that extends the classification known for translation invariant systems in terms of their band structure. No translation invariance is assumed. The classification is complete in the sense that two walks have the same indices if and only if they can be connected by a norm continuous path along which symmetries are respected and  a gap around eigenvalues at symmetry protected points is preserved. Of the three indices we identify, two are related to the asymptotic behaviour far to the right and far to the left, respectively. These are also stable under compact perturbations. The third index is sensitive to those compact perturbations which cannot be contracted to a trivial one. We also obtain a rigorous bulk-edge-correspondence-principle if two walks with different indices are joined.