Thematic Semester on Mathematical Challenges in Many-Body Physics and Quantum Information

September – December 2018

International Scientific Advisory Committee: Joseph Avron (Technion), Svetlana Jitomirskaya (UC Irvine), Mathieu Lewin (Paris-Dauphine), Bruno Nachtergaele (UC Davis), Claude-Alain Pillet (Toulon), Robert Seiringer (IST Austria), Armen Shirikyan (Cergy-Pontoise), Barry Simon (Caltech)

Local Organizing Committee: Jacques Hurtubise (McGill), Dmitry Jakobson (McGill), Vojkan Jakšić (McGill), Dmitry Korotkin (Concordia), Luc Vinet (Montréal)

The opening events of the thematic program are the XIX International Congress on Mathematical Physics and its satellite meetings. They will be followed by five workshops held at the CRM and a joint CRM–Princeton workshop held at Princeton. Long-term participants will give daily seminars and mini-courses between the workshops.

AISENSTADT CHAIRS

Svetlana Jitomirskaya (UC Irvine)
Jitomirskaya’s lectures will be a part of the workshop on Spectral Theory of Quasi-Periodic and Random Operators (November 12–16). The first two lectures will take place on November 12 and the third one on November 13.

Robert Seiringer (IST Austria)
Seiringer’s lectures will be a part of the workshop on Many-Body Quantum Mechanics (September 10–14) and will take place on September 10, 11, and 13, respectively.

WORKSHOPS

Many-Body Quantum Mechanics
September 10–14, 2018
Organizers: Rupert Frank (Caltech), Mathieu Lewin (Paris-Dauphine), Benjamin Schlein (Zürich)

In the last years, there has been substantial progress in the mathematical analysis of many-body quantum systems. The main goal of this workshop is to bring together researchers working on different questions connected with many-body quantum mechanics, to discuss recent developments, exchange ideas and propose new challenges and research directions. In particular the workshop will focus on the following central topics:

  1. derivation of effective equations;
  2. disordered many body systems;
  3. open quantum systems in and out of equilibrium;
  4. quantum spin systems.

Entanglement, Integrability and Topology in Many-Body Systems
September 17–21, 2018
Organizers: Paul Fendley (Oxford), Israel Klich (Virginia)

Concepts from quantum information theory have found fertile ground in many-body condensed-matter physics and have become ubiquitous in the past decade. In particular, concepts related to entanglement have emerged as an important theoretical tool in tackling one of the central goals of condensed matter physics: the understanding and characterization of phase transitions. As such, these have spurred a rich activity that has brought much interest in and interaction with ideas in pure and applied mathematics. For example, subjects long studied in abstract topology and topological field theory have found a new venue of investigations with the identification of the quantum dimension as a universal sub-leading term in the entanglement entropy of topological systems. The log scaling of entanglement expected in conformal field theories has been verified and derived in classes of integrable systems. Functional and Fourier analysis tools, such as Szegő, Fisher–Hartwig, and Widom asymptotics for the scaling of traces of Toeplitz operators and their generalizations, have found immediate use in entanglement studies. Indeed, the renewed interest in the Widom conjecture in the context of entanglement calculations has led to substantial development in the field and a rigorous mathematical proof by Sobolev.

It is evident that these activities are just scratching the surface of possible connections. This workshop will bring practitioners from mathematics and physics to exchange information and ideas related to these advances. A particular attention will be given to exactly solvable systems where the behaviour of entanglement may be explored, with the potential to yield rigorous mathematical results.

Quantum Information and Quantum Statistical Mechanics
October 15–19, 2018
Organizers: Fernando Brandão (Caltech), Bruno Nachtergaele (UC Davis), Claude-Alain Pillet (Toulon), Michael Wolf (TU München)

In Quantum Information Theory (QIT) one studies quantum systems consisting of a finite (potentially large) number of elementary systems with a finite-dimensional state space. The most common case is that of systems of qubits, each of which has a two-dimensional state space. The aim is to understand the potential of such systems for the storage and processing of information. QIT is made interesting and essentially different from the classical theory of information and computation (based on the Turing machine) through the existence of entangled states in quantum mechanics.

In Quantum Statistical Mechanics (QSM) one studies quantum many-body systems with a large number of identical particles or spins. Understanding the detailed properties of states that describe such systems, at zero temperature, in thermal equilibrium, or out of equilibrium, is at the core of condensed matter physics. The structure of these states is often very complex because of entanglement. On the one hand, many of the most interesting phenomena involve specific quantum effects and entanglement, such as quantum phase transitions, spin fractionalization, and topological order. On the other hand, when one wants to study these systems, whether analytically or numerically, entanglement is often what makes the problem hard.

It is clear then that researchers in QIT and QSM share an interest in developing techniques to quantify, analyze, and understand entanglement in quantum many-body systems. Over the past several years, very fruitful interactions between researchers in quantum information, statistical mechanics, and condensed matter physics have already taken place and we expect this activity to continue and to intensify over the next years. This workshop will bring together leading researchers in QIT and QSM focusing on the following topics: gapped ground state phases; dynamics and equilibration; area laws; entanglement and many-body localization.

Joint CRM–Princeton Workshop: Critical Phenomena
Dates to be confirmed (first or second week of October 2018)
Organizers: Michael Aizenman (Princeton), David Brydges (British Columbia), Igor Klebanov (Princeton)
This workshop will be held at the Princeton Center for Theoretical Science (PCTS, Jadwin Hall, Princeton University).

The workshop is conceived as a meeting ground for “Mathematical Physicists”’ and “Theoretical Physicists” over topics that are of current research interest. Interesting and novel results are currently being developed through different tools and perspectives in the field. The organizers will strive to gather participants from different intellectual communities so that they start conversations about their respective tools and perspectives during the workshop.

Entropic Fluctuation Relations in Mathematics and Physics
October 29–November 2, 2018
Organizers: Vojkan Jakšić (McGill), Christian Maes (KU Leuven), Claude-Alain Pillet (Toulon)

Statistical mechanics starts as a theory of fluctuations. Its foundations are found in the theory of large deviations. Its development for non-equilibrium purposes requires a fine combination of probabilistic and analytical methods. Moreover mathematical precision is extremely helpful for understanding some subtle conceptual points related to the origin of irreversibility and to the precise properties of entropy (production) functionals. For genuine non-equilibrium situations, the workshop will clarify the mathematical properties of a linear response theory and of the canonical structure of static and dynamical fluctuations. Applications are found in the theory of complex systems, where the mathematical structure of a diversity of problems can be related to the structure of fluctuations and the response. Specific topics include:

  1. discussions on a rigorous fluctuation theory in perturbative order around non-equilibrium and its relationship to the response of the system;
  2. systematic mathematical corrections to local equilibrium in the derivation of hydrodynamic and kinetic equations.

Spectral Theory of Quasi-Periodic and Random Operators
November 12–16, 2018
Organizers: Jonathan Breuer (Hebrew University), David Damanik (Rice), Milivoje Lukic (Rice), Simone Warzel (TU München)

The spectral theory of complex systems has long been a topic of central interest in mathematical physics. In this context, the study of random and quasi-periodic Schrödinger operators reveals the consequences of complex long-range order or the lack thereof. While considerable progress has been made on some issues (e.g., conditions ensuring spectral and dynamical localization), our understanding of many other issues is partial at best. Examples include the issues of dynamical and spectral properties of operators with weak extensive disorder, multiparticle systems and asymptotic spectral properties of finite-volume truncations. One particularly fascinating aspect is the recently realized multifaceted connection to random matrix theory, a connection that is made both through analogy and certain conjectures of “universal” behaviour.

This workshop plans to bring together leading researchers in closely related fields of quasi-periodic spectral theory, random spectral theory, and many-body localization, in the hope that this interaction may shed light on the outstanding open problems in their respective fields and forge new directions of research.