Overview
Les Houches schools on mathematical physics have shaped generations of mathematical physicists. The proposed school is to continue this tradition, with a main focus on the mathematical analysis of quantum systems with very many or infinitely many degrees of freedom.
The last Les Houches school with this focus took place in 2010 under the title Quantum theory from small to large scales. It brought together many of the best doctoral students and postdoctoral researchers in the field and from all over the world and gave them a perspective beyond their specific thesis and research work. It further led to lasting research connections, friendships and a sense of community. Many of those that attended the 2010 school as PhD students and postdoctoral fellows have since been appointed as faculty at major research universities, including the first two of the present organizers. We are convinced that it is now time to provide the same service for the next generation of mathematical physicists.
It is also time to highlight some significant progress made in the last decade, and to indicate interesting open questions for the future. Indeed, quantum physics and its mathematical aspects will continue to draw the attention of physics and mathematics research for many years to come. – Among the highlights of mathematical quantum theory in the last decades are rigorous results on the large-scale dynamics of interacting quantum many-particle systems beyond the mean-field approximation. Moreover, old open questions have seen significant progress, such as those revolving around thermalization in closed and open systems, and new questions have been addressed, as for example the study of entanglement patterns in topological condensed matter systems. These topics continue to feed a fruitful bidirectional relationship between mathematics and theoretical physics, which we wish to encourage in the proposed school.
Topics in Quantum Physics
-
Topological Quantum Matter
The topological approach in condensed matter physics, pioneered in the 80s by Thouless and Laughlin, provided spectacular results, most prominently with the understanding of the quantization of the quantum Hall conductance. The last decade has seen a second era of dramatic progress with new mathematical techniques that have allowed to tackle interacting systems. Quantization of the Hall conductance has been understood for correlated electrons and much progress has been made towards the analysis of the fractional Hall effect, such as incompressibility estimates, or the emergence of anyons in two dimensions. A related research direction which has seen explosive growth is the study of ground state entanglement structures and the classification(s) of topological phases of matter. New mathematical tools have been introduced, some from analysis, others from algebraic topology, and with a field maturing, it is time to provide young researcher with a solid overview of the topic.
-
Open Quantum Systems
The topic open quantum system has witnessed significant progress on conceptual questions in the last decade or so and the surge of interest for Quantum Engineering has triggered new directions of research in this topic. The onset of (non-)equilibrium steady states for microscopic models of dynamical quantum systems coupled to one reservoir (or more), the decay of correlations validating the Born-Markov approximations and the entropy production accompagnying the process are some fundamental questions that were addressed successfully in certain cases. Questions triggered by Quantum Engineering considerations include the need to quantify the effect of the environment on quantum control strategies, in order to master decoherence phenomena and preserve key entanglement properties. Also, the need to include measurement protocols in quantum dynamical processes, to gather instantaneous information on the reference system through its environnement, induced new questions. These trends are evidenced by renewed efforts in the derivation and analysis of effective dynamics in various regimes, beyond the Born-Markovian approximation, and by the appearance of various discrete time models of quantum channels that gave rise to quantum trajectory models and non equilibrium thermodynamical considerations.
-
Integral Representations for Quantum Theory and the Renormalization Group
Starting with the Feynman-Kac formula, integral representations of quantum amplitudes have provided both intuition and calculational tools for quantum theory. For systems with many degrees of freedom, such as quantum many-body and quantum field theoretical systems, this involves integration over function or distribution spaces. The analysis, and often already the definition, of these functional integrals requires multiscale techniques, such as the renormalization group (RG). The RG has become ubiquitous in theoretical physics and its mathematically rigorous version has been used in the proofs of several important results in statistical mechanics, constructive quantum field theory, and many-body theory, in particular in condensed matter physics. For instance, the understanding of Fermi liquids has been put on a firm footing. Multiscale methods have also been applied to quantum transport and diffusion. Some highlights in the last decade were the application of the RG to topological phases in interacting fermion systems, and non-Fermi liquid behaviour in systems with nested or singular Fermi surfaces. The RG is also one of the approaches to the (still elusive) proof of Bose-Einstein condensation in the thermodynamic limit at sufficiently low temperatures. There, the Bogoliubov ansatz corresponds to mean-field theory, and the functional integral for the fluctuations is a complex oscillatory integral, which poses new challenges and has inspired new methods. Further methodical developments include wavelet methods in renormalization, tensor renormalization, and geometric flows.
-
Quantum Physics and Randomness
Besides the randomness intrinsic to measurement processes in quantum mechanics, many physical situations include some degree of randomness in their description and modelling. First, randomness is inherent in many physical systems, e.g. due to defects in crystals or irregularities of doping in alloys, a famous example being the Anderson Hamiltonian, which is the prototypical model for the metal-insulator transition. Second, Hamiltonians with randomness can be seen as generic, hence allow to test the stability of certain phenomena, especially for many degrees of freedom. (The ubiquitous use of random matrices in theoretical physics was motivated by the analysis of typical nuclear spectra.) Third, continuous measurement can be modelled by stochastic processes on Hilbert space. True to this spirit, approaches making use of probabilistic methods and/or random matrix theory have been successfully applied to random versions of entanglement structures in topological phases of matter and are at the root of recent progress in quantum information. Several effective random and deterministic dynamical models of open quantum systems have been addressed in the perspective of large deviations theory, and the analysis of transport properties of quantum systems in a disordered environment remains an important challenge of current mathematical physics. Finally, quantum trajectories have become a ripe subject for rigorous mathematical analysis.
Macroscopic Quantum Systems: Beyond Mean-Field Descriptions
A typical approach used in physics to understand the macroscopic behaviour of interacting quantum systems is by means of effective theories, which are heuristically understood in terms of averaging mechanisms taking place at a mesoscopic scale. Celebrated examples are the Gross-Pitaevskii and Bogoliubov theories for bosonic systems and the Hartree-Fock and Bardeen-Cooper-Schrieffer theories for fermionic systems. Remarkably, new analytic methods developed in the past fifteen years have allowed to investigate quantum systems beyond their mean-field description. In suitable scaling regimes, it has become possible to identify the range of applicability of effective theories and to describe fluctuations around the effective limiting behaviour. These advances have laid the groundwork for addressing new mathematical questions related to situations where quantum particles are strongly correlated, as in the case of large volume limits, positive temperature phases and non-dilute regimes. In this context the interplay among different methods and expertises has proved to be of fundamental importance to go beyond the current state of the art. We aim to introduce young researchers to the novel methods and ideas developed so far, and to challenge them to the long-term ambition of understanding the emergence of scaling laws and universality in quantum systems, as well as to investigate the occurrence of quantum phase transitions.
Entanglement, Entropy and Spacetime
Quantum entanglement leads to correlations of quantum particles that have no counterpart for classical particles. After its significance for quantum information and computation was realized, it has been studied extensively for quantum systems on finite-dimensional Hilbert spaces, especially systems of qbits. More recently, important results have been achieved for continuous quantum systems and quantum field theories, where interesting new questions arise, in particular with respect to spatial and causal structure, and the presence of horizons. For the school we plan to have a review of progress on entanglement entropy in quantum field theoretical systems on curved spacetimes, as well as lectures both by theoretical and experimental physicists on the simulation of curved space-times in cold quantum gases, and studies of entanglement in that context.
