Quantum Theory on All Scales

Lectures

Long Courses

• Connected Components of the Space of Gapped Ground States

  Martin Fraas (UC Davis)

  These lectures discuss the classification of gapped quantum phases from an analytic viewpoint. The central question is when two systems can be connected by a continuous path of gapped Hamiltonians, placing the emphasis on the connected components of the space of gapped ground states rather than on topological invariants. I will outline the basic analytic framework for constructing such paths and illustrate it in selected examples. I will also discuss open questions and conjectures.

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• Approach to Equilibrium for translationally invariant lattice fermionic and quantum spin systems

  Vojkan Jaksic (Milano)

  The phenomena of the macroscopic world are irreversible: heat flows from hot to cold, entropy increases, and isolated systems approach equilibrium. Explaining these phenomena from the time-reversible laws of classical and quantum mechanics has been one of the central problems of statistical mechanics since the works of Maxwell and Boltzmann. The lectures will discuss this problem from the perspective of modern mathematical physics and focus on what has recently been called the “Minus First Law” of thermodynamics: an isolated macroscopic system, left to itself, evolves toward thermal equilibrium.
  The lectures will present a research program devoted to the mathematical theory of approach to equilibrium in infinitely extended classical and quantum systems. The main emphasis will be on translation-invariant quantum spin and fermionic systems. We will discuss both the kinematical thermodynamic formalism — Gibbs and KMS states, variational principles, entropy, phases, and constants of motion — and the corresponding dynamical formalism describing irreversible relaxation toward equilibrium.
  A central theme will be the relation between entropy increase, conservation laws, locality, and scattering. We will explain why finite systems cannot exhibit genuine thermalization, how infinitely extended systems allow irreversible behavior through escape of correlations to infinity, and how equilibration is related to restoration of locality in the large-time limit. Integrable and non-integrable quantum spin systems, including Ising, XY, and XYZ models, will serve as the main examples and sources of open problems.
  The lectures will include recent results, conjectures, and open problems concerning dynamical Gibbs variational principles, weak Gibbsianity, constants of motion, dephasing, and thermalization in many-body quantum systems.

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• Renormalization group and quantum transport

  Marcello Porta (SISSA)

  The goal of these lectures is to introduce a rigorous renormalization group approach to interacting fermionic systems on a lattice. A primary application is the analysis of transport properties in situations where no spectral gap is present. The motivation comes from the experimentally observed universality of transport in a variety of metallic and semimetallic systems, including graphene, Weyl semimetals, and one-dimensional quantum systems. We will discuss how renormalization group methods can be employed to compute transport coefficients and to place on rigorous grounds the predictions obtained from the analysis of the scaling limits of these models. A central role in the quantitative comparison between lattice models and their scaling limits, as well as in the proof of universality, is played by Ward identities, which follow from the conservation laws of both the lattice models and the effective field theories describing their scaling limits. I will discuss ongoing research directions and open problems.

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• Facets of quantum glasses

  Simone Warzel (TUM)

  Spin glasses are disordered magnetic systems that have played an important role in probability theory, statistical mechanics, and mathematical physics for several decades. Parisi’s theory of classical spin glasses like the one of Sherrington–Kirkpatrick (SK) is by now also mathematically fairly well understood. These lectures do not assume familiarity with that theory, but will import some of its results and techniques.
  Studying the fate of spin glass physics with respect to quantum effects induced by e.g. a transversal field has been a topic of continuing interest in the physics community ever since the 1990's. In the past 10 years, this subject received an additional boost due to its relevance as a testing ground for quantum algorithms or Fock space localization. One of the more striking quantum effects is the emergence of a phase transition at zero temperature for transversal field models.
  In comparison to the classical spin glass theory much less is known about its quantum counterpart, where random disorder competes with quantum fluctuations.
  These lectures will give an introduction and overview over of recent mathematical progress for transversal field models as the quantum SK (QSK). I will explain the probabilistic representation of the Gibbs state, which allow one to reformulate these quantum systems in terms of classical, continuous vector spin glasses. This then enables a Parisi variational formulation of their free energy. I will then survey mathematical techniques that enabled progress to understand the phase diagram of the quantum SK.

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• Macroscopic Quantum Systems: Mean-Field Description and Beyond

  Chiara Saffirio (U Basel & Freiburg)

  A standard approach in physics to understanding the macroscopic behavior of interacting quantum systems is through effective theories, which are heuristically justified by averaging mechanisms occurring at mesoscopic scales. Prominent examples include the Gross-Pitaevskii and Bogoliubov theories for bosonic systems, and the Hartree-Fock and Bardeen-Cooper-Schrieffer theories for fermionic systems.
  Remarkably, analytic methods developed over the past fifteen years have made it possible to investigate quantum many-body systems at the level of their mean-field description and beyond. In suitable scaling regimes, these methods allow one to rigorously establish the validity of effective theories, determine their range of applicability, and describe fluctuations around the effective limiting dynamics.
  In this course, we aim at introducing young researchers to these recent developments, with a particular focus on the Fermi Coulomb gas and its dynamical behavior in the mean-field and semiclassical regimes. Long-standing open problems, as well as new mathematical challenges arising in different scaling regimes, will also be discussed.

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• Quantum Information and Gravity

  Stefan Hollands (U Leipzig)

  These lectures provide an exposition to a circle of ideas at the interface between high energy physics and gravity that involve directly or indirectly ideas about entropy and quantum information. Emphasis is in particular on the connection to operator algebras. My exposition will be informal for the most part, assuming a knowledge of the standard formalism of quantum theory, basic quantum field theory, and basic notions related to entropy as would be taught in a standard course on statistical physics, but no prior knowledge of operator algebra theory. My aim is to introduce some methods and notions from operator algebras that can be useful also for someone with only a casual interest in the technicalities of the subject. In particular, I want to highlight recent advances related to Bekenstein-type bounds, the quantum null energy condition, and modular theory of von Neumann algebras.

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Short Courses

In addition to the long courses, the school also offers short courses:

• Mixing times of open quantum spin systems

  Angela Capel

  Open quantum spin systems provide a fundamental framework for understanding non-equilibrium dynamics in many-body quantum physics, with dissipation induced by the environment. In this course, we will study the notion of mixing times in such systems, focusing on how rapidly the dynamics of open quantum spin systems converge to their stationary states under Markovian evolution.
  We will introduce the mathematical formalism of quantum dynamical semigroups and Lindblad generators for spin systems on lattices, and we will study their sets of fixed points. The course will cover key techniques for bounding mixing times, including functional inequalities. When the fixed point is unique and corresponds to the Gibbs state of the Hamiltonian of the isolated system, we will show that there is a correspondence between static properties of decay of correlations on such a Gibbs state and dynamical properties of the relaxation of the semigroup towards the equilibrium.
  We will conclude the course with a review of the current state-of-the-art on mixing times of open quantum spin systems and with some applications of these results in various problems in quantum information theory and condensed matter theory.

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• Many-Body Localization

  Francois Huveneers

  This course explores Many-Body Localization (MBL). This is a phenomenon where isolated interacting quantum systems at positive temperature, meaning far from the ground state, fail to thermalize. We will investigate what MBL truly means, developing its definition from both conceptual and mathematical viewpoints. The lectures will focus on the rigorous mathematical foundations of the field alongside the core physical mechanisms that enable or prevent localization. From a physical point of view, we will discuss how rare regions can destabilize the MBL phase (avalanche theory). Finally, we will explore how this behavior extends to systems that are not MBL strictly speaking, but where the same phenomenology is still present.

• Some applications of functional integration in quantum mechanics

  Antti Knowles

  These lectures will present an overview of functional integration techniques in quantum mechanics. I will start with a review of Gaussian measures on infinite-dimensional spaces, in particular Brownian motion. I will then present three applications: the construction of simple Euclidean quantum field theories, a rigorous functional integral representation for a system of interacting bosons, and quantitative estimates for Green functions of Schrödinger operators.

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• Order in Disorder: Theory and Applications of Ergodic Quantum Processes

  Jeffrey Schenker

  This lecture series provides an exploration of the mathematical structures that emerge in quantum systems subjected to extensive disorder. To begin, we develop the theory of Ergodic Quantum Processes (EQP): sequences of quantum channels and states driven by underlying measure-preserving transformations. In this context, we present foundational convergence theorems demonstrating how decoherence drives disordered channel compositions toward a unique stochastic equilibrium. We then transition to the many-body setting to introduce Ergodic Matrix Product States (EMPS), providing a systematic generalization of Fannes-Nachtergaele-Werner results to the regime of extensive disorder. By utilizing the machinery of EQP and Banach bundles, we characterize finitely correlated states of spin chains with homogeneous disorder. Finally, we investigate the physical consequences of these theories, including the construction of frustration-free parent Hamiltonians and the asymptotic purification of quantum trajectories under stationary noise.

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• Bogoliubov theory for dilute quantum gases

  Benjamin Schlein

  In this mini-course, I am going to discuss some recent progress in the mathematical analysis of many-body quantum systems. I will present a recently developed rigorous version of Bogoliubov theory and I will explain how it can be used to study equilibrium and non-equilibrium properties of quantum gases.
  First, I will consider a system of N bosons confined in a volume of order one in the so-called mean-field regime. In this limit, I will derive precise estimates on the ground state energy and on the low-energy excitation spectrum of the Hamilton operator. Furthermore, I will discuss how to approximate the many-body time-evolution through an effective dynamics for the Bose-Einstein condensate and a quadratic evolution for its orthogonal excitations.
  Afterwards, I will switch to a more realistic but mathematically more challenging regime, known as the Gross-Pitaevskii regime, in which N particles interact through a repulsive potential with scattering length of the order 1/N. I will discuss some mathematical tools that have been developed in the last years to control correlations among particles, that are crucial in this limit, and to obtain precise estimates on the low-energy spectrum.
  Finally, I will explain how similar techniques can also be applied to infer precise estimates on the ground state energy of dilute Bose gases in the thermodynamic limit, at fixed but small density.

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• TBA

  Nalini Anantharaman

Colloquia

• Open quantum system dynamics and the mean force Gibbs state

  Janet Anders

  This colloquium discusses thermal equilibrium states of open quantum systems where a quantum or nanoscale system interacts with a large scale bath. When the system-bath coupling is not negligible, the equilibrium state of the system starts to deviate from the standard Gibbsian form. The mathematical characterisation of such environment corrected mean force Gibbs states is of much interest for strong coupling thermodynamics, where the impact of the environment can give rise to additional heat, work and entropy contributions not present in standard thermodynamics. Another key question is whether the dynamical steady state of the open system indeed matches the static mean force Gibbs state. Furthermore, for the example of the spin-boson model, we will discuss the quantum-classical transition of the mean force Gibbs state, and coupling regimes (from weak to ultrastrong) in the quantum and in the classical setting.

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• Events, States and Evolution in Quantum Mechanics

  Jürg Fröhlich

  In this talk I present a critical analysis of the notions of Events, States and (Time-)Evolution in Quantum Mechanics. With the purpose of understanding the quantum-mechanical time-evolution of isolated open systems I identify a fundamental mechanism of dissipation at work in systems of matter coupled to the quantized electromagnetic field (in a limit where the velocity of light tends to \infty). This mechanism is dubbed “Principle of Declining Potentialities.” When combined with a “State-Selection Postulate” it yields a precise law for the stochastic time-evolution of states of individual isolated open systems superseding Schroedinger evolution. This law furnishes a precise description of many phenomena, such as fluorescence of charged matter, and yields a solution of the infamous “Measurement Problem” in Quantum Mechanics.

• TBA

  Markus Oberthaler

• Public Lecture

  Nalini Anantharaman