The classification of states of quantum lattice systems is a well -defined mathematical endeavour which started with the discovery of the quantum Hall effect. In this talk, I will discuss the topology of a simple cl ass, the so-called invertible states, which I will define. It is by definition a connected set, and we shall […]

We prove a thermodynamically stable Lieb-Robinson bound (LRB) for bosons with long-range interaction on lattices. The condition is that the initial state admits (i) uniformly bounded density from above and (ii) no particle in the region separating the initial supports of the observables entering the LRB. Furthermore, if the initial state has controlled density from […]

We introduce a new approach to justify mean-field limits for first- and second-order particle systems with singular interactions. It is based on a duality approach combined with the analysis of linearized dual correlations, and it allows to cover for the first time arbitrary square-integrable interaction forces at possibly vanishing temperature. This is joint work with […]

In this talk, I will try to motivate the subject of constructive quantum field theory which was born in the 70’s as an attempt to give rigorous constructions of quantum field theory models on Minkowski space and also describe scaling limits of spin systems. We will focus on some examples which give a taste of […]

The talk reviews the state of affairs in the mathematically rigorous foundations of the special-relativistic Vlasov-Maxwell equations. The progress is made possible by a recent formulation of a well-posed Lorentz co-variant initial value problem for the joint evolution of charged point particles and their electromagnetic Maxwell fields in a Bopp–Land ́e–Thomas–Podolsky (BLTP) vacuum.

In the setting of lattice gauge theories with finite (possibly non-Abelian) gauge groups at weak coupling, we prove exponential decay of correlations for a wide class of gauge invariant functions, which in particular includes arbitrary functions of Wilson loop observables. Based on joint work with Sky Cao.

In this talk – based on a joint work with J. Lampart, N. Leopold, and D. Mitrouskas – I will talk about the mean field limit of the renormalized Nelson model, in which a large number of bosonic particles is weakly coupled with a large number of coherent excitations of the scalar field. We prove […]

I will present a new approach to finding the asymptotic states of Nonlinear Wave Equations with general initial data. In particular, we show for a large class of equations, that all asymptotic states are linear combinations of free wave, localized parts (solitons, breathers..) and a possibility of self-similar solutions as well in some cases. These […]

The homogeneous electron gas (jellium) where electrons interact with each other and with a positive background charge is one of the simplest model system in condensed matter physics. Still, the precise determination of the zero temperature phase diagram remains challenging. In the talk I will review some recent progress from a computational perspective concerning the […]

We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose-Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables. We prove that in the limit N to infinity, […]

In this talk, we describe the kinetic equation for the Bogoliubov excitations of the Bose-Einstein Condensate. We find three collisional processes: One of them describes the 1↔2 interactions between the condensate and the excited atoms. The other two describe the 2↔2 and 1↔3 interactions between the excited atoms themselves. This is a joint work with […]

Propagation and generation of “chaos” is an important ingredient in rigorous control of applicability of kinetic theory, in general. Chaos can here be understood as sufficient statistical independence of random variables related to the “kinetic” obser- vables of the system. Cumulant hierarchy of these random variables thus often gives a way of controlling the evolution […]

We consider Markovian open quantum dynamics (MOQD) in continuous space. We show that, up to small-probability tails, the supports of quantum states evolving under such dynamics propagate with finite speed in any finite-energy subspace. More precisely, we prove that if the initial quantum state is localized in space, then any finite-energy part of the solution […]

In this talk I will describe the vacuum sector of the Weinberg-Salam (WS) model of electroweak forces. In the vacuum sector the WS model yields the U(2)-Yang-Mills-Higgs equations. We show that at large constant magnetic fields the translational symmetry of the equations is broken spontaneously. Namely, there are solutions, which in the plane orthogonal to […]

We consider two-dimensional unbounded magnetic Dirac operators, either defined on the whole plane, or with infinite mass boundary conditions on a half-plane. Our main results use techniques from elliptic PDEs and integral operators, while their topological consequences are presented as corollaries of some more general identities involving magnetic derivatives of local traces of fast decaying […]

Unraveling the origin of unconventional superconductivity is one of the driving forces behind quantum simulations with Fermions in optical lattices. In these strongly correlated materials, the necessary pairing of charge carriers is often assumed to be related to the interplay of antiferromagnetic correlations and dopant motion. Despite impressive recent progress in the numerical treatment of […]

We consider the infrared problem in translation-invariant Nelson-type models describing a single quantum mechanical particle linearly coupled to a field of scalar bosons at fixed total momentum. Physical examples include the non- and semi-relativistic Nelson models. If the bosons are massless, then the model is infrared divergent and the infimum of the spectrum is not […]

Recent progress in the development of quantum technologies has enabled the direct investigation of dynamics of increasingly complex quantum many-body systems. This motivates the study of the complexity of classical algorithms for this problem in order to benchmark quantum simulators and to delineate the regime of quantum advantage. Here we present classical algorithms for approximating […]

A mathematical understanding of the mechanism of metallic ferromagnetism still needs to be completed. In this talk, the following three fundamental theorems on metallic ferromagnetism will be first outlined: the Marshall-Lieb-Mattis theorem, the Lieb theorem, and the stability theorem of Lieb ferrimagnetism. Next, I will outline a mathematical framework within which these theorems can be […]

We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature T=0) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed […]

The classification of states of quantum lattice systems is a well -defined mathematical endeavour which started with the discovery of the quantum Hall effect. In this talk, I will discuss the topology of a simple cl ass, the so-called invertible states, which I will define. It is by definition a connected set, and we shall […]

We prove a thermodynamically stable Lieb-Robinson bound (LRB) for bosons with long-range interaction on lattices. The condition is that the initial state admits (i) uniformly bounded density from above and (ii) no particle in the region separating the initial supports of the observables entering the LRB. Furthermore, if the initial state has controlled density from […]

We introduce a new approach to justify mean-field limits for first- and second-order particle systems with singular interactions. It is based on a duality approach combined with the analysis of linearized dual correlations, and it allows to cover for the first time arbitrary square-integrable interaction forces at possibly vanishing temperature. This is joint work with […]

In this talk, I will try to motivate the subject of constructive quantum field theory which was born in the 70’s as an attempt to give rigorous constructions of quantum field theory models on Minkowski space and also describe scaling limits of spin systems. We will focus on some examples which give a taste of […]

The talk reviews the state of affairs in the mathematically rigorous foundations of the special-relativistic Vlasov-Maxwell equations. The progress is made possible by a recent formulation of a well-posed Lorentz co-variant initial value problem for the joint evolution of charged point particles and their electromagnetic Maxwell fields in a Bopp–Land ́e–Thomas–Podolsky (BLTP) vacuum.

In the setting of lattice gauge theories with finite (possibly non-Abelian) gauge groups at weak coupling, we prove exponential decay of correlations for a wide class of gauge invariant functions, which in particular includes arbitrary functions of Wilson loop observables. Based on joint work with Sky Cao.

In this talk – based on a joint work with J. Lampart, N. Leopold, and D. Mitrouskas – I will talk about the mean field limit of the renormalized Nelson model, in which a large number of bosonic particles is weakly coupled with a large number of coherent excitations of the scalar field. We prove […]

I will present a new approach to finding the asymptotic states of Nonlinear Wave Equations with general initial data. In particular, we show for a large class of equations, that all asymptotic states are linear combinations of free wave, localized parts (solitons, breathers..) and a possibility of self-similar solutions as well in some cases. These […]

The homogeneous electron gas (jellium) where electrons interact with each other and with a positive background charge is one of the simplest model system in condensed matter physics. Still, the precise determination of the zero temperature phase diagram remains challenging. In the talk I will review some recent progress from a computational perspective concerning the […]

We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose-Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables. We prove that in the limit N to infinity, […]

In this talk, we describe the kinetic equation for the Bogoliubov excitations of the Bose-Einstein Condensate. We find three collisional processes: One of them describes the 1↔2 interactions between the condensate and the excited atoms. The other two describe the 2↔2 and 1↔3 interactions between the excited atoms themselves. This is a joint work with […]

Propagation and generation of “chaos” is an important ingredient in rigorous control of applicability of kinetic theory, in general. Chaos can here be understood as sufficient statistical independence of random variables related to the “kinetic” obser- vables of the system. Cumulant hierarchy of these random variables thus often gives a way of controlling the evolution […]

We consider Markovian open quantum dynamics (MOQD) in continuous space. We show that, up to small-probability tails, the supports of quantum states evolving under such dynamics propagate with finite speed in any finite-energy subspace. More precisely, we prove that if the initial quantum state is localized in space, then any finite-energy part of the solution […]

In this talk I will describe the vacuum sector of the Weinberg-Salam (WS) model of electroweak forces. In the vacuum sector the WS model yields the U(2)-Yang-Mills-Higgs equations. We show that at large constant magnetic fields the translational symmetry of the equations is broken spontaneously. Namely, there are solutions, which in the plane orthogonal to […]

We consider two-dimensional unbounded magnetic Dirac operators, either defined on the whole plane, or with infinite mass boundary conditions on a half-plane. Our main results use techniques from elliptic PDEs and integral operators, while their topological consequences are presented as corollaries of some more general identities involving magnetic derivatives of local traces of fast decaying […]

Unraveling the origin of unconventional superconductivity is one of the driving forces behind quantum simulations with Fermions in optical lattices. In these strongly correlated materials, the necessary pairing of charge carriers is often assumed to be related to the interplay of antiferromagnetic correlations and dopant motion. Despite impressive recent progress in the numerical treatment of […]

We consider the infrared problem in translation-invariant Nelson-type models describing a single quantum mechanical particle linearly coupled to a field of scalar bosons at fixed total momentum. Physical examples include the non- and semi-relativistic Nelson models. If the bosons are massless, then the model is infrared divergent and the infimum of the spectrum is not […]

Recent progress in the development of quantum technologies has enabled the direct investigation of dynamics of increasingly complex quantum many-body systems. This motivates the study of the complexity of classical algorithms for this problem in order to benchmark quantum simulators and to delineate the regime of quantum advantage. Here we present classical algorithms for approximating […]

A mathematical understanding of the mechanism of metallic ferromagnetism still needs to be completed. In this talk, the following three fundamental theorems on metallic ferromagnetism will be first outlined: the Marshall-Lieb-Mattis theorem, the Lieb theorem, and the stability theorem of Lieb ferrimagnetism. Next, I will outline a mathematical framework within which these theorems can be […]

We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature T=0) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed […]