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Speakers
University of Bari
Spectral properties of quantum channels and GKLS generators
[video] [slides]The evolution of an open quantum system is described by a quantum channel, i.e. a completely positive trace-preserving map. Under the Markovian approximation, the continuous dynamics of an open quantum system is given by a semigroup with a Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) generator. In this talk we discuss some spectral properties of both quantum channels and GKLS generators for finite dimensional systems.
IRMAR - Université de Rennes I
Derivation of the Newton-Maxwell equation of classical electrodynamics from non-relativistic QED
[video] [slides]Despite the conflicting relationship between classical and quantum electrodynamics, one can assert that the quantum and classical dynamics of non-relativistic extended charges interacting with their fields are consistent and coincide in the classical limit. In this talk, I will present a rigorous mathematical method that supports this claim. Joint work with Marco Falconi and Fumio Hiroshima.
Università di Trieste
Asymptotic properties of open quantum spin chains
[video] [slides]The reduced master equation describing the dissipative dynamics of an XX quantum spin chain of arbitrary length with the end spins coupled to two Bosonic thermal baths will be derived in the so-called global approach where one does not neglect the inter-spin interactions. Then, asymptotic properties as spin currents and entanglement will be studied with respect to the manifold of the open chain stationary states.
Università di Milano
Effective Dynamics of Interacting Fermions: Hartree-Fock Theory and Beyond
[video] [slides]Almost 100 years after the publication of the Schrödinger equation, the quantum many-body problem still remains a mathematical challenge. The derivation of effective evolution equations provides a tool for an approximate solution, deriving physical predictions in certain scaling limits. I will discuss the coupled mean-field and semiclassical scaling limit for high-density fermionic systems, and sketch the derivation of the time-dependent Hartree-Fock equation in this limit. I will also present more recent results that can be seen as a next-order correction to Hartree-Fock theory, based on bosonization.
Morris Brooks
Morris Brooks
IST Austria
Validity of Bogoliubov’s approximation for translation-invariant Bose gases
We will discuss Bogoliubov’s approximation for translation invariant Bose gases in the mean field regime, i.e. we show that the subleading correction to the Hartree energy is given by the ground state energy of the corresponding Bogoliubov operator. As an intermediate result we obtain the existence of approximate ground states, exhibiting complete Bose–Einstein condensation with respect to one of the Hartree minimizers.
Ikerbasque & BCAM/UPV Bilbao
From Short-Range to Mean-Field Models in Lattice Quantum Systems
[video] [slides]Realistic interparticle interactions of quantum many-body systems are largely seen as short-range. By contrast, mean-field models come from different approximations or Ansätze but are technically advantageous, by allowing explicit computations while capturing surprisingly well many physical phenomena. We will establish a precise mathematical relation between mean-field and short-range models, by using the long-range limit called the Kac limit.
Gran Sasso Science Institute
A new second order upper bound for the ground state energy of dilute Bose gases
[video] [slides]Back in 2009 H.-T. Yau and J. Yin established a second order upper bound for the ground state energy of dilute Bose gases in the thermodynamic limit, finally in agreement with the celebrated Lee-Huang-Yang formula. In this talk we describe recent ideas allowing to establish the same result for a larger class of potentials, and via a simpler proof. We also discuss the difficulties of extending the actual proofs to obtain a second order upper bound valid for non-integrable interactions, like hard-sphere potentials. Based on a joint work with G. Basti and B. Schlein.
Politecnico di Milano
Dirichlet forms on von Neumann algebras associated to thermal states
[video] [slides]We introduce a natural construction of Dirichlet forms associated to point eigenvalues of the modular Hamiltonian of a normal non tracial state of a von Neumann algebra, we then analyze their structure in terms of derivations and derive natural coercivity bounds from which superboundedness of the associated Markovian semigroups follows. We apply this setting to the construction of generalized quantum Ornstein-Uhlenbeck semigroups associated to Gibbs states on Type I von Neumann algebras. This is a joint work with B. Zegarlinski.
University of Zurich
Effective dynamics for weakly interacting Bose gases at positive temperature
[video] [slides]We study the dynamics of a trapped, weakly interacting Bose gas at positive temperature after the trapping potential has been switched off. The one-particle density matrix of the initial condition is to leading order, as the particle number goes to infinity, given by the one of the ideal gas, where the condensate wave function has been replaced by the minimizer of the Hartree energy functional. Our main result is a proof showing that this structure is preserved by the time-evolution of the Schrödinger equation in the following sense: The one-particle density matrix of the time-evolved state is to leading order given by the one of the ideal gas propagated with the free time evolution, where the condensate wave function has been replaced by the solution to the time-dependent Hartree equation. This is joint work with Marco Caporaletti and Benjamin Schlein.
LMU München
Dispersive Estimates for Nonlinear Schrödinger Equations with External Potentials
[video] [slides]We consider the long time dynamics of nonlinear Schrödinger equations with an external potential. More precisely, we look at Hartree type equations in three or higher dimensions with small initial data. We prove an optimal decay estimate, which is comparable to the decay of free solutions. Our proof relies on good control on a high Sobolev norm of the solution to estimate the terms in Duhamel's formula.
Università di Bari
KAM stability of quantum symmetries
[video] [slides]We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries, small perturbations can lead to large deviations over long times, while for robust symmetries, their expectation values remain close to their initial values for all times. This is in analogy with the celebrated Kolmogorov-Arnold-Moser theorem in classical mechanics.
Politecnico di Milano
Gaussian Quantum Markov Semigroups
[video] [slides]Gaussian (quasi-free) quantum Markov semigroups describe the quantum open system evolution of bosons under the markovian approximation. Moreover, they generalize bosonic quadratic Hamiltonians because their formal GKLS generator has a quadratic Hamiltonian in bosonic annihilation and creation operators. In this talk we outline the construction of the semigroup and some properties of the Markovian dynamics such as irreducibility, ergodicity, existence of invariant states and the structure of the decoherence-free subalgebra in which the reduced evolution is homomorphic.
"Sapienza" Università di Roma
Hamiltonian for a system of N bosons with regularized zero-range interactions
[video] [slides]We discuss a three-dimensional gas of N bosons interacting via a regularized contact interaction only with an impurity of different mass. This regularization is built adopting a proper three-body repulsion. The so-called Minlos-Faddeev extension of the free Hamiltonian that encodes such a regularization has proven to be self-adjoint and bounded from below. The typical Thomas collapse that occurs in these three-dimensional bosonic systems is therefore healed with the use of a three-body interaction.
LMU München
The dilute Fermi gas via Bogoliubov theory
[video] [slides]We consider N spin 1/2 fermions interacting with a positive and regular enough potential in three dimensions. We compute the ground state energy of the system in the dilute regime making use of the almost-bosonic nature of the low-energy excitations of the systems.
Università di Roma Tre
Spontaneous magnetization in the classical 3D Heisenberg model: asymptotic nature of the low temperature expansion
[video] [slides]In 1976, in one of the most influential papers in mathematical statistical mechanics, Frohlich-Simon-Spencer (FSS) proved the existenceof orientational long range order for the classical XY and Heisenberg models in three or more dimensions. In 1981, Bricmont-Fontaine-Lebowitz-Lieb-Spencer (BFLLS) extended the FSS result for the 3D rotator model, proving that the formal low temperature expansion for its magnetization is asymptotic; i.e., any truncation at finite order n of such a formal series is close to the actual magnetization up to an error o(T^n) at low temperatures. The generalization of this result to the Heisenberg model remained open since, for a difficulty to extend the BFLLS analysis to the case of non-abelian rotational symmetry. Recently, in collaboration with S. Ott, we succeeded in overcoming some of the technical limitations of the BFLLS proof, preventing them to apply their ideas to the Heisenberg model. In this talk I will report the statement of our main result on the asympotic nature of the low-temperature expansion for the spontaneous magnetization of the 3D Heisenberg model and discuss the main ideas of the proof. I will comment on the state-of-the-art and possible extensions to the quantum setting.
LMU München
Boundary superconductivity in the BCS model
[video] [slides]We consider a linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary.This questions the validity of the often employed phenomenological Ginzburg-Landau theory in the presence of boundaries.
McGill University
Phases of the number theoretic spin chain
[video] [slides]The number theoretic spin chain was introduced in 1993 by Andreas Knauf in connection with Riemann zeta function, and has been intensely studied since. In this talk I will describe certain novel structural properties of this model and using them resolve the open problem regarding the nature of its phases at the critical temperature. This talk is based on a joint work with T. Benoist, N. Cuneo, and C-A. Pillet.
LMU München
Convergence towards the Vlasov-Poisson equation from the N-fermionic Schrödinger equation
[video] [slides]We consider the quantum dynamics of N interacting fermions in the large N limit. The particles in the system interact with each other via repulsive interaction that is regularized Coulomb potential with a polynomial cutoff with respect to N. From the quantum system, we derive the Vlasov-Poisson system by simultaneously estimating the semiclassical and mean-field residues in terms of the Husimi measure.
Universität Basel
Mean-field dynamics for particles in interaction with quantized radiation fields
[video] [slides]In this talk I will outline results on the derivation of effective evolution equations for quantum mechanical systems in which a large number of particles weakly couple to a second quantized radiation field. The presentation includes the derivation of the Landau-Pekar equations and the Maxwell-Schrödinger equations in a mean-field limit of many bosons as well as the derivation of the Schrödinger-Klein-Gordon equations from the Nelson model with ultraviolet cutoff in a mean-field limit of many fermions.The talk is based on joint works with Marco Falconi, David Mitrouskas, Soren Petrat, Peter Pickl and Robert Seiringer.
Uppsala Universitet
Quantum statistics transmutation via magnetic flux attachment
[video] [slides]Quantum statistics refers to the ways that identical quantum particles may distribute themselves into different energy configurations or probability distributions: Bose-Einstein statistics obeyed by bosons, respectively Fermi-Dirac statistics obeyed by fermions, the latter being subject to Pauli's exclusion principle. In recent joint work with Gaultier Lambert and Nicolas Rougerie we consider a situation in two dimensions and a strong transverse magnetic field in which we may achieve statistics transmutation: by capturing magnetic flux, fermions turn into bosons and vice versa. Our method uses estimates for the characteristic polynomial of the Ginibre ensemble of random matrices. The result also sheds some light on the emergence of intermediate/fractional statistics in the fractional quantum Hall effect, and exotic quasi-particles called "anyons".
Memorial University of Newfoundland
Correlation decay and markovianity in open systems
[video] [slides]A finite quantum system S is coupled to a thermal, bosonic reservoir R. Initial SR states are possibly correlated, obtained by applying a quantum operation taken from a large class, to the uncoupled equilibrium state. We show that the full system-reservoir dynamics is given by a markovian term plus a correlation term, plus a remainder small in the coupling constant λ uniformly for all times t ≥ 0. The correlation term decays polynomially in time, at a speed independent of λ. After this, the markovian term becomes dominant, where the system evolves according to the completely positive, trace-preserving semigroup generated by the Davies generator, while the reservoir stays stationary in equilibrium. This shows that (a) after initial SR correlations decay, the SR dynamics enters a regime where both the Born and Markov approximations are valid, and (b) the reduced system dynamics is markovian for all times, even for correlated SR initial states.
University of Zurich
Second order upper bound for the ground state energy of hard-core bosons in the Gross-Pitaevskii regime
[video] [slides]I will present a result on the ground state energy of N bosons interacting through a hard-core potential. We take the radius of the interaction to be proportional to N^{-1} , and we consider particles confined to the three dimensional unit torus (Gross-Pitaevskii regime).We prove an upper bound for the ground state energy up to an error which vanishes as N grows. Our result matches the known expression for the energy in the case of integrable potentials, which depends universally on the scattering length of the interaction, thus confirming the prediction of Bogoliubov’s theory.Joint work with G. Basti (GSSI), S. Cenatiempo (GSSI), G. Pasqualetti (University of Zürich), and B. Schlein (University of Zürich).
"Sapienza" Università di Roma
Spin conductivity in the Quantum Spin Hall effect
[video] [slides]We consider some foundational issues in the quantum theory of spin transport, in the general case when the unperturbed Hamiltonian operator H_0 does not commute with the spin operator in view of Rashba interactions, as in the typical models for the Quantum Spin Hall effect. A gapped periodic one-particle Hamiltonian H_0 is perturbed by adding a constant electric field of intensity ε ≪ 1 in the j-th direction, and the linear response in terms of a S-current in the i-th direction is computed, where S is a generalized spin operator. We derive a general formula for the spin conductivity that covers both the choice of the conventional and of the proper spin current operator. We investigate the independence of the spin conductivity from the choice of the fundamental cell (Unit Cell Consistency), and we isolate a subclass of discrete periodic models where the conventional and the proper S-conductivity agree, thus showing that the controversy about the choice of the spin current operator is immaterial as far as models in this class are concerned. As a consequence of the general theory, we obtain that whenever the spin is (almost) conserved, the spin conductivity is (approximately) equal to the spin-Chern number. The method relies on the characterization of a non-equilibrium almost-stationary state (NEASS), which well approximates the physical state of the system (in the sense of space-adiabatic perturbation theory). This technique can be applied in a general framework, which includes both discrete and continuum models.The talk is based on a joint work with G. Marcelli and S. Teufel.
Università di Roma Tor Vergata
Stability of gapped quantum lattice systems under small perturbations
[video] [slides]We consider a family of quantum lattice systems that has attracted much interest amongst people studying topological phases of matter. Their Hamiltonians are perturbations, by interactions of short range, of a Hamiltonian consisting of on-site terms and with a strictly positive energy gap above its ground-state energy. We prove stability of the spectral gap, uniformly in the size of the lattice.In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the lattice. We can treat fermions and bosons on the same footing, and our technique does not face a large field problem, even though bosons are involved. Furthermore the method can be extended to complex Hamiltonians obtained by considering complex values of the coupling constant.Joint works with S. Del Vecchio, J. Fröhlich, and S. Rossi.
SISSA Trieste
Multi-Channel Luttinger Liquids at the Edge of Quantum Hall Systems
[video] [slides]We consider the edge transport properties of a generic class of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the large-scale behavior of the edge correlation functions is effectively described by the multi-channel Luttinger model. In particular, we prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. The proof is based on rigorous renormalization group methods, that allow to fully take into account the effect of backscattering at the edge. Universality arises as a consequence of the integrability of the emergent multi-channel Luttinger liquid combined with lattice Ward identities for the microscopic 2d theory. Joint work with Vieri Mastropietro.
Barbara Roos
Barbara Roos
IST Austria
Two-Particle Bound States at Interfaces and Corners
We study two interacting quantum particles forming a bound state in d-dimensional free space, and constrain the particles to half-space in k directions, with Neumann boundary conditions. First, we show that the particles stick to the corner where all boundary planes intersect. Second, we prove that the resulting Hamiltonian, after removing the free part of the kinetic energy, has only finitely many eigenvalues below the essential spectrum. This generalizes the work of Egger, Kerner and Pankrashkin (J. Spectr. Theory 10(4):1413--1444, 2020) to dimensions d>1 and was obtained in collaboration with Robert Seiringer.
Universität Basel
From the many-body fermionic dynamics to the Vlasov equation with singular interactions
[video] [slides]We will consider the many-body evolution of initially confined fermions interacting through a singular potential. In a joint mean-field and semiclassical scaling and using second quantization techniques, we will show that, for mixed states enjoying a semiclassical structure, the many-body dynamics can be approximated in Schatten norms by the time-dependent Hartree-Fock equation.Moreover, in the semiclassical limit, the dynamics is close to the one described by the Vlasov equation with singular interactions, thus providing a two-step approximation of the many-body evolution. If time permits we will discuss the extension of these results to the relativistic setting.Based on joint works with J. Chong and L. Laflèche, and with N. Leopold.
"Sapienza" Università di Roma
Three-body Hamiltonian with regularized zero-range interactions
[video] [slides]We consider the Hamiltonian for a system of three identical bosons in dimension three interacting via zero-range forces. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide.
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