Niels Benedikter

Fortnightly seminar organized with Politecnico di Milano and University of Insubria. See quantumseminar.unimi.it/.

Rigorous Renormalization Group Analysis of Collective Phenomena in Fermionic Quantum Systems
Interacting fermions are a prime example of complex systems in quantum physics: the microscopic Schrödinger equation gives rise to a baffling variety of collective behavior, ranging from collective plasmon excitations in Fermi liquids to the emergence of anyonic quasiparticles in the quantum Hall effect. Despite these effects having been spectacularly confirmed in experiment (e.g., the 1998 Nobel Prize for Laughlin, Störmer, and Tsui "for their discovery of a new form of quantum fluid with fractionally charged excitations"), their rigorous understanding continues to pose major challenges to mathematicians. The last few years however have seen formidable progress of the mathematical theory, such as the development of approximate bosonization methods and rigorous renormalization group (multiscale) methods. The school aims to introduce mathematicians and physicists to renormalization group methods, and initiate the development of synergies with Bogoliubov-type methods.

Organizing committee:
Niels Benedikter, Dipartimento di Matematica, Università degli Studi di Milano
Marco Falconi, Dipartimento di Matematica, Politecnico di Milano
Chiara Boccato, Dipartimento di Matematica, Università degli Studi di Milano

Schedule: Main courses by Vieri Mastropietro (Università degli Studi di Milano) and Marcello Porta (SISSA Trieste). Background will be provided in introductory lectures by Matteo Gallone (SISSA Trieste).

Details and Registration: feqs.lakecomoschool.org

Session title: Functional Analytic Methods in Quantum Many-Body Theory

Location: Università degli Studi di Palermo, Sicily (Italy)

Details: Invited speakers also need to register and pay the registration fee. We hope to reimburse speakers' expenses. Early bird registration until 18 April 2024. See also the Special Session Webpage and note the introductory and strategy meeting on Monday July 22 at 20:00.

Organizers: Niels Benedikter, Serena Cenatiempo (GSSI L'Aquila), Ian Jauslin (Rutgers, New Jersey)

Tentative topics:

• Entanglement
• Quantum Spin Systems

Online course: 7 March - 2 May 2024 (approximately one two-hour lecture every other week, in total 7 lectures)

Student projects: 2 May - 31 May 2024 (posters to be presented at the summer school)

Masterclass: June 24-28, 2024 summer school at University of Copenhagen, Denmark

Details: Registration (open until March 1, 2024). Under supervisor approval, indicate "yes"

Admission criteria: All students can participate, but funds for travel reimbursement are limited. If a selection becomes necessary, it will be based on grades and the Bachelor's thesis. Just register through the webpage of Copenhagen, if documents for the selection become necessary you will be contacted by e-mail.

Quantum and Dynamical Christmas 2023 - Workshop
The workshop is composed by two parts, one mainly devoted to dynamical systems that will occupy the days 19 and 20 December and one on quantum systems and many-body problems that will occupy the days 21 and 22 December. The main focus of the part on dynamical systems is on the stability/instability properties of finite and infinite dimensional Hamiltonian systems, but we are more generally interested in discussing recent advances in the theory of dynamical systems and in applications to some important problems coming from applications. The part on quantum systems will primarily focus on interacting Fermi and Bose gases, effective theories, and emergent phenomena and phase transitions. It is our aim to explore touching points of the fields and develop synergies between them.

Conference webpage with schedule and venues

Organizing committee:
Dario Bambusi, Niels Benedikter, Chiara Boccato, Beatrice Langella, Sascha Lill, Ngoc Nhi Nguyen, Laurent Niederman, Simone Paleari, Shulamit Terracina

This is a one-semester course in blended modality (starting in March 2022, about 2 hours every second week) for students from the 4EU+ European University Alliance followed by a one-week summer school in Copenhagen, June 13-18, 2022. Students of the University of Milan can be reimbursed for participation cost (but travel and accomodation have to be booked individually). 3 CFU may be obtained by students of mathematics and physics (laurea magistrale). For more information see https://qmath.ku.dk/teaching/4eu/.

We are currently witnessing the second quantum revolution and with it the advent of quantum technological devices for information processing purposes. Understanding these systems and their capabilities as well as developing robust algorithms for them, requires a fundamental understanding of complex quantum many-body systems as well as ways to characterize their properties efficiently. Within this course, we will help the participants to obtain proficiency in all of these aspects of quantum theory. Topics are:

• Complex quantum many-body systems:
  • Bosonization methods for the fermion gas
  • Renormalization group methods for fermions
• Identification and certification of quantum systems
  • Quantum state tomography
  • Self-testing

For each of these topics, we will provide introductory lectures to get students acquainted with the topics before entering into the in-person phase, where experts and practitioners of these fields will give further insight.

Workshop on mathematical physics from many-body quantum systems to normal forms. For more information, please go to the meeting webpage. Some slides are available for download.

Functional Integrals for Bose Ensembles

I discuss rigorously defined coherent-state functional integral representations for the partition function and correlation functions of many-boson systems, both for the canonical and the grand-canonical ensemble, and the relation of these representations to ensembles of interacting random walks. I will highlight a few essential differences between the canonical and grand-canonical ensemble, outline the proof of equivalence of different functional integral representations in the time-continuum limit, and discuss expansions for the correlation functions.

Lectures by: Marcin Napiórkowski (University of Warsaw), Benjamin Schlein (University of Zurich), Robert Seiringer (IST Austria), Jan Philip Solovej (University of Copenhagen). Virtual poster session for all participants.

For more information and the obligatory registration, please go to the meeting webpage.

Speakers: Ai-Hua Fan, Jens Wittsten, Jesper Lykke Jacobsen, Wojciech De Roeck.

On the website of the University of Lund you can find the full program with abstracts.

Lund organization team: Erik Wahlén, Copenhagen organization team: Niels Benedikter

Speakers: Marcello Porta (Zurich), Niels Martin Møller (Copenhagen), Jacob Christiansen (Lund), Marcus Carlsson (Lund).

You can get the full program with abstracts as a pdf.

Copenhagen organization team: Niels Benedikter, Lund organization team: Erik Wahlén

Speakers: Malcolm Brown (Cardiff University), Gian Michele Graf (ETH Zürich), Gerd Grubb (University of Copenhagen), Bernard Helffer (Université de Nantes), Elmar Schrohe (Leibniz Universität Hannover), Joachim Toft (Linnaeus University, Växjö-Kalmar), Mikael P. Sundqvist (Lund University).

On the website of the University of Lund you can find the full program with abstracts.

Lund organization team: Erik Wahlén, Copenhagen organization team: Niels Benedikter and Fabian Portman

On November 16, 2016 we had this fall's Øresundseminar (joint seminar with Lund in Sweden) with talks by Alessandro Giuliani, Matthias Christandl, Yacin Ameur and Mark Groves.

Copenhagen organization team: Niels Benedikter and Fabian Portman, Lund organization team: Erik Wahlén

The excitation spectrum of Bose gases interacting through singular potentials

We consider systems of $N$ bosons in a box with periodic boundary conditions, interacting through a repulsive two-body potential of the form $\kappa N^{3\beta-1} V(N^\beta x)$. For all $0 < \beta < 1$, and for sufficiently small coupling constant $\kappa > 0$, we establish the validity of Bogoliubov theory, identifying the ground state energy and the low-lying excitation spectrum up to errors that vanish in the limit of large $N$. Joint work with Christian Brennecke, Serena Cenatiempo and Benjamin Schlein.

On the dynamics of multi-component condensates: mixtures and spinors

I will present two results concerning the derivation of effective equations for multicomponent Bose-Einstein condensates; these can be easily tough of as systems with distinguishable components each of which is in a condensed state. The two cases we consider are mixture condensates, namely systems where two different families of bosons are present, and bosons with spin. In both cases the dynamics is ruled by a coupled system of non-linear Schrödinger equations, one for each component; the results hold equally in the mean-field or Gross-Pitaevskii regime. Time permitting, I will sketch the proof for mixtures in the mean-field case.

Feynman-Kac formulas for the ultra-violet renormalized Nelson model

We explain the structure and sketch the derivation of Feynman-Kac formulas representing the semi-group generated by the ultra-violet renormalized Nelson Hamiltonian. Furthermore, ignoring the Pauli principle, we present a new lower bound on the spectrum of the renormalized Nelson Hamiltonian and indicate how the Feynman-Kac formulas can be used to prove ergodicity of its semi-group.
The talk is based on recent joint work with Jacob Schach Møller.

Universality in interacting topological insulators

In this talk I will consider the charge transport properties of two-dimensional interacting fermionic systems, in the zero temperature and infinite volume limit. In the first part of the talk, I will discuss the stability of the integer quantum Hall effect for general weakly interacting gapped fermionic systems. In the second part of the talk, I will focus on the topological phase transitions of the Haldane-Hubbard model. I will present a rigorous construction of the interacting topological phase diagram, and prove the universality of conductivity at criticality.

Periodically driven impurity in the XX quantum spin chain

Consider a XX quantum spin chain with a periodic forcing, acting only on one site. Some interesting features of this system will be discussed, both on the mathematical and physical viewpoint. In particular we will show that, if the driving frequency is sufficiently high, the state of the system becomes asymptotically a periodic function of time.
A joint work with L. Corsi.

Quantum many-body fluctuations around nonlinear Schrödinger dynamics

We consider a system of bosons interacting through a two-body potential $N^{3\beta-1} V(N^\beta x)$, scaling with the number of particles $N$. When a very large number of particles is involved, it is difficult to calculate physical quantities from the time-dependent Schrödinger equation. However, it is possible to approximate the dynamics by a one-particle nonlinear Schrödinger equation; the reliability of this approximation is the problem we want to investigate. Our work focuses on fluctuations around the nonlinear Schrödinger dynamics. In a second-quantized setting, for $0 < \beta < 1$, we construct a unitary evolution with quadratic generator, which, in the limit of large $N$, allows us to approximate the fully evolved state in Fock space norm.
Joint work with Serena Cenatiempo and Benjamin Schlein.

A Pfaffian formula for monomer-dimer partition functions

In this talk, I will present a formula for the partition function of the monomer-dimer problem on an arbitrary planar graph, in terms of a Pfaffian, provided the monomers are restricted to the boundary of the graph.
This is joint work with A. Giuliani and E. H. Lieb.

Fluctuations in the half-heavy tailed Wigner ensemble

We will consider a Wigner matrix with entries whose cumulative distribution decays as $x^{-\alpha}$ with $2 < \alpha < 4$ for large $x$. We prove that the fluctuations of the linear spectral statistics for nice test functions have order $N^{-\alpha/4}$. The behavior of such fluctuations has been understood for both heavy-tailed matrices (i.e. $\alpha < 2$) and light-tailed matrices (i.e. $\alpha > 4$). Our result fills in the gap of understanding for $2 < \alpha < 4$.
This is based on joint work with Florent Benaych-Georges.