# Niels Benedikter

2021–: Seminar "Itinerant Quantum Math Meetings"

This is a bi-weekly seminar organized together with Politecnico di Milano and the University of Insubria at Como. See quantumseminar.unimi.it/.

Summer 2022: 4EU+ summer school "Quantum Mechanics from Condensed Matter to Computing"

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.

December 20-22, 2021: 3-day workshop "Quantum before Christmas"

June 30, 2021: Manfred Salmhofer (University of Heidelberg)

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.

June 19+22, 2020 (Friday and Monday): Applications of Bogoliubov Theory

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.

May 3, 2017: Øresund Seminar in Lund

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

December 8, 2016: Øresund Seminar in Copenhagen

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

May 30-31, 2016: Øresund Seminar in Lund

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

16 November 2015: Øresund Seminar in Copenhagen

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

Below is an archive of talks in mathematical physics that I organized in Copenhagen in the Quantum Lunch format.

August 16, 2017: Chiara Boccato (University of Zurich)

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.

February 3, 2017: Alessandro Olgiati (SISSA Trieste)

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.

January 18, 2017: Oliver Matte (Aarhus University)

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.

December 7, 2016: Marcello Porta (University of Zurich)

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.

September 28, 2016: Giuseppe Genovese (University of Zurich)

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.

March 16, 2016: Chiara Boccato (University of Zurich)

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.

January 13, 2016: Ian Jauslin (University Roma Tre)

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.

15 June 2015: Anna Maltsev (University of Bristol)

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.