Georgios
Athanasopoulos
Title: About
the exact solutions of the 2D classical and 1D
quantum Ising models
Abstract:
I will begin by
discussing the exact solution of the 2D classical
Ising model using the Kac--Ward method. In joint
work with Daniel Ueltschi, we have extended this
method to systems with negative coupling
constants. As a consequence, we observe that the
Cimasoni--Duminil-Copin--Li equation for the
critical temperature, originally proved for
general bi-periodic planar lattices with positive
couplings, remains valid in a broader setting.
Finally, I will explain how this method can be
used to rigorously compute the free energy of the
1D quantum Ising model.
Domenico Cafiero
Title: Homogenization regime
for singular perturbations of quantum
Hamiltonians
Abstract:
We study the homogenization limit
for singular electric and magnetic perturbations
of quantum Hamiltonians. Consider a
non-relativistic quantum particle in Rd, d = 2, 3,
subject to N point-like δ-interactions. In a
suitable scaling regime where the intensities of
the single interactions and the distance between
the points go to zero, we show that the
point-interactions Hamiltonian converges (in the
sense of Γ−convergence of quadratic forms) to a
Schrödinger operator with a regular electric
potential. Next, we discuss the case of a
non-relativistic quantum particle in presence of N
Aharonov-Bohm fluxes in R2. Under an appropriate
rescaling of each flux intensity, we establish the
Γ−convergence of the Friedrichs Hamiltonian.
Based on joint works with Michele Correggi and
Davide Fermi (Politecnico di Milano).
Fabrizio Caragiulo
Title: QHE and
quasi-periodic deformations
Abstract: Integer Quantum Hall Effect is a
prominent macroscopic quantum effet: for 2D
insulators, at very small temperature and very
high magnetic field, the transverse conductivity
is exactly quantized. Its experimental discovery
spurred whole new directions of research in
Mathematical Physics.
While th picture in translation
invariant and non-interacting models is well
understood, a challenge is to understand models
where both interactions and some kind of
disorder, breaking translation invariance, are
present.
I would like to present some
progress in the analysis of a non-interacting
model in the presence of quasi-periodic
disorder. The methods are based on Second
Quantization and Renormalization Group, which
are already well suited to also add interactions
in the future.
Michele
Fantechi
Title:
Mean-field limit dynamics
of quantum systems coupled to symmetric
environments
Abstract: The reduced density matrix
dynamics of a quantum system interacting by a
mean-field coupling with a symmetric
environment of N particles converges in the
limit of large N to unitary dynamics generated
by an effective time-dependent Hamiltonian.
The effective dynamics can describe properties
of the reduced system such as the protection
of intra-system entanglement, the appearence
of bound states, and non-Markovian behaviour
of the evolution when the environment initial
state is given by consistent families of
bosonic density matrices.
Florian Haberberger
Title: Free energy of the dilute 2D
Bose gas
Abstract: We study dilute, interacting
Bose gases in two dimensions at positive
temperature, aiming to rigorously prove Popov's
formula for the free energy in the thermodynamic
limit. This work extends recent results obtained
in three dimensions. Using Bogoliubov
transformations and the Jastrow ansatz, we
address the challenges posed by the singular
scaling in two dimensions.
This is joint work with Lukas
Junge.
Michal Jex
Title: Quantum systems at the
brink: critical potentials and finite
moments
Abstract: One of the crucial properties of
a quantum system is the existence of bound
states. The existence of eigenvalues below the
essential spectrum is well understood. They
exhibit exponential decay, and their existence
is linked to the energy gap. However, the
situation at the threshold is much more subtle.
There are two challenging problems for the
states at the threshold-their existence and
asymptotic behaviour.
We present necessary and
sufficient conditions for Schrödinger operators
to have a zero-energy bound state. Our sharp
criteria show that the existence and
non-existence of zero-energy ground states
depends strongly on the dimension and the
asymptotic behaviour of the potential. There is
a spectral phase transition with dimension four
being critical. Furthermore, we present
necessary and sufficient condition for the
threshold states to have finite k-th momentum.
Marcel
Majocha
Title: Ground state energy of a
dense Bose gas
Abstract: I’ll talk about the formula for
the ground state energy of a dense bose gas in
the thermodynamic limit and I’ll sketch the
proof of it. Additionally I’ll mention the most
interesting problem during derivation and how to
deal with it.
Diwakar
Naidu
Title: Momentum distribution
of a Fermi gas in mean-field regime
under Random Phase approximation
Abstract: I will talk about the momentum
distribution of an interacting Fermi gas on a 3D
torus in the mean field regime. The key tool for
deriving the distribution is a rigorous
bosonization method. I will start with the
construction of a natural trial state and then
show the implementation of the bosonization
procedure. Finally, I will sketch how we obtain
the momentum distribution in the mean field
approximation, along with the novel bootstrap
technique. The expression for the momentum
distribution contains the contributions of
collective excitations above the Fermi-surface
going beyond the precision of Hartree-Fock
theory. This result is an extension of the
previous result for the momentum distribution by
Benedikter-Lill.
References:
[BL24] Niels Benedikter and
Sascha Lill. Momentum distribution of a fermi
gas in the random phase approximation, 2024.
[BNP+21] Niels Benedikter, Phan
Th`anh Nam, Marcello Porta, Benjamin Schlein,
and Robert Seiringer. Correlation energy of a
weakly interacting fermi gas. Inventiones
mathematicae, 225(3):885–979, May 2021.
[CHN23] Martin Ravn
Christiansen, Christian Hainzl, and Phan Th`anh
Nam. The random phase approximation for
interacting fermi gases in the mean-field
regime, 2023.
Oskar Olander
Title: Exchangeability in quantum
spin systems
Abstract:Stochastic processes that are
permutation invariant can be decomposed as a
mixture of i.i.d. processes. This makes them easy to study. However, a
finite symmetric process does not always extend
to an infinite one. There is a quantum version
of this problem relevant for studying
spin system. Given a quantum system consisting
of finitely many symmetric subsystems, we would
like to easily tell if there exists an
infinite extension.
Jakob Oldenburg
Title: Third Order Upper Bound
for the Ground State Energy of the Dilute
Bose Gas
Abstract: We consider a dilute Bose gas in
the thermodynamic limit. We prove an upper bound
for the ground state energy per unit volume,
capturing the expected thirs order term, as
predicted by Wu, Hugenholtz-Pines and Sawada.
This is joint work with Morris
Brooks, Diane Saint Aubin and Benjamin Schlein.
Andreas Schaefer
Title: Dynamical Localization
of Quantum Walks on the hexagonal lattice
in the regime of strong disorder
Abstract: We study Quantum Walks (QW) on
the hexagonal lattice. The random unitary
operator describing the QW is given by the
composition of a shift and coin operator,
together with a random phase that depends on the
lattice site the Walker is in. We will prove
dynamical localization under the condition that
the coin matrix used to define the coin operator
is close enough to the permutation matrices that
correspond to the permutations 𝜎 = (1 2 3) or
𝜎 = (1 3 2) and induce full localization. Under
dynamical localization we understand the
property that the probability to move from a
lattice site x to another site y decreases on
average exponentially in the distance |x - y|,
independently of how many steps the Quantum
Walker may take. We will prove dynamical
localization by showing exponential decay of the
fractional moments of the resolvent. A special
emphasis is placed on the type of randomness
used.
Tobias Schmidt
Title: Enhanced binding in
one-particle Polaron models
Abstract: A quantum particle coupled to a
quantised field behaves as if it were
effectively heavier than its actual mass.
Enhanced binding refers to the phenomenon that
due to this effective mass of the particle, the
system admits a ground state, unlike the
uncoupled system. Feynman-Kac formulas allow for
a probabilistic interpretation of the problem:
one studies a pinned Brownian motion with a
Gibbs-type reweighting. The reweighting
incorporates two factors: a reward for paths
that stay close to the origin and a reward for
paths that look locally the same. The strength
of the second reward is determined by a system
parameter called the coupling strength. We are
interested in the behaviour of the path at t=0
(taking the pinning at -T and T) for T large.
Our main contribution is that for large enough
coupling, these distributions do not vanish in a
neighbourhood around 0. This can be interpreted
as the particle localising, which implies the
existence of a ground state in the quantum
system. Joint work with Volker Betz and Mark
Sellke.
François
Visconti
Title: Hilbert-Schmidt norm estimates
for fermionic reduced density matrices
Abstract:
It was conjectured by
Carlen-Lieb-Reuvers (2016,2018) that the
Hilbert-Schmidt norm of fermionic reduced density
matrices is maximised by Slater determinants.
Though this is easy to see for 1-particle reduced
density matrices, it remains an open problem for
higher order density matrices. Recently,
Christiansen (2024) proved that the
Hilbert-Schmidt norm of a 2-particle reduced
density matrix of an N-body fermionic state is
bounded by \sqrt{5/2}N
YSS Titles and Abstracts
, which is of the same order as that of Slater
determinants. In this talk, I will discuss a
generalisation of this result to higher order
density matrices, namely that the Hilbert-Schmidt
norm of a k-particle reduced density matrix is
bounded by C_kN^{k/2}, which also matches the
scaling behaviour of Slater determinants. These
bounds imply strong bounds on the von Neumann
entropy of fermionic reduced density matrices.