211 research outputs found
The Bose gas in a box with Neumann boundary conditions
No full textWe consider a gas of bosonic particles confined in a box with Neumann
boundary conditions. We prove Bose-Einstein condensation in the
Gross-Pitaevskii regime, with an optimal bound on the condensate depletion. Our
lower bound for the ground state energy in the box implies (via Neumann
bracketing) a lower bound for the ground state energy of the Bose gas in the
thermodynamic limit.Comment: 45 page
Validity of spin-wave theory for the quantum Heisenberg model
No full textSpin-wave theory is a key ingredient in our comprehension of quantum spin systems,
and is used successfully for understanding a wide range of magnetic phenomena, including
magnon condensation and stability of patterns in dipolar systems. Nevertheless, several decades of
research failed to establish the validity of spin-wave theory rigorously, even for the simplest models
of quantum spins. A rigorous justification of the method for the three-dimensional quantum
Heisenberg ferromagnet at low temperatures is presented here. We derive sharp bounds on its free
energy by combining a bosonic formulation of the model introduced by Holstein and Primakoff
with probabilistic estimates and operator inequalities
Periodic Striped Ground States in Ising Models with Competing Interactions
No full textWe consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent p of the long range interaction is larger than d + 1, with d the space dimension, this happens for all values of J smaller than a critical value Jc(p), beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for p > 2d and J in a left neighborhood of Jc(p). In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes (d = 2) or slabs (d = 3), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity
Bose gases in the Gross-Pitaevskii limit: a survey of some rigorous results
No full textWe review some mathematical work on the Bose gas in the Gross-Pitaevskii
regime. We start with the classical results by Lieb, Seiringer and Yngvason on
the ground state energy and by Lieb and Seiringer on the existence of
Bose-Einstein condensation. Afterwards, we discuss some more recent progress,
based on a rigorous version of Bogoliubov theory.Comment: 26 page
Validity of the Spin-Wave Approximation for the Free Energy of the Heisenberg Ferromagnet
No full textWe consider the quantum ferromagnetic Heisenberg model in three dimensions,
for all spins S ≥ 1/2. We rigorously prove the validity of the spin-wave approximation
for the excitation spectrum, at the level of the first non-trivial contribution to the
free energy at low temperatures. Our proof comes with explicit, constructive upper and
lower bounds on the error term. It uses in an essential way the bosonic formulation of the
model in terms of the Holstein–Primakoff representation. In this language, the model
describes interacting bosons with a hard-core on-site repulsion and a nearest-neighbor
attraction. This attractive interaction makes the lower bound on the free energy particularly
tricky: the key idea there is to prove a differential inequality for the two-particle
density, which is thereby shown to be smaller than the probability density of a suitably
weighted two-particle random process on the lattice
Correlation Energy of a Weakly Interacting Fermi Gas with Large Interaction Potential
No full textRecently the leading order of the correlation energy of a Fermi gas in a
coupled mean-field and semiclassical scaling regime has been derived, under the
assumption of an interaction potential with a small norm and with compact
support in Fourier space. We generalize this result to large interaction
potentials, requiring only . Our
proof is based on approximate, collective bosonization in three dimensions.
Significant improvements compared to recent work include stronger bounds on
non-bosonizable terms and more efficient control on the bosonization of the
kinetic energy.Comment: 50 pages; v4: Lemma 7.2 simplified (log-corrections removed
Bosonization of Fermionic Many-Body Dynamics
No full textWe consider the quantum many-body evolution of a homogeneous Fermi gas in
three dimensions in the coupled semiclassical and mean-field scaling regime. We
study a class of initial data describing collective particle-hole pair
excitations on the Fermi ball. Using a rigorous version of approximate
bosonization, we prove that the many-body evolution can be approximated in Fock
space norm by a quasifree bosonic evolution of the collective particle-hole
excitations.Comment: 31 pages, 1 figur
Ubiquity of bound states for the strongly coupled polaron
No full textWe study the spectrum of the Fröhlich Hamiltonian for the polaron at fixed total momentum. We prove the existence of excited eigenvalues between the ground state energy and the essential spectrum at strong coupling. In fact, our main result shows that the number of excited energy bands diverges in the strong coupling limit. To prove this we derive upper bounds for the min-max values of the corresponding fiber Hamiltonians and compare them with the bottom of the essential spectrum, a lower bound on which was recently obtained by Brooks and Seiringer (Comm. Math. Phys. 404:1 (2023), 287–337). The upper bounds are given in terms of the ground state energy band shifted by momentum-independent excitation energies determined by an effective Hamiltonian of Bogoliubov type
CIME School on Quantum Many Body Systems
No full textThe book is based on the lectures given at the CIME school "Quantum many body systems" held in the summer of 2010. It provides a tutorial introduction to recent advances in the mathematics of interacting systems, written by four leading experts in the field: V. Rivasseau illustrates the applications of constructive Quantum Field Theory to 2D interacting electrons and their relation to quantum gravity; R. Seiringer describes a proof of Bose-Einstein condensation in the Gross-Pitaevski limit and explains the effects of rotating traps and the emergence of lattices of quantized vortices; J.-P. Solovej gives an introduction to the theory of quantum Coulomb systems and to the functional analytic methods used to prove their thermodynamic stability; finally, T. Spencer explains the supersymmetric approach to Anderson localization and its relation to the theory of random matrices. All the lectures are characterized by their mathematical rigor combined with physical insights
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