@unpublished{7524,
abstract = {We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density $\rho$ and inverse temperature $\beta$ differs from the one of the non-interacting system by the correction term $4 \pi \rho^2 |\ln a^2 \rho|^{-1} (2 - [1 - \beta_{\mathrm{c}}/\beta]_+^2)$. Here $a$ is the scattering length of the interaction potential, $[\cdot]_+ = \max\{ 0, \cdot \}$ and $\beta_{\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. The result is valid in the dilute limit
$a^2\rho \ll 1$ and if $\beta \rho \gtrsim 1$.},
author = {Deuchert, Andreas and Mayer, Simon and Seiringer, Robert},
booktitle = {arXiv:1910.03372},
pages = {61},
publisher = {ArXiv},
title = {{The free energy of the two-dimensional dilute Bose gas. I. Lower bound}},
year = {2019},
}
@inproceedings{11,
abstract = {We report on a novel strategy to derive mean-field limits of quantum mechanical systems in which a large number of particles weakly couple to a second-quantized radiation field. The technique combines the method of counting and the coherent state approach to study the growth of the correlations among the particles and in the radiation field. As an instructional example, we derive the Schrödinger–Klein–Gordon system of equations from the Nelson model with ultraviolet cutoff and possibly massless scalar field. In particular, we prove the convergence of the reduced density matrices (of the nonrelativistic particles and the field bosons) associated with the exact time evolution to the projectors onto the solutions of the Schrödinger–Klein–Gordon equations in trace norm. Furthermore, we derive explicit bounds on the rate of convergence of the one-particle reduced density matrix of the nonrelativistic particles in Sobolev norm.},
author = {Leopold, Nikolai K and Pickl, Peter},
location = {Munich, Germany},
pages = {185 -- 214},
publisher = {Springer},
title = {{Mean-field limits of particles in interaction with quantised radiation fields}},
doi = {10.1007/978-3-030-01602-9_9},
volume = {270},
year = {2018},
}
@article{554,
abstract = {We analyse the canonical Bogoliubov free energy functional in three dimensions at low temperatures in the dilute limit. We prove existence of a first-order phase transition and, in the limit (Formula presented.), we determine the critical temperature to be (Formula presented.) to leading order. Here, (Formula presented.) is the critical temperature of the free Bose gas, ρ is the density of the gas and a is the scattering length of the pair-interaction potential V. We also prove asymptotic expansions for the free energy. In particular, we recover the Lee–Huang–Yang formula in the limit (Formula presented.).},
author = {Napiórkowski, Marcin M and Reuvers, Robin and Solovej, Jan},
issn = {00103616},
journal = {Communications in Mathematical Physics},
number = {1},
pages = {347--403},
publisher = {Springer},
title = {{The Bogoliubov free energy functional II: The dilute Limit}},
doi = {10.1007/s00220-017-3064-x},
volume = {360},
year = {2018},
}
@article{446,
abstract = {We prove that in Thomas–Fermi–Dirac–von Weizsäcker theory, a nucleus of charge Z > 0 can bind at most Z + C electrons, where C is a universal constant. This result is obtained through a comparison with Thomas-Fermi theory which, as a by-product, gives bounds on the screened nuclear potential and the radius of the minimizer. A key ingredient of the proof is a novel technique to control the particles in the exterior region, which also applies to the liquid drop model with a nuclear background potential.},
author = {Frank, Rupert and Phan Thanh, Nam and Van Den Bosch, Hanne},
journal = {Communications on Pure and Applied Mathematics},
number = {3},
pages = {577 -- 614},
publisher = {Wiley-Blackwell},
title = {{The ionization conjecture in Thomas–Fermi–Dirac–von Weizsäcker theory}},
doi = {10.1002/cpa.21717},
volume = {71},
year = {2018},
}
@article{295,
abstract = {We prove upper and lower bounds on the ground-state energy of the ideal two-dimensional anyon gas. Our bounds are extensive in the particle number, as for fermions, and linear in the statistics parameter (Formula presented.). The lower bounds extend to Lieb–Thirring inequalities for all anyons except bosons.},
author = {Lundholm, Douglas and Seiringer, Robert},
journal = {Letters in Mathematical Physics},
number = {11},
pages = {2523--2541},
publisher = {Springer},
title = {{Fermionic behavior of ideal anyons}},
doi = {10.1007/s11005-018-1091-y},
volume = {108},
year = {2018},
}
@article{400,
abstract = {We consider the two-dimensional BCS functional with a radial pair interaction. We show that the translational symmetry is not broken in a certain temperature interval below the critical temperature. In the case of vanishing angular momentum, our results carry over to the three-dimensional case.},
author = {Deuchert, Andreas and Geisinge, Alissa and Hainzl, Christian and Loss, Michael},
journal = {Annales Henri Poincare},
number = {5},
pages = {1507 -- 1527},
publisher = {Springer},
title = {{Persistence of translational symmetry in the BCS model with radial pair interaction}},
doi = {10.1007/s00023-018-0665-7},
volume = {19},
year = {2018},
}
@article{455,
abstract = {The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, and widely used in contexts such as superconductivity, nuclear physics, Bose–Einstein condensation and quantum chemistry. We reformulate the Dirac–Frenkel approximation principle in terms of reduced density matrices and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov–de Gennes and Hartree–Fock–Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the class of quasifree states. Furthermore, we prove well-posedness of the Bogoliubov–de Gennes equations in energy space and discuss conserved quantities},
author = {Benedikter, Niels P and Sok, Jérémy and Solovej, Jan},
journal = {Annales Henri Poincare},
number = {4},
pages = {1167 -- 1214},
publisher = {Birkhäuser},
title = {{The Dirac–Frenkel principle for reduced density matrices and the Bogoliubov–de Gennes equations}},
doi = {10.1007/s00023-018-0644-z},
volume = {19},
year = {2018},
}
@article{5983,
abstract = {We study a quantum impurity possessing both translational and internal rotational degrees of freedom interacting with a bosonic bath. Such a system corresponds to a “rotating polaron,” which can be used to model, e.g., a rotating molecule immersed in an ultracold Bose gas or superfluid helium. We derive the Hamiltonian of the rotating polaron and study its spectrum in the weak- and strong-coupling regimes using a combination of variational, diagrammatic, and mean-field approaches. We reveal how the coupling between linear and angular momenta affects stable quasiparticle states, and demonstrate that internal rotation leads to an enhanced self-localization in the translational degrees of freedom.},
author = {Yakaboylu, Enderalp and Midya, Bikashkali and Deuchert, Andreas and Leopold, Nikolai K and Lemeshko, Mikhail},
issn = {2469-9950},
journal = {Physical Review B},
number = {22},
publisher = {American Physical Society},
title = {{Theory of the rotating polaron: Spectrum and self-localization}},
doi = {10.1103/physrevb.98.224506},
volume = {98},
year = {2018},
}
@phdthesis{52,
abstract = {In this thesis we will discuss systems of point interacting fermions, their stability and other spectral properties. Whereas for bosons a point interacting system is always unstable this ques- tion is more subtle for a gas of two species of fermions. In particular the answer depends on the mass ratio between these two species. Most of this work will be focused on the N + M model which consists of two species of fermions with N, M particles respectively which interact via point interactions. We will introduce this model using a formal limit and discuss the N + 1 system in more detail. In particular, we will show that for mass ratios above a critical one, which does not depend on the particle number, the N + 1 system is stable. In the context of this model we will prove rigorous versions of Tan relations which relate various quantities of the point-interacting model. By restricting the N + 1 system to a box we define a finite density model with point in- teractions. In the context of this system we will discuss the energy change when introducing a point-interacting impurity into a system of non-interacting fermions. We will see that this change in energy is bounded independently of the particle number and in particular the bound only depends on the density and the scattering length. As another special case of the N + M model we will show stability of the 2 + 2 model for mass ratios in an interval around one. Further we will investigate a different model of point interactions which was discussed before in the literature and which is, contrary to the N + M model, not given by a limiting procedure but is based on a Dirichlet form. We will show that this system behaves trivially in the thermodynamic limit, i.e. the free energy per particle is the same as the one of the non-interacting system.},
author = {Moser, Thomas},
pages = {115},
publisher = {IST Austria},
title = {{Point interactions in systems of fermions}},
doi = {10.15479/AT:ISTA:th_1043},
year = {2018},
}
@article{180,
abstract = {In this paper we define and study the classical Uniform Electron Gas (UEG), a system of infinitely many electrons whose density is constant everywhere in space. The UEG is defined differently from Jellium, which has a positive constant background but no constraint on the density. We prove that the UEG arises in Density Functional Theory in the limit of a slowly varying density, minimizing the indirect Coulomb energy. We also construct the quantum UEG and compare it to the classical UEG at low density.},
author = {Lewi, Mathieu and Lieb, Élliott and Seiringer, Robert},
journal = {Journal de l'Ecole Polytechnique - Mathematiques},
pages = {79 -- 116},
publisher = {Ecole Polytechnique},
title = {{Statistical mechanics of the uniform electron gas}},
doi = {10.5802/jep.64},
volume = {5},
year = {2018},
}