@article{6649, abstract = {While Hartree–Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree–Fock state given by plane waves and introduce collective particle–hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann–Brueckner–type upper bound to the ground state energy. Our result justifies the random-phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials. }, author = {Benedikter, Niels P and Nam, Phan Thành and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, pages = {2097–2150}, publisher = {Springer Nature}, title = {{Optimal upper bound for the correlation energy of a Fermi gas in the mean-field regime}}, doi = {10.1007/s00220-019-03505-5}, volume = {374}, year = {2020}, } @article{80, abstract = {We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross–Pitaevskii (GP) limit where the trap frequency ω, the temperature T, and the particle number N are related by N∼ (T/ ω) 3→ ∞ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.}, author = {Deuchert, Andreas and Seiringer, Robert and Yngvason, Jakob}, journal = {Communications in Mathematical Physics}, number = {2}, pages = {723--776}, publisher = {Springer}, title = {{Bose–Einstein condensation in a dilute, trapped gas at positive temperature}}, doi = {10.1007/s00220-018-3239-0}, volume = {368}, year = {2019}, } @article{5856, abstract = {We give a bound on the ground-state energy of a system of N non-interacting fermions in a three-dimensional cubic box interacting with an impurity particle via point interactions. We show that the change in energy compared to the system in the absence of the impurity is bounded in terms of the gas density and the scattering length of the interaction, independently of N. Our bound holds as long as the ratio of the mass of the impurity to the one of the gas particles is larger than a critical value m∗ ∗≈ 0.36 , which is the same regime for which we recently showed stability of the system.}, author = {Moser, Thomas and Seiringer, Robert}, issn = {14240637}, journal = {Annales Henri Poincare}, number = {4}, pages = {1325–1365}, publisher = {Springer}, title = {{Energy contribution of a point-interacting impurity in a Fermi gas}}, doi = {10.1007/s00023-018-00757-0}, volume = {20}, year = {2019}, } @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{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}, issn = {2270-518X}, 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}, } @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}, issn = {2663-337X}, pages = {115}, publisher = {Institute of Science and Technology Austria}, title = {{Point interactions in systems of fermions}}, doi = {10.15479/AT:ISTA:th_1043}, 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{6002, abstract = {The Bogoliubov free energy functional is analysed. The functional serves as a model of a translation-invariant Bose gas at positive temperature. We prove the existence of minimizers in the case of repulsive interactions given by a sufficiently regular two-body potential. Furthermore, we prove the existence of a phase transition in this model and provide its phase diagram.}, author = {Napiórkowski, Marcin M and Reuvers, Robin and Solovej, Jan Philip}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {3}, pages = {1037--1090}, publisher = {Springer Nature}, title = {{The Bogoliubov free energy functional I: Existence of minimizers and phase diagram}}, doi = {10.1007/s00205-018-1232-6}, volume = {229}, year = {2018}, } @article{154, abstract = {We give a lower bound on the ground state energy of a system of two fermions of one species interacting with two fermions of another species via point interactions. We show that there is a critical mass ratio m2 ≈ 0.58 such that the system is stable, i.e., the energy is bounded from below, for m∈[m2,m2−1]. So far it was not known whether this 2 + 2 system exhibits a stable region at all or whether the formation of four-body bound states causes an unbounded spectrum for all mass ratios, similar to the Thomas effect. Our result gives further evidence for the stability of the more general N + M system.}, author = {Moser, Thomas and Seiringer, Robert}, issn = {15729656}, journal = {Mathematical Physics Analysis and Geometry}, number = {3}, publisher = {Springer}, title = {{Stability of the 2+2 fermionic system with point interactions}}, doi = {10.1007/s11040-018-9275-3}, volume = {21}, year = {2018}, } @article{399, abstract = {Following an earlier calculation in 3D, we calculate the 2D critical temperature of a dilute, translation-invariant Bose gas using a variational formulation of the Bogoliubov approximation introduced by Critchley and Solomon in 1976. This provides the first analytical calculation of the Kosterlitz-Thouless transition temperature that includes the constant in the logarithm.}, author = {Napiórkowski, Marcin M and Reuvers, Robin and Solovej, Jan}, journal = {EPL}, number = {1}, publisher = {IOP Publishing Ltd.}, title = {{Calculation of the critical temperature of a dilute Bose gas in the Bogoliubov approximation}}, doi = {10.1209/0295-5075/121/10007}, volume = {121}, year = {2018}, } @article{1079, abstract = {We study the ionization problem in the Thomas-Fermi-Dirac-von Weizsäcker theory for atoms and molecules. We prove the nonexistence of minimizers for the energy functional when the number of electrons is large and the total nuclear charge is small. This nonexistence result also applies to external potentials decaying faster than the Coulomb potential. In the case of arbitrary nuclear charges, we obtain the nonexistence of stable minimizers and radial minimizers.}, author = {Nam, Phan and Van Den Bosch, Hanne}, issn = {13850172}, journal = {Mathematical Physics, Analysis and Geometry}, number = {2}, publisher = {Springer}, title = {{Nonexistence in Thomas Fermi-Dirac-von Weizsäcker theory with small nuclear charges}}, doi = {10.1007/s11040-017-9238-0}, volume = {20}, year = {2017}, } @article{1120, abstract = {The existence of a self-localization transition in the polaron problem has been under an active debate ever since Landau suggested it 83 years ago. Here we reveal the self-localization transition for the rotational analogue of the polaron -- the angulon quasiparticle. We show that, unlike for the polarons, self-localization of angulons occurs at finite impurity-bath coupling already at the mean-field level. The transition is accompanied by the spherical-symmetry breaking of the angulon ground state and a discontinuity in the first derivative of the ground-state energy. Moreover, the type of the symmetry breaking is dictated by the symmetry of the microscopic impurity-bath interaction, which leads to a number of distinct self-localized states. The predicted effects can potentially be addressed in experiments on cold molecules trapped in superfluid helium droplets and ultracold quantum gases, as well as on electronic excitations in solids and Bose-Einstein condensates. }, author = {Li, Xiang and Seiringer, Robert and Lemeshko, Mikhail}, issn = {24699926}, journal = {Physical Review A}, number = {3}, publisher = {American Physical Society}, title = {{Angular self-localization of impurities rotating in a bosonic bath}}, doi = {10.1103/PhysRevA.95.033608}, volume = {95}, year = {2017}, } @article{739, abstract = {We study the norm approximation to the Schrödinger dynamics of N bosons in with an interaction potential of the form . Assuming that in the initial state the particles outside of the condensate form a quasi-free state with finite kinetic energy, we show that in the large N limit, the fluctuations around the condensate can be effectively described using Bogoliubov approximation for all . The range of β is expected to be optimal for this large class of initial states.}, author = {Nam, Phan and Napiórkowski, Marcin M}, issn = {00217824}, journal = {Journal de Mathématiques Pures et Appliquées}, number = {5}, pages = {662 -- 688}, publisher = {Elsevier}, title = {{A note on the validity of Bogoliubov correction to mean field dynamics}}, doi = {10.1016/j.matpur.2017.05.013}, volume = {108}, year = {2017}, } @article{741, abstract = {We prove that a system of N fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical m*. The value of m* is independent of N and turns out to be less than 1. This fact has important implications for the stability of the unitary Fermi gas. We also characterize the domain of the Hamiltonian of this model, and establish the validity of the Tan relations for all wave functions in the domain.}, author = {Moser, Thomas and Seiringer, Robert}, issn = {00103616}, journal = {Communications in Mathematical Physics}, number = {1}, pages = {329 -- 355}, publisher = {Springer}, title = {{Stability of a fermionic N+1 particle system with point interactions}}, doi = {10.1007/s00220-017-2980-0}, volume = {356}, year = {2017}, } @article{484, abstract = {We consider the dynamics of a large quantum system of N identical bosons in 3D interacting via a two-body potential of the form N3β-1w(Nβ(x - y)). For fixed 0 = β < 1/3 and large N, we obtain a norm approximation to the many-body evolution in the Nparticle Hilbert space. The leading order behaviour of the dynamics is determined by Hartree theory while the second order is given by Bogoliubov theory.}, author = {Nam, Phan and Napiórkowski, Marcin M}, issn = {10950761}, journal = {Advances in Theoretical and Mathematical Physics}, number = {3}, pages = {683 -- 738}, publisher = {International Press}, title = {{Bogoliubov correction to the mean-field dynamics of interacting bosons}}, doi = {10.4310/ATMP.2017.v21.n3.a4}, volume = {21}, year = {2017}, } @article{1198, abstract = {We consider a model of fermions interacting via point interactions, defined via a certain weighted Dirichlet form. While for two particles the interaction corresponds to infinite scattering length, the presence of further particles effectively decreases the interaction strength. We show that the model becomes trivial in the thermodynamic limit, in the sense that the free energy density at any given particle density and temperature agrees with the corresponding expression for non-interacting particles.}, author = {Moser, Thomas and Seiringer, Robert}, issn = {03779017}, journal = {Letters in Mathematical Physics}, number = {3}, pages = { 533 -- 552}, publisher = {Springer}, title = {{Triviality of a model of particles with point interactions in the thermodynamic limit}}, doi = {10.1007/s11005-016-0915-x}, volume = {107}, year = {2017}, } @article{1478, abstract = {We consider the Tonks-Girardeau gas subject to a random external potential. If the disorder is such that the underlying one-particle Hamiltonian displays localization (which is known to be generically the case), we show that there is exponential decay of correlations in the many-body eigenstates. Moreover, there is no Bose-Einstein condensation and no superfluidity, even at zero temperature.}, author = {Seiringer, Robert and Warzel, Simone}, journal = {New Journal of Physics}, number = {3}, publisher = {IOP Publishing Ltd.}, title = {{Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas}}, doi = {10.1088/1367-2630/18/3/035002}, volume = {18}, year = {2016}, } @article{1545, abstract = {We provide general conditions for which bosonic quadratic Hamiltonians on Fock spaces can be diagonalized by Bogoliubov transformations. Our results cover the case when quantum systems have infinite degrees of freedom and the associated one-body kinetic and paring operators are unbounded. Our sufficient conditions are optimal in the sense that they become necessary when the relevant one-body operators commute.}, author = {Nam, Phan and Napiórkowski, Marcin M and Solovej, Jan}, journal = {Journal of Functional Analysis}, number = {11}, pages = {4340 -- 4368}, publisher = {Academic Press}, title = {{Diagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations}}, doi = {10.1016/j.jfa.2015.12.007}, volume = {270}, year = {2016}, } @article{1422, abstract = {We study the time-dependent Bogoliubov–de-Gennes equations for generic translation-invariant fermionic many-body systems. For initial states that are close to thermal equilibrium states at temperatures near the critical temperature, we show that the magnitude of the order parameter stays approximately constant in time and, in particular, does not follow a time-dependent Ginzburg–Landau equation, which is often employed as a phenomenological description and predicts a decay of the order parameter in time. The full non-linear structure of the equations is necessary to understand this behavior.}, author = {Frank, Rupert and Hainzl, Christian and Schlein, Benjamin and Seiringer, Robert}, journal = {Letters in Mathematical Physics}, number = {7}, pages = {913 -- 923}, publisher = {Springer}, title = {{Incompatibility of time-dependent Bogoliubov–de-Gennes and Ginzburg–Landau equations}}, doi = {10.1007/s11005-016-0847-5}, volume = {106}, year = {2016}, } @inproceedings{1428, abstract = {We report on a mathematically rigorous analysis of the superfluid properties of a Bose- Einstein condensate in the many-body ground state of a one-dimensional model of interacting bosons in a random potential.}, author = {Könenberg, Martin and Moser, Thomas and Seiringer, Robert and Yngvason, Jakob}, booktitle = {Journal of Physics: Conference Series}, location = {Shanghai, China}, number = {1}, publisher = {IOP Publishing Ltd.}, title = {{Superfluidity and BEC in a Model of Interacting Bosons in a Random Potential}}, doi = {10.1088/1742-6596/691/1/012016}, volume = {691}, year = {2016}, }