TY - JOUR AB - 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. AU - Lundholm, Douglas AU - Seiringer, Robert ID - 295 IS - 11 JF - Letters in Mathematical Physics TI - Fermionic behavior of ideal anyons VL - 108 ER - TY - JOUR AB - 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. AU - Deuchert, Andreas AU - Geisinge, Alissa AU - Hainzl, Christian AU - Loss, Michael ID - 400 IS - 5 JF - Annales Henri Poincare TI - Persistence of translational symmetry in the BCS model with radial pair interaction VL - 19 ER - TY - JOUR AB - 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. AU - Moser, Thomas AU - Seiringer, Robert ID - 154 IS - 3 JF - Mathematical Physics Analysis and Geometry SN - 13850172 TI - Stability of the 2+2 fermionic system with point interactions VL - 21 ER - TY - JOUR AB - 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 AU - Benedikter, Niels P AU - Sok, Jérémy AU - Solovej, Jan ID - 455 IS - 4 JF - Annales Henri Poincare TI - The Dirac–Frenkel principle for reduced density matrices and the Bogoliubov–de Gennes equations VL - 19 ER - TY - JOUR AB - 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. AU - Frank, Rupert AU - Phan Thanh, Nam AU - Van Den Bosch, Hanne ID - 446 IS - 3 JF - Communications on Pure and Applied Mathematics TI - The ionization conjecture in Thomas–Fermi–Dirac–von Weizsäcker theory VL - 71 ER - TY - JOUR AB - 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. AU - Yakaboylu, Enderalp AU - Midya, Bikashkali AU - Deuchert, Andreas AU - Leopold, Nikolai K AU - Lemeshko, Mikhail ID - 5983 IS - 22 JF - Physical Review B SN - 2469-9950 TI - Theory of the rotating polaron: Spectrum and self-localization VL - 98 ER - TY - JOUR AB - 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. AU - Napiórkowski, Marcin M AU - Reuvers, Robin AU - Solovej, Jan Philip ID - 6002 IS - 3 JF - Archive for Rational Mechanics and Analysis SN - 0003-9527 TI - The Bogoliubov free energy functional I: Existence of minimizers and phase diagram VL - 229 ER - TY - THES AB - 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. AU - Moser, Thomas ID - 52 SN - 2663-337X TI - Point interactions in systems of fermions ER - TY - JOUR AB - 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. AU - Lewi, Mathieu AU - Lieb, Élliott AU - Seiringer, Robert ID - 180 JF - Journal de l'Ecole Polytechnique - Mathematiques SN - 2429-7100 TI - Statistical mechanics of the uniform electron gas VL - 5 ER - TY - JOUR AB - 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. AU - Nam, Phan AU - Napiórkowski, Marcin M ID - 484 IS - 3 JF - Advances in Theoretical and Mathematical Physics SN - 10950761 TI - Bogoliubov correction to the mean-field dynamics of interacting bosons VL - 21 ER - TY - JOUR AB - We consider a 2D quantum system of N bosons in a trapping potential |x|s, interacting via a pair potential of the form N2β−1 w(Nβ x). We show that for all 0 < β < (s + 1)/(s + 2), the leading order behavior of ground states of the many-body system is described in the large N limit by the corresponding cubic nonlinear Schrödinger energy functional. Our result covers the focusing case (w < 0) where even the stability of the many-body system is not obvious. This answers an open question mentioned by X. Chen and J. Holmer for harmonic traps (s = 2). Together with the BBGKY hierarchy approach used by these authors, our result implies the convergence of the many-body quantum dynamics to the focusing NLS equation with harmonic trap for all 0 < β < 3/4. AU - Lewin, Mathieu AU - Nam, Phan AU - Rougerie, Nicolas ID - 632 IS - 6 JF - Proceedings of the American Mathematical Society TI - A note on 2D focusing many boson systems VL - 145 ER - TY - JOUR AB - 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. AU - Moser, Thomas AU - Seiringer, Robert ID - 1198 IS - 3 JF - Letters in Mathematical Physics SN - 03779017 TI - Triviality of a model of particles with point interactions in the thermodynamic limit VL - 107 ER - TY - JOUR AB - 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. AU - Li, Xiang AU - Seiringer, Robert AU - Lemeshko, Mikhail ID - 1120 IS - 3 JF - Physical Review A SN - 24699926 TI - Angular self-localization of impurities rotating in a bosonic bath VL - 95 ER - TY - JOUR AB - 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. AU - Nam, Phan AU - Van Den Bosch, Hanne ID - 1079 IS - 2 JF - Mathematical Physics, Analysis and Geometry SN - 13850172 TI - Nonexistence in Thomas Fermi-Dirac-von Weizsäcker theory with small nuclear charges VL - 20 ER - TY - JOUR AB - 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. AU - Moser, Thomas AU - Seiringer, Robert ID - 741 IS - 1 JF - Communications in Mathematical Physics SN - 00103616 TI - Stability of a fermionic N+1 particle system with point interactions VL - 356 ER - TY - JOUR AB - 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. AU - Nam, Phan AU - Napiórkowski, Marcin M ID - 739 IS - 5 JF - Journal de Mathématiques Pures et Appliquées SN - 00217824 TI - A note on the validity of Bogoliubov correction to mean field dynamics VL - 108 ER - TY - JOUR AB - Recently it was shown that molecules rotating in superfluid helium can be described in terms of the angulon quasiparticles (Phys. Rev. Lett. 118, 095301 (2017)). Here we demonstrate that in the experimentally realized regime the angulon can be seen as a point charge on a 2-sphere interacting with a gauge field of a non-abelian magnetic monopole. Unlike in several other settings, the gauge fields of the angulon problem emerge in the real coordinate space, as opposed to the momentum space or some effective parameter space. Furthermore, we find a topological transition associated with making the monopole abelian, which takes place in the vicinity of the previously reported angulon instabilities. These results pave the way for studying topological phenomena in experiments on molecules trapped in superfluid helium nanodroplets, as well as on other realizations of orbital impurity problems. AU - Yakaboylu, Enderalp AU - Deuchert, Andreas AU - Lemeshko, Mikhail ID - 997 IS - 23 JF - Physical Review Letters SN - 0031-9007 TI - Emergence of non-abelian magnetic monopoles in a quantum impurity problem VL - 119 ER - TY - JOUR AB - We consider a many-body system of fermionic atoms interacting via a local pair potential and subject to an external potential within the framework of Bardeen-Cooper-Schrieffer (BCS) theory. We measure the free energy of the whole sample with respect to the free energy of a reference state which allows us to define a BCS functional with boundary conditions at infinity. Our main result is a lower bound for this energy functional in terms of expressions that typically appear in Ginzburg-Landau functionals. AU - Deuchert, Andreas ID - 912 IS - 8 JF - Journal of Mathematical Physics SN - 00222488 TI - A lower bound for the BCS functional with boundary conditions at infinity VL - 58 ER - TY - JOUR AB - We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive nonlinear Schrödinger functional whose quartic term is proportional to the scattering length of the interparticle interaction potential. We propose a new derivation of this limit problem, with a method that bypasses some of the technical difficulties that previous derivations had to face. The new method is based on a combination of Dyson\'s lemma, the quantum de Finetti theorem and a second moment estimate for ground states of the effective Dyson Hamiltonian. It applies equally well to the case where magnetic fields or rotation are present. AU - Nam, Phan AU - Rougerie, Nicolas AU - Seiringer, Robert ID - 1143 IS - 2 JF - Analysis and PDE TI - Ground states of large bosonic systems: The gross Pitaevskii limit revisited VL - 9 ER - TY - JOUR AB - We consider the Bogolubov–Hartree–Fock functional for a fermionic many-body system with two-body interactions. For suitable interaction potentials that have a strong enough attractive tail in order to allow for two-body bound states, but are otherwise sufficiently repulsive to guarantee stability of the system, we show that in the low-density limit the ground state of this model consists of a Bose–Einstein condensate of fermion pairs. The latter can be described by means of the Gross–Pitaevskii energy functional. AU - Bräunlich, Gerhard AU - Hainzl, Christian AU - Seiringer, Robert ID - 1259 IS - 2 JF - Mathematical Physics, Analysis and Geometry TI - Bogolubov–Hartree–Fock theory for strongly interacting fermions in the low density limit VL - 19 ER -