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 - TY - JOUR AB - We give a simplified proof of the nonexistence of large nuclei in the liquid drop model and provide an explicit bound. Our bound is within a factor of 2.3 of the conjectured value and seems to be the first quantitative result. AU - Frank, Rupert AU - Killip, Rowan AU - Nam, Phan ID - 1267 IS - 8 JF - Letters in Mathematical Physics TI - Nonexistence of large nuclei in the liquid drop model VL - 106 ER - TY - JOUR AB - We 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. AU - Giuliani, Alessandro AU - Seiringer, Robert ID - 1291 IS - 3 JF - Communications in Mathematical Physics TI - Periodic striped ground states in Ising models with competing interactions VL - 347 ER - TY - CONF AB - 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. AU - Könenberg, Martin AU - Moser, Thomas AU - Seiringer, Robert AU - Yngvason, Jakob ID - 1428 IS - 1 T2 - Journal of Physics: Conference Series TI - Superfluidity and BEC in a Model of Interacting Bosons in a Random Potential VL - 691 ER - TY - JOUR AB - 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. AU - Frank, Rupert AU - Hainzl, Christian AU - Schlein, Benjamin AU - Seiringer, Robert ID - 1422 IS - 7 JF - Letters in Mathematical Physics TI - Incompatibility of time-dependent Bogoliubov–de-Gennes and Ginzburg–Landau equations VL - 106 ER - TY - JOUR AB - We study the time evolution of a system of N spinless fermions in R3 which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schrödinger's equation with the time-dependent Hartree-Fock approximation, and we give an estimate for the accuracy of this approximation in terms of the kinetic energy of the system. This leads, in turn, to bounds in terms of the initial total energy of the system. AU - Bach, Volker AU - Breteaux, Sébastien AU - Petrat, Sören P AU - Pickl, Peter AU - Tzaneteas, Tim ID - 1436 IS - 1 JF - Journal de Mathématiques Pures et Appliquées TI - Kinetic energy estimates for the accuracy of the time-dependent Hartree-Fock approximation with Coulomb interaction VL - 105 ER - TY - JOUR AB - 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. AU - Seiringer, Robert AU - Warzel, Simone ID - 1478 IS - 3 JF - New Journal of Physics TI - Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas VL - 18 ER - TY - JOUR AB - We review recent results concerning the mathematical properties of the Bardeen-Cooper-Schrieffer (BCS) functional of superconductivity, which were obtained in a series of papers, partly in collaboration with R. Frank, E. Hamza, S. Naboko, and J. P. Solovej. Our discussion includes, in particular, an investigation of the critical temperature for a general class of interaction potentials, as well as a study of its dependence on external fields. We shall explain how the Ginzburg-Landau model can be derived from the BCS theory in a suitable parameter regime. AU - Hainzl, Christian AU - Seiringer, Robert ID - 1486 IS - 2 JF - Journal of Mathematical Physics TI - The Bardeen–Cooper–Schrieffer functional of superconductivity and its mathematical properties VL - 57 ER - TY - JOUR AB - We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schrödinger equation for fermionic many-particle systems in quantum mechanics. The method is an adaption of the method used in Pickl (Lett. Math. Phys. 97 (2) 151–164 2011) for bosonic systems to fermionic systems. It is based on a Gronwall type estimate for a suitable measure of distance between the microscopic solution and an antisymmetrized product state. We use this method to treat a new mean-field limit for fermions with long-range interactions in a large volume. Some of our results hold for singular attractive or repulsive interactions. We can also treat Coulomb interaction assuming either a mild singularity cutoff or certain regularity conditions on the solutions to the Hartree(-Fock) equations. In the considered limit, the kinetic and interaction energy are of the same order, while the average force is subleading. For some interactions, we prove that the Hartree(-Fock) dynamics is a more accurate approximation than a simpler dynamics that one would expect from the subleading force. With our method we also treat the mean-field limit coupled to a semiclassical limit, which was discussed in the literature before, and we recover some of the previous results. All results hold for initial data close (but not necessarily equal) to antisymmetrized product states and we always provide explicit rates of convergence. AU - Petrat, Sören P AU - Pickl, Peter ID - 1493 IS - 1 JF - Mathematical Physics, Analysis and Geometry TI - A new method and a new scaling for deriving fermionic mean-field dynamics VL - 19 ER - TY - JOUR AB - We study the ground state of a trapped Bose gas, starting from the full many-body Schrödinger Hamiltonian, and derive the non-linear Schrödinger energy functional in the limit of a large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive non-linear Schrödinger ground state. AU - Lewin, Mathieu AU - Nam, Phan AU - Rougerie, Nicolas ID - 1491 IS - 9 JF - Transactions of the American Mathematical Society TI - The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases VL - 368 ER - TY - JOUR AB - 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. AU - Nam, Phan AU - Napiórkowski, Marcin M AU - Solovej, Jan ID - 1545 IS - 11 JF - Journal of Functional Analysis TI - Diagonalization of bosonic quadratic Hamiltonians by Bogoliubov transformations VL - 270 ER - TY - JOUR AB - We consider the Bardeen–Cooper–Schrieffer free energy functional for particles interacting via a two-body potential on a microscopic scale and in the presence of weak external fields varying on a macroscopic scale. We study the influence of the external fields on the critical temperature. We show that in the limit where the ratio between the microscopic and macroscopic scale tends to zero, the next to leading order of the critical temperature is determined by the lowest eigenvalue of the linearization of the Ginzburg–Landau equation. AU - Frank, Rupert AU - Hainzl, Christian AU - Seiringer, Robert AU - Solovej, Jan ID - 1620 IS - 1 JF - Communications in Mathematical Physics TI - The external field dependence of the BCS critical temperature VL - 342 ER - TY - JOUR AB - We prove analogues of the Lieb–Thirring and Hardy–Lieb–Thirring inequalities for many-body quantum systems with fractional kinetic operators and homogeneous interaction potentials, where no anti-symmetry on the wave functions is assumed. These many-body inequalities imply interesting one-body interpolation inequalities, and we show that the corresponding one- and many-body inequalities are actually equivalent in certain cases. AU - Lundholm, Douglas AU - Nam, Phan AU - Portmann, Fabian ID - 1622 IS - 3 JF - Archive for Rational Mechanics and Analysis TI - Fractional Hardy–Lieb–Thirring and related Inequalities for interacting systems VL - 219 ER - TY - JOUR AB - We 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. AU - Correggi, Michele AU - Giuliani, Alessandro AU - Seiringer, Robert ID - 1572 IS - 1 JF - Communications in Mathematical Physics TI - Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet VL - 339 ER - TY - JOUR AB - We present a new, simpler proof of the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy in ℝ3. One of the main tools in our analysis is the quantum de Finetti theorem. Our uniqueness result is equivalent to the one established in the celebrated works of Erdos, Schlein, and Yau. AU - Chen, Thomas AU - Hainzl, Christian AU - Pavlović, Nataša AU - Seiringer, Robert ID - 1573 IS - 10 JF - Communications on Pure and Applied Mathematics TI - Unconditional uniqueness for the cubic gross pitaevskii hierarchy via quantum de finetti VL - 68 ER - TY - JOUR AB - Given a convex function (Formula presented.) and two hermitian matrices A and B, Lewin and Sabin study in (Lett Math Phys 104:691–705, 2014) the relative entropy defined by (Formula presented.). Among other things, they prove that the so-defined quantity is monotone if and only if (Formula presented.) is operator monotone. The monotonicity is then used to properly define (Formula presented.) for bounded self-adjoint operators acting on an infinite-dimensional Hilbert space by a limiting procedure. More precisely, for an increasing sequence of finite-dimensional projections (Formula presented.) with (Formula presented.) strongly, the limit (Formula presented.) is shown to exist and to be independent of the sequence of projections (Formula presented.). The question whether this sequence converges to its "obvious" limit, namely (Formula presented.), has been left open. We answer this question in principle affirmatively and show that (Formula presented.). If the operators A and B are regular enough, that is (A − B), (Formula presented.) and (Formula presented.) are trace-class, the identity (Formula presented.) holds. AU - Deuchert, Andreas AU - Hainzl, Christian AU - Seiringer, Robert ID - 1704 IS - 10 JF - Letters in Mathematical Physics TI - Note on a family of monotone quantum relative entropies VL - 105 ER - TY - JOUR AB - We study a double Cahn-Hilliard type functional related to the Gross-Pitaevskii energy of two-components Bose-Einstein condensates. In the case of large but same order intercomponent and intracomponent coupling strengths, we prove Γ-convergence to a perimeter minimisation functional with an inhomogeneous surface tension. We study the asymptotic behavior of the surface tension as the ratio between the intercomponent and intracomponent coupling strengths becomes very small or very large and obtain good agreement with the physical literature. We obtain as a consequence, symmetry breaking of the minimisers for the harmonic potential. AU - Goldman, Michael AU - Royo-Letelier, Jimena ID - 1807 IS - 3 JF - ESAIM - Control, Optimisation and Calculus of Variations TI - Sharp interface limit for two components Bose-Einstein condensates VL - 21 ER - TY - JOUR AB - We investigate the relation between Bose-Einstein condensation (BEC) and superfluidity in the ground state of a one-dimensional model of interacting bosons in a strong random potential. We prove rigorously that in a certain parameter regime the superfluid fraction can be arbitrarily small while complete BEC prevails. In another regime there is both complete BEC and complete superfluidity, despite the strong disorder AU - Könenberg, Martin AU - Moser, Thomas AU - Seiringer, Robert AU - Yngvason, Jakob ID - 1880 JF - New Journal of Physics TI - Superfluid behavior of a Bose-Einstein condensate in a random potential VL - 17 ER - TY - JOUR AB - We study the spectrum of a large system of N identical bosons interacting via a two-body potential with strength 1/N. In this mean-field regime, Bogoliubov's theory predicts that the spectrum of the N-particle Hamiltonian can be approximated by that of an effective quadratic Hamiltonian acting on Fock space, which describes the fluctuations around a condensed state. Recently, Bogoliubov's theory has been justified rigorously in the case that the low-energy eigenvectors of the N-particle Hamiltonian display complete condensation in the unique minimizer of the corresponding Hartree functional. In this paper, we shall justify Bogoliubov's theory for the high-energy part of the spectrum of the N-particle Hamiltonian corresponding to (non-linear) excited states of the Hartree functional. Moreover, we shall extend the existing results on the excitation spectrum to the case of non-uniqueness and/or degeneracy of the Hartree minimizer. In particular, the latter covers the case of rotating Bose gases, when the rotation speed is large enough to break the symmetry and to produce multiple quantized vortices in the Hartree minimizer. AU - Nam, Phan AU - Seiringer, Robert ID - 2085 IS - 2 JF - Archive for Rational Mechanics and Analysis TI - Collective excitations of Bose gases in the mean-field regime VL - 215 ER - TY - JOUR AB - We prove that nonlinear Gibbs measures can be obtained from the corresponding many-body, grand-canonical, quantum Gibbs states, in a mean-field limit where the temperature T diverges and the interaction strength behaves as 1/T. We proceed by characterizing the interacting Gibbs state as minimizing a functional counting the free-energy relatively to the non-interacting case. We then perform an infinite-dimensional analogue of phase-space semiclassical analysis, using fine properties of the quantum relative entropy, the link between quantum de Finetti measures and upper/lower symbols in a coherent state basis, as well as Berezin-Lieb type inequalities. Our results cover the measure built on the defocusing nonlinear Schrödinger functional on a finite interval, as well as smoother interactions in dimensions d 2. AU - Lewin, Mathieu AU - Phan Thanh, Nam AU - Rougerie, Nicolas ID - 473 JF - Journal de l'Ecole Polytechnique - Mathematiques TI - Derivation of nonlinear gibbs measures from many-body quantum mechanics VL - 2 ER - TY - CONF AB - We present a rigorous derivation of the BCS gap equation for superfluid fermionic gases with point interactions. Our starting point is the BCS energy functional, whose minimizer we investigate in the limit when the range of the interaction potential goes to zero. AU - Bräunlich, Gerhard AU - Hainzl, Christian AU - Seiringer, Robert ID - 1516 T2 - Proceedings of the QMath12 Conference TI - On the BCS gap equation for superfluid fermionic gases ER - TY - JOUR AB - We review recent progress towards a rigorous understanding of the Bogoliubov approximation for bosonic quantum many-body systems. We focus, in particular, on the excitation spectrum of a Bose gas in the mean-field (Hartree) limit. A list of open problems will be discussed at the end. AU - Seiringer, Robert ID - 1821 IS - 7 JF - Journal of Mathematical Physics TI - Bose gases, Bose-Einstein condensation, and the Bogoliubov approximation VL - 55 ER - TY - JOUR AU - Jakšić, Vojkan AU - Pillet, Claude AU - Seiringer, Robert ID - 1822 IS - 7 JF - Journal of Mathematical Physics TI - Introduction VL - 55 ER - TY - JOUR AB - We study translation-invariant quasi-free states for a system of fermions with two-particle interactions. The associated energy functional is similar to the BCS functional but also includes direct and exchange energies. We show that for suitable short-range interactions, these latter terms only lead to a renormalization of the chemical potential, with the usual properties of the BCS functional left unchanged. Our analysis thus represents a rigorous justification of part of the BCS approximation. We give bounds on the critical temperature below which the system displays superfluidity. AU - Bräunlich, Gerhard AU - Hainzl, Christian AU - Seiringer, Robert ID - 1889 IS - 7 JF - Reviews in Mathematical Physics TI - Translation-invariant quasi-free states for fermionic systems and the BCS approximation VL - 26 ER - TY - JOUR AB - We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation with a time-dependent potential and we show the existence of the wave operator in Schatten spaces. AU - Frank, Rupert AU - Lewin, Mathieu AU - Lieb, Élliott AU - Seiringer, Robert ID - 1904 IS - 7 JF - Journal of the European Mathematical Society TI - Strichartz inequality for orthonormal functions VL - 16 ER - TY - JOUR AB - As the nuclear charge Z is continuously decreased an N-electron atom undergoes a binding-unbinding transition. We investigate whether the electrons remain bound and whether the radius of the system stays finite as the critical value Zc is approached. Existence of a ground state at Zc is shown under the condition Zc < N-K, where K is the maximal number of electrons that can be removed at Zc without changing the energy. AU - Bellazzini, Jacopo AU - Frank, Rupert AU - Lieb, Élliott AU - Seiringer, Robert ID - 1918 IS - 1 JF - Reviews in Mathematical Physics TI - Existence of ground states for negative ions at the binding threshold VL - 26 ER - TY - JOUR AB - We consider Ising models in d = 2 and d = 3 dimensions with nearest neighbor ferromagnetic and long-range antiferromagnetic interactions, the latter decaying as (distance)-p, p > 2d, at large distances. If the strength J of the ferromagnetic interaction is larger than a critical value J c, then the ground state is homogeneous. It has been conjectured that when J is smaller than but close to J c, the ground state is periodic and striped, with stripes of constant width h = h(J), and h → ∞ as J → Jc -. (In d = 3 stripes mean slabs, not columns.) Here we rigorously prove that, if we normalize the energy in such a way that the energy of the homogeneous state is zero, then the ratio e 0(J)/e S(J) tends to 1 as J → Jc -, with e S(J) being the energy per site of the optimal periodic striped/slabbed state and e 0(J) the actual ground state energy per site of the system. Our proof comes with explicit bounds on the difference e 0(J)-e S(J) at small but positive J c-J, and also shows that in this parameter range the ground state is striped/slabbed in a certain sense: namely, if one looks at a randomly chosen window, of suitable size ℓ (very large compared to the optimal stripe size h(J)), one finds a striped/slabbed state with high probability. AU - Giuliani, Alessandro AU - Lieb, Élliott AU - Seiringer, Robert ID - 1935 JF - Communications in Mathematical Physics SN - 0010-3616 TI - Formation of stripes and slabs near the ferromagnetic transition VL - 331 ER - TY - JOUR AB - Spin-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. AU - Correggi, Michele AU - Giuliani, Alessandro AU - Seiringer, Robert ID - 2029 IS - 2 JF - EPL TI - Validity of spin-wave theory for the quantum Heisenberg model VL - 108 ER - TY - JOUR AB - We prove the existence of scattering states for the defocusing cubic Gross-Pitaevskii (GP) hierarchy in ℝ3. Moreover, we show that an exponential energy growth condition commonly used in the well-posedness theory of the GP hierarchy is, in a specific sense, necessary. In fact, we prove that without the latter, there exist initial data for the focusing cubic GP hierarchy for which instantaneous blowup occurs. AU - Chen, Thomas AU - Hainzl, Christian AU - Pavlović, Nataša AU - Seiringer, Robert ID - 2186 IS - 7 JF - Letters in Mathematical Physics TI - On the well-posedness and scattering for the Gross-Pitaevskii hierarchy via quantum de Finetti VL - 104 ER - TY - JOUR AB - We review recent progress towards a rigorous understanding of the excitation spectrum of bosonic quantum many-body systems. In particular, we explain how one can rigorously establish the predictions resulting from the Bogoliubov approximation in the mean field limit. The latter predicts that the spectrum is made up of elementary excitations, whose energy behaves linearly in the momentum for small momentum. This property is crucial for the superfluid behavior of the system. We also discuss a list of open problems in this field. AU - Seiringer, Robert ID - 10814 JF - Jahresbericht der Deutschen Mathematiker-Vereinigung KW - General Medicine SN - 0012-0456 TI - The excitation spectrum for Bose fluids with weak interactions VL - 116 ER - TY - CONF AB - Many questions concerning models in quantum mechanics require a detailed analysis of the spectrum of the corresponding Hamiltonian, a linear operator on a suitable Hilbert space. Of particular relevance for an understanding of the low-temperature properties of a system is the structure of the excitation spectrum, which is the part of the spectrum close to the spectral bottom. We present recent progress on this question for bosonic many-body quantum systems with weak two-body interactions. Such system are currently of great interest, due to their experimental realization in ultra-cold atomic gases. We investigate the accuracy of the Bogoliubov approximations, which predicts that the low-energy spectrum is made up of sums of elementary excitations, with linear dispersion law at low momentum. The latter property is crucial for the superfluid behavior the system. AU - Seiringer, Robert ID - 8044 SN - 9788961058063 T2 - Proceeding of the International Congress of Mathematicans TI - Structure of the excitation spectrum for many-body quantum systems VL - 3 ER -