TY - JOUR AB - We study the Fröhlich polaron model in R3, and establish the subleading term in the strong coupling asymptotics of its ground state energy, corresponding to the quantum corrections to the classical energy determined by the Pekar approximation. AU - Brooks, Morris AU - Seiringer, Robert ID - 14441 JF - Communications in Mathematical Physics SN - 0010-3616 TI - The Fröhlich Polaron at strong coupling: Part I - The quantum correction to the classical energy VL - 404 ER - TY - JOUR AB - We consider the large polaron described by the Fröhlich Hamiltonian and study its energy-momentum relation defined as the lowest possible energy as a function of the total momentum. Using a suitable family of trial states, we derive an optimal parabolic upper bound for the energy-momentum relation in the limit of strong coupling. The upper bound consists of a momentum independent term that agrees with the predicted two-term expansion for the ground state energy of the strongly coupled polaron at rest and a term that is quadratic in the momentum with coefficient given by the inverse of twice the classical effective mass introduced by Landau and Pekar. AU - Mitrouskas, David Johannes AU - Mysliwy, Krzysztof AU - Seiringer, Robert ID - 13178 JF - Forum of Mathematics TI - Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron VL - 11 ER - TY - DATA AB - We introduce a stochastic cellular automaton as a model for culture and border formation. The model can be conceptualized as a game where the expansion rate of cultures is quantified in terms of their area and perimeter in such a way that approximately round cultures get a competitive advantage. We first analyse the model with periodic boundary conditions, where we study how the model can end up in a fixed state, i.e. freezes. Then we implement the model on the European geography with mountains and rivers. We see how the model reproduces some qualitative features of European culture formation, namely that rivers and mountains are more frequently borders between cultures, mountainous regions tend to have higher cultural diversity and the central European plain has less clear cultural borders. AU - Klausen, Frederik Ravn AU - Lauritsen, Asbjørn Bækgaard ID - 12869 TI - Research data for: A stochastic cellular automaton model of culture formation ER - TY - JOUR AB - We introduce a stochastic cellular automaton as a model for culture and border formation. The model can be conceptualized as a game where the expansion rate of cultures is quantified in terms of their area and perimeter in such a way that approximately geometrically round cultures get a competitive advantage. We first analyze the model with periodic boundary conditions, where we study how the model can end up in a fixed state, i.e., freezes. Then we implement the model on the European geography with mountains and rivers. We see how the model reproduces some qualitative features of European culture formation, namely, that rivers and mountains are more frequently borders between cultures, mountainous regions tend to have higher cultural diversity, and the central European plain has less clear cultural borders. AU - Klausen, Frederik Ravn AU - Lauritsen, Asbjørn Bækgaard ID - 12890 IS - 5 JF - Physical Review E SN - 2470-0045 TI - Stochastic cellular automaton model of culture formation VL - 108 ER - TY - JOUR AB - This paper establishes new connections between many-body quantum systems, One-body Reduced Density Matrices Functional Theory (1RDMFT) and Optimal Transport (OT), by interpreting the problem of computing the ground-state energy of a finite-dimensional composite quantum system at positive temperature as a non-commutative entropy regularized Optimal Transport problem. We develop a new approach to fully characterize the dual-primal solutions in such non-commutative setting. The mathematical formalism is particularly relevant in quantum chemistry: numerical realizations of the many-electron ground-state energy can be computed via a non-commutative version of Sinkhorn algorithm. Our approach allows to prove convergence and robustness of this algorithm, which, to our best knowledge, were unknown even in the two marginal case. Our methods are based on a priori estimates in the dual problem, which we believe to be of independent interest. Finally, the above results are extended in 1RDMFT setting, where bosonic or fermionic symmetry conditions are enforced on the problem. AU - Feliciangeli, Dario AU - Gerolin, Augusto AU - Portinale, Lorenzo ID - 12911 IS - 4 JF - Journal of Functional Analysis SN - 0022-1236 TI - A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature VL - 285 ER - TY - JOUR AB - It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature Ξ and the critical temperature Tc is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density and high density. The goal of this short note is to extend the universal behavior to lower dimensions d=1,2 and give an exemplary proof in the weak coupling limit. AU - Henheik, Sven Joscha AU - Lauritsen, Asbjørn Bækgaard AU - Roos, Barbara ID - 14542 JF - Reviews in Mathematical Physics SN - 0129-055X TI - Universality in low-dimensional BCS theory ER - TY - JOUR AB - We consider a class of polaron models, including the Fröhlich model, at zero total momentum, and show that at sufficiently weak coupling there are no excited eigenvalues below the essential spectrum. AU - Seiringer, Robert ID - 14662 IS - 3 JF - Journal of Spectral Theory SN - 1664-039X TI - Absence of excited eigenvalues for Fröhlich type polaron models at weak coupling VL - 13 ER - TY - JOUR AB - Recently 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 |⋅|V^∈ℓ1(Z3). 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. AU - Benedikter, Niels P AU - Porta, Marcello AU - Schlein, Benjamin AU - Seiringer, Robert ID - 13225 IS - 4 JF - Archive for Rational Mechanics and Analysis SN - 0003-9527 TI - Correlation energy of a weakly interacting Fermi gas with large interaction potential VL - 247 ER - TY - JOUR AB - We consider the ground state and the low-energy excited states of a system of N identical bosons with interactions in the mean-field scaling regime. For the ground state, we derive a weak Edgeworth expansion for the fluctuations of bounded one-body operators, which yields corrections to a central limit theorem to any order in 1/N−−√. For suitable excited states, we show that the limiting distribution is a polynomial times a normal distribution, and that higher-order corrections are given by an Edgeworth-type expansion. AU - Bossmann, Lea AU - Petrat, Sören P ID - 13226 IS - 4 JF - Letters in Mathematical Physics SN - 0377-9017 TI - Weak Edgeworth expansion for the mean-field Bose gas VL - 113 ER - TY - JOUR AB - For the Fröhlich model of the large polaron, we prove that the ground state energy as a function of the total momentum has a unique global minimum at momentum zero. This implies the non-existence of a ground state of the translation invariant Fröhlich Hamiltonian and thus excludes the possibility of a localization transition at finite coupling. AU - Lampart, Jonas AU - Mitrouskas, David Johannes AU - Mysliwy, Krzysztof ID - 14192 IS - 3 JF - Mathematical Physics, Analysis and Geometry KW - Geometry and Topology KW - Mathematical Physics SN - 1385-0172 TI - On the global minimum of the energy–momentum relation for the polaron VL - 26 ER - TY - JOUR AB - We consider N trapped bosons in the mean-field limit with coupling constant λN = 1/(N − 1). The ground state of such systems exhibits Bose–Einstein condensation. We prove that the probability of finding ℓ particles outside the condensate wave function decays exponentially in ℓ. AU - Mitrouskas, David Johannes AU - Pickl, Peter ID - 14715 IS - 12 JF - Journal of Mathematical Physics SN - 0022-2488 TI - Exponential decay of the number of excitations in the weakly interacting Bose gas VL - 64 ER - TY - JOUR AB - Abstract We 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. AU - Mitrouskas, David Johannes AU - Seiringer, Robert ID - 14854 IS - 4 JF - Pure and Applied Analysis KW - General Medicine SN - 2578-5885 TI - Ubiquity of bound states for the strongly coupled polaron VL - 5 ER - TY - JOUR AB - In [10] Nam proved a Lieb–Thirring Inequality for the kinetic energy of a fermionic quantum system, with almost optimal (semi-classical) constant and a gradient correction term. We present a stronger version of this inequality, with a much simplified proof. As a corollary we obtain a simple proof of the original Lieb–Thirring inequality. AU - Seiringer, Robert AU - Solovej, Jan Philip ID - 14254 IS - 10 JF - Journal of Functional Analysis SN - 0022-1236 TI - A simple approach to Lieb-Thirring type inequalities VL - 285 ER - TY - CHAP AB - In this chapter we first review the Levy–Lieb functional, which gives the lowest kinetic and interaction energy that can be reached with all possible quantum states having a given density. We discuss two possible convex generalizations of this functional, corresponding to using mixed canonical and grand-canonical states, respectively. We present some recent works about the local density approximation, in which the functionals get replaced by purely local functionals constructed using the uniform electron gas energy per unit volume. We then review the known upper and lower bounds on the Levy–Lieb functionals. We start with the kinetic energy alone, then turn to the classical interaction alone, before we are able to put everything together. A later section is devoted to the Hohenberg–Kohn theorem and the role of many-body unique continuation in its proof. AU - Lewin, Mathieu AU - Lieb, Elliott H. AU - Seiringer, Robert ED - Cances, Eric ED - Friesecke, Gero ID - 14992 SN - 3005-0286 T2 - Density Functional Theory TI - Universal Functionals in Density Functional Theory ER - TY - JOUR AB - Ongoing development of quantum simulators allows for a progressively finer degree of control of quantum many-body systems. This motivates the development of efficient approaches to facilitate the control of such systems and enable the preparation of nontrivial quantum states. Here we formulate an approach to control quantum systems based on matrix product states (MPSs). We compare counterdiabatic and leakage minimization approaches to the so-called local steering problem that consists in finding the best value of the control parameters for generating a unitary evolution of the specific MPS in a given direction. In order to benchmark the different approaches, we apply them to the generalization of the PXP model known to exhibit coherent quantum dynamics due to quantum many-body scars. We find that the leakage-based approach generally outperforms the counterdiabatic framework and use it to construct a Floquet model with quantum scars. We perform the first steps towards global trajectory optimization and demonstrate entanglement steering capabilities in the generalized PXP model. Finally, we apply our leakage minimization approach to construct quantum scars in the periodically driven nonintegrable Ising model. AU - Ljubotina, Marko AU - Roos, Barbara AU - Abanin, Dmitry A. AU - Serbyn, Maksym ID - 12276 IS - 3 JF - PRX Quantum KW - General Medicine TI - Optimal steering of matrix product states and quantum many-body scars VL - 3 ER - TY - JOUR AB - We consider a gas of N bosons with interactions in the mean-field scaling regime. We review the proof of an asymptotic expansion of its low-energy spectrum, eigenstates, and dynamics, which provides corrections to Bogoliubov theory to all orders in 1/ N. This is based on joint works with Petrat, Pickl, Seiringer, and Soffer. In addition, we derive a full asymptotic expansion of the ground state one-body reduced density matrix. AU - Bossmann, Lea ID - 11783 IS - 6 JF - Journal of Mathematical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0022-2488 TI - Low-energy spectrum and dynamics of the weakly interacting Bose gas VL - 63 ER - TY - JOUR AB - We study the many-body dynamics of an initially factorized bosonic wave function in the mean-field regime. We prove large deviation estimates for the fluctuations around the condensate. We derive an upper bound extending a recent result to more general interactions. Furthermore, we derive a new lower bound which agrees with the upper bound in leading order. AU - Rademacher, Simone Anna Elvira AU - Seiringer, Robert ID - 11917 JF - Journal of Statistical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0022-4715 TI - Large deviation estimates for weakly interacting bosons VL - 188 ER - TY - JOUR AB - We consider the many-body time evolution of weakly interacting bosons in the mean field regime for initial coherent states. We show that bounded k-particle operators, corresponding to dependent random variables, satisfy both a law of large numbers and a central limit theorem. AU - Rademacher, Simone Anna Elvira ID - 12083 IS - 8 JF - Journal of Mathematical Physics SN - 0022-2488 TI - Dependent random variables in quantum dynamics VL - 63 ER - TY - THES AB - The scope of this thesis is to study quantum systems exhibiting a continuous symmetry that is broken on the level of the corresponding effective theory. In particular we are going to investigate translation-invariant Bose gases in the mean field limit, effectively described by the Hartree functional, and the Fröhlich Polaron in the regime of strong coupling, effectively described by the Pekar functional. The latter is a model describing the interaction between a charged particle and the optical modes of a polar crystal. Regarding the former, we assume in addition that the particles in the gas are unconfined, and typically we will consider particles that are subject to an attractive interaction. In both cases the ground state energy of the Hamiltonian is not a proper eigenvalue due to the underlying translation-invariance, while on the contrary there exists a whole invariant orbit of minimizers for the corresponding effective functionals. Both, the absence of proper eigenstates and the broken symmetry of the effective theory, make the study significantly more involved and it is the content of this thesis to develop a frameworks which allows for a systematic way to circumvent these issues. It is a well-established result that the ground state energy of Bose gases in the mean field limit, as well as the ground state energy of the Fröhlich Polaron in the regime of strong coupling, is to leading order given by the minimal energy of the corresponding effective theory. As part of this thesis we identify the sub-leading term in the expansion of the ground state energy, which can be interpreted as the quantum correction to the classical energy, since the effective theories under consideration can be seen as classical counterparts. We are further going to establish an asymptotic expression for the energy-momentum relation of the Fröhlich Polaron in the strong coupling limit. In the regime of suitably small momenta, this asymptotic expression agrees with the energy-momentum relation of a free particle having an effectively increased mass, and we find that this effectively increased mass agrees with the conjectured value in the physics literature. In addition we will discuss two unrelated papers written by the author during his stay at ISTA in the appendix. The first one concerns the realization of anyons, which are quasi-particles acquiring a non-trivial phase under the exchange of two particles, as molecular impurities. The second one provides a classification of those vector fields defined on a given manifold that can be written as the gradient of a given functional with respect to a suitable metric, provided that some mild smoothness assumptions hold. This classification is subsequently used to identify those quantum Markov semigroups that can be written as a gradient flow of the relative entropy. AU - Brooks, Morris ID - 12390 SN - 2663-337X TI - Translation-invariant quantum systems with effectively broken symmetry ER - TY - JOUR AB - We study the BCS energy gap Ξ in the high–density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential V on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3, 2022) on the critical temperature Tc at high densities, we prove the universality of the ratio of the energy gap and the critical temperature. AU - Henheik, Sven Joscha AU - Lauritsen, Asbjørn Bækgaard ID - 11732 JF - Journal of Statistical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0022-4715 TI - The BCS energy gap at high density VL - 189 ER - TY - JOUR AB - The Lieb–Oxford inequality provides a lower bound on the Coulomb energy of a classical system of N identical charges only in terms of their one-particle density. We prove here a new estimate on the best constant in this inequality. Numerical evaluation provides the value 1.58, which is a significant improvement to the previously known value 1.64. The best constant has recently been shown to be larger than 1.44. In a second part, we prove that the constant can be reduced to 1.25 when the inequality is restricted to Hartree–Fock states. This is the first proof that the exchange term is always much lower than the full indirect Coulomb energy. AU - Lewin, Mathieu AU - Lieb, Elliott H. AU - Seiringer, Robert ID - 12246 IS - 5 JF - Letters in Mathematical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0377-9017 TI - Improved Lieb–Oxford bound on the indirect and exchange energies VL - 112 ER - TY - THES AB - The polaron model is a basic model of quantum field theory describing a single particle interacting with a bosonic field. It arises in many physical contexts. We are mostly concerned with models applicable in the context of an impurity atom in a Bose-Einstein condensate as well as the problem of electrons moving in polar crystals. The model has a simple structure in which the interaction of the particle with the field is given by a term linear in the field’s creation and annihilation operators. In this work, we investigate the properties of this model by providing rigorous estimates on various energies relevant to the problem. The estimates are obtained, for the most part, by suitable operator techniques which constitute the principal mathematical substance of the thesis. The first application of these techniques is to derive the polaron model rigorously from first principles, i.e., from a full microscopic quantum-mechanical many-body problem involving an impurity in an otherwise homogeneous system. We accomplish this for the N + 1 Bose gas in the mean-field regime by showing that a suitable polaron-type Hamiltonian arises at weak interactions as a low-energy effective theory for this problem. In the second part, we investigate rigorously the ground state of the model at fixed momentum and for large values of the coupling constant. Qualitatively, the system is expected to display a transition from the quasi-particle behavior at small momenta, where the dispersion relation is parabolic and the particle moves through the medium dragging along a cloud of phonons, to the radiative behavior at larger momenta where the polaron decelerates and emits free phonons. At the same time, in the strong coupling regime, the bosonic field is expected to behave purely classically. Accordingly, the effective mass of the polaron at strong coupling is conjectured to be asymptotically equal to the one obtained from the semiclassical counterpart of the problem, first studied by Landau and Pekar in the 1940s. For polaron models with regularized form factors and phonon dispersion relations of superfluid type, i.e., bounded below by a linear function of the wavenumbers for all phonon momenta as in the interacting Bose gas, we prove that for a large window of momenta below the radiation threshold, the energy-momentum relation at strong coupling is indeed essentially a parabola with semi-latus rectum equal to the Landau–Pekar effective mass, as expected. For the Fröhlich polaron describing electrons in polar crystals where the dispersion relation is of the optical type and the form factor is formally UV–singular due to the nature of the point charge-dipole interaction, we are able to give the corresponding upper bound. In contrast to the regular case, this requires the inclusion of the quantum fluctuations of the phonon field, which makes the problem considerably more difficult. The results are supplemented by studies on the absolute ground-state energy at strong coupling, a proof of the divergence of the effective mass with the coupling constant for a wide class of polaron models, as well as the discussion of the apparent UV singularity of the Fröhlich model and the application of the techniques used for its removal for the energy estimates. AU - Mysliwy, Krzysztof ID - 11473 SN - 2663-337X TI - Polarons in Bose gases and polar crystals: Some rigorous energy estimates ER - TY - JOUR AB - We study a class of polaron-type Hamiltonians with sufficiently regular form factor in the interaction term. We investigate the strong-coupling limit of the model, and prove suitable bounds on the ground state energy as a function of the total momentum of the system. These bounds agree with the semiclassical approximation to leading order. The latter corresponds here to the situation when the particle undergoes harmonic motion in a potential well whose frequency is determined by the corresponding Pekar functional. We show that for all such models the effective mass diverges in the strong coupling limit, in all spatial dimensions. Moreover, for the case when the phonon dispersion relation grows at least linearly with momentum, the bounds result in an asymptotic formula for the effective mass quotient, a quantity generalizing the usual notion of the effective mass. This asymptotic form agrees with the semiclassical Landau–Pekar formula and can be regarded as the first rigorous confirmation, in a slightly weaker sense than usually considered, of the validity of the semiclassical formula for the effective mass. AU - Mysliwy, Krzysztof AU - Seiringer, Robert ID - 10564 IS - 1 JF - Journal of Statistical Physics SN - 0022-4715 TI - Polaron models with regular interactions at strong coupling VL - 186 ER - TY - JOUR AB - We study two interacting quantum particles forming a bound state in d-dimensional free space, and constrain the particles in k directions to (0, ∞)k ×Rd−k, with Neumann boundary conditions. First, we prove that the ground state energy strictly decreases upon going from k to k+1. This shows that the particles stick to the corner where all boundary planes intersect. Second, we show that for all k the resulting Hamiltonian, after removing the free part of the kinetic energy, has only finitely many eigenvalues below the essential spectrum. This paper generalizes the work of Egger, Kerner and Pankrashkin (J. Spectr. Theory 10(4):1413–1444, 2020) to dimensions d > 1. AU - Roos, Barbara AU - Seiringer, Robert ID - 10850 IS - 12 JF - Journal of Functional Analysis KW - Analysis SN - 0022-1236 TI - Two-particle bound states at interfaces and corners VL - 282 ER - TY - JOUR AB - We provide a definition of the effective mass for the classical polaron described by the Landau–Pekar (LP) equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by LP (1948 J. Exp. Theor. Phys. 18 419–423). AU - Feliciangeli, Dario AU - Rademacher, Simone Anna Elvira AU - Seiringer, Robert ID - 10755 IS - 1 JF - Journal of Physics A: Mathematical and Theoretical SN - 1751-8113 TI - The effective mass problem for the Landau-Pekar equations VL - 55 ER - TY - JOUR AB - Recently it was shown that anyons on the two-sphere naturally arise from a system of molecular impurities exchanging angular momentum with a many-particle bath (Phys. Rev. Lett. 126, 015301 (2021)). Here we further advance this approach and rigorously demonstrate that in the experimentally realized regime the lowest spectrum of two linear molecules immersed in superfluid helium corresponds to the spectrum of two anyons on the sphere. We develop the formalism within the framework of the recently experimentally observed angulon quasiparticle AU - Brooks, Morris AU - Lemeshko, Mikhail AU - Lundholm, Douglas AU - Yakaboylu, Enderalp ID - 10585 IS - 4 JF - Atoms KW - anyons KW - quasiparticles KW - Quantum Hall Effect KW - topological states of matter TI - Emergence of anyons on the two-sphere in molecular impurities VL - 9 ER - TY - JOUR AB - We consider a gas of interacting bosons trapped in a box of side length one in the Gross–Pitaevskii limit. We review the proof of the validity of Bogoliubov’s prediction for the ground state energy and the low-energy excitation spectrum. This note is based on joint work with C. Brennecke, S. Cenatiempo and B. Schlein. AU - Boccato, Chiara ID - 7685 IS - 1 JF - Reviews in Mathematical Physics SN - 0129-055X TI - The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime VL - 33 ER - TY - JOUR AB - We consider the Fröhlich polaron model in the strong coupling limit. It is well‐known that to leading order the ground state energy is given by the (classical) Pekar energy. In this work, we establish the subleading correction, describing quantum fluctuation about the classical limit. Our proof applies to a model of a confined polaron, where both the electron and the polarization field are restricted to a set of finite volume, with linear size determined by the natural length scale of the Pekar problem. AU - Frank, Rupert AU - Seiringer, Robert ID - 8603 IS - 3 JF - Communications on Pure and Applied Mathematics SN - 00103640 TI - Quantum corrections to the Pekar asymptotics of a strongly coupled polaron VL - 74 ER - TY - JOUR AB - Studies on the experimental realization of two-dimensional anyons in terms of quasiparticles have been restricted, so far, to only anyons on the plane. It is known, however, that the geometry and topology of space can have significant effects on quantum statistics for particles moving on it. Here, we have undertaken the first step toward realizing the emerging fractional statistics for particles restricted to move on the sphere instead of on the plane. We show that such a model arises naturally in the context of quantum impurity problems. In particular, we demonstrate a setup in which the lowest-energy spectrum of two linear bosonic or fermionic molecules immersed in a quantum many-particle environment can coincide with the anyonic spectrum on the sphere. This paves the way toward the experimental realization of anyons on the sphere using molecular impurities. Furthermore, since a change in the alignment of the molecules corresponds to the exchange of the particles on the sphere, such a realization reveals a novel type of exclusion principle for molecular impurities, which could also be of use as a powerful technique to measure the statistics parameter. Finally, our approach opens up a simple numerical route to investigate the spectra of many anyons on the sphere. Accordingly, we present the spectrum of two anyons on the sphere in the presence of a Dirac monopole field. AU - Brooks, Morris AU - Lemeshko, Mikhail AU - Lundholm, D. AU - Yakaboylu, Enderalp ID - 9005 IS - 1 JF - Physical Review Letters SN - 00319007 TI - Molecular impurities as a realization of anyons on the two-sphere VL - 126 ER - TY - JOUR AB - We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle numbers and a suitably small coupling, we show that the dynamics of the system is approximately described by the Landau–Pekar equations. These describe a Bose–Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order. AU - Leopold, Nikolai K AU - Mitrouskas, David Johannes AU - Seiringer, Robert ID - 9246 JF - Archive for Rational Mechanics and Analysis SN - 00039527 TI - Derivation of the Landau–Pekar equations in a many-body mean-field limit VL - 240 ER - TY - JOUR AB - We consider the ferromagnetic quantum Heisenberg model in one dimension, for any spin S≥1/2. We give upper and lower bounds on the free energy, proving that at low temperature it is asymptotically equal to the one of an ideal Bose gas of magnons, as predicted by the spin-wave approximation. The trial state used in the upper bound yields an analogous estimate also in the case of two spatial dimensions, which is believed to be sharp at low temperature. AU - Napiórkowski, Marcin M AU - Seiringer, Robert ID - 9256 IS - 2 JF - Letters in Mathematical Physics SN - 03779017 TI - Free energy asymptotics of the quantum Heisenberg spin chain VL - 111 ER - TY - JOUR AB - We consider a system of N bosons in the mean-field scaling regime for a class of interactions including the repulsive Coulomb potential. We derive an asymptotic expansion of the low-energy eigenstates and the corresponding energies, which provides corrections to Bogoliubov theory to any order in 1/N. AU - Bossmann, Lea AU - Petrat, Sören P AU - Seiringer, Robert ID - 9318 JF - Forum of Mathematics, Sigma TI - Asymptotic expansion of low-energy excitations for weakly interacting bosons VL - 9 ER - TY - JOUR AB - We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state φ0⊗ξα, where φ0 is the electron ground state of the Pekar energy functional and ξα the associated coherent state of the phonons, can be approximated by a global phase for times small compared to α2. In the present note we prove that a similar approximation holds for t=O(α2) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to α−2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order α2, while the phonon fluctuations around the coherent state ξα can be described by a time-dependent Bogoliubov transformation. AU - Mitrouskas, David Johannes ID - 9333 JF - Letters in Mathematical Physics SN - 03779017 TI - A note on the Fröhlich dynamics in the strong coupling limit VL - 111 ER - TY - JOUR AB - We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates that are consistent with central limit theorems that have been established in the last years. AU - Kirkpatrick, Kay AU - Rademacher, Simone Anna Elvira AU - Schlein, Benjamin ID - 9351 JF - Annales Henri Poincare SN - 1424-0637 TI - A large deviation principle in many-body quantum dynamics VL - 22 ER - TY - JOUR AB - We consider the stochastic quantization of a quartic double-well energy functional in the semiclassical regime and derive optimal asymptotics for the exponentially small splitting of the ground state energy. Our result provides an infinite-dimensional version of some sharp tunneling estimates known in finite dimensions for semiclassical Witten Laplacians in degree zero. From a stochastic point of view it proves that the L2 spectral gap of the stochastic one-dimensional Allen-Cahn equation in finite volume satisfies a Kramers-type formula in the limit of vanishing noise. We work with finite-dimensional lattice approximations and establish semiclassical estimates which are uniform in the dimension. Our key estimate shows that the constant separating the two exponentially small eigenvalues from the rest of the spectrum can be taken independently of the dimension. AU - Brooks, Morris AU - Di Gesù, Giacomo ID - 9348 IS - 3 JF - Journal of Functional Analysis SN - 0022-1236 TI - Sharp tunneling estimates for a double-well model in infinite dimension VL - 281 ER - TY - JOUR AB - We consider a system of N trapped bosons with repulsive interactions in a combined semiclassical mean-field limit at positive temperature. We show that the free energy is well approximated by the minimum of the Hartree free energy functional – a natural extension of the Hartree energy functional to positive temperatures. The Hartree free energy functional converges in the same limit to a semiclassical free energy functional, and we show that the system displays Bose–Einstein condensation if and only if it occurs in the semiclassical free energy functional. This allows us to show that for weak coupling the critical temperature decreases due to the repulsive interactions. AU - Deuchert, Andreas AU - Seiringer, Robert ID - 9462 IS - 6 JF - Journal of Functional Analysis SN - 0022-1236 TI - Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons VL - 281 ER - TY - JOUR AB - Extending on ideas of Lewin, Lieb, and Seiringer [Phys. Rev. B 100, 035127 (2019)], we present a modified “floating crystal” trial state for jellium (also known as the classical homogeneous electron gas) with density equal to a characteristic function. This allows us to show that three definitions of the jellium energy coincide in dimensions d ≥ 2, thus extending the result of Cotar and Petrache [“Equality of the Jellium and uniform electron gas next-order asymptotic terms for Coulomb and Riesz potentials,” arXiv: 1707.07664 (2019)] and Lewin, Lieb, and Seiringer [Phys. Rev. B 100, 035127 (2019)] that the three definitions coincide in dimension d ≥ 3. We show that the jellium energy is also equivalent to a “renormalized energy” studied in a series of papers by Serfaty and others, and thus, by the work of Bétermin and Sandier [Constr. Approximation 47, 39–74 (2018)], we relate the jellium energy to the order n term in the logarithmic energy of n points on the unit 2-sphere. We improve upon known lower bounds for this renormalized energy. Additionally, we derive formulas for the jellium energy of periodic configurations. AU - Lauritsen, Asbjørn Bækgaard ID - 9891 IS - 8 JF - Journal of Mathematical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0022-2488 TI - Floating Wigner crystal and periodic jellium configurations VL - 62 ER - TY - JOUR AB - We investigate the Fröhlich polaron model on a three-dimensional torus, and give a proof of the second-order quantum corrections to its ground-state energy in the strong-coupling limit. Compared to previous work in the confined case, the translational symmetry (and its breaking in the Pekar approximation) makes the analysis substantially more challenging. AU - Feliciangeli, Dario AU - Seiringer, Robert ID - 10224 IS - 3 JF - Archive for Rational Mechanics and Analysis SN - 0003-9527 TI - The strongly coupled polaron on the torus: Quantum corrections to the Pekar asymptotics VL - 242 ER - TY - JOUR AB - We 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 quasi-free bosonic evolution of the collective particle–hole excitations. AU - Benedikter, Niels P AU - Nam, Phan Thành AU - Porta, Marcello AU - Schlein, Benjamin AU - Seiringer, Robert ID - 10537 JF - Annales Henri Poincaré SN - 1424-0637 TI - Bosonization of fermionic many-body dynamics ER - TY - JOUR AB - We derive rigorously the leading order of the correlation energy of a Fermi gas in a scaling regime of high density and weak interaction. The result verifies the prediction of the random-phase approximation. Our proof refines the method of collective bosonization in three dimensions. We approximately diagonalize an effective Hamiltonian describing approximately bosonic collective excitations around the Hartree–Fock state, while showing that gapless and non-collective excitations have only a negligible effect on the ground state energy. AU - Benedikter, Niels P AU - Nam, Phan Thành AU - Porta, Marcello AU - Schlein, Benjamin AU - Seiringer, Robert ID - 7901 JF - Inventiones Mathematicae SN - 0020-9910 TI - Correlation energy of a weakly interacting Fermi gas VL - 225 ER - TY - JOUR AB - Hartree–Fock theory has been justified as a mean-field approximation for fermionic systems. However, it suffers from some defects in predicting physical properties, making necessary a theory of quantum correlations. Recently, bosonization of many-body correlations has been rigorously justified as an upper bound on the correlation energy at high density with weak interactions. We review the bosonic approximation, deriving an effective Hamiltonian. We then show that for systems with Coulomb interaction this effective theory predicts collective excitations (plasmons) in accordance with the random phase approximation of Bohm and Pines, and with experimental observation. AU - Benedikter, Niels P ID - 7900 IS - 1 JF - Reviews in Mathematical Physics SN - 0129-055X TI - Bosonic collective excitations in Fermi gases VL - 33 ER - TY - JOUR AB - We review old and new results on the Fröhlich polaron model. The discussion includes the validity of the (classical) Pekar approximation in the strong coupling limit, quantum corrections to this limit, as well as the divergence of the effective polaron mass. AU - Seiringer, Robert ID - 10852 IS - 01 JF - Reviews in Mathematical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0129-055X TI - The polaron at strong coupling VL - 33 ER - TY - JOUR AB - The Landau–Pekar equations describe the dynamics of a strongly coupled polaron. Here, we provide a class of initial data for which the associated effective Hamiltonian has a uniform spectral gap for all times. For such initial data, this allows us to extend the results on the adiabatic theorem for the Landau–Pekar equations and their derivation from the Fröhlich model obtained in previous works to larger times. AU - Feliciangeli, Dario AU - Rademacher, Simone Anna Elvira AU - Seiringer, Robert ID - 9225 JF - Letters in Mathematical Physics SN - 03779017 TI - Persistence of the spectral gap for the Landau–Pekar equations VL - 111 ER - TY - GEN AB - We investigate the Fröhlich polaron model on a three-dimensional torus, and give a proof of the second-order quantum corrections to its ground-state energy in the strong-coupling limit. Compared to previous work in the confined case, the translational symmetry (and its breaking in the Pekar approximation) makes the analysis substantially more challenging. AU - Feliciangeli, Dario AU - Seiringer, Robert ID - 9787 T2 - arXiv TI - The strongly coupled polaron on the torus: Quantum corrections to the Pekar asymptotics ER - TY - JOUR AB - We prove an adiabatic theorem for the Landau–Pekar equations. This allows us to derive new results on the accuracy of their use as effective equations for the time evolution generated by the Fröhlich Hamiltonian with large coupling constant α. In particular, we show that the time evolution of Pekar product states with coherent phonon field and the electron being trapped by the phonons is well approximated by the Landau–Pekar equations until times short compared to α2. AU - Leopold, Nikolai K AU - Rademacher, Simone Anna Elvira AU - Schlein, Benjamin AU - Seiringer, Robert ID - 10738 IS - 7 JF - Analysis and PDE SN - 2157-5045 TI - The Landau–Pekar equations: Adiabatic theorem and accuracy VL - 14 ER - TY - GEN AB - This paper establishes new connections between many-body quantum systems, One-body Reduced Density Matrices Functional Theory (1RDMFT) and Optimal Transport (OT), by interpreting the problem of computing the ground-state energy of a finite dimensional composite quantum system at positive temperature as a non-commutative entropy regularized Optimal Transport problem. We develop a new approach to fully characterize the dual-primal solutions in such non-commutative setting. The mathematical formalism is particularly relevant in quantum chemistry: numerical realizations of the many-electron ground state energy can be computed via a non-commutative version of Sinkhorn algorithm. Our approach allows to prove convergence and robustness of this algorithm, which, to our best knowledge, were unknown even in the two marginal case. Our methods are based on careful a priori estimates in the dual problem, which we believe to be of independent interest. Finally, the above results are extended in 1RDMFT setting, where bosonic or fermionic symmetry conditions are enforced on the problem. AU - Feliciangeli, Dario AU - Gerolin, Augusto AU - Portinale, Lorenzo ID - 9792 T2 - arXiv TI - A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature ER - TY - JOUR AB - We consider the Fröhlich Hamiltonian with large coupling constant α. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm approximation of the evolution, valid up to times of order α2. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. This allows us to show that the Landau-Pekar equations approximately describe the evolution of the electron- and one-phonon reduced density matrices under the Fröhlich dynamics up to times of order α2. AU - Leopold, Nikolai K AU - Mitrouskas, David Johannes AU - Rademacher, Simone Anna Elvira AU - Schlein, Benjamin AU - Seiringer, Robert ID - 14889 IS - 4 JF - Pure and Applied Analysis SN - 2578-5893 TI - Landau–Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron VL - 3 ER - TY - JOUR AB - We consider a system of N interacting bosons in the mean-field scaling regime and construct corrections to the Bogoliubov dynamics that approximate the true N-body dynamics in norm to arbitrary precision. The N-independent corrections are given in terms of the solutions of the Bogoliubov and Hartree equations and satisfy a generalized form of Wick's theorem. We determine the n-point correlation functions of the excitations around the condensate, as well as the reduced densities of the N-body system, to arbitrary accuracy, given only the knowledge of the two-point functions of a quasi-free state and the solution of the Hartree equation. In this way, the complex problem of computing all n-point correlation functions for an interacting N-body system is essentially reduced to the problem of solving the Hartree equation and the PDEs for the Bogoliubov two-point functions. AU - Bossmann, Lea AU - Petrat, Sören P AU - Pickl, Peter AU - Soffer, Avy ID - 14890 IS - 4 JF - Pure and Applied Analysis SN - 2578-5893 TI - Beyond Bogoliubov dynamics VL - 3 ER - TY - THES AB - This thesis is the result of the research carried out by the author during his PhD at IST Austria between 2017 and 2021. It mainly focuses on the Fröhlich polaron model, specifically to its regime of strong coupling. This model, which is rigorously introduced and discussed in the introduction, has been of great interest in condensed matter physics and field theory for more than eighty years. It is used to describe an electron interacting with the atoms of a solid material (the strength of this interaction is modeled by the presence of a coupling constant α in the Hamiltonian of the system). The particular regime examined here, which is mathematically described by considering the limit α →∞, displays many interesting features related to the emergence of classical behavior, which allows for a simplified effective description of the system under analysis. The properties, the range of validity and a quantitative analysis of the precision of such classical approximations are the main object of the present work. We specify our investigation to the study of the ground state energy of the system, its dynamics and its effective mass. For each of these problems, we provide in the introduction an overview of the previously known results and a detailed account of the original contributions by the author. AU - Feliciangeli, Dario ID - 9733 SN - 2663-337X TI - The polaron at strong coupling ER - TY - GEN AB - We provide a definition of the effective mass for the classical polaron described by the Landau-Pekar equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by Landau and Pekar. AU - Feliciangeli, Dario AU - Rademacher, Simone Anna Elvira AU - Seiringer, Robert ID - 9791 T2 - arXiv TI - The effective mass problem for the Landau-Pekar equations ER - TY - JOUR AB - 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. AU - Benedikter, Niels P AU - Nam, Phan Thành AU - Porta, Marcello AU - Schlein, Benjamin AU - Seiringer, Robert ID - 6649 JF - Communications in Mathematical Physics SN - 0010-3616 TI - Optimal upper bound for the correlation energy of a Fermi gas in the mean-field regime VL - 374 ER - TY - JOUR AB - In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of N d-dimensional bosons for large N. The bosons initially form a Bose–Einstein condensate and interact with each other via a pair potential of the form (N−1)−1Ndβv(Nβ·)forβ∈[0,14d). We derive a sequence of N-body functions which approximate the true many-body dynamics in L2(RdN)-norm to arbitrary precision in powers of N−1. The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution. AU - Bossmann, Lea AU - Pavlović, Nataša AU - Pickl, Peter AU - Soffer, Avy ID - 7508 JF - Journal of Statistical Physics SN - 0022-4715 TI - Higher order corrections to the mean-field description of the dynamics of interacting bosons VL - 178 ER - TY - JOUR AB - 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 𝜌 and inverse temperature 𝛽 differs from the one of the noninteracting system by the correction term 𝜋𝜌𝜌𝛽𝛽 . Here, is the scattering length of the interaction potential, and 𝛽 is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. The result is valid in the dilute limit 𝜌 and if 𝛽𝜌 . AU - Deuchert, Andreas AU - Mayer, Simon AU - Seiringer, Robert ID - 7790 JF - Forum of Mathematics, Sigma TI - The free energy of the two-dimensional dilute Bose gas. I. Lower bound VL - 8 ER - TY - JOUR AB - We consider systems of N bosons in a box of volume one, interacting through a repulsive two-body potential of the form κN3β−1V(Nβx). For all 0<β<1, and for sufficiently small coupling constant κ>0, we establish the validity of Bogolyubov theory, identifying the ground state energy and the low-lying excitation spectrum up to errors that vanish in the limit of large N. AU - Boccato, Chiara AU - Brennecke, Christian AU - Cenatiempo, Serena AU - Schlein, Benjamin ID - 8042 IS - 7 JF - Journal of the European Mathematical Society SN - 14359855 TI - The excitation spectrum of Bose gases interacting through singular potentials VL - 22 ER - TY - JOUR AB - In the setting of the fractional quantum Hall effect we study the effects of strong, repulsive two-body interaction potentials of short range. We prove that Haldane’s pseudo-potential operators, including their pre-factors, emerge as mathematically rigorous limits of such interactions when the range of the potential tends to zero while its strength tends to infinity. In a common approach the interaction potential is expanded in angular momentum eigenstates in the lowest Landau level, which amounts to taking the pre-factors to be the moments of the potential. Such a procedure is not appropriate for very strong interactions, however, in particular not in the case of hard spheres. We derive the formulas valid in the short-range case, which involve the scattering lengths of the interaction potential in different angular momentum channels rather than its moments. Our results hold for bosons and fermions alike and generalize previous results in [6], which apply to bosons in the lowest angular momentum channel. Our main theorem asserts the convergence in a norm-resolvent sense of the Hamiltonian on the whole Hilbert space, after appropriate energy scalings, to Hamiltonians with contact interactions in the lowest Landau level. AU - Seiringer, Robert AU - Yngvason, Jakob ID - 8091 JF - Journal of Statistical Physics SN - 00224715 TI - Emergence of Haldane pseudo-potentials in systems with short-range interactions VL - 181 ER - TY - JOUR AB - We prove an upper bound on the free energy of a two-dimensional homogeneous Bose gas in the thermodynamic limit. We show that for a2ρ ≪ 1 and βρ ≳ 1, the free energy per unit volume differs from the one of the non-interacting system by at most 4πρ2|lna2ρ|−1(2−[1−βc/β]2+) to leading order, where a is the scattering length of the two-body interaction potential, ρ is the density, β is the inverse temperature, and βc is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. In combination with the corresponding matching lower bound proved by Deuchert et al. [Forum Math. Sigma 8, e20 (2020)], this shows equality in the asymptotic expansion. AU - Mayer, Simon AU - Seiringer, Robert ID - 8134 IS - 6 JF - Journal of Mathematical Physics SN - 00222488 TI - The free energy of the two-dimensional dilute Bose gas. II. Upper bound VL - 61 ER - TY - JOUR AB - One of the hallmarks of quantum statistics, tightly entwined with the concept of topological phases of matter, is the prediction of anyons. Although anyons are predicted to be realized in certain fractional quantum Hall systems, they have not yet been unambiguously detected in experiment. Here we introduce a simple quantum impurity model, where bosonic or fermionic impurities turn into anyons as a consequence of their interaction with the surrounding many-particle bath. A cloud of phonons dresses each impurity in such a way that it effectively attaches fluxes or vortices to it and thereby converts it into an Abelian anyon. The corresponding quantum impurity model, first, provides a different approach to the numerical solution of the many-anyon problem, along with a concrete perspective of anyons as emergent quasiparticles built from composite bosons or fermions. More importantly, the model paves the way toward realizing anyons using impurities in crystal lattices as well as ultracold gases. In particular, we consider two heavy electrons interacting with a two-dimensional lattice crystal in a magnetic field, and show that when the impurity-bath system is rotated at the cyclotron frequency, impurities behave as anyons as a consequence of the angular momentum exchange between the impurities and the bath. A possible experimental realization is proposed by identifying the statistics parameter in terms of the mean-square distance of the impurities and the magnetization of the impurity-bath system, both of which are accessible to experiment. Another proposed application is impurities immersed in a two-dimensional weakly interacting Bose gas. AU - Yakaboylu, Enderalp AU - Ghazaryan, Areg AU - Lundholm, D. AU - Rougerie, N. AU - Lemeshko, Mikhail AU - Seiringer, Robert ID - 8769 IS - 14 JF - Physical Review B SN - 2469-9950 TI - Quantum impurity model for anyons VL - 102 ER - TY - JOUR AB - We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross–Pitaevskii limit, where the scattering length a is so small that the interaction energy is of the same order of magnitude as the spectral gap of the Laplacian, and for temperatures that are comparable to the critical temperature of the ideal gas. We show that the difference between the specific free energy of the interacting system and the one of the ideal gas is to leading order given by 4πa(2ϱ2−ϱ20). Here ϱ denotes the density of the system and ϱ0 is the expected condensate density of the ideal gas. Additionally, we show that the one-particle density matrix of any approximate minimizer of the Gibbs free energy functional is to leading order given by the one of the ideal gas. This in particular proves Bose–Einstein condensation with critical temperature given by the one of the ideal gas to leading order. One key ingredient of our proof is a novel use of the Gibbs variational principle that goes hand in hand with the c-number substitution. AU - Deuchert, Andreas AU - Seiringer, Robert ID - 7650 IS - 6 JF - Archive for Rational Mechanics and Analysis SN - 0003-9527 TI - Gross-Pitaevskii limit of a homogeneous Bose gas at positive temperature VL - 236 ER - TY - JOUR AB - We study the dynamics of a system of N interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to an interval of length of order ε. The interaction is non-negative and scaled in such a way that its scattering length is of order ε/N, while its range is proportional to (ε/N)β with scaling parameter β∈(0,1]. We consider the simultaneous limit (N,ε)→(∞,0) and assume that the system initially exhibits Bose–Einstein condensation. We prove that condensation is preserved by the N-body dynamics, where the time-evolved condensate wave function is the solution of a two-dimensional non-linear equation. The strength of the non-linearity depends on the scaling parameter β. For β∈(0,1), we obtain a cubic defocusing non-linear Schrödinger equation, while the choice β=1 yields a Gross–Pitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential. AU - Bossmann, Lea ID - 8130 IS - 11 JF - Archive for Rational Mechanics and Analysis SN - 0003-9527 TI - Derivation of the 2d Gross–Pitaevskii equation for strongly confined 3d Bosons VL - 238 ER - TY - JOUR AB - We consider the Fröhlich model of a polaron, and show that its effective mass diverges in thestrong coupling limit. AU - Lieb, Elliott H. AU - Seiringer, Robert ID - 7235 JF - Journal of Statistical Physics SN - 0022-4715 TI - Divergence of the effective mass of a polaron in the strong coupling limit VL - 180 ER - TY - JOUR AB - We consider a system of N bosons in the limit N→∞, interacting through singular potentials. For initial data exhibiting Bose–Einstein condensation, the many-body time evolution is well approximated through a quadratic fluctuation dynamics around a cubic nonlinear Schrödinger equation of the condensate wave function. We show that these fluctuations satisfy a (multi-variate) central limit theorem. AU - Rademacher, Simone Anna Elvira ID - 7611 JF - Letters in Mathematical Physics SN - 0377-9017 TI - Central limit theorem for Bose gases interacting through singular potentials VL - 110 ER - TY - THES AB - We study the interacting homogeneous Bose gas in two spatial dimensions in the thermodynamic limit at fixed density. We shall be concerned with some mathematical aspects of this complicated problem in many-body quantum mechanics. More specifically, we consider the dilute limit where the scattering length of the interaction potential, which is a measure for the effective range of the potential, is small compared to the average distance between the particles. We are interested in a setting with positive (i.e., non-zero) temperature. After giving a survey of the relevant literature in the field, we provide some facts and examples to set expectations for the two-dimensional system. The crucial difference to the three-dimensional system is that there is no Bose–Einstein condensate at positive temperature due to the Hohenberg–Mermin–Wagner theorem. However, it turns out that an asymptotic formula for the free energy holds similarly to the three-dimensional case. We motivate this formula by considering a toy model with δ interaction potential. By restricting this model Hamiltonian to certain trial states with a quasi-condensate we obtain an upper bound for the free energy that still has the quasi-condensate fraction as a free parameter. When minimizing over the quasi-condensate fraction, we obtain the Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity, which plays an important role in our rigorous contribution. The mathematically rigorous result that we prove concerns the specific free energy in the dilute limit. We give upper and lower bounds on the free energy in terms of the free energy of the non-interacting system and a correction term coming from the interaction. Both bounds match and thus we obtain the leading term of an asymptotic approximation in the dilute limit, provided the thermal wavelength of the particles is of the same order (or larger) than the average distance between the particles. The remarkable feature of this result is its generality: the correction term depends on the interaction potential only through its scattering length and it holds for all nonnegative interaction potentials with finite scattering length that are measurable. In particular, this allows to model an interaction of hard disks. AU - Mayer, Simon ID - 7514 SN - 2663-337X TI - The free energy of a dilute two-dimensional Bose gas ER - TY - JOUR AB - Inspired by the possibility to experimentally manipulate and enhance chemical reactivity in helium nanodroplets, we investigate the effective interaction and the resulting correlations between two diatomic molecules immersed in a bath of bosons. By analogy with the bipolaron, we introduce the biangulon quasiparticle describing two rotating molecules that align with respect to each other due to the effective attractive interaction mediated by the excitations of the bath. We study this system in different parameter regimes and apply several theoretical approaches to describe its properties. Using a Born–Oppenheimer approximation, we investigate the dependence of the effective intermolecular interaction on the rotational state of the two molecules. In the strong-coupling regime, a product-state ansatz shows that the molecules tend to have a strong alignment in the ground state. To investigate the system in the weak-coupling regime, we apply a one-phonon excitation variational ansatz, which allows us to access the energy spectrum. In comparison to the angulon quasiparticle, the biangulon shows shifted angulon instabilities and an additional spectral instability, where resonant angular momentum transfer between the molecules and the bath takes place. These features are proposed as an experimentally observable signature for the formation of the biangulon quasiparticle. Finally, by using products of single angulon and bare impurity wave functions as basis states, we introduce a diagonalization scheme that allows us to describe the transition from two separated angulons to a biangulon as a function of the distance between the two molecules. AU - Li, Xiang AU - Yakaboylu, Enderalp AU - Bighin, Giacomo AU - Schmidt, Richard AU - Lemeshko, Mikhail AU - Deuchert, Andreas ID - 8587 IS - 16 JF - The Journal of Chemical Physics KW - Physical and Theoretical Chemistry KW - General Physics and Astronomy SN - 0021-9606 TI - Intermolecular forces and correlations mediated by a phonon bath VL - 152 ER - TY - JOUR AB - We consider the Pekar functional on a ball in ℝ3. We prove uniqueness of minimizers, and a quadratic lower bound in terms of the distance to the minimizer. The latter follows from nondegeneracy of the Hessian at the minimum. AU - Feliciangeli, Dario AU - Seiringer, Robert ID - 9781 IS - 1 JF - SIAM Journal on Mathematical Analysis KW - Applied Mathematics KW - Computational Mathematics KW - Analysis SN - 0036-1410 TI - Uniqueness and nondegeneracy of minimizers of the Pekar functional on a ball VL - 52 ER - TY - JOUR AB - We consider the quantum mechanical many-body problem of a single impurity particle immersed in a weakly interacting Bose gas. The impurity interacts with the bosons via a two-body potential. We study the Hamiltonian of this system in the mean-field limit and rigorously show that, at low energies, the problem is well described by the Fröhlich polaron model. AU - Mysliwy, Krzysztof AU - Seiringer, Robert ID - 8705 IS - 12 JF - Annales Henri Poincare SN - 1424-0637 TI - Microscopic derivation of the Fröhlich Hamiltonian for the Bose polaron in the mean-field limit VL - 21 ER - TY - JOUR AB - We give the first mathematically rigorous justification of the local density approximation in density functional theory. We provide a quantitative estimate on the difference between the grand-canonical Levy–Lieb energy of a given density (the lowest possible energy of all quantum states having this density) and the integral over the uniform electron gas energy of this density. The error involves gradient terms and justifies the use of the local density approximation in the situation where the density is very flat on sufficiently large regions in space. AU - Lewin, Mathieu AU - Lieb, Elliott H. AU - Seiringer, Robert ID - 14891 IS - 1 JF - Pure and Applied Analysis SN - 2578-5893 TI - The local density approximation in density functional theory VL - 2 ER - TY - JOUR AB - We consider systems of bosons trapped in a box, in the Gross–Pitaevskii regime. We show that low-energy states exhibit complete Bose–Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in Boccato et al. (Commun Math Phys 359(3):975–1026, 2018), removing the assumption of small interaction potential. AU - Boccato, Chiara AU - Brennecke, Christian AU - Cenatiempo, Serena AU - Schlein, Benjamin ID - 6906 JF - Communications in Mathematical Physics SN - 0010-3616 TI - Optimal rate for Bose-Einstein condensation in the Gross-Pitaevskii regime VL - 376 ER - TY - JOUR AB - The interaction among fundamental particles in nature leads to many interesting effects in quantum statistical mechanics; examples include superconductivity for charged systems and superfluidity in cold gases. It is a huge challenge for mathematical physics to understand the collective behavior of systems containing a large number of particles, emerging from known microscopic interactions. In this workshop we brought together researchers working on different aspects of many-body quantum mechanics to discuss recent developments, exchange ideas and propose new challenges and research directions. AU - Hainzl, Christian AU - Schlein, Benjamin AU - Seiringer, Robert AU - Warzel, Simone ID - 15072 IS - 3 JF - Oberwolfach Reports SN - 1660-8933 TI - Many-body quantum systems VL - 16 ER - TY - JOUR AB - 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. AU - Deuchert, Andreas AU - Seiringer, Robert AU - Yngvason, Jakob ID - 80 IS - 2 JF - Communications in Mathematical Physics TI - Bose–Einstein condensation in a dilute, trapped gas at positive temperature VL - 368 ER - TY - JOUR AB - We consider the Nelson model with ultraviolet cutoff, which describes the interaction between non-relativistic particles and a positive or zero mass quantized scalar field. We take the non-relativistic particles to obey Fermi statistics and discuss the time evolution in a mean-field limit of many fermions. In this case, the limit is known to be also a semiclassical limit. We prove convergence in terms of reduced density matrices of the many-body state to a tensor product of a Slater determinant with semiclassical structure and a coherent state, which evolve according to a fermionic version of the Schrödinger–Klein–Gordon equations. AU - Leopold, Nikolai K AU - Petrat, Sören P ID - 6788 IS - 10 JF - Annales Henri Poincare SN - 1424-0637 TI - Mean-field dynamics for the Nelson model with fermions VL - 20 ER - TY - JOUR AB - We discuss thermodynamic properties of harmonically trapped imperfect quantum gases. The spatial inhomogeneity of these systems imposes a redefinition of the mean-field interparticle potential energy as compared to the homogeneous case. In our approach, it takes the form a 2N2 ωd, where N is the number of particles, ω—the harmonic trap frequency, d—system’s dimensionality, and a is a parameter characterizing the interparticle interaction. We provide arguments that this model corresponds to the limiting case of a long-ranged interparticle potential of vanishingly small amplitude. This conclusion is drawn from a computation similar to the well-known Kac scaling procedure, which is presented here in a form adapted to the case of an isotropic harmonic trap. We show that within the model, the imperfect gas of trapped repulsive bosons undergoes the Bose–Einstein condensation provided d > 1. The main result of our analysis is that in d = 1 the gas of attractive imperfect fermions with a = −aF < 0 is thermodynamically equivalent to the gas of repulsive bosons with a = aB > 0 provided the parameters aF and aB fulfill the relation aB + aF = . This result supplements similar recent conclusion about thermodynamic equivalence of two-dimensional (2D) uniform imperfect repulsive Bose and attractive Fermi gases. AU - Mysliwy, Krzysztof AU - Napiórkowski, Marek ID - 6840 IS - 6 JF - Journal of Statistical Mechanics: Theory and Experiment TI - Thermodynamics of inhomogeneous imperfect quantum gases in harmonic traps VL - 2019 ER - TY - JOUR AB - We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting froman interacting N-particle system of bosons. We consider the interaction potential to be given either by Wβ(x)=N−1+2βW(Nβx), for any β>0, or to be given by VN(x)=e2NV(eNx), for some spherical symmetric, nonnegative and compactly supported W,V∈L∞(R2,R). In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics. AU - Jeblick, Maximilian AU - Leopold, Nikolai K AU - Pickl, Peter ID - 7100 IS - 1 JF - Communications in Mathematical Physics SN - 0010-3616 TI - Derivation of the time dependent Gross–Pitaevskii equation in two dimensions VL - 372 ER - TY - JOUR AB - We consider Bose gases consisting of N particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of order N−1 (Gross–Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as N→∞. Our results confirm Bogoliubov’s predictions. AU - Boccato, Chiara AU - Brennecke, Christian AU - Cenatiempo, Serena AU - Schlein, Benjamin ID - 7413 IS - 2 JF - Acta Mathematica SN - 0001-5962 TI - Bogoliubov theory in the Gross–Pitaevskii limit VL - 222 ER - TY - JOUR AB - 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. AU - Moser, Thomas AU - Seiringer, Robert ID - 5856 IS - 4 JF - Annales Henri Poincare SN - 14240637 TI - Energy contribution of a point-interacting impurity in a Fermi gas VL - 20 ER - TY - GEN AB - 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 $\beta \rho \gtrsim 1$. AU - Deuchert, Andreas AU - Mayer, Simon AU - Seiringer, Robert ID - 7524 T2 - arXiv:1910.03372 TI - The free energy of the two-dimensional dilute Bose gas. I. Lower bound ER - TY - JOUR AU - Jaksic, Vojkan AU - Seiringer, Robert ID - 7226 IS - 12 JF - Journal of Mathematical Physics SN - 00222488 TI - Introduction to the Special Collection: International Congress on Mathematical Physics (ICMP) 2018 VL - 60 ER - TY - JOUR AB - We modify the "floating crystal" trial state for the classical homogeneous electron gas (also known as jellium), in order to suppress the boundary charge fluctuations that are known to lead to a macroscopic increase of the energy. The argument is to melt a thin layer of the crystal close to the boundary and consequently replace it by an incompressible fluid. With the aid of this trial state we show that three different definitions of the ground-state energy of jellium coincide. In the first point of view the electrons are placed in a neutralizing uniform background. In the second definition there is no background but the electrons are submitted to the constraint that their density is constant, as is appropriate in density functional theory. Finally, in the third system each electron interacts with a periodic image of itself; that is, periodic boundary conditions are imposed on the interaction potential. AU - Lewin, Mathieu AU - Lieb, Elliott H. AU - Seiringer, Robert ID - 7015 IS - 3 JF - Physical Review B SN - 2469-9950 TI - Floating Wigner crystal with no boundary charge fluctuations VL - 100 ER - TY - CONF AB - 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. AU - Leopold, Nikolai K AU - Pickl, Peter ID - 11 TI - Mean-field limits of particles in interaction with quantised radiation fields VL - 270 ER - TY - JOUR AB - 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.). AU - Napiórkowski, Marcin M AU - Reuvers, Robin AU - Solovej, Jan ID - 554 IS - 1 JF - Communications in Mathematical Physics SN - 00103616 TI - The Bogoliubov free energy functional II: The dilute Limit VL - 360 ER - TY - JOUR AB - 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. AU - Napiórkowski, Marcin M AU - Reuvers, Robin AU - Solovej, Jan ID - 399 IS - 1 JF - EPL TI - Calculation of the critical temperature of a dilute Bose gas in the Bogoliubov approximation VL - 121 ER - 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 -