@article{9462, abstract = {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.}, author = {Deuchert, Andreas and Seiringer, Robert}, issn = {1096-0783}, journal = {Journal of Functional Analysis}, number = {6}, publisher = {Elsevier}, title = {{Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons}}, doi = {10.1016/j.jfa.2021.109096}, volume = {281}, year = {2021}, } @article{9891, abstract = {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.}, author = {Lauritsen, Asbjørn Bækgaard}, issn = {1089-7658}, journal = {Journal of Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {8}, publisher = {AIP Publishing}, title = {{Floating Wigner crystal and periodic jellium configurations}}, doi = {10.1063/5.0053494}, volume = {62}, year = {2021}, } @article{10224, abstract = {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.}, author = {Feliciangeli, Dario and Seiringer, Robert}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {3}, pages = {1835–1906}, publisher = {Springer Nature}, title = {{The strongly coupled polaron on the torus: Quantum corrections to the Pekar asymptotics}}, doi = {10.1007/s00205-021-01715-7}, volume = {242}, year = {2021}, } @article{10537, abstract = {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.}, author = {Benedikter, Niels P and Nam, Phan Thành and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert}, issn = {1424-0637}, journal = {Annales Henri Poincaré}, publisher = {Springer Nature}, title = {{Bosonization of fermionic many-body dynamics}}, doi = {10.1007/s00023-021-01136-y}, year = {2021}, } @article{7901, abstract = {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.}, author = {Benedikter, Niels P and Nam, Phan Thành and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, pages = {885--979}, publisher = {Springer}, title = {{Correlation energy of a weakly interacting Fermi gas}}, doi = {10.1007/s00222-021-01041-5}, volume = {225}, year = {2021}, } @article{7900, abstract = {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.}, author = {Benedikter, Niels P}, issn = {1793-6659}, journal = {Reviews in Mathematical Physics}, number = {1}, publisher = {World Scientific}, title = {{Bosonic collective excitations in Fermi gases}}, doi = {10.1142/s0129055x20600090}, volume = {33}, year = {2021}, } @article{10852, abstract = { 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.}, author = {Seiringer, Robert}, issn = {1793-6659}, journal = {Reviews in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {01}, publisher = {World Scientific Publishing}, title = {{The polaron at strong coupling}}, doi = {10.1142/s0129055x20600120}, volume = {33}, year = {2021}, } @article{9225, abstract = {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.}, author = {Feliciangeli, Dario and Rademacher, Simone Anna Elvira and Seiringer, Robert}, issn = {15730530}, journal = {Letters in Mathematical Physics}, publisher = {Springer Nature}, title = {{Persistence of the spectral gap for the Landau–Pekar equations}}, doi = {10.1007/s11005-020-01350-5}, volume = {111}, year = {2021}, } @unpublished{9787, abstract = {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.}, author = {Feliciangeli, Dario and Seiringer, Robert}, booktitle = {arXiv}, title = {{The strongly coupled polaron on the torus: Quantum corrections to the Pekar asymptotics}}, year = {2021}, } @article{10738, abstract = {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.}, author = {Leopold, Nikolai K and Rademacher, Simone Anna Elvira and Schlein, Benjamin and Seiringer, Robert}, issn = {1948-206X}, journal = {Analysis and PDE}, number = {7}, pages = {2079--2100}, publisher = {Mathematical Sciences Publishers}, title = {{ The Landau–Pekar equations: Adiabatic theorem and accuracy}}, doi = {10.2140/APDE.2021.14.2079}, volume = {14}, year = {2021}, } @unpublished{9792, abstract = {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.}, author = {Feliciangeli, Dario and Gerolin, Augusto and Portinale, Lorenzo}, booktitle = {arXiv}, title = {{A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature}}, doi = {10.48550/arXiv.2106.11217}, year = {2021}, } @article{14889, abstract = {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.}, author = {Leopold, Nikolai K and Mitrouskas, David Johannes and Rademacher, Simone Anna Elvira and Schlein, Benjamin and Seiringer, Robert}, issn = {2578-5885}, journal = {Pure and Applied Analysis}, number = {4}, pages = {653--676}, publisher = {Mathematical Sciences Publishers}, title = {{Landau–Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron}}, doi = {10.2140/paa.2021.3.653}, volume = {3}, year = {2021}, } @article{14890, abstract = {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.}, author = {Bossmann, Lea and Petrat, Sören P and Pickl, Peter and Soffer, Avy}, issn = {2578-5885}, journal = {Pure and Applied Analysis}, number = {4}, pages = {677--726}, publisher = {Mathematical Sciences Publishers}, title = {{Beyond Bogoliubov dynamics}}, doi = {10.2140/paa.2021.3.677}, volume = {3}, year = {2021}, } @phdthesis{9733, abstract = {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.}, author = {Feliciangeli, Dario}, issn = {2663-337X}, pages = {180}, publisher = {Institute of Science and Technology Austria}, title = {{The polaron at strong coupling}}, doi = {10.15479/at:ista:9733}, year = {2021}, } @unpublished{9791, abstract = {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.}, author = {Feliciangeli, Dario and Rademacher, Simone Anna Elvira and Seiringer, Robert}, booktitle = {arXiv}, title = {{The effective mass problem for the Landau-Pekar equations}}, year = {2021}, } @article{6649, abstract = {While Hartree–Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree–Fock state given by plane waves and introduce collective particle–hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann–Brueckner–type upper bound to the ground state energy. Our result justifies the random-phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials. }, author = {Benedikter, Niels P and Nam, Phan Thành and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, pages = {2097–2150}, publisher = {Springer Nature}, title = {{Optimal upper bound for the correlation energy of a Fermi gas in the mean-field regime}}, doi = {10.1007/s00220-019-03505-5}, volume = {374}, year = {2020}, } @article{7508, abstract = {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.}, author = {Bossmann, Lea and Pavlović, Nataša and Pickl, Peter and Soffer, Avy}, issn = {1572-9613}, journal = {Journal of Statistical Physics}, pages = {1362--1396}, publisher = {Springer Nature}, title = {{Higher order corrections to the mean-field description of the dynamics of interacting bosons}}, doi = {10.1007/s10955-020-02500-8}, volume = {178}, year = {2020}, } @article{7790, abstract = {We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density 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 .}, author = {Deuchert, Andreas and Mayer, Simon and Seiringer, Robert}, issn = {20505094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{The free energy of the two-dimensional dilute Bose gas. I. Lower bound}}, doi = {10.1017/fms.2020.17}, volume = {8}, year = {2020}, } @article{8042, abstract = {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.}, author = {Boccato, Chiara and Brennecke, Christian and Cenatiempo, Serena and Schlein, Benjamin}, issn = {14359855}, journal = {Journal of the European Mathematical Society}, number = {7}, pages = {2331--2403}, publisher = {European Mathematical Society}, title = {{The excitation spectrum of Bose gases interacting through singular potentials}}, doi = {10.4171/JEMS/966}, volume = {22}, year = {2020}, } @article{8091, abstract = {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.}, author = {Seiringer, Robert and Yngvason, Jakob}, issn = {15729613}, journal = {Journal of Statistical Physics}, pages = {448--464}, publisher = {Springer}, title = {{Emergence of Haldane pseudo-potentials in systems with short-range interactions}}, doi = {10.1007/s10955-020-02586-0}, volume = {181}, year = {2020}, }