@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 = {1573-0530}, 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}, } @article{9246, abstract = {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.}, author = {Leopold, Nikolai K and Mitrouskas, David Johannes and Seiringer, Robert}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, pages = {383--417}, publisher = {Springer Nature}, title = {{Derivation of the Landau–Pekar equations in a many-body mean-field limit}}, doi = {10.1007/s00205-021-01616-9}, volume = {240}, year = {2021}, } @article{9256, abstract = {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.}, author = {Napiórkowski, Marcin M and Seiringer, Robert}, issn = {1573-0530}, journal = {Letters in Mathematical Physics}, number = {2}, publisher = {Springer Nature}, title = {{Free energy asymptotics of the quantum Heisenberg spin chain}}, doi = {10.1007/s11005-021-01375-4}, volume = {111}, year = {2021}, } @article{9318, abstract = {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.}, author = {Bossmann, Lea and Petrat, Sören P and Seiringer, Robert}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Asymptotic expansion of low-energy excitations for weakly interacting bosons}}, doi = {10.1017/fms.2021.22}, volume = {9}, year = {2021}, } @article{9333, abstract = {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.}, author = {Mitrouskas, David Johannes}, issn = {1573-0530}, journal = {Letters in Mathematical Physics}, publisher = {Springer Nature}, title = {{A note on the Fröhlich dynamics in the strong coupling limit}}, doi = {10.1007/s11005-021-01380-7}, volume = {111}, year = {2021}, } @article{9348, abstract = {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.}, author = {Brooks, Morris and Di Gesù, Giacomo}, issn = {1096-0783}, journal = {Journal of Functional Analysis}, number = {3}, publisher = {Elsevier}, title = {{Sharp tunneling estimates for a double-well model in infinite dimension}}, doi = {10.1016/j.jfa.2021.109029}, volume = {281}, year = {2021}, } @article{9351, abstract = {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. }, author = {Kirkpatrick, Kay and Rademacher, Simone Anna Elvira and Schlein, Benjamin}, issn = {1424-0637}, journal = {Annales Henri Poincare}, pages = {2595--2618}, publisher = {Springer Nature}, title = {{A large deviation principle in many-body quantum dynamics}}, doi = {10.1007/s00023-021-01044-1}, volume = {22}, year = {2021}, } @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}, } @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}}, doi = {10.48550/arXiv.2101.12566}, 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}}, doi = {10.48550/arXiv.2107.03720}, 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{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}, } @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}, } @article{14891, abstract = {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.}, author = {Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert}, issn = {2578-5885}, journal = {Pure and Applied Analysis}, number = {1}, pages = {35--73}, publisher = {Mathematical Sciences Publishers}, title = {{ The local density approximation in density functional theory}}, doi = {10.2140/paa.2020.2.35}, volume = {2}, year = {2020}, } @article{7611, abstract = {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.}, author = {Rademacher, Simone Anna Elvira}, issn = {1573-0530}, journal = {Letters in Mathematical Physics}, pages = {2143--2174}, publisher = {Springer Nature}, title = {{Central limit theorem for Bose gases interacting through singular potentials}}, doi = {10.1007/s11005-020-01286-w}, volume = {110}, year = {2020}, } @article{7650, abstract = {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.}, author = {Deuchert, Andreas and Seiringer, Robert}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {6}, pages = {1217--1271}, publisher = {Springer Nature}, title = {{Gross-Pitaevskii limit of a homogeneous Bose gas at positive temperature}}, doi = {10.1007/s00205-020-01489-4}, volume = {236}, 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 = {2050-5094}, 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{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 = {1572-9613}, 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}, } @article{8130, abstract = {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.}, author = {Bossmann, Lea}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {11}, pages = {541--606}, publisher = {Springer Nature}, title = {{Derivation of the 2d Gross–Pitaevskii equation for strongly confined 3d Bosons}}, doi = {10.1007/s00205-020-01548-w}, volume = {238}, year = {2020}, } @article{8134, abstract = {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.}, author = {Mayer, Simon and Seiringer, Robert}, issn = {0022-2488}, journal = {Journal of Mathematical Physics}, number = {6}, publisher = {AIP Publishing}, title = {{The free energy of the two-dimensional dilute Bose gas. II. Upper bound}}, doi = {10.1063/5.0005950}, volume = {61}, year = {2020}, }