@unpublished{9792,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{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},
}

@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},
}

@article{8587,
  abstract     = {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.},
  author       = {Li, Xiang and Yakaboylu, Enderalp and Bighin, Giacomo and Schmidt, Richard and Lemeshko, Mikhail and Deuchert, Andreas},
  issn         = {1089-7690},
  journal      = {The Journal of Chemical Physics},
  keywords     = {Physical and Theoretical Chemistry, General Physics and Astronomy},
  number       = {16},
  publisher    = {AIP Publishing},
  title        = {{Intermolecular forces and correlations mediated by a phonon bath}},
  doi          = {10.1063/1.5144759},
  volume       = {152},
  year         = {2020},
}

@article{8705,
  abstract     = {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.},
  author       = {Mysliwy, Krzysztof and Seiringer, Robert},
  issn         = {1424-0637},
  journal      = {Annales Henri Poincare},
  number       = {12},
  pages        = {4003--4025},
  publisher    = {Springer Nature},
  title        = {{Microscopic derivation of the Fröhlich Hamiltonian for the Bose polaron in the mean-field limit}},
  doi          = {10.1007/s00023-020-00969-3},
  volume       = {21},
  year         = {2020},
}

@article{8769,
  abstract     = {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.},
  author       = {Yakaboylu, Enderalp and Ghazaryan, Areg and Lundholm, D. and Rougerie, N. and Lemeshko, Mikhail and Seiringer, Robert},
  issn         = {2469-9969},
  journal      = {Physical Review B},
  number       = {14},
  publisher    = {American Physical Society},
  title        = {{Quantum impurity model for anyons}},
  doi          = {10.1103/physrevb.102.144109},
  volume       = {102},
  year         = {2020},
}

@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{6906,
  abstract     = {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.},
  author       = {Boccato, Chiara and Brennecke, Christian and Cenatiempo, Serena and Schlein, Benjamin},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  pages        = {1311--1395},
  publisher    = {Springer},
  title        = {{Optimal rate for Bose-Einstein condensation in the Gross-Pitaevskii regime}},
  doi          = {10.1007/s00220-019-03555-9},
  volume       = {376},
  year         = {2020},
}

@article{7235,
  abstract     = {We consider the Fröhlich model of a polaron, and show that its effective mass diverges in thestrong coupling limit.},
  author       = {Lieb, Elliott H. and Seiringer, Robert},
  issn         = {1572-9613},
  journal      = {Journal of Statistical Physics},
  pages        = {23--33},
  publisher    = {Springer Nature},
  title        = {{Divergence of the effective mass of a polaron in the strong coupling limit}},
  doi          = {10.1007/s10955-019-02322-3},
  volume       = {180},
  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},
}

@phdthesis{7514,
  abstract     = {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.},
  author       = {Mayer, Simon},
  issn         = {2663-337X},
  pages        = {148},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{The free energy of a dilute two-dimensional Bose gas}},
  doi          = {10.15479/AT:ISTA:7514},
  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         = {1435-9855},
  journal      = {Journal of the European Mathematical Society},
  number       = {7},
  pages        = {2331--2403},
  publisher    = {EMS Press},
  title        = {{The excitation spectrum of Bose gases interacting through singular potentials}},
  doi          = {10.4171/JEMS/966},
  volume       = {22},
  year         = {2020},
}

@article{15072,
  abstract     = {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.},
  author       = {Hainzl, Christian and Schlein, Benjamin and Seiringer, Robert and Warzel, Simone},
  issn         = {1660-8933},
  journal      = {Oberwolfach Reports},
  number       = {3},
  pages        = {2541--2603},
  publisher    = {EMS Press},
  title        = {{Many-body quantum systems}},
  doi          = {10.4171/owr/2019/41},
  volume       = {16},
  year         = {2020},
}