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 -