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 - 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 - 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 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 - 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 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
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 - 00221236
TI - Sharp tunneling estimates for a double-well model in infinite dimension
VL - 281
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 - 00209910
TI - Correlation energy of a weakly interacting Fermi gas
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 - 00221236
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 - 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 - 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 - 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 - 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 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 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 -