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 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 -
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 - 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 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 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 -