@article{7685,
abstract = {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.},
author = {Boccato, Chiara},
issn = {0129-055X},
journal = {Reviews in Mathematical Physics},
number = {1},
publisher = {World Scientific},
title = {{The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime}},
doi = {10.1142/S0129055X20600065},
volume = {33},
year = {2021},
}
@article{7900,
abstract = {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.},
author = {Benedikter, Niels P},
issn = {0129-055X},
journal = {Reviews in Mathematical Physics},
number = {1},
publisher = {World Scientific},
title = {{Bosonic collective excitations in Fermi gases}},
doi = {10.1142/s0129055x20600090},
volume = {33},
year = {2021},
}
@article{9005,
abstract = {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.},
author = {Brooks, Morris and Lemeshko, Mikhail and Lundholm, D. and Yakaboylu, Enderalp},
issn = {10797114},
journal = {Physical Review Letters},
number = {1},
publisher = {American Physical Society},
title = {{Molecular impurities as a realization of anyons on the two-sphere}},
doi = {10.1103/PhysRevLett.126.015301},
volume = {126},
year = {2021},
}
@article{9225,
abstract = {The Landau–Pekar equations describe the dynamics of a strongly coupled polaron.
Here, we provide a class of initial data for which the associated effective Hamiltonian
has a uniform spectral gap for all times. For such initial data, this allows us to extend the
results on the adiabatic theorem for the Landau–Pekar equations and their derivation
from the Fröhlich model obtained in previous works to larger times.},
author = {Feliciangeli, Dario and Rademacher, Simone Anna Elvira and Seiringer, Robert},
issn = {15730530},
journal = {Letters in Mathematical Physics},
publisher = {Springer Nature},
title = {{Persistence of the spectral gap for the Landau–Pekar equations}},
doi = {10.1007/s11005-020-01350-5},
volume = {111},
year = {2021},
}
@article{8603,
abstract = {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.},
author = {Frank, Rupert and Seiringer, Robert},
issn = {10970312},
journal = {Communications on Pure and Applied Mathematics},
number = {3},
pages = {544--588},
publisher = {Wiley},
title = {{Quantum corrections to the Pekar asymptotics of a strongly coupled polaron}},
doi = {10.1002/cpa.21944},
volume = {74},
year = {2021},
}
@article{9246,
abstract = {We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle numbers and a suitably small coupling, we show that the dynamics of the system is approximately described by the Landau–Pekar equations. These describe a Bose–Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order.},
author = {Leopold, Nikolai K and Mitrouskas, David Johannes and Seiringer, Robert},
issn = {14320673},
journal = {Archive for Rational Mechanics and Analysis},
pages = {383--417},
publisher = {Springer Nature},
title = {{Derivation of the Landau–Pekar equations in a many-body mean-field limit}},
doi = {10.1007/s00205-021-01616-9},
volume = {240},
year = {2021},
}
@article{9318,
abstract = {We consider a system of N bosons in the mean-field scaling regime for a class of interactions including the repulsive Coulomb potential. We derive an asymptotic expansion of the low-energy eigenstates and the corresponding energies, which provides corrections to Bogoliubov theory to any order in 1/N.},
author = {Bossmann, Lea and Petrat, Sören P and Seiringer, Robert},
issn = {20505094},
journal = {Forum of Mathematics, Sigma},
publisher = {Cambridge University Press},
title = {{Asymptotic expansion of low-energy excitations for weakly interacting bosons}},
doi = {10.1017/fms.2021.22},
volume = {9},
year = {2021},
}
@article{9333,
abstract = {We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state φ0⊗ξα, where φ0 is the electron ground state of the Pekar energy functional and ξα the associated coherent state of the phonons, can be approximated by a global phase for times small compared to α2. In the present note we prove that a similar approximation holds for t=O(α2) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to α−2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order α2, while the phonon fluctuations around the coherent state ξα can be described by a time-dependent Bogoliubov transformation.},
author = {Mitrouskas, David Johannes},
issn = {15730530},
journal = {Letters in Mathematical Physics},
publisher = {Springer Nature},
title = {{A note on the Fröhlich dynamics in the strong coupling limit}},
doi = {10.1007/s11005-021-01380-7},
volume = {111},
year = {2021},
}
@article{9351,
abstract = {We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates that are consistent with central limit theorems that have been established in the last years. },
author = {Kirkpatrick, Kay and Rademacher, Simone Anna Elvira and Schlein, Benjamin},
issn = {1424-0637},
journal = {Annales Henri Poincare},
publisher = {Springer Nature},
title = {{A large deviation principle in many-body quantum dynamics}},
doi = {10.1007/s00023-021-01044-1},
year = {2021},
}
@article{9348,
abstract = {We consider the stochastic quantization of a quartic double-well energy functional in the semiclassical regime and derive optimal asymptotics for the exponentially small splitting of the ground state energy. Our result provides an infinite-dimensional version of some sharp tunneling estimates known in finite dimensions for semiclassical Witten Laplacians in degree zero. From a stochastic point of view it proves that the L2 spectral gap of the stochastic one-dimensional Allen-Cahn equation in finite volume satisfies a Kramers-type formula in the limit of vanishing noise. We work with finite-dimensional lattice approximations and establish semiclassical estimates which are uniform in the dimension. Our key estimate shows that the constant separating the two exponentially small eigenvalues from the rest of the spectrum can be taken independently of the dimension.},
author = {Brooks, Morris and Di Gesù, Giacomo},
issn = {10960783},
journal = {Journal of Functional Analysis},
number = {3},
publisher = {Elsevier},
title = {{Sharp tunneling estimates for a double-well model in infinite dimension}},
doi = {10.1016/j.jfa.2021.109029},
volume = {281},
year = {2021},
}
@article{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 = {14359855},
journal = {Journal of the European Mathematical Society},
number = {7},
pages = {2331--2403},
publisher = {European Mathematical Society},
title = {{The excitation spectrum of Bose gases interacting through singular potentials}},
doi = {10.4171/JEMS/966},
volume = {22},
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 = {15729613},
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 = {0003-9527},
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 = {00222488},
journal = {Journal of Mathematical Physics},
number = {6},
publisher = {AIP},
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 = {0021-9606},
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 = {2469-9950},
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{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 = {0022-4715},
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 = {IST Austria},
title = {{The free energy of a dilute two-dimensional Bose gas}},
doi = {10.15479/AT:ISTA:7514},
year = {2020},
}