@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},
publisher = {World Scientific},
title = {{Bosonic collective excitations in Fermi gases}},
doi = {10.1142/s0129055x20600090},
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},
}
@unpublished{7901,
abstract = {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.},
author = {Benedikter, Niels P and Nam, Phan Thành and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert},
booktitle = {ArXiv},
pages = {56},
publisher = {ArXiv},
title = {{Correlation energy of a weakly interacting Fermi gas}},
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},
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},
year = {2020},
}
@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 = {0129055X},
journal = {Reviews in Mathematical Physics},
publisher = {World Scientific},
title = {{The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime}},
doi = {10.1142/S0129055X20600065},
year = {2020},
}
@article{7611,
abstract = {We consider a system of N bosons in the limit N→∞, interacting through singular potentials. For initial data exhibiting Bose–Einstein condensation, the many-body time evolution is well approximated through a quadratic fluctuation dynamics around a cubic nonlinear Schrödinger equation of the condensate wave function. We show that these fluctuations satisfy a (multi-variate) central limit theorem.},
author = {Rademacher, Simone Anna Elvira},
issn = {0377-9017},
journal = {Letters in Mathematical Physics},
publisher = {Springer Nature},
title = {{Central limit theorem for Bose gases interacting through singular potentials}},
doi = {10.1007/s11005-020-01286-w},
year = {2020},
}
@article{6649,
abstract = {While Hartree–Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree–Fock state given by plane waves and introduce collective particle–hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann–Brueckner–type upper bound to the ground state energy. Our result justifies the random-phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials.
},
author = {Benedikter, Niels P and Nam, Phan Thành and Porta, Marcello and Schlein, Benjamin and Seiringer, Robert},
issn = {1432-0916},
journal = {Communications in Mathematical Physics},
pages = {2097–2150},
publisher = {Springer Nature},
title = {{Optimal upper bound for the correlation energy of a Fermi gas in the mean-field regime}},
doi = {10.1007/s00220-019-03505-5},
volume = {374},
year = {2020},
}
@article{7650,
abstract = {We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross–Pitaevskii limit, where the scattering length a is so small that the interaction energy is of the same order of magnitude as the spectral gap of the Laplacian, and for temperatures that are comparable to the critical temperature of the ideal gas. We show that the difference between the specific free energy of the interacting system and the one of the ideal gas is to leading order given by 4πa(2ϱ2−ϱ20). Here ϱ denotes the density of the system and ϱ0 is the expected condensate density of the ideal gas. Additionally, we show that the one-particle density matrix of any approximate minimizer of the Gibbs free energy functional is to leading order given by the one of the ideal gas. This in particular proves Bose–Einstein condensation with critical temperature given by the one of the ideal gas to leading order. One key ingredient of our proof is a novel use of the Gibbs variational principle that goes hand in hand with the c-number substitution.},
author = {Deuchert, Andreas and Seiringer, Robert},
issn = {0003-9527},
journal = {Archive for Rational Mechanics and Analysis},
number = {6},
pages = {1217--1271},
publisher = {Springer Nature},
title = {{Gross-Pitaevskii limit of a homogeneous Bose gas at positive temperature}},
doi = {10.1007/s00205-020-01489-4},
volume = {236},
year = {2020},
}
@article{7790,
abstract = {We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density 𝜌 and inverse temperature 𝛽 differs from the one of the noninteracting system by the correction term 𝜋𝜌𝜌𝛽𝛽 . Here, is the scattering length of the interaction potential, and 𝛽 is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. The result is valid in the dilute limit 𝜌 and if 𝛽𝜌 .},
author = {Deuchert, Andreas and Mayer, Simon and Seiringer, Robert},
issn = {20505094},
journal = {Forum of Mathematics, Sigma},
publisher = {Cambridge University Press},
title = {{The free energy of the two-dimensional dilute Bose gas. I. Lower bound}},
doi = {10.1017/fms.2020.17},
volume = {8},
year = {2020},
}
@article{80,
abstract = {We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross–Pitaevskii (GP) limit where the trap frequency ω, the temperature T, and the particle number N are related by N∼ (T/ ω) 3→ ∞ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.},
author = {Deuchert, Andreas and Seiringer, Robert and Yngvason, Jakob},
journal = {Communications in Mathematical Physics},
number = {2},
pages = {723--776},
publisher = {Springer},
title = {{Bose–Einstein condensation in a dilute, trapped gas at positive temperature}},
doi = {10.1007/s00220-018-3239-0},
volume = {368},
year = {2019},
}
@article{6840,
abstract = {We discuss thermodynamic properties of harmonically trapped
imperfect quantum gases. The spatial inhomogeneity of these systems imposes
a redefinition of the mean-field interparticle potential energy as compared
to the homogeneous case. In our approach, it takes the form a
2N2 ωd, where
N is the number of particles, ω—the harmonic trap frequency, d—system’s
dimensionality, and a is a parameter characterizing the interparticle interaction.
We provide arguments that this model corresponds to the limiting case of
a long-ranged interparticle potential of vanishingly small amplitude. This
conclusion is drawn from a computation similar to the well-known Kac scaling
procedure, which is presented here in a form adapted to the case of an isotropic
harmonic trap. We show that within the model, the imperfect gas of trapped
repulsive bosons undergoes the Bose–Einstein condensation provided d > 1.
The main result of our analysis is that in d = 1 the gas of attractive imperfect
fermions with a = −aF < 0 is thermodynamically equivalent to the gas of
repulsive bosons with a = aB > 0 provided the parameters aF and aB fulfill
the relation aB + aF = . This result supplements similar recent conclusion
about thermodynamic equivalence of two-dimensional (2D) uniform imperfect
repulsive Bose and attractive Fermi gases.},
author = {Mysliwy, Krzysztof and Napiórkowski, Marek},
issn = {1742-5468},
journal = {Journal of Statistical Mechanics: Theory and Experiment},
number = {6},
publisher = {IOP Publishing},
title = {{Thermodynamics of inhomogeneous imperfect quantum gases in harmonic traps}},
doi = {10.1088/1742-5468/ab190d},
volume = {2019},
year = {2019},
}
@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},
publisher = {Springer},
title = {{Divergence of the effective mass of a polaron in the strong coupling limit}},
doi = {10.1007/s10955-019-02322-3},
year = {2019},
}
@article{7413,
abstract = {We consider Bose gases consisting of N particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of order N−1 (Gross–Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as N→∞. Our results confirm Bogoliubov’s predictions.},
author = {Boccato, Chiara and Brennecke, Christian and Cenatiempo, Serena and Schlein, Benjamin},
issn = {0001-5962},
journal = {Acta Mathematica},
number = {2},
pages = {219--335},
publisher = {International Press of Boston},
title = {{Bogoliubov theory in the Gross–Pitaevskii limit}},
doi = {10.4310/acta.2019.v222.n2.a1},
volume = {222},
year = {2019},
}
@article{6788,
abstract = {We consider the Nelson model with ultraviolet cutoff, which describes the interaction between non-relativistic particles and a positive or zero mass quantized scalar field. We take the non-relativistic particles to obey Fermi statistics and discuss the time evolution in a mean-field limit of many fermions. In this case, the limit is known to be also a semiclassical limit. We prove convergence in terms of reduced density matrices of the many-body state to a tensor product of a Slater determinant with semiclassical structure and a coherent state, which evolve according to a fermionic version of the Schrödinger–Klein–Gordon equations.},
author = {Leopold, Nikolai K and Petrat, Sören P},
issn = {1424-0661},
journal = {Annales Henri Poincare},
number = {10},
pages = {3471–3508},
publisher = {Springer Nature},
title = {{Mean-field dynamics for the Nelson model with fermions}},
doi = {10.1007/s00023-019-00828-w},
volume = {20},
year = {2019},
}
@article{7015,
abstract = {We modify the "floating crystal" trial state for the classical homogeneous electron gas (also known as jellium), in order to suppress the boundary charge fluctuations that are known to lead to a macroscopic increase of the energy. The argument is to melt a thin layer of the crystal close to the boundary and consequently replace it by an incompressible fluid. With the aid of this trial state we show that three different definitions of the ground-state energy of jellium coincide. In the first point of view the electrons are placed in a neutralizing uniform background. In the second definition there is no background but the electrons are submitted to the constraint that their density is constant, as is appropriate in density functional theory. Finally, in the third system each electron interacts with a periodic image of itself; that is, periodic boundary conditions are imposed on the interaction potential.},
author = {Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert},
issn = {2469-9950},
journal = {Physical Review B},
number = {3},
publisher = {APS},
title = {{Floating Wigner crystal with no boundary charge fluctuations}},
doi = {10.1103/physrevb.100.035127},
volume = {100},
year = {2019},
}
@article{7100,
abstract = {We present microscopic derivations of the defocusing two-dimensional cubic nonlinear Schrödinger equation and the Gross–Pitaevskii equation starting froman interacting N-particle system of bosons. We consider the interaction potential to be given either by Wβ(x)=N−1+2βW(Nβx), for any β>0, or to be given by VN(x)=e2NV(eNx), for some spherical symmetric, nonnegative and compactly supported W,V∈L∞(R2,R). In both cases we prove the convergence of the reduced density corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential VN we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.},
author = {Jeblick, Maximilian and Leopold, Nikolai K and Pickl, Peter},
issn = {1432-0916},
journal = {Communications in Mathematical Physics},
number = {1},
pages = {1--69},
publisher = {Springer Nature},
title = {{Derivation of the time dependent Gross–Pitaevskii equation in two dimensions}},
doi = {10.1007/s00220-019-03599-x},
volume = {372},
year = {2019},
}
@article{5856,
abstract = {We give a bound on the ground-state energy of a system of N non-interacting fermions in a three-dimensional cubic box interacting with an impurity particle via point interactions. We show that the change in energy compared to the system in the absence of the impurity is bounded in terms of the gas density and the scattering length of the interaction, independently of N. Our bound holds as long as the ratio of the mass of the impurity to the one of the gas particles is larger than a critical value m∗ ∗≈ 0.36 , which is the same regime for which we recently showed stability of the system.},
author = {Moser, Thomas and Seiringer, Robert},
issn = {14240637},
journal = {Annales Henri Poincare},
number = {4},
pages = {1325–1365},
publisher = {Springer},
title = {{Energy contribution of a point-interacting impurity in a Fermi gas}},
doi = {10.1007/s00023-018-00757-0},
volume = {20},
year = {2019},
}
@article{6906,
abstract = {We consider systems of bosons trapped in a box, in the Gross–Pitaevskii regime. We show that low-energy states exhibit complete Bose–Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in Boccato et al. (Commun Math Phys 359(3):975–1026, 2018), removing the assumption of small interaction potential.},
author = {Boccato, Chiara and Brennecke, Christian and Cenatiempo, Serena and Schlein, Benjamin},
issn = {0010-3616},
journal = {Communications in Mathematical Physics},
publisher = {Springer},
title = {{Optimal rate for Bose-Einstein condensation in the Gross-Pitaevskii regime}},
doi = {10.1007/s00220-019-03555-9},
year = {2019},
}
@article{7226,
author = {Jaksic, Vojkan and Seiringer, Robert},
issn = {00222488},
journal = {Journal of Mathematical Physics},
number = {12},
publisher = {AIP},
title = {{Introduction to the Special Collection: International Congress on Mathematical Physics (ICMP) 2018}},
doi = {10.1063/1.5138135},
volume = {60},
year = {2019},
}
@unpublished{7524,
abstract = {We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density $\rho$ and inverse temperature $\beta$ differs from the one of the non-interacting system by the correction term $4 \pi \rho^2 |\ln a^2 \rho|^{-1} (2 - [1 - \beta_{\mathrm{c}}/\beta]_+^2)$. Here $a$ is the scattering length of the interaction potential, $[\cdot]_+ = \max\{ 0, \cdot \}$ and $\beta_{\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. The result is valid in the dilute limit
$a^2\rho \ll 1$ and if $\beta \rho \gtrsim 1$.},
author = {Deuchert, Andreas and Mayer, Simon and Seiringer, Robert},
booktitle = {arXiv:1910.03372},
pages = {61},
publisher = {ArXiv},
title = {{The free energy of the two-dimensional dilute Bose gas. I. Lower bound}},
year = {2019},
}