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
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 - 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
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 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
JF - Reviews in Mathematical Physics
SN - 0129055X
TI - The excitation spectrum of the Bose gas in the Gross-Pitaevskii regime
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 - 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
JF - Reviews in Mathematical Physics
SN - 0129-055X
TI - Bosonic collective excitations in Fermi gases
ER -
TY - GEN
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
T2 - ArXiv
TI - Correlation energy of a weakly interacting Fermi gas
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 - 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
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
JF - Archive for Rational Mechanics and Analysis
SN - 0003-9527
TI - Derivation of the 2d Gross–Pitaevskii equation for strongly confined 3d Bosons
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
AU - Jaksic, Vojkan
AU - Seiringer, Robert
ID - 7226
IS - 12
JF - Journal of Mathematical Physics
SN - 00222488
TI - Introduction to the Special Collection: International Congress on Mathematical Physics (ICMP) 2018
VL - 60
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
ER -
TY - JOUR
AB - 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.
AU - Boccato, Chiara
AU - Brennecke, Christian
AU - Cenatiempo, Serena
AU - Schlein, Benjamin
ID - 7413
IS - 2
JF - Acta Mathematica
SN - 0001-5962
TI - Bogoliubov theory in the Gross–Pitaevskii limit
VL - 222
ER -
TY - GEN
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 $\rho$ 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 $\beta \rho \gtrsim 1$.
AU - Deuchert, Andreas
AU - Mayer, Simon
AU - Seiringer, Robert
ID - 7524
T2 - arXiv:1910.03372
TI - The free energy of the two-dimensional dilute Bose gas. I. Lower bound
ER -
TY - JOUR
AB - 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.
AU - Moser, Thomas
AU - Seiringer, Robert
ID - 5856
IS - 4
JF - Annales Henri Poincare
SN - 14240637
TI - Energy contribution of a point-interacting impurity in a Fermi gas
VL - 20
ER -
TY - JOUR
AB - 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.
AU - Leopold, Nikolai K
AU - Petrat, Sören P
ID - 6788
IS - 10
JF - Annales Henri Poincare
SN - 1424-0637
TI - Mean-field dynamics for the Nelson model with fermions
VL - 20
ER -
TY - JOUR
AB - 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.
AU - Mysliwy, Krzysztof
AU - Napiórkowski, Marek
ID - 6840
IS - 6
JF - Journal of Statistical Mechanics: Theory and Experiment
TI - Thermodynamics of inhomogeneous imperfect quantum gases in harmonic traps
VL - 2019
ER -