Asymptotic expansion of low-energy excitations for weakly interacting bosons

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 $1/N$ .

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Low-energy excitations from interacting tunneling units in the mean-field approximation

Journal of Non-Crystalline Solids ◽

10.1016/0022-3093(91)90306-q ◽

1991 ◽

Vol 131-133 ◽

pp. 225-228

Author(s):

Peter Nielaba ◽

Michael W Klein

Keyword(s):

Mean Field ◽

Field Approximation ◽

Low Energy ◽

Mean Field Approximation ◽

The Mean

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η-PAIRING SUPERCONDUCTIVITY IN THE NEGATIVE-U HUBBARD MODEL: EFFECT OF FLUCTUATIONS AND DISORDER

International Journal of Modern Physics B ◽

10.1142/s0217979294000841 ◽

1994 ◽

Vol 08 (15) ◽

pp. 2041-2058

Author(s):

JÜRGEN STEIN

Keyword(s):

Phase Diagram ◽

Asymptotic Expansion ◽

Hubbard Model ◽

Strong Coupling ◽

Mean Field ◽

Higher Order ◽

Field Phase ◽

Effective Pair ◽

The Mean ◽

Pair Hopping

We have studied the influence of singular fluctuations around the mean-field solution as well as higher order contributions to the geometry controlled asymptotic expansion of the propagators on η-pairing superconductivity in the strong coupling negative-U Hubbard model in the presence of unpaired electrons. The modifications of the mean-field results due to the introduction of disorder and allowance for finite U are also calculated. Besides the singularities already present in the O(2) nonlinear σ-model we find a filling-depending singular depression of the repulsive effective pair-hopping interaction which strongly alters the mean-field phase diagram and appears to suppress the η-pairing phase near the band edges.

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Equilibration of weakly nonlinear salt fingers

Journal of Fluid Mechanics ◽

10.1017/s0022112009992552 ◽

2010 ◽

Vol 645 ◽

pp. 121-143 ◽

Cited By ~ 14

Author(s):

TIMOUR RADKO

Keyword(s):

Asymptotic Expansion ◽

Linear Growth ◽

Mean Field ◽

Vertical Shear ◽

Weakly Nonlinear ◽

Salinity Gradients ◽

Salt Fingers ◽

Wide Range ◽

Shear Effects ◽

The Mean

An analytical model is developed to explain the equilibration mechanism of the salt finger instability in unbounded temperature and salinity gradients. The theory is based on the weakly nonlinear asymptotic expansion about the point of marginal instability. The proposed solutions attribute equilibration of salt fingers to a combination of two processes: (i) the triad interaction and (ii) spontaneous development of the mean vertical shear. The non-resonant triad interactions control the equilibration of linear growth for moderate and large values of Prandtl number (Pr) and for slightly unstable parameters. For small Pr and/or rigorous instabilities, the mean shear effects become essential. It is shown that, individually, neither the mean field nor the triad interaction models can accurately describe the equilibrium patterns of salt fingers in all regions of the parameter space. Therefore, we propose a new hybrid model, which represents both stabilizing effects in a single framework. The resulting solutions agree with the fully nonlinear numerical simulations over a wide range of governing parameters.

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Bogoliubov corrections and trace norm convergence for the Hartree dynamics

Reviews in Mathematical Physics ◽

10.1142/s0129055x19500247 ◽

2019 ◽

Vol 31 (08) ◽

pp. 1950024 ◽

Cited By ~ 4

Author(s):

David Mitrouskas ◽

Sören Petrat ◽

Peter Pickl

Keyword(s):

Time Evolution ◽

Convergence Rates ◽

Fock Space ◽

Mean Field ◽

Optimal Rate ◽

Trace Norm ◽

Mean Field Limit ◽

Bogoliubov Theory ◽

The Mean ◽

The One

We consider the dynamics of a large number [Formula: see text] of nonrelativistic bosons in the mean field limit for a class of interaction potentials that includes Coulomb interaction. In order to describe the fluctuations around the mean field Hartree state, we introduce an auxiliary Hamiltonian on the [Formula: see text]-particle space that is similar to the one obtained from Bogoliubov theory. We show convergence of the auxiliary time evolution to the fully interacting dynamics in the norm of the [Formula: see text]-particle space. This result allows us to prove several other results: convergence of reduced density matrices in trace norm with optimal rate, convergence in energy trace norm, and convergence to a time evolution obtained from the Bogoliubov Hamiltonian on Fock space with expected optimal rate. We thus extend and quantify several previous results, e.g., by providing the physically important convergence rates, including time-dependent external fields and singular interactions, and allowing for more general initial states, e.g., those that are expected to be ground states of interacting systems.

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FLAVOR MIXING AND THE UA(1) ANOMALY

Modern Physics Letters A ◽

10.1142/s0217732389001969 ◽

1989 ◽

Vol 04 (18) ◽

pp. 1737-1743 ◽

Cited By ~ 7

Author(s):

R. ALKOFER ◽

I. ZAHED

Keyword(s):

Mean Field ◽

Low Energy ◽

Bulk Properties ◽

Qcd Vacuum ◽

Field Level ◽

Large N ◽

Generic Character ◽

Flavor Mixing ◽

The Mean ◽

Complex Origin

The relevance of the UA(1) anomaly is discussed in the context of models with constituent quarks. It is argued that at the mean-field level (large Nc), the inclusion of the UA(1)anomaly does not necessarily lead to flavor mixing as generally believed. Arguments are presented for the generic character of flavor dynamics at low energy and the complex origin of flavor mixing in relation to the bulk properties of the QCD vacuum.

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CRITICAL TEMPERATURE OF A WEAKLY INTERACTING BOSE GAS IN A POWER-LAW POTENTIAL

International Journal of Modern Physics B ◽

10.1142/s0217979203018442 ◽

2003 ◽

Vol 17 (12) ◽

pp. 2439-2446 ◽

Cited By ~ 3

Author(s):

HIDENORI SUZUKI ◽

MASUO SUZUKI

Keyword(s):

Critical Temperature ◽

Chemical Potential ◽

Particle Density ◽

Dimensional Space ◽

Mean Field ◽

Bose Gas ◽

Ideal Gas ◽

Mean Field Approximation ◽

Weakly Interacting ◽

The Mean

The critical temperature T c of a weakly interacting Bose gas in an isotropic power-low potential is investigated in the mean-field approximation by taking into account the fact that the particle density distribution function appearing in the mean-field depends on the chemical potential. We derive the general formula of the shift of T c from that of the ideal gas to the lowest order of an interaction. In three-dimensional space, we show that the shift of T c changes its sign from a negative value for n < 3 to a positive one for n > 3, where n is the exponent of the power-low potential.

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On the Diffusive-Mean Field Limit for Weakly Interacting Diffusions Exhibiting Phase Transitions

Archive for Rational Mechanics and Analysis ◽

10.1007/s00205-021-01648-1 ◽

2021 ◽

Author(s):

Matias G. Delgadino ◽

Rishabh S. Gvalani ◽

Grigorios A. Pavliotis

Keyword(s):

Diffusion Processes ◽

Mean Field ◽

Rates Of Convergence ◽

Kuramoto Model ◽

Field Energy ◽

Field Limit ◽

Mean Field Limit ◽

Weakly Interacting ◽

The Mean ◽

Field Plane

AbstractThe objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say, if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature.

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