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Robert Seiringer

Publications and source records attributed to Robert Seiringer.

At least 55 records · Page 3Linked to original sources
arXiv2021

Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons

We consider a system of N trapped bosons with repulsive interactions in a combined semiclassical mean-field limit at positive temperature. We show that the free energy is well approximated by the minimum of the Hartree free energy functional - a natural extension of the Hartree energy functional to positive temperatures. The Hartree free energy functional converges in the same limit to a semiclassical free energy functional, and we show that the system displays Bose-Einstein condensation if and only if it occurs in the semiclassical free energy functional. This allows us to show that for weak coupling the critical temperature decreases due to the repulsive interactions.

math-ph↗
arXiv2021

Correlation Energy of a Weakly Interacting Fermi Gas

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.

math-ph↗
arXiv2021

Free energy asymptotics of the quantum Heisenberg spin chain

We consider the ferromagnetic quantum Heisenberg model in one dimension, for any spin $S\geq 1/2$. We give upper and lower bounds on the free energy, proving that at low temperature it is asymptotically equal to the one of an ideal Bose gas of magnons, as predicted by the spin-wave approximation. The trial state used in the upper bound yields an analogous estimate also in the case of two spatial dimensions, which is believed to be sharp at low temperature.

math-ph↗
arXiv2021

Landau-Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron

We consider the Fröhlich Hamiltonian with large coupling constant $α$. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm approximation of the evolution, valid up to times of order $α^2$. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. This allows us to show that the Landau-Pekar equations approximately describe the evolution of the electron- and one-phonon reduced density matrices under the Fröhlich dynamics up to times of order $α^2$.

math-ph↗
arXiv2021

Derivation of the Landau-Pekar equations in a many-body mean-field limit

We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle number and 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.

math-ph↗
arXiv2020

A Quantum Impurity Model for Anyons

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/vortices to it and thereby converts it into an Abelian anyon. The corresponding quantum impurity model, first, provides a new approach to the numerical solution of the many-anyon problem, along with a new concrete perspective of anyons as emergent quasiparticles built from composite bosons or fermions. More importantly, the model paves the way towards 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 are impurities immersed in a two-dimensional weakly interacting Bose gas.

cond-mat.quant-gas↗
arXiv2020

Persistence of the spectral gap for the Landau-Pekar equations

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 Froehlich model obtained in [8, 7] to larger times.

math-ph↗
arXiv2020

The Polaron at Strong Coupling

We review old and new results on the Fröhlich polaron model. The discussion includes the validity of the (classical) Pekar approximation in the strong coupling limit, quantum corrections to this limit, as well as the divergence of the effective polaron mass.

math-ph↗
arXiv2020

Emergence of Haldane pseudo-potentials in systems with short-range interactions

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 \cite{LS}, 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.

math-ph↗
arXiv2020

The free energy of the two-dimensional dilute Bose gas. II. Upper bound

We prove an upper bound on the free energy of a two-dimensional homogeneous Bose gas in the thermodynamic limit. We show that for $a^2 ρ\ll 1$ and $βρ\gtrsim 1$ the free energy per unit volume differs from the one of the non-interacting system by at most $4 πρ^2 |\ln a^2 ρ|^{-1} (2 - [1 - β_{\mathrm{c}}/β]_+^2)$ to leading order, where $a$ is the scattering length of the two-body interaction potential, $ρ$ is the density, $β$ the inverse temperature and $β_{\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. In combination with the corresponding matching lower bound proved in \cite{DMS19} this shows equality in the asymptotic expansion.

math-ph↗
arXiv2020

The free energy of the two-dimensional dilute Bose gas. I. Lower bound

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 non-interacting system by the correction term $4 πρ^2 |\ln a^2 ρ|^{-1} (2 - [1 - β_{\mathrm{c}}/β]_+^2)$. Here $a$ is the scattering length of the interaction potential, $[\cdot]_+ = \max\{ 0, \cdot \}$ and $β_{\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. The result is valid in the dilute limit $a^2ρ\ll 1$ and if $βρ\gtrsim 1$.

math-ph↗
arXiv2020

Quantum corrections to the Pekar asymptotics of a strongly coupled polaron

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.

math-ph↗
arXiv2020

Gross-Pitaevskii Limit of a Homogeneous Bose Gas at Positive Temperature

We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross-Pitaevskii (GP) 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 $4 πa \left( 2 \varrho^2 - \varrho_0^2 \right)$. Here $\varrho$ denotes the density of the system and $\varrho_0$ 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.

math-ph↗
arXiv2019

The Local Density Approximation in Density Functional Theory

We give the first mathematically rigorous justification of the Local Density Approximation in Density Functional Theory. We provide a quantitative estimate on the difference between the grand-canonical Levy-Lieb energy of a given density (the lowest possible energy of all quantum states having this density) and the integral over the Uniform Electron Gas energy of this density. The error involves gradient terms and justifies the use of the Local Density Approximation in the situation where the density is very flat on sufficiently large regions in space.

math-ph↗
arXiv2019

Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet

We consider the quantum ferromagnetic Heisenberg model in three dimensions, for all spins $S\geq 1/2$. We rigorously prove the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial contribution to the free energy at low temperatures. Our proof comes with explicit, constructive upper and lower bounds on the error term. It uses in an essential way the bosonic formulation of the model in terms of the Holstein-Primakoff representation. In this language, the model describes interacting bosons with a hard-core on-site repulsion and a nearest-neighbor attraction. This attractive interaction makes the lower bound on the free energy particularly tricky: the key idea there is to prove a differential inequality for the two-particle density, which is thereby shown to be smaller than the probability density of a suitably weighted two-particle random process on the lattice.

math-ph↗
arXiv2019

Floating Wigner crystal with no boundary charge fluctuations

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.

cond-mat.str-el↗
arXiv2019

Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime

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.

math-ph↗