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Robert Seiringer

Publications and source records attributed to Robert Seiringer.

At least 163 records · Page 9Linked to original sources
arXiv2004

Bose-Einstein Quantum Phase Transition in an Optical Lattice Model

Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an external potential, such as that presented by an optical lattice. We present a model of this phenomenon which we are able to analyze rigorously. The system is a hard core lattice gas at half-filling and the optical lattice is modeled by a periodic potential of strength $λ$. For small $λ$ and temperature, BEC is proved to occur, while at large $λ$ or temperature there is no BEC. At large $λ$ the low-temperature states are in a Mott insulator phase with a characteristic gap that is absent in the BEC phase. The interparticle interaction is essential for this transition, which occurs even in the ground state. Surprisingly, the condensation is always into the $p=0$ mode in this model, although the density itself has the periodicity of the imposed potential.

cond-mat.stat-mech↗
arXiv2004

Equivalent forms of the Bessis-Moussa-Villani conjecture

The BMV conjecture for traces, which states that $Tr exp(A -λB)$ is the Laplace transform of a positive measure, is shown to be equivalent to two other statements: (i) The polynomial $λ\mapsto Tr(A+λB)^p$ has only non-negative coefficients for all $A, B \geq 0$, $p\in N$ and (ii) $λ\mapsto \Tr (A+λB)^{-p}$ is the Laplace transform of a positive measure for $A, B \geq 0$, $p>0$.

math-ph↗
arXiv2003

Mass Renormalization and Energy Level Shift in Non-Relativistic QED

Using the Pauli-Fierz model of non-relativistic quantum electrodynamics, we calculate the binding energy of an electron in the field of a nucleus of charge $Z$ and in presence of the quantized radiation field. We consider the case of small coupling constant $α$, but fixed $Zα$ and ultraviolet cut-off $Λ$. We prove that after renormalizing the mass the binding energy has, to leading order in $α$, a finite limit as $Λ$ goes to infinity; i.e., the cut-off can be removed. The expression for the ground state energy shift thus obtained agrees with Bethe's formula for small values of $Zα$, but shows a different behavior for bigger values.

math-ph↗
arXiv2003

One-Dimensional Behavior of Dilute, Trapped Bose Gases

Recent experimental and theoretical work has shown that there are conditions in which a trapped, low-density Bose gas behaves like the one-dimensional delta-function Bose gas solved years ago by Lieb and Liniger. This is an intrinsically quantum-mechanical phenomenon because it is not necessary to have a trap width that is the size of an atom -- as might have been supposed -- but it suffices merely to have a trap width such that the energy gap for motion in the transverse direction is large compared to the energy associated with the motion along the trap. Up to now the theoretical arguments have been based on variational - perturbative ideas or numerical investigations. In contrast, this paper gives a rigorous proof of the one-dimensional behavior as far as the ground state energy and particle density are concerned. There are four parameters involved: the particle number, $N$, transverse and longitudinal dimensions of the trap, $r$ and $L$, and the scattering length $a$ of the interaction potential. Our main result is that if $r/L\to 0$ and $N\to\infty$ the ground state energy and density can be obtained by minimizing a one-dimensional density functional involving the Lieb-Liniger energy density with coupling constant $\sim a/r^2$.

math-ph↗
arXiv2003

One-dimensional Bosons in Three-dimensional Traps

Recent experimental and theoretical work has indicated conditions in which a trapped, low-density Bose gas ought to behave like the 1D delta-function Bose gas solved by Lieb and Liniger. Up to now the theoretical arguments have been based on variational - perturbative ideas or numerical investigations. There are 4 parameters: density, transverse and longitudinal dimensions, and scattering length. In this paper we explicate 5 parameter regions in which various types of 1D or 3D behavior occur. Our discussion is based on a rigorous analysis of the many-body Schroedinger equation. This version of the article is a combination of the earlier 4-page arXiv version and the revised 4-page Physical Review Letters version.

cond-mat.stat-mech↗
arXiv2003

Ground State Asymptotics of a Dilute, Rotating Gas

We investigate the ground state properties of a gas of interacting particles confined in an external potential in three dimensions and subject to rotation around an axis of symmetry. We consider the so-called Gross-Pitaevskii (GP) limit of a dilute gas. Analyzing both the absolute and the bosonic ground state of the system we show, in particular, their different behavior for a certain range of parameters. This parameter range is determined by the question whether the rotational symmetry in the minimizer of the GP functional is broken or not. For the absolute ground state, we prove that in the GP limit a modified GP functional depending on density matrices correctly describes the energy and reduced density matrices, independent of symmetry breaking. For the bosonic ground state this holds true if and only if the symmetry is unbroken.

math-ph↗
arXiv2003

The Ground State of the Bose Gas

Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, $ρ$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is $2π\hbar^2 a ρ/m$ where $a$ is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange $N^{7/5}$ law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.

math-ph↗
arXiv2002

Bose-Einstein Condensation of Dilute Gases in Traps

The ground state of a gas of Bosons confined in an external trap potential and interacting via repulsive two-body forces has recently been shown to exhibit complete Bose-Einstein condensation in the dilute limit, yielding for the first time a rigorous proof of this phenomenon in a physically realistic setting. We give here an account of this work about the Gross-Pitaevskii limit where the particle number $N$ goes to infinity with $Na$ fixed, where $a$ is the scattering length of the interparticle interaction, measured in units of the trap size.

math-ph↗
arXiv2002

Superfluidity in Dilute Trapped Bose Gases

A commonly used theoretical definition of superfluidity in the ground state of a Bose gas is based on the response of the system to an imposed velocity field or, equivalently, to twisted boundary conditions in a box. We are able to carry out this program in the case of a dilute interacting Bose gas in a trap. We prove that a gas with repulsive interactions is 100% superfluid in the dilute limit in which the Gross-Pitaevskii equation is exact. This is the first example in an experimentally realistic continuum model in which superfluidity is rigorously verified.

cond-mat↗
arXiv2002

General Decomposition of Radial Functions on R^n and Applications to N-Body Quantum Systems

We present a generalization of the Fefferman-de la Llave decomposition of the Coulomb potential to quite arbitrary radial functions $V$ on $R^n$ going to zero at infinity. This generalized decomposition can be used to extend previous results on $N$-body quantum systems with Coulomb interaction to a more general class of interactions. As an example of such an application we derive the high density asymptotics of the ground state energy of jellium with Yukawa interaction in the thermodynamic limit, using a correlation estimate by Graf and Solovej.

math-ph↗
arXiv2002

Gross-Pitaevskii Theory of the Rotating Bose Gas

We study the Gross-Pitaevskii functional for a rotating two-dimensional Bose gas in a trap. We prove that there is a breaking of the rotational symmetry in the ground state; more precisely, for any value of the angular velocity and for large enough values of the interaction strength, the ground state of the functional is not an eigenfunction of the angular momentum. This has interesting consequences on the Bose gas with spin; in particular, the ground state energy depends non-trivially on the number of spin components, and the different components do not have the same wave function. For the special case of a harmonic trap potential, we give explicit upper and lower bounds on the critical coupling constant for symmetry breaking.

math-ph↗
arXiv2002

Proof of Bose-Einstein Condensation for Dilute Trapped Gases

The ground state of bosonic atoms in a trap has been shown experimentally to display Bose-Einstein condensation (BEC). We prove this fact theoretically for bosons with two-body repulsive interaction potentials in the dilute limit, starting from the basic Schroedinger equation; the condensation is 100% into the state that minimizes the Gross-Pitaevskii energy functional. This is the first rigorous proof of BEC in a physically realistic, continuum model.

math-ph↗
arXiv2000

On the ordering of energy levels in homogeneous magnetic fields

We study the energy levels of a single particle in a homogeneous magnetic field and in an axially symmetric external potential. For potentials that are superharmonic off the central axis, we find a general ``pseudoconcave'' ordering of the ground state energies of the Hamiltonian restricted to the sectors with fixed angular momentum. The physical applications include atoms and ions in strong magnetic fields. There the energies are monotone increasing and concave in angular momentum. In the case of a periodic chain of atoms the pseudoconcavity extends to the entire lowest band of Bloch functions.

math-ph↗
arXiv2000

A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-Dimensional Bose Gas

We consider the ground state properties of an inhomogeneous two-dimensional Bose gas with a repulsive, short range pair interaction and an external confining potential. In the limit when the particle number $N$ is large but $\barρa^2$ is small, where $\barρ$ is the average particle density and $a$ the scattering length, the ground state energy and density are rigorously shown to be given to leading order by a Gross-Pitaevskii (GP) energy functional with a coupling constant $g\sim 1/|\ln(\barρa^2)|$. In contrast to the 3D case the coupling constant depends on $N$ through the mean density. The GP energy per particle depends only on $Ng$. In 2D this parameter is typically so large that the gradient term in the GP energy functional is negligible and the simpler description by a Thomas-Fermi type functional is adequate.

cond-mat.stat-mech↗
arXiv2000

Bosons in a Trap: Asymptotic Exactness of the Gross-Pitaevskii Ground State Energy Formula

Recent experimental breakthroughs in the treatment of dilute Bose gases have renewed interest in their quantum mechanical description, respectively in approximations to it. The ground state properties of dilute Bose gases confined in external potentials and interacting via repulsive short range forces are usually described by means of the Gross-Pitaevskii energy functional. In joint work with Elliott H. Lieb and Jakob Yngvason its status as an approximation for the quantum mechanical many-body ground state problem has recently been rigorously clarified. We present a summary of this work, for both the two- and three-dimensional case.

math-ph↗
arXiv2000

A discrete density matrix theory for atoms in strong magnetic fields

This paper concerns the asymptotic ground state properties of heavy atoms in strong, homogeneous magnetic fields. In the limit when the nuclear charge Z tends to infinity with the magnetic field B satisfying B >> Z^{4/3} all the electrons are confined to the lowest Landau band. We consider here an energy functional, whose variable is a sequence of one-dimensional density matrices corresponding to different angular momentum functions in the lowest Landau band. We study this functional in detail and derive various interesting properties, which are compared with the density matrix (DM) theory introduced by Lieb, Solovej and Yngvason. In contrast to the DM theory the variable perpendicular to the field is replaced by the discrete angular momentum quantum numbers. Hence we call the new functional a discrete density matrix (DDM) functional. We relate this DDM theory to the lowest Landau band quantum mechanics and show that it reproduces correctly the ground state energy apart from errors due to the indirect part of the Coulomb interaction energy.

math-ph↗
arXiv2000

Atoms with bosonic "electrons" in strong magnetic fields

We study the ground state properties of an atom with nuclear charge $Z$ and $N$ bosonic ``electrons'' in the presence of a homogeneous magnetic field $B$. We investigate the mean field limit $N\to\infty$ with $N/Z$ fixed, and identify three different asymptotic regions, according to $B\ll Z^2$, $B\sim Z^2$, and $B\gg Z^2$. In Region 1 standard Hartree theory is applicable. Region 3 is described by a one-dimensional functional, which is identical to the so-called Hyper-Strong functional introduced by Lieb, Solovej and Yngvason for atoms with fermionic electrons in the region $B\gg Z^3$; i.e., for very strong magnetic fields the ground state properties of atoms are independent of statistics. For Region 2 we introduce a general {\it magnetic Hartree functional}, which is studied in detail. It is shown that in the special case of an atom it can be restricted to the subspace of zero angular momentum parallel to the magnetic field, which simplifies the theory considerably. The functional reproduces the energy and the one-particle reduced density matrix for the full $N$-particle ground state to leading order in $N$, and it implies the description of the other regions as limiting cases.

math-ph↗