arXiv subjects

Jakob Yngvason

Publications and source records attributed to Jakob Yngvason.

At least 37 records · Page 2Linked to original sources

arXiv2013

Quantum Hall states of bosons in rotating anharmonic traps

We study a model of bosons in the lowest Landau level in a rotating trap where the confinement potential is a sum of a quadratic and a quartic term. The quartic term improves the stability of the system against centrifugal deconfinement and allows to consider rotation frequencies beyond the frequency of the quadratic part. The interactions between particles are modeled by a Dirac delta potential. We derive rigorously conditions for ground states of the system to be strongly correlated in the sense that they are confined to the kernel of the interaction operator, and thus contain the correlations of the Bose-Laughlin state. Rigorous angular momentum estimates and trial state arguments indicate a transition from a pure Laughlin state to a state containing in addition a giant vortex at the center of the trap (Laughlin quasi-hole). There are also indications of a second transition where the density changes from a flat profile in a disc or an annulus to a radial Gaussian confined to a thin annulus.

cond-mat.quant-gas↗

arXiv2013

Disordered Bose Einstein Condensates with Interaction

We study the effects of random scatterers on the ground state of the one-dimensional Lieb-Liniger model of interacting bosons on the unit interval in the Gross-Pitaevskii regime. We prove that Bose Einstein condensation survives even a strong random potential with a high density of scatterers. The character of the wave function of the condensate, however, depends in an essential way on the interplay between randomness and the strength of the two-body interaction. For low density of scatterers or strong interactions the wave function extends over the whole interval. High density of scatterers and weak interaction, on the other hand, leads to localization of the wave function in a fragmented subset of the interval.

math-ph↗

arXiv2012

Giant vortex phase transition in rapidly rotating trapped Bose-Einstein condensates

A Bose-Einstein condensate of cold atoms is a superfluid and thus responds to rotation of its container by the nucleation of quantized vortices. If the trapping potential is su ciently strong, there is no theoretical limit to the rotation frequency one can impose to the fluid, and several phase transitions characterized by the number and distribution of vortices occur when it is increased from zero to infinity. In this note we focus on a regime of very large rotation velocity where vortices disappear from the bulk of the fluid, gathering in a central hole of low matter density induced by the centrifugal force.

cond-mat.quant-gas↗

arXiv2012

Critical Rotational Speeds for Superfluids in Homogeneous Traps

We present an asymptotic analysis of the effects of rapid rotation on the ground state properties of a superfluid confined in a two-dimensional trap. The trapping potential is assumed to be radial and homogeneous of degree larger than two in addition to a quadratic term. Three critical rotational velocities are identified, marking respectively the first appearance of vortices, the creation of a `hole' of low density within a vortex lattice, and the emergence of a giant vortex state free of vortices in the bulk. These phenomena have previously been established rigorously for a `flat' trap with fixed boundary but the `soft' traps considered in the present paper exhibit some significant differences, in particular the giant vortex regime, that necessitate a new approach. These differences concern both the shape of the bulk profile and the size of vortices relative to the width of the annulus where the bulk of the superfluid resides. Close to the giant vortex transition the profile is of Thomas-Fermi type in `flat' traps, whereas it is gaussian for soft traps, and the `last' vortices to survive in the bulk before the giant vortex transition are small relative to the width of the annulus in the former case but of comparable size in the latter.

math-ph↗

arXiv2012

Rotating superfluids in anharmonic traps: From vortex lattices to giant vortices

We study a superfluid in a rotating anharmonic trap and explicate a rigorous proof of a transition from a vortex lattice to a giant vortex state as the rotation is increased beyond a limiting speed determined by the interaction strength. The transition is characterized by the disappearance of the vortices from the annulus where the bulk of the superfluid is concentrated due to centrifugal forces while a macroscopic phase circulation remains. The analysis is carried out within two-dimensional Gross-Pitaevskii theory at large coupling constant and reveals significant differences between 'soft' anharmonic traps (like a quartic plus quadratic trapping potential) and traps with a fixed boundary: In the latter case the transition takes place in a parameter regime where the size of vortices is very small relative to the width of the annulus whereas in 'soft' traps the vortex lattice persists until the width of the annulus becomes comparable to the vortex cores. Moreover, the density profile in the annulus where the bulk is concentrated is, in the 'soft' case, approximately gaussian with long tails and not of the Thomas-Fermi type like in a trap with a fixed boundary.

cond-mat.quant-gas↗

arXiv2009

The Yrast Line of a Rapidly Rotating Bose Gas: Gross-Pitaevskii Regime

We consider an ultracold rotating Bose gas in a harmonic trap close to the critical angular velocity so that the system can be considered to be confined to the lowest Landau level. With this assumption we prove that the Gross-Pitaevskii energy functional accurately describes the ground state energy of the corresponding $N$-body Hamiltonian with contact interaction provided the total angular momentum $L$ is much less than $N^2$. While the Gross-Pitaevskii energy is always an obvious variational upper bound to the ground state energy, a more refined analysis is needed to establish it as an exact lower bound. We also discuss the question of Bose-Einstein condensation in the parameter range considered. Coherent states together with inequalities in spaces of analytic functions are the main technical tools.

cond-mat.quant-gas↗

arXiv2008

Bosons in Rapid Rotation

Some recent progress in the mathematical physics of rapidly rotating, dilute Bose gases in anharmonic traps is reviewed.

math-ph↗

arXiv2007

Bose-Einstein Condensation and Spontaneous Symmetry Breaking

After recalling briefly the connection between spontaneous symmetry breaking and off-diagonal long range order for models of magnets a general proof of spontaneous breaking of gauge symmetry as a consequence of Bose-Einstein Condensation is presented. The proof is based on a rigorous validation of Bogoliubov's $c$-number substitution for the ${\bf k}={\bf 0}$ mode operator $a_{\bf 0}$.

math-ph↗

arXiv2006

The Mathematics of the Bose Gas and its Condensation

This book surveys results about the quantum mechanical many-body problem of the Bose gas that have been obtained by the authors over the last seven years. These topics are relevant to current experiments on ultra-cold gases; they are also mathematically rigorous, using many analytic techniques developed over the years to handle such problems. Some of the topics treated are the ground state energy, the Gross-Pitaevskii equation, Bose-Einstein condensation, superfluidity, one-dimensional gases, and rotating gases. The book also provides a pedagogical entry into the field for graduate students and researchers.

cond-mat.stat-mech↗

arXiv2005

Justification of c-Number Substitutions in Bosonic Hamiltonians

The validity of substituting a c-number $z$ for the $k=0$ mode operator $a_0$ is established rigorously in full generality, thereby verifying one aspect of Bogoliubov's 1947 theory. This substitution not only yields the correct value of thermodynamic quantities like the pressure or ground state energy, but also the value of $|z|^2$ that maximizes the partition function equals the true amount of condensation in the presence of a gauge-symmetry breaking term -- a point that had previously been elusive.

math-ph↗

arXiv2004

The Role of Type III Factors in Quantum Field Theory

One of von Neumann's motivations for developing the theory of operator algebras and his and Murray's 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type ${\rm III}_{1}$ factor in Connes' classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e., the way they are embedded into each other.

math-ph↗

arXiv2004

The Quantum-mechanical Many Body Problem: The Bose Gas

This article gives a detailed presentation of the authors' recent results on the ground state properties of the Bose gas. It is a much expanded version of a talk given by one of the authors (E.H.L.) at the conference "Perspectives in Analysis" at the KTH, Stockholm, June 23, 2003. It is based on, but supersedes, the article math-ph/0204027.

math-ph↗

arXiv2004

String-localized quantum fields from Wigner representations

In contrast to the usual representations of of the Poincaré group of finite spin or helicity the Wigner representations of mass zero and infinite spin are known to be incompatible with pointlike localized quantum fields. We present here a construction of quantum fields associated with these representations that are localized in semi-infinite, space-like strings. The construction is based on concepts outside the realm of Lagrangian quantization with the potential for further applications.

math-ph↗

arXiv2004

Poincare Inequalities in Punctured Domains

The classic Poincare inequality bounds the $L^q$-norm of a function $f$ in a bounded domain $Ω\subset \R^n$ in terms of some $L^p$-norm of its gradient in $Ω$. We generalize this in two ways: In the first generalization we remove a set $Γ$ from $Ω$ and concentrate our attention on $Λ= Ω\setminus Γ$. This new domain might not even be connected and hence no Poincare inequality can generally hold for it, or if it does hold it might have a very bad constant. This is so even if the volume of $Γ$ is arbitrarily small. A Poincare inequality does hold, however, if one makes the additional assumption that $f$ has a finite $L^p$ gradient norm on the whole of $Ω$, not just on $Λ$. The important point is that the Poincare inequality thus obtained bounds the $L^q$-norm of $f$ in terms of the $L^p$ gradient norm on $Λ$ (not $Ω$) plus an additional term that goes to zero as the volume of $Γ$ goes to zero. This error term depends on $Γ$ only through its volume. Apart from this additive error term, the constant in the inequality remains that of the `nice' domain $Ω$. In the second generalization we are given a vector field $A$ and replace $\nabla $ by $\nabla +i A(x)$ (geometrically, a connection on a U(1) bundle). Unlike the A=0 case, the infimum of $\|(\nabla +i A)f\|_p $ over all $f$ with a given $\|f\|_q$ is in general not zero. This permits an improvement of the inequality by the addition of a term whose sharp value we derive. We describe some open problems that arise from these generalizations.

math.FA↗

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↗

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

Asymptotic Exactness of Magnetic Thomas-Fermi Theory

We consider the grand canonical pressure for Coulombic matter with nuclear charges $\sim Z$ in a magnetic field $B$ and at nonzero temperature. We prove that its asymptotic limit as $Z\to\infty$ with $B/Z^3\to 0$ can be obtained by minimizing a Thomas-Fermi type pressure functional.

math-ph↗