arXiv subjects

J. Yngvason

Publications and source records attributed to J. Yngvason.

18 recordsLinked to original sources

arXiv2014

Superfluid behavior of a Bose-Einstein condensate in a random potential

We investigate the relation between Bose-Einstein condensation (BEC) and superfluidity in the ground state of a one-dimensional model of interacting Bosons in a strong random potential. We prove rigorously that in a certain parameter regime the superfluid fraction can be arbitrarily small while complete BEC prevails. In another regime there is both complete BEC and complete superfluidity, despite the strong disorder.

cond-mat.quant-gas↗

arXiv2012

Vortex Phases of Rotating Superfluids

We report on the first mathematically rigorous proofs of a transition to a giant vortex state of a superfluid in rotating anharmonic traps. The analysis is carried out within two-dimensional Gross-Pitaevskii theory at large coupling constant and large rotational velocity and is based on precise asymptotic estimates on the ground state energy. An interesting aspect is a significant difference between 'soft' anharmonic traps (like a quartic plus quadratic trapping potential) and traps with a fixed boundary. In the former case vortices persist in the bulk until the width of the annulus becomes comparable to the size of the vortex cores. In the second case the transition already takes place in a parameter regime where the size of vortices is very small relative to the width of the annulus. Moreover, the density profiles in the annulus are different in the two cases. In both cases rotational symmetry of the density in a true ground state is broken, even though a symmetric variational ansatz gives an excellent approximation to the energy.

cond-mat.quant-gas↗

arXiv2012

Disordered Bose Einstein Condensates with Interaction in One Dimension

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

Massless, String Localized Quantum Fields for Any Helicity

For any massless, irreducible representation of the covering of the proper, orthochronous Poincaré group we construct covariant, free quantum fields that generate the representation space from the vacuum and are localized in semi-infinite strings in the sense of commutation or anti-commutation of the field operators at space-like separation of the strings.

math-ph↗

arXiv2011

Critical Rotational Speeds in the Gross-Pitaevskii Theory on a Disc with Dirichlet Boundary Conditions

We study the two-dimensional Gross-Pitaevskii theory of a rotating Bose gas in a disc-shaped trap with Dirichlet boundary conditions, generalizing and extending previous results that were obtained under Neumann boundary conditions. The focus is on the energy asymptotics, vorticity and qualitative properties of the minimizers in the parameter range $|\log\eps| \ll Ω\lesssim \eps^{-2}|\log\eps|^{-1}$ where $ Ω$ is the rotational velocity and the coupling parameter is written as $ \eps^{-2} $ with $ \eps \ll 1 $. Three critical speeds can be identified. At $ Ω= Ω_{\mathrm{c_1}} \sim|\log\eps| $ vortices start to appear and for $ |\log\eps| \ll Ω< Ω_{\mathrm{c_2}} \sim \eps^{-1} $ the vorticity is uniformly distributed over the disc. For $ Ω\geq Ω_{\mathrm{c_2}} $ the centrifugal forces create a hole around the center with strongly depleted density. For $ Ω\ll \eps^{-2}|\log\eps|^{-1} $ vorticity is still uniformly distributed in an annulus containing the bulk of the density, but at $ Ω= Ω_{\mathrm{c_3}} \sim \eps^{-2}|\log\eps|^{-1} $ there is a transition to a giant vortex state where the vorticity disappears from the bulk. The energy is then well approximated by a trial function that is an eigenfunction of angular momentum but one of our results is that the true minimizers break rotational symmetry in the whole parameter range, including the giant vortex phase.

math-ph↗

arXiv2010

The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate

We study the Gross-Pitaevskii (GP) energy functional for a fast rotating Bose-Einstein condensate on the unit disc in two dimensions. Writing the coupling parameter as $ 1 / \eps^2 $ we consider the asymptotic regime $ \eps \to 0 $ with the angular velocity $Ω$ proportional to $ (\eps^2|\log\eps|)^{-1} $. We prove that if $ Ω= Ω_0 (\eps^2|\log\eps|)^{-1} $ and $ Ω_0 > 2(3π)^{-1} $ then a minimizer of the GP energy functional has no zeros in an annulus at the boundary of the disc that contains the bulk of the mass. The vorticity resides in a complementary `hole' around the center where the density is vanishingly small. Moreover, we prove a lower bound to the ground state energy that matches, up to small errors, the upper bound obtained from an optimal giant vortex trial function, and also that the winding number of a GP minimizer around the disc is in accord with the phase of this trial function.

math-ph↗

arXiv2008

Energy and Vorticity in Fast Rotating Bose-Einstein Condensates

We study a rapidly rotating Bose-Einstein condensate confined to a finite trap in the framework of two-dimensional Gross-Pitaevskii theory in the strong coupling (Thomas-Fermi) limit. Denoting the coupling parameter by $1/\eps^2$ and the rotational velocity by $Ω$, we evaluate exactly the next to leading order contribution to the ground state energy in the parameter regime $|\log\eps|\ll Ω\ll 1/(\eps^2|\log\eps|)$ with $\eps\to 0$. While the TF energy includes only the contribution of the centrifugal forces the next order corresponds to a lattice of vortices whose density is proportional to the rotational velocity.

math-ph↗

arXiv2007

Rapidly Rotating Bose-Einstein Condensates in Homogeneous Traps

We extend the results of a previous paper on the Gross-Pitaevskii description of rotating Bose-Einstein condensates in two-dimensional traps to confining potentials of the form V(r) = r^s, $2<s <\infty$. Writing the coupling constant as $1/ε^2$ we study the limit $ε\to 0$. We derive rigorously the leading asymptotics of the ground state energy and the density profile when the rotation velocity Ωtends to infinity as a power of $1/ε$. The case of asymptotically homogeneous potentials is also discussed.

math-ph↗

arXiv2007

The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps

Starting from the full many body Hamiltonian we derive the leading order energy and density asymptotics for the ground state of a dilute, rotating Bose gas in an anharmonic trap in the ` Thomas Fermi' (TF) limit when the Gross-Pitaevskii coupling parameter and/or the rotation velocity tend to infinity. Although the many-body wave function is expected to have a complicated phase, the leading order contribution to the energy can be computed by minimizing a simple functional of the density alone.

math-ph↗

arXiv2007

Rapidly Rotating Bose-Einstein Condensates in Strongly Anharmonic Traps

We study a rotating Bose-Einstein Condensate in a strongly anharmonic trap (flat trap with a finite radius) in the framework of 2D Gross-Pitaevskii theory. We write the coupling constant for the interactions between the gas atoms as $1/ε^2$ and we are interested in the limit $ε\to 0$ (TF limit) with the angular velocity $Ω$ depending on $ε$. We derive rigorously the leading asymptotics of the ground state energy and the density profile when $Ω$ tends to infinity as a power of $1/ε$. If $Ω(ε)=Ω_0/ε$ a ``hole'' (i.e., a region where the density becomes exponentially small as $1/ε\to\infty$) develops for $Ω_0$ above a certain critical value. If $Ω(ε)\gg 1/ε$ the hole essentially exhausts the container and a ``giant vortex'' develops with the density concentrated in a thin layer at the boundary. While we do not analyse the detailed vortex structure we prove that rotational symmetry is broken in the ground state for ${\rm const.}|\logε|<Ω(ε)\lesssim \mathrm{const.}/ε$.

math-ph↗

arXiv2006

Bosons in Disc-Shaped Traps: From 3D to 2D

We present a mathematically rigorous analysis of the ground state of a dilute, interacting Bose gas in a three-dimensional trap that is strongly confining in one direction so that the system becomes effectively two-dimensional. The parameters involved are the particle number, $N\gg 1$, the two-dimensional extension, $\bar L$, of the gas cloud in the trap, the thickness, $h\ll \bar L$ of the trap, and the scattering length $a$ of the interaction potential. Our analysis starts from the full many-body Hamiltonian with an interaction potential that is assumed to be repulsive, radially symmetric and of short range, but otherwise arbitrary. In particular, hard cores are allowed. Under the premisses that the confining energy, $\sim 1/h^2$, is much larger than the internal energy per particle, and $a/h\to 0$, we prove that the system can be treated as a gas of two-dimensional bosons with scattering length $a_{\rm 2D}= h\exp(-(\hbox{\rm const.)}h/a)$. In the parameter region where $a/h\ll |\ln(\barρh^2)|^{-1}$, with $\barρ\sim N/\bar L^2$ the mean density, the system is described by a two-dimensional Gross-Pitaevskii density functional with coupling parameter $\sim Na/h$. If $|\ln(\barρh^2)|^{-1}\lesssim a/h$ the coupling parameter is $\sim N |\ln(\barρh^2)|^{-1}$ and thus independent of $a$. In both cases Bose-Einstein condensation in the ground state holds, provided the coupling parameter stays bounded.

math-ph↗

arXiv2005

String-localized Quantum Fields and Modular Localization

We study free, covariant, quantum (Bose) fields that are associated with irreducible representations of the Poincaré group and localized in semi-infinite strings extending to spacelike infinity. Among these are fields that generate the irreducible representations of mass zero and infinite spin that are known to be incompatible with point-like localized fields. For the massive representation and the massless representations of finite helicity, all string-localized free fields can be written as an integral, along the string, of point-localized tensor or spinor fields. As a special case we discuss the string-localized vector fields associated with the point-like electromagnetic field and their relation to the axial gauge condition in the usual setting.

math-ph↗

arXiv2004

Bose-Einstein Condensation as a Quantum Phase Transition in an Optical Lattice

One of the most remarkable recent developments in the study of ultracold Bose gases is the observation of a reversible transition from a Bose Einstein condensate to a state composed of localized atoms as the strength of a periodic, optical trapping potential is varied. In \cite{ALSSY} a model of this phenomenon has been analyzed rigorously. The gas is a hard core lattice gas and the optical lattice is modeled by a periodic potential of strength $λ$. For small $λ$ and temperature Bose-Einstein condensation (BEC) is proved to occur, while at large $λ$ BEC disappears, even in the ground state, which is a Mott-insulator state with a characteristic gap. The inter-particle interaction is essential for this effect. This contribution gives a pedagogical survey of these results.

cond-mat.stat-mech↗

arXiv2000

On the PCT--Theorem in the Theory of Local Observables

We review the PCT-theorem and problems connected with its demonstration. We add a new proof of the PCT-theorem in the theory of local observables which is similar to that one of Jost in Wightman quantum field theory. We also look at consequences in case the PCT-symmetry is given on the algebraic level. At the end we present some examples which answer general questions and throw some light on open problems.

math-ph↗

arXiv1998

Modular Groups of Quantum Fields in Thermal States

For a quantum field in a thermal equilibrium state we discuss the group generated by time translations and the modular action associated with an algebra invariant under half-sided translations. The modular flows associated with the algebras of the forward light cone and a space-like wedge admit a simple geometric description in two dimensional models that factorize in light-cone coordinates. At large distances from the domain boundary compared to the inverse temperature the flow pattern is essentially the same as time translations, whereas the zero temperature results are approximately reproduced close to the edge of the wedge and the apex of the cone. Associated with each domain there is also a one parameter group with a positive generator, for which the thermal state is a ground state. Formally, this may be regarded as a certain converse of the Unruh-effect.

math-ph↗

arXiv1997

Thomas-Fermi Calculations of Atoms and Matter in Magnetic Neutron Stars II: Finite Temperature Effects

We present numerical calculations of the equation of state for dense matter in high magnetic fields, using a temperature dependent Thomas-Fermi theory with a magnetic field that takes all Landau levels into account. Free energies for atoms and matter are also calculated as well as profiles of the electron density as a function of distance from the atomic nucleus for representative values of the magnetic field strength, total matter density, and temperature. The Landau shell structure, which is so prominent in cold dense matter in high magnetic fields, is still clearly present at finite temperature as long as it is less than approximately one tenth of the cyclotron energy. This structure is reflected in an oscillatory behaviour of the equation of state and other thermodynamic properties of dense matter and hence also in profiles of the density and pressure as functions of depth in the surface layers of magnetic neutron stars. These oscillations are completely smoothed out by thermal effects at temperatures of the order of the cyclotron energy or higher.

astro-ph↗

arXiv1995

The Ground States of Large Quantum Dots in Magnetic Fields

The quantum mechanical ground state of a 2D $N$-electron system in a confining potential $V(x)=Kv(x)$ ($K$ is a coupling constant) and a homogeneous magnetic field $B$ is studied in the high density limit $N\to\infty$, $K\to \infty$ with $K/N$ fixed. It is proved that the ground state energy and electronic density can be computed {\it exactly} in this limit by minimizing simple functionals of the density. There are three such functionals depending on the way $B/N$ varies as $N\to\infty$: A 2D Thomas-Fermi (TF) theory applies in the case $B/N\to 0$; if $B/N\to{\rm const.}\neq 0$ the correct limit theory is a modified $B$-dependent TF model, and the case $B/N\to\infty$ is described by a ``classical'' continuum electrostatic theory. For homogeneous potentials this last model describes also the weak coupling limit $K/N\to 0$ for arbitrary $B$. Important steps in the proof are the derivation of a new Lieb-Thirring inequality for the sum of eigenvalues of single particle Hamiltonians in 2D with magnetic fields, and an estimation of the exchange-correlation energy. For this last estimate we study a model of classical point charges with electrostatic interactions that provides a lower bound for the true quantum mechanical energy.

cond-mat↗

arXiv1994

Quantumdots

Atomic-like systems in which electronic motion is two dimensional are now realizable as ``quantum dots''. In place of the attraction of a nucleus there is a confining potential, usually assumed to be quadratic. Additionally, a perpendicular magnetic field $B$ may be present. We review some recent rigorous results for these systems. We have shown that a Thomas-Fermi type theory for the ground state is asymptotically correct when $N$ and $B$ tend to infinity. There are several mathematically and physically novel features. 1. The derivation of the appropriate Lieb-Thirring inequality requires some added effort. 2. When $B$ is appropriately large the TF ``kinetic energy'' term disappears and a peculiar ``classical'' continuum electrostatic theory emerges. This is a two dimensional problem, but with a three dimensional Coulomb potential. 3. Corresponding to this continuum theory is a discrete ``classical'' electrostatic theory. The former provides an upper bound and the latter a lower bound to the true quantum energy; the problem of relating the two classical energies offers an amusing exercise in electrostatics.

cond-mat↗