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Jakob Yngvason

Publications and source records attributed to Jakob Yngvason.

At least 19 recordsLinked to original sources
arXiv2024

Many-body physics and resolvent algebras

Some advantages of the algebraic approach to many body physics, based on resolvent algebras, are illustrated by the simple example of non-interacting bosons which are confined in compact regions with soft boundaries. It is shown that the dynamics of these systems converges to the spatially homogeneous dynamics for increasing regions and particle numbers and a variety of boundary forces. The corresponding correlation functions of thermal equilibrium states also converge in this limit. Depending on the filling of the regions with particles, the limits can either be spatially homogeneous, including the Bose-Einstein condensates, or they become inhomogeneous with varying, but finite local particle densities. In case of this spontaneous breakdown of the spatial symmetry, the presence of condensates can be established by exhibiting temporal correlations over large temporal distances (memory effects).

math-ph↗
arXiv2022

Quantum Hall states in higher Landau levels

The unitary correspondence between Quantum Hall states in higher Landau levels and states in the lowest Landau level is discussed together with the resulting transformation formulas for particle densities and interaction potentials. This correspondence leads in particular to a representation of states in arbitrary Landau levels in terms of holomorphic functions. Some important special Quantum Hall states in higher Landau levels are also briefly discussed.

cond-mat.mes-hall↗
arXiv2022

Quantum Hall Phases of Cold Bose Gases

Cold atomic gases of interacting bosons subject to rapid rotation and confined in anharmonic traps can theoretically exhibit analogues of the fractional quantum Hall effect for electrons in strong magnetic fields. In this setting the Coriolis force due to the rotation mimics the Lorentz force on charged particles but artificial gauge fields can also be obtained by coupling the internal structure of the atoms to light fields. The chapter discusses mathematical aspects of transitions to different strongly correlated phases that appear when the parameters of a model Hamiltonian are varied.

math-ph↗
arXiv2022

A Direct Road to Entropy and the Second Law of Thermodynamics

The entropy of classical thermodynamics is uniquely determined by the relation of adiabatical accessibilty between equilibrium states of thermodynamical systems. This review outlines the logical path leading to this results and the challenges that have to be faced on the way.

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

Holomorphic quantum Hall states in higher Landau levels

Eigenstates of the planar magnetic Laplacian with homogeneous magnetic field form degenerate energy bands, the Landau levels. We discuss the unitary correspondence between states in higher Landau levels and those in the lowest Landau level, where wave functions are holomorphic. We apply this correspondence to many-body systems, in particular we represent effective Hamiltonians and particle densities in higher Landau levels by corresponding quantities in the lowest Landau level.

cond-mat.mes-hall↗
arXiv2018

Rigidity of the Laughlin liquid

We consider general N-particle wave functions that have the form of a product of the Laughlin state with filling factor $1/\ell$ and an analytic function of the N variables. This is the most general form of a wave function that can arise through a perturbation of the Laughlin state by external potentials or impurities, while staying in the lowest Landau level and maintaining the strong correlations of the original state. We show that the perturbation can only shift or lower the 1-particle density but nowhere increase it above a maximum value. Consequences of this bound for the response of the Laughlin state to external fields are discussed.

math-ph↗
arXiv2018

Bose-Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature

We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross-Pitaevskii (GP) limit where the trap frequency $ω$, the temperature $T$ and the particle number $N$ are related by $N \sim (T / ω)^{3} \to\infty$ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose-Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.

math-ph↗
arXiv2018

Local incompressibility estimates for the Laughlin phase

We prove sharp density upper bounds on optimal length-scales for the ground states of classical 2D Coulomb systems and generalizations thereof. Our method is new, based on an auxiliary Thomas-Fermi-like variational model. Moreover, we deduce density upper bounds for the related low-temperature Gibbs states. Our motivation comes from fractional quantum Hall physics, more precisely, the perturbation of the Laughlin state by external potentials or impurities. These give rise to a class of many-body wave-functions that have the form of a product of the Laughlin state and an analytic function of many variables. This class is related via Laughlin's plasma analogy to Gibbs states of the generalized classical Coulomb systems we consider. Our main result shows that the perturbation of the Laughlin state cannot increase the particle density anywhere, with implications for the response of FQHE systems to external perturbations.

math-ph↗
arXiv2017

The Laughlin liquid in an external potential

We study natural perturbations of the Laughlin state arising from the effects of trapping and disorder. These are N-particle wave functions that have the form of a product of Laughlin states and analytic functions of the N variables. We derive an upper bound to the ground state energy in a confining external potential, matching exactly a recently derived lower bound in the large N limit. Irrespective of the shape of the confining potential, this sharp upper bound can be achieved through a modification of the Laughlin function by suitably arranged quasi-holes.

math-ph↗
arXiv2016

Incompressibility estimates for the Laughlin phase, part II

We consider fractional quantum Hall states built on Laughlin's original N-body wave-functions, i.e., they are of the form holomorphic times gaussian and vanish when two particles come close, with a given polynomial rate. Such states appear naturally when looking for the ground state of 2D particles in strong magnetic fields, interacting via repulsive forces and subject to an external potential due to trapping and/or disorder. We prove that all functions in this class satisfy a universal local density upper bound, in a suitable weak sense. Such bounds are useful to investigate the response of fractional quantum Hall phases to variations of the external potential. Contrary to our previous results for a restricted class of wave-functions, the bound we obtain here is not optimal, but it does not require any additional assumptions on the wave-function, besides analyticity and symmetry of the pre-factor modifying the Laughlin function.

math-ph↗
arXiv2014

Incompressibility Estimates for the Laughlin Phase

This paper has its motivation in the study of the Fractional Quantum Hall Effect. We consider 2D quantum particles submitted to a strong perpendicular magnetic field, reducing admissible wave functions to those of the Lowest Landau Level. When repulsive interactions are strong enough in this model, highly correlated states emerge, built on Laughlin's famous wave function. We investigate a model for the response of such strongly correlated ground states to variations of an external potential. This leads to a family of variational problems of a new type. Our main results are rigorous energy estimates demonstrating a strong rigidity of the response of strongly correlated states to the external potential. In particular we obtain estimates indicating that there is a universal bound on the maximum local density of these states in the limit of large particle number. We refer to these as incompressibility estimates.

cond-mat.quant-gas↗
arXiv2014

Entropy Meters and the Entropy of Non-extensive Systems

In our derivation of the second law of thermodynamics from the relation of adiabatic accessibility of equilibrium states we stressed the importance of being able to scale a system's size without changing its intrinsic properties. This leaves open the question of defining the entropy of macroscopic, but unscalable systems, such as gravitating bodies or systems where surface effects are important. We show here how the problem can be overcome, in principle, with the aid of an `entropy meter'. An entropy meter can also be used to determine entropy functions for non-equilibrium states and mesoscopic systems.

math-ph↗
arXiv2014

The entropy concept for non-equilibrium states

In earlier work we presented a foundation for the Second Law of Classical Thermodynamics in terms of the Entropy Principle. More precisely, we provided an empirically accessible axiomatic derivation of an entropy function defined on all equilibrium states of all systems that has the appropriate additivity and scaling properties and whose increase is a necessary and sufficient condition for an adiabatic process between two states to be possible. Here, after a brief review of this approach, we address the question of defining entropy for non-equilibrium states. Our conclusion is that it is generally not possible to find a unique entropy that has all relevant physical properties. We do show, however, that one can define two entropy functions, called $S_-$ and $S_+$, which, taken together, characterize the range of adiabatic processes that can occur between non-equilibrium states. The concept of {\it comparability} of states with respect to adiabatic changes plays an important role in our reasoning.

math-ph↗
arXiv2014

Topics in the Mathematical Physics of Cold Bose Gases

In these notes of six lectures on selected topics in the theory of cold, dilute Bose gases, presented at the 5th Warsaw School of Statistical Physics in June 2013, the following topics are discussed: 1) The concept of BEC, 2) the ground state energy of a dilute Bose gas with short range interactions, 3) Gross-Pitaevskii theory and BEC in trapped gases, 4) Bose gases in rotating traps and quantized vortices, and 5) strongly correlated phases in the lowest Landau level generated by rapid rotation.

cond-mat.quant-gas↗
arXiv2014

Localization and Entanglement in Relativistic Quantum Physics

The combination of quantum theory and special relativity leads to structures that differ in several respects from non-relativistic quantum mechanics of particles. These differences are quite familiar to practitioners of Algebraic Quantum Field Theory but less well known outside this community. The paper is intended as a concise survey of some selected aspects of relativistic quantum physics, in particular regarding localization and entanglement.

quant-ph↗
arXiv2013

Quantum Hall phases and plasma analogy in rotating trapped Bose gases

A bosonic analogue of the fractional quantum Hall eff ect occurs in rapidly rotating trapped Bose gases: There is a transition from uncorrelated Hartree states to strongly correlated states such as the Laughlin wave function. This physics may be described by eff ective Hamiltonians with delta interactions acting on a bosonic N-body Bargmann space of analytic functions. In a previous paper [N. Rougerie, S. Serfaty, J. Yngvason, Phys. Rev. A 87, 023618 (2013)] we studied the case of a quadratic plus quartic trapping potential and derived conditions on the parameters of the model for its ground state to be asymptotically strongly correlated. This relied essentially on energy upper bounds using quantum Hall trial states, incorporating the correlations of the Bose-Laughlin state in addition to a multiply quantized vortex pinned at the origin. In this paper we investigate in more details the density of these trial states, thereby substantiating further the physical picture described in [N. Rougerie, S. Serfaty, J. Yngvason, Phys. Rev. A 87, 023618 (2013)], improving our energy estimates and allowing to consider more general trapping potentials. Our analysis is based on the interpretation of the densities of quantum Hall trial states as Gibbs measures of classical 2D Coulomb gases (plasma analogy). New estimates on the mean- field limit of such systems are presented.

math-ph↗