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Ultracold Quantum Gases beyond Equilibrium Natal, Brasil, September 27 – October 1, 2010

Dynamic properties of interacting bosons and magnons

Peter Kopietz, Universität Frankfurt collaboration:  A. Kreisel, T. Kloss, L. Bartosch, F. Sauli, J. Hick (Frankfurt)

A. Sinner (Augsburg)    N. Hasselmann (Natal)A. Serga, B. Hillebrands (Kaiserslautern, experiment)

●Eur. Phys. J. B 71, 59 (2009)●Phys. Rev. B 81,104308 (2010) ●arXiv:1007.3200 and 1008.4521●Phys. Rev. Lett. 102, 120601 (2009)  

references:

 outline: 1. Towards a non­equilibrium many­body theory for YIG2. Spectral function of interacting bosons in 2D  

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1.Towards a non-equilibrium many-body theory for YIG

●Motivation:

collaboration with experimental group of B. Hillebrands (Kaiserslautern)

non-equilibrium dynamics of interacting magnons in YIG

●Experiment:

microwave-pumping of magnons in YIG

measurement of magnon distriubution via Brillouin light scattering

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Parallel pumping of magnons in YIG by microwaves

●what is parallel pumping of magnons?

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Parallel pumping of magnons in YIG by microwaves

●what is parallel pumping of magnons?

two pumps in parallel:

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Parallel pumping of magnons in YIG by microwaves

●what is parallel pumping of magnons?

two pumps in parallel:

●in context of YIG: oscillating magnetic field is parallel to magnetization:

Hamiltonian in Bogoliubov basis: time-dependent off-diagonal terms:

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Parametric resonance

●what is parametric resonance? ●classical harmonic oscillator with harmonic frequency modulation:

●resonance condition:

oscillator absorbs energy at a rate proportional to the energy it already has!

●history:●discovered: Melde experiment, 1859

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●theoretically explained: Rayleigh 1883

excite oscillations of string by periodicallyvarying its tension at twice its resonance frequency

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History: parametric resonance in YIG

H. Suhl, 1957, E. Schlömann et al, 1960s,V. E. Zakharov, V. S. L’vov, S. S. Starobionets, 1970s

●minimal model:

●“S-theory”: time-dependent self-consistent Hartree- Fock approximation for magnon distributions functions

weak points: ●no microscopic description of dissipation and damping●possibility of BEC not included!

●goals:

●consistent quantum kinetic theory for magnons in YIG beyond Hartree-Fock●include time-evolution of Bose-condendsate●develop functional renormalization group for non-equilibrium

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Toy model for parametric resonance

T. Kloss, A. Kreisel, PK, PRB 2010

●anharmonic oscillator with off-diagonal pumping:

●rotating reference frame:

●instability of non-interacting system for large pumping:

●for oscillator has negative mass●non-interacting Hamiltonian not positive definite

interactions stabilize ground state for large pumping

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Time-dependent Hartree-Fock approximation (“S-theory”)

●order parameter:Hamiltonian dynamicsin effective potential(Hartree-Fock)

●order parameter Gross-Pitaevskii equation:

●connected correlation functions:

kinetic equations:

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Functional integral formulation of theKeldysh technique

T. Kloss, PK, in preparation

●6 types of non-equilibrium Green functions:

●Keldysh component at equal times gives distribution function

●retarded:●advanced:

●Keldysh:

●functional integral formulation (Kamenev 2004)

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Non-perturbative method: functional renormalization group

●exact equation for change of generating functional of irreducible vertices as IR cutoff is reduced (Wetterich 1993)

●exact RG flow equations for all vertices

● flow of self-energy:

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FRG for bosons out of equilibrium

●self-energy defines collision integral in quantum kinetic equation:

●introduce RG flow parameter which cuts off long-time behavior

●exact RG flow equation for non-equilibrium self-energy:

see also Gasenzer, Pawlowski 2008

eventually

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Out-scattering rate cutoff scheme

●solution: non-equilibrium dynamics from the functional RG!

●problem: perturbation theory fails for long time dynamics (secular terms: U*t)

●RG flow parameter: out scattering rate:

make infinitesimal imaginary part finite and use it as flowparameter for the FRG

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Comparison of exact solution of toy model with FRG

●toy model can be solved numerically exactly by solving time-dependent Schrödinger equation

●comparison with various approximations:

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2. Interacting bosons: Bogoliubov theory (1947)

●Bogoliubov-shift:

●Bogoliubov mean-field Hamiltonian:

●condensate density:

●excitation energy:

●long wavelength excitations: sound!

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Beyond mean-field: infrared divergencies

●Bogoliubov mean-field Hamiltonian:

normal self-energy:

anomalous self-energy:

●mean-field fails due to infrared divergent fluctuation corrections:

for Ginzburg scale

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Exact result:Nepomnyashi-identity (1975)

●anomalous self-energy vanishes at zero momentum/frequency:

●Bogoliubov approximation wrong:

●need non-perturbative methods!

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Origin of infrared divergencies

●reason for IR divergence:  coupling between transverse and  longitudinal fluctuations   and resulting divergence of  longitudinal susceptibility

(Patashinskii+Pokrovskii, JETP 1973)

(Kreisel, Hasselmann, PK, PRL 2007)

●consequence: critical continuum in longitudinal part of spectral function(for bosons not directly measurable!)

●experimentally accessible: longitudinal spin structure factor   in quantum antiferromagnets: BEC of magnons!

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FRG flow equations for order parameter and self-energies

●order parameter:

●normal self-energy: ●anomalous self-energy:

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Low-density truncation: retain only two-body interactions●bare action:

●ansatz for generating functional of irreducible vertices:

●depends on condensate density

momentum-dependent functions

and two frequency- and

and

analytic non-analytic

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Parametrization of three- and four-point vertices:

●normal and anomalous self-energy:

Hugenholtz-Pines relation

Nepomnyashchy identity

●three-leggedvertices:

●effective interaction:

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Results: spectral function and quasi-particle damping in 2D

●spectral function:●spectral function: ●quasi-particle damping:

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What about experiments?

Cold atom experiments can now measure renormalizedexcitation spectrum of strongly interacting bosons:

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Summary+Outlook

● magnon gas in yttrium-iron garnet: model system for non-equilibrium physics of interacting bosons

● functional RG out of equilibrium: non-perturbative calculation of time-evolution

● functional RG in equilibrium: non-perturbative calculation of spectral function of interacting bosons