boseeinstein condensation of magnons and spinwave …kopietz/documents/talk07_spi_rutherford.… ·...
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1. Some notsowellknown aspects of BEC2. New formulation of spinwave expansion for QAF3. QAF in a uniform magnetic field: QEC of magnons
BoseEinstein condensation of magnons and spinwave interactions
in quantum antiferromagnets
Peter Kopietz, Universität Frankfurtcollaborators: Nils Hasselmann, Andreas Kreisel, Sascha Ledowski,Florian Schütz, Andreas Sinner
S. Ledowski, N. Hasselmann, PK, Phys Rev A 69, 061601(R) (2004)A. Sinner, F. Schütz, PK, Phys Rev A 74, 023608 (2006)N. Hasselmann and PK, Europhys. Lett. 74, 1067 (2006)A. Kreisel, N. Hasselmann, and PK, Phys. Rev. Lett. 98, 067203 (2007)
Talk at Rutherford Appleton Lab, March 13, 2007
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1.Some notsowell known aspectsof BoseEinstein condensation
Hamiltonian of the interacting Bose gas:
(BEC):
is of order unity
Spontaneous symmetry breaking (SSB): expectation value of is of order unity
Thermodynamic limit for
Finite systems: no SSB but BEC possible!
BoseEinstein condensation
expectation value of
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a) noninteracting bosons: Landau functions A. Sinner, F. Schütz, PK, Phys Rev A 74, 023608 (2006)
probability distributions of eigenvaluesand of operators and
canonical partition function:
Landau functions develop minima if relevant dimensioless density
is sufficiently large.
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b) effect of interactions on the critical temperature for BEC G. Baym et al, PRL 1999, functional RG: S. Ledowski, N. Hasselmann, PK, PRA 2004
Critical temperature for BEC in the free Bose gas (D=3):
Question: Do interactions increase or decrease the critical temperature?
This is a nonperturbative problem! Need sophisticated fieldtheoreticalmethods, such as functional renormalization group flow equations:
Answer:
swave scattering length
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c) interacting bosons at T=0: Bogoliubov theory
Euclidian action for interacting Bose gas (constant chemical potential):
symmetrized interaction:
Bogoliubovshift:
(N. Bogoliubov, 1947)
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...Bogoliubov continued...
dispersion of singleparticle excitations:
condensate density at zero temperature:
velocity of Goldstone mode:
Correlation functions:
longitudinal transverse
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(d) beyond Bogoliubov C. Castellani, C. Di Castro, F. Pistolesi, G. C. Strinati, PRL 1997, PRB 2004
●in dimensions d <= 3 Bogoliubov fixed point is unstable towards a different fixed point characterized by the divergence of thelongitudinal correlation function
●lots of divergencies in perturbation theory
●lots of cancellations controlled by Ward identities
●critical contiuum in longitudinal structure factor
●observable? Yes: in BEC of magnons!
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2.New formulation of the spinwaveexpansion for antiferromagnets
(a) Warmup: Heisenberg ferromagnetN. Hasselmann, PK, Europhys. Lett. 74, 1067 (2006)
Hamiltonianfor nearest neighbors
Map to Bosonproblem via HolsteinPrimakoff transformation:
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Why not BEC of HolsteinPrimakoff bosons in ferromagnets?
Fourier transform of twobody boson interaction:
effective interaction vanishes at long wavelenths:
ferromagnetic spinwaves (Goldstone bosons) interact too weak at long wavelengths!
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(b) how about antiferromagnetic magnons?Look at conventional spinwave expansion for QAF:(Anderson 1952, Kubo 1952, Oguchi 1960)
3 Transformations:
1.) HolsteinPrimakoff: mapping onto bosonproblem
2.) Fourier transformation in sublattice basis:on sublattice A: on sublattice B:
3.)Bogoliubov transformation:
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...spinwave theory for QAF...
quadratic part of boson hamiltonian is diagonal
spinwave interactions: very complicated!(Harris,Kumar,Halperin, Hohenberg, PRB 1971, PK, PRB 1990)
interaction vertices are infraredsingular in some limits:
miracle: all divergencies cancelin physical quantities!
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effective action for antiferromagnetic spin fluctuations
alternative description of antiferromagnetic spinfluctuations:nonlinear sigma model (Chakravarty, Halperin, Nelson, PRL and PRB 1988)
represents slowly fluctuating part of staggered magnetization
●interactions between AF spin fluctations involve derivativesand therefore vanish at long wavelengths
●what is relation between NLsM and conventional spinwave theory?
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c) hermitian operator approach
(Anderson 1952, N. Hasselmann and PK, 2006)
●introduce symmetric and antisymmetric combinations of Bogoliubov operators:
●express each of them in terms of two canonically conjugate hermitian operators:
●quadratic part of spinwave hamiltonian:
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advantages of hermitian operator approach
●simple relation to staggered and uniform components of spin
●precise relation between nonlinear sigmamodel and SWT established:
●allows to derive new effective action of staggered fluctuations alone
●very convenient to understand spinwaves in finite systems
●suppression of effective interaction between AF magnons at long
wavelengths becomes manifest
●for AF in uniform field: relation to divergencies in the interacting Bose gas!(A. Kreisel, N. Hasselmann, PK, PRL 2007)
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zero modes in finite quantum antiferromagnets (Anderson, 1952)
●Contribution of mode to quadratic spinwave Hamiltonian:
where
●Groundstate has
u
●Uncertainly principle:
●Staggered magnetization diverges
spinwave approach appears inconsistent!?
●Solution: for Gaussian wavepackt with spin wave approach
remains consistent for macroscopic times:
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spinwave interactions in quantum antiferromagnets
important: effective interaction between staggered spin fluctuations vanishesat long wavelenths (see ferromagnet!):
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3.BEC of magnons: realized in QAFs in a uniform magnetic field
(Matsubara+Matsuda 1956, Batyev+Braginsky 1984,...)A. Kreisel, N. Hasselmann, PK, PRL 2007: connection with anomalouslongitudinal fluctuations in the interacting Bose gas.
classical ground state:
experiment: Cs_2CuCl_4
(Radu et al, PRL 2005)
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(a) mapping onto interacting boson problemfor large external magnetic field: ground state is saturated ferromagnet:
spinwave expansion usual forquantum ferromagnets:
●HolsteinPrimakoff:
●Fourier trafo in sublattice basis
●Two magnon branches in reduced BZ:
gapless: gapped:
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(b) BEC of magnons and hermitian field parametrization
●retain only gapless mode and perform continuum limit
two body interaction has finite longwavelength limit
spinHamiltonian is mapped on interacting Bose gas:
B●BEC of magnons: for ( i.e. ) field acquires finite expectation value:
Bogoliubov approximation:
●Physical meaning: condendsate density corresponds to staggered magnetization
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hermitian field parameterization and correlation functions What are spincorrelation functions in condensed phase?Idea: Use nonperturbative results by Castellani and coworkers!
●hermitian operator parametrization
●physical meaning in underlying spinproblem:
components of staggered magnetization
●correlation functions in linear spinwave theory:
qualitatively wrong!
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c) critical continuum in longitudinal structure factorTrue asymptotics of correlation functions:
critical continuum in longitudinal structure factor:
Ginzburg scale:D<3
D=3
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Summary: Interacting bosons and magnons
●Some notsowell known facts about the Bose gas●BEC is not the same as spontaneous symmetry breaking●Landau functions for free Bose gas●shift in the critical temperature due to interactions●corrections to Bogoliubov approximation large in D <= 3
●Spinwave interactions in quantum antiferromagnets
●QAF in a uniform magnetic field: BEC of magnons
●Usual spinwave expansion leads to rather nasty interaction vertices ●hermitianfield parameterization has many advantages:
●weak interactions between Golstone modes manifest●relation between SWT and nonlinear sigma model precise●zero modes and spinwaves infinite systems