novel magnetism in ultracold atomic gases
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faculty.virginia.edu/austen/
Novel magnetism in ultracoldatomic gases
Austen Austen LamacraftLamacraft
Virginia, February 2010Virginia, February 2010
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faculty.virginia.edu/austen/
Novel magnetism in ultracoldatomic gases
Austen Austen LamacraftLamacraft
Virginia, February 2010Virginia, February 2010
DynamicsDynamics and Statistical Mechanicsand Statistical Mechanics of of Multicomponent Multicomponent Quantum FluidsQuantum Fluids
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Outline
Magnetism in Bose condensates– Phases of spin 1 bosons– Ferromagnetic and polar states
The dynamics of the Bose ferromagnet– Superfluid flow– Equations of motion– Dipolar interactions
Statistical mechanics of the polar state– Vortices, domain walls, and phase transitions
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Exotic magnetism - solid state
ZnCr2O4
– Ji et al., PRL (2009)
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Exotic magnetism - atomic gas
87Rb is a boson with I=3/2, S=1/2– Possible total spin F=1 or 2
What are magnetic properties of F=1 or 2 Bose gas?
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Recent observation of Fermi magnet
– Science 325, 1521-1524 (2009)
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Ultracold atomic gases
Fe becomes ferromagnetic at T=1043 K Ultracold atomic physics takes place at <10-6 K
Quantum effects determine collective (i.e. material)properties when
thermal wavelength ≈ interparticle separation
Atomic gases are heavier and less dense than gasof electrons in Fe
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So what’s new?
In the solid state we (mostly) care about the quantummechanics of electrons. These are fermions
By contrast, atoms (considered as particles) may bebosons or fermions
Possibility of Bose-Einstein condensation - bosonsaccumulate in lowest energy state
– Nobel prize 2001
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Outline
Magnetism in Bose condensates– Phases of spin 1 bosons– Ferromagnetic and polar states
The dynamics of the Bose ferromagnet– Superfluid flow– Equations of motion– Dipolar interactions
Statistical mechanics of the polar state– Vortices, domain walls, and phase transitions
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Magnetism in Bose gases
BEC: (nearly) all atoms sit in same quantum state– This state is called the condensate wavefunction
But what if lowest energy state is degenerate?
Condensate wavefunction is a spin vector (spinor) and must pick a direction in spin space
Bose condensates with spin are always magnets
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Why higher spin is fun
Spin 1/2 (e.g. of electron) points in some direction
– To make electron magnetism more interesting caninvoke non-trivial arrangements on lattice (e.g. Néel)
Spin 1 doesn’t necessarily “point” anywhere
– spin-1 matrices– Yet evidently there is still a director or nematic axis!
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Which spin state wins?
Must consider interatomic interactions Atoms can collide with total spin 0 or 2
– Total spin 1? Antisymmetric and blocked by Bose statistics
spin 2 spin 0
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Spin dependent interactions
Energy of state includes a piece
– For c2 < 0 (e.g. 87Rb): maximize Ferromagnet
– For c2 > 0 (e.g. 23Na): minimize Polar state
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Mean field ground states: spin 1
Work in cartesian components where
Ferromagnet– maximal for– form orthonormal triad
Polar state– minimal for
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The Bose ferromagnet: 87Rb
– Stamper-Kurn group, Berkeley
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Outline
Magnetism in Bose condensates– Phases of spin 1 bosons– Ferromagnetic and polar states
The dynamics of the Bose ferromagnet– Superfluid flow– Equations of motion– Dipolar interactions
Statistical mechanics of the polar state– Vortices, domain walls, and phase transitions
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– Onsager-Feynman relation
Circulation quantized in normalsuperfluids
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Superfluid velocity in the Ferromagnet
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The Mermin-Ho relation
– Mermin & Ho (1976)
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Geometrical meaning
Stokes’ theorem
“…a result apparently due to Gauss himself”
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Unwinding a vortex
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Superfluids and bicycle wheels
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Spherical triangle
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Incompressible flowIncompressible flow
Normal fluids approximately incompressible at low Mach number
Scalar superfluids
Leaves only possibility of isolated vortex lines
In the spinor case this limit is non-trivial!
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Equations of motion of Bose Ferromagnet
AL, PRA 77 63622 (2008)
Spinwaves Spinwaves have quadratic dispersion around uniform statehave quadratic dispersion around uniform state
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Relevance of dipolar forces?
M. M. Vengalttore Vengalttore et al.et al. arXiv:0901.3800
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Easily include dipolar forces
Larmor frequency dwarfs other scales– Average dipole-dipole energy over rapid precession
– q=0 part is easy axis anisotropy(exercise in demagnetizing factors)
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Effect on spinwaves
Boundary between stability and instability
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Other kinds of magnetic order
Can also have Antiferromagnetism (e.g. MnO)
Néel order Nematic order a.k.a. “Moment free magnetism”
Chiral order
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The holy grail: no order at all!
Quantum fluctuations keep spins disordered at T=0
Quantum spin liquid [Artist’s impression]
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Outline
Magnetism in Bose condensates– Phases of spin 1 bosons– Ferromagnetic and polar states
The dynamics of the Bose ferromagnet– Superfluid flow– Equations of motion– Dipolar interactions
Statistical mechanics of the polar state– Vortices, domain walls, and phase transitions
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Recall spin dependent interactions
Polar state
– minimal for– Convenient to write
Polar condensate
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Polar condensate - a spin nematic
As far as spin is concerned equivalent
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Half vortices and disclinations
Notice that and are the same!
– Possibility of half vortex / disclinations
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Disclinations in a nematic liquid crystal
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Picturing the space
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Kosterlitz-Thouless transition
Free energy to add a vortex
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Domain Walls
Quadratic Zeeman effect aligns parallel to field
FieldField
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Conjectured phase diagram
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Summary
• Dynamics of spinor condensates
– Instabilities, dipole-dipole interactions, Berkeley experiment
– Phys. Rev. Lett. 98, 160404 (2007)– Phys. Rev. A 77, 063622 (2008)– arXiv:0909.5620 (2009) [concerns higher spin]
• Statistical mechanics of polar condensates
– Novel transitions driven by 1/2 vortices and domain walls
– Current work with Andrew James