quasi-1d antiferromagnets - ucsb physicsweb.physics.ucsb.edu/~balents/talks/nordita07a.pdf · 2007....
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Quasi-1d Antiferromagnets
Leon Balents, UCSBMasanori Kohno, NIMS, Tsukuba
Oleg Starykh, U. Utah
“Quantum Fluids”, Nordita 2007
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OutlineMotivation:
Quantum magnetism and the search for spin liquids
Neutron scattering from Cs2CuCl4 and spinons in two dimensionsLow energy properties of quasi-1d antiferromagnets and Cs2CuCl4 in particular
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Quantum AntiferromagnetsHeisenberg model:
“Classical” Neel state is modified by quantum fluctuations
In most cases, Neel order survives1000’s of ordered antiferromagnets
Spin flip terms
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MagnonsBasic excitation: spin flip
Carries “Sz”=± 1
Periodic Bloch states: spin wavesQuasi-classical picture: small precession
MnF2
Image: B. Keimer
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Inelastic neutron scatteringNeutron can absorb or emit magnon
La2CuO4
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One dimensionHeisenberg model is a spin liquid
No magnetic orderPower law correlations of spins and dimers
Excitations are s=1/2 spinonsGeneral for 1d chainsCartoon
Ising anisotropy
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Spinons by neutronsBethe ansatz:
Spinon energySpin-1 states
2-particle continuum
Theory versus experiment for KCuF3, with spatial exchange anisotropy of 30 (very 1d)
B. Lake et al, HMI
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Spinons in d>1?Resonating Valence Bond theories (Anderson…)
Spin “liquid” of singlets
Broken singlet “releases” 2 spinons
Many phenomenological theoriesNo solid connection to experiment
+ + …
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Cs2CuCl4: a 2d spin liquid?
J’/J ≈ 0.3
Couplings: J’ ≈ 0.3 JD ≈ 0.05 J
J ≈ 0.37 meV
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Inelastic Neutron ResultsColdea et al, 2001,2003
Very broad spectra similar to 1d (in some directions of k space). Roughly fits to power law
Fit of “peak” dispersion to spin wave theory requires adjustment of J,J’ by ≈ 40% - in opposite directions!
Note asymmetry
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2d theoriesArguments for 2d:
J’/J = 0.3 not very smallTransverse dispersion
Exotic theories:
Spin waves:
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Back to 1dFrustration enhances one-dimensionality
First order energy correction vanishes due to cancellation of effective fieldNumerical evidence: J’/J <0.7 is “weak”
Numerical phase diagram contrasted with spin wave theory Very small inter-chain correlations
Wen
get al, 2
006
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Excitations for J’>0Coupling J’ is not frustrated for excited statesPhysics: transfer of spin 1
Spinons can hop in pairsExpect spinon binding to lower energySpin bound state=“triplon” clearly disperses transverse to chains
y
y+1
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Effective Schrödinger equationStudy two spinon subspace
Momentum conservation: 1d Schrödinger equation in ε space
Crucial matrix elements known exactly Bougourzi et al, 1996
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Structure FactorSpectral Representation
Can obtain closed-form “RPA-like” expression for 2d S(k,ω) in 2-spinon approximation
Weight in 1d:73% in 2 spinon states99% in 2+4 spinons
J.S. Caux et al
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Types of behaviorBehavior depends upon spinon interaction
Bound “triplon” Identical to 1D Upward shift of spectral weight. Broad resonance in continuum or anti-bound state (small k)
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Broad lineshape: “free spinons”“Power law” fits well to free spinon result
Fit determines normalization
J’(k)=0 here
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Bound stateCompare spectra at J’(k)<0 and J’(k)>0:
Curves: 2-spinon theory w/ experimental resolutionCurves: 4-spinon RPA w/ experimental resolution
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Transverse dispersion
Bound state and resonance
Solid symbols: experimentNote peak (blue diamonds) coincides with bottom edge only for J’(k)<0
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Spectral asymmetryComparison:
Vertical lines: J’(k)=0.
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Conclusion (spectra)Simple theory works well for frustrated quasi-1d antiferromagnets
Frustration actually simplifies problem by enhancing one-dimensionality and reducing modifications to the ground state
“Mystery” of Cs2CuCl4 solvedNeed to look elsewhere for 2d spin liquids!
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Low Temperature BehaviorCs2CuCl4 orders at 0.6K into weakly incommensurate coplanar spiralOrder evolves in complex way in magnetic field
cone
Several phases with field in triangular plane
Note break in scale: zero field phase destroyed by “weak”field
One phase with field normal to triangular plane
Zero field order enhanced slightly in field
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Low energy theoryStrategy:
Identify instability of weakly coupled chains (science)Try to determine the outcome (art)
InstabilitiesRenormalization group view: relevant couplings
relevant
irrelevant
Subtleties:More than 1 relevant
couplingSome relevant couplings
absent due to frustration
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RG picturesCompeting relevant operators
• Smaller initial couplings favor more relevant direction
• can be compensated by initial conditions
Perturbativeregime
“Accidentally” zero couplings (frustration)
• non-linearities bend RG flow lines• relevant operators generated by fluctuations
Frustrated line
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What are the couplings?Single chain: a conformal field theory
Primary fields:
operator scaling dimension h=0 h→ hsat
Interchain couplings composed from thesey
Further chain couplings just as relevant but smaller
more XY-like in field
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Zero fieldAllowed operators strongly restricted by reflections
reflections
marginal
relevant
Leads to very weak instability in J-J’ modelBroken SU(2) from DM interaction more important
Instability leads to spiral state
Relevant
Generated at O[(J’/J)4]
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Transverse (to plane) FieldXY spin symmetry preserved
DM term becomes more relevant
b-c spin components remain commensurate: XY coupling of “staggered” magnetizations still cancels by frustration (reflection symmetry)Spiral (cone) state just persists for all fields.
Experiment:
h
Order increases with h here due to increasing relevance of DM term
Order decreases with h here due to vanishing amplitude as hsat is approached
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Longitudinal FieldField breaks XY symmetry:
Competes with DM term and eliminates this instability for H & D
Other weaker instabilities take hold
Naïve theoretical phase diagram
h/hsat0 ∼ 0.1 10.9
Weak “collinear” SDWpolarized
(DM) “cycloid”
“cone”
T
?
Commensurate AF stateScycloidExpt. AF state differs from theory (J2?)
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Magnetization PlateauBeyond the naïve: commensurate SDW state unstable to plateau formation
Strongest locking at M=Msat/3Gives “uud” state which also occurs in spin wave theory (Chubukov)
h/hsat0 ∼ 0.1 10.9
“collinear” SDW
polarized(DM)
“cycloid”“cone”
T
?
uud
Magnetization plateau observed in Cs2CuBr4
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SummaryOne-dimensional methods are very powerful for quasi-1d frustrated magnets, even when inter-chain coupling is not too smallFor the future:
Behavior of spectra in a fieldQuasi-1d conductorsOther materials, geometries
Whether a quasi-1d material can ever exhibit a true 2d quantum spin liquid ground state is an open question