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Quasi-1d Frustrated Antiferromagnets
Leon Balents, UCSBMasanori Kohno, NIMS, Tsukuba
Oleg Starykh, U. Utah
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OutlineFrustration in quasi-1d systemsExcitations: magnons versus spinons
Neutron scattering from Cs2CuCl4 and spinonsin two dimensions
Low energy properties of quasi-1d antiferromagnets and Cs2CuCl4 in particular
Renormalization group technique
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OutlineFrustration in quasi-1d systemsExcitations: magnons versus spinons
Neutron scattering from Cs2CuCl4 and spinonsin two dimensions
Low energy properties of quasi-1d antiferromagnets and Cs2CuCl4 in particular
Renormalization group technique
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What is frustration?Competing interactions
Can’t satisfy all interactions simultaneouslyOptimization is “frustrating”
“People need trouble – a little frustration to sharpen the spirit on, toughen it. Artists do; I don't mean you need to live in a rat hole or gutter, but you have to learn fortitude, endurance. Only vegetables are happy.” – William Faulkner
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Checkerboard lattice
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Quasi-1d systemsWeakly coupled chains
Single Heisenberg chain well understoodExact solution (Bethe 1932…) gives energies, wavefunctions, some correlationsLow energy bosonization theory
J’/J gives expansion parameter
J
J’J’ ¿ J
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Frustration in quasi-1D systemsWeakly coupled chains
FrustrationDominant antiferromagnetic correlations incompatible between chains
Broadened domain of J’/J expansion
J
J’J’ ¿ J
Zero net exchange field from one chain upon another
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OutlineFrustration in quasi-1d systemsExcitations: magnons versus spinons
Neutron scattering from Cs2CuCl4 and spinonsin two dimensions
Low energy properties of quasi-1d antiferromagnets and Cs2CuCl4 in particular
Renormalization group technique
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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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OutlineFrustration in quasi-1d systemsExcitations: magnons versus spinons
Neutron scattering from Cs2CuCl4 and spinonsin two dimensions
Low energy properties of quasi-1d antiferromagnets and Cs2CuCl4 in particular
Renormalization group technique
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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
effective “potential”
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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
Bougourzi et al,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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OutlineFrustration in quasi-1d systemsExcitations: magnons versus spinons
Neutron scattering from Cs2CuCl4 and spinonsin two dimensions
Low energy properties of quasi-1d antiferromagnets and Cs2CuCl4 in particular
Renormalization group technique
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Frustration: Low energy physicsRecall: no naïve (leading order) preference for inter-chain ordering
Q: How is the degeneracy resolved in the ground state?
Magnetic order? What sort?Dimerization?Spin liquid?
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Experimental BehaviorCs2CuCl4 orders at 0.6K into weakly incommensurate coplanar spiralOrder evolves in complex way in magnetic fieldField normal to plane:
Only one phaseOrder slightly enhancedin field
0 10B(T)
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Experimental BehaviorCs2CuCl4 orders at 0.6K into weakly incommensurate coplanar spiralOrder evolves in complex way in magnetic fieldField parallel to plane:
Several phasesZero field state weakened by field
0 10B(T)
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Renormalization Group theoryStrategy:
Identify instability of weakly coupled chains (science)Try to determine the outcome (art)
InstabilitiesRenormalization group view: relevant couplings
relevant
irrelevant= decoupled chain
fixed point
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What are the couplings?Inter-chain couplings are composed from scaling operators of individual chain theory, e.g. in zero field:
Staggered magnetization Staggered dimerization
Can order these by range and relevance
y Further chain couplings just as relevant but smaller
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Example: Zero field J-J’ modelAllowed operators strongly restricted by reflections
First and second neighbor chains:
marginal gtw∼ J’/J
Relevant but generated by fluctuations at O[(J’/J)4]
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RG Subtleties (1)“Accidentally” zero couplings
E.g. staggered magnetization coupling gN=0
Frustrated line
Fluctuations generate relevant operatorNon-linearities bend RG flow lines
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RG Subtleties (2)Competing Relevant Operators
Fluctuations generate several relevant couplings that compete (gN,gε)
Two factors:More relevant operators grow faster under RGLarger bare values can compensate
Perturbativeregime
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Result in Zero FieldPure J-J’ model:
Staggered magnetization coupling gNdominates and induces collinear magnetic order
Very weak instability occurs only below energy scale ∼ (J’)4/J3
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Result in Zero FieldDzyaloshinskii-Moriya interaction
Cannot be neglected since it is large compared to fluctuation-generated coupling
Result: non-collinear spiral state
relevant
≈ Agrees with neutron experiments
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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 state?cycloidExpt. AF state differs from theory (J2?)
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Magnetization Plateau“Umklapp” (dangerously irrelevant operator): 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 smallIntegrability allows access to high energy spectral propertiesSystematic RG methods describe low energy physics for
Triangular latticeCheckerboard latticeSpatially anisotropic frustrated square lattice
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For the FutureQuasi-1d conductorsSpectra in magnetic fieldOther geometries
kagome basket, Shojoshin-in temple, Koyasan