Onedimensional view of frustrated magnets

1
One-dimensional view of frustrated magnets
Oleg Starykh , University of Utah
In collaboration with Leon Balents PRL 93,
127202 (2004) and
Akira Furusaki (work in progress).
2
Outline

• Phases of frustrated magnets Introduction

• One-dimensional spin liquid - properties of
  S1/2 Heisenberg chain

• Coupled chains - confinement of spinons and
  spontaneous dimerization

• Spatially anisotropic J1-J2 antiferromagnet
  columnar dimer phase

• Pyrochlore and its 2D projection crossed-dimer
  phase

• Conclusions and experimental hints

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Phases of frustrated antiferromagnet

• Neel phase

staggered magnetization
gapless magnons, S1
(Neel, Landau 1933 but Pomeranchuk - neutral
fermions 1941)
gapped magnons (triplons), S1

• Valence bond solid

(Sachdev,Read 1990)
Dimer long-range order, broken translational
symmetry
crystal of singlets
deconfined massive spinons, S1/2
No local order parameter

• Spin liquid (RVB)

(Anderson 1973,1987)

• Luttinger liquid (d1)

Algebraic correlations
deconfined massless spinons, S1/2
(Bethe 1931,Takhajyan,Faddeev 1974)
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The strategy

• Build up the lattice from spin chains (exchange
  J gtgt J1 , J2) stability analysis
• of the gapless Luttinger phase

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D1 Breaking the (spin) waves
Create spin-flip and evolve with
Energy cost J
Segment between two domain walls has opposite to
initial orientation. Energy is size independent.
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S1/2 chain non-Neel critical ground state
1) Spinons propagating domain walls in AFM
background, carries S1/2
domain of opposite Neel orientation
Dispersion
2) Domain walls are created in pairs (but one
kink per bond gt fermions)
3) Single spin-flip two domain walls spin-1
wave breaks into pairs of deconfined spin-1/2
spinons.
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Continuum of excitations
Upper boundary
Energy
Lower boundary
Low-energy sector QAFMp
Neutron scattering on Cs2CuCl4 (Coldea et al.
2001)
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Staggered dimerization
Measure of bond strength

same as
Critical correlations
Lieb-Schultz-Mattis theorem either critical or
doubly degenerate ground state
Frustration leads to spontaneous dimerization
(Majumdar,Ghosh 1970)
J1-J2 spin chain - spontaneously dimerized ground
state (J2 gt 0.24J1)
Spin singlet
J2 gt 0
OR
Kink between dimerization patterns - massive but
free S1/2 spinon!
Shastry,Sutherland 1981
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Key variables

• Quantum order uniform magnetization M MR ML
  ,
• staggered magnetization N and staggered
  dimerization e
• components of Wess-Zumino-Witten-Novikov matrix
  G N-i tr(G.s), etr(G)

• Hamiltonian H MRMR MLML

• Operator product expansion

(similar to commutation relations)
,

• Scaling dimension 1/2 (relevant)

• Scaling dimension 1 (marginal)

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Two chains - spin ladder - confined spinons
(Shelton,Nersesyan,Tsvelik 1996)
Inter-chain coupling J1 destroys spinons
confinement due to J1N1 N2
Energy cost of wrong domain size of the
domain linear potential J1 x between spinons!
Two S1/2 spinons form S1 triplet (magnon) and
(higher-energy) singlet.
S1 spin chain

• J1 lt 0 Haldane phase (FM)

(Haldane 1984)
all excitations are massive

• J1 gt 0 Rung-singlet phase (AFM)

Massive S1 excitations only
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Frustrated ladder I
(Allen,Essler,Nersesyan 2000)
chain 1
chain 2
Marginally relevant preserves spinons lMJ12J2

• lN0 no relevant interactions (energy cost 0)
  gt line of deconfined spinons?

J2J1/2
J2
Haldane
Rung-singlet
J1

• Can be extended to two-dimensional model
  (Nersesyan,Tsvelik 2003)

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Frustrated ladder II
1
J1
J2
2

• Inter-chain Hamiltonian

• Continuum limit

allowed by symmetry!
J1/2
0
J12J2
J1-2J2

• Accidental degeneracy lNle0 at J12J2 ?

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Quantum fluctuations generate dimerization !

• Classical degeneracy line J12J2

• Second-order perturbation theory OPE

dimerization
Fuse fields from the same chain

• Renormalized inter-chain interaction

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Frustrated ladder III
(OS, Balents 2004)
1
J1
J2
2
generated by fluctuations
SU2(2) transition, c3/2
propagating S1 triplet
physical trajectory
Z2 transition, c1/2
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( Nersesyan, Tsvelik 2003 OS, Balents 2004)
Deconfined spinons le3lN

• First order direct transition between Haldane
  and rung-singlet phases
• (exponentially narrow 1/ln lN,e ltlt lJ ) gt
  mixed state

on the 1st order transition
rung-singlet
Haldane

• Excitations massive S1/2 spinons
  domain walls between two co-existing states

• Spinons are mobile version of end-chain S1/2
  moments of S1 chain

S1
JimpuritygtgtJ
S0
S0
Non-magnetic (S0) impurity
Localized S1/2
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Spatially anisotropic square lattice AFM
(OS,Balents 2004)
m1
m
m-1

• Chain mean-field approximation

Maps onto sine-Gordon action

• Self-consistent equations

(exact free energy -- Lukyanov, Zamolodchikov
1997)

• Predicts three phases separated by first-order
  transitions at

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(OS,Balents 2004)
Phase diagram of the 2d model
2d J1-J2 model
J2/J
Classical phase boundary J2J1/2
Frustrated square lattice AFM
(p,0)
Columnar Dimer phase
(p,p)
J1/J

• Magnetically ordered phases AFM or FM

• Columnar dimer phase

Staggered dimerization
Spin gap

• Phase diagram agrees with large-N approach
  (Sachdev,Read 1990, 1991) J1J

Series expansion (Singh 1990) J1J, exact
diagonalization (Sindzingre 2003) J1 lt J
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Most frustrated antiferromagnet pyrochlore
Corner-sharing 3d lattice of tetrahedra (4 spins
in each)
X
Hence, on every tetrahedron
Y
Extensive degeneracy tetrahedra
(Moessner, Chalker, PRB 58, 12049 (1998))
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Two-dimensional pyrochlore checkerboard
antiferromagnet
Corner-sharing geometry but
non-equivalent J1 , J2 bonds
?
J1/J2
0
1
4/3
Chains J2gtgtJ1
Neel (square lattice of J1 bonds)
Plaquette phase
Singh,OS,Freitas 1998 Canals 2002
4-spin singlet
(classically decoupled)
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Checkerboard antiferromagnet - spin liquid?
(OS,Singh,Levine 2002)

• Random phase approximation predicts 1d critical
  phase - sliding SU(2) Luttinger liquid

gt No instability!
but
T0
J1J2 , T0.01J2 , w 0.1J2
V
3

• Local current-current interaction is irrelevant

1
H
2
HinterJ1(S1S2).(S3S4) gt J1 MH MV
4
Scaling dimension2 gt 1
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Maybe not?!
(OS, Balents, Furusaki 2004)

• Keep irrelevant terms!

• Expand in J1 and apply OPE

• Generate interaction of staggered dimerizations
  on crossing chains

(respects all lattice symmetries)
Y

• Chain mean-field staggered dimer order on
  parallel chains

m1
m
horizontal
vertical
X
n
n1
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Checkerboard antiferromagnet - valence bond solid
(OS,Balents,Furusaki)

• Crossed dimer phase

Spin gap
J1 ltlt J2
O(3)
O(3)
O(3)
J1/J2
Neel
0
1.3
Crossed dimer
Plaquette
LRO phase
0.8
Tchernyshyov et al Brenig et al
Exact diagonalization Sindzingre et al. 2002
Large-N (Bernier et al. PRB 69, 214427 (2004))
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Extension to three dimensions
S1/2 pyrochlore GeCu2O4, T.Yamada et al.
J.Phys.Soc.Japan 69, 1477 (2000)
J1
J2
J/J 0.16 but seems to order
D3 crossed-dimer phase for J ltlt J ?
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Conclusions

• One-dimensional strategy offers perturbative
  approach to spontaneously
• dimerized phases of frustrated
  antiferromagnets
• columnar dimer phase in frustrated ladder
• spatially anisotropic J1-J2 model (extends all
  way to 2D limit)
• two-dimensional pyrochlore antiferromagnet (and
  its 3D extention)

• Disordered phases of collinear antiferromagnet
  are spontaneously
• dimerized. Thus, confining -- in agreement
  with gauge theory formulation

Thanks to A. Abanov, I. Affleck, F. Essler, D.
Haldane, P. Lecheminant, R. Moessner, A.
Nersesyan, O. Tchernyshyov, S. Sachdev, P.
Sindzingre, A. Tsvelik, A. Vishwanath
Supported by Research Corporation and KITP
Scholarship.