Download Ground states and excitations of spatially anisotropic quantum antiferromagnets Oleg Starykh

Ground states and excitations of
spatially anisotropic quantum
antiferromagnets
Oleg Starykh, University of Utah
Leon Balents, KITP
Masanori Kohno, NIMS, Tsukuba
Jason Alicea, Caltech/UC Irvine
Andrey Chubukov, U Wisconsin
Ground states and excitations of
spatially anisotropic triangular
lattice antiferromagnets
Oleg Starykh, University of Utah
Leon Balents, KITP
Masanori Kohno, NIMS, Tsukuba
Jason Alicea, Caltech/UC Irvine
Andrey Chubukov, U Wisconsin
Main idea: exploit spatial anisotropy J’/J << 1
• Approach from the limit of decoupled chains
(frustration helps -- it greatly enhances spatial anisotropy)
• Allow for ALL symmetry-allowed inter-chain interactions to develop
• Most relevant perturbations of decoupled chains drive spin order
• High energy excitations: not very sensitive to details of the ground state
J’/J
Decoupled chains J’=0
critical (infinitely degenerate) state
Coupled chains J’/J = 0.34
quantum collinear order
(OAS and Balents, PRL 98 077205 (2007))
How small should J’/J be ?
• Frustration greatly enhances region of “small” interchain J’
• Example: spatially anisotropic triangular AFM
• Numerics: no interchain correlations for J’/J < 0.6 - 0.7
Weng et al 2006, Hayashi Ogata 2007, Heidarian Sorella Becca 2009
interchain spin correlations,
J’=0.6 J;
exponential decay
Weng et al 2006
Collinear AFM state,
generated by quantum fluctuations,
coupling between NN chains (J’/J)4
(J’)4/J3
Pardini Singh 2008 - no; Bishop et al 2008 - yes
Starykh, Balents 2007
Cs2CuCl4: structure and parameters
- +
moderate frustration f = 6
J = 0.374 meV
J’/J = 0.34
D/J = 0.05
J”/J= 0.05
transverse to chain
Along the chain
J’
J
k
Very unusual response: broad and strong continuum!
Numerous 2D theories
• Arguments for 2D:
 J’/J = 0.3 not very small
 Significant transverse dispersion
★ Exotic theories:
★ Series expansion:
• W. Zheng, J. Fjaerestad,
R. R. P. Singh, R. H. McKenzie, R. Coldea 2006.
• S. Yunoki and S. Sorella, 2006.
• M. Q. Weng, D. N. Sheng, Z. Y. Weng, R. J. Bursill, 2006
★
★ d=1: Bocquet, Essler, Tsvelik, Gogolin,
PRB 64, 094425 (2001)
Spin waves:
Two-spinon continuum
Spinon energy
S=1 excitation
Energy ε
Upper boundary
Variables:
kx1 and kx2
or
ε and Qx
Q
Lower
boundary
x
Low-energy sector QAFM=π
Dynamic structure factor of copper pyrazine dinitirate (CuPzN)
Stone et al,
PRL 91, 037205 (2003)
No single
particle peaks!
Effective Schrödinger equation in
two-spinon basis
• Study two spinon subspace
(two spinons on chain y with Sz tot =+1)
– Momentum conservation: 1d Schrödinger
equation in ε space (k = (kx, ky))
• Crucial matrix elements known exactly
Bougourzi et al, 1996
Structure Factor
• Spectral Representation
J.S. Caux et al, 2006
Weight in 1d:
73% in 2 spinon states
99% in 2+4 spinons
– Can obtain closed-form “RPA-like” expression
for 2d S(k,ω) in 2-spinon approximation
1 e
!
Θ[ωkU − ω]Θ[ωkL − ω]
2
2
2π ωU (k) − ω
−I(ρ(k,ω))
S1d (k, ω) =
I(ρ) =
!
0
t
∞
dt
e cosh(2t) cos(4ρt) − 1
t
cosh t sinh(2t)
ρ(k, ω) =
4
acosh
π
!
2 (k) − ω 2 (k)
ωU
L
2 (k)
ω 2 − ωL
Types of behavior
• Behavior depends upon spinon interaction J’(kx,ky)
Bound “triplon”
Identical to 1D
J’(k) = 4J1’cos2[kx/2] cos[ky]
J’2 = J’3 = J’1/2
Upward shift of spectral
weight. Broad resonance
in continuum or antibound state (small k)
Details of triplon dispersion
• Energy separation from the continuum δE ~ [J’(k)]2
• Spectral weight in the triplon pole Z ~ |J’(k)|
bound
k = (π/4,π)
• Anti-bound triplon when J’(k) > J’critical(k) and
• Expect at small k~0 where continuum is narrow.
• Always of finite width due to 4-spinon contributions.
Cs2CuCl4 -- Broad lineshape: “free spinons”
• “Power law” fits well to free spinon result
– Fit determines normalization
J’(k)=0 here !
scan G
triangular geometry:
kx’ = kx , ky’ = kx + 2ky
Triplon: S=1 bound state of two spinons
• Effect of finite experimental resolution:
‣ Dashed curve:
theoretical line
shape
scan E
scan F
 Curves: 2-spinon RPA with experimental resolution
 Curves: 4-spinon RPA with experimental resolution
Transverse dispersion
J’(k)=0
Bound state and resonance
Solid symbols: experiment
Note that peak (blue diamonds) coincides
with bottom edge only for J’(k)<0
Kohno, Starykh, Balents, Nature Physics 2007
Spectral asymmetry
• Comparison:

Vertical green lines: J’(k)=4J’ cos[k’x/2] cos[k’y/2] = 0.
“Spinons unchained”
Summary (I)
❖ Mystery of Cs2CuCl4 is (almost) solved:
• not a 2d spin liquid
• Dynamic response: high-energy elementary
excitations are descendants of 1d spinons
- (kinematically) bind in S=1 pairs (triplons)
- significant variation of spectral weight with twodimensional momentum (kx,ky)
★ Geometric frustration preserves 1d features,
promotes multi-particle excitations
Low-energy properties
• Ordered ground state
– understood as an instability of weakly coupled chains
– need to follow renormalization of all (bare and dynamically
generated) symmetry-permitted interactions
• Examples:
– dimerized phases in quasi-1d J1-J2 and checkerboard models
– non-coplanar in quasi-1d kagome
– collinear AF (CoAF) in triangular antiferromagnet
• Here: ground states of Cs2CuCl4 and Cs2CuBr4 in magnetic field
Heisenberg spin chain via free Dirac fermions
• Spin-1/2 AFM chain = half-filled (1 electron per site, kF=π/2a ) fermion chain
• Spin-charge
separation
 q=0 fluctuations: right- and left- spin currents
 2kF (= π/a) fluctuations: charge density wave ε
Staggered
Magnetization N
Staggered
Dimerization
ε = (-1) S S
x
x
, spin density wave N
Spin flip ΔS=1
-kF
kF
Susceptibility
1/q
1/q
ΔS=0
-kF
x+a
• Must be careful: often spin-charge separation must be enforced by hand
kF
1/q
S=1/2 AFM Chain in a Field
• Field-split Fermi momenta:
 Uniform magnetization
 Half-filled condition
•
Affleck and Oshikawa, 1999
1
Sz
component (ΔS=0) peaked at
scaling dimension
increases
1/2
• Sx,y components (ΔS=1) remain at
scaling dimension
decreases
π
• Derived for free electrons but correct always - Luttinger Theorem
0
1/2
M
h/hsat
1
hsat=2J
0
• XY AF correlations grow with h and remain commensurate
• Ising “SDW” correlations decrease with h and shift from π
h/hsat
1
Weakly coupled Heisenberg chains in magnetic field
z z
y y
x x
!
!
!
+
N
N
S
·
S
→
N
!
• non - frustrated inter-chain coupling r r
r Nr ! + Nr Nr !
r r
most relevant
less relevant
2πR2 < 1/(2πR2)
J’
spins order in the plane perpendicular
to the direction of magnetic field (z):
umbrella / cone / spin-flop states
•
frustrated inter-chain coupling
y+1
y
y
z
z
x
!x,y ·(S
!x,y+1 +S
!x+1,y+1 ) → Nyx ∂x Ny+1
(y+1)
(y)Sπ+2δ
+sin(δ)Sπ−2δ
+Nyy ∂x Ny+1
S
less relevant
most relevant (small to intermediate fields)
1/(2πR2)
1+2πR2 >
★ frustration promotes collinear SDW order
Real material: relevant inter-chain DM interaction, B||D
•
Even though D=0.05 : DM beats CoAF and dimerization zero-field instabilities
[y = chain index]
relevant: dim = 2πR2
•
DM allows relevant coupling of Nx and Ny on neighboring chains
– immediately stabilizes spiral state
•
orthogonal spins on neighboring chains
c
b
–
- + -+
- + -+
small J’ perturbatively makes spiral weakly incommensurate
Dzyaloshinskii-Moriya interaction (DM)
controls phase diagram of Cs2CuCl4 for
B || D (B along crystal a axis)
Finite D, but J’=0
Finite D and J’
Transverse Field: B || D
B || a
•
DM term becomes more relevant
•
b-c spin components (XY) remain commensurate: spin simply tilt in the
direction of the field
•
Spiral (cone) state just persists for all fields.
Experiment vs Theory
Tc
Order increases with h here
due to increasing relevance of
DM term
h
Order decreases with h here
due to vanishing amplitude
as hsat is approached
(density of magnons -> 0 )
physics is controlled by weak (~1/20 of J)
Dzyaloshinskii-Moriya interaction
BEC
Coldea et al 2002;
Radu et al 2005;
Veillette et al 2006;
Kovrizhin et al 2006
Longitudinal Field: B
D removes DM
DM term involves Sz (at π − 2δ) and Sx (at π):
•
 Leads to momentum mis-match for h>0: DM “irrelevant” for h > D
•
With DM killed, sub-dominant instabilities take hold
•
Two important couplings for h>0: kF ↓ − kF ↑ = 2δ = 2πM
Magnetic field relieves frustration!
dim 1/2πR2: 1 -> 2
dim 1+2πR2: 2 -> 3/2
“collinear” SDW
spiral “cone” state
• “Critical point”: 1+2πR2 = 1/2πR2
at M = 0.3
1
Tc
sdw
gives
cone
M
1/2
h/hsat
(somewhat naïve) Phase Diagram
T
Theory
“collinear” SDW
(DM)
“cycloid”
“cone”
?
0
polarized
0.9 1
D/J ~ 0.1
h/hsat
Experiment
“commensurate” AF order ?
DM
cycloid
sdw
cone
?
?
? “Commensurate Collinear” order of some sort has been observed recently:
- seems to have orthogonal spins on neighboring chains (Coldea // Veillette Chalker 2006)
RG length:
shortest wins
sdw / cone
cone
ComAF
sdw
h
Our theory does predict fluctuations-induced
term of right sign, but too small an amplitude
+
Beyond the naïve: commensurate locking
• “Collinear” SDW state locks to the lattice at low-T
-“irrelevant” (1d) umklapp terms become relevant once SDW order is present
(when commensurate): multiparticle umklapp scattering
-strongest locking is at M=1/3 Msat
Observed in Cs2CuBr4 (Ono 2004, Tsuji 07, Fortune 09)
• down-spins at the centers of hexagons
T
(DM)
“cycloid”
0
!
Ψ†R ΨL
"n
“collinear” SDW
Cs2CuBr4 Fortune et al PRL 2009
“cone”
polarized
?
0.9 1
uud
~ 0.1
h/hsat
→ (π − 2δ)n = 2πm → 2M = 1 − 2m/n
n
3
4
5
5
6
m
1
1
1
2
1
2M
1/3
1/2 3/5 1/5 2/3
1/3
~1/2 5/9 2/3
M=1/3 magnetization plateau
★ Observed in Cs2CuBr4 (Ono 2004, Tsuji 2007) J’/J = 0.75
but not Cs2CuCl4 [J’/J = 0.34]
S=1/2
J’
J
• UUD (up-up-down) structure -- down-spins at the centers of hexagons;
commensurate structure -- one down spin per every triangle
★ “up-up-down” state predicted by large-S expansion for
spatially isotropic triangular lattice antiferromagnet
(Chubukov, Golosov 1991): special to J=J’ situation
Classical isotropic Δ AFM in magnetic field
•
Spiral state in the absence of the field
•
Apply field H: two degenerate states
✦ planar - magnetic field is in the plane of spin spiral
✦ cone (umbrella) - field perpendicular to the spiral plane, spins tilt out of the plane
towards the field direction
•
Energies (and susceptibilities) of the two states are equal for J=J’: Eclass = - c J S2
(accidental degeneracy)
•
1/3 plateau is possible at collinear point (φ=0): develops into finite interval by thermal
and quantum fluctuations
Planar
plateau is possible at “collinear” point
Umbrella (cone)
No plateau possible
Order-by-Disorder mechanism
plateau
RbFe(MoO4)2:
S=5/2 Fe3+
Svistov et al PRB (2003)
Smirnov et al PRB (2007)
Tsuji et al (2007)
NMR spectra: Fujii et al (2004)
Cs2CuBr4: S=1/2
Ono et al 2003
δh
The problem: spatial anisotropy stabilizes umbrella !
Spatially anisotropic model: classical analysis fails
H = ∑ J ijSi ⋅ S j − h ∑ S
〈 ij 〉
!
J != J
•
z
i
J’
i
hsat
0
0
J
1/3-plateau
hsat
h
h
Umbrella state:
favored classically
Planar states: favored by
quantum fluctuations
J’/J = 0.75 not particularly small: OK to use semiclassical spin
wave expansion
• Technical formulation: spatial anisotropy J-J’ causes softening
of interacting (including 1/S correction) spin waves
Our semiclassical approach: treat spatial anisotropy
(J-J’) as a perturbation to interacting spin waves
• single dimensionless parameter δ=(40/3)S(1 - J’/J)2:
classical anisotropy energy ~ (J - J’)/J
fully polarized state
spin-wave gap ~ J/S
h
distorted
umbrella (2)
planar
hc2
UUD plateau
BEC k = 0
hc1
commensurate
BEC k != 0
roton condensation
2 low-energy gapped modes
planar
distorted
umbrella (1)
shaded: local minimum only;
pulsed field measurements?
incommensurate
zero-field spiral
1
Cs2CuBr4
2
Alicea, Chubukov, Starykh PRL 102, 137201 (2009)
3
Cs2CuCl4
4
! ~ S(1 - J’/J)
2
Summary (II)
✦ Rich, interesting physics: much can be understood by viewing the
problem/material from 1d perspective
✦ Good for weakly ordered states: abundance of multi-particle
excitations
✦ Delicate details of 3d ordering are determined by minute, and
often anisotropic, sub-leading interactions
• T-H phase diagram of Cs2CuCl4 and Cs2CuBr4 remain to be
understood
– commensurate AF state in Cs2CuCl4
– 2/3 and other fractional states in Cs2CuBr4
• Evolution of high-energy continuum into low-energy spin waves?
NMR data: persistence of planar
structures
RG for quantum spins:
•
– collinear tendency persists among orthogonal
to the field components (Sx,y)
top view:
umbrella
– generates collinear coupling between nextnearest chains
(J’/J)4 [ S+x,y S-x,y+2 + h.c.]
•
Observed in magnetic field in Cs2CuCl4
(M. Takigawa, presentation at HFM-08):
2-sublattice collinear order between chains in the
plane perpendicular to the applied magnetic field
Tokiwa et al, 2007
2-sublattice
collinear
Magnetization measurements
 M(h) is smooth: not sensitive to low-energy (long-distance) fluctuations. Determined by uncorrelated
(but magnetized) chains.
Tokiwa et al, 2006
“Molecular” field
J’=
0
J’=0.34
J
Error in saturation field is (J’)2/2J ≈ 2%
 dM/dh delineates phase boundaries: divergent derivative = phase transition
Cs2CuBr4
• Isostructural to Cs2CuCl4 but believed to be less quasi-1d: J’/J = 0.75
T. Ono et al, 2004
• Magnetization plateau at
M=1/3 Msat observed for
longitudinal but not transverse
fields
(additional feature at 2/3 Msat) - ?
Alicea, Chubukov, Starykh 2008
Order-by-Disorder: Quantum fluctuations (1/S)
•
fluctuation spectra of different spin structures are different: E = Eclass + ΔEsw
•
quantum fluctuations prefer planar arrangement
sw
=
∆Eplanar
S!
S!
sw
=
ωplanar (k) < ∆Eumbrella
ωumbrella (k)
2
2
k
k
•
prefer collinear configuration even more, when possible: state with maximum number of soft
modes wins.
•
plateau is a quantum effect, width δh = 1.8 J/S (hsaturation = 9 J)
•
plateau is the effect of interactions (hence, width ~ 1/S) between spin waves
Tsuji et al (2007)
Chubukov, Golosov (1991)
NMR spectra: Fujii et al (2004)
δh
The problem: spatial anisotropy stabilizes umbrella
Continuum of excitations is a high-energy feature!
Coldea et al,
PRB 2003
J
J’
Low-energy degrees of freedom
• Quantum triad: uniform magnetization M = JR + JL ,
staggered magnetization N and staggered dimerization ε = (-1)x Sx Sx+1
Components of Wess-Zumino-Witten-Novikov SU(2) matrix
• Hamiltonian H ~ JRJR + JLJL + γbs JRJL
marginal perturbation
• Operator product expansion
• Scaling dimension 1/2 (relevant)
• Scaling dimension 1 (marginal)
(similar to commutation relations)
EXPERIMENT: Longitudinal Field ,Tc vs B; SDW, …, cone
• Very different behavior for B along b, c axes (both orthogonal to DM direction a)
• Additional anisotropies in the problem?
 Nature of AFM and AFM’ phases - the biggest puzzle?
AFM
FM
Cone states
AFM’
R. Coldea et al, 2001;
T. Radu et al, 2005;
Y. Tokiwa et al, 2006
FM