Upload
imaxsw
View
223
Download
0
Embed Size (px)
Citation preview
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
1/33
Spatially anisotropic S=1/2
Heisenberg Kagomeantiferromagnet
Oleg Starykh, University of Utah
Andreas Schnyder, KITP
Leon Balents, KITP and UCSB
Thanks to J.-S. Caux for numerical data
PRB 78, 174420 (2008)
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
2/33
Kagome spin-1/2 antiferromagnet:
from model to experiments
herbersmithite
volborthite
vesignieite
Hiroi et al. 2001, Shores et al. 2005, Okamoto et al. 2009
no order for T
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
3/33
Outline
Motivation
- strong spatial anisotropy offers well controlled analysis
- relevant experimentally
Description of the J
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
4/33
Volborthite Cu3V2O7(OH)2 2H2O
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
5/33
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
6/33
Volborthite Cu3V2O7(OH)2 2H2O
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
7/33
k,
jk
,j
k,
j"
JJJ!!!
J
J'
2i 2i+1 2i+2
2y
2y+1
2y-1
J'
Spatially anisotropic geometry
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
8/33
The model
Chain spins S (black dots), exchange interactionJ (blue lines)
Interstitial (interchain) spins (red dots), exchange J
Quasi-one-dimensional limit: J
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
9/33
How small should J/J be ? Frustration greatly enhances region of small interchain J
Numerics: no interchain correlations for J/J < 0.6 - 0.7Weng et al 2006, Hayashi Ogata 2007, Heidarian Sorella Becca 2009
Example: spatially anisotropic triangularAFM
Collinear AFM state,generated by quantum fluctuations,coupling between NN chains (J/J)4
Pardini Singh 2008 - no; Bishop et al 2008 - yes
(J)4/J3
Starykh, Balents 2007
interchain spin correlations,
J=0.6 J;
exponential decay
Weng et al 2006
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
10/33
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
11/33
The decoupled limit, J 0, is very singular
collection ofindependentspin chains and interstitial spins
JS S
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
12/33
Separation of scales
local Kondo coupling J (marginal, TK ~ e-c J/J )
RKKY interaction between interstitial spins (via chains)
Tinterstitial ~ (J)2/J
The biggest non-frustrated interaction energy
Tinterstitial >> Tchains >> TK
Interstitial-mediated coupling between spin chainsTchains ~ (J)4/J3
=>
Ny+1
Ny
y+1
y
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
13/33
Frustrated vs Non-Frustrated geometry
Imagine non-frustrated rectangulargeometry:then Tinterstitial ~ Tchain ~ (J)2/J
=>no clear separation of energy scales,
difficult to analyze.e.g. Kondo necklace, Essler Kuzmenko Zaliznyak 2007
=>
Frustrated Kagome geometry:
Ny+1
Ny
y+1
y
(S2n + S2n+1)y My + x Ny
V(1)ch =
y
dx Nx Ny x Ny+1 + M My My+1
(J)2/J order: marginal coupling
(J)4/J order: relevant coupling
V(2)ch =
y dx N Ny Ny+1 + yy+1
M xN
=>
uniform magnetization
staggered magnetization
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
14/33
RKKY interaction between interstitial spins
a polarizes spin chain, which in turn couples to b,c: interactionbetween s connected to the same spin chain.
JS
J
S
c
b
a
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
15/33
RKKY interaction between interstitial spins
Spatially anisotropic triangular lattice formed byK1 and K2 bonds, withfurther-neighbor interactions K3,4 etc.
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
16/33
RKKY interaction between interstitial spins
Spatially anisotropic triangular lattice formed byK1 and K2 bonds, withfurther-neighbor interactions K3,4 etc.
K1: nearest-neighbor interaction between s
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
17/33
RKKY interaction between interstitial spins
Spatially anisotropic triangular lattice formed byK1 and K2 bonds, withfurther-neighbor interactions K3,4 etc.
K1: nearest-neighbor interaction between s
K2: next nearest-neighbor interaction
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
18/33
RKKY interaction between interstitial spins
Spatially anisotropic triangular lattice formed byK1 and K2 bonds, withfurther-neighbor interactions K3,4 etc.
K1: nearest-neighbor interaction between s
K2: next nearest-neighbor interaction
K3: further-neighbor interactions
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
19/33
RKKY interaction between interstitial spins
Inter- interactions are determined by dynamical structure factor of critical spin-1/2 chain
K1: nearest-neighbor interaction between s
K2: next nearest-neighbor interaction
K1 = 2(J)2A(1)
K2 = 4(J)2A(2)
A(r) =8
0
d
0
dqS(q, )
cos2(q
2) cos(qr)
< 0 ferro> 0 antiferro
K1
K2K2
K1 0.7
because (J/J)4
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
20/33
: Spatially anisotropic triangular lattice
K1: nearest-neighbor interaction
K2: next nearest-neighbor interaction
K1 = 2(J)2A(1)
K2 = 4(J)2A(2)
< 0 ferro
> 0 antiferro
K1
K2
H =
q
K(q)q q
K(q) = 2K1 cos qx cos qy + K2 cos(2qx) + 2K3 cos(3qx)cos qy + K4 cos(4qx)
2-spinon approximation for A(r) : rotating spiral ground state, q=2(0.08,0)
ABACUS database (N=500 site chain): ferromagnetic ground state, q=0
J.S.Caux, U Amsterdam
= s0[x cos(qx) + y sin(qx)]
2d magnetic orderamongst
interstitial spins!
Note: s0 ~ O(1)
consider both
cases!
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
21/33
Ordering of chain spins I
Chain spins are subject to: external rotating field due to ordered s:
marginal backscattering term in every chain
(J/J)4 fluctuation-generated interchain interactions
Expect response to hx :static magnetization in x-yplane, of magnitude O(J/J)
hx = 2s0 cos(q/2)J[x cos(qx) + y sin(qx)]
V(2)ch =
y
dx N Ny Ny+1 + yy+1
Hbs = gbs
dx MR ML
q > 0
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
22/33
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
23/33
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
24/33
Ordering of chain spins IV
Order in a rotated basis: Nx
y = (1)yM , M = c(J/J)2
M
a
R/L=
y= 0
Order in the original basis: S+x+1/2,y = h2 + d2
2vcos e
iqx
Szx+1/2,y = (1)x+y
M
yx
z
a
x
z
Non-coplanar order (q > 0)
x,y
O(1) Sx,y
O(J
/J) Sz
O(J
/J)2
~ J/J
~ (J/J)2
q > 0
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
25/33
(b)xz + + + +
+ + + + +
+ + + + +
+ + + +
-yx
Ordering of chain spins: top viewinterstitial spins form spiral,
chain spins are locally anti-parallel to it,
with small staggered component normal to the spiral plane
basic physics: s=1/2 chain subject to magnetic field -components orthogonal to the field are most relevant
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
26/33
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
27/33
Ordering of chain spins: q=0 case
Ferromagnetic order among interstitial spins,
predominantly ferromagnetic ordering among chain spins,
with weaker antiferromagnetic order along, and between, chains:
coplanar state, ferrimagnet.
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
28/33
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
29/33
Comparison
Wang, Vishwanath, Kim 2007:q=0 state
Yavorskii, Apel, Everts 2007:extended region of incommensurate order q>0
among inter-chain spins, disordered chains
1d limit: J=1, J>>1
large-S semiclassical analysis large-N Sp(N)
very similar to our q=0 state
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
30/33
Conclusions
Surprise: spatially anisotropic kagomeantiferromagnet is magnetically ordered
The order is non-coplanar (q > 0)- coplanar with q=0 is possible (ferrimagnetic state)
Interesting hierarchy of scales
- interstitial spins order at Tinterstitial ~ (J)2/J
- chain spins order at Tchain ~ (J)4/J3
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
31/33
Heisenberg spin chain via free Dirac fermions
Spin-1/2 AFM chain = half-filled (1 electron per site, kF=!/2a ) fermion chain
Spin-charge separation
! 2kF(= !/a) fluctuations: charge density wave" , spin density wave N
Spin flip #S=1
#S=0
Staggered
Magnetization N
Staggered
Dimerization
"= (-1)x Sx Sx+a
Susceptibility
1/q
1/q
1/q
kF-kF
kF-kF
! q=0 fluctuations: right- and left- spin currents
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
32/33
Quantumtriad: uniform magnetization M = JR
+ JL
,
staggered magnetization N and staggered dimerization != (-1)x Sx Sx+1!Components of Wess-Zumino-Witten-Novikov SU(2) matrix
Hamiltonian H ~ JRJR
+ JLJL+ "
bsJRJL
Operator product expansion
Scaling dimension 1/2 (relevant)
Scaling dimension 1 (marginal)
Low-energy degrees of freedom
(similar to commutation relations)
marginal perturbation
8/3/2019 Oleg Starykh, Andreas Schnyder and Leon Balents- Spatially anisotropic S=1/2 Heisenberg Kagome antiferromagnet
33/33
S=1/2 AFM Chain in a Field
1
1/2
0 h/hsat1
M
1/2
XY AF correlations grow with hand remain commensurate Ising SDW correlations decrease with h and shift from !
Affleck and Oshikawa, 1999
Field-split Fermi momenta:
! Uniform magnetization
! Half-filled condition
Sz component ("S=0) peaked at
scaling dimension
increases
Sx,y components ("S=1) remain at !scaling dimension
decreases
Derived for free electrons but correct always - Luttinger Theorem
10 h/hsat
hsat=2J