kagome spin liquid
DESCRIPTION
Kagome Spin Liquid. Assa Auerbach Ranny Budnik Erez Berg. Triangular. Kagome. a. a. b. c. b. O(3)xO(2)/O(2) -> O(4) critical pt. b. a. c. b. Three sublattice N’eel state Huse, Singh. Macroscopic degeneracy. Classical Heisenberg AFM. Experiments . S=3/2 layered Kagome. ‘90. - PowerPoint PPT PresentationTRANSCRIPT
Kagome Spin LiquidAssa AuerbachRanny Budnik
Erez Berg
Classical Heisenberg AFM
Macroscopic degeneracy
Kagome
O(3)xO(2)/O(2) -> O(4) critical pt
Three sublattice N’eel stateHuse, Singh
Triangular
ac b
ab
c
a b
b
Experiments
Strong quantum spin fluctuations (spin gap?)
S=3/2 layered Kagome
‘90
‘90
However: Large low T specific heat
2TC
S=1/2 Kagome: Numerical Results
1 .Short range spin correlations : Zheng & Elser ’90; Chalker & Eastmond ‘92
Spin gap
0.06J
2 .Finite spin gap
E(Smin+1)-E(Smin)=
Lots of Low Energy Singlets
Mambrini & Mila
Finite T=0 entropy?
energy
Log
(# s
tate
s)
Number of sites
Misguich&Lhuillier
Log
(# s
tate
s)
2k1k
Massless nonmagnetic modes?
S=0 S=1
E
RVB on the Kagome Mambrini & Mila, EPJB 2000
Weak bonds
strong bonds
6-site singlet “dimer”
Perturbation theory in weak/strong bonds .
1 .Number of dimer coverings is N
15361.2 .Dimers (10-5 of all singlets N=36) exhaust low energy spectrum.
Contractor Renormalization (CORE)C.Morningstar, M.Weinstein, PRD 54, 4131 (1996).
E. Altman and A. A, PRB 65, 104508, (2002).
Details: Ehud Altman's Ph.D. Thesis.
Truncate small longer range interactions
2. Interactions range N
subclus
iiii
renNN
NN
NhHh
'
',...
,...,,),...(1
111
From exact diagonalization of clusters
2. Effective Hamiltonian (exact)
..
...ijk
ijkijk
ijki ij
ijieff hhhhH
Kagome CORE step 1 Triangles on a triangular superlattice
s l
States of
2/
z zx ˆˆ 23
21
S
zx ˆˆ 23
21 ze
2
el ˆ
e
Dominant range 2 interactions
2 triangles
cb
21212112 SSJhclhblSSJH se
)ˆ)(ˆ(
HeisenbergDimerization field
TEST Supertriangle has 4-fold degeneracyFor Heisenberg, and CORE range 2
supertriangle
Range 3 corrections
Effective Bond Interactions
Large Dimerization fields. Contributions will cancel
for uniform <SS!<
cb
0.953
0.2111
0.053
0.1079
0.2805
0.0598
0.038
21 SS
2121
yy llSS
clblSS ˆˆ 2121
2121 llSS
clblSS ˆˆ 1221
clbl ˆˆ 21
21yy ll
)( corrh12
01 zcbl ˆˆˆ
Variational theory
Columnar dimers win!
Barrier between ground states is 0.66/site
Spin OrderE = -0.134/site
2021 .SS
Columnar Dimers. E=-0.2035/site
1243
21 SS
Energies of dimer configurationsDefect in Columnar state: 02720.E
Flipping dimers using yy
yy llJ 21
0.038
Quantum Dimer Model (Rokhsar, Kivelson)
H = -t +V
0.038 -0.0272
Quantum Dimer Model
Quantum Dimer Model (Rokhsar, Kivelson)
H = -t +V
0.038 -0.0272
Moessner& Sondhi:For t/V=1: an exponentially disordereddimer liquid phase!
Here t/V<0.
Long Wavelength GL Theory
/expexp
)cos(
3636
6
020
0222
21
mmm
mxddS low
2+1 dimensional N=6 Clock model ,
Exponentially suppressed mas gap.Extremely close to the 2+1 D O(2) modelCv ~ T2
The triangular Heisenberg Antiferromagnet
2121212112 SSllJSSJhclhclSSJH yyyyse
)ˆ)(ˆ(
Comparison to the Kagome:1. Je, and h are smaller.2. Jyy is negative!3. Variationally: Triangular Heisenberg also prefers Columnar Dimers.
Kagome Triangular
Iterated Core Transformations
Second Renormalization
0.081
0.005
0.039-
0.112
0.1
-0.018
0.004
21 SS
2121
yy llSS
clblSS ˆˆ 2121
2121 llSS
clblSS ˆˆ 1221
clbl ˆˆ 21
21yy ll
)( corrh12
0.039-
0.005
0.037-
0.038
0.05
-0.03
-0.05
21 SS
2121
yy llSS
clblSS ˆˆ 2121
2121 llSS
clblSS ˆˆ 1221
clbl ˆˆ 21
21yy ll
Kagome triangular)( corrh12
Dominant “ferromagnetic” interaction. Leads to <ly> > 0 in the ground state
Pseudospins align ferromagnetically in xz plane
Proposed RG flow
3 sublattice Neelspinwaves
O(2)-spin liquidMassless singlets
triangular Kagome
hJ yy
0yl 0bl ˆ
0
50.Spin gap, 6 sites
090.20.18 sites
54 sites
Conclusions• Using CORE, we derived effective low energy models for the
Kagome and Triangular AFM. • The Kagome model, describes local singlet formation, and a spin
gap.• We derive the Quantum Dimer Model parameters and find the
Kagome to reside in the columnar dimer phase. • Low excitations are described by a Quantum O(2) field theory, with
a 6-fold Clock model mass term. This leads to an exponentially small mass gap in the spinwaves.
• The triangular lattice flows to chiral symmetry breaking, probably the 3 sublattice Neel phase.
• Future: Investigations of the quantum phase transition in the effective Hamiltonian by following the RG flow.
Contractor Renormalization (CORE)C.Morningstar, M.Weinstein, PRD 54, 4131 (1996).
E. Altman and A. A, PRB 65, 104508, (2002).
Details: Ehud Altman's Ph.D. Thesis.
Step I: Divide lattice to disjoint blocks. Diagonalize H on each Block.
block excitations are the ''atoms'' (composite particles)
Truncate:
M lowest states per block
Niii 2 1 ,,
Reduced Hilbert space: ( dim= MN )
CORE Step II: The Effective Hamiltonian on a particular cluster
1. Diagonalize H on the connected cluster.
NN HHEPHH ',..'|..')('|,..
10
001
1
Old perturbative RG
n
M
nnn
renN
N
H ~~1
,,1
2. Project on reduced Hilbert space
nnn ~,
nnnnnnn P
'''
~~~nZ
1
3. Orthonormalize from ground state up. (Gramm-Schmidt)
CORE Step III: The Cluster Expansion
subclus
iiii
renNN
NN
NhHh
'
',...
,...,,),...(1
111 Effective Interactions:
2. CORE Exact Identity:
..
...ijk
ijkijk
ijki ij
ijieff hhhhH
+ + + + d>1: only rectangular shapes!
E. Altman's thesis.
3. If long range interactions are sufficiently small, truncate Heff at finite range.
coherence
4. is the size ("coherence length") of the renormalized degrees of freedom. Note:
Heff is not perturbative in hi j,and not a variational approximation.All the error is in the discarded longer range interactions.
pseudospin S=1/2
Tetrahedra Psedospins
2 JS=1 S=1 S=1
S=2
S=0 S=0
E
tetrahedron =
super-tetrahedron
pseudospin S=1/2
E. Berg, E. Altman and A.A,
cond-mat/0206384, PRL (03)
10-2
Cubic
16-site singlets
2 CORE Steps to Ground State
pyrochlore
1
E/J Heisenberg antiferromagnet
Fcc
10-1
CORE step 1
Anisotropic spin half model: frustrated
CORE step 2
Ising like model: not frustrated
spinJHhex
/.4150
Variational comparison (S=1/2)
Hexagons Versus Supertetrahedra
spinJHST
/.4440
What do experiments say?
Ground state
Moessner, Tshernyshyov, Sondhi
Domain wall singlet excitations
The Checkerboard )(.ˆˆ. SSSSJSSJH
i
ziijj
ijiji
effCORE
25050
Palmer and Chalker (2001)
Geometrical Frustration on Pyrochlores
2D Checkerboard3D Pyrochlore
constJJHtet
tetij
ji 2SSS
Non dispersive zero energy modes.
Spinwave theory is poorly controlled
Villain (79); Moessner and Chalker (98);
free hexagons Free plaquettes
Insufficient Renormalization!
Remaining Mean-Field zero energy modes
Perturbative Expansions+spinwave theory
Harris, Berlinsky,Bruder (92), Tsunetsugu (02)
Pseudospins defined on a FCC lattice
Range 3 CORE
+0.4 J (
0.1 J
Interactions between pseudospins
10-1
Fcc
pyrochlore
1No order!
Macroscopic degeneracy!
Spin-½ Pyrochlore AntiferromagnetE/J Mean Field OrderEffective model
4 sublattice “order”:Harris, Berlinsky,Bruder (92)
Pseudospins
Macroscopic degeneracy!
10-2
Cubic
Ising-like AFM: not frustrated
),( mqS
),( qS
)( qS
)( qS
)( qS
CORE:
Correlations: Theory vs Experiment
Ansatz:
Theory:
S=3/2
S=1/2
E. Berg AA.,, to be published
Tchernyshyov et.al.
S.H. Lee et. al.
magnon gap
fixed q
1 meV