Transcript Kagome Spin Liquid - University of British Columbia

Kagome Spin Liquid
Assa Auerbach
Ranny Budnik
Erez Berg
Classical Heisenberg AFM
Triangular
a
a
c
b
Kagome
b
b
a
c
b
Three sublattice N’eel state
Macroscopic degeneracy
Huse, Singh
O(3)xO(2)/O(2) -> O(4) critical pt
Experiments
S=3/2 layered Kagome
‘90
Strong quantum spin fluctuations (spin gap?)
‘90
However: Large low T specific heat
C T
2
S=1/2 Kagome: Numerical Results
1. Short range spin correlations : Zheng & Elser ’90; Chalker & Eastmond ‘92
2. Finite spin gap
E(Smin+1)-E(Smin)= Spin gap
0.06J
Lots of Low Energy Singlets
E
Misguich&Lhuillier
Log (# states)
Mambrini & Mila
S=1
Log (# states)
S=0
Number of sites
energy
 k
2
 k
Massless nonmagnetic modes?
Finite T=0 entropy?
1
RVB on the Kagome
Mambrini & Mila, EPJB 2000
Weak bonds
strong bonds
6-site singlet “dimer”
1. Number of dimer coverings is
1.1536
N
2. Dimers (10-5 of all singlets N=36) exhaust low energy spectrum.
Perturbation theory in weak/strong bonds.
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.
2. Interactions range N
h( 1,...N )  H
ren
1,,N 
subclus
h

i1 ,...i N '
i1 ,...i N '  N
From exact diagonalization of clusters
2. Effective Hamiltonian (exact)
H eff   hi   hij   hijk   hijk ...
i
ij
ijk
ijk ..
Truncate small longer range interactions
Kagome CORE step 1
Triangles on a triangular superlattice

l

s
States of

S
   /
eˆ 
zˆ


ˆ
l e  2
2
1
2
xˆ 
3
2
zˆ
 12 xˆ 
3
2
zˆ
eˆ   zˆ
Dominant range 2 interactions

  
 
H12  J e S1  S2 ( l1  bˆ  h )( l2  cˆ  h )  J s S1  S2
Dimerization field
Heisenberg
2 triangles
bˆ
supertriangle
cˆ
TEST
Supertriangle has 4-fold degeneracy
For Heisenberg, and CORE range 2
Range 3 corrections
Effective Bond Interactions
( corr )
12
h
  
 0.953 S1  S2l1  bˆ l2  cˆ
  
0.2111 S  S l  l
1
21 2
0.053
0.1079
  y y
S1  S2l 1  l 2
 
S1  S2

0.2805

  
ˆ
S1  S2 l2  b  l1  cˆ
0.0598
 
l1  bˆ l2  cˆ
0.038
l 1 l
y
y
2
bˆ

cˆ
Large Dimerization fields.
Contributions will cancel
for uniform <SS>!



l1  bˆ  cˆ  zˆ  0
Variational theory
Columnar Dimers.
E=-0.2035/site
 
S1  S 2   34  12
×
Spin Order
E = -0.134/site
 
S1  S 2  0.2
Columnar dimers win!
Barrier between ground states is 0.66/site
Energies of dimer configurations
Defect in Columnar state:E
 0.0272
0.038
Flipping dimers using
Quantum Dimer Model (Rokhsar, Kivelson)
0.038
H = -t
-0.0272
+V
y y
yy 1 2
J l l
Quantum Dimer Model
Quantum Dimer Model (Rokhsar, Kivelson)
0.038
H = -t
-0.0272
+V
Moessner& Sondhi:
For t/V=1: an exponentially disordered
dimer liquid phase!
Here t/V<0.
Long Wavelength GL Theory
2+1 dimensional N=6 Clock model,
S low  12  dd 2 x       m0 cos( 6 )
2


2

m0
m0
2
m
exp  36  
exp  36 /  


Exponentially suppressed mas gap.
Extremely close to the 2+1 D O(2) model
Cv ~ T2

The triangular Heisenberg Antiferromagnet

  
 
 
H12  J e S1  S2 ( l1  cˆ  h )( l2  cˆ  h )  J s S1  S2  J yyl yl y S1  S2
Comparison to the Kagome:
1. Je, and h are smaller.
2. Jyy is negative!
3. Variationally: Triangular Heisenberg also prefers Columnar Dimers.
Iterated Core Transformations
Kagome
Triangular
Second Renormalization
Kagome
h12( corr ) 
h12( corr ) 
  
0.081 S  S l  bˆ l  cˆ
1
21
2
  
0.005 S  S l  l
1
21 2
0.039 0.112
0.1
  y y
S1  S2l 1  l 2
 
S1  S2


  
ˆ
S1  S2 l2  b  l1  cˆ
triangular
  
0.039- S  S l  bˆ l  cˆ
1
21
2
  
0.005 S  S l  l
1
21 2
0.037 -

0.038
  y y
S1  S2l 1  l 2
 
S1  S2

0.05

  
ˆ
S1  S2 l2  b  l1  cˆ
-0.018
 
l1  bˆ l2  cˆ
-0.03
 
l1  bˆ l2  cˆ
0.004
l y1  l y 2
-0.05
l y1  l y 2
Pseudospins
align ferromagnetically in xz plane
Dominant “ferromagnetic” interaction.
Leads to <ly> > 0 in the ground state

Proposed RG flow
  0.5
18 sites   0.2
54 sites   0.09
Spin gap, 6 sites
triangular
l
y
Kagome
0
3 sublattice Neel
spinwaves
0

l bˆ  0
O(2)-spin liquid
Massless singlets
J yy
h
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.
Truncate:
M lowest states per block
block excitations are the
''atoms'' (composite particles)
Reduced Hilbert space: ( dim= MN )
   i 1 , i 2 , i N
CORE Step II:
The Effective Hamiltonian on a particular cluster
2.
1. Orthonormalize
3.
Project
Diagonalize
on from
reduced
H on
ground
the
Hilbert
state
connected
up.
space
(Gramm-Schmidt)
cluster.
~
n , n  
n
MN
1 

~
~
~
n 
P  n    n'  n'  n 

Zn 
n' n

~ ~
H ren
1,, N     n  n  n
n 1
( 1  P0 )
 1 ,.. N | H 0  H'
H' .. | '1 ,..' N
E  H0
Old perturbative RG
CORE Step III: The Cluster Expansion
h( 1,...N )  H
Effective Interactions:
ren
1,,N 

subclus
h
i1 ,...i N '
i1 ,...i N '  N
2. CORE Exact Identity:
H eff   hi   hij   hijk   hijk ...
i

ij
coherence
+
+
ijk
ijk ..
+
+
d>1: only rectangular shapes!
E. Altman's thesis.
3. If long range interactions are sufficiently
small, truncate Heff at finite range.
Heff is not perturbative in hi j,
4.  is the size ("coherence length")
of athe
renormalized
degrees of freedom.
and not
variational
approximation.
Note: All the error is in the discarded longer
range interactions.
Tetrahedra Psedospins
E. Berg, E. Altman and A.A,
E
cond-mat/0206384, PRL (03)
S=2
tetrahedron
=
S=1
S=1
S=1
2J
S=0
pseudospin S=1/2
super-tetrahedron
pseudospin S=1/2
S=0
2 CORE Steps to Ground State
E/J
Heisenberg antiferromagnet
pyrochlore
1
CORE step 1
Anisotropic spin half model: frustrated
Fcc
10-1
CORE step 2
Ising like model: not frustrated
Cubic
10-2
16-site singlets
Hexagons Versus Supertetrahedra
Variational comparison (S=1/2)
H
H
hex
ST
 0.415J / spin
 0.444J / spin
What do experiments say?
H
eff
CORE
The
 Checkerboard


ˆ
 0.5 J  S i  
ij

ij


ˆ  0.25 J S  ( SSS )
S j 

ij
z
i
i
Palmer and Chalker (2001)
Ground wall
statesinglet excitations
Domain
Moessner, Tshernyshyov, Sondhi
Geometrical Frustration on Pyrochlores
2
H  J  Si  S j  J  Stet
 const
ij
3D Pyrochlore
free hexagons
tet
2D Checkerboard
Free plaquettes
Villain (79);
Moessner and Chalker (98);
Non dispersive zero energy modes.
Spinwave theory is poorly controlled
Interactions between pseudospins
Pseudospins defined on
a FCC lattice
Range 3 CORE
0.1 J
+0.4 J (
Perturbative Expansions+spinwave theory
Harris, Berlinsky,Bruder (92), Tsunetsugu (02)
Remaining Mean-Field zero energy modes
Insufficient Renormalization!
Spin-½ Pyrochlore Antiferromagnet
E/J
Effective model
Mean Field Order
pyrochlore
1
No order!
Macroscopic degeneracy!
4 sublattice “order”:
Harris, Berlinsky,Bruder (92)
10-1
Fcc
Macroscopic degeneracy!
Pseudospins
Cubic
10-2
Ising-like AFM: not frustrated
Correlations: Theory vs
Experiment
S=3/2

S( q , )
fixed q
S.H. Lee et. al.
S ( q , m )
1 meV

Ansatz:

Tchernyshyov et.al. S ( q )

S( q )
Theory: CORE:
E. Berg AA.,, to be published

S( q )
S=1/2
magnon gap