Samiyeh mahmoudian Abdollah langari

Samiyeh mahmoudian
Abdollah langari
introduction
y Anisotropic Kondo necklace model
y One dimension
y Two and three dimension
y Spin ladder geometry
y Odd leg ladders
y
y
Mean field approach
Even leg ladders
y
Spin wave approach
Kondo-necklace model
S
y Isotropic
J
t
τ
Η KN = t ∑ (τ ixτ ix+1 + τ iyτ iy+1 ) + J ∑ si .τ i
i
i
y Anisotropic
Η KN = t ∑ (τ τ +τ τ + δτ τ ) + J ∑ (τ s + τ s + Δτ s )
<i , j >
x x
i j
y y
i j
z z
i j
x x
i i
i
y y
i i
z z
i i
Anisotropic case
Spin ladder geometry
nl
H = t∑
∑ (τ
n =1 < m, m' >
τ
x
x
nm nm'
nl −1
y
y
z
z
'
x
x
y
y
z
z
+ τ nm
τ nm
' + δτ nmτ
' ) + t ∑∑ (τ nmτ n +1m + τ nmτ n +1m + δτ nmτ n +1m )
nm
n =1 m
nl
x
x
y
y
z
z
+ J ∑ ∑ (τ nm
s nm
+ τ nm
s nm
+ Δ τ nm
s nm
)
nl =the number of legs
n =1 m
t'
J
t
t
Mean field approach
y Bond operator method
s nm ,α =
1 +
+
+
( s nm t nm ,α + t nm
,α s nm − i ε αβγ t nm , β t nm , γ )
2
1
2
+
+
+
τ nm ,α = (− snm
t nm ,α − t nm
,α s nm − iε αβγ t nm , β t nm ,γ )
+
< snm >=< snm
>= s
y J/t→∞
tk x k y ,β =
If
N t δ k x k y ;Q x Q y + η k x k y , β
δ >1 → β=z
&
If
δ <1 → β=x,y
y Ordering parameter
M =< t >< s >
Even & odd leg ladder
t2 =
5
1
J
−
−
4 2 Dδt 4 N
∑
1
k x ,k y
1+ (
γx +γ y
D
−
)
1
4N
∑
1
k x ,k y
1+ (
γx +γ y
)
δD
∞
t2 =
γα =cos(kα )
Kx =2π/N
J
5
1
−
−
4 2 D δt 4 N
∑
1
k x ,k y
1+ (
γx +γ y
D
−
)
1
4N
∑
1
k x ,k y
1+
(γ x + γ y )δ
D
D=2 min value of (γx + γy)
N=0,1,2,…,N-1
Ky =0,π/3,2π/3 (three legs ladder)
Ky =0,π (two legs ladder )
Odd leg ladders
y Mean field approach works correctly
y By increasing the number of legs we get thermodynamic limit
y For δ>1 increasing the anisotropy term will grow ordering
y For δ<1 increasing the anisotropy term will disrupt ordering
m
δ=0.6
δ=2
t/J
(t/J)c =0.8
(t/J)c =0.34
Even leg ladders
y Mean
field approach doesn’t work correctly
•δ <1
y We
wavea approach
for½two
ladder
If can
J→0use
we spin
will have
two legs spin
XXZlegs
Hamiltonian
1/2
1
τ
The model is known to be in the gapped phase
If J→∞ the ground state is the direct product of singlets
S
τ
J
The model is again in disordered gapped phase
Spin wave approach
y Two legs ladder
sA
τA
sB
τB
y J/t→0
t'
t
H =t
+ t′
L/2
∑∑
xA xB
yA yB
zA zB
(
τ
τ
+
τ
τ
+
δτ
∑ nm nm +σ nm nm + σ
nm τ nm + σ )
n =1 , 2 m =1 σ
L
∑ ∑
n ≠ n ′=1, 2 m =1
+J
xA
yA
(τ nm
τ nxB′m + τ nm
τ nyB′m + δτ
L/2
∑∑
n =1 , 2 m =1
zA
nm
A
B
B
A
(τ nm
. s nm
+ τ nm
. s nm
)
τ nzA′m )
Para-unitary transformation
H = ∑ ΦMΦ
y We perform the Holstein-Primakov transformation
+
k
y We use the
J
⎡
′
δ
(
+
)
+
t
t
translational
⎢
2
⎢
⎢
0
M =⎢
⎢ (t ′ + tγ ( k ) )
⎢
2
⎢
J
⎢
2
⎣
tγ ( k )
0invariance
(t ′ +
)
2
J
J
2
2
J
J
δ (t + t ′) +
2
2
J
0
2
J⎤
symmetry
2⎥
⎥
0⎥
⎥
0⎥
⎥
⎥
0⎥
⎦
The Hamiltonian can be diagonalized if M is positive definite
Behavior of anisotropic 2 legs
ladder
y t=t’
δ
0.76
0.8
0.9
1
1.1
Jc
2.2
4
6.5
8.3
10.2
with increasing δ parameter, ordering will survive for larger
J
y t≠t’
t’=t
δc =0.76
4t
0.91
6t
8t
0.93
0.95
the ordering is mainly created by the anisotropy, not the hopping
Conclusion
y Odd leg ladders
Their behavior is similar to the two and three dimension
y Even leg ladders
Their behavior is similar to one dimension so the only
ordering that can be accrued is the result of anisotropy