# The Spin 1 SU(2)-invariant model B. Lees Supervisor: D. Ueltschi March 21, 2014

advertisement ```The Spin 1 SU(2)-invariant model
B. Lees
Supervisor: D. Ueltschi
University of Warwick
March 21, 2014
B. Lees Supervisor: D. Ueltschi
The Spin 1 SU(2)-invariant model
The model
We work on a finite lattice Λ ⊂ Zd , with a set of edges E ⊂ Λ &times; Λ.
For concreteness we have nearest neighbour edges on
(
)
(
)
L1
Ld
Ld
L1
&times; ... &times; − + 1, ...,
.
Λ = − + 1, ...,
2
2
2
2
1
2
3
we have
the usual
spin operators S , S , S . We let
S = S 1 , S 2 , S 3 and Sxi be the operator that applies S i to site
x ∈ Λ and leaves other sites unchanged.
B. Lees Supervisor: D. Ueltschi
The Spin 1 SU(2)-invariant model
Spin-1: the Phase diagram
For S = 1 the most general rotation-invariant interaction is
HΛ = −
X J1 Sx &middot; Sy + J2 Sx &middot; Sy
2 .
{x ,y }∈E
The phase diagram has been partially completed, for example for
J2 = 0 and J1 &lt; 0 large enough, this is the result of Dyson, Lieb
and Simon. However, for some regions very little is known, for
example J2 &lt; 0.
Spin-1: the Phase diagram
B. Lees Supervisor: D. Ueltschi
The Spin 1 SU(2)-invariant model
The case J1 = 0, J2 &gt; 0
For the case J1 = 0, J2 &gt; 0 the Hamiltonian can be written as
nem
HΛ,
h
X
= −2
2
(Sx &middot; Sy ) −
{x ,y }∈E
however if we let U =
HΛ,h = −2
X
!
hx (Sx3 )2
x ∈Λ
2
− 1
3
e i πSx and
2
Q
(Sx1 Sy1
X
x ∈ΛB
−
Sx2 Sy2
{x ,y }∈E
+
Sx3 Sy3 )2
−
X
!
hx (Sx3 )2
x ∈Λ
2
− 1
3
nem
we have the useful relation U −1 HΛ,h U = HΛ,
. Writing the
h
Hamiltonian in this form allows us to show the model is reflection
positive.
B. Lees Supervisor: D. Ueltschi
The Spin 1 SU(2)-invariant model
Long range order
Theorem
Let S = 1. Assume h = 0 and L1 , ..., Ld are even. Then we have
the bounds

qD
E


2
1
1 3 1 3

X

1
 9 − √3 Jd Sq0 S0 Se1 Se1
lim lim
ρ(x ) ≥ 
D
E


β→∞ Li →∞ |Λ|

 ρ(e1 ) − √1 Id S01 S03 Se11 Se31 .
x ∈Λ
3
where
*
ρ(x ) =
(S03 )2
B. Lees Supervisor: D. Ueltschi
2
−
3
!
(Sx3 )2
2
−
3
!+
.
The Spin 1 SU(2)-invariant model
The two integrals in the theorem are given by
1
Id =
(2π)d
Z
1
(2π)d
Z
Jd =
s
[−π,π]d
s
[−π,π]d
ε(k + π)
ε(k )

 d

 1 X

cos ki  dk ,
d
i =1
+
(1)
ε(k + π)
dk .
ε(k )
By relating these correlations to the probability of nearest
neighbours being in the same loop in the loop model [Ueltschi ’13]
we show that one of these bounds is satisfied if Id Jd ≤ 4/27, this is
satisfied in d ≥ 8.
B. Lees Supervisor: D. Ueltschi
The Spin 1 SU(2)-invariant model
references
M. Aizenman and B. Nachtergaele, Geometric aspects of
quantum spin states, Commun. Math. Phys. 164, 17–63 (1994).
 F.J. Dyson, E.H. Lieb, B. Simon, Phase transitions in quantum
spin systems with isotropic and non- isotropic interactions, J.
Statist. Phys. 18, 335–383 (1978)
 T. Kennedy, E.H. Lieb, B.S. Shastry, Existence of Néel order in
some spin- 12 Heisenberg antiferromagnets, J. Statist. Phys. 53,
1019–1030 (1988)
 D. Ueltschi, Random loop representations for quantum spin
systems, J. Math. Phys. 54, 083301 (2013)
B. Lees Supervisor: D. Ueltschi
The Spin 1 SU(2)-invariant model
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