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 × Λ.
For concreteness we have nearest neighbour edges on
(
)
(
)
L1
Ld
Ld
L1
× ... × − + 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 · Sy + J2 Sx · Sy
2 .
{x ,y }∈E
The phase diagram has been partially completed, for example for
J2 = 0 and J1 < 0 large enough, this is the result of Dyson, Lieb
and Simon. However, for some regions very little is known, for
example J2 < 0.
Spin-1: the Phase diagram
B. Lees Supervisor: D. Ueltschi
The Spin 1 SU(2)-invariant model
The case J1 = 0, J2 > 0
For the case J1 = 0, J2 > 0 the Hamiltonian can be written as
nem
HΛ,
h
X
= −2
2
(Sx · 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
[1]M. Aizenman and B. Nachtergaele, Geometric aspects of
quantum spin states, Commun. Math. Phys. 164, 17–63 (1994).
[2] 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)
[3] 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)
[4] 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