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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