Orbital Compass Model

Directional Exchange Frustration — Jx σᵢˣσⱼˣ + Jy σᵢʸσⱼʸ — Z₂ Symmetry Breaking
Orbital configuration (color = σˣ component)
Correlation functions & order parameter
Jx = 1.00
Jy = 1.00
kT = 0.50
h = 0.00
N = 16
Compass Hamiltonian:
H = −Jx Σ_{⟨ij⟩ₓ} σᵢˣσⱼˣ
−Jy Σ_{⟨ij⟩ᵧ} σᵢʸσⱼʸ
− h Σᵢ σᵢᶻ
Key properties:
• Directional bonds:
x-bonds couple σˣ
y-bonds couple σʸ
• Extensively frustrated
• No LRO in 2D (Mermin)
• Z₂ QPT at Jx = Jy
• String order parameter
The orbital compass model is a frustrated quantum magnet where exchange interactions are bond-directional: only σˣ components couple along x-bonds, only σʸ along y-bonds. This creates extensive frustration — each plaquette is frustrated simultaneously. Despite no conventional long-range order, there is a quantum phase transition at Jx = Jy (dual symmetry) with Z₂ symmetry breaking and non-local string order.