Order-by-disorder and long-range interactions in the antiferromagnetic transverse-field Ising model on the triangular lattice -- A perturbative point of view

We study the low-field ground-state (GS) properties of the antiferromagnetic transverse-field Ising model with long-range interactions (afLRTFIM) on the triangular lattice. We use the method of perturbative continuous unitary transformations (pCUT) to derive an effective model for the degenerate GS space of the antiferromagnetic nearest-neighbour (NN) Ising model on a finite system, by treating the transverse-field (TF) and the long-range interactions (LRI) as a perturbation. We determine a level-crossing between the plain stripe phase at small TF and the clock-ordered phase at intermediate TF at $h\cong 0.129$ for $\alpha =6$, $N=36$ spins in order three perturbation theory. We discuss the qualitative layout of the quantum phase diagram of the afLRTFIM on the triangular lattice.