Projective cluster-additive transformation for quantum lattice models

Abstract

We construct a projection-based cluster-additive transformation that block-diagonalizes wide classes of lattice Hamiltonians \mathcal{H}=\mathcal{H}_0 +Vℋ=ℋ0+V. Its cluster additivity is an essential ingredient to set up perturbative or non-perturbative linked-cluster expansions for degenerate excitation subspaces of \mathcal{H}_0ℋ0. Our transformation generalizes the minimal transformation known amongst others under the names Takahashi’s transformation, Schrieffer-Wolff transformation, des Cloiseaux effective Hamiltonian, canonical van Vleck effective Hamiltonian or two-block orthogonalization method. The effective cluster-additive Hamiltonian and the transformation for a given subspace of \mathcal{H}ℋ, that is adiabatically connected to the eigenspace of \mathcal{H}_0ℋ0 with eigenvalue e_0^ne0n, solely depends on the eigenspaces of \mathcal{H}ℋ connected to e_0^me0m with e_0^m\leq e_0^ne0m≤e0n. In contrast, other cluster-additive transformations like the multi-block orthogonalization method or perturbative continuous unitary transformations need a larger basis. This can be exploited to implement the transformation efficiently both perturbatively and non-perturbatively. As a benchmark, we perform perturbative and non-perturbative linked-cluster expansions in the low-field ordered phase of the transverse-field Ising model on the square lattice for single spin-flips and two spin-flip bound-states.