Non-perturbative effects in two-dimensional lattice O(N) models

Mark B. Evans

Edwin Langmann A

We derive a low energy effective model for a two dimensional (2D) system of spinless fermions on a square lattice with repulsive short-range interactions and away from half filling. This model is a natural 2D analogue of the Luttinger model which describes interacting fermions and can be treated exactly by bosonization. It also includes fermion degrees of freedom which cannot be bosonized but, as we argue, can be treated adequately by mean field theory. Our results generalize to 2D a standard approach to 1D lattice fermion systems making them amenable to non-perturbative computation methods by a suitable continuum limit.

N. Dupuis, K. Sengupta

Murray T. Batchelor, John Cardy

Max Hörmann, Kai Phillip Schmidt

We construct a projection-based cluster-additive transformation that block-diagonalizes wide classes of lattice Hamiltonians \mathcal{H}=\mathcal{H}_0 +V ℋ = ℋ 0 + 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}_0 ℋ 0 . 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}_0 ℋ 0 with eigenvalue e_0^n e 0 n , solely depends on the eigenspaces of \mathcal{H} ℋ connected to e_0^m e 0 m with e_0^m\leq e_0^n e 0 m ≤ e 0 n . 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.

S. Jain