Compass model on a ladder and square clusters

We obtained exact heat capacities of the quantum compass model on the square L×L clusters with L = 2, 3, 4, 5 using Kernel Polynomial Method and compare them with heat capacity of a large compass ladder. Intersite correlations found in the ground state for these systems demonstrate that the quantum compass model differs from its classical version. Quantum compass model originating from the orbital (Kugel-Khomskii) superexchange in transition metal oxides has been recently studied both using analytical [1] and numerical [2, 3] approach. In spite of remarkable progress in numerical methods for two–dimensional (2D) Ising– like models [4, 5], exact solutions are necessary to investigate the structure of excited states. The one–dimensional (1D) compass model is exactly solvable by mapping to quantum Ising model [6] and exhibits interesting properties while approaching to the quantum critical point at zero temperature [7]. In addition, its ladder version, first considered by Douçot et al. [1] is solvable in a similar way and its partition function [8] can be obtained exactly in case of a large (but finite) system. There is no exact solution for the 2D compass model but the latest Monte Carlo data [3] prove that the model exhibits a phase transition at finite temperature both in quantum and classical version with symmetry breaking between x and z part of the Hamiltonian. This paper suggests a scenario for a phase transition with increasing cluster size by the behavior of the specific heat obtained via Kernel Polynomial Method (KPM) [9]. The Hamiltonian of the quantum compass model on a square L× L lattice is given by