Exact solution for a quantum compass ladder

Abstract

We introduce a spin ladder with antiferromagnetic Ising ZZ interactions along the legs, and interactions on the rungs which interpolate between the Ising ladder and the quantum compass ladder. We show that the entire energy spectrum of the ladder may be determined exactly for finite number of spins 2N by mapping to the quantum Ising chain and using Jordan-Wigner transformation in invariant subspaces. We also demonstrate that subspaces with spin defects lead to excited states using finite size scaling, and the ground state corresponds to the quantum Ising model without defects. At the quantum phase transition to maximally frustrated interactions of the compass ladder, the ZZ spin correlation function on the rungs collapses to zero and the ground state degeneracy increases by 2. We formulate a systematic method to calculate the partition function for a mesoscopic system, and employ it to demonstrate that fragmentation of the compass ladder by kink defects increases with increasing temperature. The obtained heat capacity of a large compass ladder consisting of 2N=104 spins reveals two relevant energy scales and has a broad maximum due to dense energy spectrum. The present exact results elucidate the nature of the quantum phase transition from ordered to disordered ground state found in the compass model in two dimensions.Comment: 15 pages, 11 figures, to appear in Phys. Rev.

Compass models: Theory and physical motivations

“…In particular, compass models appear in effective low energy description of quantum magnets that have a chiral degree of freedom (Budnik and Auerbach, 2004;Capponi et al, 2004;Ferrero et al, 2003;Mila et al, 2007). In these systems, the degree of freedom plays the role of the pseudo-spin with non-trivial directional dependence of the coupling.…”

“…To date, two variants of the kagome lattice antiferromagnet were investigated in their low energy sector. These are the trimerized kagome lattice antiferromagnet (Ferrero et al, 2003) and the uniform kagome antiferromagnet (Budnik and Auerbach, 2004). Both of these systems were investigated for a spin S = 1/2 rendition of the original antiferromagnet.…”

“…Several groups (Budnik and Auerbach, 2004;Capponi et al, 2004) employed the "contractor renormalization method" (CORE) to investigate kagome antiferromagnets. This method has been invoked to find an effective low energy Hamiltonian for the uniform kagome antiferromagnet wherein all exchange couplings are the same.…”

“…Rather the system is solved on larger size units and effective Hamiltonians involving the more primitive basic units are constructed. In (Budnik and Auerbach, 2004), a related yet, by comparison to (Capponi et al, 2004;Ferrero et al, 2003), different definition of the spin and chiral degrees of free-dom is employed. Rather explicitly, with s an S z eigenvalue of a spin operator S and ⇑ and ⇓ denoting states of eigenvalues ±1/2 of a pseudo-spin operator T, (Budnik and Auerbach, 2004) |s, ⇑ = |s ↑↓ − |s ↓↑ √ 2 ,…”

Quantum phase transitions in exactly solvable one-dimensional compass models

“…In the latter case the operators include just one of the orthogonal pseudospin components at each bond and are Ising-like. This form of interactions is found as well in the compass models [28][29][30][31][32][33][34][35][36][37][38][39], and in the Kitaev model on the honeycomb lattice [40][41][42]. The interactions that are considered here are defined by the pseudospin operators T γ i for two active orbitals (for T = 1/2), and we define them as linear combinations of the Pauli matrices {σ These operators define the generalized compass model (GCM) considered in this paper.…”