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arXiv:0906.4462 · Physical Review B

Exact solution for a quantum compass ladder

Wojciech Brzezicki, Andrzej M. Oleś

Published Jul 2, 2009• 30 citations
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Abstract

We present 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 two. 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.

DOI
10.1103/physrevb.80.014405

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We present a spin ladder with antiferromagnetic Ising ZZZZZZ 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 2N2N2N 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 ZZZZZZ spin-correlation function on the rungs collapses to zero and the ground-state degeneracy increases by two. 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=1042N=1042N=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.
We introduce a spin ladder with Ising interactions along the legs and intrinsically frustrated Heisenberg-like ferromagnetic interactions on the rungs. The model is solved exactly in the subspaces relevant for the ground state by mapping to the quantum Ising model, and we show that a first order quantum phase transition separates the classical from quantum regime, with the spin correlations on the rungs being either ferromagnetic or antiferromagnetic, and different spin excitations in both regimes. The present case resembles the quantum phase transition found in the compass model in one and two dimensions.
Abstract. 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
We have studied the exact solution of the extended cluster compass ladder, which is equivalent to the extended quantum compass model with cluster interaction between next-nearest-neighbor spins, by using the Jordan–Wigner transformation. We show that this model is always gapful except at the critical surfaces where the energy gap disappears. We obtain the analytic expressions of all critical couplings which drive quantum phase transitions. This model shows a rich phase diagram which includes spin-flop, strip antiferromagnetic, and topological ordered on the legs, in addition to the phase with antiparallel ordering of the spin |yyy| component on the rungs. We also study the universality and scaling properties of the three point correlation function derivatives in different regions to confirm the results obtained using the energy gap analysis. On the other hand, we have replaced the cluster interaction with the usual form and using the Lanczos method a numerical experiment is done. Analyzing the numerical results, we show that the effect of the cluster interaction between next-nearest-neighbor spins is completely different from the usual form.
The quantum spin-1/2 two-leg ladder with an anisotropic X Y Z Heisenberg intra-rung interaction and Ising inter-rung interactions is treated by means of a rigorous approach based on the unitary transformation.The particular case of the considered model with X -X intra-rung interaction resembles a quantum compass ladder with additional frustrating diagonal Ising interactions.Using an appropriately chosen unitary transformation, the model under investigation may be reduced to a transverse Ising chain with composite spins, and one may subsequently find the ground state quite rigorously.We obtain a ground-state phase diagram and analyze the interplay of the competition between several factors: the X Y Z anisotropy in the Heisenberg intra-rung coupling, the Ising interaction along the legs, and the frustrating diagonal Ising interaction.The investigated model shows extraordinarily diverse ground-state phase diagrams including several unusual quantum ordered phases, two different disordered quantum paramagnetic phases, as well as discontinuous or continuous quantum phase transitions between those phases.
We present an exact solution for a class of one-dimensional compass models which stand for interacting orbital degrees of freedom in a Mott insulator. By employing the Jordan-Wigner transformation we map these models on noninteracting fermions and discuss how spin correlations, high degeneracy of the ground state, and Z2{Z}_{2}Z2​ symmetry in the quantum compass model are visible in the fermionic language. Considering a zigzag chain of ions with singly occupied eg{e}_{g}eg​ orbitals (eg{e}_{g}eg​ orbital model) we demonstrate that the orbital excitations change qualitatively with increasing transverse field, and that the excitation gap closes at the quantum phase transition to a polarized state. This phase transition disappears in the quantum compass model with maximally frustrated orbital interactions which resembles the Kitaev model. Here we find that the finite transverse field destabilizes the orbital-liquid ground state with macroscopic degeneracy, and leads to peculiar behavior of the specific heat and orbital susceptibility at finite temperature. We show that the entropy and the cooling rate at finite temperature exhibit quite different behavior near the critical point for these two models.
The ground-state behaviour of the frustrated quantum spin-1/2 two-leg ladder with the Heisenberg intra-rung and Ising inter-rung interactions is examined in detail. The investigated model is transformed into the quantum Ising chain with composite spins in an effective transverse and longitudinal field by employing either the bond-state representation or the unitary transformation. It is shown that the ground state of the Heisenberg–Ising ladder can be descended from three exactly solvable models: the quantum Ising chain in a transverse field, the ‘classical’ Ising chain in a longitudinal field or the spin-chain model in a staggered longitudinal–transverse field. The last model serves in evidence of the staggered bond phase with alternating singlet and triplet bonds on the rungs of a two-leg ladder, which appears at moderate values of the external magnetic field and consequently leads to a fractional plateau at a half of the saturation magnetization. The ground-state phase diagram totally consists of five ordered and one quantum paramagnetic phase, which are separated from each other either by the lines of discontinuous or continuous quantum phase transitions. The order parameters are exactly calculated for all five ordered phases and the quantum paramagnetic phase is characterized through different short-range spin–spin correlations.