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arXiv:1001.5457 · Physical Review B

Spontaneous symmetry breaking in a generalized orbital compass model

Łukasz Cincio, Jacek Dziarmaga, Andrzej M. Oleś

Published Sep 10, 2010• 39 citations
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Abstract

We introduce a generalized two-dimensional orbital compass model, which interpolates continuously from the classical Ising model to the orbital compass model with frustrated quantum interactions, and investigate it using the multiscale entanglement renormalization ansatz (MERA). The results demonstrate that increasing frustration of exchange interactions triggers a second-order quantum phase transition to a degenerate symmetry broken state which minimizes one of the interactions in the orbital compass model. Using boson expansion within the spin-wave theory we unravel the physical mechanism of the symmetry-breaking transition as promoted by weak quantum fluctuations and explain why this transition occurs only surprisingly close to the maximally frustrated interactions of the orbital compass model. The spin waves remain gapful at the critical point, and both the boson expansion and MERA do not find any algebraically decaying spin-spin correlations in the critical ground state.

DOI
10.1103/physrevb.82.104416

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We introduce a generalized two-dimensional orbital compass model, which interpolates continuously from the classical Ising model to the orbital compass model with frustrated quantum interactions, and investigate it using the multiscale entanglement renormalization ansatz (MERA). The results demonstrate that increasing frustration of exchange interactions triggers a second-order quantum phase transition to a degenerate symmetry broken state which minimizes one of the interactions in the orbital compass model. Using boson expansion within the spin-wave theory we unravel the physical mechanism of the symmetry-breaking transition as promoted by weak quantum fluctuations and explain why this transition occurs only surprisingly close to the maximally frustrated interactions of the orbital compass model. The spin waves remain gapful at the critical point, and both the boson expansion and MERA do not find any algebraically decaying spin-spin correlations in the critical ground state.
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.
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 one-dimensional model which interpolates between the Ising model and the quantum compass model with frustrated pseudospin interactions ${\ensuremath{\sigma}}_{i}^{z}{\ensuremath{\sigma}}_{i+1}^{z}$ and ${\ensuremath{\sigma}}_{i}^{x}{\ensuremath{\sigma}}_{i+1}^{x}$, alternating between even and/or odd bonds, and present its exact solution by mapping to quantum Ising models. We show that the nearest-neighbor pseudospin correlations change discontinuously and indicate divergent correlation length at the first-order quantum phase transition. At this transition, one finds the disordered ground state of the compass model with high degeneracy $2\ifmmode\times\else\texttimes\fi{}{2}^{N∕2}$ in the limit of $N\ensuremath{\rightarrow}\ensuremath{\infty}$.
We study the one-dimensional quantum compass model with two independent parameters by means of an exact mapping to the quantum Ising model. This allows us to uncover hidden features of the quantum phase transition in the ordinary one-parameter quantum compass model, showing that it occurs at a multicritical point where a line of first-order transitions intersects a line of second-order symmetry-breaking transitions of Ising type. We calculate the concurrence and the block entanglement entropy in the four ground-state phases and find that these entanglement measures accurately signal the second-order, but not the first-order, transitions.
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We study an orbital compass model on a checkerboard lattice where orbital degree of freedom is represented by the pseudospin operator. Competition arises from an Ising interaction for the zzz component of pseudospins along the vertical/horizontal bonds and an Ising interaction for the xxx component along diagonal bonds. Classical and quantum compass models are analyzed by utilizing several analytical methods and numerical simulations. At a fully frustrated point where the two Ising interactions compete with each other, a macroscopic number of orbital configurations are degenerate in a classical ground state. This degeneracy is lifted by thermal and quantum fluctuations, and a staggered long-range order of the zzz component of the pseudospin is realized. A tricritical point for this order appears due to competition between the bond dependent Ising interactions. Roles of geometrical frustration on excitation dynamics are also examined.