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arXiv:1008.2312 · Physical Review B

Hidden dimer order in the quantum compass model

Wojciech Brzezicki, Andrzej M. Oleś

Published Aug 12, 2010• 33 citations
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Abstract

We introduce an exact spin transformation that maps frustrated ${Z}_{i,j}{Z}_{i,j+1}$ and ${X}_{i,j}{X}_{i+1,j}$ spin interactions along the rows and columns of the quantum compass model (QCM) on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice to $(L\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(L\ensuremath{-}1)$ quantum spin models with $2(L\ensuremath{-}1)$ classical spins. Using the symmetry properties we unravel the hidden dimer order in the QCM, with equal two-dimer correlations $⟨{X}_{i,i}{X}_{i+1,i}{X}_{k,l}{X}_{k+1,l}⟩$ and $⟨{X}_{i,i}{X}_{i+1,i}{X}_{l,k}{X}_{l+1,k}⟩$ in the ground state, which is independent of the actual interactions. This order coexists with Ising-type spin correlations which decay with distance.

DOI
10.1103/physrevb.82.060401

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I show the symmetries of the compass model and spin transformations making the Hamiltonian block‐diagonal. I present the new form of the Hamiltonian and explain how the diagonal blocks are related by translational symmetry and by the isotropy of interactions. I reveal the hidden symmetry of the lowest‐energy block and resulting identities in four‐point dimer‐dimer correlators. Using exact diagonalization I show that the ground state has classical order with local quantum fluctuation vanishing in a long range and that the energy spectrum consists of discrete and continuous part.
We consider the square lattice S=1/2S=1/2S=1/2 quantum compass model (QCM) parametrized by Jx,Jz{J}_{x},{J}_{z}Jx​,Jz​, under a field, h\mathbf{h}h, in the $x\text{\ensuremath{-}}z$ plane. At the special field value (hx★,hz★)=2S(Jx,Jz)({h}_{x}^{★},{h}_{z}^{★})=2S({J}_{x},{J}_{z})(hx★​,hz★​)=2S(Jx​,Jz​), we show that the QCM Hamiltonian may be written in a form such that two simple product states can be identified as exact ground states, below a gap. Exact excited states can also be found. We use the staggered vector chirality to characterize the exact ground states at (hx★,hz★)({h}_{x}^{★},{h}_{z}^{★})(hx★​,hz★​) and states obtained for fields within the vicinity of (hx★,hz★)({h}_{x}^{★},{h}_{z}^{★})(hx★​,hz★​). This gapped phase occupies most of the in-plane field phase diagram. For some values of hx>hz{h}_{x}>{h}_{z}hx​>hz​ and hz>hx{h}_{z}>{h}_{x}hz​>hx​ at the edges of the phase diagram, we have found transitions between this gapped phase and phases of weakly coupled Ising-chain states. In zero field, the QCM is known to have an emergent subextensive ground-state degeneracy. As the field is increased from zero, we find that this degeneracy is partially lifted, resulting in bond-oriented spin-stripe states, which are each separated from one another and the gapped phase by first-order transitions. Our findings are important for understanding the field dependent phase diagram of materials with predominantly directionally dependent Ising interactions.
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 introduce a one-dimensional model which interpolates between the Ising model and the quantum compass model with frustrated pseudospin interactions σizσi+1zσ_i^zσ_{i+1}^zσiz​σi+1z​ and σixσi+1xσ_i^xσ_{i+1}^xσix​σi+1x​, alternating between even/odd bonds, and present its exact solution by mapping to quantum Ising models. We show that the nearest neighbor pseudospin correlations change discontinuosly 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 2×2N/22\times 2^{N/2}2×2N/2 in the limit of N→∞N\to\inftyN→∞.
Quantum spin models with spatially dependent interactions, known as compass models, play an important role in the study of frustrated quantum magnetism. One example is the Kitaev model on the honeycomb lattice with spin-liquid (SL) ground states and anyonic excitations. Another example is the geometrically frustrated quantum 120° model on the same lattice whose ground state has not been unambiguously established. To generalize the Kitaev model beyond the exactly solvable limit and connect it with other compass models, we propose a new model, dubbed 'the tripod model', which contains a continuum of compass-type models. It smoothly interpolates the Ising model, the Kitaev model, and the quantum 120° model by tuning a single parameter , the angle between the three legs of a tripod in the spin space. Hence it not only unifies three paradigmatic spin models, but also enables the study of their quantum phase transitions. We obtain the phase diagram of the tripod model numerically by tensor networks in the thermodynamic limit. We show that the ground state of the quantum 120° model has long-range dimer order. Moreover, we find an extended spin-disordered (SL) phase between the dimer phase and an antiferromagnetic phase. The unification and solution of a continuum of frustrated spin models as outline here may be useful to exploring new domains of other quantum spin or orbital models.
A frustrated spin-1/2 model is constructed on the square lattice, the Hamiltonian of which contains five types of interactions: nearest-neighbor (NN), diagonal (D), next-nearest-neighbor (NNN), knight's-move-distance away (KM), and further-neighbor diagonal (FND). The NN bonds in a particular direction, horizontal or vertical, are of alternating coupling strengths 7J and J. The other interaction strengths are: J (for NN and D), (J), J/2 (for NNN and KM), and J/4 (for FND). We prove that the exact ground state of the Hamiltonian is the columnar-dimer state.