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arXiv:1203.3683 · Physical Review B

Ordering and excitation in orbital compass model on a checkerboard lattice

Joji Nasu, Synge Todo, Sumio Ishihara

Published May 29, 2012• 7 citations
arXivHTMLPDF

Abstract

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 $z$ component of pseudospins along the vertical/horizontal bonds and an Ising interaction for the $x$ 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 $z$ 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.

DOI
10.1103/physrevb.85.205141

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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.
Compass models are theories of matter in which the couplings between the internal spin (or other relevant field) components are inherently spatially (typically, direction) dependent. Compass-type interactions appear in diverse physical systems including Mott insulators with orbital degrees of freedom (where interactions sensitively depend on the spatial orientation of the orbitals involved), the low energy effective theories of frustrated quantum magnets, systems with strong spin-orbit couplings (such as the iridates), vacancy centers, and cold atomic gases. Kitaev's models, in particular the compass variant on the honeycomb lattice, realize basic notions of topological quantum computing. The fundamental inter-dependence between internal (spin, orbital, or other) and external (i.e., spatial) degrees of freedom which underlies compass models generally leads to very rich behaviors including the frustration of (semi-)classical ordered states on non-frustrated lattices and to enhanced quantum effects prompting, in certain cases, the appearance of zero temperature quantum spin liquids. As a consequence of these frustrations, new types of symmetries and their associated degeneracies may appear. These intermediate symmetries lie midway between the extremes of global symmetries and local gauge symmetries and lead to effective dimensional reductions. We review compass models in a unified manner, paying close attention to exact consequences of these symmetries, and to thermal and quantum fluctuations that stabilize orders via order out of disorder effects. We review non-trivial statistics and the appearance of topological quantum orders in compass systems in which, by virtue of their intermediate symmetry standard orders do not arise. This is complemented by a survey of numerical results. Where appropriate theoretical and experimental results are compared.
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→∞.
Compass models are theories of matter in which the couplings between the internal spin (or other relevant field) components are inherently spatially (typically, direction) dependent. A simple illustrative example is furnished by the 90\ifmmode^\circ\else\textdegree\fi{} compass model on a square lattice in which only couplings of the form ${\ensuremath{\tau}}_{i}^{x}{\ensuremath{\tau}}_{j}^{x}$ (where ${{\ensuremath{\tau}}_{i}^{a}{}}_{a}$ denote Pauli operators at site iii) are associated with nearest-neighbor sites iii and jjj separated along the xxx axis of the lattice while ${\ensuremath{\tau}}_{i}^{y}{\ensuremath{\tau}}_{j}^{y}$ couplings appear for sites separated by a lattice constant along the yyy axis. Similar compass-type interactions can appear in diverse physical systems. For instance, compass models describe Mott insulators with orbital degrees of freedom where interactions sensitively depend on the spatial orientation of the orbitals involved as well as the low-energy effective theories of frustrated quantum magnets, and a host of other systems such as vacancy centers, and cold atomic gases. The fundamental interdependence between internal (spin, orbital, or other) and external (i.e., spatial) degrees of freedom which underlies compass models generally leads to very rich behaviors, including the frustration of (semi-)classical ordered states on nonfrustrated lattices, and to enhanced quantum effects, prompting, in certain cases, the appearance of zero-temperature quantum spin liquids. As a consequence of these frustrations, new types of symmetries and their associated degeneracies may appear. These intermediate symmetries lie midway between the extremes of global symmetries and local gauge symmetries and lead to effective dimensional reductions. In this article, compass models are reviewed in a unified manner, paying close attention to exact consequences of these symmetries and to thermal and quantum fluctuations that stabilize orders via order-out-of-disorder effects. This is complemented by a survey of numerical results. In addition to reviewing past works, a number of other models are introduced and new results established. In particular, a general link between flat bands and symmetries is detailed.
Transition metal compounds have been central to the study of quantum magnetism in recent years. The multi-orbital nature of these compounds together with the interplay between electron-electron interactions, crystal field effects and spin-orbit coupling leads to spin-orbit entangled wave functions, known as pseudospins. Pseudospins depend highly on the electron filling of the transition metal, and are highly spatially anisotropic which leads to frustrated, bond-dependent exchange interactions. These interactions manifest themselves in a variety of non-trivial ways, depending on the structure and dimensionality of the underlying lattice, which we explore in this thesis. In one dimension we show that CoNb2O6, thought to be the best known candidate for the 1D transverse field Ising model, is an example of a rare 1D Kitaev chain where the bond-dependent pseudospin interactions are responsible for the transverse field Ising behaviour. On the two dimensional honeycomb lattice we propose a mechanism to generate the bond-dependent, exactly solvable, Yao-Lee interaction among others. We show at the exactly solveable point of our model, the ground state is a quantum spin-orbital liquid with emergent fermionic octupolar excitations. Finally, we study the three dimensional 5d double-perovskites (DPs), which have been proposed to be the first d-orbital candidates exhibiting hidden octupolar order. We develop a microscopic theory for these materials, and determine several microscopic mechanisms that may be responsible for stabilizing octupolar order.
Compass models provide insights into the properties of Mott-insulating materials that host bond-dependent anisotropic interactions between their pseudospin degrees of freedom. In this article, we explore the classical and quantum ground-state properties of one such model relevant to certain layered perovskite materials akin to Ba2IrO4{\mathrm{Ba}}_{2}{\mathrm{IrO}}_{4}Ba2​IrO4​---namely, the Heisenberg-compass model on the square lattice. We first investigate the ground-state phase diagram of this model using classical Monte Carlo simulations. These reveal that the low-temperature classical phase diagram is divided into six different classes of long-range ordered phases, including four phases that exhibit an order by disorder selection and two phases that are stabilized energetically. This model admits a special duality transformation, known as the Klein duality, conveniently allowing to map one region of coupling parameters onto another and constraining the phase diagram, and which we exploit in our study. From the analysis of the zero-point energy and the free energy of the spin waves, we find that order by quantum disorder at zero temperature and order by thermal disorder select the same orderings as those found from classical Monte Carlo simulations. We further investigate the quantum ground states of this model using numerical exact diagonalization on small clusters by exploiting the translational symmetry of the square lattice. We obtain a ground-state phase diagram bearing close resemblance to that found from the classical analysis.