Compass models: Theory and physical motivations

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° compass model on a square lattice in which only couplings of the form ${\tau }_i^x{\tau }_j^x$ (where $\{{\tau }_i^a{\}}_a$ denote Pauli operators at site $i$) are associated with nearest-neighbor sites $i$ and $j$ separated along the $x$ axis of the lattice while ${\tau }_i^y{\tau }_j^y$ couplings appear for sites separated by a lattice constant along the $y$ 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.

Figure 1

The planar 90° compass model on a square lattice: The interaction of (pseudo)spin degrees of freedom $\mathit{\tau }=({\tau }^x,{\tau }^y)$ along horizontal bonds that are connected by the unit vector ${\mathit{e}}_x$ is ${\tau }_{\mathit{r}}^x{\tau }_{\mathit{r}+{\mathit{e}}_x}^x$. Along vertical bonds ${\mathit{e}}_y$ it is ${\tau }_{\mathit{r}}^y{\tau }_{\mathit{r}+{\mathit{e}}_y}^y$.

Figure 2

The 90° compass model on a cubic lattice: The interaction of (pseudo)spin degrees of freedom $\mathit{\tau }=({\tau }^x,{\tau }^y,{\tau }^z)$ along horizontal bonds that are connected by the unit vector ${\mathit{e}}_x$ is $J{\tau }_i^x{\tau }_{i+{\mathit{e}}_x}^x$. On bonds connected by ${\mathit{e}}_y$ it is $J{\tau }_i^y{\tau }_{i+{\mathit{e}}_y}^y$ and along the vertical bonds it is $J{\tau }_i^z{\tau }_{i+{\mathit{e}}_z}^z$.

Figure 3

Frustration in the 90° compass model on a cubic lattice. The interactions between pseudospins $\mathit{\tau }$ are such that the pseudospins tend to align their components ${\tau }^x$, ${\tau }^y$, and ${\tau }^z$ along the $x$, $y$, and $z$ axes, respectively. This causes mutually exclusive ordering patterns.

Figure 4

Kitaev’s compass model on a honeycomb lattice: the interaction of (pseudo)spin degrees of freedom $\mathit{\tau }=({\tau }^x,{\tau }^y,{\tau }^z)$ along the three bonds that each site is connected to are ${\tau }_{\mathit{r}}^x{\tau }_{\mathit{r}+{\mathit{e}}_1}^x$, ${\tau }_{\mathit{r}}^y{\tau }_{\mathit{r}+{\mathit{e}}_2}^y$, and ${\tau }_{\mathit{r}}^z{\tau }_{\mathit{r}+{\mathit{e}}_3}^z$, where the bond vectors of the honeycomb lattice $\{{\mathit{e}}_1,{\mathit{e}}_2,{\mathit{e}}_3\}$ are $\{{\mathit{e}}_x,-{\mathit{e}}_x/2+\sqrt{3}{\mathit{e}}_y/2,-{\mathit{e}}_x/2-\sqrt{3}{\mathit{e}}_y/2\}$, respectively.

Figure 5

Schematic representation of the $XXZ$ honeycomb compass model. From [159].

Figure 6

The 120° compass model on a cubic lattice: The interaction of (pseudo)spin degrees of freedom $\mathit{\tau }=({\tau }^x,{\tau }^y,{\tau }^z)$ along the three bonds that each site is connected to are ${\widehat{\pi }}_{\mathit{r}}^1{\widehat{\pi }}_{\mathit{r}+{\mathit{e}}_x}^1$, ${\widehat{\pi }}_{\mathit{r}}^2{\widehat{\pi }}_{\mathit{r}+{\mathit{e}}_y}^2$, and ${\widehat{\pi }}_{\mathit{r}}^3{\widehat{\pi }}_{\mathit{r}+{\mathit{e}}_z}^3$, where the different components $\{{\widehat{\pi }}^1,{\widehat{\pi }}^2,{\widehat{\pi }}^3\}$ of the vector $\widehat{\mathit{\pi }}=\mathbf{(}{\tau }^x,(-{\tau }^x+\sqrt{3}{\tau }^y)/2,(-{\tau }^x-\sqrt{3}{\tau }^y)/2\mathbf{)}$ interact along the different bonds $\{{\mathit{e}}_x,{\mathit{e}}_y,{\mathit{e}}_z\}$.

Figure 7

Left: A unit disk with three uniformly spaced vectors, the building blocks for the 120° model with $n=2$, on, for instance, a 3D cubic or the 2D honeycomb lattice. Right: Generalization to higher dimensions with four uniformly spaced vectors on the $n=3$-dimensional unit sphere, relevant to a 4D hypercubic lattice, or the 3D diamond lattice.

Figure 8

Left: The configuration underlying the definition of the plaquette orbital model. Here the $x$ components of the spins are coupled over the solid edges and the $z$ components are coupled over the dashed edges. From [10]. Right: A schematic representation for the orbital compass model on a checkerboard lattice. From [143].

Figure 9

The five orthogonal $d$ orbitals. Crystal field effects lift the fivefold degeneracy of the $d$ atomic orbitals into an $e_g$ doublet (top) and a $t_{2g}$ triplet of states.

Figure 10

Result of the rotations of the $e_g$ orbital spinor by an angle $\varphi /2=2\pi /3$.

Figure 11

Superexchange between spin-$1/2$ electrons, resulting into the effective antiferromagnetic Heisenberg Hamiltonian $H=J\sum _{i,j}({\mathbf{S}}_i·{\mathbf{S}}_j-1/4)$, with $J=4t^2/U$.

Figure 12

Hopping amplitudes between $e_g$ orbitals along the $\widehat{z}$ axis: the hopping matrix is $t_{\alpha \beta }^{\widehat{z}}=t{\delta }_{\alpha ,2}{\delta }_{\beta ,2}$. Three matrix elements vanish because of the symmetry of the $x^2-y^2$ orbitals, with a wave function on adjacent lobes that has opposite sign.

Figure 13

Top: $H_{⬡}^{120°}$ models orbital-orbital interactions in $R{\mathrm{Fe}}_2{\mathrm{O}}_4$ (a) A pair of triangular planes and (b) three Fe-O bond directions in a triangular lattice in $R{\mathrm{Fe}}_2{\mathrm{O}}_4$. Below: Schematic of the charge and spin structures in the $2{\mathrm{Fe}}^{2+}\text{−}{\mathrm{Fe}}^{3+}$ plane (right) and in the ${\mathrm{Fe}}^{2+}\text{−}2{\mathrm{Fe}}^{3+}$ plane (left) for $R{\mathrm{Fe}}_2{\mathrm{O}}_4$. Filled and open circles represent ${\mathrm{Fe}}^{3+}$ and ${\mathrm{Fe}}^{2+}$, respectively. At sites surrounded by dotted circles, spin directions are not uniquely determined due to frustration. From [142].

Figure 14

Left: The crystal structure of ${\mathrm{NaNiO}}_2$. Right: A plaquette in the $\alpha \beta$ plane $(\alpha ,\beta =x,y,z)$ formed by two nearest-neighbor Ni ions, 1 and 2, and two oxygen ions, ${\mathrm{O}}_1$ and ${\mathrm{O}}_2$. From [129].

Figure 15

Hopping amplitudes between $t_{2g}$ orbitals along the $\widehat{x}$ axis, assuming that the hopping is via a ligand intermediate $p$ state (not shown here), for instance, of an oxygen atom between two TM ions; see Fig. 24.

Figure 16

Jahn-Teller distortions of $e_g$ symmetry, $Q_2$ and $Q_3$, of a transition metal–oxygen octahedron. The electronic orbital degree of freedom is locked to the distortion $(Q_3,Q_2)$.

Figure 17

The trimerized kagome lattice. The solid and dashed lines indicate the antiferromagnetic coupling $J$ and $J^{\prime }$, respectively. The numbers 1, 2, and 3 indicate the site indexing inside the elementary triangles which defines the gauge. From [56].

Figure 18

Triangular lattice on which the effective Hamiltonian is defined. The unit vector for the bond is indicated by solid lines (${\mathit{e}}_{ij}={\mathit{e}}_1$), dashed lines (${\mathit{e}}_{ij}={\mathit{e}}_2$), and dotted lines (${\mathit{e}}_{ij}={\mathit{e}}_3$). From [56].

Figure 19

(a) The 90° square lattice compass model. The action of the $d=1$ symmetry operation of Eq. (84) when the “plane” $P$ is chosen to lie along the vertical axis. (b) A $d=0$ (local) gauge symmetry. Defects within a gauge theory cost a finite amount of energy. Local symmetries such as the one depicted above for an Ising lattice gauge theory cannot be broken. (c) A defect in a semiclassical ground state of the two-dimensional orbital compass model. Defects such as this do not allow for a finite on-site magnetization. The energy penalty for this defect is finite (there is only one bad bond—the dashed line) whereas, precisely as in $d=1$ Ising systems, the entropy associated with such defects is monotonically increasing with system size. From [158].

Figure 20

The symmetries of Eq. (87) applied to a uniform ground state (top left).

Figure 21

Left: Pseudospin configuration for ${\theta }^{*}=0$. Right: Configuration obtained by $\pm \delta \theta$ rotations of pseudospins in each zigzag chain. From [142].

Figure 22

The fully packed oriented loop configurations in which $\tau$ vectors lie in directions of $\varphi =\pm 30°,\pm 90°$, and $\pm 150°$. (a) The closest-packed loop configuration with all the loops in the same chirality. (b) The $p$-orbital configuration for one closed loop in (a). The azimuthal angles of the $p$ orbitals are 45°, 105°, 165°, 225°, 285°, and 345°. From [229].

Figure 23

The triangular lattice formed in the [111] plane. Shown is a disordered mean-field ground state, in which the isospins form lines parallel to the unit vector ${\mathit{e}}_{xy}$, such that $⟨T_j^z⟩$ is the same on all lattice sites, while the sign of $⟨T_j^x⟩$ varies arbitrarily from line to line. From [129].

Figure 24

The anisotropic hopping amplitudes leading to the KK Hamiltonian after [6]. The spins are indicated by rods. Following [69], the four-lobed states denote the $3d$ orbitals of a TM ion while the intermediate small $p$ orbitals are oxygen orbitals through which the superexchange process occurs. Because of orthogonality with intermediate oxygen $p$ states, in any orbital state $|\gamma ⟩$ (e.g., $|c⟩\equiv |xy⟩$ above), hopping is disallowed between sites separated along the cubic $\gamma$ ($c$ above) axis. The ensuing KK Hamiltonian has a $d=2$ SU(2) symmetry that corresponds to a uniform rotation of all spins whose orbital state is $|\gamma ⟩$ in any plane orthogonal to the cubic direction $\gamma$. Such a rotation in the $xy$ plane is indicated by the spins in the figure.

Figure 25

Left: Quantum corrections for the cubic lattice 120° model system as functions of rotation angle $\theta$ for the renormalized order parameter $\mathrm{\Delta }{T}^z$ (full lines) and the ground-state energy $\mathrm{\Delta }E/J$ (dashed lines). From [209]. Right: The gap $\mathrm{\Delta }$ as a function of $1/(2S)$ (solid curve) and the square root behavior at small $1/(2S)$ given by ${\mathrm{\Delta }}^2/(2SJ{)}^2=0.49/(2S)$, for pseudospin $S$ (dashed curve). From [101].

Figure 26

Top: Some of the pseudospin configurations where the honeycomb lattice is covered by nearest-neighbor (NN) bonds with the minimum bond energy. One of the $q=1$ states in (a) and one of the $q=-1$ in (b). In NN bonds surrounded by ellipses, the bond energy is the lowest. Bottom: One example for the two pseudospin configurations where a resonance state is possible due to the off-diagonal matrix element. From [142].

Figure 27

A configuration of the pseudovectors on the diamond lattice and its mapping to the spin-ice state on the dual pyrochlore lattice. The pseudovector assumes only six different values $⟨{\mu }_i⟩=\pm \widehat{x},\pm \widehat{y},\pm \widehat{z}$ in the ground states, corresponding to ($p_y\pm p_z$), ($p_z\pm p_x$), and ($p_x\pm p_y$) orbitals, respectively. These six orbital configurations are mapped to the six two-in-two-out ice state on a tetrahedron. From [33].