Discrete sliding symmetries, dualities, and self-dualities of quantum orbital compass models and $p+ip$ superconducting arrays

We study the spin-$1∕2$ two- and three-dimensional orbital compass models relevant to the problem of orbital ordering in transition metal oxides. We show that these systems display self-dualities and gauge-like discrete sliding symmetries. An important and surprising consequence is that these models are dual to (seemingly unrelated) recently studied models of $p+ip$ superconducting arrays. The duality transformations are constructed by means of a path-integral representation in discretized imaginary time and considering its ${\mathbb{Z}}_2$ spatial reflection symmetries and space-time discrete rotations, we obtain, in a transparent unified geometrical way, several dualities. We also introduce an alternative construction of the duality transformations using operator identities. We discuss the consequences of these dualities for the order parameters and phase transitions of the orbital compass model and its generalizations, and apply these ideas to a number of related systems.

Figure 1

The classical Euclidean action corresponding to the Hamiltonian of Eq. (6) at zero temperature in a basis quantized along the ${\sigma }^z$ direction. The transverse field leads to bonds parallel to the imaginary time axis while the four plaquette interactions become replicated along the imaginary time axis. Taking an equal time slice of this system we find the four spin term of Eq. (6) and the on-site magnetic field term. If we interchange $\tau$ with $z$, we find the anisotropic planar orbital compass model of Eq. (3) in the basis quantized along the ${\sigma }^x$ direction.

Figure 2

The lattice and a dual lattice site (marked by an asterisk “$*$” at a plaquette center). Here we illustrate the representation of ${\sigma }^z$ as on a given dual plaquette site as the product of ${\tau }^x$ operators placed on all four bonds composing the plaquette of the original lattice.

Figure 3

A graphical representation of ${\sigma }^x$ the string product of ${\tau }^z$ on all vertical bonds from the boundary up to the dual lattice site.

Figure 4

The product of $J_x{\sigma }_{\mathbf{r}}^x{\sigma }_{\mathbf{r}+{\widehat{\mathbf{e}}}_{\mathbf{x}}}^x$ becomes $J_x{\tau }^z$ on a single vertical bond on the right-hand side of the dual plaquette site corresponding to $\mathbf{r}$. The product $J_z{\sigma }_{\mathbf{r}}^z{\sigma }_{\mathbf{r}+{\widehat{\mathbf{e}}}_{\mathbf{z}}}^z$ becomes in the dual representation the product of all ${\tau }^x$ operators forming the outer shell of a vertical domino multiplied by $J_z$. Putting all of the pieces together, the Hamiltonian becomes the sum of $J_z$ multiplying a domino shell of ${\tau }^x$ on bonds augmented by $J_x$ multiplying a single vertical bond on which ${\tau }^z$ is placed.

Figure 5

Choosing a gauge in which ${\tau }^x=1$ on all horizontal bonds, identifying the centers of the vertical bonds as sites, we find that the Hamiltonian corresponds to the product of four ${\sigma }^x$ operators on the vertices of a plaquette $\left[K{s}_{{\mathbf{r}}^{*}}^x{s}_{{\mathbf{r}}^{*}+{\widehat{\mathbf{e}}}_{\mathbf{z}}}^x{s}_{{\mathbf{r}}^{*}+{\widehat{\mathbf{e}}}_{\mathbf{z}}-{\widehat{\mathbf{e}}}_{\mathbf{x}}}^x{s}_{{\mathbf{r}}^{*}-{\widehat{\mathbf{e}}}_{\mathbf{x}}}^x\right]$ augmented by an external transverse field giving rise to $h{s}_{{\mathbf{r}}^{*}}^z$. Here, $h=J_x$ and $K=J_y$. Thus, the planar orbital compass model is dual to the superconducting array system of Refs. 21 and 22.