${\mathbb{Z}}_{3}$ topological order in the face-centered-cubic quantum plaquette model

$Z_{3}$ topological order in the face-centered-cubic quantum plaquette model

Trithep Devakul

Phys. Rev. B 97, 155111 – Published 6 April 2018

Abstract

We examine the topological order in the resonating singlet valence plaquette (RSVP) phase of the hard-core quantum plaquette model (QPM) on the face centered cubic (FCC) lattice. To do this, we construct a Rohksar-Kivelson type Hamiltonian of local plaquette resonances. This model is shown to exhibit a $Z_{3}$ topological order, which we show by identifying a $Z_{3}$ topological constant (which leads to a $3^{3}$ -fold topological ground state degeneracy on the 3-torus) and topological pointlike charge and looplike magnetic excitations which obey $Z_{3}$ statistics. We also consider an exactly solvable generalization of this model, which makes the geometrical origin of the $Z_{3}$ order explicitly clear. For other models and lattices, such generalizations produce a wide variety of topological phases, some of which are novel fracton phases.

Received 18 January 2018
Revised 27 February 2018

DOI:https://doi.org/10.1103/PhysRevB.97.155111

Physics Subject Headings (PhySH)

Anyons Fractionalization Geometric & topological phases Magnetic interactions Spin liquid Topological phases of matter

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Trithep Devakul

Department of Physics, Princeton University, Princeton, New Jersey 08540, USA

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Vol. 97, Iss. 15 — 15 April 2018

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Figure 1
A unit cell of the face centered cubic lattice. Nearest neighbor pairs are connected by gray lines. Triangles on which trimers may occupy are formed by three mutually nearest neighbor sites. Regular polyhedra formed by the triangular faces include octahedra (one shown in red) and tetrahedra (one shown in green).
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Figure 2
Illustration of a few terms in the Hamiltonian, which we describe by sets of triangles $C = C_{A} \cup C_{B}$ , where the orange and blue triangles indicate $C_{A}$ and $C_{B}$ . All $| C | = 4$ terms are loop terms of the form (a) or (b). (c) and (d) show terms involving a larger number of triangles. The term (c) involves flipping between configurations with local “divergence” $\pm 3$ (as described in the text), and (d) is an example of a $| C | = 8$ length loop term.
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Figure 3
The convention for assigning directions to trimer configurations. The top row shows the configuration (for example) in state $| C_{A} 〉$ , and the bottom shows the flipped state $| C_{B} 〉$ ; the red arrows indicate the direction assignment. Configurations along looplike paths are assigned a direction as shown in (a). Terms which involve flips along nonloop paths include triangles with local “divergence” $\pm 3$ , as shown in (b). Finally, (c) shows how a monomer (an untrimerized site) may be moved along a path via trimer flips.
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Figure 4
A sample trimer configuration in an $x y$ plane specified by $z$ -coordinate $z_{0}$ , which includes triangles spanning the site layers $z_{0}$ and $z_{0} + 1 / 2$ . Upwards facing trimers are shown in orange, while downwards facing trimers are shown in blue. The topologically conserved “winding number” is the difference between the number of upwards facing trimers ( $N_{▵}$ ) and downwards facing trimers ( $N_{▿}$ ) modulo 3 [Eq. (5)].
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Figure 5
Pictorial representation of the terms in the Hamiltonian for (a) the square lattice $N$ -GDM, (b) the triangular lattice $N$ -GDM, and (c) the triangular lattice $N$ -GPM. Blue and orange bonds/triangles indicate operators involved in the kinetic terms in the Hamiltonian ( $Z$ and $Z^{†}$ ), and red indicates those involved in the site constraint ( $X$ ). Only one of three possible rhombus orientations is shown for the kinetic term in the triangular lattice $N$ -GDM (b).
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Figure 6
The corner-sharing octahedra lattice, on which the $N$ -GPM shows a $Z_{N}$ X-cube fracton phase. (a) shows of the corner-sharing octahedra lattice, where sites from the three sublattices are colored red, green, and blue. The centers of the octahedra form into a simple cubic lattice, with lattice constant taken to be 1. Sites from each sublattice themselves also sit an offset simple cubic lattice. (b) shows two types of octahedron flips $| C_{oct} | = 4$ , and (c) shows a cuboctahedron flip $| C_{cuboct} | = 8$ . (d) shows a portion of the $W_{z} (x_{0}, y_{0})$ operator, which measures a $Z_{N}$ topologically conserved quantity. Blue triangles indicates $Z$ operators and orange indicates $Z^{†}$ operators.
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Physical Review B

covering condensed matter and materials physics

$Z_{3}$ topological order in the face-centered-cubic quantum plaquette model

Trithep Devakul

Phys. Rev. B 97, 155111 – Published 6 April 2018

Abstract

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Physical Review B

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Z3 topological order in the face-centered-cubic quantum plaquette model

Trithep Devakul

Phys. Rev. B 97, 155111 – Published 6 April 2018

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$Z_{3}$ topological order in the face-centered-cubic quantum plaquette model