Fracton topological phases from strongly coupled spin chains

TL;DR

The paper develops a general framework to realize fracton topological phases by strongly coupling intersecting spin chains, focusing on an eight-coordinated lattice with four spin-1/2 flavors per site. In the strong-coupling limit, the system yields a commuting-projector fracton phase with the effective Hamiltonian $\tilde{H} = \sum_{\mathbf{r}} W_{\mathbf{r}}$, where each $W_{\mathbf{r}}$ is a high-order product of spin operators, and excitations exhibit restricted mobility along lines or planes. An exact parton construction with eight Majorana fermions per site, subject to overlapping gauge constraints, captures the low-energy physics and reveals excitations that move along specific $\langle 1\,1\,1\rangle$ directions, tying spin-model realizations to parton pictures. The work couples spin-chain intuition to layered- and parton-based viewpoints, uses only two-spin interactions to approach experimental realizations, and clarifies connections to known fracton models (e.g., X-cube) and coupled-layer constructions.

Abstract

We provide a new perspective on fracton topological phases, a class of three-dimensional topologically ordered phases with unconventional fractionalized excitations that are either completely immobile or only mobile along particular lines or planes. We demonstrate that a wide range of these fracton phases can be constructed by strongly coupling mutually intersecting spin chains and explain via a concrete example how such a coupled-spin-chain construction illuminates the generic properties of a fracton phase. In particular, we describe a systematic translation from each coupled-spin-chain construction into a parton construction where the partons correspond to the excitations that are mobile along lines. Remarkably, our construction of fracton phases is inherently based on spin models involving only two-spin interactions and thus brings us closer to their experimental realization.

Figures (4)

• Figure 1: Fracton spin model on the BCC lattice characterized by cubic lattice vectors $\mathbf{a}_{1,2,3}$ and bond vectors $\mathbf{b}_{1,2,3,4}$. (a) Nearest-neighbor terms of the model Hamiltonian $H$. Each term corresponding to a $j = 1$ (red), $j = 2$ (green), $j = 3$ (blue), or $j = 4$ (purple) bond acts on spin flavor $j$ via $\sigma^x$ at the tail and via $\sigma^y$ at the head of the bond arrow. (b) Effective Hamiltonian $\tilde{H}$ in the strong-coupling limit. Each term $W_{\mathbf{r}}$ in $\tilde{H}$ is induced by nearest-neighbor terms (colored lines) in degenerate perturbation theory and is a product of eight spin operators $Y_{\tilde{\mathbf{r}}}$ at the corners of the basic BCC cube as well as six spin operators $X_{\tilde{\mathbf{r}}}$ at the apices of the square pyramids based on the faces of this cube. All sites $\tilde{\mathbf{r}}$ are marked by appropriate labels.
• Figure 2: One-dimensional (a) and zero-dimensional (b) excitations in the $(1 \, 0 \, \bar{1})$ plane of our BCC model. In each case, the excitations are (schematically) located within the red circles and are created by the operator $\prod_{\mathbf{r} \in A, B} Z_{\mathbf{r}}$ over the sites $\mathbf{r} \in A, B$ marked by black dots.
• Figure 3: Exact parton construction of our BCC model. (a) Each spin $\Sigma_{\mathbf{r}}$ (large white sphere) is decomposed into eight Majorana fermions (colored dots) at the bonds around the site $\mathbf{r}$: four $\gamma_{\mathbf{r}, j}$ above $\mathbf{r}$ and four $\hat{\gamma}_{\mathbf{r}, j}$ below $\mathbf{r}$ with flavors $j = 1$ (red), $j = 2$ (green), $j = 3$ (blue), and $j = 4$ (purple). Each bond is occupied by two Majorana fermions $\gamma_{\mathbf{r}, j}$ and $\hat{\gamma}_{\mathbf{r}', j}$ that are in a state characterized by the bond-fermion operator $i \gamma_{\mathbf{r}, j} \hat{\gamma}_{\mathbf{r}', j} = \pm 1$ (black line). (b) Decomposition of each term $W_{\mathbf{r}}$ in the effective spin Hamiltonian $\tilde{H}$ [see Fig. \ref{['fig-1']}(b)] into a product of bond-fermion operators (black lines).
• Figure 4: Coupled-spin-chain constructions (a) and effective strong-coupling Hamiltonians (b) capturing type-I fracton phases on two different lattices. The notation is taken from Fig. \ref{['fig-1']}. For the second construction, the strong-coupling Hamiltonian has three independent terms and corresponds to the X-cube model Vijay-2016.