Fracton Topological Order from Nearest-Neighbor Two-Spin Interactions and Dualities

TL;DR

The work constructs a realistic 3D lattice model that realizes X-cube fracton order with only nearest-neighbor two-spin interactions by coupling two orthogonal stacks of Kitaev honeycomb layers. Through degenerate perturbation theory, it shows how XY/XZ planes host decoupled 2+1D $Z_2$ topological orders and how strong inter-layer coupling generates X-cube fracton order, with a higher-limit extended model yielding 3+1D $Z_2$ topological order. It provides two dual descriptions of phase transitions out of the X-cube phase, each mapping to stacks of 1+1D or 2+1D Ising models, and supports the first-order nature of these transitions via arguments and Monte Carlo hints. An extended model further reveals 3+1D topological order and a global duality structure, connecting to prior coupled-layer constructions while emphasizing the minimal two-stack realization and its universal aspects. The results offer a path toward experimental realizations in solid-state or synthetic quantum systems and motivate future exploration of continuous fracton transitions and exact solvability analogs.

Abstract

Fracton topological order describes a remarkable phase of matter which can be characterized by fracton excitations with constrained dynamics and a ground state degeneracy that increases exponentially with the length of the system on a three-dimensional torus. However, previous models exhibiting this order require many-spin interactions which may be very difficult to realize in a real material or cold atom system. In this work, we present a more physically realistic model which has the so-called X-cube fracton topological order but only requires nearest-neighbor two-spin interactions. The model lives on a three-dimensional honeycomb-based lattice with one to two spin-1/2 degrees of freedom on each site and a unit cell of 6 sites. The model is constructed from two orthogonal stacks of $Z_2$ topologically ordered Kitaev honeycomb layers, which are coupled together by a two-spin interaction. It is also shown that a four-spin interaction can be included to instead stabilize 3+1D $Z_2$ topological order. We also find dual descriptions of four quantum phase transitions in our model, all of which appear to be discontinuous first order transitions.

Figures (13)

• Figure 1: Two series of condensation driven phase transitions with dual descriptions ((b) and (c)), which are described in Sec. \ref{['sec:dualities']} and Sec. \ref{['sec:global duality']}. All four phase transitions appear to be first order when generic perturbations are added (Appendix \ref{['app:MC']}). The transition from XY & XZ plane 2+1D $Z_2$ quantum spin liquid (QSL) to X-cube fracton topological order is driven by a condensation of YZ-plane composite flux loops (red loop in (b)), which are excitations of a loop of XY and XZ--plane 2+1D toric code flux operators ($[\tilde{\sigma}]^{\square_\text{XY}}_{p}$ and $[\tilde{\tau}]^{\square_\text{XZ}}_{p}$ in Sec. \ref{['sec:duality']}). In the dual theory, the YZ-plane flux loop is mapped to a YZ-plane domain wall excitation of Ising spins (green cones in (b), $\eta^z_{\hat{\iota}}$ in Sec. \ref{['sec:duality']}). The transition from X-cube order to a YZ-plane 2+1D $Z_2$ QSL is driven by a condensation of dimension-1 X-cube particle excitations (red cross in (c), $[\hat{\rho}]^{+_\text{XY}}_{\iota}$ and $[\hat{\rho}]^{+_\text{XZ}}_{\iota}$ in Sec. \ref{["sec:duality'"]}) which are bound to 1D lines parallel to the x-axis. In the dual theory, these x-axis particle excitations are mapped to excitations (light green cone $|\leftarrow \rangle$ in (c), $\mu^x_\iota$ in Sec. \ref{["sec:duality'"]}) of 1+1D Ising paramagnets, which spontaneously break a $Z_2$ Ising symmetry when they condense since they are conserved modulo two. The transition from XY & XZ plane QSL to 3+1D QSL is driven by condensing a composite of XY and XZ--plane 2+1D toric code charge excitations (red cross in (c)), which is also dual to a 1+1D Ising paramagnet excitation. The transition from 3+1D QSL to YZ-plane QSL is is driven by condensing YZ-plane flux loops (red in (b)) in 3+1D toric code, which are dual to 2+1D Ising domain walls. These dualities also describe phase transitions shown in Fig. \ref{['fig:phaseDiagram']} and Fig. \ref{['fig:extendedPhaseDiagram']}.
• Figure 2: The lattice that we consider: orthogonal stacks of honeycomb lattices along the XY and XZ planes which intersect on the dark-blue links, which we refer to as shared links in the text. $\sigma^\mu$ and $\tau^\mu$ spin-1/2 degrees of freedom reside on sites of the XY and XZ honeycomb lattices, respectively. Thus, the shared sites on either end of a dark-blue link host both a $\sigma^\mu$ and $\tau^\mu$ spin. The unit cell consists of 8 spins: the spins on both sides of a $\sigma$-link, a $\tau$-link, and a shared-link which has two spins on each side. On each honeycomb lattice we impose a Kitaev honeycomb model (Eq. (\ref{['eq:H_K']})) where the links are colored red ($x$-link), green ($y$-link), or (dark) blue ($z$-link) to indicate the kind of coupling in Eq. (\ref{['eq:H_K']}): $\sigma_i^x \sigma_j^x$, $\sigma_i^y \sigma_j^y$, $\sigma_i^z \sigma_j^z$ respectively (and similar for $\tau^\mu$). To induce X-cube fracton topological order, we place a strong $\sigma_i^z \tau_i^z$ coupling on every shared site.
• Figure 3: Rough phase diagram of the Hamiltonian in Eq. (\ref{['eq:H']}) as a function of $t$ (horizontal axis) and $K_x = K_y$ (vertical axis); $K_z$ will be used to set the energy scale. When $t=0$, the model consists of two orthogonal stacks (along the XY and XZ planes) of decoupled honeycomb lattices (Fig. \ref{['fig:lattice']}), where each honeycomb lattice will host a quantum spin liquid (QSL) with $Z_2$ topological order, which will be gapped when $K_x = K_y < K_z/2$ and gapless when $K_x = K_y \ge K_z/2$KitaevHoneycomb. These QSL phases are stable to a small coupling $t$ which couples the orthogonal honeycomb lattices together. (The gapless QSL is stable since $t$ preserves time reversal symmetry; see e.g. Sec. 6.1 of Ref. KitaevHoneycomb.) In Sec. \ref{['sec:X-cube']} and Appendix \ref{['app:X-cube']} we argue that X-cube fracton topological order VijayXCube results when $\max(K_x, K_y) \ll \min(K_z, t)$. We do not know what phase(s) occur in the white region. In the bottom left corner, a phase transition between the QSL and fracton orders occurs near $t \sim K_x^2 K_y^2 / K_z^3$ (in the $\max(K_x,K_y) \ll K_z$ limit), which can be inferred from Fig. \ref{['fig:largeKzDiagram']} in the appendix. In Sec. \ref{['sec:duality']} we propose that this transition is dual to a stack 2+1D Ising transitions; in Appendix \ref{['app:MC']} we find that the transition is likely first order and that (depending on the sign of certain perturbations) a topologically ordered intermediate phase is also possible.
• Figure 4: An XY plane honeycomb lattice: a constant $z$ slice of Fig. \ref{['fig:lattice']}. There is a $\sigma^\mu$ degree of freedom on every site, and also a $\tau^\mu$ on every shared-site: i.e. every site next to a dark-blue shared-link, which overlap gray lines. Red, green, and blue links host a $K_x \sigma_i^x \sigma_j^x$, $K_y \sigma_i^y \sigma_j^y$, or $K_z \sigma_i^z \sigma_j^z$ coupling, respectively. When $\max(K_x,K_y) \ll K_z$, degenerate perturbation theory will produce the operators $[\sigma]^{\square_\text{XY}}_{p}$ (top-left) and $[\sigma]^{+_\text{XY}}_{\iota}$ (top-right), which both take the form $\sigma_{i_1}^z \sigma_{i_2}^y \sigma_{i_3}^x \sigma_{i_4}^z \sigma_{i_5}^y \sigma_{i_6}^x$ for sites around a hexagon, as shown above. The low energy Hilbert space will have $\sigma^z_i \sigma^z_j = 1$ across all blue links; and so we will perform a local change of basis (and projection) $\sigma \rightarrow \tilde{\sigma}$ given in Eq. (\ref{['eq:toric basis']}) to describe the effective low energy Hilbert space; the new $\tilde{\sigma}^\mu$ operators are centered on the blue links. In the $\tilde{\sigma}$ basis, $[\sigma]^{\square_\text{XY}}_{p} \rightarrow [\tilde{\sigma}]^{\square_\text{XY}}_{p}$ (bottom-left) and $[\sigma]^{+_\text{XY}}_{\iota} \rightarrow [\tilde{\sigma}]^{+_\text{XY}}_{\iota}$ (bottom-right) take the form of toric code flux and charge operators, as shown above. The gray lines indicate the rectangular lattice of the toric code. The XZ planes are similar, but with the replacement $\sigma \leftrightarrow \tau$. See Fig. \ref{['fig:cubicLattice']} for a 3 dimensional version.
• Figure 5: The lattice that we consider (Fig. \ref{['fig:lattice']}), except we also draw gray lines (as in Fig. \ref{['fig:toricCode']}) to indicate the intersecting rectangular lattices of the toric code (Eq. (\ref{['eq:H toric']})) on XY and XZ planes.