Topological Fracton Quantum Phase Transitions by Tuning Exact Tensor Network States

Gapped fracton phases of matter generalize the concept of topological order and broaden our fundamental understanding of entanglement in quantum many-body systems. However, their analytical or numerical description beyond exactly solvable models remains a formidable challenge. Here we employ an exact 3D quantum tensor-network approach that allows us to study a ${\mathbb{Z}}_N$ generalization of the prototypical X cube fracton model and its quantum phase transitions between distinct topological states via fully tractable wave function deformations. We map the (deformed) quantum states exactly to a combination of a classical lattice gauge theory and a plaquette clock model, and employ numerical techniques to calculate various entanglement order parameters. For the ${\mathbb{Z}}_N$ model we find a family of (weakly) first-order fracton confinement transitions that in the limit of $N\to \infty$ converge to a continuous phase transition beyond the Landau-Ginzburg-Wilson paradigm. We also discover a line of 3D conformal quantum critical points (with critical magnetic flux loop fluctuations) which, in the $N\to \infty$ limit, appears to coexist with a gapless deconfined fracton state.

Figure 1

Phase diagram of tuning the exact quantum TN state between the stacked 2D toric codes, 3D toric code, X cube, and the trivial paramagnet (PM) for ${\mathbb{Z}}_N$ gauge group. The TN states are illustrated in dual cubes, where the black dots denote the virtual variables and the red (green) arrow denotes physical spin ${\sigma }^z({\mu }^z)$, satisfying ${\mu }^z={\omega }^{n_1-n_2-n_3+n_4}$, ${\sigma }^z={\omega }^{n_4-n_3}$. The classical TNs lying at the QPTs show the wave function norm.

Figure 2

QPT of $m$-loop condensation in the ${\mathbb{Z}}_N$ model for (a),(b) $N=2$ and (c),(d) $N=5$. (a),(c) Inverse of the $m$-loop condensation length scale. (b),(d) Deconfined charge amplitude. Insets fit the critical exponents ${\xi }_m\propto |h_c-h{|}^{-{\nu }_m}$ and $⟨e|e⟩\propto |h_c-h{|}^{\beta }$. Data are computed by iPEPS of bond dimension $(D=3,\chi =72)$ for $N=2$ and $(D=2,\chi =80)$ for $N=5$. Dashed vertical line denotes the critical point $h_c\approx 0.7614$ for $N=2$ and $h_c\approx 2.2$ for $N=5$.

Figure 3

Fracton confinement transition in the ${\mathbb{Z}}_N$ model for $N=24$. $t_c\approx 1.5788(13)$. (a) Left axis: inverse fracton-confinement lengthscale; right axis: monopole condensate fraction. (b) Left axis: fracton quadrupole amplitude; right axis: logarithm of the wave function norm mapped to the classical free energy density. Data are computed with iPEPS with bond dimension $D=2$, $\chi =24$.

Figure 4

Large-$N$ limit. (a) Schematic phase diagram. The XC fracton phase shrinks to a quantum critical line at $\tanh (h_c)=1$, on which deconfined fractons are glued by critical fluctuating $m$ strings. $h$ perturbation drives a partial confinement that binds fractons into a fracton dipole, adiabatically connected to the $m$ particle in 2D TCs. $t>t_c$ drives complete confinement that leaves no deconfined fracton dipoles. (b) $N$-scaling for the first-order QPTs of ${\mathbb{Z}}_N$ plaquette model: $t_c$ converges to $t_c\approx 1.5775(25)$. The jump of the monopole condensate fraction remains finite and largely independent of $N$. The fracton confining length ${\xi }_f$ for $t_c^{+}$ is proportional to $N$. The jump of fracton quadrupole amplitude vanishes like $1/N^2$. Data are computed with bond dimension $D=2$ for $N=2,\dots ,15,16,24$.