Fracton Topological Order at Finite Temperature

As new kinds of stabilizer code models, fracton models have been promising in realizing quantum memory or quantum hard drives. However, it has been shown that the fracton topological order of many 3D fracton models occurs only at zero temperature, such as the 3+1d X-cube model or Haah code. In this work, we show that higher dimensional fracton models can support a fracton topological order below a nonzero critical temperature Tc. Focusing on a typical 4D X- cube model, we show that there is a finite critical temperature Tc by analyzing its free energy from duality. We also obtained the expectation value of the ’t Hooft loops in the 4D X-cube model, which directly shows a confinement-deconfinement phase transition at finite temperature. This finite-temperature phase transition can be understood as spontaneously breaking the Z₂ one-form subsystem symmetry. Moreover, we propose a new no-go theorem for finite-temperature quantum fracton topological order.