Exceptionally Slow, Long Range, and Non-Gaussian Critical Fluctuations Dominate the Charge Density Wave Transition
$(TaSe_4)_2I$ is a well-studied quasi-one-dimensional compound long-known to have a charge-density wave (CDW) transition around 263 K. We argue that the critical fluctuations of the pinned CDW order parameter near the transition can be inferred from the resistance noise on account of their coupling to the dissipative normal carriers. Remarkably, the critical fluctuations of the CDW order parameter are slow enough to survive the thermodynamic limit and dominate the low-frequency resistance noise. The noise variance and relaxation time show rapid growth (critical opalescence and critical slowing down) within a temperature window of $ \varepsilon \approx \pm 0.1$, where $\varepsilon$ is the reduced temperature. This is very wide but consistent with the Ginzburg criterion. We further show that this resistance noise can be quantitatively used to extract the associated critical exponents. Below $|\varepsilon | \lesssim 0.02$, we observe a crossover from mean-field to a fluctuation-dominated regime with the critical exponents taking anomalously low values. The distribution of fluctuations in the critical transition region is skewed and strongly non-Gaussian. This non-Gaussianity is interpreted as the breakdown of the validity of the central limit theorem as the diverging coherence volume becomes comparable to the macroscopic sample size. The large magnitude critical fluctuations observed over an extended temperature range, as well as the crossover from the mean-field to the fluctuation-dominated regime highlight the role of the quasi-one dimensional character in controlling the phase transition.