Untangling Disorder and Fluctuations in Charge Density Waves

December 17, 2023

Matthew O'Brien

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Charge density waves (CDWs) are a phenomenon found in materials where charges such as electrons arrange themselves in a periodic pattern. This is a form of long-range order that can persist over thousands of atoms. This long-range order has two enemies. First, random impurities &mdash present in every metal—can disrupt this periodic pattern. Secondly, heat has a general tendency to discourage order—think of the regular crystal structure of a solid melting into a disorganized liquid. When both thermal fluctuations and disorder are strong, untangling their effects is theoretically challenging. Moreover, understanding the physics of CDWs is of immediate practical importance, as superconductivity in many high-temperature superconductors is experimentally observed to emerge from charge order as the material is cooled. For these reasons, understanding the effects of disorder on CDWs has remained a fundamental puzzle in theoretical physics.

With my PhD advisor, Prof. Eduardo Fradkin, I have made significant progress in understanding the interaction between fluctuations and random disorder in systems with a continuous order parameter, publishing our results in The Journal of Statistical Mechanics: Theory and Experiment (JSTAT). Our work sheds light on the destruction of long-range CDW order and unveils a novel crossover between weakly and strongly disordered regimes.

To address the complex interactions between disorder and thermal fluctuations, it was necessary to devise a model of a CDW which is exactly solvable. Specifically, we focused on the case where the wave's periodicity does not necessarily align with the material's crystal lattice, because the thermal fluctuations of the CDW phase are much stronger if the lattice is not providing a stabilizing influence. We constructed the model from two N-component complex vectors and represented the CDW order as the phase difference between these two vectors. This allowed us to apply a powerful theoretical tool known as the large-N technique, with which we mapped out the behavior of the model as a function of the coupling between the two vectors which create the CDW and the strength of the disruption due to disorder.

The large-N analysis revealed two distinct regimes as a function of the disorder strength. In the regime of strong disorder, we observed a phase with short-range order, as we expected. Surprisingly, in the weaker disorder regime, the large-N approximation predicted the disorder should have no effect at all and that long-range CDW order should exist. This finding appeared to contradict the well-known Imry-Ma theorem, which guarantees that any amount of disorder—no matter how small—should destroy long-range order in dimensions below four (i.e., any physical dimension of space). It seemed that our initial approach was overly crude, and the culprit was clear: In the large-N approximation, the two vectors from which the CDW is constructed are allowed to fluctuate, but any composites of the vectors, such as the CDW itself, are held fixed.

To resolve the discrepancy with the Imry-Ma theorem, we looked in more detail at the thermal fluctuations of the CDW order. We used the spatial correlations of the CDW order to diagnose the existence of short or long-range order. In the strong disorder regime we confirmed that the CDW order is destroyed by comparing our results with earlier studies of the random field Ising model, where the Imry-Ma theorem is well understood. Then, upon analyzing the corrections necessitated by the thermal fluctuations of the CDW, we discovered that the correlations actually had the same behavior in both the weak and strong disorder regimes, effectively reconciling the disagreement with the Imry-Ma theorem. This implies the existence of a subtle crossover between the two regimes which had not previously been predicted.

While our study focused on a purely classical theory suitable for describing static correlations in insulating materials with weak quantum effects, we plan to extend our approach to include quantum fluctuations. Addressing the rich behavior brought about by disorder and fluctuations in systems such as charge density waves also has broader implications. It can provide valuable insights into a wide range of physical systems, including conducting systems, where disorder plays a significant, yet poorly understood, role. Moreover, we plan to use our results to help interpret experimental measurements of CDWs, particularly the work done by UIUC colleagues in the Abbamonte lab.