Surface probes such as scanning tunnelling microscopy have detected complex electronic patterns at the nanoscale in many high-temperature superconductors. In cuprates, the pattern formation is associated with the pseudogap phase, a precursor to the high-temperature superconducting state. Rotational symmetry breaking of the host crystal in the form of electronic nematicity has recently been proposed as a unifying theme of the pseudogap phase. However, the fundamental physics governing the nanoscale pattern formation has not yet been identifed. Here we introduce a new set of methods for analysing strongly correlated electronic systems, including the effects of both disorder and broken symmetry. We use universal cluster properties extracted from scanning tunnelling microscopy studies of cuprate superconductors to identify the fundamental physics controlling the complex pattern formation. Because of a delicate balance between disorder, interactions, and material anisotropy, we find that the electron nematic is fractal in nature, and that it extends throughout the bulk of the material.
We report the first application of critical cluster techniques to the Mott metal-insulator transition in vanadium dioxide. We show that the geometric properties of the metallic and insulating puddles observed by scanning near-field infrared microscopy are consistent with the system passing near criticality of the random field Ising model as temperature is varied. The resulting large barriers to equilibrium may be the source of the unusually robust hysteresis phenomena associated with the metal-insulator transition in this system.
Inside the metals, semiconductors, and magnets of our everyday experience, electrons are uniformly distributed throughout the material. By contrast, electrons often form clumpy patterns inside of strongly correlated electronic systems (SCES) such as colossal magnetoresistance materials and high temperature superconductors. In copper-oxide-based high temperature superconductors, scanning tunneling microscopy (STM) has detected an electron nematic on the surface of the material in which the electrons form nanoscale structures which break the rotational symmetry of the host crystal. These structures may hold the key to unlocking the mystery of high temperature superconductivity in these materials, but only if the nematic also exists throughout the entire bulk of the material. Using newly developed methods for decoding these surface structures, we find that the nematic indeed persists throughout the bulk of the material. We furthermore find that the intricate pattern formation is set by a delicate balance among disorder, interactions, and material anisotropy, leading to a fractal nature of the cluster pattern. The methods we have developed can be extended to many other surface probes and materials, enabling surface probes to determine whether surface structures are confined only to the surface or whether they extend throughout the material.
We consider tunneling of vortices across a superconducting film that is both narrow and short ͑and connected to bulk superconducting leads at the ends͒. We find that in the superconducting state the resistance, at low values of the temperature ͑T͒ and current, does not follow the power-law dependence on T characteristic of longer samples but is exponential in 1 / T. The coefficient of 1 / T in the exponent depends on the length or, equivalently, the total normal-state resistance of the sample. These conclusions persist in the one-dimensional limit, which is similar to the problem of quantum phase slips in an ultranarrow short wire.
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