Quantum criticality

classified as the study of quantum matter. The aim is to describe the phases of large numbers of interacting particles at temperatures low enough that quantum mechanics plays a crucial role in determining the system’s distinguishing characteristics. For electrons in solids, the needed “low” temperatures can be even higher than room temperature. Gases of trapped atoms require ultracold temperatures in the nanokelvin range. And through collisions of heavy nuclei, groups at particle accelerators are pursuing a quantum plasma of quarks with temperatures approaching those realized soon after the Big Bang. Remarkably, a common set of ideas on the phases of quantum matter applies across that wide range of energy scales. Electron systems in solids are one of the best places to study quantum matter. With modern fabrication techniques, one can make an almost infinite variety of crystals, which provide a rich test bed for experimental investigations and theoretical descriptions of quantum phases. This article focuses on certain phase transitions the electrons undergo at the absolute zero of temperature. There are no thermal fluctuations at absolute zero. Instead, the transitions are driven by quantum fluctuations demanded by Heisenberg’s uncertainty principle. The quantum critical point, where the transitions occur, is present only at absolute zero, but its influence nevertheless is felt in a broad regime of “quantum criticality” at nonzero temperatures, and it is the key to understanding a variety of experiments. The most common quantum phases of electrons should be familiar to most readers: metals, with electrons occupying mobile plane-wave states; superconductors, with electrons forming Cooper pairs that can transport charge without dissipation; insulators, in which charge transport requires the exciting of electrons across an energy gap, which rarely happens at room temperatures; and semiconductors, which are essentially insulators with a smaller energy gap. Such a classification focuses on the motion of the charge carried by the electron. However, the electron also has spin, and a study of the spin configuration in the electron wavefunction allows a more subtle classification of quantum phases. The rich variety of magnetic phases includes ferromagnets and antiferromagnets. The magnetic phases can be metals, superconductors, insulators, or semiconductors in their charge degrees of freedom. Much recent experimental work has focused on so-called correlated-electron materials. In such materials, the electrons can occupy the atomic d or f orbitals, whose smaller spatial extent increases the importance of the Coulomb repulsion between the electrons, so the electrons’ motion must be correlated to ensure that they stay apart from each other. Although most correlated-electron materials realize one of the common quantum phases above, many can be tuned between two or