The electronic phase diagrams of many highly correlated systems, and, in particular, the cuprate high temperature superconductors, are complex, with many different phases appearing with similar (sometimes identical) ordering temperatures even as material properties, such as dopant concentration, are varied over wide ranges. This complexity is sometimes referred to as “competing orders.” However, since the relation is intimate, and can even lead to the existence of new phases of matter such as the putative “pair-density wave,” the general relation is better thought of in terms of “intertwined orders.” Some of the experiments in the cuprates which suggest that essential aspects of the physics are reflected in the intertwining of multiple orders, not just in the nature of each order by itself, are selectively analyzed. Several theoretical ideas concerning the origin and implications of this complexity are also summarized and critiqued.

This article discusses fluctuating order in a quantum disordered phase proximate to a quantum critical point, with particular emphasis on fluctuating stripe order. Optimal strategies are derived for extracting information concerning such local order from experiments, with emphasis on neutron scattering and scanning tunneling microscopy. These ideas are tested by application to two model systems—an exactly solvable one-dimensional (1D) electron gas with an impurity, and a weakly interacting 2D electron gas. Experiments on the cuprate high-temperature superconductors which can be analyzed using these strategies are extensively reviewed. The authors adduce evidence that stripe correlations are widespread in the cuprates. They compare and contrast the advantages of two limiting perspectives on the high-temperature superconductor: weak coupling, in which correlation effects are treated as a perturbation on an underlying metallic (although renormalized) Fermi-liquid state, and strong coupling, in which the magnetism is associated with well-defined localized spins, and stripes are viewed as a form of micro phase separation. The authors present quantitative indicators that the latter view better accounts for the observed stripe phenomena in the cuprates.

We study the phase diagram of lattice gauge theories coupled to fixed-length scalar (Higgs) fields. We consider several gauge groups: $Z_2$, U(1), and $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$. We find that when the Higgs fields transform like the fundamental representation of the gauge group the Higgs and confining phases are smoothly connected, i.e., they are not separated by a phase boundary. When the Higgs fields transform like some representation other than the fundamental, a phase boundary may exist. This is the case for $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ with all the Higgs fields in the adjoint representation and for U(1) with all the Higgs fields in the charge-$N(N>\mathrm{}1\mathrm{})\mathrm{}$ representation. We present an argument due to Wegner that indicates the stability of the pure gauge transition. Another phase, free charge or Coulomb, is generally present. In this regime, the spectrum of the theory contains massless gauge bosons (for continuous groups) and finite-energy states that represent free charges.

Correlated electron fluids can exhibit a startling array of complex phases, among which one of the more surprising is the electron nematic, a translationally invariant metallic phase with a spontaneously generated spatial anisotropy. Classical nematics generally occur in liquids of rod-like molecules; given that electrons are point like, the initial theoretical motivation for contemplating electron nematics came from thinking of the electron fluid as a quantum melted electron crystal, rather than a strongly interacting descendent of a Fermi gas. Dramatic transport experiments in ultra-clean quantum Hall systems in 1999 and in Sr₃Ru₂O₇in a strong magnetic field in 2007 established that such phases exist in nature. In this article, we briefly review the theoretical considerations governing nematic order, summarize the quantum Hall and Sr₃Ru₂O₇experiments that unambiguously establish the existence of this phase, and survey some of the current evidence for such a phase in the cuprate and Fe-based high temperature superconductors.

We present a theory of the fractional quantum Hall effect (FQHE) based on a second-quantized fermion path-integral approach. We show that the problem of interacting electrons moving on a plane in the presence of an external magnetic field is equivalent to a family of systems of fermions bound to an even number of fluxes described by a Chern-Simons gauge field. The semiclassical approximation of this system has solutions that describe incompressible-liquid states, Wigner crystals, and solitonlike defects. The liquid states belong to the Laughlin sequence and to the first level of the hierarchy. We give a brief description of the FQHE for bosons and anyons in this picture. The semiclassical spectrum of collective modes of the FQHE states has a gap to all excitations. We derive an effective action for the Gaussian fluctuations and study the hydrodynamic regime. The dispersion curve for the magnetoplasmon is calculated in the low-momentum limit. We find a nonzero gap at ${\mathrm{\omega }}_{\mathit{c}}$. The fractionally quantized Hall conductance is calculated and argued to be exact in this approximation. We also give an explicit derivation of the polarization tensor in the integer Hall regime and show that it is transverse.