Abstract:
It is now widely accepted that the cuprate superconductors are characterized by the same long-range order as that present in the Bardeen-Cooper-Schrieffer (BCS) theory: that associated with the condensation of Cooper pairs. We argue that many physical properties of the cuprates require interplay with additional order parameters associated with a proximate Mott insulator. We review a classification of Mott insulators in two dimensions, and contend that the experimental evidence so far shows that the class appropriate to the cuprates has collinear spin correlations, bond order, and confinement of neutral, spin S=1/2 excitations. Proximity to second-order quantum phase transitions associated with these orders, and with the pairing order of BCS, has led to systematic predictions for many physical properties. We use this context to review the results of recent neutron scattering, fluxoid detection, nuclear magnetic resonance, and scanning tunnelling microscopy experiments.

Abstract:
Close to optimal doping, the copper oxide superconductors show 'strange metal' behavior, suggestive of strong fluctuations associated with a quantum critical point. Such a critical point requires a line of classical phase transitions terminating at zero temperature near optimal doping inside the superconducting 'dome'. The underdoped region of the temperature-doping phase diagram from which superconductivity emerges is referred to as the 'pseudogap' because evidence exists for partial gapping of the conduction electrons, but so far there is no compelling thermodynamic evidence as to whether the pseudogap is a distinct phase or a continuous evolution of physical properties on cooling. Here we report that the pseudogap in YBCO cuprate superconductors is a distinct phase, bounded by a line of phase transitions. The doping dependence of this line is such that it terminates at zero temperature inside the superconducting dome. From this we conclude that quantum criticality drives the strange metallic behavior and therefore superconductivity in the cuprates.

Abstract:
We investigate the interplay between lattice-symmetry breaking and superconducting order in a two-dimensional model of doped antiferromagnets, with long-range Coulomb interactions and Sp(2N) spin symmetry, in the large-N limit. Our results motivate the outline of a global phase diagram for the cuprate superconductors. We describe the quantum transitions between the phases, the evolution of their fermion excitation spectrum, and the experimental implications.

Abstract:
We present a general introduction to the non-zero temperature dynamic and transport properties of low-dimensional systems near a quantum phase transition. Basic results are reviewed in the context of experiments on the spin-ladder compounds, insulating two-dimensional antiferromagnets, and double-layer quantum Hall systems. Recent large N computations on an extended t-J model (cond-mat/9906104) motivate a global scenario of the quantum phases and transitions in the high temperature superconductors, and connections are made to numerous experiments.

Abstract:
Geometrical Berry phase is recognized as having profound implications for the properties of electronic systems. Over the last decade, Berry phase has been essential to our understanding of new materials, including graphene and topological insulators. The Berry phase can be accessed via its contribution to the phase mismatch in quantum oscillation experiments, where electrons accumulate a phase as they traverse closed cyclotron orbits in momentum space. The high-temperature cuprate superconductors are a class of materials where the Berry phase is thus far unknown despite the large body of existing quantum oscillations data. In this report we present a systematic Berry phase analysis of Shubnikov - de Haas measurements on the hole-doped cuprates YBa$_2$Cu$_3$O$_{y}$, YBa$_2$Cu$_4$O$_8$, HgBa$_2$CuO$_{4 + \delta}$, and the electron-doped cuprate Nd$_{2-x}$Ce$_x$CuO$_4$. For the hole-doped materials, a trivial Berry phase of 0 mod $2\pi$ is systematically observed whereas the electron-doped Nd$_{2-x}$Ce$_x$CuO$_4$ exhibits a significant non-zero Berry phase. These observations set constraints on the nature of the high-field normal state of the cuprates and points towards contrasting behaviour between hole-doped and electron-doped materials. We discuss this difference in light of recent developments related to charge density-wave and broken time-reversal symmetry states.

Abstract:
Confinement effects on the phase transitions in antiferromagnets are studied as a function of the surface coupling v and the surface field h for bcc(110) films. Unusual topologies for the phase diagram are attained for particular combinations of v and h. It is shown that some of the characteristics of the finite-temperature behavior of the system are driven by its low-temperature properties and consequently can be explained in terms of a ground-state analysis. Cluster variation free energies are used for the investigation of the finite temperature behavior.

Abstract:
Motivated by the strong, low temperature damping of nodal quasiparticles observed in some cuprate superconductors, we study quantum phase transitions in d_{x^2-y^2} superconductors with a spin-singlet, zero momentum, fermion bilinear order parameter. We present a complete, group-theoretic classification of such transitions into 7 distinct cases (including cases with nematic order) and analyze fluctuations by the renormalization group. We find that only 2, the transitions to d_{x^2-y^2}+is and d_{x^2-y^2} + i d_{xy} pairing, possess stable fixed points with universal damping of nodal quasiparticles; the latter leaves the gapped quasiparticles along (1,0), (0,1) essentially undamped.

Abstract:
The phase diagrams of cuprate superconductors and of QCD at non-zero baryon chemical potential are qualitatively similar. The Neel phase of the cuprates corresponds to the chirally broken phase of QCD, and the high-temperature superconducting phase corresponds to the color superconducting phase. In the SO(5) theory for the cuprates the $SO(3)_s$ spin rotational symmetry and the $U(1)_{em}$ gauge symmetry of electromagnetism are dynamically unified. This suggests that the $SU(2)_L \otimes SU(2)_R \otimes U(1)_B$ chiral symmetry of QCD and the $SU(3)_c$ color gauge symmetry may get unified to SO(10). Dynamical enhancement of symmetry from $SO(2)_s \otimes \Z(2)$ to $SO(3)_s$ is known to occur in anisotropic antiferromagnets. In these systems the staggered magnetization flops from an easy 3-axis into the 12-plane at a critical value of the external magnetic field. Similarly, the phase transitions in the SO(5) and SO(10) models are flop transitions of a ``superspin''. Despite this fact, a renormalization group flow analysis in $4-\epsilon$ dimensions indicates that a point with full SO(5) or SO(10) symmetry exists neither in the cuprates nor in QCD.

Abstract:
Scanning tunneling spectroscopy studies on Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ suggest the presence of electronic inhomogeneity with a large spatial variation in gap size. Andersen {\it et al} have modelled this variation by assuming a spatially-varying pairing interaction. We show that their calculated specific heat is incompatible with the experimental data which exhibit narrow transitions. This calls into question the now-common assumption of gap and pairing inhomogeneity.

Abstract:
Recent theoretical and experimental studies on the critical properties of frustrated antiferromagnets with the noncollinear spin order, including stacked-triangular antiferromagnets and helimagnets, are reviewed. Particular emphasis is put on the novel critical and multicritical behaviors exhibited by these magnets, together with an important role played by the `chirality'.