This article reviews the effort to understand the physics of high temperature superconductors from the point of view of doping a Mott insulator. The basic electronic structure of the cuprates is reviewed, emphasizing the physics of strong correlation and establishing the model of a doped Mott insulator as a starting point. A variety of experiments are discussed, focusing on the region of the phase diagram close to the Mott insulator (the underdoped region) where the behavior is most anomalous. The normal state in this region exhibits the pseudogap phenomenon. In contrast, the quasiparticles in the superconducting state are well defined and behave according to theory. We introduce Anderson's idea of the resonating valence bond (RVB) and argue that it gives a qualitative account of the data. The importance of phase fluctuation is discussed, leading to a theory of the transition temperature which is driven by phase fluctuation and thermal excitation of quasiparticles. However, we argue that phase fluctuation can only explain the pseudogap phenomenology over a limited temperature range, and some additional physics is needed to explain the onset of singlet formation at very high temperatures. We then describe the numerical method of projected wavefunction which turns out to be a very useful technique to implement the strong correlation constraint, and leads to a number of predictions which are in agreement with experiments. The remainder of the paper deals with an analytic treatment of the t-J model, with the goal of putting the RVB idea on a more formal footing. The slave-boson is introduced to enforce the constraint of no double occupation. The implementation of the local constraint leads naturally to gauge theories. We follow the historical order and first review the U (1) formulation of the gauge theory. Some inadequacies of this formulation for underdoping are discussed, leading to the SU (2) formulation. Here we digress with a rather thorough discussion of the role of gauge theory in describing the spin liquid phase of the undoped Mott insulator. We emphasize the difference between the high energy gauge group in the formulation of the problem versus the low energy gauge group which is an emergent phenomenon. Several possible routes to deconfinement based on different emergent gauge groups are discussed, which lead to the physics of fractionalization and spin-charge separation. We next describe the extension of the SU (2) formulation to nonzero doping. We focus on a part of the mean field phase diagram called the staggered flux liquid phase. We show that inclusion of gauge fluctuation provides a reasonable description of the pseudogap phase. We emphasize that d-wave superconductivity can be considered as evolving from a stable U (1) spin liquid. We apply these ideas to the high Tc cuprates, and discuss their implications for the vortex structure and the phase diagram. A possible test of the topological structure of the pseudogap phase is discussed.

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We perform a Gutzwiller projected-wave-function study for the spin-1/2 Heisenberg model on the Kagomé lattice to compare energies of several spin-liquid states. The result indicates that a U(1)-Dirac spin-liquid state has the lowest energy. Furthermore, even without variational parameters, the energy turns out to be very close to that found by exact diagonalization. We show that such a U(1)-Dirac state represents a quantum phase whose low-energy physics is governed by four flavors of two-component Dirac fermions coupled to a U(1) gauge field. These results are discussed in the context of recent experiments on ZnCu(3)(OH)(6)Cl(2).

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We present a theory of the thermal Hall effect in insulating quantum magnets, where the heat current is totally carried by charge-neutral objects such as magnons and spinons. Two distinct types of thermal Hall responses are identified. For ordered magnets, the intrinsic thermal Hall effect for magnons arises when certain conditions are satisfied for the lattice geometry and the underlying magnetic order. The other type is allowed in a spin liquid which is a novel quantum state since there is no order even at zero temperature. For this case, the deconfined spinons contribute to the thermal Hall response due to Lorentz force. These results offer a clear experimental method to prove the existence of the deconfined spinons via a thermal transport phenomenon.