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.

A large class of topological orders can be understood and classified using the string-net condensation picture. These topological orders can be characterized by a set of data (N, di, F ijk lmn , δ ijk ). We describe a way to detect this kind of topological order using only the ground state wave function. The method involves computing a quantity called the "topological entropy" which directly measures the quantum dimension D = P i d 2 i . [3,4], and edge excitations [1]. In this paper, we demonstrate that topological order is manifest not only in these dynamical properties but also in the basic entanglement of the ground state wave function. We hope that this characterization of topological order can be used as a theoretical tool to classify trial wave functions -such as resonating dimer wave functions [5], Gutzwiller projected states, [6][7][8][9][10] or quantum loop gas wave functions [11]. In addition, it may be useful as a numerical test for topological order. Finally, it demonstrates definitively that topological order is a property of a wave function, not a Hamiltonian. The classification of topologically ordered states is nothing but a classification of complex functions of thermodynamically large numbers of variables.Main Result: We focus on the (2 + 1) dimensional case (though the result can be generalized to any dimension). Let Ψ be an arbitrary wave function for some two dimensional lattice model. For any subset A of the lattice, one can compute the associated quantum entanglement entropy S A .[12] The main result of this paper is that one can determine the "quantum dimension" D of Ψ by computing the entanglement entropy S A of particular regions

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We show that quantum systems of extended objects naturally give rise to a large class of exotic phases -namely topological phases. These phases occur when the extended objects, called "string-nets", become highly fluctuating and condense. We derive exactly soluble Hamiltonians for 2D local bosonic models whose ground states are string-net condensed states. Those ground states correspond to 2D parity invariant topological phases. These models reveal the mathematical framework underlying topological phases: tensor category theory. One of the Hamiltoniansa spin-1/2 system on the honeycomb lattice -is a simple theoretical realization of a fault tolerant quantum computer. The higher dimensional case also yields an interesting result: we find that 3D string-net condensation naturally gives rise to both emergent gauge bosons and emergent fermions. Thus, string-net condensation provides a mechanism for unifying gauge bosons and fermions in 3 and higher dimensions.

Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which is protected by U (1) and time reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain antiunitary time reversal symmetry) can be labeled by the elements in H 1+d [G, UT (1)] -the Borel (1 + d)-group-cohomology classes of G over the G-module UT (1). Our theory, which leads to explicit ground state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological θ-term in continuous non-linear σ-models to lattice non-linear σ-models. The boundary excitations of the non-trivial SPT phases are described by lattice non-linear σ-models with a non-local Lagrangian term that generalizes the Wess-ZuminoWitten term for continuous non-linear σ-models. As a result, the symmetry G must be realized as a non-on-site symmetry for the low energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use] to obtain interacting bosonic topological insulators (protected by time reversal Z T 2 and boson number conservation), which contain one non-trivial phases in 1D or 2D, and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time reversal symmetry only), in term of, which contain one non-trivial phase in odd spatial dimensions and none for even. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use H 1+d [U (1) × Z T 2 , UT (1)] to construct the SPT phases of integer spin systems with time reversal and U (1) spin rotation symmetry, which contain three non-trivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: GH , GΨ, H 1+d [GΨ, UT (1)] , where GH is the symmetry group of the Hamiltonian and GΨ the symmetry group of the ground states.

A quantum spin liquid is an exotic quantum state of matter in which spins are highly entangled and remain disordered down to zero temperature. Such a state of matter is potentially relevant to high-temperature superconductivity and quantum-information applications, and experimental identification of a quantum spin liquid state is of fundamental importance for our understanding of quantum matter. Theoretical studies have proposed various quantum-spin-liquid ground states 1-4 , most of which are characterized by exotic spin excitations with fractional quantum numbers (termed 'spinon'). Here, we report neutron scattering measurements that reveal broad spin excitations covering a wide region of the Brillouin zone in a triangular antiferromagnet YbMgGaO 4. The observed diffusive spin excitation persists at the lowest measured energy and shows a clear upper excitation edge, which is consistent with the particle-hole excitation of a spinon Fermi surface. Our results therefore point to a QSL state with a spinon Fermi surface in YbMgGaO 4 that has a perfect spin-1/2 triangular lattice as in the original proposal 4 of quantum spin liquids. In 1973, Anderson proposed the pioneering idea of the quantum spin liquid (QSL) in the study of the triangular lattice Heisenberg antiferromagnet 4. This idea was revived after the discovery in 1986 of high-temperature superconductivity 5. A QSL, as currently understood, does not fit into Landau's conventional paradigm of symmetry breaking phases 1,2,6,7 , and is 2 instead an exotic state of matter characterized by spinon excitations and emergent gauge structures 1-3,6. The search for QSLs in models and materials 8-12 has been partly facilitated by the Oshikawa-Hastings-Lieb-Schultz-Mattis (OHLSM) theorem that may hint at the possibility of QSLs in Mott insulators with odd electron fillings and a global U(1) spin rotational symmetry 13-15. Indeed, a continuum of spin excitations has been observed in a kagome-lattice material ZnCu 3 (OD) 6 Cl 2 (refs 12,16). However, the requirement of the U(1) spin rotational symmetry, prevents the application of OHLSM theorem in strong spin-orbit-coupled (SOC) Mott insulators in which the spin rotational symmetry is completely absent. A recent theory addressed this limitation of the OHLSM theorem, arguing that, as long as time-reversal symmetry is preserved, the ground state of an SOC Mott insulator with odd electron fillings must be exotic 17. The newly discovered triangular antiferromagnet YbMgGaO 4 (refs 18,19) displays no indication of magnetic ordering or symmetry breaking at temperatures as low as 30 mK despite approximately 4 K for the spin interaction energy scale. Because of the strong SOC of the Yb electrons, YbMgGaO 4 was the first QSL to be proposed beyond the OHLSM theorem 19. The thirteen 4 f electrons of the Yb 3+ ion form the spin-orbit-entangled Kramers doublets that are split by the D 3d crystal electric fields 20-22. At temperatures considerably lower than the crystal field gap (∼ 420 K), the magnetic properties are captured by the g...

We study a new kind of ordering -topological order -in rigid states (the states with no local gapless excitations). We concentrate on characterization of the different topological orders. As an example we discuss in detail chiral spin states of 2+1 dimensional spin systems. Chiral spin states are described by the topological Chern-Simons theories in the continuum limit. We show that the topological orders can be characterized by a nonAbelian gauge structure over the moduli space which parametrizes a family of the model Hamiltonians supporting topologically ordered ground states. In 2+1 dimensions, the non-Abelian gauge structure determines possible fractional statistics of the quasi-particle excitations over the topologically ordered ground states. The dynamics of the low lying global excitations is shown to be independent of random spatial dependent perturbations.The ground state degeneracy and the non-Abelian gauge structures discussed in this paper are very robust, even against those perturbations that break translation symmetry. We also discuss the symmetry properties of the degenerate ground states of chiral spin states.We find that some degenerate ground states of chiral spin states on torus carry non-trivial quantum numbers of the 90 • rotation.

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