### Broadbrush view of correlated electrons and quantum magnetism

### Conventional Mott Insulators - Background

Band theory predicts that crystals with an odd number of electrons per unit cell will be metallic. When the atomic orbitals contributing to the conduction band are s-like, as in the noble metals or p-like as in aluminum, this expectation is fulfilled. But when the electrons at the Fermi energy are from partially filled atomic d or f shells, this simplicity often breaks down. Since the d- or f-shell electrons on neighboring atoms have a small overlap (because the inter atomic spacing is determined by filled orbitals with larger radial quantum numbers), correlation effects are important and can self-localize the electrons on their respective atoms forming a Mott insulating state. In transition metal oxides with partially filled d-shells, Mott insulating behavior is commonplace. The partially filled f-shell electrons in many rare earth compounds are so ``heavy" that they form local moments. Exchange coupled local moments usually magnetically order at low temperatures in such Mott insulators, forming an enlarged unit cell with an even number of electrons. These quantum phases are adiabatically connected to weak coupling spin-density-wave phases which form as an instability of a nearly nested Fermi surface. Spin Peierls or spontaneous dimerization is equally effective at enlarging the unit cell, and can be viewed as a Fermi surface energy density wave instability.

### Spin Liquid Insulators in 2d

Phil Anderson suggested over 30 years ago that upon cooling such Mott insulators another scenario is possible - the local moments remain in a "liquid" phase down to absolute zero, no symmetries are broken and the unit cell is not enlarged. If this occurs novelty is guaranteed. Specifically, it can be shown that in a 2d system with s=1/2 per unit cell (no symmetry breaking doubling) and periodic boundary conditions the gap between the (unique) ground state and the lowest energy excited state vanishes algebraically in the thermodynamic limit. There are then a number of possibilities. One is the presence of ``topological order", with a degenerate ground state in the thermodynamic limit (the multiplicity determined by the topology of the 2d closed manifold). The ground state wavefunctions cannot be distinguished by any local measurement but are topologically distinct. During the past five years remarkable progress has been achieved in understanding the general features of such topologically ordered phases. These states generally have an emergent gauge structure which is in its deconfined regime, and supports gapped particle-like excitations which carry ``electric" or ``magnetic" gauge charges. It is possible to (partially) classify such spin liquid phases in terms of the symmetry group of the emergent gauge theory. The simplest example is the discrete Z2 gauge group. Many microscopic models have now been constructed which have a Z2 spin liquid ground state.

Unfortunately, we have yet to unambiguously identify any topologically ordered phases in any real material (outside of the quantum Hall effect). This is a dire situation, and frankly is "egg on the face" for the theoretical community. Some limited progress has been made in identifying possible ingredients which tend to drive spin liquid physics. Frustration or competing interactions, such as in a triangle of three antiferromagnetically coupled spins, are favorable in suppressing magnetic order. Multi-particle ring exchange terms are also effective at destroying magnetic and other types of order, and can tend to drive spin liquid physics. Insulators which are near the Mott transition will have large multi-particle ring exchanges and are good candidate systems to explore in the search for topological order.

### 2d Gapless Spin liquids

The theorem that featureless 2d Mott insulators with periodic boundary conditions and an odd number of electrons per unit cell must have a vanishing gap between the ground and lowest energy excited state in the thermodynamic limit leaves a second possibility besides topologial order. As emphasized by X.G. Wen, one can envisage spin liquid phases which are ``critical" with no fine tuning needed. Such spin liquids support gapless excitations, but these low lying excitations do not behave as weakly interacting quasiparticles. Indeed, all local observables are expected to exhibit power law correlators with non-trivial exponents. In a sense, these gapless spin liquid phases are analogous to 2d interacting quantum critical points, but there are no relevant perturbations whatsoever, so it is not necessary to fine tune to access them. These ``critical phases" are the 2d analog of the Luttinger liquid phase of 1d interacting fermions or bosons.

Our understanding of such critical phases in 2d is currently very rudimentary The most surprising aspect of the gapless spin liquids is the absence of any relevant perturbations. A phase with gapless interacting excitations is exceedingly fragile, and naively unstable. So, in order to establish the existence of such phases it is necessary to have a field theoretic approach which is powerful enough to allow an analysis of all symmetry allowed perturbations.

Currently, the two best experimental candidates for 2d spin liquid insulating behavior are Cs_2CuCl_4 and an organic material K-BEDT, which are both s=1/2 triangular lattice antiferromagnets. Despite the increased complexity of the algebraic spin liquids compared with their boring simple minded cousins (the topologically ordered spin liquids), the existing data on these two materials appear to be more consistent with the former class.

### 2d "Algebraic Spin liquids" and "Spin Bose-Metals"

Effective field theory approaches such as slave particle gauge theories or vortex dualities, while unable to solve any particular Hamiltonian, do indicate the possibility of stable gapless 2D spin liquid phases. Such gapless 2D spin liquids generically exhibit spin correlations that decay as a power law in space, perhaps with anomalous exponents, and which can oscillate at particular wavevectors. The location of these dominant singularities in momentum space provides a convenient characterization of gapless spin liquids. In the ``algebraic" spin liquids these wavevectors are limited to a finite discrete set, often at high symmetry points in the Brilloin zone, and their effective field theories can often exhibit a relativistic structure. But the singularities can also occur along surfaces in momentum space, as they do in the Gutzwiller-projected spinon Fermi sea state, a 2D "Spin Bose-Metal" (SBM) phase. It must be stressed that it is the spin (i.e., bosonic) correlation functions that possess such singular surfaces -- there are no fermions in the system -- and the low energy excitations cannot be described in terms of weakly interacting quasiparticles. A related "Bose-Metal" phase in a system of hard core bosons hopping on a square lattice with a frustrating four-site term was discussed in D-wave correlated Critical Bose Liquids in two dimensions.

Remarkably, it is possible to access such Bose-Metals by systematically approaching 2D from a sequence of quasi-1D ladder models (see Spin Bose-Metal phase in a spin-1/2 model with ring exchange on a two-leg triangular strip). On a ladder the quantized transverse momenta cut through the 2D surface, leading to a quasi-1D descendant state with a set of low-energy modes whose number grows with the number of legs and whose momenta are inherited from the 2D Bose surfaces. These quasi-1D descendant states can be accessed in a controlled fashion by analyzing the 1D ladder models using numerical and analytical approaches. These multi-mode quasi-1D liquids constitute a new and previously unanticipated class of quantum states interesting in their own right. But more importantly they carry a distinctive quasi-1D ``fingerprint" of the parent 2D state.

### Spin Liquid Insulators in 3d

There are various frustrated lattices in three dimensions, such as the pyrochlore lattice (a lattice of corner sharing tetrahedra). A number of experimental Mott insulators have the pyrochlore lattice structure, and give some indications of novel spin physics at low temperatures. Some of these might host novel 3d spin liquid physics. Theoretically, the simplest model to consider is the near neighbor $s=1/2$ Heisenberg antiferromagnet on the pyrochlore lattice, but this model is largely intractable. A number of generalizations have been considered, such as a strong easy-axis exchange anisotropy, where the model has an emergent $U(1)$ gauge symmetry. Upon addition of an extra local interaction term, the model supports a novel fractionalized 3d $U(1)$ spin liquid phase. This 3d spin liquid is stable to all perturbations, and supports gapped $S^z = 1/2$ spinons, a gapped topological point defect or ``magnetic'' monopole, and a gapless ``photon''. The presence of this ``photon mode" is particularly exciting, since it would lead to significant signatures in the thermodynamics, such as $T^3$ specific heat which could swamp the usual phonon contribution. Ultracold atoms subjected to an optical lattice formed by laser standing waves, might offer an exciting new arena in which to explore such exotic 3d quantum phases

### Non-Fermi Liquid Conductors in 2d

The unusual behavior of the cuprate ``strange metal" state at optimal doping was one of the first striking departures from standard Fermi liquid theory, and even today remains mysterious. A theory for this strange metal is exceedingly challenging, since it must incorporate both the anomalous charge transport at optimal doping and the unusual spin physics of the pseudo-gap.

Very thin superconducting films when placed in a perpundicular magnetic field appear to exhibit extremely anomalous behavior which is also suggestive of a 2d non-Fermi liquid conducting phase. Experiments by Kapitulnik and coworkers on the MoGe films in magnetic fields well below $H_{c2}$ and at $T << T_c$, reveal a ``metallic" (temperature independent) resistance which can be more than a thousand times smaller than the normal state resistance. Evidently, the field induced vortices are mobile, destroying the superconductivity, but have not driven the film normal.

Understanding these two systems will likely be near impossible within a standard Fermi liquid approach, and it seems plausible that one or both might be examples of exotic non-Fermi liquid conducting phases. Currently, the theoretical understanding of such novel quantum phases is almost nil, although there has been scattered progress on various toy models. In my view, this field is as wide open as it is challenging.