We develop a new theoretical framework for describing and analyzing exotic phases of strongly correlated electrons which support excitations with fractional quantum numbers. Starting with a class of microscopic models believed to capture much of the essential physics of the cuprate superconductors, we derive a new gauge theory---based upon a discrete Ising or
Z2 symmetry---which interpolates naturally between an antiferromagnetic Mott insulator and a conventional d-wave superconductor. We explore the intervening regime, and demonstrate the possible existence of an exotic fractionalized insulator, the nodal liquid, as well as various more conventional insulating phases exhibiting broken lattice symmetries. A crucial role is played by vortex configurations in the
Z2 gauge field. Fractionalization is obtained if they are uncondensed. Within the insulating phases, the dynamics of these
Z2 vortices in two dimensions is described, after a duality transformation, by an Ising model in a transverse field, the Ising spins representing the
Z2 vortices. The presence of an unusual Berry's phase term in the gauge theory leads to a doping-dependent ``frustration'' in the dual Ising model, being fully frustrated at half filling. The
Z2 gauge theory is readily generalized to a variety of different situations, in particular, it can also describe three-dimensional insulators with fractional quantum numbers. We point out that the mechanism of fractionalization for
d>1 is distinct from the well-known one-dimensional spin--charge separation. Other interesting results include a description of an exotic fractionalized superconductor in two or higher dimensions.
We demonstrate that spin-charge separation can occur in two dimensions and note its confluence with superconductivity, topology, gauge theory, and fault-tolerant quantum computation. We construct a microscopic Ising-like model and, at a special coupling constant value, find its exact ground state as well as neutral spin-
21 (spinon), spinless charge e (holon), and
Z2 vortex (vison) states and energies. The fractionalized excitations reflect the topological order of the ground state which is evinced by its fourfold degeneracy on the torus---a degeneracy which is unrelated to translational or rotational symmetry---and is described by a
Z2 gauge theory. A magnetic moment coexists with the topological order. Our model is a member of a family of topologically ordered models, one of which is integrable and realizes the toric quantum error correction code but does not conserve any component of the spin. We relate our model to a dimer model which could be a spin
SU(2) symmetric realization of topological order and its concomitant quantum number fractionalization.
We describe the phases of a solvable $t\text{\ensuremath{-}}J$ model of electrons with infinite-range, and random, hopping, and exchange interactions, similar to those in the Sachdev-Ye-Kitaev models. The electron fractionalizes, as in an ``orthogonal metal,'' into a fermion
f, which carries both the electron spin and charge, and a boson $\ensuremath{\phi}$. Both
f and $\ensuremath{\phi}$ carry emergent
Z2 gauge charges. The model has a phase in which the $\ensuremath{\phi}$ bosons are gapped, and the
f fermions are gapless and critical, and so the electron spectral function is gapped. This phase can be considered as a toy model for the underdoped cuprates, without spatial structure. The model also has an extended, critical, ``quasi-Higgs'' phase where both $\ensuremath{\phi}$ and
f are gapless, and the electron operator $\ensuremath{\sim}f\ensuremath{\phi}$ has a Fermi liquid-like $1/\ensuremath{\tau}$ propagator in imaginary time, $\ensuremath{\tau}$. So while the electron spectral function has a Fermi liquid form, other properties are controlled by
Z2 fractionalization and the anomalous exponents of the
f and $\ensuremath{\phi}$ excitations. This quasi-Higgs phase is proposed as a toy model of the overdoped cuprates. We also describe the critical state separating these two phases.
We study the electron spectral function of various zero-temperature spin-charge separated phases in two dimensions. In these phases, the electron is not a fundamental excitation of the system, but rather ``decays'' into a spin-1/2 chargeless fermion (the spinon) and a spinless charge e boson (the chargon). Using low-energy effective theories for the spinons (d-wave pairing plus possible N\'eel order) and the chargons (condensed or quantum-disordered bosons), we explore three phases of possible relevance to the cuprate superconductors: (1)
AF∗, a fractionalized antiferromagnet where the spinons are paired into a state with long-ranged N\'eel order and the chargons are 1/2-filled and (Mott) insulating; (2) the nodal liquid, a fractionalized insulator where the spinons are d-wave paired and the chargons are uncondensed; and (3) the d-wave superconductor, where the chargons are condensed and the spinons retain a d-wave gap. Working within the
Z2 gauge theory of such fractionalized phases, our results should be valid at scales below the energy gap of the vison---the basic vortex excitation in the theory. However, on a phenomenological level, our results should apply to any spin-charge separated system where the excitations have these low-energy effective forms. Comparison with angle-resolved photoemission spectroscopy data in the undoped, pseudogapped, and superconducting regions is made.
We investigate the continuous quantum phase transition from an antiferromagnetic metal to a heavy-fermion liquid based on the Kondo lattice model in two dimensions. We propose that antiferromagnetic spin fluctuations and conduction electrons fractionalize into neutral bosonic spinons and charged spinless fermions at the quantum critical point. This deconfined quantum criticality leads us to establish a critical field theory in terms of the fractionalized fields interacting via emergent U(1) gauge fields. The critical field theory not only predicts non-Fermi liquid physics near the quantum critical point but also recovers Fermi liquid physics away from the quantum critical point.