Topological states of matter are, generally, quantum liquids of conserved topological defects. We establish this by constructing and analyzing topological field theories which introduce gauge fields to describe the dynamics of singularities in the original field configurations. Homotopy groups are utilized to identify topologically protected singularities, and the conservation of their protected number is captured by a topological action term that unambiguously obtains from the given set of symmetries. Stable phases of these theories include quantum liquids with emergent massless Abelian and non-Abelian gauge fields, as well as topological orders with long-range quantum entanglement, fractional excitations, boundary modes, and unconventional responses to external perturbations. This paper focuses on the derivation of topological field theories and basic phenomenological characterization of topological orders associated with homotopy groups ${\ensuremath{\pi}}_{n}({S}^{n}), n\ensuremath{\ge}1$. These homotopies govern monopole and hedgehog topological defects in
d=n+1 dimensions, and enable the generalization of both weakly interacting and fractional quantum Hall liquids of vortices to
d>2. Hedgehogs have not been in the spotlight so far, but they are particularly important defects of magnetic moments because they can be stimulated in realistic systems with spin-orbit coupling, such as chiral magnets and
d=3 topological materials. We predict topological orders in systems with $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{Spin}(d)$ symmetry in which fractional electric charge attaches to hedgehogs. Monopoles, the analogous defects of charge or generic U(1) currents, may bind to hedgehogs via Zeeman effect, or effectively emerge in purely magnetic systems. The latter can lead to spin liquids with different topological orders than that of the resonant valence bond spin liquid. Charge fractionalization of quarks in atomic nuclei is also seen as possibly arising from the charge-hedgehog attachment.
A magnetic monopole is a hypothetical elementary particle with an isolated magnetic pole. Its existence would directly lead to the quantization of electric charges. In recent years, analogs of magnetic monopoles, represented by topological defects in parameter spaces, have been studied in a wide range of physical systems. These works mainly focused on Abelian Dirac monopoles in spin-1/2 or non-Abelian Yang monopoles in spin-3/2 systems. Here we propose to realize three types of spin-1 topological monopoles and study their geometric properties using the parameter space formed by three hyperfine states of ultracold atoms coupled by radio-frequency fields. These spin-1 monopoles, characterized by different monopole charges, possess distinct Berry curvature fields and spin textures, which are directly measurable in experiments. The topological phase transitions between different monopoles are accompanied by the emergence of a spin ``vortex'' and can be intuitively visualized using Majorana's stellar representation. We show how to determine the Berry curvature, hence the geometric phase and monopole charge from dynamical effects. Our scheme provides a simple and highly tunable platform for observing and manipulating spin-1 topological monopoles, paving the way for exploring new topological quantum matter.
Monopole operators are studied in a large family of quantum critical points between Dirac spin liquids and topological quantum spin liquids (QSLs): chiral and Z 2 QSLs.These quantum phase transitions are described by conformal field theories (CFTs): quantum electrodynamics in 2 þ 1 dimensions with 2N flavors of two-component massless Dirac fermions and a four-fermion interaction term.For the transition to a chiral spin liquid, it is the Gross-Neveu interaction (QED 3 -GN), while for the transitions to Z 2 QSLs, it is a superconducting pairing term with general spin and valley structure (generalized QED 3 -Z 2 GN).Using the state-operator correspondence, we obtain monopole scaling dimensions to subleading order in 1=N.For monopoles with a minimal topological charge q ¼ 1=2, the scaling dimension is 2N × 0.26510 at leading order, with the quantum correction being 0.118911(7) for the chiral spin liquid, and 0.102846(9) for the simplest Z 2 case (the expression is also given for a general pairing term).Although these two anomalous dimensions are nearly equal, the underlying quantum fluctuations possess distinct origins.The analogous result in QED 3 is also obtained, and we find a subleading contribution of -0.038138ð5Þ, which is slightly different from the value -0.0383 first obtained in the literature.The scaling dimension of a QED 3 -GN monopole with minimal charge is very close to the scaling dimensions of other operators predicted to be equal by a conjectured duality between QED 3 -GN with 2N ¼ 2 flavors and the CP 1 model.Additionally, nonminimally charged monopoles with equal charges on both sides of the duality have similar scaling dimensions.By studying the large-q asymptotics of the scaling dimensions in QED 3 , QED 3 -GN, and QED 3 -Z 2 GN, we verify that the constant Oðq 0 Þ coefficient precisely matches the universal nonperturbative prediction for CFTs with a global U(1) symmetry.Finally, we identify numerous open questions regarding the fate of monopoles and their hierarchies at transitions to spin liquids and ordered phases.
We propose doping
SU(3)-symmetric spin liquids as a route toward charge-
6e superconductivity. This generalizes the idea of constructing charge-
4e superconductivity from doped
SU(4)-symmetric phases. As a concrete platform, we study a bilayer triangular-lattice Hubbard model with
SU(3) spin symmetry and interlayer antiferromagnetic exchange. Using complementary parton constructions, we analyze doped
Z3 quantum spin liquid and
SU(3)-related chiral spin liquids. Doping a
Z3 quantum spin liquid can produce an orthogonal metal with a gauge invariant fermi surface of charge-
3e fermionic trions. Pairing these trions gives a time-reversal-symmetric charge-
6e superconductor. Doping Abelian
SU(3)1 and
SU(6)1 chiral spin liquids yields chiral charge-
6e superconductors with and without residual Abelian topological order, respectively. Doping a non-Abelian
SU(3)2 chiral spin liquid leads to a non-Abelian chiral charge-
6e superconductor intertwined with
SO(3)−3 topological order and supporting non-Abelian
h/(6e) superconducting vortices. We also identify several other phases, including
Z3 orthogonal metal, quantum anomalous Hall (crystal) phases enriched by
Z3 or
Z2 topological order,
SU(3)-breaking charge-
2e superconductors, composite fermi liquid coupled to non-Abelian gauge field, and descendant chiral spin liquids. Our results identify doped
SU(3) spin liquids as a natural setting where symmetry, fractionalization, and topology cooperate to produce charge-
6e superconductivity.
Triangular-lattice quantum spin-liquid insulators are observed to undergo transitions to superconductivity under pressure or doping, and exhibit enhanced terahertz conductivity when driven by mid-infrared light. We present a general theoretical framework for the emergence of superconducting and charge-ordered phases from a U(1) Dirac spin liquid with fermionic spinons, as well as from its gapped
Z2 and chiral descendants. Numerical studies have provided substantial evidence for these spin-liquid states. The spin-liquid phase hosts fractionalized Dirac spinons coupled to an emergent gauge field, whereas the superconducting and charge-ordered phases are conventional, with neither fractionalized excitations nor emergent gauge dynamics. The transition between these phases is driven by the Higgs condensation of spinless charge-
e bosonic chargons (``doublons'' and ``holons''). We show that the projective symmetry group of the Dirac spinons uniquely determines the symmetry and dispersion of the chargons, allowing us to construct an effective low-energy theory near the chargon band minima. Gauge-invariant composites of the chargon Higgs fields provide the order parameters characterizing the phases. The resulting phase diagram contains a rich variety of ordered states, including
d+id superconductivity, charge-density waves, bond-density waves, and pair-density waves.
Quantum fluctuations of the Néel state of the square lattice antiferromagnet are usually described by a CP 1 theory of bosonic spinons coupled to a U(1) gauge field, and with a global SU(2) spin rotation symmetry.Such a theory also has a confining phase with valence bond solid (VBS) order, and upon including spin-singlet charge-2 Higgs fields, deconfined phases with Z 2 topological order possibly intertwined with discrete broken global symmetries.We present dual theories of the same phases starting from a mean-field theory of fermionic spinons moving in π flux in each square lattice plaquette.Fluctuations about this π-flux state are described by (2 þ 1)-dimensional quantum chromodynamics (QCD 3 ) with a SU(2) gauge group and N f ¼ 2 flavors of massless Dirac fermions.It has recently been argued by Wang et al. [Deconfined Quantum Critical Points: Symmetries and Dualities, Phys. Rev. X 7, 031051 (2017).]that this QCD 3 theory describes the Néel-VBS quantum phase transition.We introduce adjoint Higgs fields in QCD 3 and obtain fermionic dual descriptions of the phases with Z 2 topological order obtained earlier using the bosonic CP 1 theory.We also present a fermionic spinon derivation of the monopole Berry phases in the U(1) gauge theory of the VBS state.The global phase diagram of these phases contains multicritical points, and our results imply new boson-fermion dualities between critical gauge theories of these points.