Two-dimensional (2d) electronic systems on a lattice at fractional filling
ν=p/q exhibit a competition between charge ordered insulators, called Wigner-Mott insulators (WMIs), at large Coulomb repulsion and Fermi-liquid metals at large electronic kinetic energy. When those two energy scales are roughly equal, insulating states that restore the lattice translation symmetry, which we call quantum charge liquids (QCLs), may emerge. When gapped, these QCLs must exhibit topological order. In this work, we show that the allowed topological ordered phases that are proximate to the WMI strongly depend on the charge ordering in the WMI. In particular, we show that when
q is even, no direct transition exists between a WMI with the smallest allowed unit cell size from filling constraints, i.e., the "minimal" WMI, and the topological order with the smallest ground state degeneracy on a torus allowed by filling constraints, i.e., the "minimal" TO. Furthermore, we describe the quantum melting transition of the WMIs to the proximate QCLs in terms of the proliferation of the topological defects of the WMIs. The field theory of this transition in terms of the topological defects reveals their role as precursors to the anyon excitations in the QCLs.
We provide a route to constructing an electronic quantum charge liquid (QCL), a state made up of fermions at fractional filling of a lattice that does not break translation. Starting with spinless fermions at filling
ν=3/2 we pair them to get bosons at filling
ν=3/4 per unit cell. The tetramer model, a generalization of the dimer model, on the square lattice is evaluated as a candidate bosonic QCL at filling
ν=3/4. It is shown that these models exhibit a local
Z4 symmetry. Upon numerical study of a family of tetramer wavefunctions it is found that while one is gapless due to
U(1)3 symmetry at least one other can be definitively shown to be gapped. The gapped nature of this state, along with its
Z4 symmetry, leads us to propose that it is an example of the elusive bosonic QCL displaying the minimal
Z4 topological order. We conclude by discussing possible extensions to other lattice geometries, electronic QCLs, and to Rydberg atoms.
We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with
Z2 topological order, that is, on gapped
Z2 spin liquids. We find that the fractionalization class of each anyon is an equivalence class of projective representations of the symmetry group, corresponding to elements of the cohomology group
H2(G,Z2). This result leads us to a symmetry classification of gapped
Z2 spin liquids, such that two phases in different symmetry classes cannot be connected without breaking symmetry or crossing a phase transition. Symmetry classes are defined by specifying a fractionalization class for each type of anyon. The fusion rules of anyons play a crucial role in determining the symmetry classes. For translation and internal symmetries, braiding statistics plays no role, but can affect the classification when point group symmetries are present. For square lattice space group, time-reversal, and
SO(3) spin rotation symmetries, we find $2\phantom{\rule{0.16em}{0ex}}098\phantom{\rule{0.16em}{0ex}}176\ensuremath{\approx}{2}^{21}$ distinct symmetry classes. Our symmetry classification is not complete, as we exclude, by assumption, permutation of the different types of anyons by symmetry operations. We give an explicit construction of symmetry classes for square lattice space group symmetry in the toric code model. Via simple examples, we illustrate how information about fractionalization classes can, in principle, be obtained from the spectrum and quantum numbers of excited states. Moreover, the symmetry class can be partially determined from the quantum numbers of the four degenerate ground states on the torus. We also extend our results to arbitrary Abelian topological orders (limited, though, to translations and internal symmetries), and compare our classification with the related projective symmetry group classification of parton mean-field theories. Our results provide a framework for understanding and probing the sharp distinctions among symmetric
Z2 spin liquids and are a first step toward a full classification of symmetric topologically ordered phases.
An exotic feature of the fractional quantum Hall effect is the emergence of anyons, which are quasiparticle excitations with fractional statistics. In the presence of a symmetry, such as
U(1) charge conservation, it is well known that anyons can carry fractional symmetry quantum numbers. In this paper we reveal a different class of symmetry realizations, i.e., anyons can ``breed'' in multiples under symmetry operation. We focus on the global Ising (
Z2) symmetry and show examples of these unconventional symmetry realizations in Laughlin-type fractional quantum Hall states. One remarkable consequence of such an Ising symmetry is the emergence of anyons on the Ising symmetry domain walls. We also provide a mathematical framework which generalizes this phenomenon to any Abelian topological orders.
A system with charge conservation and lattice translation symmetry has a well-defined filling ν, which is a real number representing the average charge per unit cell.We show that if ν is fractional (i.e., not an integer), this imposes very strong constraints on the low-energy theory of the system and give a framework to understand such constraints in great generality, vastly generalizing the Luttinger and Lieb-Schultz-Mattis theorems.The most powerful constraint comes about if ν is continuously tunable (i.e., the system is charge compressible), in which case, we show that the low-energy theory must have a very large emergent symmetry group-larger than any compact Lie group.An example is the Fermi surface of a Fermi liquid, where the charge at every point on the Fermi surface is conserved.We expect that in many, if not all, cases, even exotic non-Fermi liquids will have the same emergent symmetry group as a Fermi liquid, even though they could have very different dynamics.We call a system with this property an ersatz Fermi liquid.We show that ersatz Fermi liquids share a number of properties in common with Fermi liquids, including Luttinger's theorem (which is thus extended to a large class of non-Fermi liquids) and periodic "quantum oscillations" in the response to an applied magnetic field.We also establish versions of Luttinger's theorem for the composite Fermi liquid in quantum Hall systems and for spinon Fermi surfaces in Mott insulators.Our work makes a connection between filling constraints and the theory of symmetry-protected topological phases, in particular through the concept of "'t Hooft anomalies."
In this paper, we propose a generalization of the
S-duality of four-dimensional quantum electrodynamics (
QED4) to
QED4 with fractionally charged excitations, the fractional
S-duality. Such
QED4 can be obtained by gauging the
U(1) symmetry of a topologically ordered state with fractional charges. When time-reversal symmetry is imposed, the axion angle ($\ensuremath{\theta}$) can take a nontrivial but still time-reversal-invariant value $\ensuremath{\pi}/{t}^{2}$ ($t\ensuremath{\in}\mathbb{Z}$). Here,
1/t specifies the minimal electric charge carried by bulk excitations. Such states with time-reversal and
U(1) global symmetry (fermion number conservation) are fractional topological insulators (FTIs). We propose a topological quantum field theory description, which microscopically justifies the fractional
S-duality. Then, we consider stacking operations (i.e., a direct sum of Hamiltonians) among FTIs. We find that there are two topologically distinct classes of FTIs: type I and type II. Type I ($t\ensuremath{\in}{\mathbb{Z}}_{\mathrm{odd}}$) can be obtained by directly stacking a noninteracting topological insulator and a fractionalized gapped fermionic state with minimal charge
1/t and vanishing $\ensuremath{\theta}$. But type II ($t\ensuremath{\in}{\mathbb{Z}}_{\mathrm{even}}$) cannot be realized through any stacking. Finally, we study the surface topological order of fractional topological insulators.