We establish a generalization of Luttinger's theorem that applies to fractionalized Fermi liquids, i.e. Fermi liquids coexisting with symmetry enriched topological order. We find that, in the linear relation between the Fermi volume and the density of fermions, the contribution of the density is changed by the filling fraction associated with the topologically ordered sector, which is determined by how the symmetries fractionalize. Conversely, this places constraints on the allowed symmetry enriched topological orders that can manifest in a fractionalized Fermi liquid with a given Fermi volume and density of fermions.We establish a generalization of Luttinger's theorem that applies to fractionalized Fermi liquids, i.e. Fermi liquids coexisting with symmetry enriched topological order. We find that, in the linear relation between the Fermi volume and the density of fermions, the contribution of the density is changed by the filling fraction associated with the topologically ordered sector, which is determined by how the symmetries fractionalize. Conversely, this places constraints on the allowed symmetry enriched topological orders that can manifest in a fractionalized Fermi liquid with a given Fermi volume and density of fermions.

arXiv:1601.07902Jan 28, 20167 citations

Fractionalization at a Critical Point

Bogdan Andrei Bernevig, Domenico Giuliano, R. B. Laughlin

APSMar 1, 20040 citations

Fractional S -duality, classification of fractional topological insulators, and surface topological order

Peng Ye, Meng Cheng, Eduardo Fradkin

In this paper, we propose a generalization of the $S$ -duality of four-dimensional quantum electrodynamics ( ${\text{QED}}_{4}$ ) to ${\text{QED}}_{4}$ with fractionally charged excitations, the fractional $S$ -duality. Such ${\text{QED}}_{4}$ can be obtained by gauging the $\text{U(1)}$ 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 $\text{U(1)}$ 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.In this paper, we propose a generalization of the $S$-duality of four-dimensional quantum electrodynamics (${\text{QED}}_{4}$) to ${\text{QED}}_{4}$ with fractionally charged excitations, the fractional $S$-duality. Such ${\text{QED}}_{4}$ can be obtained by gauging the $\text{U(1)}$ 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 $\text{U(1)}$ 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.

arXiv:1701.05559Physical review. B./Physical review. BAug 18, 201731 citations

In this paper, we propose a generalization of the

S

-duality of four-dimensional quantum electrodynamics (

{\text{QED}}_{4}

) to

{\text{QED}}_{4}

with fractionally charged excitations, the fractional

S

-duality. Such

{\text{QED}}_{4}

can be obtained by gauging the

\text{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

\text{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.