Extending Luttinger’s theorem to $Z_2$ fractionalized phases of matter

Luttinger’s theorem for Fermi liquids equates the volume enclosed by the Fermi surface in momentum space to the electron filling, independent of the strength and nature of interactions. Motivated by recent momentum balance arguments that establish this result in a nonperturbative fashion [M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000)], we present extensions of this momentum balance argument to exotic systems which exhibit quantum number fractionalization focusing on $Z_2$ fractionalized insulators, superfluids and Fermi liquids. These lead to nontrivial relations between the particle filling and some intrinsic property of these quantum phases, and hence may be regarded as natural extensions of Luttinger’s theorem. We find that there is an important distinction between fractionalized states arising naturally from half filling versus those arising from integer filling. We also note how these results can be useful for identifying fractionalized states in numerical experiments.

Figure 1

Schematic figure showing flux threading in a cylinder geometry, with flux ${\mathrm{\Phi }}_0=hc∕Q$.

Figure 2

Evolution of energy levels upon flux threading in a conventional insulator with two-fold broken translational symmetry on a cylinder. The two levels which become degenerate ground states in the thermodynamic limit carry momenta 0, $\pi$. (a) The two levels cross upon threading flux along $\widehat{y}$ in a geometry with $L_x=\text{even}$, $L_y=\text{odd}$. (b) The two levels return to themselves upon threading flux along $\widehat{y}$ in a geometry with $L_x=\text{even}$, $L_y=\text{even}$.

Figure 3

(a) Schematic figure showing a vison threading the hole of a cylinder in the absence of vison tunneling terms. The dark (light) bonds correspond to ${\sigma }_{ij}^z=-1({\sigma }_{ij}^z=+1)$. We can detect the presence of the vison by evaluating the Wilson loop operator ${\mathrm{\Pi }}_C{\sigma }_{ij}^z$ along the contour $C$ taken around the cylinder. (b) The translation operator along the $\widehat{x}$ direction moves the dark $({\sigma }_{ij}^z=-1)$ bonds by one lattice spacing from ($L_x$, 1) to (1, 2) at each $y$. This is accomplished equivalently by acting with the gauge transformation operator $G_i$ (which changes the sign of ${\sigma }_{ij}^z$ on all bonds emanating from $i$) acting on each of the circled sites.

Figure 4

Threading a vortex through the hole of a torus. (a) A vortex (arrow coming out) – antivortex (arrow going in) pair is created, and separated by the translation operator $T_y$. (b) When the vortex is taken all the way around the torus and then annihilated with the antivortex, the resulting state has one unit of circulation about the hole of the torus as shown.

Figure 5

Schematic figure showing the filling dependence [$\nu =0$ (1) is the empty (full) band] of the Hall conductivity in a conventional Fermi liquid or $\mathcal{F}{\mathcal{L}}_{\mathit{\text{even}}}^{*}$ (solid line), for electrons within a simple Drude-like picture, contrasted with the behavior expected in the $\mathcal{F}{\mathcal{L}}_{\mathit{\text{odd}}}^{*}$ phase (dashed line). Similar results are expected for the conventional superfluid or $\mathcal{S}{\mathcal{F}}_{\mathit{\text{even}}}^{*}$ phase compared to the $\mathcal{S}{\mathcal{F}}_{\mathit{\text{odd}}}^{*}$ phase in a vortex liquid phase.