Composite Fermi Liquid at Zero Magnetic Field in Twisted ${\mathrm{MoTe}}_2$

The pursuit of exotic phases of matter outside of the extreme conditions of a quantizing magnetic field is a long-standing quest of solid state physics. Recent experiments have observed spontaneous valley polarization and fractional Chern insulators in zero magnetic field in twisted bilayers of ${\mathrm{MoTe}}_2$, at partial filling of the topological valence band ($\nu =-2/3$ and $-3/5$). We study the topological valence band at half filling, using exact diagonalization and density matrix renormalization group calculations. We discover a composite Fermi liquid (CFL) phase even at zero magnetic field that covers a large portion of the phase diagram near twist angle $\sim 3.6°$. The CFL is a non-Fermi liquid phase with metallic behavior despite the absence of Landau quasiparticles. We discuss experimental implications including the competition between the CFL and a Fermi liquid, which can be tuned with a displacement field. The topological valence band has excellent quantum geometry over a wide range of twist angles and a small bandwidth that is, remarkably, reduced by interactions. These key properties stabilize the exotic zero field quantum Hall phases. Finally, we present an optical signature involving “extinguished” optical responses that detects Chern bands with ideal quantum geometry.

Figure 1

The top valence band has favorable conditions for fractionalized topological phases. Band structure as seen from (a) charge neutrality and (b) from $\nu =-1$ computed from self-consistent Hartree-Fock. (c) Quantum geometry in terms of trace condition $T$ and Berry curvature deviation $\sigma [\mathrm{\Omega }]$. (d) Bare and SCHF bandwidths and (e) the many-body gap of FCIs at $\nu =-1/3$ and $\nu =-2/3$ as a function of twist angle. The FCI gaps are obtained from ED with $N_e=8$ and 16, respectively. (f) and (g): ED spectrum for 14 particles at half filling for Coulomb interaction in lowest Landau level and screened Coulomb interaction in twisted ${\text{MoTe}}_2$, respectively. Parameters: $(\theta ,{\epsilon }_r,d)=(3.7°,15,300\AA )$ unless otherwise noted.

Figure 2

Numerical identification of the composite Fermi liquid (CFL) from iDMRG. (a) Occupations $n(\mathit{k})$ in the Brillouin zone at $L_y=8$ for the Fermi liquid (FL, left side) versus the CFL (right side). (b) Connected structure factor $S(\mathit{q})=⟨{\widehat{\rho }}_{\mathit{q}}{\widehat{\rho }}_{-\mathit{q}}⟩-⟨{\widehat{\rho }}_{\mathit{q}}⟩⟨{\widehat{\rho }}_{-\mathit{q}}⟩$ at $L_y=8$. Characteristic features of a Fermi surface are visible for both the FL and CFL: near-vanishing weight outside $|\mathit{q}|\approx 2k_F$, and peaks corresponding to momentum transfers inside that radius. (c) Cuts of $S(\mathit{q})$ at constant $q_y$ for $L_y=5$ for the CFL. Each peak or inflection in $S(\mathit{q})$ quantitatively matches scattering events across the almost-circular composite Fermi surface (Inset). Parameters match Fig. 1 with ${\epsilon }_r=15$ (100) for the CFL (FL).

Figure 3

Many-body phase diagrams and optical responses. (a) Phase diagram at $\nu =-1$ with $\theta =3.7°$ showing a transition from $C=1$ layer-unpolarized state to a $C=0$ layer polarized state. (b) Phase diagram of the topological regime at $\nu =-1/2$: The CFL phase is shown in red, whereas the green region corresponds to the FL phase. Here LP indicates a layer polarization instability determined from $\nu =-1$ SCHF. (c) Direct optical probe of almost-ideal quantum geometry via an “extinguished” valence-valence optical responses in ${\sigma }^{-}$, Inset: the Haldane model at $(t,t_2)=(1,0.05)$ has nonideal geometry. Parameters match Fig. 1.