Adiabatic continuity of the spinful quantum Hall states

By using the extended Hubbard model of anyons, we numerically demonstrate the adiabatic deformation of the spinful quantum Hall (QH) states by transmutation of statistical fluxes. While the ground state is always spin-polarized in a series of $\nu =1$ integer QH systems, the adiabatic continuity between the singlet QH states at $\nu =2$ and $\nu =2/5$ is confirmed. These results are consistent with the composite fermion theory with spin. The many-body Chern number of the ground-state multiplet works as an adiabatic invariant and also explains the discrete change of the topological degeneracy during the evolution. The generalized Středa formula of spinful systems is justified.

Figure 1

Each line represents the family of the $\nu =1$ IQH system (red) and the family of $\nu =2$ IQH system (blue). These paths are given by Eq. (6) with $p=1$ and 2, respectively. B and F in the parentheses stand for bosons and fermions, respectively. The red line contains the lattice analog of the Halperin $lll$ state [28] at $\nu =1/l$ with $l=1,2,3$ (systems at $\nu <0$ in this family are trivially mapped to that at $\nu >0$). The blue line contains the lattice analogues of the Halperin 110 state, the Halperin 332 state, $\nu =-2$ (equivalently $\nu =2$) bosonic IQH state [58, 59], and the singlet $\nu =-2/3$ (equivalently $\nu =2/3$) FQH state [33].

Figure 2

Energy gaps as functions of $1/\nu$ for (a) $V=0$ and (b) $V=5$. We set $N_p=4$ and $N_x\times N_y=9\times 9$. We plot the lowest 15 energies within the $S_z^{\text{tot}}=0$ sector at each $1/\nu$. The number of plots looks less than 15 because of the topological degeneracy. The ground-state degeneracy for each $1/\nu$ is shown in Fig. 3. Since its number increases as $1/\nu$ increases, only the first excited energy is plotted in the region $1<1/\nu$. The vertical dashed lines represent fermionic systems.

Figure 3

(a) Ground-state degeneracy $N_D$ divided by the spin degeneracy $2S_{\text{tot}}+1$ ($=5$ in this case) as a function of $1/\nu$. (b) Many-body Chern number. (c) Ground-state degeneracy $N_D$ divided by the denominator of $\theta /\pi$. (d,e) Spectral flows at (d) $1/\nu =3$ of fermions and (e) $1/\nu =0$ of bosons. In the all panels, the same setting as Fig. 2 is used.

Figure 4

Energy gaps as functions of $1/\nu$ for (a) $V=0$ and (b) $V=5$. We set $N_p=4$ and $N_x\times N_y=8\times 8$. We plot the lowest 25 energies within the $S_z^{\text{tot}}=0$ sector at each $1/\nu$. The number of plots looks less than 25 because of the topological degeneracy. The ground-state degeneracy for each $1/\nu$ is shown in Fig. 6. The vertical dashed lines represents fermionic systems.

Figure 5

Energy gaps as functions of $1/\nu$ for $V=5$. We set $N_p=6$ and $N_x\times N_y=6\times 5$. We plot the lowest 15 energies within the $S_z^{\text{tot}}=0$ sector at each at each $1/\nu$. The vertical dashed lines represent fermionic systems.

Figure 6

(a) Ground-state degeneracy $N_D$ divided by the spin degeneracy $2S_{\text{tot}}+1$ ($=1$ in this case) as a function of $1/\nu$. (b) Many-body Chern number. (c) Ground-state degeneracy $N_D$ divided by the denominator of $\theta /\pi$. (d,e) Spectral flows at (d) $1/\nu =5/2$ of fermions and (e) $1/\nu =0$ with $\theta /\pi =3/2$. In the all panels, the same setting as Fig. 4 is used.