Spin separation in the half-filled fractional topological insulator

All topological insulators observed so far are the lattice analogs of the integer quantum Hall states with time-reversal symmetry, composed of two decoupled copies of the Chern insulator with opposite chiralities for different spins. The fractional topological insulator (FTI) has been similarly envisioned as being composed of two decoupled copies of the fractional Chern insulator (FCI), which is in turn the lattice analog of the fractional quantum Hall state (FQHS). An important question is if such a vision can be realized for the Coulomb interaction, whose strength is irrespective of spin. To address this question, we investigate the effects of the interspin correlation in the spin-holomorphic Landau levels, where electrons with one spin reside in the usual holomorphic lowest Landau level, while those with the other in the antiholomorphic counterpart. By performing exact diagonalization of the Coulomb interaction Hamiltonian in the spin-holomorphic Landau levels, here, we show that no fractionally filled states in the spin-holomorphic Landau levels can occur as two decoupled copies of the FQHS, suggesting that no FTIs can occur as those of the FCI in the lattice either. Fractionally filled states in this system are generally compressible except at half filling, where a transport gap develops with spontaneous breaking of the space rotational symmetry in the thermodynamic limit, leading to the spatial separation of different spins, i.e., spin separation. It is predicted that there is a novel bulk-edge correspondence at half filling, representing the hallmark of the half-filled spin-separated FTI.

Figure 1

Formation of the spin-holomorphic Landau levels. (a) The red and blue (mixed-color) circles denote the (coincidental) energy levels of spin up and down electrons, respectively, in the two-dimensional harmonic oscillator as a function of the $z$-component angular momentum eigenvalue $l_z$. Note that $\hslash {\omega }_0$ is subtracted from the original energy eigenvalues for convenience. For a guide to eye, the circles are threaded by the respectively colored lines. The energy levels are independent of spin in the absence of spin-orbit coupling. (b) With addition of spin-orbit coupling, the energy levels of spin up and down electrons evolve differently. Specifically, the energy levels of spin up and down electrons, threaded by the red and blue lines, rotate around each circle at $l_z=0$ to the opposite directions, as depicted by the respective arrows. In the figure, the spin-orbit coupling constant $\alpha$ is set to be ${\omega }_0/2$. Note that the sign of the spin-orbit coupling constant is not important since different signs just interchange the role of spin up and down electrons. (c) At an appropriate value of the spin-orbit coupling constant, i.e., $\alpha ={\omega }_0$, the energy levels form the effective Landau levels with opposite magnetic fields for different spins. The lowest energy levels among these effective Landau levels are called the spin-holomorphic Landau levels.

Figure 2

Electron density as a function of the residual confining potential strength $\gamma$ at half filling in the spin-holomorphic disk geometry. $\gamma ={\omega }_0-\alpha$ with ${\omega }_0$ being the natural frequency of the parabolic potential well and $\alpha$ being the spin-orbit coupling constant. (a) $\hslash \gamma =0.25$, (b) 0.125, (c) 0.105, and (d) 0.0 in units of $e^2/\epsilon l_0$, where $l_0=\sqrt{\hslash /2m_e{\omega }_0}$ is the natural length scale of the system. Scale bars in the figure denote $l_0$, providing a measure for the overall size of the electron droplet. Also, $N=N_{\uparrow }+N_{\downarrow }=16$ and $m_{\max }=15$.

Figure 3

Exact energy spectra as a function of the total angular momentum quantum number $L_{\mathrm{tot}}$ at half filling in the spin-holomorphic spherical geometry. The particle number $N$ is varied with (a) $N=10$, (b) 12, (c) 14, and (d) 16. Here, the $z$-component total angular momentum quantum number, $L_{\mathrm{tot},z}$, is set equal to zero. Similar to the usual spherical geometry, there is a $2L_{\mathrm{tot}}+1$ degeneracy for each $L_{\mathrm{tot}}$ with $L_{\mathrm{tot},z}$ ranging from $-L_{\mathrm{tot}}$ to $L_{\mathrm{tot}}$.

Figure 4

Pair correlation functions in the half-filled spin-holomorphic Landau levels. (a) Intraspin pair correlation function, $g_{\uparrow \uparrow }\left(\mathbf{r}\right)$, and (b) interspin pair correlation function, $g_{\uparrow \downarrow }\left(\mathbf{r}\right)$, in the spin-holomorphic spherical geometry for the state at $L_{\mathrm{tot}}=64$ and $L_{\mathrm{tot},z}=0$ with $N=16$. Note that the reference electron is chosen to be spin up and placed at the location indicated by the red arrow on the equator. Similar pair correlation functions, (c) $g_{\uparrow \uparrow }\left(\mathbf{r}\right)$ and (d) $g_{\uparrow \downarrow }\left(\mathbf{r}\right)$, in the spin-holomorphic disk geometry with $N=16$. Again, the spin-up reference electron is placed at the location indicated by the red arrow. Here, the residual confining potential strength is set to be $\hslash \gamma =0.105e^2/\epsilon l_0$ to ensure the uniform electron density.

Figure 5

Intraspin pair correlation functions $g_{\uparrow \uparrow }\left(\mathbf{r}\right)$ [(a)–(e)] and interspin pair correlation functions $g_{\uparrow \downarrow }\left(\mathbf{r}\right)$ [(f)–(j),] in the spin-holomorphic spherical geometry for various values of the flux shift $\mathcal{S}$ with the flux-particle relation $2Q=N+\mathcal{S}$. Note that the particle number is $N=14$ in this figure.

Figure 6

Pair correlation functions in the spin-holomorphic disk geometry under a strong confining potential. (a) Intraspin pair correlation function $g_{\uparrow \uparrow }\left(\mathbf{r}\right)$ and (b) interspin pair correlation function $g_{\uparrow \downarrow }\left(\mathbf{r}\right)$, in the spin-holomorphic disk geometry with $N=16$ and $\hslash \gamma =0.25e^2/\epsilon l_0$, which corresponds to the situation in Fig. 2.

Figure 7

Squares of overlap, ${\mathcal{O}}^2={\left|\langle {\mathrm{\Psi }}_{H\left(\lambda \right)}|{\mathrm{\Psi }}_{H\left(0\right)}\rangle \right|}^2$ and $|\langle {\mathrm{\Psi }}_{H\left(\lambda \right)}|{\mathrm{\Psi }}_{H\left(1\right)}\rangle {|}^2$, as a function of the tuning parameter $\lambda$. Here, ${\mathrm{\Psi }}_{H\left(\lambda \right)}$ represents the exact ground state of the model Hamiltonian $H\left(\lambda \right)$ at half filling in the spin-holomorphic spherical geometry. Exact diagonalization is performed for (a) $N=12$ and (b) 14 in the $L_{\mathrm{tot},z}=0$ sector.

Figure 8

Ratio between the interspin pair correlation functions at the origin and its antipode, $g_{\uparrow \downarrow }(r=2R)/g_{\uparrow \downarrow }(r=0)$ as a function of the tuning parameter $\lambda$. The insets show $g_{\uparrow \downarrow }\left(\mathbf{r}\right)$ directly plotted on the sphere at three different values of $\lambda =0.01,0.78$, and 0.8025, whose locations are indicated by the arrows. This result is obtained from exact diagonalization results obtained in the $N=16$ system.

Figure 9

Incompressibility of the half-filled fractional topological insulator in the spin-holomorphic Landau levels. (a) Transport gap as a function of inverse particle number $1/N$ in the spin-holomorphic spherical geometry. (b) Schematic diagram showing the helical edge states in the spin-holomorphic disk geometry. (c) Electron density difference between before and after adding two extra electrons with both spins in the system with $N=14$. Here, the residual confining potential strength $\hslash \gamma$ is set to be $0.105e^2/\epsilon l_0$.

Figure 10

Transport gap at $1/3$ filling in the spin-holomorphic spherical geometry as a function of inverse particle number $1/N$. Similar to half filling, $1/3$ filling is defined as ${\nu }_{\uparrow }={\nu }_{\downarrow }=1/3$ and thus ${\nu }_{\mathrm{tot}}=2/3$. The flux shift is set to be exactly the same as that of the Laughlin state, i.e., $2Q=3N/2-3$. The insets show the exact energy spectra for $N=8$, 10, and 12, whose corresponding transport gaps are indicated by the respective arrows. The transport gap is computed via $\mathrm{\Delta }=E_{N+1,Q}+E_{N-1,Q}-2E_{N,Q}$. Here, only the spin-unpolarized states are considered, i.e., $N_{\uparrow }=N_{\downarrow }=N/2$.

Figure 11

Ground-state energy as a function of the spin polarization, $\mathcal{P}=(N_{\uparrow }-N_{\downarrow })/(N_{\uparrow }+N_{\downarrow })$, at half filling in the spin-holomorphic spherical geometry. Note that $\mathcal{P}=0$ and $\pm 1$ indicate the unpolarized and fully polarized situations, respectively. The particle number $N$ is set to be (a) 14 and (b) 16.