Interplay of topology and interactions in quantum Hall topological insulators: U(1) symmetry, tunable Luttinger liquid, and interaction-induced phase transitions

We consider a class of quantum Hall topological insulators: topologically nontrivial states with zero Chern number at finite magnetic field, in which the counterpropagating edge states are protected by a symmetry (spatial or spin) other than time-reversal. HgTe-type heterostructures and graphene are among the relevant systems. We study the effect of electron interactions on the topological properties of the system. We particularly focus on the vicinity of the topological phase transition, marked by the crossing of two Landau levels, where the system is a strongly interacting quantum Hall ferromagnet. We analyze the edge properties using the formalism of the nonlinear $\sigma$-model. We establish the symmetry requirement for the topological protection in this interacting system: effective continuous U(1) symmetry with respect to uniaxial isospin rotations must be preserved. If U(1) symmetry is preserved, the topologically nontrivial phase persists; its edge is a helical Luttinger liquid with highly tunable effective interactions. We obtain explicit analytical expressions for the parameters of the Luttinger liquid in the quantum-Hall-ferromagnet regime. However, U(1) symmetry may be broken, either spontaneously or by U(1)-asymmetric interactions. In either case, interaction-induced transitions occur to the respective topologically trivial phases with gapped edge charge excitations.

Figure 1

One possible type of the Landau level (LL) structure of a quantum Hall topological insulator (QHTI). At the single-particle topological phase transition point, LLs of different symmetries, labeled $a$ (red) and $b$ (blue), cross. The spectrum shown was calculated for the Bernevig-Hughes-Zhang model [2, 18] with a perpendicular orientation of the magnetic field; in this case, LLs $a$ are $b$ are distinguished by the spatial inversion (or, equivalently, reflection) parity, even and odd.

Figure 2

(a) Half-infinite sample occupying the region $x\le 0$. (b) and (c) Schematics of the edge spectrum of a QHTI in the Landau gauge, including only the two intersecting LLs $a$ and $b$ of interest, see Fig. 1. The single-particle states are labeled by the conserved 1D momentum $p$ in the $y$ direction. The states located in the bulk and at the edge correspond to the values $p\lesssim 0$ and $p\gtrsim 0$, respectively. In the topologically nontrivial (TnT) phase at lower fields $B<B^{*}$ (b), the edge states cross and are gapless. In the topologically trivial (TT) phase at higher fields $B^{*}<B$ (c), the edge states do not cross and are gapped.

Figure 3

Quantum Hall ferromagnet (QHFM) state realized at the crossing of LLs at half-filling. For each momentum $p$, one electron occupies the state $|\mathbf{n}\rangle$ with given isospin $\mathbf{n}$, see Eqs. (1.1), (1.2), and (1.3) and Fig. 4.

Figure 4

Bloch sphere of the isospin $\mathbf{n}$ in the 2D space of $a$ and $b$ LLs [Eqs. (1.1), (1.2), and (1.3)]. The north $|+{\mathbf{n}}_z\rangle =|a\rangle$ and south $|-{\mathbf{n}}_z\rangle =|b\rangle$ poles correspond to the occupation of the $a$ and $b$ LLs, respectively. Any other state is a coherent mixture of the $|a\rangle$ and $|b\rangle$ states.

Figure 5

Bulk phase diagram for the $\text{U}\left(1\right)$-symmetric $\sigma$ model obtained by minimizing the energy function $\mathcal{E}\left(n_z\right)$ [Eq. (3.6)]. The plot shows the dependence $n_z^{\infty }\left({\overline{h}}_z\right)$ of the optimal isospin projection on the normalized field ${\overline{h}}_z=h_z/u$; the color code depicts the weight of the $a$ (red) and $b$ (blue) LL states. The Bloch spheres depict the corresponding isospin orders (4.2), (4.3), and (4.4).

Figure 6

The angle functions ${\theta }_0\left(x\right)={\theta }_0(\overline{x},{\overline{h}}_z)$ [Eqs. (5.11), (5.12), and (5.13)] for the ground-state solution (5.5) for a system with an edge. Three cases according to the three phases [Eqs. (4.2), (4.3), and (4.4)] are shown. At $x\to -\infty$, the solutions approach the asymptotic values ${\theta }^{\infty }\left({\overline{h}}_z\right)$ for the bulk ground states. At $x=0$, the solutions satisfy the boundary condition (5.8) imposed by the edge. The functions are color-coded according to the $n_z=cos{\theta }_0\left(x\right)$ projection, depicting the weight of the $a$ (red) and $b$ (blue) LL states.

Figure 7

The ground-state geodesic paths ${\mathbf{n}}_0\left(x\right|{\phi }_0)$ [Eq. (5.5)] on the isospin Bloch sphere for the ${\mathbf{n}}^{\infty }={\mathbf{n}}_z$ (left) and ${\mathbf{n}}^{\infty }={\mathbf{n}}_0^{*}\left({\phi }_0\right)$ (right) phases, realized at $u<h_z$ and $-h_z<u<h_z$, respectively.

Figure 8

The function $F\left({\overline{h}}_z\right)$ [Eq. (5.22)], which determines the dependence of the ground-state domain-wall energy $d{E}_0^{\text{1D}}(u,h_z)$ [Eq. (5.21)] on the normalized field ${\overline{h}}_z=h_z/u$ in the ${\mathbf{n}}^{\infty }={\mathbf{n}}_z$ phase at $u<h_z$.

Figure 9

The unit-charge isospin configuration ${\mathbf{n}}_0^1\left(\mathbf{r}\right)$ in the TnT ${\mathbf{n}}^{\infty }={\mathbf{n}}_z$ phase with preserved U(1) symmetry at $u<h_z$, obtained numerically for ${\overline{h}}_z=1.02$; the numerical solution accurately agrees with the exact analytical expressions (6.7), (6.9), and (6.11). The arrows depict the 2D components $(n_x\left(\mathbf{r}\right),n_y\left(\mathbf{r}\right))$, while the red-blue color code represents the value of the $n_z\left(\mathbf{r}\right)$ component, in accord with Figs. 5 and 6. The paths on the Bloch sphere for varying $-\infty <x\le 0$ and several constant values of $y$ are shown in Fig. 7(left).

Figure 10

The isospin unit-charge configuration ${\mathbf{n}}_0^1\left(\mathbf{r}\right)$ in the TT ${\mathbf{n}}^{\infty }={\mathbf{n}}^{*}\left({\phi }_0\right)$ phase with spontaneously broken U(1) symmetry at $-u<h_z<u$, obtained numerically for ${\overline{h}}_z=0.5$. (top) The arrows depict the 2D components $(n_x\left(\mathbf{r}\right),n_y\left(\mathbf{r}\right))$, while the red-blue color code represents the value of the $n_z\left(\mathbf{r}\right)$ component. (bottom) The paths on the Bloch sphere ${\mathbf{n}}_0^1(x,y)$ for varying $-\infty <x\le 0$ and several constant values of $y$.

Figure 11

(Top) The dependence of the gap ${\overline{\mathrm{\Delta }}}^1\left({\overline{h}}_z\right)$ of the edge charge excitations ${\mathbf{n}}_0^1\left(\mathbf{r}\right)$ on the normalized field ${\overline{h}}_z$, calculated numerically for various sample sizes. The black solid lines in the $1<{\overline{h}}_z$ region are the exact gap dependencies (6.10) for a finite-size system. The black horizontal solid line in the $-1<{\overline{h}}_z$ region is the analytic value (6.16) of the gap, given by the energy of a free skyrmion. The black dashed line in the $-1<{\overline{h}}_z<1$ region is the asymptotic gap dependence (6.21) with the fitted parameter $\text{ln}C=4.27$. (Bottom) Zoomed-in region around the phase transition point ${\overline{h}}_z=1$.

Figure 12

The angle functions ${\theta }_0^1(x,y=0)$ and ${\phi }_0^1(x=-0,y)$ of the unit-charge edge configurations ${\mathbf{n}}_0^1\left(\mathbf{r}\right)$ in the intermediate phase ${\mathbf{n}}^{\infty }={\mathbf{n}}^{*}\left({\phi }_0\right)$ at $-u<h_z<u$, obtained numerically for $L_x=L_y=10\sqrt{2}{l}_u$ sample for various values of $h_z$. The size of the configuration decreases with decreasing $h_z$.

Figure 13

The function ${\partial }_{\theta }^2\mathcal{E}\left({\theta }_0\left(x\right)\right)$ [Eq. (7.8)], which serves as an effective potential energy in the operator ${\widehat{U}}_x$ [Eq. (7.7)], which describes the quadratic fluctuations $\delta \theta$ [Eq. (7.1)] about the ground-state configuration ${\theta }_0\left(x\right)$ [Eq. (5.11)] in the ${\mathbf{n}}^{\infty }={\mathbf{n}}_z$ phase at $u<h_z$. The operator ${\widehat{U}}_x$ has only a continuous spectrum, marked by the shaded region, that starts at the bulk asymptotic value ${\partial }_{\theta }^2\mathcal{E}\left({\theta }_0(x=-\infty )\right)=h_z-u$, equal to the gap of the bulk isospin wave excitations, and there are no discrete bound states.

Figure 14

The functions $F_t\left({\overline{h}}_z\right)$ [Eqs. (7.19) and (7.25)] and $F_y\left({\overline{h}}_z\right)$ [Eqs. (7.20) and (7.26)] that determine the coefficients of the time- and coordinate-derivative terms in the Luttinger liquid model (7.18) for the edge excitations of the TnT ${\mathbf{n}}^{\infty }={\mathbf{n}}_z$ phase. The asymptotic functions (7.27) and (7.28) at the transition point ${\overline{h}}_z=1$ and at large ${\overline{h}}_z\gg 1$ are indicated.

Figure 15

The velocity $v\left({\overline{h}}_z\right)$ [Eq. (7.34)] and interaction parameter $\mathcal{K}\left({\overline{h}}_z\right)$ [Eq. (7.35)] of the Luttinger liquid model (7.18) for the edge excitations of the TnT ${\mathbf{n}}^{\infty }={\mathbf{n}}_z$ phase. The asymptotic functions (7.36) and (7.37) at the transition point ${\overline{h}}_z=1$ and at large ${\overline{h}}_z\gg 1$ are indicated.

Figure 16

Schematic plots of the magnetic-field dependencies of (a) the velocity $v\left(B\right)$ and (b) interaction parameters $\mathcal{K}\left(B\right)$ of the helical Luttinger liquid describing the edge excitations of the TnT phase of a QHTI for U(1)-symmetric interactions with easy-plane anisotropy $u>0$. At small fields $B\ll B^{*}$, the effective interactions are weak, $\mathcal{K}\left(B\right)\approx 1$, and the velocity $v\left(B\right)\approx v_0\left(B\right)$ is close to its bare value. In the QHFM regime close to the single-particle topological phase transition, the effective interactions are strong, $\mathcal{K}\left(B\right)\ll 1$. The effective interactions are thus highly tunable. (c) Summarized properties of the edge excitations. The TT phase with gapped edge excitations ensues at lower magnetic fields upon switching on the interactions and lowering their symmetry.