Coulomb Instabilities of a Three-Dimensional Higher-Order Topological Insulator

Topological insulators (TIs) are an exciting discovery because of their robustness against disorder and interactions. Recently, second-order TIs have been attracting increasing attention, because they host topologically protected 1D hinge states in 3D or 0D corner states in 2D. A significantly critical issue is whether the second-order TIs also survive interactions, but it is still unexplored. We study the effects of weak Coulomb interactions on a 3D second-order TI, with the help of renormalization-group calculations. We find that the 3D second-order TIs are always unstable, suffering from two types of topological phase transitions. One is from second-order TI to TI, the other is to normal insulator. The first type is accompanied by emergent time-reversal and inversion symmetries and has a dynamical critical exponent $\kappa =1$. The second type does not have the emergent symmetries but has nonuniversal dynamical critical exponents $\kappa <1$. Our results may inspire more inspections on the stability of higher-order topological states of matter and related novel quantum criticalities.

Figure 1

Schematic of a typical 3D second-order TI (left) that hosts 3D gapped bulk states in the interior but topologically protected 2D massive (gapped) Dirac cones on the surfaces and gapless 1D chiral hinge states. It may turn to a TI (right top) or a NI (right bottom) in the presence of the Coulomb interactions.

Figure 2

(a)–(c) The renormalized $D$, $m$, and $\alpha$ as functions of the running scale parameter $\ell$. $D$ and $m$ protect the second- and first-order topological properties, respectively. The vanishing $D$ and increasing $m$ mean a topological phase transition from second-order TI to TI. (d) The scale dependence of the dynamic exponent $\kappa$, whose value is obtained by fixing $v$ as scale invariant. The solutions are obtained by fixing the initial values $m_0=B_{\perp }^0=1$, $B_z^0=0.5$, ${\alpha }_0=0.1={\gamma }_0^2=0.1$ while varying $D_0$. (a)–(d) share the same legends. In (b), the $m(\ell )$ curves of $D_0=5$, 10, and 20 are shifted vertically by 1 for clarity.

Figure 3

(a),(b),(d) The renormalized $m$, $D$, and $\alpha$ as functions of the running scale parameter $\ell$. $D$ and $m$ protect the second-order and first-order topological properties, respectively. The finite $D$ at the sign change of $m$ means a topological phase transition from 3D second-order TI to NI. (c) The scale dependence of $\kappa$. The solutions are obtained by varying the initial value of $D$ while fixing those of other parameters as $m_0=0.2$, $B_{\perp }^0=2B_z^0=2$, ${\alpha }_0=0.5$, and ${\gamma }_0^2=0.1$. (a)–(d) Legends are the same. The orange dashed lines in (a)–(c) label the values of $\ell$ at the phase transition points. (b),(c) The crossing points of the orange dashed lines with the red, black, blue lines indicate the values of $D$ and $\kappa$ when the phase transition occurs. From left to right, $D=0.43$, 0.72, 0.74 in (b) and $\kappa =0.79$, 0.81, 0.83 in (c).

Figure 4

(a),(b) The renormalized ${\mathrm{\Delta }}_M$ and ${\mathrm{\Delta }}_C$. The different curves are obtained by fixing the initial values $m_0=B_{\perp }^0=B_z^0={\gamma }_0^2=0.1$, ${\alpha }_0=0.5$ while varying the initial values of ${\mathrm{\Delta }}_M$ in (a) and ${\mathrm{\Delta }}_C$ in (b). (c)–(f) Phase diagrams in the $m–\alpha$, ${\mathrm{\Delta }}_C–\alpha$, $D–\alpha$, and $B–\alpha$ planes, respectively. SOTI and CDM stand for the second-order topological insulator and compressible diffusive metal, respectively. In (c), the black, red, and blue lines represent the boundaries for the cases without disorder, with random mass ${\mathrm{\Delta }}_M^0=0.3$ and with random chemical potential ${\mathrm{\Delta }}_C^0=0.3$, respectively. No disorder in (e) and (f). In (f), we take $B_{\perp }^0=B_z=B$. The other parameters are fixed at (c)–(e) $B_{\perp }^0=B_z^0=1$; (d)–(f) $m_0=0.1$; (c),(d),(f) $D_0=1$, 1, 0.1, respectively; ${\gamma }_0^2=0.1$ for all diagrams.