Type-II quadrupole topological insulators

Modern theory of electric polarization is formulated by the Berry phase, which, when quantized, leads to topological phases of matter. Such a formulation has recently been extended to higher electric multipole moments, through the discovery of the so-called quadupole topological insulator. It has been established by a classical electromagnetic theory that in a two-dimensional material the quantized properties for the quadupole topological insulator should satisfy a basic relation. Here we discover a new type of quadrupole topological insulator (dubbed type-II) that violates this relation due to the breakdown of the correspondence that a Wannier band and an edge energy spectrum close their gaps simultaneously. We find that, similarly to the previously discovered (referred to as type-I) quadrupole topological insulator, the type-II hosts topologically protected corner states carrying fractional corner charges. However, the edge polarizations only occur at a pair of boundaries in the type-II insulating phase, leading to the violation of the classical constraint. We demonstrate that such new topological phenomena can appear from quench dynamics in non-equilibrium systems, which can be experimentally observed in ultracold atomic gases. We also propose an experimental scheme with electric circuits to realize such a new topological phase of matter. The existence of the new topological insulating phase means that new multipole topological insulators with distinct properties can exist in broader contexts beyond classical constraints.

Figure 1

Schematics of edge polarizations and corner charges. (a) Edge dipole moments exist at all boundaries in type-I QTI and (b) they exist only at the boundaries perpendicular to $y$ in type-II QTIs. Corner charges $Q^{\mathrm{corner}}=\pm e/2$ (marked by different colors) appear in both phases.

Figure 2

Schematics of our model, phase diagram and topological properties. (a) Schematics of the tunneling in our tight-binding model. (b) Phase diagram with respect to a system parameter $\gamma$. The quadrupole moment evaluated for a $80\times 80$ system is plotted as a blue line. In the phase diagram, we observe the topologically trivial insulator, the type-I QTI, the type-I anomalous quadrupole topological insulator (AQTI) (anomalousness exists in the Wannier band ${\nu }_x$) and the type-II AQTI. The subsets display the Wannier spectrum ${\nu }_x$ (${\nu }_y$) in a cylinder geometry with periodic boundaries along $x$ ($y$) and open ones along $y$ ($x$) with the isolated Wannier centers highlighted by red circles. The edge polarization $(p_x^{\mathrm{edge}},p_y^{\mathrm{edge}})$ calculated using the formula (6), the Wannier-sector polarization $(p_y^{{\nu }_x},p_x^{{\nu }_y})$, the number of edge states of the Wannier Hamiltonian $N_{\nu }\equiv (N_{{\nu }_x}^0,N_{{\nu }_x}^{\pi },N_{{\nu }_y}^0,N_{{\nu }_y}^{\pi })$ are also shown. The vertical dashed lines represent the critical points where the bulk energy gap, the edge energy gap, or the Wannier spectrum gap vanish. The light red and blue regions denote the type-I AQTI(xy) (anomalousness exists in both Wannier band ${\nu }_x$ and ${\nu }_y$) and type-I AQTI(x). Richer phase diagram for the topologically trivial phase can be found in Appendix pp3. (c) The energy spectrum as a function of $\gamma$ for open boundary conditions along both $x$ and $y$ directions with zero-energy corner modes being highlighted by a red line. (d) The electron density distribution in a typical type-II AQTI phase with the zero-energy corner modes marked by the green square in (c). Here, a very small $\delta$ is imposed so that two corner states are occupied. [(e) and (f)] The edge polarization profiles for a type-I AQTI state at $\gamma =-0.1$ and a type-II AQTI state at $\gamma =0.2$, respectively.

Figure 3

The edge polarizations calculated using the formula (15) by choosing a gauge such that $W_{{\nu }_x}^{\epsilon =\pi }({\gamma }_0=-1)=W_{{\nu }_y}^{\epsilon =\pi }({\gamma }_0=-1)=0$. The edge polarization $p_x^{\text{edge}}$ and $p_y^{\text{edge}}$ in units of $e$ are plotted as the red and blue lines, respectively.

Figure 4

[(a) and (c)] The energy spectra of the Hamiltonian $H_{\text{NNN}}$. [(b) and (d)] The Berry phase of the occupied band for the corresponding Hamiltonian. In (a) and (b) $b_2=0.8$ and in (c) and (d) $b_2=1.2$. The green lines show the value of $\gamma$ where zero-energy edge modes arise, and the blue lines show the value of $\gamma$ where the Berry phase is equal to $\pi$. These lines do not coincide, implying that the correspondence between $C_1=\pi$ and the existence of zero-energy edge states breaks down.

Figure 5

(a) Phase diagram for the Hamiltonian $H_{II}$ with respect to a system parameter $\gamma$, where the quadrupole moment is plotted as a blue line. In the phase diagram, we observe the topologically trivial insulator, the type-I QTI and the type-II QTI. The subsets display the same quantities as in Fig. 2. (b) The energy spectrum as a function of $\gamma$ for open boundary conditions along all directions with zero-energy corner modes being highlighted by a red line. (c) The edge polarizations calculated based on the formula (15) by choosing a gauge such that $W_{{\nu }_x}^{\epsilon =\pi }({\gamma }_0=-0.9)=W_{{\nu }_y}^{\epsilon =\pi }({\gamma }_0=-0.9)=0$ given that the phase is topologically trivial when $\gamma =-0.9$. (d) The spatial profiles of the edge polarizations in a type-II QTI with $\gamma =-0.35$. (e) $|{\gamma }_{nc}-{\gamma }_c|$ with respect to the system size $N$ in the logarithmic scale, showing $|{\gamma }_{nc}-{\gamma }_c|\propto N^{-\delta }$ with $\delta =0.636$. ${\gamma }_c$ and ${\gamma }_{nc}$ denote the point where the edge energy gap closes and the phase transition point determined by the quandruple moment for a finite system, respectively. Here $b_2=0.8$. See Appendix pp5 for the phase diagram for $b_2=1.2$.

Figure 6

(a) The energy spectrum versus $\gamma$ for the Hamiltonian $H_{IV}$ with open boundaries in all directions. (b) The Wannier band ${\nu }_x$ and ${\nu }_y$ versus $\gamma$ under open boundary conditions along $y$ and $x$, respectively. The red lines depict the edge states in the Wannier spectrum. (c) The quadrupole moment $q_{xy}$, the edge polarizations $p_{\mu }^{\text{edge}}$ ($\mu =x,y$) computed using the formula (15) and the winding numbers $W_{{\nu }_{\mu }}^{\epsilon =\pi }$ ($\mu =x,y$) computed using the formula (13) versus $\gamma$. When calculating the edge polarizations and the winding number, we choose a gauge such that $W_{{\nu }_x}^{\epsilon =\pi }({\gamma }_0=-0.8)=W_{{\nu }_y}^{\epsilon =\pi }({\gamma }_0=-0.8)=0$ because the phase is topologically trivial when $\gamma =-0.8$. The figure demonstrates the existence of a new topological phase with nonzero $p_y^{\text{edge}}$ but without quadrupole moments and zero-energy corner modes when $-0.45<\gamma <-0.22$ (see the inset for the spatial distribution of the edge polarizations). Note that $p_x^{\text{edge}}$ jumps to 0.5 at $\gamma =-0.22$ due to the $y$-normal edge gap closing. $q_{xy}$ and $p_x^{\text{edge}}$ are hidden behind the blue line when $\gamma <-0.22$. The small discrepancy between the quadrupole moment transition point ($N=80$ for evaluation of the quadrupole moment) and the edge gap closing point is due to the finite-size effects. Here $b_2=1.2$.

Figure 7

The connection between the edge energy spectrum and Wannier spectrum. (a) Visualization of the edge potential for $V_0$ in the left column as well as $V_L$ with $M=1$ and $M=5$ in the middle and right columns, respectively. (b) The energy bands of $H_e^0$ ($H_e^L$) with $V_0\left(y\right)\left[V_L\left(y\right)\right]$ versus $k_x$ in the left column (in the middle and right columns) for $\gamma =0.34$. The edge energy spectrum encoded in $H_e^0$ has a vanishing gap, while the Wannier spectrum encoded in $H_e^L$ for a moderate or large $M$ (e.g., $M=5$) has a finite gap. (c) The same spectrum as in (b) for $\gamma =0.12$. The edge energy spectrum has a finite gap, which gradually vanishes as $M$ is increased for $V_L\left(y\right)$, leading to a gapless Wannier spectrum. (d) The energy spectrum of $H_e^0$ ($H_e^L$) with $V_0\left(x\right)$ ($V_L\left(x\right)$) versus $k_y$ in the left column (in the middle and right columns). In this scenario, both the edge energy spectrum and Wannier spectrum are gapless. In the middle and right columns for (b)–(d), the Wannier spectra are plotted as the red lines in comparison with the energy spectrum of $H_e^L$, showing that the former agrees very well with the latter for $M=5$. For (b)–(d), the middle and right columns correspond to the results for $M=1$ and $M=5$, respectively.

Figure 8

The transport of the quadrupole moment $q_{xy}$, corner charge $Q^{\mathrm{corner}+\mathrm{x},+\mathrm{y}}$ and edge polarization $p_{x,y}^{\mathrm{edge}\alpha }$ over a full cycle. (a) The cycle refers to the evolution of a system from a topologically trivial phase to the type-II AQTI and finally return. Note that the green line is hidden behind the red one. (b) The cycle refers to the evolution of a system from the type-II phase to the type-I AQTI and then return. Note that the green and black lines are hidden behind the red one. The units of all the quantities are $e$.

Figure 9

(a) Phase diagram of the type-I Hamiltonian, where the yellow region corresponds to the type-I QTI and other regions to topologically trivial insulating phases. Note that while the Berry phase of the Wannier bands along one direction in the blue regions is nonzero, there are no edge polarizations since the Wannier bands close their gaps at either ${\nu }_x=0$ or ${\nu }_y=0$. The quench dynamics are performed by suddenly tuning system parameters from ${\gamma }_x={\gamma }_y,\lambda =0$ to ${\gamma }_x=0.6,{\gamma }_y=1.2,\lambda =1$ and to ${\gamma }_x=0.6,{\gamma }_y=0.2,\lambda =1$, respectively (see the red and black arrows). (b1) Entanglement spectra, (c1) Wannier spectra, and (d1) quadrupole moments and edge polarizations as time evolves, for the quench dynamics denoted by the red arrow. [(b2)–(d2)] display the same physical quantities as (b1)–(d1), but for the quench dynamics denoted by the black arrow. In (b1) and (b2), ${\text{ES}}_x$ (${\text{ES}}_y$) refers to the entanglement spectrum in the left (bottom) subsystem and ${\text{ES}}_{xy}$ the nested entanglement spectrum (see Appendix pp7). The edge polarizations in (d1) and (d2) are calculated based on the formula (15) by choosing a gauge such that $W_{{\nu }_x}^{\epsilon =\pi }(t=0)=W_{{\nu }_y}^{\epsilon =\pi }(t=0)=0$ because the initial states are topologically trivial. In (b1)–(d1), the light blue region displays the type-II QTI (specifically, in this region, when $t<1.22$, it corresponds to the type-II AQTI, and when $t>1.22$, it corresponds to the type-II normal QTI) and the light red region displays the new topological phase with only quantized $p_x^{\text{edge}}$. In (b2)–(d2), the light green region shows the type-I AQTI(x) and the dashed blue lines label the time at which the gap of ${\text{ES}}_y$ vanishes. The region between the two dashed blue lines around $t=4$ corresponds to the type-II AQTI phase. In (d1), $q_{xy}=0$ when $t<1.1$ or $t>2.43$, and in (d2), $q_{xy}=0$ when $t<1.04$ or $1.84<t<3.94$ or $t>4.6$. In (d1), the slight discrepancy between $q_{xy}$ and $p_y^{\text{edge}}$ around $t=2.5$ is caused by finite-size effects while calculating $q_{xy}$ in real space.

Figure 10

A scheme to realize our Hamiltonian in electric circuits. (a) The electric device configuration to simulate the term $h_{\left(00\right)}$ within a unit cell. Here, the box labeled by $Z_a$ or $Z_b$ can be either a capacitor or a inductor depending on the sign of $\gamma$ with their impedances being $Z_a=-1/\left(i\gamma \right)$ and $Z_b=1/\left(i\gamma \right)$, respectively. The device in the red box denotes a negative impedance converter with current inversion (INIC), the impedance of which depends on the direction of the current. The resistance of the INIC should be taken as $R=1/\mathrm{\Delta }$. The boxes labeled by $Z_{a=1,2,3,4}^{\prime }$ represent device configurations to eliminate the unnecessary onsite terms (see Appendix pp8). (b) A suitable electric device is applied to connect two nodes: $(\mathbf{R},\alpha )$ and $(\mathbf{R}+\mathbf{d},\beta )$ for distinct unit cells. The device can be either a capacitor, or an inductor or an INIC with the impedance being $Z_{\mathbf{d},\alpha \beta }=i/h_{\left(d_x{d}_y\right)}^{\alpha \beta }$, as shown in the lower right corner figure. Here, $\mathbf{d}=d_x{\mathbf{e}}_x+d_y{\mathbf{e}}_y$.

Figure 11

The relationship between the type-II AQTI and two simplest models without and with quadruopole moments. (a) Schematics of a simplest model of a trivial insulator with zero quadrupole moments and (d) schematics of a simplest model of type-I QTI with quantized nonzero quadrupole moment. In (a) and (d), the solid and dashed red lines represent the hopping terms with positive and negative signs, respectively. The index $\alpha =1,2,3,4$ denotes the sites within a unit cell. (b) The energy spectrum (obtained under periodic boundary conditions along both $x$ and $y$ directions) and (c) the quadrupole moment as we change a system from the trivial insulating phase in a to a type-II anomalous quadrupole topological insulating phase. Here the quadrupole moment changes suddenly from zero to $e/2$ when the bulk energy gap vanishes, implying that the type-II AQTI is topologically distinct from the trivial insulator. This also distinguishes the type-II AQTI from the trivial insulator attached with a pair of SSH models. (e) The energy spectrum (obtained under open boundary conditions along both $x$ and $y$ directions) and f, the quadrupole moment as we change a system from the type-I QTI in (d) to a type-II AQTI. Here the quadrupole moment remains unchanged and both bulk and edge energy gaps remain open and the zero-energy corner states exist during the whole process. It shows that the type-II AQTI possesses the same quadrupole moment as the type-I QTI.

Figure 12

Distinct regions in topologically trivial insulators. The subsets display the Wannier spectrum ${\nu }_x$ (${\nu }_y$) for periodic boundary conditions along $x$ ($y$) and open ones along $y$ ($x$) with the isolated Wannier centers highlighted by red circles. The edge polarization $(p_x^{\text{edge}},p_y^{\text{edge}})$, the Wannier-sector polarization $(p_y^{{\nu }_x},p_x^{{\nu }_y})$, the number of edge states of the Wannier Hamiltonian $N_{\nu }\equiv (N_{{\nu }_x}^0,N_{{\nu }_x}^{\pi },N_{{\nu }_y}^0,N_{{\nu }_y}^{\pi })$ are also shown. The vertical dashed lines represent the critical points where the Wannier spectrum gap closes.

Figure 13

(a) Phase diagram for the Hamiltonian $H_{II}$ (with $b_2=1.2$ and $g_0=-0.29$) versus $\gamma$, where the quadrupole moment is plotted as a blue line. It displays the topologically trivial insulator, the type-II QTI and the new phase with nonzero edge polarizations but without quadrupole moments and zero-energy corner modes (displayed in the light red region). The subsets display the same quantities as in Fig. 2. The small discrepancy between the edge gap closing points and the quadrupole moment transition points is caused by the finite-size effects for calculating the quadrupole moment in a finite-size system with $N=240$. (b) The energy spectrum with respect to $\gamma$ for open boundary conditions along all directions with zero-energy corner modes being highlighted by a red line. (c) The edge polarizations calculated based on the formula (15) by choosing a gauge such that $W_{{\nu }_x}^{\epsilon =\pi }({\gamma }_0=-0.8)=W_{{\nu }_y}^{\epsilon =\pi }({\gamma }_0=-0.8)=0$ because it is topologically trivial when $\gamma =-0.8$. (d) The spatial distribution of the edge polarization in a type-II QTI with $\gamma =-0.4$.

Figure 14

(a1) Phase diagram for the Hamiltonian $H_{III}$ (with $b_2=1.2$ and $g_0=0.65$) versus $\gamma$, where the quadrupole moment is plotted as a blue line. The light red region shows the new topological phase with nonzero edge polarizations but without quadrupole moments and zero-energy corner modes. The subsets display the same quantities as in Fig. 2. (b1) The energy spectrum with respect to $\gamma$ for open boundary conditions along all directions with zero-energy corner modes being denoted by a red line. (c1) The edge polarizations calculated based on the formula (15) by choosing a gauge such that $W_{{\nu }_x}^{\epsilon =\pi }({\gamma }_0=-1)=W_{{\nu }_y}^{\epsilon =\pi }({\gamma }_0=-1)=0$ because it is topologically trivial when $\gamma =-1$. (d1) The spatial profiles of the edge polarization in a type-II QTI. Panels (a2)–(d2) display the same quantities as (a1)–(d1) but for the Hamiltonian $H_{III}+\alpha sin{k}_x{\sigma }_1\otimes {\sigma }_1$ (with $b_2=1.2$, $g_0=0.65$, and $\alpha =0.2$). We use $N=80$ for evaluation of the quadrupole moment.

Figure 15

[(a1) and (a2)] The energy spectrum versus $\gamma$ in a geometry with open boundaries along all directions. [(b1) and (b2)] The Wannier band ${\nu }_x$ and ${\nu }_y$ versus $\gamma$. [(c1) and (c2)] The quadrupole moment $q_{xy}$, the edge polarizations $p_{\mu }^{\text{edge}}$ ($\mu =x,y$) computed using the formula (15) and the winding numbers $W_{{\nu }_{\mu }}^{\epsilon =\pi }$ ($\mu =x,y$) computed using the formula (13) versus $\gamma$. When calculating the edge polarizations and the winding number, we choose a gauge such that $W_{{\nu }_x}^{\epsilon =\pi }({\gamma }_0=-0.8)=W_{{\nu }_y}^{\epsilon =\pi }({\gamma }_0=-0.8)=0$ because the phase is topologically trivial when $\gamma =-0.8$. $q_{xy}$ and $p_x^{\text{edge}}$ are hidden behind the blue line when $\gamma <-0.22$. The small discrepancy between the quadrupole moment transition point and the edge gap closing point is caused by the finite-size effects. Here we use $N=80$ for evaluation of the quadrupole moment. In (a1)–(c1) and (a2)–(c2), we consider the Hamiltonian $H_{IV}+\alpha sin{k}_x{\sigma }_1\otimes {\sigma }_1$ with $b_2=1.2$ and $\alpha =0.2$, and the Hamiltonian $H_{IV}+\alpha sin{k}_x{\sigma }_3\otimes {\sigma }_3$ with $b_2=1.2$ and $\alpha =0.2$, respectively.

Figure 16

The energy spectrum and Wannier spectrum during the pumping process. (a) The cycle from a topologically trivial phase to a type-II AQTI and back to the trivial phase. (b) The cycle from a type-II AQTI to a type-I AQTI and back to the type-II AQTI. For both (a) and (b) the first column corresponds to the energy spectrum with open boundary along $x$ and $y$ and the second (third) column correspond to the Wannier spectrum ${\nu }_x$ (${\nu }_y$) with open boundary along $y$ ($x$) and periodic boundary along $x$ ($y$).

Figure 17

The entanglement spectrum ${\text{ES}}_x$ in (a) and ${\text{ES}}_y$ in (b) are evaluated by partitioning a system into two subsystems $A$ and $B$ and tracing out the subsystem $B$. (c) The nested entanglement spectrum ${\text{ES}}_{xy}$ is obtained by further tracing out the subsystem $A_2$.