Many-body order parameters for multipoles in solids (2024)

Authorization Required

×

×

• 

  Figure 1

  Charge configuration in real space with multipole moments.

  Reuse & Permissions

• 

  Figure 2

  Interpretation of partition functions with the insertion of ${\widehat{U}}_a$. ${\widehat{U}}_a$ acts at $\tau =0$ followed by the imaginary time evolution along $\tau$. Periodic boundary condition is used along the $\tau$ direction.

  Reuse & Permissions

• 

  Figure 3

  (a)Corner charges of quadrupolar insulator derived from the effective action. Charges are localized only at four corners of the rectangular region $\mathcal{M}$, which has a nonzero quadrupole moment $Q_{xy}\ne 0$. Note that the vacuum has a trivial quadrupole moment $Q_{xy}=0$. (b)Charge current configuration derived from the effective action under the Thouless pumping. In this case charges flow only along the boundary of $\mathcal{M}$, and their directions are denoted as arrows.

  Reuse & Permissions

• 

  Figure 4

  (a)The phase diagram of the quadrupole insulator (21) when $\delta =0$ and ${\lambda }_x={\lambda }_y=1$. The shaded region denotes topologically nontrivial phase (quadrupole moment equals 0.5), and black dots denote bulk energy gap closing points. (b)and (c)Evaluation of $Q_{xy}$ (2) for the quadrupole insulator (21). We set ${\lambda }_x={\lambda }_y=1$ and $\delta =0$, and evaluate $Q_{xy}$ (b)along the cut ${\gamma }_x={\gamma }_y$ and (c)along the cut ${\gamma }_x=0.5$. In both cases there is a sharp change at ${\gamma }_y=1$, which is consistent with the phase diagram (a). (d)Anomalous topological quadrupole insulator [19]. Here the ground state is known to be topological if $\frac{V_z}{\sqrt{{\mathrm{\Delta }}^2+{\mu }^{\prime }{}^2}}>1$ and trivial if $\frac{V_z}{\sqrt{{\mathrm{\Delta }}^2+{\mu }^{\prime }{}^2}}<1$, and this is well captured by ${\widehat{U}}_2$. See the text for details on the model and the numerical values of the parameters.

  Reuse & Permissions

• 

  Figure 5

  (a)The complex phase of $\langle {\widehat{U}}_3\rangle$ for topological octupole insulator (23). We set ${\lambda }_x={\lambda }_y={\lambda }_z=1.0$ and tune $\gamma \equiv {\gamma }_x={\gamma }_y={\gamma }_z\in [0,2]$. When $\gamma <1.0$ ($\gamma >1.0$), the ground state is in a topological (trivial) octupole insulator phase, which is indeed captured by $\langle {\widehat{U}}_3\rangle$ up to a finite-size effect. (b)Edge-localized polarization model (24) with $(\gamma ,{\lambda }_x,{\lambda }_y)=(0.1,1.0,0.5)$ and change $\delta \in [0,0.6]$. The corner charge comes solely from the boundary localized polarization, and the bulk quadrupole moment vanishes. We indeed see that the complex phase of $\langle {\widehat{U}}_2\rangle$ is trivial up to finite-size effects.

  Reuse & Permissions

• 

  Figure 6

  Evaluation of the phase of $\langle {\widehat{V}}_{a=1,2}\left(l\right)\rangle$ of Eq.(19) and Eq.(20) for various values of $l$ in the case of (a)the Su-Schrieffer-Heeger chain (25) and (b)the quadrupole insulator (21). We set $\gamma =1.0$ (${\gamma }_x={\gamma }_y=1.0$) and $\delta =0.2$ and change $\lambda$ ($={\lambda }_x={\lambda }_y$).

  Reuse & Permissions

• 

  Figure 7

  Comparison between three different physical quantities, complex phase of $\langle {\widehat{U}}_2\rangle$, nested Wilson quadrupole moment $Q_{xy}^{\omega }$, and corner charge $q_{\text{c}}$ for (a)isotropic Thouless pumping (27) and (b)anisotropic Thouless pumping (28). While the complex phase of $\langle {\widehat{U}}_2\rangle$ and the corner charge $q_{\text{c}}$ agree with each other with almost no discernible differences, $Q_{xy}^{\omega }$ given in terms of the Wannier-sector polarizations agrees with the two only at $\theta =\pi /2,3\pi /2$, i.e., $\delta =0$ and the quadrupole moment is quantized.

  Reuse & Permissions

• 

  Figure 8

  (a)Tight-binding model of topological quadrupole insulator with $\pi /2$-flux per unit cell. The tight-binding hopping parameters are shown with arrows, which specify the direction of the hopping. (b)Comparison between several different physical quantities, the many-body order parameter $Q_{xy}$ using $\langle {\widehat{U}}_2\rangle$ with respect to the periodic and open boundary condition, the physical quadrupole moment via the bulk-boundary correspondence (26) $p_x^{\mathrm{edge}}+p_y^{\mathrm{edge}}-q_c$, and corner charge $q_{\text{c}}$ during the Thouless pumping (29). We see a remarkable agreement between the many-body order parameter $Q_{xy}$ using ${\widehat{U}}_2$ under a periodic boundary condition and the physical quadrupole moment. The many-body order parameter $Q_{xy}$ using ${\widehat{U}}_2$ under an open boundary condition agrees with the corner charge $q_{\text{c}}$ up to a finite-size error. (c)The system size dependence of the many-body order parameter $Q_{xy}$ using ${\widehat{U}}_2$ under an open boundary condition at $\theta =\pi /4$ in Eq.(29) and the convergence to the corner charge in the thermodynamic limit.

  Reuse & Permissions

• 

  Figure 9

  Scaling of many-body order parameters $|\langle {\widehat{U}}_{n=1,2}\rangle |$ versus the linear system size $L$ in the case of (a)SSH chain (C3) with $(\gamma ,\lambda ,\delta )=(1.0,2.0,0.1)$ and (b)topological quadrupole insulator (C5) with $({\gamma }_x,{\gamma }_y,{\lambda }_x,{\lambda }_y,\delta )=(1.0,1.0,2.0,2.0,0.1)$. (a)The modulus of $\langle {\widehat{U}}_1\rangle$ converges to 1 as $L\to \infty$, as expected in the case of a gapped insulator. (b)The modulus of $\langle {\widehat{U}}_2\rangle$ decays exponentially in $L$ even in the case of a gapped insulator.

  Reuse & Permissions

• 

  Figure 10

  Comparison between the Wannier gap and the modulus of $|\langle {\widehat{U}}_2\rangle |$ for a topological quadrupole insulator (C4). In panels (a)and (b)we fix $({\gamma }_x,{\gamma }_y,{\lambda }_y,{\lambda }_x,{\lambda }_y)=(0.75,1.0,1.0,1.0)$ and tune $\delta \in [-0.2,0.2]$. When $\delta =0$ (a)the Wannier gap closes and (b)$|\langle {\widehat{U}}_2\rangle |$ is the smallest. In panels (c)and (d)we fix $({\gamma }_x,{\lambda }_x,{\lambda }_y,\delta )=(0.75,1.0,1.0,0.0)$ and tune ${\gamma }_y\in [0.75,1.25]$. (c)The Wannier gap associated with the Wannier band ${\nu }_x\left(k_y\right)$ closes at ${\gamma }_y=1.0$ and (d)$|\langle {\widehat{U}}_2\rangle |$ is the smallest around ${\gamma }_y=1.0$. In all cases the Wannier gap is well defined in the thermodynamic limit while $|\langle {\widehat{U}}_2\rangle |$ vanishes in the thermodynamic limit.

  Reuse & Permissions

• 

  Figure 11

  Many-body order parameter based on ${\widehat{V}}_2\left(l\right)$ applied to Eq.(C4) for various $l$ with $({\gamma }_x,{\lambda }_x={\lambda }_y,\delta )=(0.5,1.0,0.0)$ while changing ${\gamma }_y$, which is the same as the cut used in Fig.4. The curve is identical for $l=19,15$, and 11, showing that even when $l$ is an order of $L/2$, the many-body order parameter based on ${\widehat{V}}_2\left(l\right)$ correctly reproduces the phase diagram along the cut.

  Reuse & Permissions

• 

  Figure 12

  The complex phase of $\langle {\widehat{V}}_1(l,d)\rangle$ and $\langle {\widehat{V}}_2(l,d)\rangle$ for various $l$ as a function of $d$. Panels (a)and (c)correspond to the SSH chain (C3) where we set $(\gamma ,\lambda )=(1.0,2.0)$ and (c)$\delta =0$ and (d)$\delta =0.1$. Panels (b)and (d)correspond to the topological quadrupole insulator (C5) where we set $({\gamma }_x,{\gamma }_y,{\lambda }_x,{\lambda }_y)=(1.0,1.0,2.0,2.0)$ and (c)$\delta =0$ and (d)$\delta =0.1$. While the ground state of panels (a)and (b)are topologically nontrivial, for some $l$ and $d$, the complex phase of $\langle {\widehat{V}}_n(l,d)\rangle$ becomes trivial. In panels (c)and (d)as a function of $d$, we see “lumps.”

  Reuse & Permissions

• 

  Figure 13

  (a)$\frac{1}{2\pi }\mathrm{Im}log\langle {\widehat{U}}_2\rangle$ as a function of hopping parameter $t$ in Eq.(E1) with system sizes $L=16$ and $L=17$. For $L=16$, we take into account additional $(-1)$ factor coming from background charge distribution. Note that both $L=16$ and $L=17$ show dependency in $t$, while the fractional corner charge is independent of $t\in [0,0.25]$, hence indicating an unstable nature of the $C_4$ insulator with respect to ${\widehat{U}}_2$. (b)The Wannier band ${\nu }_x\left(k_y\right)$ of the ground state of Eq.(E1) at $3/8$ filling. We have one doubly degenerate and one nondegenerate flat bands, so in total zero net polarization.

  Reuse & Permissions

• 

  Figure 14

  (a)$\frac{1}{2\pi }\mathrm{Im}log\langle {\widehat{U}}_2\rangle$ as a function of onsite $C_4$-breaking parameter $\delta$ in Eq.(E5) with $t=0.3$. Because of finite $\delta$, the corner charge changes smoothly from a quantized value 0.5. Note that there is a mismatch between the corner charge and $\frac{1}{2\pi }\mathrm{Im}log\langle {\widehat{U}}_2\rangle$, showing an unstable nature of the $C_4$-symmetric insulator with respect to ${\widehat{U}}_2$. (b)The Wannier band ${\nu }_x\left(k_y\right)$ of the ground state of Eq.(E5) with $t=0.3$ and $\delta =0.3$ and at $1/2$ filling. The Wannier band becomes gapless at $k_y=0,\pi$.

  Reuse & Permissions

×