Interacting stochastic topology and Mott transition from light response

We develop a stochastic description of the topological properties in an interacting Chern insulator. We confirm the Mott transition's first-order nature in the interacting Haldane model on the honeycomb geometry from a mean-field variational approach supported by density matrix renormalization group results and Ginzburg-Landau arguments. From the Bloch sphere, we make predictions for circular dichroism of light related to the quantum Hall conductivity on the lattice and in the presence of interactions. This analysis shows that the topological number can be measured from the light response at the Dirac points. Electron-electron interactions can also produce a substantial number of particle-hole pairs above the band gap, which leads us to propose a stochastic Chern number as an interacting measure of the topology. The stochastic Chern number can describe disordered situations with a fluctuating staggered potential, and we build an analogy between interaction-induced particle-hole pairs and temperature effects. Our stochastic approach is physically intuitive, easy to implement, and leads the way to further studies of interaction effects.

Figure 1

(a) Honeycomb lattice in real space with sublattices A and B, lattice vectors, and (next-)nearest neighbor displacements. (b) Honeycomb lattice in momentum space, marking the Brillouin zone, reciprocal lattice vectors, and high symmetry points.

Figure 2

(a) Band structure (blue pair of bands) for small $t_2<0.2$ with $t_1=1$. The low-energy physics is centered around the Dirac points, where the stochastic approach applies. If we allow for fluctuations at $V>0$, the sampling of ${\varphi }^z$ corresponds to the creation of quasistates that change the band gap at the K and K′ points. (b) Haldane band structure for large $t_2>0.2$ such that the low-energy regime is located at the M points.

Figure 3

(a) $V-t_2$ mean-field phase diagram from the method. The transition marks the condensation of the CDW order parameter ${\varphi }^z$. (b) Same phase diagram obtained with iDMRG. (c) Absolute value of the self-consistent ${\varphi }^r$ variable as a function of $V$ for $t_2=0.1$. (d) Hall conductivity and $\langle n_A-n_B\rangle$ from iDMRG.

Figure 4

CDW order parameter from mean-field theory (${\varphi }^z$, top) and iDMRG ($L_y=6, \chi =200$, bottom) as a function of the interaction strength $V/t$ for different values of the next-nearest-neighbor hopping amplitude $t_2$. In both cases, the smaller $t_2$ the smaller the jump in the order parameter (in agreement with known results in the literature for $t_2=0$ [43]). In the mean-field diagram (left), we computed a solution to the self-consistent equations in small incremental steps of $\mathrm{\Delta }V=0.0005$ to show clearly the jump in the order parameter for the values of $t_2$ under investigation.

Figure 5

(a) Energy of the CI and CDW Mott phases obtained from mean-field theory at $t_2=0.2$. The curves cut in one point, forcing the CDW order parameter to jump as the system abruptly prefers to change the phase to minimize energy. (b) Energy landscape around the mean-field solution at $t_2=0.2$ as a function of the CDW order parameter ${\varphi }^z$ at the phase transition. The coexistence of local minima indicates a first-order transition according to Ginzburg-Landau theory.

Figure 6

Energy of the mean-field ground state as function of $V$ (red line). Also shown is the energy distribution of quasistates obtained from sampling $({\varphi }^x,{\varphi }^y,{\varphi }^z)=\mathit{\varphi }$ around the saddle point. This creates quasiexcited states at energies higher than the mean-field ground state. Each quasistate can be attributed a Chern number $C\left(\mathit{\varphi }\right)$ which will be either one (green) or zero (blue).

Figure 7

(a)–(c) Ground-state depletion rate ${\mathrm{\Gamma }}_{\pm }\equiv {\sum }_{\mathit{k}\in \mathrm{BZ}}{\mathrm{\Gamma }}_{l\to u}^{\pm }({\omega }_{\mathit{k}},\mathit{k})$ as a function of frequency for $t_2=0.1$ and different fixed values of the interaction strength $V$. (d) Stochastic frequency-integrated depletion rate ${\mathrm{\Gamma }}_{\pm }\equiv \frac{1}{2}{\rho }^{-1}\int d\omega {\sum }_{\mathit{k}=\mathit{K},{\mathit{K}}^{\prime }}{\mathrm{\Gamma }}_{l\to u}^{\pm }({\omega }_{\mathit{k}},\mathit{k})$ as a function of $V$.

Figure 8

(a) Evaluation of the ground-state Chern number $C_{\mathrm{gs}}$ and stochastic Chern number $C_{\mathrm{st}}$ as a function of $V$. The grey curve comes from the analytical formula in Eq. (6.8). (b) $C_{\mathrm{th}}$ from Eq. (6.11) at $V=0$ as a function of $k_{B}T$.

Figure 9

iDMRG convergence. In (a), the figure shows the charge-density-wave order parameter. (b) shows the Hall conductivity calculated by adiabatically threading a flux ${\mathrm{\Phi }}_y/{\mathrm{\Phi }}_0\in [0,2\pi ]$ in units of the flux quantum ${\mathrm{\Phi }}_0$ through the cylinder geometry (inset). Within the Chern insulator or CDW phase, one or zero charges are pumped, leading to ${\sigma }_{xy}=e^2/h$ or ${\sigma }_{xy}=0$, respectively. (c) shows the entanglement entropy $S$ and (d) shows the correlation length $\xi /a$ in units of the lattice constant $a$. (d) the charge-density-wave order parameter. Except for ${\sigma }_{xy}$ which converges at $\chi =100$, these quantities are calculated for bond dimensions $\chi \in [50,1200]$ and convergence is achieved for $\chi \sim 200$.

Figure 10

Phase diagrams. (a) CDW order parameter and (b) ${\sigma }_{xy}$ calculated in iDMRG for $L_y=6, \chi =200$. The phase boundary for both order parameters fall on top of each other, indicating that there is no coexistence of a finite CDW order with a nonzero Hall conductivity.

Figure 11

Order of the phase transition in iDMRG. (a) shows the correlation length at the transition ${\xi }_c$ in units of the lattice constant $a$. A second-order phase transition would show a divergent correlation length at the transition. Instead the correlation length seems to saturate with increasing $\chi$. (b) Entanglement entropy at the transition $S_c$ plotted as a function of the correlation length at the transition ${\xi }_c$ in units of lattice constant $a$. The entanglement saturates as a function of correlation length, deviating from the critical scaling law $S_c=c/6ln({\xi }_c/a)+s_0$.

Figure 12

Fine-size effects. We show the CDW order parameter (left) and correlation length in units of the lattice constant ($\xi /a$, right) calculated in iDMRG for $L_y=6$ and $L_y=12$, and $\chi =1200$. The phase boundary for the CDW order parameter does not change and the correlation length drops when increasing the system size. These indicate that large system sizes lead to qualitatively the same results.

Figure 13

Finite-temperature version of the topological index. Green: Definition from Ref. [34] as in Eq. (B1). Blue: Based on Eq. (B2). Red: Circular dichroism of light as presented in Ref. [30], with a momentum-integral over the entire Brillouin zone.