Robustness of the topological quantization of the Hall conductivity for correlated lattice electrons at finite temperatures

Electrons on a two-dimensional lattice which is exposed to a strong uniform magnetic field show intriguing physical phenomena. The spectrum of such systems exhibits a complex (multi)band structure known as Hofstadter's butterfly. For fillings at which the system is a band insulator, one observes a quantized integer-valued Hall conductivity ${\sigma }^{xy}$ corresponding to a topological invariant, the first Chern number ${\mathcal{C}}_1$. This is robust against many-body interactions as long as no changes in the gap structure occur. Strictly speaking, this stability holds only at zero temperatures $T$, while for $T>0$ correlation effects have to be taken into account. In this paper, we address this question by presenting a dynamical mean-field theory (DMFT) study of the Hubbard model in a uniform magnetic field. The inclusion of local correlations at finite temperature leads to (i) a shrinking of the integer plateaus of ${\sigma }^{xy}$ as a function of the chemical potential and (ii) eventually to a deviation from these integer values. We demonstrate that these effects can be related to a correlation-driven narrowing and filling of the band gap, respectively.

Figure 1

Division of the lattice into magnetic unit cells (for $B={\mathrm{\Phi }}_0/2$) for two different versions of the Landau gauge. The left figure corresponds to the gauge $\mathbf{A}\left(\mathbf{r}\right)=-B(y,0,0)$ and the right one to the choice $\tilde{\mathbf{A}}\left(\mathbf{r}\right)=B(0,x,0)$.

Figure 2

The dispersion relation of the Hofstadter model for $p/q=1/3$ and $p/q=1/4$.

Figure 3

Gaps of the Hofstadter's butterfly colored correspondingly to their Hall conductivity as a function $\varepsilon$ of $p/q$. Readapted from Ref. [45].

Figure 4

Hall conductivity vs chemical potential for three different values of $U$ (= 0, 1, and 3 from above to below) and $T$. The insets in the right upper corner show the electron density vs the chemical potential at $T=0.025$ for DMFT (red) and the Hartree approximation (gray, see Sec. 3b). For the two interacting cases ($U\ne 0$), the imaginary part of self-energy at half filling is also presented for $T=0.025$ in the left bottom insets.

Figure 5

Density of states at different values of $U$ for $T=0.025$. Here, we fixed the value $\mu -\text{Re}\mathrm{\Sigma }(\omega =0)$ to keep the effective Fermi level close to the middle of the first gap.

Figure 6

Quasiparticle weight $Z={(1-\partial \text{Re}\mathrm{\Sigma }/\partial \omega {|}_{\omega =0})}^{-1}$ (left $y$ axis) and inverse quasiparticle lifetime $\gamma =-\text{Im}\mathrm{\Sigma }(\omega =0)$ (right $y$ axis) as a function of chemical potential $\mu$, for $U=1$ (upper panel) and $U=3$ (lower panel), respectively. Vertical lines indicate the location of the first (band) gap. The inset shows $\gamma$ for $U=3$ inside the first band gap on a smaller scale.

Figure 7

Dependence of ${\sigma }^{xy}$ and $A(\omega =0)$ on the interaction strength $U$ at $T=0.025$.

Figure 8

The temperature dependence of the Hall conductivity ${\sigma }^{xy}$ and the spectral function $A(\omega =0)$.