Quantum geometric contributions to the BKT transition: Beyond mean field theory

We study quantum geometric contributions to the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature $T_{\text{BKT}}$ in the presence of fluctuations beyond BCS theory. Because quantum geometric effects become progressively more important with stronger pairing attraction, a full understanding of 2D multiorbital superconductivity requires the incorporation of preformed pairs. We find it is through the effective mass of these pairs that quantum geometry enters the theory and this suggests that the quantum geometric effects are present in the nonsuperconducting pseudogap phase as well. Increasing these geometric contributions tends to raise $T_{\text{BKT}}$, which then competes with fluctuation effects that generally depress it. We argue that a way to physically quantify the magnitude of these geometric terms is in terms of the ratio of the pairing onset temperature $T^{*}$ to $T_{\text{BKT}}$. Our paper calls attention to an experimental study demonstrating how both temperatures and, thus, their ratio may be currently accessible. They can be extracted from the same voltage-current measurements, which are generally used to establish BKT physics. We use these observations to provide rough preliminary estimates of the magnitude of the geometric contributions in, for example, magic angle twisted bilayer graphene.

Figure 1

Behavior of calculated (a) $T_{\text{BKT}}$ (labeled “Present theory”) and (b) $\{n_{\text{B}}/n,M_{\text{B}}\}$, (c) decomposition of $T_{\text{BKT}}$ (“Tot”) into conventional (“Conv”) and geometric contributions (“Geom”) for topological bands, and (d) $T_{\text{BKT}}$ for a nontopological system, as a function of $U/t$, all with $\mathcal{F}=0.2$. In comparison, also plotted in (a) and (d) are $T^{*}$ and $T_{\text{BKT}}$ (“BCS MF”) calculated using the MF phase stiffness.

Figure 2

(a) Characteristic temperatures for the topological $\mathcal{F}=0.01$ superconductor, and comparison with lower bound of $T_{\text{BKT}}$ in the isolated flat band limit (“Isol. Flat. Lim”), obtained using Eqs. (10) and (12). This bound nearly coincides with the calculated $T_{\text{BKT}}$ for the range $0.4\lesssim U/t\lesssim 2$, where the system is in the BEC regime and $T_{\text{BKT}}$ is nearly completely geometric. (b) Comparison between the $T$ dependence of $n_{\text{B}}/M_{\text{B}}$ and that of BCS MF $D_s$ at $U/t=0.5$. For the sake of clarity, only geometric contributions are included.

Figure 3

(a) Calculated $T^{*}/T_{\text{BKT}}$ as a function of $U$ and (b) relative magnitude of the geometric terms plotted as ${(n_{\text{B}}/M_{\text{B}})}^{\text{geom}}$ over ${(n_{\text{B}}/M_{\text{B}})}^{\text{tot}}$ as a function of $T^{*}/T_{\text{BKT}}$, for the topological $\mathcal{F}=0.2$ and $\mathcal{F}=0.01$ cases. Arrows correspond to where $T^{*}/T_{\text{BKT}}=4$, deduced from the experiments in the inset of (b). (Inset) $V\text{−}I$ curves measured at different $T$ for magic-angle TBLG by Cao et al. [2]

Figure 4

Results from the pairing fluctuation theory for $T_{\text{BKT}}$ and $T^{*}$, for both the topological $\mathcal{F}=0.2$ band (solid blue and dark-green dash-dotted lines) and $\mathcal{F}=0.01$ bands (red and violet dashed lines). For a better comparison to the Monte-Carlo results [26], only data for $U/t$ up to about 4 are shown. (Inset) Zoomed view of $T_{\text{BKT}}$.

Figure 5

(a) The tight binding model for $H_{\text{K}}$. $\{A,B\}$ denote two different sub-lattices, resulting from a staggered $\pi$ flux. The NN hopping amplitudes are $t{e}^{i\pi /4}$ for spin $\uparrow$ along the direction depicted by the arrows. Black dashed and blue dotted lines show the second NN bond with which the associated hopping amplitudes are $t_2$ and $-t_2$, respectively. There is also a uniform hopping between the fifth NN sites, which is not shown for clarity. (b) Fermi surfaces (FS), in blue, for the band flatness ratio $\mathcal{F}=0.2$ at electron density $n=0.3$ per site. The regime bounded by the red dashed lines defines the reduced Brillouin zone (RBZ). (c) Corresponding band structure for $\mathcal{F}=0.2$. In the vertical axis, ${\epsilon }_{\mathbf{k}}=h_0\left(\mathbf{k}\right)\pm \left|\mathbf{h}\left(\mathbf{k}\right)\right|$. (d) Band structure for $\mathcal{F}=0.01$.

Figure 6

Decomposition of the BCS MF $T_{\text{BKT}}$ into the conventional (“Conv”) and geometric (“Geom”) contributions for the topological $\mathcal{F}=0.2$ band. $n=0.3$.

Figure 7

Results for the topological $\mathcal{F}=0.01$ band. (a) $\{T_{\text{BKT}},T^{*}\}$ and (b) $\{n_{\text{B}}/n,M_{\text{B}}\}$ plotted as a function of $U/t$. “Iso. Flat. Lim.” stands for the lower bound on $T_{\text{BKT}}$ calculated from the lower bound of $n_{\text{B}}/M_{\text{B}}$ in the isolated flat band limit, given by the last line of Eq. (C5). Inset in (b): zoomed view of $\{n_{\text{B}}/n,M_{\text{B}}\}$ at small $U/t$. $M_{\text{B}}$ is plotted in units of $t{a}_{\text{L}}^2$, where $a_{\text{L}}$ is the square lattice spacing. [(c) and (d)] $n_{\text{B}}/M_{\text{B}}$ and BCS MF $D_s$ plotted as a function of $T/t$ for $U/t=0.5$. (c) and (d) show the total and conventional contributions, respectively.