Quantum Geometric Contributions to the BKT Transition: Beyond Mean Field Theory


Abstract

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 multi-orbital superconductivity requires the incorporation of preformed pairs. We find it is through the effective mass of these pairs that quantum geometry enters the theory. Increasing these geometric contributions tends to raise \(T_{\text{BKT}}\)​ which then competes with fluctuation effects which generally depress it. We quantify the magnitude of the geometric terms through the ratio of the pairing onset temperature \(T^*\)​ to \(T_{\text{BKT}}\)​. Both temperatures can be extracted from the same voltage-current measurements, thereby providing an important characterization of a given superconductor.

Introduction The recent discovery of superconducting phases in twisted bilayer graphene (TBLG) at first magic angle has attracted much attention [1]–[14]. The excitement surrounding this material is driven largely by the flatness of the energy bands, which effectively enhances the importance of electron-electron interactions. This stronger interaction effect is consistent with the observed high superconducting transition temperatures [2] and has been speculated to place TBLG somewhere in the crossover between the BCS and the Bose-Einstein condensation (BEC) regimes [2], [15], [16]. Because of its two dimensionality (2D) this superconductivity is associated with a BKT instability, in which the transition temperature \(T_{\text{BKT}}\)​ is directly proportional to the superfluid phase stiffness  [17]–[19]. In a single flat band this stiffness vanishes; however, in multi-orbital band models, it was shown that the inclusion of quantum geometric effects may reinstate a finite transition temperature [20]–[24].

This physical picture of flat-band superconductivity has been established within BCS mean field (MF) theory, which is known to be problematic in 2D. Moreover, quantum geometric effects become most apparent outside the BCS regime, where non-condensed pairs, neglected in MF theory, play an important role in the phase stiffness.

In this paper we present a theory which addresses these shortcomings through studies of the interplay of preformed pairs with quantum geometric effects. We determine \(T_{\text{BKT}}\)​, in 2D superconductors using a simple two-band tight-binding model [25], [26] that captures some key ingredients in common with its TBLG counterpart, including potentially nontrivial band topology. The model has some formal similarities to a spin-orbit coupled Fermi gas Hamiltonian, where the nature of (albeit, three dimensional) pairing fluctuations within the BCS-BEC crossover is well studied [27]–[31]. Built on the BCS-Leggett ground state [32], our approach yields results for \(T_{\text{BKT}}\)​ that are consistent with the mean field literature at weak attraction, precisely where the MF theory is expected to work.

A major contribution of this paper is to establish the important competition: bosonic excitations lead to a decrease in the effective phase stiffness, whereas, geometric effects generally cause an increase. These latter become more appreciable as the bands become flatter. As a result, \(T_{\text{BKT}}\)​ remains substantial, even though it is reduced by beyond mean-field fluctuations. An important finding is that geometric contributions appear through the inverse pair mass, \(1/M_{\text{B}}\)​ which necessarily depends on the fermionic excitation gap. Because \(M_{\text{B}}\)​ enters the excitation spectrum of the pairs, the effect of geometry must be present in a host of general characteristics beyond the superfluid stiffness including transport and thermodynamics[33], persisting even into the pseudogap phase. Here “pseudogap phase" refers to the non-superconducting state with preformed pairs at \(T_{\text{BKT}} < T < T^*\)​. We reserve the term”normal state" for a non-interacting system without pairing.

To physically understand the relation between the pair mass and geometry, note that an increased magnitude of the quantum metric reflects an increased spatial extent of the normal state Wannier orbitals [34], [35]. This increase leads to larger pairs, which have a bigger overlap, leading to higher pair mobility (smaller \(M_{\text{B}}\)​). Nontrivial normal state band topology enhances these effects, which become most apparent in the so-called "isolated flat band limit" [21], where the conventional contributions to the pair mobility are negligible. In analogy with earlier findings  [20], [21] we demonstrate that a nontrivial band topology provides a lower bound for \(1/M_\text{B}\)​ in this limit.

Finally, it is important to determine the size of the geometric contributions using experimentally accessible quantities. We find that the ratio of the pairing onset temperature, \(T^*\)​, and \(T_{\text{BKT}}\)​ allows quantification of the geometric contributions and characterization of a given 2D superconductor more generally. We demonstrate how both temperatures can be determined from the same voltage-current measurements [36].

Model Our tight-binding model [25], [26] is defined on a square lattice, which splits into two sublattices, \(\{A,B\}\)​, due to a staggered \(\pi\)​ magnetic flux. The flux is opposite for opposite spins with preserved time reversal symmetry. This symmetry and the absence of spin-orbit coupling reduces the four band pairing problem, including sublattices and spin, to a two-band system with sub-lattices only and we henceforth drop the spin. Note that we only consider zero center-of-mass momentum and spin singlet pairing.

As a result we have a simple normal state Hamiltonian  [25] in \(\mathbf{k}\)​ space, \[\begin{equation}\begin{gathered} H_{\text{N}}(\mathbf{k}) = h_0(\mathbf{k}) +\mathbf{h}(\mathbf{k}) \cdot \mathbf{s} -\mu_\text{F}, \end{gathered}\label{eq:-32HK}\end{equation}\] written in the basis \((c_{A}^\dagger (\mathbf{k}), \; c^\dagger_{B}(\mathbf{k}) )\)​. Here \(\mathbf{s} =(s_x, s_y,s_z)\)​ are Pauli matrices defined for the sublattice space, \(h_0 = -2 t_5 [ \cos 2(k_x+ k_y) +\cos 2(k_x -k_y)]\)​, \(h_z = - 2 t_2 [ \cos(k_x + k_y) - \cos(k_x-k_y) ]\)​, \(h_x + i \, h_y = -2 t [e^{ i (- \phi - k_y)} \cos k_y + e^{i (\phi -k_y)} \cos k_x]\)​, with \(\phi= \pi/4\)​, and \(\mu_\text{F}\)​ is the fermionic chemical potential. We set the lattice constant \(a_\text{L}^{}=1\)​. Diagonalizing \(H_{\text{N}}(\mathbf{k})\)​ gives two energy bands, \(\xi_{\pm}(\mathbf{k}) = h_0 (\mathbf{k}) \pm |\mathbf{h}(\mathbf{k}) | -\mu_\text{F}\)​, with a nonzero Chern number \(C = \mp 1\)​.

For definiteness, following Ref. [25] we consider two sets of hopping parameters: (1) \((t, t_2, t_5) = (1, 1/\sqrt{2}, (1-\sqrt{2}))/4\)​ and (2) \((t, t_2, t_5)=(1, 1/\sqrt{2}, 0)\)​, corresponding, respectively, to a lower band width \(W\approx 0.035 t\)​ and \(0.83t\)​, and to a band flatness (ratio) \(\mathcal{F} \equiv W/E_g \approx 0.01\)​ and 0.2. Both sets have a band gap \(E_g= 4 t\)​. Throughout the paper we consider electron density \(n=0.3\)​ per square lattice site so that the lower band is only partially filled. Theory Our approach is based on a finite temperature formalism built on the BCS ground state, which can readily be extended to include stronger pairing correlations [32]. This approach has been used to address pairing and pseudogap phenomena in Fermi gases and the cuprates [37]–[39] as well as the effects of spin-orbit coupling on ultracold Fermi gases [27]–[31], and most recently to address the two dimensional BKT transition [16], [40] in several simple cases. In 2D, the natural energy scale parameter, \(n_{\text{B}}/M_{\text{B}}\)​, enters to describe \(T_{\text{BKT}}\)​, where \(n_{\text{B}}\)​ is the areal density of the preformed pairs [16].

To determine \(n_\text{B}\)​ and \(M_\text{B}\)​ we begin with the pair susceptibility \(\chi(Q)\)​. We presume that \(\chi(Q)\)​ assumes a special form (involving one dressed and one bare Green’s function) such that the \(Q=0\)​ pole of the many body T-matrix \(t_\text{pg}\)[37],

\[\begin{equation}t_{ \text{pg} }(Q) = \frac{-U}{1 - U \chi(Q)}\,, \label{eq:-32tpgdef}\end{equation}\] yields the usual BCS gap equation for the pairing gap \(\Delta_{\text{pg}}\)​ in the fermionic excitation energy spectrum, \(E_{\pm}(\mathbf{k}) =\sqrt{\xi_\pm(\mathbf{k})^2 + \Delta_{\text{pg}}^2 }\)​. This \(\Delta_{\text{pg}}\)​ is to be distinguished from the superconducting order parameter \(\Delta_{\text{sc}}\)​, which vanishes at any finite \(T\)​ in 2D. Here \(U>0\)​ is the strength of a local attractive Hubbard interaction. \(Q\equiv (i\Omega_m,\mathbf{q})\)​ with \(\Omega_m= 2 m \pi T\)​ the bosonic Matsubara frequency. Expressions for \(\chi(Q), t_{\text{pg}}(Q)\)​, and details of the following derivations can be found in the Supplemental Materials [41].

Within "the pseudogap approximation" [27], [28], [31], \(t_{ \text{pg} }(Q)\)​ is sharply peaked near \(Q=0\)​, close to an instability, so that [37] \[\begin{equation}\begin{gathered} \Delta_{ \text{pg} }^2 \equiv - T \sum_{Q \ne 0} t_{ \text{pg} }(Q). \end{gathered}\label{eq:-32DeltaPg}\end{equation}\]

Following Refs. [16], [37], [40], for small \(Q\)​, we Taylor-expand \(t_\text{pg}^{-1}(Q) = {\mathcal{Z}}^{-1}(i \Omega_m -\mathbf{q}^2/(2 M_{\text{B}}) + \mu_{\text{B}})\)​, where

\[\begin{equation}\frac{ \mu_{\text{B}}}{ {\mathcal{Z}}}= -\frac{1}{U}+\chi(0) = -\frac{1}{U}+ \sum_{\mathbf{k} \in \text{RBZ} } \sum_{ \alpha =\pm} \frac{\tanh ( \beta E_\alpha/2) }{2 E_\alpha}. \label{eq:gap}\end{equation}\]

For brevity, we have suppressed the \(\mathbf{k}\)​ dependence on the r.h.s. "RBZ" stands for reduced Brillouin zone. \(\mu_{\text{B}}\)​ is the bosonic pair chemical potential. When \(\mu_{\text{B}}\)​ is zero Eq. \(\ref{eq:gap}\) can be recognized as the BCS gap equation, but for the present purposes we must include non-vanishing \(\mu_{\text{B}}\)​. Note that \(t_{\text{pg}}(Q)\)​ can be roughly viewed as a propagator for the preformed pairs with an energy \(E_\text{B}=\mathbf{q}^2/2M_{\text{B}} -\mu_\text{B}\)​. Both expressions for \({\mathcal{Z}}\)​ and \(1/M_\text{B}\)​ are obtained as functions of \(\{ \Delta_{\text{pg}}, \mu_{\text{F}} \}\)​ from the Taylor expansion.

In 2D, with a simple parabolic pair dispersion, Eq. \(\ref{eq:-32DeltaPg}\) yields [16], [40] \[\begin{gather} n_{\text{B}}\equiv \sum_{\mathbf{q}} f_{\text{B}}(E_{\text{B}}) = \mathcal{Z}^{-1}\Delta_{ \text{pg} }^2 & = -\frac{M_{\text{B}}}{2\pi \beta} \ln( 1- e^{\beta \mu_{\text{B}}^{}}), \label{eq:-32nBeqn}\end{gather}\]

where \(\beta=1/T\)​, and \(f_{\text{B}}(x)=1/(e^{\beta x} -1)\)​. Then we have

\[\begin{equation}n_\text{B} /M_{\text{B}} = \Delta_{\text{pg}}^2 /(M_\text{B} {\mathcal{Z}}) =2 \; \Delta_{\text{pg}}^2 \big( T_{\text{conv}} + T_{\text{geom}} \big), \label{eq:-32DBdef}\end{equation}\] where we have split the contributions to the inverse pair mass into two terms: \(T_{\text{conv}}\)​ is the conventional contribution that only depends on the normal state dispersion while \(T_{\text{geom}}\)​ is the geometric contribution that carries information about the normal state wavefunction. Here we present an expression for \(T_{\text{geom}}\)​ (leaving \(T_{\text{conv}}\)​ to the SM), \[\begin{gather} & T_{\text{geom} } = \sum_{\mathbf{k} \in \text{RBZ} } \sum_{ \{ \alpha, \alpha^\prime, \eta \} =\pm} \frac{ 1}{4} \bigg[1+ \eta \frac{\xi_\alpha}{ E_\alpha} \bigg] \times \nonumber \\ & \frac{n_\text{F}(\eta E_\alpha) - n_\text{F}(-\xi_{ \alpha^\prime })}{\eta E_\alpha + \xi_{\alpha^\prime}} (- \alpha \alpha^\prime) \frac{1}{4} \sum_{\mu=x,y} \partial_\mu \hat{h}\cdot \partial_\mu \hat{h}, \label{eq:-32Tgeom} \end{gather}\]

where \(n_\text{F}(x)= 1/(e^{\beta x} +1)\)​ is the Fermi-Dirac distribution, and \(\hat{h}(\mathbf{k}) \equiv \mathbf{h}(\mathbf{k})/| \mathbf{h}(\mathbf{k})|\)​. Interestingly, we see that \(T_{\text{geom} }\)​ contains both intra- and inter-band terms.

Quantum geometry enters into \(T_{\text{geom}}\)​, or equivalently \(n_\text{B}/M_\text{B}\)​, through the diagonal components of the quantum metric tensor, \(g_{\mu\nu} (\mathbf{k})\)​: \[\begin{gather} g_{\mu\nu} (\mathbf{k}) & = \frac{1}{2} \partial_\mu \hat{h}(\mathbf{k}) \cdot \partial_\nu \hat{h}(\mathbf{k}), \label{eq:-32gmunu}\end{gather}\] where \(\{\mu,\nu\}=\{x,y\}\)​. \(g_{\mu\nu}\)​ is a measure of the distance between two Bloch states in the projective normal state Hilbert space [42]. In the BEC regime, where \(n_\text{B}=n/2\)​, \(g_{\mu\nu}\)​ is directly connected to the inverse pair mass \(1/M_\text{B}\)​. We stress that in contrast to other work [43], [44] here \(1/M_\text{B}\)​ depends on the self consistently determined pairing gap.

Finally, the electrons are subject to the number constraint [16], [37], [40],

\[\begin{equation} n = \sum_{\mathbf{k} \in \text{RBZ}} \sum_{\alpha =\pm} \big[ 1- \frac{\xi_{\alpha}}{E_\alpha} \tanh ( \frac{\beta E_\alpha}{2}) \big] .\label{eq:num}\end{equation}\]

Equations \(\ref{eq:gap}\), \(\ref{eq:-32nBeqn}\), and \(\ref{eq:num}\) form a closed set that can be solved for \(\Delta_{\text{pg}}\)​ and \(\mu_\text{F}\)​, for given (\(T\)​, \(n\)​, \(U\)​), which also detemines the important ratio \(n_\text{B}/M_\text{B}\)​.

BKT criterion It was initially proposed in Ref. [45] based on experiments in Fermi gases, that the 2D BKT superconducting transition can be re-interpreted as a "quasi-condensation" of preformed Cooper pairs. The onset of quasi-condensation provides a normal state access to the BKT instability. Here the transition is approached from above, which is complementary to the superfluid phase stiffness based approach (from below). The quasi-condensation onset is quantified through the parameter \(n_\text{B} /M_{\text{B}}\)​ which provides a natural 2D energy scale. More specifically, this approach to the BKT transition builds on a Monte-Carlo study of weakly interacting bosons [46] where it was found that at the onset of quasi-condensation, i. e. \(T=T_{\text{BKT}}\)​, one has:

\[\begin{equation}\frac{n_\text{B}(T)}{ M_\text{B}(T)} =\frac{\mathcal{D}_\text{B}^\text{crit} }{2\pi} T. \label{eq:-32BKT}\end{equation}\] Here \(\mathcal{D}_{\text{B}}^{\text{crit}}\)​ is the critical value of the phase space density, \(\mathcal{D}_\text{B} (T) \equiv n_\text{B} \lambda_\text{B}^2\)​ with \(\lambda_\text{B} = \sqrt{2\pi/M_\text{B}T}\)​ the bosonic thermal de-Broglie wavelength (setting \(\hbar=k_{\text{B}}=1\)​). This BKT criterion has been supported by experimental studies on atomic Bose gases [47]–[49].

In general \(\mathcal{D}_{\text{B}}^{\text{crit}}\)​ depends on the non-universal boson-boson interaction strength \(g_\text{B}\)​. In the most general case, \(g_\text{B}\)​ is unknown for a fermionic superconductor where Cooper pairs are the emergent composite bosons. However, a small value of \(g_{\text{B}}\)​ appears consistent with the BCS ground state, as the bosonic degrees of freedom enter this wavefunction in a quasi-ideal manner. Notably the dependence of \(\mathcal{D}_{\text{B}}^{\text{crit}}\)​ on \(g_\text{B}\)​ is logarithmic and therefore weak [46]. Estimates for \(\mathcal{D}_{\text{B}}^{\text{crit}}\)​ for fermionic superfluids range from 4.9 to 6.45 [45], [50]. We choose \(\mathcal{D}_{\text{B}}^{\text{crit}} =4.9\)​ that best fits the data on Fermi gases [50].

Isolated flat band limit It is useful to arrive at some analytical insights on how \(n_\text{B}/M_\text{B}\)​ depends on the normal state band topology. This can be done in the isolated flat band limit, corresponding to \(W\ll U \ll E_g\)​ (which is also a BEC regime). In this limit, superconductivity is restricted to the lower flat band while the upper band is inactive, and Eq. \(\ref{eq:-32DBdef}\) simplifies to \[\begin{gather} \frac{n_\text{B}}{M_\text{B}} & \approx \Delta_{\text{pg}}^2 \sum_{\mathbf{k} \in \text{RBZ} } \frac{ \tanh(\beta E_-(\mathbf{k}) /2 ) }{2 E_{-}(\mathbf{k})} \frac{1}{2} \sum_{\mu=x,y} g_{\mu\mu}(\mathbf{k}). \label{eq:-32isoflatband}\end{gather}\] Using an inequality between the quantum metric tensor and the normal state band Berry curvature, one obtains \[\begin{gather} \frac{n_\text{B}}{M_\text{B}} & \ge \Delta_{\text{pg}}^2 \frac{\tanh(\beta E_- /2 )}{4 E_{-} } \frac{|C|}{ \pi}, \label{eq:-32bound}\end{gather}\] which sets a lower bound for \(n_{\text{B}}/M_{\text{B}}\)​ when \(C\ne 0\)​, i.e. when the system is topologically nontrivial. Here \(E_-\)​ is \(\mathbf{k}\)​ independent and \(C=1\)​ is the normal state conduction band Chern number. Interestingly, this lower bound is almost identical to the one derived for the MF superfluid phase stiffness in Ref. [21], provided one replaces \(\Delta_{\text{pg}}\)​ with the MF superconducting order parameter.

Numerical Results In Fig. ¿fig:fig?:-32Fig1(a) we compare the calculated \(T_{\text{BKT}}\)​ from our pairing fluctuation theory with that using the BCS MF superfluid phase stiffness \(D_{s}\)​ for \(\mathcal{F}=0.2\)​. Also plotted is the pairing onset temperature, \(T^*\)​, approximated by the mean field transition temperature. In the weak-coupling BCS limit, all three temperatures converge. However, in the strong coupling regime, pairing flucuations become important and our \(T_{\text{BKT}}\)​ is significantly reduced relative to its MF counterpart, as a consequence of an additional bosonic excitation channel. Unlike the single band theory, where there is a more dramatic \(T_{\text{BKT}}\)​ downturn near \(U/t\approx 3\)​ (see below) [16], in this multi-orbital model the geometric contribution prevents the expected strong decrease.

These features can be traced to the behavior of the pair mass, \(M_{\text{B}}\)​, which is plotted along with \(n_{\text{B}}\)​ in Fig. ¿fig:fig?:-32Fig1(b). In single band theories with conventional contributions only, due to a large suppression of pair hopping [51] and an increase of pair-pair repulsion with pair density [52], pairs tend to be localized near \(U/t \approx 3\)​, corresponding to \(M_{\text{B}} \rightarrow \infty\)​. The presence of geometric terms prevents this pair mass divergence. Figures ¿fig:fig?:-32Fig1(a) and (b) reveal that, while the small \(U\)​ behavior of \(T_{\text{BKT}}\)​ derives from variations in both \(M_{\text{B}}\)​ and \(n_{\text{B}}\)​, the behavior of \(T_{\text{BKT}}\)​ in the BEC regime reflects that of \(1/M_{\text{B}}\)​ only.

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 non-topological system, as a function of \(U/t\), all with \(\mathcal{F} = 0.2\). In comparison, also plotted in (a,d) are \(T^*\) and \(T_{\text{BKT}}\) (”BCS MF") calculated using the MF phase stiffness.

To see the importance of the geometric contributions more clearly, in Fig. ¿fig:fig?:-32Fig1(c) we present a decomposition of \(T_{\text{BKT}}\)​ in terms of the conventional and geometric components, by separating the total \(n_\text{B}/M_\text{B}\)​ into two terms, \((n_\text{B}/M_\text{B})^{\text{conv}} \equiv 2 \Delta_{\text{pg}}^2 T_{\text{conv}}\)​ and \((n_\text{B}/M_\text{B})^{\text{geom}} \equiv 2 \Delta_{\text{pg}}^2 T_{\text{geom}}\)​. We then apply the BKT criterion in Eq. ??? BKT to each of \(\{n_\text{B}/M_\text{B} , (n_\text{B}/M_\text{B})^{\text{conv}} , (n_\text{B}/M_\text{B})^{\text{geom}} \}\)​ to arrive at the three curves in Fig. ¿fig:fig?:-32Fig1(c). Here we see that \(T_{\text{BKT}}\)​ is almost completely geometric at \(U/t \gtrsim 3\)​. The conventional contribution in Fig. ¿fig:fig?:-32Fig1(c) exhibits a dome-like dependence on \(U\)​ with a maximum at \(U \sim W\)​. Its contribution to \(T_{\text{BKT}}\)​ in the pairing fluctuation theory falls precipitously to almost zero at \(U/t \approx 3\)​ and remains extremely small at larger \(U\)​, resulting from a cancellation between pair hopping and inter-pair repulsion effects [41].

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. \(\ref{eq:-32BKT}\) and \(\ref{eq:-32bound}\). 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.

It is instructive to compare with a non-topological superconductor, as shown in Fig. ¿fig:fig?:-32Fig1(d). Our non-topological bands are constructed by adding a staggered on-site potential to the topologically nontrivial Hamiltonian \(H_{\text{N}}\)​ in Eq. \(\ref{eq:-32HK}\). For a meaningful comparison the trivial band structure is so chosen that both its conduction band width \(W\)​ and band gap \(E_g\)​ are comparable to the nontrivial \(\mathcal{F}=0.2\)​ case. This insures that the conventional contributions to \(T_{\text{BKT}}\)​, as well as the \(U\)​ dependence of \(\Delta_\text{pg}\)​ and \(\mu_\text{F}\)​, are more or less the same in both cases. Comparison of \(T_{\text{BKT}}\)​ in Fig. ??? Fig1(d) and Fig. ¿fig:fig?:-32Fig1(a) at \(U/t \gtrsim 4\)​, where the geometric component dominates, demonstrates that the geometric contribution to \(T_{\text{BKT}}\)​ is significantly enhanced in the non-trivial case.

In Fig. ¿fig:fig?:-32Fig2(a) we present a comparison between the MF and present theory for a nearly flat conduction band, with \(\mathcal{F} \approx 0.01\)​. Just as in Fig. ¿fig:fig?:-32Fig1(a), pairing fluctuations suppress significantly the transition temperature relative to the mean field result. Also important is the absence of the conventional \(T_{\text{BKT}}\)​ peak, seen in Fig. ??? Fig1(a). There is a small residual feature at \(U \sim W =0.035 t\)​ from the conventional term, which, however, is invisible in the plot. In this nearly flat band limit, \(T_{\text{BKT}}\)​ is essentially purely geometric for the entire range of \(U/t\)​ displayed. Notably, even a very small attraction (\(U/t \approx 0.3\)​) puts the system in the BEC regime, where \(n_\text{B}/n\)​ reaches \(1/2\)​ [41].

Also plotted in Fig. ¿fig:fig?:-32Fig2(a) are the pairing onset temperature \(T^*\)​ (dot-dashed) along with the lower bound of \(T_{\text{BKT}}\)​ in the isolated flat band limit (black dotted line), which is obtained by applying the BKT criterion in Eq. \(\ref{eq:-32BKT}\) to the r.h.s. of Eq. \(\ref{eq:-32bound}\). Interestingly this bound is almost saturated by our calculated \(T_{\text{BKT}}\)​ when \(0.4\lesssim U/t \lesssim 2\)[41].

Even with the reduction of \(T_{\text{BKT}}\)​ relative to the BCS MF result, in the isolated flat band limit, \(n_\text{B}/M_\text{B}\)​ is essentially equal to its BCS MF counterpart \(D_s\)​ at \(T=T_{\text{BKT}}\)​ and even for higher temperatures, provided \(T \ll T^*\)​. This can be seen through the comparison in Fig. ¿fig:fig?:-32Fig2(b) between our \(n_\text{B}/M_\text{B}\)​ in Eq. \(\ref{eq:-32isoflatband}\) and that of the MF \(D_s\)​, where for clarity we have dropped the small but nonzero conventional term.

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: \(V-I\) curves measured at different \(T\) for magic-angle TBLG by Cao et. al. [2].

We turn finally to the physical implications of our calculations for a given 2D superconductor. We quantify the relative size of the geometric terms by use of the dimensionless ratio \(T^*/T_{\text{BKT}}\)​ which is measurable in voltage current (\(V -I\)​) experiments [36] with consistency checks from STM data. As shown in Figure ¿fig:fig?:-32Fig3(a), \(T^*/T_{\text{BKT}}\)​ increases monotonically with interaction strength \(U\)​ for both the topological \(\mathcal{F}=0.2\)​ and \(\mathcal{F}=0.01\)​ cases, with an even more rapid increase as the system approaches the BEC regime. The fractional contribution of the geometric terms, \((n_\text{B}/M_\text{B})^{\text{geom}}/ (n_\text{B}/M_\text{B})^{\text{tot}}\)​, is plotted in Fig. ¿fig:fig?:-32Fig3(b). Once in the BEC regime, \(T_{\text{BKT}}\)​ is dominantly geometric.

To connect to experiments on TBLG, we present the experimental \(V-I\)​ curves for an optimal example [2], in the inset of ¿fig:fig?:-32Fig3(b). At \(T=T_{\text{BKT}}\)​ the \(V-I\)​ curve follows a power law, \(V\propto R_{\text{N}} I_c (I/I_c)^\alpha\)​ with \(\alpha=3\)​; \(I_c\)​ is the critical current and \(R_{\text{N}}\)​ is the normal state resistance [53]–[58]. Importantly, when \(T\)​ reaches \(T^*\)​ the \(V-I\)​ curve fully recovers its normal state Ohmic behavior, \(V \propto R_{\text{N}} I\)​.

From the \(V-I\)​ characteristics by Cao et al. [2], we estimate \(T^*\approx 4 \text{K}\)​ and \(T_{\text{BKT}} \approx 1 \text{K}\)​,  [2], which yield \(T^*/T_{\text{BKT}} =4\)​. At this ratio, the normalized geometric contribution is about \(70\%\)​ and 50% for \(\mathcal{F}=0.01\)​ and 0.2, respectively, in Fig. ¿fig:fig?:-32Fig3(b). This suggests that the system is in the intermediate BCS-BEC crossover regime, and has not yet passed into the BEC regime. However, we note that the \(T^*/T_{\text{BKT}}\)​ ratio inferred from the \(V-I\)​ measurements is somewhat variable in different experiments  [59]–[61] [41]. Further experiments are needed to firmly settle where magic angle TBLG is in the BCS-BEC spectrum. In future data, it would be useful to have a more continuous variation of the temperature scales to establish the Ohmic recovery point, \(T^*\)​, more accurately.

In summary, we have established the quantum geometric contribution to superfluidity in a pair-fluctuation theory, where these contributions modify the pair mass. In general the quantum geometric contribution plays a dominant role in the strong coupling BEC regime. It restricts pairs from becoming infinitely heavy in a perfectly flat band, making them more mobile. We further show how to quantify the magnitude of the geometric contributions in a multi-orbital 2D superconductor in terms of the \(T^*/T_{\text{BKT}}\)​ ratio. Our analysis was based on important experimental observations [36] which have shown that the two temperature scales characterizing a 2D superconductor (\(T_{\text{BKT}}\)​ and \(T^*\)​) can be extracted from \(V-I\)​ plots. We have ended this paper by presenting speculations on magic angle TBLG, concerning the size of the geometric terms and the location of this exotic superconductor within the BCS-BEC crossover. Despite our oversimplified band structure, our identification of these temperatures and their (measurable) ratio sets up a template which should be broadly useful in future to both theoretical and experimental communities.

Acknowledgments We thank M. Levin for a useful discussion. This work was primarily funded by the University of Chicago Materials Research Science and Engineering Center, and the National Science Foundation under Grant No. DMR-1420709. Q.C. was supported by NSF of China (Grant No. 11774309).

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