Bounds on the Superconducting Transition Temperature: Applications to Twisted Bilayer Graphene and Cold Atoms

Understanding the material parameters that control the superconducting transition temperature $T_c$ is a problem of fundamental importance. In many novel superconductors phase fluctuations determine $T_c$, rather than the collapse of the pairing amplitude. We derive rigorous upper bounds on the superfluid phase stiffness for multiband systems, valid in any dimension. This in turn leads to an upper bound on $T_c$ in two dimensions, which holds irrespective of pairing mechanism, interaction strength, or order-parameter symmetry. Our bound is particularly useful for the strongly correlated regime of low-density and narrow-band systems, where mean-field theory fails. For a simple parabolic band in 2D with Fermi energy $E_F$, we find that $k_B{T}_c\le E_F/8$, an exact result that has direct implications for the 2D BCS-BEC crossover in ultracold Fermi gases. Applying our multiband bound to magic-angle twisted bilayer graphene, we find that band structure results constrain the maximum $T_c$ to be close to the experimentally observed value. Finally, we discuss the question of deriving rigorous upper bounds on $T_c$ in 3D.

Popular Summary

Below a threshold known as the transition temperature ($T_c$), the electric current in a superconducting material flows unimpeded without resistance. This leads to applications ranging from energy-efficient magnets used in MRI machines to superconducting devices used in quantum information processing and computation. At ambient pressure, all known superconductors have $T_c$ values well below room temperature and understanding the material parameters that limit $T_c$ has been a long-standing problem.

Here, we obtain upper bounds on the superconducting $T_c$ for two-dimensional (2D) systems. While these bounds are always valid, they turn out to be particularly useful in strongly interacting systems with a low density of carriers or very narrow electronic bands. For a 2D system with quadratic dispersion, we show that $T_c$ can never exceed one-eighth of the Fermi temperature, no matter how strong the pairing interaction. This prediction can be tested in existing experiments on 2D ultracold atomic gases. We also apply our results to the recent surprising discovery of superconductivity in magic-angle twisted bilayer graphene. We find that, using only the noninteracting electronic band structure for this 2D material, we can obtain bounds on its superconducting $T_c$ that are close to observed values.

We expect that our findings will pave the way for obtaining upper bounds on the superconducting $T_c$ in 3D systems as well.

Figure 1

$\tilde{D}$ as a function of doping from the charge neutrality point (CNP) in magic-angle twisted bilayer graphene (MA TBG), calculated using the band structure of Ref. [17] at $T=0$. $(2\pi e^2/{\hslash }^2)\tilde{D}$ is the integrated optical spectral weight and $\pi \tilde{D}/2$ is an upper bound on the SC $T_c$ in MA TBG.

Figure 2

$T_c$ for the 2D attractive Hubbard model at density $n=0.7$ with QMC results from Ref. [30]. The BCS mean-field $T_c^{\mathrm{MFT}}$ controls $T_c$ at weak coupling. Phase fluctuations, estimated using our upper bound $T_c^{\text{bound}}$ (see text), dominate at intermediate and strong coupling, where we also show the $t^2/|U|$ asymptotics of our bound.

Figure 3

Energy dispersion for MA TBG along high-symmetry lines in the moiré Brillouin zone (BZ) for the continuum model dispersion [15] that is accurately described by the tight-binding model of Koshino et al. [17]. The bands shown in red and blue correspond to the two valleys of the original BZ and are related by time reversal.

Figure 4

Comparison of (a) the band structure and (b) the integrated spectral weight $\tilde{D}$ for the models in Ref. [17] (in black) and Ref. [18] (in red).