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Superconductivity is boundless

Johannes Stephan Hofmann, Debanjan Chowdhury, Steven Allan Kivelson, Erez Berg

Published May 19, 2021• 0 citations
OpenAlex

Abstract

What limits the value of the superconducting transition temperature ($T_c$) is a question of great fundamental and practical importance. Various upper bounds on $T_c$ have been proposed, expressed as fractions of the Fermi temperature, $T_F$, the zero-temperature superfluid stiffness, $\rho_s(0)$, or a characteristic Debye frequency, $\omega_0$. We show that while these bounds are physically motivated and are certainly useful in many relevant situations, none of them serve as a rigorous bound on $T_c$. To demonstrate this, we provide explicit models where $T_c/T_F$ (with an appropriately defined $T_F$), $T_c/\rho_s(0)$, and $T_c/\omega_0$ are unbounded.

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Upper bound on the superfluid stiffness in disordered Bose and Fermi systems

Arun Paramekanti, Mohit Randeria, Nandini Trivedi

•Jan 1, 1997•0 citations
What limits the value of the superconducting transition temperature (TcT_cTc​) is a question of great fundamental and practical importance. Various upper bounds on TcT_cTc​ have been proposed, expressed as fractions of the Fermi temperature, TFT_FTF​, the zero-temperature superfluid stiffness, ρs(0)\rho_s(0)ρs​(0), or a characteristic Debye frequency, ω0\omega_0ω0​. We show that while these bounds are physically motivated and are certainly useful in many relevant situations, none of them serve as a rigorous bound on TcT_cTc​. To demonstrate this, we provide explicit models where Tc/TFT_c/T_FTc​/TF​ (with an appropriately defined TFT_FTF​), Tc/ρs(0)T_c/\rho_s(0)Tc​/ρs​(0), and Tc/ω0T_c/\omega_0Tc​/ω0​ are unbounded.
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 narrowband 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 ≤ 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.
I summarize recent progress on obtaining rigorous upper bounds on superconducting transition temperature [Formula: see text] in two dimensions independent of pairing mechanism or interaction strength. These results are derived by finding a general upper bound for the superfluid stiffness for a multi-band system with arbitrary interactions, with the only assumption that the external vector potential couples to the kinetic energy and not to the interactions. This bound is then combined with the universal relation between the superfluid stiffness and the Berezinskii–Kosterlitz–Thouless [Formula: see text] in 2D. For parabolic dispersion, one obtains the simple result that [Formula: see text], which has been tested in recent experiments. More generally, the bounds are expressed in terms of the optical spectral weight and lead to stringent constraints for the [Formula: see text] of low-density, strongly correlated superconductors. Results for [Formula: see text] bounds for models of flat-band superconductors, where the kinetic energy vanishes and the vector potential must couple to interactions, are briefly summarized. Upper bounds on [Formula: see text] in 3D remains an open problem, and I describe how questions of universality underlie the challenges in 3D.
Migdal-Eliashberg theory of boson-mediated superconductivity contains a $\sqrt{\ensuremath{\lambda}}$ divergence in the critical temperature Tc{T}_{c}Tc​ for strong electron-boson coupling $\ensuremath{\lambda}$. In the conventional Migdal-Eliashberg theory, the strong-coupling regime can be accessed only in the limit that ${\ensuremath{\lambda}}_{E}=\ensuremath{\lambda}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{D}/{\ensuremath{\varepsilon}}_{F}\ensuremath{\ll}1$, where ${\ensuremath{\omega}}_{D}$ is the Debye frequency and ${\ensuremath{\varepsilon}}_{F}$ is the Fermi energy. Here we go beyond this restriction in the context of the two-dimensional Yukawa-SYK (Y-SYK) model, which is solvable for arbitrary values of ${\ensuremath{\lambda}}_{E}$. We find that ${T}_{c}\ensuremath{\approx}0.18\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{D}\sqrt{\ensuremath{\lambda}}$ for large $\ensuremath{\lambda}$, provided ${\ensuremath{\lambda}}_{E}$ remains small, and crosses over to a universal value of ${T}_{c}\ensuremath{\approx}0.04\phantom{\rule{0.16em}{0ex}}{\ensuremath{\varepsilon}}_{F}$ for large ${\ensuremath{\lambda}}_{E}$. The saturation of Tc{T}_{c}Tc​ is due to a self-consistent account of the boson dynamics for large ${\ensuremath{\lambda}}_{E}$ and remains valid provided the vertex corrections are negligible. Depending on the value of $\ensuremath{\lambda}$, this self-consistent approach leads to pairing that describes multiple classes of quantum critical electronic systems. These results demonstrate how the $\sqrt{\ensuremath{\lambda}}$ growth of Tc{T}_{c}Tc​ in Migdal-Eliashberg theory saturates to a universal value independent of $\ensuremath{\lambda}$ and ${\ensuremath{\omega}}_{D}$, providing an upper bound on the critical temperature at strong electron-boson coupling.
We discuss standard and tighter upper bounds on the critical temperature TcT_cTc​ of two-dimensional superconductors and superfluids versus particle density n or filling factor ννν, under the assumption that the transition from the normal to the superconducting (superfluid) phase is governed by the Berezinskii-Kosterlitz-Thouless (BKT) mechanism of vortex-antivortex binding and a direct relation between the superfluid density tensor and TcT_cTc​ exists. The standard critical temperature upper bound Tcup1T_c^{\rm up1}Tcup1​ is obtained from the Glover-Ferrell-Tinkham sum rule for the optical conductivity, which constrains the superfluid density tensor components. However, we show that Tcup1T_c^{\rm up1}Tcup1​ is only useful in the limit of low particle/carrier density, where it may be close to the critical temperature supremum TcsupT_c^{\rm sup}Tcsup​ . For intermediate and high particle/carrier densities, Tcup1T_c^{\rm up1}Tcup1​ is far beyond TcsupT_c^{\rm sup}Tcsup​ for any given interaction strength. We demonstrate that it is imperative to consider at least the full effects of phase fluctuations of the order parameter for superconductivity (superfluidity) to establish tighter bounds over a wide range of densities. Using the renormalization group, we obtain the critical temperature supremum for phase fluctuations TcT_cTc​ and show that it is a much tighter upper bound to TcsupT_c^{\rm sup}Tcsup​ than Tcup1T_c^{\rm up1}Tcup1​ for all particle/carrier densities. We conclude by indicating that if the TcT_cTc​ is exceeded in experiments involving single band systems, then a non-BKT mechanism must be invoked.
We further analyze the recently proposed criterion, according to which a superfluid transition occurs whenever the r.m.s. displacement of a particle from its initial position in imaginary time reaches a certain fraction of the interparticle distance. The critical temperature ${T}_{\ensuremath{\lambda}}$ is expressed in terms of the single particle mobility. Within the Drude approximation for the mobility ${T}_{\ensuremath{\lambda}}$ is determined by the friction coefficient (or by the viscosity for dense liquids). To check the validity of the criterion the resulting formula is applied to nonideal Bose gas, to liquid helium and to ${}^{3}\mathrm{He}{\ensuremath{-}}^{4}\mathrm{He}$ mixtures. A reasonable qualitative agreement with available experiments is obtained in all cases. In the case of the Bose gas the present approach results also in an upper bound on the peak value of ${T}_{\ensuremath{\lambda}}{/T}_{\ensuremath{\lambda}}^{0},$ where ${T}_{\ensuremath{\lambda}}^{0}$ is the critical temperature for the ideal gas.