It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature [Formula: see text] and the critical temperature [Formula: see text] is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density and high density. The goal of this short note is to extend the universal behavior to lower dimensions [Formula: see text] and give an exemplary proof in the weak coupling limit.
It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature [Formula: see text] and the critical temperature [Formula: see text] is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density and high density. The goal of this short note is to extend the universal behavior to lower dimensions [Formula: see text] and give an exemplary proof in the weak coupling limit.
We derive upper and lower bounds on the critical temperature Tc and the energy gap $\ensuremath{\Xi}$ (at zero temperature) for the BCS gap equation, describing spin-21 fermions interacting via a local two-body interaction potential $\ensuremath{\lambda}V(x)$. At weak coupling $\ensuremath{\lambda}⪡1$ and under appropriate assumptions on V(x), our bounds show that ${T}_{c}\ensuremath{\sim}A\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}B∕\ensuremath{\lambda})$ and $\ensuremath{\Xi}\ensuremath{\sim}C\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}B∕\ensuremath{\lambda})$ for some explicit coefficients A, B, and C depending on the interaction V(x) and the chemical potential $\ensuremath{\mu}$. The ratio A∕C turns out to be a universal constant, independent of both V(x) and $\ensuremath{\mu}$. Our analysis is valid for any $\ensuremath{\mu}$; for small $\ensuremath{\mu}$, or low density, our formulas reduce to well-known expressions involving the scattering length of V(x).
Abstract We study the BCS energy gap Ξ Ξ in the high–density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential V on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3, 2022) on the critical temperature Tc T c at high densities, we prove the universality of the ratio of the energy gap and the critical temperature.
Abstract We show that the energy gap for the BCS gap equation is Ξ=μ(8e−2+o(1))exp(2μaπ) Ξ = μ 8 e - 2 + o ( 1 ) exp π 2 μ a in the low density limit μ→0 μ → 0 . Together with the similar result for the critical temperature by Hainzl and Seiringer (Lett Math Phys 84: 99–107, 2008), this shows that, in the low density limit, the ratio of the energy gap and critical temperature is a universal constant independent of the interaction potential V. The results hold for a class of potentials with negative scattering length a and no bound states.
We develop a general description of the superconductivity of lattice fermions based on the BCS theory. We propose a modeling of the density of states (DOS) of lattice fermions, where divergent and semi-metallic structures are described by asymptotic expansions around the Fermi energy. This modeling leads to a unified representation of the transition temperature Tc at half filling, which reproduces asymptotic forms of Tc derived in several lattices, such as the square, honeycomb, and Lieb lattices, for the weakly interacting limit. The derived asymptotic forms of Tc are categorized into four types, which is attributed to the different responses of the degenerate fermions depending on the DOS structures. The DOS with a delta-functional singularity induces the highest Tc in the weakly interacting region, where Tc is linearly proportional to the pairing interaction U. Three kinds of universal ratios defined in the BCS theory no longer reduce to constants independent of the system parameters but can be parameterized with a certain variable that characterizes the singular structures of the DOS. We find universal relationship among thermodynamic quantities that holds for all parameter regions. Further, we numerically demonstrate that in multi-band systems the correlation effects can induce an effective delta-functional singularity. This phenomenon generally appears in multi-energy (or multi-gap) systems and may provide a plausible guideline for material designs of high-Tc superconductor.
Following an approach similar to that of Miyake or Randeria, Duan, and Shieh in two dimensions, we study a three-dimensional many-fermion gas at zero temperature interacting via some short-ranged two-body potential. To accommodate a possible singularity (e.g., the Coulomb repulsion) in the interaction, the potential is eliminated in favor of the two-body scattering t-matrix, the low-energy form of which is expressible in terms of the s-wave scattering length as. The BCS gap equation for s-wave pairing is then solved simultaneously with the number equation in order to self-consistently obtain the zero-temperature BCS gap \ensuremath{\Delta} as well as the chemical potential \ensuremath{\mu} as functions of the dimensionless coupling variable \ensuremath{\lambda}\ensuremath{\equiv}kFas, where kF is the Fermi momentum. Results are valid for arbitrary coupling strength, and in the weak coupling limit reproduce the standard BCS results. Finally, root-mean-square pair sizes are obtained as a function of \ensuremath{\lambda} and compared with experimental values.
A four-component Fermi gas in one dimension with a short-range four-body interaction is shown to exhibit a one-dimensional analog of the BCS-BEC crossover. Its low-energy physics is governed by a Tomonaga-Luttinger liquid with three spin gaps. The spin gaps are exponentially small in the weak coupling (BCS) limit where they arise from the charge-density-wave instability, and become large in the strong coupling (BEC) limit because of the formation of tightly bound tetramers. We investigate the ground-state energy, the sound velocity, and the gap spectrum in the BCS-BEC crossover and discuss exact relationships valid in our system. We also show that a one-dimensional analog of the Efimov effect occurs for five bosons while it is absent for fermions. Our work opens up a very rich field of universal few-body and many-body physics in one dimension.