Abstract
What limits the value of the superconducting transition temperature (T_{c}) is a question of great fundamental and practical importance. Various heuristic upper bounds on T_{c} have been proposed, expressed as fractions of the Fermi temperature, T_{F}, the zerotemperature superfluid stiffness, ρ_{s}(0), or a characteristic Debye frequency, ω_{0}. We show that while these bounds are physically motivated and are certainly useful in many relevant situations, none of them serve as a fundamental bound on T_{c}. To demonstrate this, we provide explicit models where T_{c}/T_{F} (with an appropriately defined T_{F}), T_{c}/ρ_{s}(0), and T_{c}/ω_{0} are unbounded.
Introduction
While superconducting transition temperatures are nonuniversal properties, and hence not generally amenable to a simple theoretical analysis, understanding what physics determines T_{c} is of selfevident importance. One approach to this challenge is to focus on a key physical process that contributes to the development of superconducting order, and to formulate an upper bound – either rigorous or heuristic – on T_{c}^{1,2,3,4,5,6,7}.
In this paper, we examine three proposed bounds on T_{c} that are expressed as a fraction of a measurable physical quantity of a given system: an appropriately defined Fermi temperature, a characteristic phonon frequency, or the zerotemperature superfluid phase stiffness. While these putative bounds are physically motivated, and provide valuable intuition in many cases of practical importance, we show by explicit counterexamples that they can be violated by an arbitrary amount. In addition to the fundamental importance of these results, we hope they suggest routes to further optimize T_{c}.
We briefly summarize our key results here:

1.
The notion of an upper bound on T_{c} in terms of an appropriately defined Fermi energy comes from the fact that, in many situations, as E_{F} → 0, the electrons have no kinetic energy. Thus, in this limit, the superfluid stiffness must seemingly go to zero. What sets T_{c} in the limit of small E_{F} is pertinent to moiré superconductors^{8,9,10,11,12,13}, where the bands can be tuned to be narrow. To make this question precise, we must define E_{F} in a strongly interacting system. We propose two such definitions of E_{F}, in terms of (i) the difference in the chemical potential between a system with a given density of electrons and a system with a vanishing density, or (ii) the energy dispersion of a single electron added to the empty system.
We show that there is no general bound on T_{c}/E_{F} by either definition, by studying two explicit models. In the first model, a flat band is separated by an energy gap from a broad band with pairhopping interaction between the two. The second model consists of a pair of perfectly flat bands with an onsite electronelectron attraction. We show explicitly that the first model violates any putative T_{c}/E_{F} bound when using the first definition of E_{F} above, while the second model violates the bound using either definition of E_{F}. Both models have been defined on a twodimensional lattice for convenience, but generalized versions of the same models in any D > 1 can be easily seen to exhibit qualitatively similar behaviors. In the context of twodimensional systems, we identify T_{c} as the Berezinskii–Kosterlitz–Thouless (BKT) transition temperature. In contrast with some earlier discussions^{14} of this topic, the topology of the flat band plays no essential role in our analysis. Specifically, in our second model, it is the nonzero spatial extent of the Wannier functions rather than any topological property that is the essential feature responsible for the nonvanishing T_{c}.

2.
In twodimensional systems where T_{c} is limited by phase fluctuations, an intuitive bound on T_{c} is given in terms of the zerotemperature phase stiffness, ρ_{s}(0). This comes from the relation^{15}\({T}_{{{{\rm{c}}}}}=\pi {\rho }_{{{{\rm{s}}}}}({T}_{{{{\rm{c}}}}}^{})/2\), and the (often physically reasonable) assumption that ρ_{s}(T) is a decreasing function of T, and hence ρ_{s}(0) ≥ ρ_{s}(T).
We construct an explicit counterexample in a twoband model of bosons (or, equivalently, tightly bound Cooper pairs), where ρ_{s}(0) can be made arbitrarily smaller than T_{c}. In this model, ρ_{s}(0) can even vanish while T_{c} > 0, implying that there is a reentrant transition into the nonsuperconducting state below T_{c}.

3.
In electronphonon superconductors, a heuristic bound on T_{c}/ω_{0} (ω_{0} being the characteristic phonon frequency) was proposed^{6,7,16}. The reasoning behind this bound is that, as the dimensionless electronphonon coupling constant λ increases past an O(1) value, the system tends to become unstable, either towards the formation of localized bipolarons or towards a charge density wave state. At the same time (and relatedly), the phonon frequency is renormalized downward as λ increases, suppressing T_{c}.
Here, we construct an explicit d − dimensional model where these strongcoupling instabilities are avoided, and T_{c} increases without bound upon increasing λ. The model includes N electronic bands interacting with N^{2} phonon modes. The model is solvable asymptotically in the largeN limit; then, the famous AllenDynes result^{17}\({T}_{{{{\rm{c}}}}}\propto {\omega }_{0}\sqrt{\lambda }\) is valid for large λ, so long as λ ≪ N, and hence T_{c}/ω_{0} is unbounded as N → ∞. Note that at the heuristic level, it is difficult to identify physical circumstances where more than a few phonons are comparably strongly coupled to the relevant electrons. Nevertheless, our analysis suggests that generically, the larger the number of phonon modes coupled to the electrons, the larger the λ at which the suppression of T_{c} onsets.
Results
Flat band superconductivity: bound on T _{c}/T _{F}?
In most conventional superconductors, T_{c} is determined by the energy scale associated with electron pairing. On the other hand, across numerous unconventional superconductors, T_{c} is more strongly sensitive to the ‘phase ordering scale’^{4}. In this context, an important recent advance is the result by Hazra, Verma, and Randeria (HVR)^{5} of a rigorous upper bound on ρ_{s}(T), the temperaturedependent superfluid phase stiffness, in terms of the integral of the optical conductivity over frequency (the optical sum rule). However, since this integral includes all the bands, this upper bound is often of the order of several electronvolts in electronic systems of interest.
At the heuristic level, this bound has been interpreted^{18} as implying a bound on T_{c}/T_{F}, where T_{F} = E_{F}/k_{B} is the Fermi energy in units of temperature. For a Galilean invariant system with a parabolic band, HVR express their bound in terms of the Fermi energy.
At the outset, it is important to define sharply what we mean by E_{F}. A particular protocol that is often adopted in experiments to estimate E_{F} is to use an effective mass, m^{*}, obtained from quantum oscillations along with an estimate of the Fermi momentum (k_{F}) from a measurement of the the carrier density n, and then defining \({E}_{{{{\rm{F}}}}}={k}_{{{{\rm{F}}}}}^{2}/(2{m}^{* })\); in two dimension, this is equivalent to determining E_{F} = πћ^{2}n/m*^{18}. This procedure is only possible when there is a nearby Fermi liquidlike state that displays quantum oscillations.
Below, we propose two different definitions of E_{F}, that we can use in settings that do not rely on any underlying assumptions (e.g., that there is a nearby Fermi liquid) and are also amenable to an experimental interpretation. We consider the case in which we add a given density, n, of electrons to an insulating reference state. We can define E_{F} as follows:

Starting from our reference state, we set the temperature to zero and consider the change in the chemical potential, μ(n, T = 0), as we fill in n electrons,
$${E}_{{{{\rm{F}}}}}^{(1)}\equiv \mu (n,T=0){\mu }_{0},$$(1)as an effective definition of E_{F}. Here \({\mu }_{0}=\mathop{\lim }\nolimits_{n\to 0}\mu (n,T=0)\). Note that the above definition of E_{F} includes all manybody corrections, which can furthermore be dependent on the density itself, and does not make any reference to any noninteracting limit; see Fig. 1a, b.

Alternatively, we can define the Fermi energy through the ‘noninteracting’ density of states, ρ(ε), for adding a single electron to the insulating system. This density of states is typically not the same as the one measured at the ‘target’ filling. Here, the Fermi energy (\({E}_{{{{\rm{F}}}}}^{(2)}\)) is defined implicitly from the expression
$$n=\int\nolimits_{{\varepsilon }_{\min }}^{{\varepsilon }_{\min }+{E}_{{{{\rm{F}}}}}^{(2)}}d\varepsilon \,\rho (\varepsilon ),$$(2)where \({\varepsilon }_{\min }\) is the energy of the ground state with one electron added on top to the insulating reference state; see Fig. 1b, c. \({E}_{{{{\rm{F}}}}}^{(2)}\) is accessible directly in e.g. STM measurements^{19}. We note that \({E}_{{{{\rm{F}}}}}^{(2)}\) is identical to \({E}_{{{{\rm{F}}}}}^{(1)}\) for noninteracting systems.
Below, we provide model Hamiltonians of interacting electrons in flat bands where the superconducting T_{c} exceeds the Fermi energy by one or both of the above definitions. Thus, these models exemplify flat band superconductivity, where T_{c} is determined entirely by the interaction scale^{20,21,22}. Analogous phenomena may also occur in semimetals^{23,24}.
Flat band superconductivity induced by a nearby dispersive band: We consider a model consisting of a nearlyflat band and a dispersive band. A closely related model^{25} has recently been studied in the context of superconductivity in twisted bilayer graphene. The singleparticle part of the Hamiltonian is given by
where \({c}_{{{{\bf{k}}}},\ell ,\sigma }^{{\dagger} }\) creates an electron with quasimomentum k in band ℓ = 1, 2 and spin polarization σ. We consider the lower band (ℓ = 1) to be a flat band with bandwidth, W_{1}, that we will ultimately take to be parametrically small (i.e. W_{1} → 0). The upper band (ℓ = 2) is separated from the flat band by an energy gap, Δ_{gap}, and has a large bandwidth, W_{2} ≫ W_{1}. The bands are topologically trivial and the Wannier functions are tightly localized on the lattice sites.
We now introduce an onsite interaction which scatters a pair of electrons between the flat band and the dispersive band:
where R labels a lattice site. Let us focus on the case where the flat band is half filled, such that the number of particles per unit cell is n = 1.
We consider the case where V ≪ Δ_{gap} ≪ W_{2}. Within meanfield theory, the superconducting transition temperature is given by (see Supplementary Methods),
where we have assumed a constant density of states per unit cell, ν_{2} = 1/W_{2}, in the dispersive band. The zerotemperature phase stiffness is given by (Supplementary Methods)
Hence ρ_{s}(0) ≫ T_{MF}, which implies that phase fluctuations are unimportant in determining T_{c}^{4}, i.e. T_{c} ≈ T_{MF}.
We now examine the Fermi energies \({E}_{{{{\rm{F}}}}}^{(1,2)}\) defined in Eqs. (1) and (2) and compare them to T_{c}. Adding a single particle to the empty system, we find that \({E}_{{{{\rm{F}}}}}^{(2)} \sim {W}_{1}\ll {T}_{{{{\rm{c}}}}}\), and hence \({T}_{{{{\rm{c}}}}}/{E}_{{{{\rm{F}}}}}^{(2)}\) can be made arbitrarily large. \({E}_{{{{\rm{F}}}}}^{(1)}\) is computed in Supplementary Methods by calculating the chemical potentials at n = 1 and n → 0. The result is \({E}_{{{{\rm{F}}}}}^{(1)}=2{T}_{{{{\rm{MF}}}}} \sim {T}_{{{{\rm{c}}}}}\). Hence, in this model, \({T}_{{{{\rm{c}}}}}/{E}_{{{{\rm{F}}}}}^{(1)}=O(1)\). An example of a different model where \({T}_{{{{\rm{c}}}}}/{E}_{{{{\rm{F}}}}}^{(1)}\) is unbounded is presented in the next section.
Flat band superconductivity induced by spatial extent of Wannier functions: We now introduce a different model for superconductivity in a narrow band. The model is defined on a twodimensional square lattice with two electronic orbitals per unit cell. The Hamiltonian is given by
Here, \({c}_{{{{\bf{k}}}}}^{{\dagger} }=({c}_{{{{\bf{k}}}},1,\uparrow }^{{\dagger} },{c}_{{{{\bf{k}}}},1,\downarrow }^{{\dagger} },{c}_{{{{\bf{k}}}},2,\uparrow }^{{\dagger} },{c}_{{{{\bf{k}}}},2,\downarrow }^{{\dagger} })\), where the operator \({c}_{{{{\bf{k}}}},\ell ,\sigma }^{{\dagger} }\) creates an electron with momentum k in orbital ℓ = 1, 2 with spin σ = ↑, ↓. We denote γ = {σ, ℓ}. The Pauli matrices τ_{x,y,z} and σ_{x,y,z} act on the orbital and spin degrees of freedom, respectively, and \(\delta {n}_{{{{\bf{r}}}},\ell }\equiv {\sum }_{\sigma }{c}_{{{{\bf{r}}}},\ell ,\sigma }^{{\dagger} }{c}_{{{{\bf{r}}}},\ell ,\sigma }^{\phantom{{\dagger} }}1\) is the number of particles at site r and orbital ℓ, relative to half filling. \(\langle {{{\bf{r}}}},{{{\bf{r}}}}^{\prime} \rangle\) denotes nearestneighbor sites. The singleparticle Hamiltonian H_{0} leads to perfectly flat bands with energies ε = ± t. The function \({\alpha }_{{{{\bf{k}}}}}\equiv \zeta (\cos {k}_{x}+\cos {k}_{y})\) controls the spatial extent of the Wannier functions in each band, tuned by the dimensionless parameter ζ. More specifically, the Wannier functions decay exponentially over a distance proportional to ζ. Note that there is no obstruction towards constructing exponentially localized Wannier functions for the above model since the bands are topologically trivial. This can be seen from the fact that the Berry curvature of the bands is identically zero, since H_{0} contains only τ_{x,y} but not τ_{z}. The strength of onsite attractive Hubbard interaction is denoted U > 0, whereas V > 0 is a nearestneighbor repulsion.
We are interested in the limit where T ≪ U, V ≪ Δ_{gap}( = 2t) and ζ ≪ 1. An extensive numerical study of the model (9) beyond this parameter regime has been analyzed in a separate publication^{26}. In this limit, we project H_{U} to the lower eigenband. The projected Hamiltonian is expanded in powers of ζ. The average density is set to quarter filling per unit cell.
For ζ = 0, the problem effectively reduces to a set of decoupled sites with a strong attractive interaction; the resulting ground state manifold is highly degenerate with local ‘Cooper pairs’ but a vanishing phase stiffness. The linear corrections in ζ vanish due to a chiral symmetry and the orbital l independent interaction strength U. At second order in ζ, the projected interaction, \(\widetilde{{H}_{U}}\), contains nearestneighbor pairhopping and densitydensity interactions:
Here, we have also introduced the pseudospin operators \({{{{\widehat{\eta }}}}}_{{{{\bf{r}}}}}^{a}=({{{\Psi }}}_{{{{\bf{r}}}}}^{{\dagger} }{\eta }^{a}{{{\Psi }}}_{{{{\bf{r}}}}}^{\phantom{{\dagger} }})/2\), where \({{{\Psi }}}_{{{{\bf{r}}}}}^{{\dagger} }=({\tilde{c}}_{{{{\bf{r}}}},\uparrow }^{{\dagger} },{\tilde{c}}_{{{{\bf{r}}}},\downarrow }^{})\), \({\tilde{c}}_{{{{\bf{r}}}},\sigma }^{{\dagger} }\) creates an electron with spin σ in a Wannierorbital localized around site r in the lower band of H_{0} (with τ_{y} = − σ for ζ = 0), and η^{a} are Pauli matrices. The J_{⊥} and J_{z} terms correspond to hopping and a nearestneighbor interaction of the Cooper pairs, respectively, and their strengths are J_{⊥} = ζ^{2}(2U + V)/8 and J_{z} = ζ^{2}U/4 − V(2 + 9ζ^{2}/8).
For V = 0, the system has an emergent SU(2) symmetry that relates the pairing and charge order parameters^{27}. This symmetry is weakly broken by terms of order (U/t)^{2}, not included in Eq. (10). For 0 < V ≪ U, the problem is equivalent to a twodimensional pseudospin ferromagnet with a weak easyplane anisotropy. Parameterizing the anisotropy by ΔJ = J_{⊥} − J_{z}, we can estimate the critical temperature of the BKT transition as^{28}
Note that in the limit ΔJ → 0 we get T_{c} → 0, as required by the MerminWagnerHohenberg theorem.
We now turn to estimating \({E}_{{{{\rm{F}}}}}^{(1,2)}\). Due to the particlehole symmetry of the effective Hamiltonian in Eq. (10), the chemical potential at n = 1 (i.e., \(\langle {{{{\widehat{\eta }}}}}_{{{{\bf{r}}}}}^{z}\rangle =0\)) is μ(n = 1) = 0. In the limit n → 0, the system consists of dilute Cooper pairs. In this limit, the interactions between the Cooper pairs can be neglected, and the chemical potential is equal to half the energy per Cooper pair: μ(n → 0) = − (J_{⊥} − J_{z}) = − ΔJ. Importantly, for J_{⊥} > J_{z}, the system does not phase separate at any density. Therefore, \({E}_{{{{\rm{F}}}}}^{(1)}=\mu (n=1)\mu (n\to 0)={{\Delta }}J\). Comparing this to Eq. (11), we find that for ΔJ ≪ J_{⊥}, \({T}_{{{{\rm{c}}}}}\gg {E}_{{{{\rm{F}}}}}^{(1)}\). We remark that the second band of the microscopic model gives rise to perturbative corrections, controlled by U/Δ_{gap}, that are of subleading order for \({E}_{{{{\rm{F}}}}}^{(1)}\), ΔJ and J_{⊥} (note that V ≠ 0); hence, the above conclusion is unchanged. Clearly, since the lower band is completely dispersionless, \({E}_{{{{\rm{F}}}}}^{(2)}=0\). We conclude that T_{c} can be made arbitrarily larger than the Fermi energy by either of the two definitions of Eqs. (1) and (2).
It is worth noting that, in the parameter regime we are considering, ρ_{s}(0) ~ J_{⊥} ~ ζ^{2}U. Hence, the delocalization of the Cooper pairs is entirely due to the interactions and the spatial overlap between the Wannier function of the active band, as in refs. ^{14,29,30,31,32,33}. The finite value of ρ_{s}(0) and the associated lower bound as derived in refs. ^{14,29,30,31} is based on the application of BCS meanfield theory. Here, however, for U ≫ V, we get ΔJ ≪ J_{⊥} and hence ρ_{s}(0) ≫ T_{c} [see Eq. (11)].
Bound on T _{c}/ρ _{s}(0)?
In this section, we turn to the question of whether the zerotemperature phase stiffness, ρ_{s}(0), sets an upper bound on T_{c} in two dimensions. ρ_{s}(0) can be extracted directly from a measurement of the London penetration depth (\({\lambda }_{{{{\rm{L}}}}}^{2}(0)\propto 1/{\rho }_{{{{\rm{s}}}}}(0)\)), or from the imaginary part of the lowfrequency optical conductivity.
As is well known, in two spatial dimensions, the phase stiffness right below T_{c} is related to T_{c} by the inequality \({\rho }_{{{{\rm{s}}}}}(T\to {T}_{{{{\rm{c}}}}}^{})\ge 2{T}_{{{{\rm{c}}}}}/\pi\). At a continuous BKT transition, \({\rho }_{{{{\rm{s}}}}}(T\to {T}_{{{{\rm{c}}}}}^{})=2{T}_{{{{\rm{c}}}}}/\pi\). However, if the transition is first order, ρ_{s} right below T_{c} can be larger than the universal BKT value. See, e.g., ref. ^{34}. However, T_{c} is not directly related to ρ_{s}(0). On physical grounds, it often makes sense to identify ρ_{s}(0) with a ‘phase ordering scale’ that sets an upper limit on T_{c}^{4}. This is justified by the fact that ρ_{s}(T) is usually a monotonically decreasing function of temperature, i.e. ρ_{s}(0) ≥ ρ_{s}(T_{c}), and therefore T_{c} can be bounded from above by ρ_{s}(0). In conventional superconductors, ρ_{s}(0) ≫ T_{c}, and T_{c} is almost entirely determined by the pairing scale. In contrast, in underdoped cuprates, ρ_{s}(0) is close to T_{c}, as illustrated by the famous Uemura plot^{35}. This suggests that in these systems, phase fluctuations play an important role in limiting T_{c}^{4}.
While this line of reasoning is likely correct in most circumstances, we will show here that—as a matter of principle—there is no bound on T_{c}/ρ_{s}(0). We outline a concrete model where ρ_{s}(0) can be made arbitrarily smaller than ρ_{s}(T_{c}) (Fig. 2).
Let us begin with a twodimensional lattice model of two species of (complex) bosons, b_{1}, b_{2},
where ε_{2}(k) is assumed to have a large bandwidth, W_{2}, and ε_{1} = ε_{2}(0) − ε_{0} forms a completely flat band at an energy ε_{0} below the bottom of the ε_{2} band, i.e. the b_{1} bosons are completely localized on individual sites. The dispersion of the two species of bosons is illustrated in Fig. 2. For the purpose of our discussion here, we can approximate ε_{2}(k) ≈ k^{2}/2m_{b} near the bottom of the broad band. \({H}_{{{{\rm{int}}}}}^{b}\) includes an onsite (Hubbard) interaction of strength U_{1,2} for the b_{1,2} bosons. We take U_{2} ≪ W_{2} whereas U_{1} → ∞, implying that the number of b_{1} bosons on each localized site can only be 0 or 1. The total average number of bosons per unit cell is chosen to be n_{b} > 1.
At temperatures near T_{c}, the chemical potential is slightly above the bottom of the broad band. Then, assuming that ε_{0} ≪ T, the average occupation of the localized sites is close to 1/2 (since the b_{1} bosons are essentially hardcore bosons at effectively ‘infinite’ temperature), so there are approximately n_{b} − 1/2 bosons per unit cell left to occupy the broad band. The critical temperature is \({T}_{{{{\rm{c}}}}} \sim \frac{{n}_{b}1/2}{2{m}_{b}}\), up to logarithmic corrections in \(\frac{{n}_{b}1/2}{2{m}_{b}}/{U}_{2}\)^{36,37,38}. Since we are in two spatial dimensions, in the absence of interactions, T_{c} = 0. The momentum distribution of particles at T ≳ T_{c} is shown schematically in Fig. 2a.
On the other hand, at T = 0, all the localized sites are filled with one boson. The density of bosons in the broad band is thus n_{b} − 1, and the superfluid stiffness is \({\rho }_{{{{\rm{s}}}}}\approx \frac{{n}_{b}1}{2{m}_{b}}\). The boson distribution function is illustrated in Fig. 2b. Clearly, the ratio T_{c}/ρ_{s}(0) can be made arbitrary large by letting n_{b} → 1^{+}. If 1/2 < n_{b} < 1, the ground state is not a superfluid, and there is a reentrant transition into a superconducting state with increasing T.
Note that in our simple model (Eq. (12)), the numbers of the two boson species are separately conserved. However, we do not expect the key results to be changed qualitatively by the addition of a weak hybridization between the two species, that breaks this separate conservation of the two boson numbers. In particular, a small hybridization generically produces a perturbative correction to T_{c} and ρ_{s}(0).
Indeed, a mild version of this sort of breakdown of the heuristic bound on T_{c}/ρ_{s}(0) has been documented experimentally in Zndoped cuprates^{39}. Here, the pristine material comes close to saturating the heuristic bound; light Zn doping suppresses T_{c} but apparently suppresses ρ_{s}(0) more rapidly, leading to a ratio that slightly exceeds the value proposed in ref. ^{4}. This was explained—likely correctly—by the authors of ref. ^{39} as being due to Zninduced inhomogeneity of the superfluid response. This explanation is spiritually close to the model discussed above: each Zn impurity destroys the superconductor in its vicinity, possibly due to pinning of local spindensitywave order^{40}. In some sense, this can be thought of as a state with localized dwave pairs near the impurities. This effect depletes the condensate at low temperature, causing a decrease in the superfluid density. However, near T_{c}, this effect weakens, as the spindensity wave order partially melts. From this perspective, it would be interesting to explore whether this violation can be made parametrically large with increasing Zn concentration  approaching the point at which T_{c} → 0.
Electronphonon superconductivity: bound on T _{c}/ω _{0}?
Recently, it has been proposed that T_{c} in an electronphonon superconductor can never exceed a certain fraction of the characteristic phonon frequency, ω_{0}^{6,7}. This putative bound implies that Migdal–Eliashberg (ME) theory^{41,42,43} must breakdown when the dimensionless electronphonon coupling λ is of order unity^{6,16}, since according to ME theory, T_{c} grows without limit with increasing λ^{17}. In general, the failure of ME theory at λ = O(1) is a result of strongcoupling effects: (i) The lattice tends to become unstable for large λ, resulting in a charge density wave (CDW) transition; (ii) electrons become tightly bound into bipolarons, whose kinetic energy is strongly quenched in the strongcoupling limit; and (iii) as λ increases, the phonon frequency renormalizes downward by an appreciable amount, Δω, suppressing T_{c}^{44}. The softening of ω_{0} due to electronphonon coupling is formally taken into account in Eliashberg theory. Often, however, the phonon spectral function is taken as given or is fit to experiment, as is done in ref. ^{17}.
These strongcoupling effects certainly play an important role in limiting T_{c} in real systems, where it is typically found never to exceed about 0.1 ω_{0} across numerous conventional superconductors^{7}. Determinant Monte Carlo simulations of the paradigmatic Holstein model reveal that ME theory indeed fails for λ = O(1), and the maximal T_{c} is significantly below ω_{0}^{6}. As we shall now show, however, this is not a rigorous bound on T_{c}.
To demonstrate this, we consider a particular large − N variant of the electronphonon problem^{45,46} with N − component electrons and N × N − component (‘matrix’) phonons, defined on a d − dimensional hypercubic lattice. The Hamiltonian is given by
Here, \({c}_{{{{\bf{r}}}},\sigma ,a}^{{\dagger} }\) creates an electron at position r with spin σ in ‘orbital’ a. The hopping parameters \({t}_{{{{\bf{r}}}}{{{\bf{r}}}}^{\prime} }\) and chemical potential μ are assumed to be identical for all orbitals. We have introduced a real, symmetric matrix of phonon displacements, \({{{{\widehat{\bf{X}}}}}}_{{{{\bf{r}}}}}\) and their canonically conjugate momenta, \({{{{\widehat{\bf{P}}}}}}_{{{{\bf{r}}}}}\), with frequency \({\omega }_{0}=\sqrt{K/M}\), assumed to be much smaller than the Fermi energy. The phonons are taken to be dispersionless for simplicity. The purely onsite electronphonon coupling constant is denoted α, with a N—dependent normalization factor that ensures that the model has a finite energy density in the N → ∞ limit. The dimensionless electronphonon coupling constant is defined as λ = α^{2}ν(0)/K, where ν(0) is the electronic density of states at the Fermi level per orbital.
We are interested in the large − N limit of the model defined in Eq. (14). Since the number of phonon degrees of freedom is much larger than the number of electron orbitals, the phonon dynamics are essentially unaffected by the coupling to the electrons, even when the electrons are strongly perturbed. This implies that the strongcoupling effects mentioned above are suppressed, even for λ ≫ 1. In particular, as we show in Supplementary Methods, there is no lattice instability or polaron formation as long as λ ≪ N, and the softening of the phonon frequency is only of the order of Δω ~ λω_{0}/N.
To zeroth order in 1/N, the equations for the electron selfenergy and the pairing vertex are exactly those given by Eliashberg theory, whereas the phonon selfenergy is of order 1/N (see Fig. 3). Thus, the selfconsistent equations for the pairing vertex are identical to those of ME theory neglecting the renormalization of the phonons, and hence the result is the same. In particular, for 1 ≪ λ ≪ N, \({T}_{{{{\rm{c}}}}}\approx 0.1827\,{\omega }_{0}\sqrt{\lambda }\)^{17}. Implicit in the fact that MigdalEliashberg theory is exact at N → ∞ is an assurance that there is no suppression of T_{c} by phase fluctuations. In the d = 2 version of our model, the BKT temperature differs from the meanfield transition temperature only by a 1/N correction. More specifically, this follows from the observation that the superfluid stiffness is \({{{\mathcal{O}}}}(N)\). Thus, T_{c} is unbounded.
The key ingredient in our model that allows us to take λ > 1 without suffering from lattice instabilities is that the different phonon modes couple to electron bilinears that do not commute with each other [see Eq. (17)]. This limits the energy gain from distorting a given set of phonon modes when forming a CDW or a polaron bound state, since the resulting perturbations to the electronic Hamiltonian cannot be diagonalized simultaneously. In contrast, the contributions of the individual phonon modes add algebraically in the total dimensionless coupling λ that enters the equation for the pairing vertex (the same dimensionless coupling also determines the resistivity in the normal state of this model^{45}).
It is worth noting that while these considerations may be of some use in searching for systems with ever higher T_{c}, as a practical matter it may be difficult to significantly violate the proposed heuristic bound. To achieve T_{c} ≈ ω_{0} requires the extremely large value of λ ≈ 25. At the same time, to avoid polaron formation requires N ≫ λ, which means the number of distinctly coupled phonon modes would have to be N^{2} ≫ λ^{2} ~ 625!
Discussion
The notion of a fundamental upper bound on T_{c} for models of interacting electrons is an attractive concept. In this paper, we have demonstrated that while there are numerous physical settings where such bounds can be formulated at a heuristic level, there exists no fundamental, universal upper bound on T_{c} in terms of the characteristic energy scales of interest to us, which include an appropriately defined T_{F}, ρ_{s}(0) and ω_{0}. We have constructed explicit counterexamples which violate these heuristic bounds by an arbitrary amount.
On the experimental front, it would be fruitful to look for candidate materials where the heuristic bounds are violated by a large amount. The fact that these bounds are usually satisfied is to be expected, since although the bounds are not rigorous, the physical reasoning behind them is quite robust. As our theoretical discussion illustrates, whenever such bounds are violated, there is an interesting underlying physical reason behind the violation; moreover, the mechanisms behind the violation of the heuristic bounds may suggest ways to optimize T_{c}. Our work provides two such examples. Flat band systems with a large spatial extent of the Wannier functions are a promising platform for increasing T_{c}. In electronphonon systems, the instabilities that limit T_{c} at large electronphonon interaction strength can be partially mitigated if the coupling is shared between several active phonon modes that couple to noncommuting electronic operators.
Methods
All analytical calculations are explicit presented in Supplementary Information.
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Acknowledgements
We thank Pablo JarilloHerrero, Mohit Randeria, and J.M. Tranquada for stimulating discussions. S.A.K. was supported, in part, by the National Science Foundation (NSF) under Grant No. DMR2000987. D.C. was supported by the faculty startup grants at Cornell University. E.B. and J.H. were supported by the European Research Council (ERC) under grant HQMAT (Grant Agreement No. 817799), the IsraelUS Binational Science Foundation (BSF), and by a Research grant from Irving and Cherna Moskowitz.
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Hofmann, J.S., Chowdhury, D., Kivelson, S.A. et al. Heuristic bounds on superconductivity and how to exceed them. npj Quantum Mater. 7, 83 (2022). https://doi.org/10.1038/s41535022004911
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DOI: https://doi.org/10.1038/s41535022004911
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