Intrinsic constraint on $T_{c}$ for unconventional superconductivity

Despite the century-long pursuit of high-temperature superconductors, the possible existence of a theoretical upper limit to their transition temperature ($T_{c}$) under ambient pressure remains unsettled ^{1; 2; 3; 4; 5}. Both mean-field and weak-coupling Eliashberg theories ⁶ predict an artificial $T_{c}$ that grows continuously with increasing pairing interaction, while experiments often find superconducting domes with maximum $T_{c}$ near the phase boundaries of some long- or short-range orders associated with spin, charge, orbital, or structural degrees of freedom ^{7; 8; 9; 10; 11; 12; 13; 14; 15; 16}. The dome implies a dual role of the competing orders, which not only provide the pairing glues but also constrain the magnitude of maximum $T_{c}$. However, they are mostly external factors associated with instabilities of other channels. One may wonder if any intrinsic constraint on $T_{c}$ may exist owing solely to the pairing instability.

Important lessons may be learned from cuprate high-temperature superconductors in the underdoped region, where strong pairing interactions relative to the renormalized effective quasiparticle hopping parameters favor short-range electron pairs ¹⁷ that in some literatures are thought to form already at high temperatures but only become superconducting when a (quasi-)long-range phase coherence is developed ^{18; 19}. This raises a few general questions: What is the true strong-coupling limit of the pairing state? How is this strong-coupling state related to the high-temperature superconductivity? Would it put any intrinsic constraint on the maximal value of $T_{c}$? Since the absolute magnitude of $T_{c}$ is determined by certain basic energy scale, such as the pairing interaction $J$, the question of maximum $T_{c}$ turns into the question of their maximum dimensionless ratio $T_{c}/J$. To address these important issues and gain insights into possible intrinsic constraints on $T_{c}$, we propose here to discard instabilities from all other channels such as magnetic or charge orders and focus only on the pairing instability, since all other instabilities are expected to compete with the superconductivity and further suppress $T_{c}$. Their contributions to the pairing can all be included phenomenologically in a pairing interaction term.

Theoretically, one may derive various ratios with respect to other measurable energy scales such as the Fermi energy and the superfluid density. However, it has been shown that these ratios may be violated in artificial models ⁴, thus preventing a useful bound for constraining $T_{c}$. To avoid such complication, instead of deriving a model-independent constraint, we first restrict ourselves to a minimal effective model that is most relevant in real correlated materials and includes only the quasiparticle hopping and nearest-neighbour spin-singlet pairing interaction. For the one band model on a square lattice, we find a plaquette state in the strong-coupling limit that breaks both the translational and time reversal symmetries and exhibits unusual spectral properties with a pseudogap or insulating-like normal state. This plaquette state may be regarded as the strong-coupling parent state of $d$-wave superconductivity, since the latter emerges as the plaquettes melt and short-range electron pairs get mobilized to attain long-distance phase coherence at a reduced pairing interaction. A tentative phase diagram is then constructed where $T_{c}/t$ reaches its maximum at the plaquette quantum critical point (QCP), resembling those often observed in experiments with other competing orders. This suggests some intrinsic constraints that prevent $T_{c}$ from exhausting all kinetic or pairing energies in order to achieve a delicate balance between pairing and phase coherence. We then extend the calculations to more general models with either nearest-neighbour or onsite pairings, single layer or two-layer structures, intralayer or interlayer pairings, and obtain a maximum $T_{c}/J\approx 0.04-0.07$. A close examination of existing experiments in known unconventional superconductors, including cuprate, iron-based, nickelate, and heavy fermion superconductors, seems to quite universally support the obtained ratio, indicating that these families, possibly except the infinite-layer nickelates, have almost reached their maximum $T_{c}$ allowed by their respective spin exchange interactions. A room-temperature superconductor would then require a much larger pairing interaction beyond 400-700 meV within the current theoretical framework, which seems unrealistic from a single mechanism in correlated electron systems under ambient pressure. Our work therefore provides a useful criterion that may help to avoid futile efforts in exploring high-temperature superconductors along wrong directions. It also points out the necessity of new pairing mechanisms, possibly combining different pairing interactions, in order to achieve the room-temperature superconductivity.

Figure 1(a) shows a typical theoretical phase diagram for the one-band square lattice model, where we have intentionally plot $T_{c}/t$ against $t/J$. A nonuniform plaquette state emerges at sufficiently strong paring interaction formed of $2\times 2$ blocks induced by high-order pair hopping in the effective action of the pairing fields after integrating out the electron degrees of freedom. A typical pairing configuration of the plaquette state is given in the inset of Fig. 1. The paring amplitudes are relatively stronger on internal bonds of the $2\times 2$ plaquettes and weaker on their links. The phases show a $d$-wave-like character along the $x$ and $y$ directions. Thus, the plaquette state breaks the lattice translational symmetry though the electron density remains uniform. Its transition temperature $T_{\scriptscriptstyle\square}$ decreases with increasing $t/J$ and diminishes at the QCP ($t/J\approx 0.27$), where the plaquettes melt completely and uniform superconductivity emerges with a maximum $T_{c}/t\approx 0.08$ for the chosen parameters (see Method). Tuning the next-nearest-neighbour hopping and the chemical potential may slightly change the ratio and the location of the QCP, but does not alter the qualitative physics. Inside the plaquette state, $T_{c}$ is greatly reduced as the pairing interaction increases. The nonmonotonic evolution of $T_{c}$ resembles typical phase diagrams observed in many unconventional superconductors with other competing orders such as long-range magnetism, charge density wave, or nematicity ^{7; 8; 9; 10; 11; 12; 13; 14; 15; 16}. However, the plaquette state reflects the internal instability in the pairing channel and may be regarded as the true strong-coupling parent state of $d$-wave superconductivity that constrains the magnitude of $T_{c}$. Near the plaquette QCP, the superconductivity also breaks the time reversal symmetry below $T_{\rm BTRS}$. At high temperatures, the normal state exhibits pseudogap-like behavior whose onset temperature $T_{p}$ follows closely the variation of $T_{c}$ or $T_{\scriptscriptstyle\square}$ ^{36; 37} determined from the specific heat $C_{\rm v}$ or the temperature derivative of the quasiparticle density of states at the Fermi energy $dN(0)/dT$. As shown in Fig. 1(b), we find peaks in the specific heat for all transitions at $T_{\scriptscriptstyle\square}$, $T_{c}$, and $T_{\rm BTRS}$, while in $dN(0)/dT$ the feature at $T_{c}$ is greatly suppressed for $t/J<0.27$. Here and after, $J$ is set as the energy unit if not explicitly noted.

At sufficiently low temperatures, the plaquette state may also develop long-distance phase coherence and exhibits unusual spectral features due to the nonuniform spatial distribution of the pairing amplitudes. As shown in Fig. 2(c) for $t/J=0.15$, its momentum-energy dependent spectral function at negative energies splits into two sets of dispersions. One dispersion resembles that of uniform superconductivity, but its back-bending vector $k_{\rm G}$ differs consistently from the Fermi vector $k_{\rm F}$, which has also been observed experimentally for possible pair density wave (PDW) state ^{38; 39; 40}. At high temperatures, the two dispersions recombine into a single curve pointing upwards even in the normal state. The gap indicates a pseudogap or insulating-like phase due to the large nearest-neighbour pairing interaction. This suggests that the normal state may also undergo a metal-insulator transition as $t/J$ decrease, a phenomenon observed in cuprate superconductors under high pressure but unexplained ⁴¹. At intermediate temperature $T_{c}<T<T_{\square}$, the superconducting phase coherence is lost and the plaquette state is in a sense similar to the fermionic quadrupling phase with broken time reversal symmetry proposed earlier in experiment ^{42; 43}.

Under time reversal operation, the phase of the pairing field changes sign so that $\delta\theta_{xy}\rightarrow-\delta\theta_{xy}$ (mod 2$\pi$). Thus, the deviation of the peak position from $\pi$ around $t/J=0.3$ indicates an intermediate region of uniform superconductivity that breaks the time reversal symmetry, with the gap function $\Delta_{\bm{k}}\propto\cos({\bm{k}}_{x})+e^{-i\delta\theta_{xy}}\cos({\bm{k}}_{y})\propto\Delta^{d}_{\bm{k}}-i\cot\frac{\delta\theta_{xy}}{2}\Delta^{s}_{\bm{k}}$, representing $d+is$ pairing with a nodeless gap. Here $\Delta^{d}_{\bm{k}}=\cos({\bm{k}}_{x})-\cos({\bm{k}}_{y})$ is the $d$-wave component and $\Delta^{s}_{\bm{k}}=\cos({\bm{k}}_{x})+\cos({\bm{k}}_{y})$ denotes an extended $s$-wave component from the nearest-neighbour pairing interaction. The onsite pairing is not excluded due to the strong Coulomb repulsion. We have therefore a two-stage transition from the plaquette to the uniform $d$-wave superconductivity, with an intermediate region that recovers the translational symmetry but still breaks the time reversal symmetry. Similar $d+is$ pairing may have been found under certain conditions in twisted double-layer cuprates ⁴⁴ and infinite-layer nickelates ^{46; 45}. In the latter case, it arises from the interplay of Kondo and superexchange interactions ⁴⁷. Here it is associated with the quasiparticle hopping, $i\rightarrow i+\hat{x}\rightarrow i+\hat{x}+\hat{y}\rightarrow i+\hat{y}\rightarrow i$. Integrating out the electron degrees of freedom leads to a term like ${\rm Re}(\Delta_{i,i+\hat{x}}\Delta_{i+\hat{y},i+\hat{x}+\hat{y}}\Delta^{*}_{i,i+\hat{y}}\Delta_{i+\hat{x},i+\hat{x}+\hat{y}}^{*})\rightarrow{\rm Re}(\Delta_{x}^{2}\Delta_{y}^{*}{}^{2})\propto\cos(2\delta\theta_{xy})$, while the second order hopping process such as $i+\hat{x}\rightarrow i\rightarrow i+\hat{y}$ contributes a term ${\rm Re}(\Delta_{x}\Delta_{y}^{*})\propto\cos\delta\theta_{xy}$. Their combined free energy may be minimized at $\delta\theta_{xy}$ away from 0 and $\pi$ ⁴⁸. Thus, time reversal symmetry breaking represents an intrinsic tendency of the superconductivity with nearest-neighbour pairing at strong coupling, where the normal state is no longer a Fermi liquid.

To further confirm the two-stage transition, Fig. 3(c) plots the gap function $\Delta(\phi)$ with $t/J$ near the nodal and antinodal directions in the momentum space deduced from the spectral function. The gap near the antinode is always finite, but varies nonmonotonically with a maximum at the plaquette QCP $t/J=0.27$, in good correspondence with the maximum $T_{c}$. By contrast, the gap near the nodal direction decreases continuously and only diminishes at $t/J\approx 0.5$, confirming a full gap for $0.27\leq t/J\leq 0.5$ consistent with the above phase analysis. The transition temperature $T_{\rm BTRS}$ of the $d+is$ phase may also be extracted from the temperature evolution of $p(\delta\theta_{xy})$. As shown in Fig. 3(d) for $t/J=0.3$, the peak in $p(\delta\theta_{xy})$ gets broadened and moves gradually to $\delta\theta_{xy}=\pi$ as the temperature increases across $T_{\rm BTRS}$. The angle-dependent gap functions are given in Fig. 3(e), showing a fully gapped $d+is$ pairing state and a nodal $d$-wave pairing state below and above $T_{\rm BTRS}$, respectively. Note that the higher-temperature $d$-wave gap contains a finite gapless region on the Fermi surface, which has also been observed previously in some experiments ⁴⁹.

The low-temperature transition coincides with the peak position of $dn_{\rm v}/dT$ also plotted in Fig. 4(a). The maximum of $dn_{\rm v}/dT$ implies a rapid development of the vortex number $n_{\rm v}$ with increasing temperature, which is a characteristic feature of the Berezinskii-Kosterlitz-Thouless (BKT) transition for two-dimensional superconductivity ^{50; 51; 52}. We thus identify this transition as the superconducting transition. The value of $T_{c}$ is examined for other lattice size and found to vary only slightly, confirming the robustness of our qualitative conclusions.

The final phase diagram is already discussed in Fig. 1(a), showing nonmonotonic variation of $T_{c}/t$ with $t/J$ and a maximum at the plaquette QCP. This evolution may also be understood from the phase difference of the pairing fields on neighbouring bonds. Figure 4(b) plots the probabilistic distribution $p(\delta\theta_{1})$ of $\delta\theta_{1}=\theta_{\bm{0}}^{x/y}-\theta_{(1,0)/(0,1)}^{x/y}$. We find two symmetric peaks around zero in the plaquette state and a single peak in the uniform superconducting state. Interestingly, as shown in Fig. 4(c), while the peak position $|\delta\theta_{1}^{\rm max}|$ decreases gradually and diminishes above $t/J=0.27$, the inverse of its fluctuation, as well as that between next-nearest-neighbour bonds, also varies nonmontonically with $t/J$ and exhibits a maximum near the plaquette QCP, in good correspondence with the evolution of $T_{c}/t$. This coincidence is unexpected at first glance but easy to understand, since a smaller fluctuation of $\delta\theta_{1}$ around zero indicates a larger phase stiffness of the pairing fields on neighbouring bonds, thus favoring larger superfluid density and $T_{c}$. Theoretically, this is usually described by the free energy ⁵³, $F=\frac{\rho_{\rm s}}{2}\int_{x}(\delta\theta)^{2}$, such that the phase fluctuation $\langle(\delta\theta)^{2}\rangle$ is inversely related to the superfluid density $\rho_{\rm s}$. This explains our observed correlation between the fluctuation of the phase difference and the magnitude of $T_{c}$ in Fig. 4(c).

Though the Bose-Einstein condensation (BEC) ⁶² has traditionally been argued to be the strong coupling limit of the superconductivity, our results suggest that it may only hold for local $s$-wave pairing with onsite attractive interaction. For unconventional superconductors with strong onsite Coulomb repulsion, onsite $s$-wave pairing is generally unfavored and the pairs tend to occupy different sites. As a result, short-range pairing emerges for nearest-neighbour spin exchange interaction and, at strong coupling, causes an ordered plaquette state that breaks the translational symmetry. This differs from the local two-particle bound state typical of the BEC. On the other hand, the plaquette state does share some similarities with the BEC, which include the U-shaped density of states near the Fermi energy, the flat dispersion around $k_{x}=0$, and the pseudogap in the normal state at high temperatures.

Our proposed plaquette state is also different from the widely-studied PDW state ^{38; 66; 39; 67}, even though both exhibit real-space modulation of the pairing fields. While the PDW may generally lead to a charge density wave, the plaquette state ideally has a homogeneous charge distribution and breaks the time reversal symmetry. The PDW is by far only found experimentally in superconducting region ^{64; 63; 65} and might arise theoretically from the interplay of magnetism and superconductivity ⁶⁸, while the plaquette state proposed here represents an intrinsic pairing instability at strong coupling and might be closely related to the $4\times 4$ structure recently observed in underdoped cuprates ^{69; 70}.

To further explore the above idea, we extend our calculations to other variations of the minimum effective model, covering nearest-neighbour or onsite pairings, single or multi-layer structures, and intralayer or interlayer pairings (see Method). It is important to note that our models do not depend on fine details of the microscopic pairing mechanism, as long as the effective pairing interaction and the low-energy Hamiltonians remain the same. Figure 5(a) shows the variations of $T_{c}/J$ versus $t/J$ in these models, where only the nearest-neighbour hopping $t$ is considered for simplicity. $J$ is the local attractive Hubbard interaction for onsite pairing, and is the interlayer superexchange interaction for interlayer pairing as discussed previously for La₃Ni₂O₇ under high pressure ^{77; 34}. We see all curves behave nonmonotonically with the pairing interaction, although they may have different strong-coupling limit (e.g., BEC for onsite pairing and preformed local interlayer pairing for bilayer nickelates), with the maximum $T_{c}/J$ lying within the interval from 0.04 to 0.07. Notably, for the attractive Hubbard model, our simulations yield consistent results compared with previous quantum Monte Carlo simulations (open down-pointing triangles) ², which reinforces the reliability of our approach, and introducing an additional conduction layer gives the same maximum ratio ^{78; 79}. Note that we have ignored long distance pairing since it is typically weaker than onsite or nearest-neighbour ones for reaching the maximum $T_{c}$. We also only focus on quasi-two-dimensional models since three dimensionality usually suppresses $T_{c}$ in experimental observations ⁸. Our results are insensitive to the chemical potential or electron fillings in reasonable parameter ranges. This is because we have ignored all other instabilities to maximize the pairing instability, and the superconductivity occurs in a much lower energy scale compared to the Fermi energy. Our phase diagram is therefore not the full phase diagram with all possible ground states of a physical model, but a phase diagram that intentionally exaggerates the superconductivity and other possible instabilities in the pairing channel, so that the derived $T_{c}/J$ could be a better estimate of its potential upper limit.

To see if the above constraint may indeed apply in real materials, Fig. 5(b) and Table 1 collect the data for a number of well-known unconventional superconductors ^{81; 80; 82; 83; 84; 85; 89; 90; 91; 86; 87; 88; 96; 92; 93; 94; 95; 99; 97; 98; 100; 20; 101; 23; 76}. The spin energy scale in cuprate, iron-based, and nickelate superconductors have been determined mainly by the spin wave fitting in inelastic neutron scattering (INS) or RIXS experiments ^{23; 24; 25; 97; 102}, where $J$ has been found to vary only slightly with doping, which differs from the renormalized one due to the feedback effect observed in low-energy measurements by INS ¹⁰³ and two-magnon extraction in Raman spectra ¹⁰⁴. In iron-based superconductors such as CaFe₂As₂, SrFe₂As₂, BaFe₂As₂, and NaFeAs, the ratios $T_{c}/J$ are less than 0.063 ^{96; 89; 94; 97; 98; 90; 91; 92; 93; 95}, where the value of $J$ is extracted from the reported $SJ$ by taking the effective spin size $S=0.69$ for SrFe₂As₂ and $S=1/2$ for all others except for FeSe following the literatures ^{81; 80; 82}. The large error bar exceeding the shaded area in Fig. 5(b) comes from the bulk FeSe ($T_{c}=8$ K), for which neutron scattering measurements reported the ratio $T/J=0.86\pm 0.35$ at $T=110$ K ⁹⁸. Unfortunately, we do not find the data for FeSe films, whose high $T_{c}$ might involve contributions from the interface. To the best of our knowledge, there is also no exact estimate of $J$ for the 1111 systems. It has been reported that SmOFeAs adopts an intermediate spin dispersion between those of NaFeAs and BaFe₂As₂ ¹⁰⁵. Assuming that the spin interaction is not sensitive to the doping, as observed in BaFe_2-xNi_xAs₂ and NaFe_1-xCo_xAs ^{97; 102}, we might roughly estimate $J\sim 80-118.4$ meV for SmO_1-xF_xFeAs and thus obtain a maximum ratio $T_{c}/J\approx 0.040-0.059$ given its maximum $T_{c}$=55 K ¹⁰⁶. While for LaO_1-xF_xFeAs, experiments only indicate an overall magnitude of $SJ\sim 40$ meV along different directions ¹⁰⁷, which yields $T_{c}/J\sim 0.046$ with its maximum $T_{c}=43$ K using $S=1/2$ ¹⁰⁸. Both fall within our proposed range.

The infinite-layer nickelate superconductors have a small maximum ratio of about 0.026, possibly due to disorder, which indicates the potential to reach a higher $T_{c}$ ^{83; 84; 86; 87; 85; 88}. RIXS measurements ¹⁰⁹ on the high-pressure high-temperature bilayer nickelate superconductor La₃Ni₂O₇ reported an interlayer spin interaction strength ($J$) of about 140 meV assuming its spin size $S=1/2$, which also seems to be confirmed by inelastic neutron measurements ¹¹⁰. Although these measurements were performed under ambient pressure, it gives a rough estimate of the magnitude of $J$. If we naively apply this value to the high pressure region where the superconductivity was reported with $T_{c}^{\rm max}\approx 80$ K, we find $T_{c}^{\rm max}/J\approx 0.05$ for the bilayer nickelate superconductors, which agrees well with our previous Monte Carlo simulations ³⁴. Recently, superconductivity has been reported also in the trilayer nickelate superconductor La₄Ni₃O₁₀ under high pressure, albeit with a much smaller $T_{c}^{\rm max}\approx 30$ K ¹¹¹. It has been proposed theoretically that competition and frustration of interlayer pairing between the inner layer and two outer layers may lead to strong superconducting fluctuations and thus reduce the maximum ratio of $T_{c}/J$ to $0.02-0.03$ ³⁵. This, together with layer imbalance and the possibly smaller interlayer $J$, may explain the much reduced $T_{c}^{\rm max}$ in the trilayer nickelate compared to those in the bilayer ones.

By contrast, the cuprate high-temperature superconductors have the highest $T_{c}^{\rm max}$ in the trilayer structure, and their overall $T_{c}^{\rm max}/J$ ratios can reach up to 0.067, as observed in HgBa₂CaCu₂O_6+δ, YBa₂Cu₄O₈, YBa₂Cu₃O_6+δ, NdBa₂Cu₃O_6+δ, Tl₂Ba₂CuO_6+δ, and Bi_2+xSr_2-xCa₂Cu₃O_10+δ ²⁰. This opposite tendency reflects an intrinsic distinction in the pairing mechanisms between multilayer nickelate and cuprate superconductors. In heavy-fermion superconductors such as CeCoIn₅ or PuCoGa₅, systematic measurements of $J$ are lacking. We therefore estimate the spin interaction energy from the coherence temperature scale, namely the Ruderman-Kittle-Kasuya-Yosida (RKKY) scale, and find the highest $T_{c}/J$ to be about $0.046$ ^{100; 99; 101}. To the best of our knowledge, a spin wave fitting has only been applied to CePd₂Si₂ and yields $J=0.61$ meV under ambient pressure ¹¹². Combining naively this value with its $T_{c}=0.43$ K at 3 GPa gives the ratio $T_{c}/J\approx 0.061$, in good alignment with our suggested constraint.

Despite the vast complexities across all these different families of unconventional superconductors far beyond our simplified models, their maximum $T_{c}/J$ values all fall within the same range of $0.04-0.07$ predicted above, suggesting that our calculations indeed capture some essence of the fundamental physics of unconventional superconductivity. Consequently, our derived maximum ratio $T_{c}/J\approx 0.04-0.07$ represents a practical constraint for some quite generic situations in real materials.

Last, we comment on the $T_{c}/t$ ratio widely used in previous literatures. Unlike $T_{c}/J$, we find the maximum $T_{c}/t$ depends more sensitively on models and may reach 0.29, 0.15, 0.105 upon tuning the hopping parameters or the chemical potential for interlayer, intralayer onsite, intralayer nearest-neighbour (nn) pairings, respectively. Its maximum typically occurs at different optimal $t/J$ compared to that for the maximum $T_{c}/J$. For the attractive Hubbard model, our derived maximum $T_{c}/t\approx 0.15$ is close to the quantum Monte Carlo result $0.17$, which confirms the validity of our estimate ². While the maximum $T_{c}/J$ ratios lie within the proposed narrow range possibly due to their similar local or short-range pairing forms at strong coupling, we ascribe the large variation of the $T_{c}/t$ ratio to the fact that the long-range phase coherence determining $T_{c}$ may rely heavily on the cooperative hopping of paired electrons and hence differ greatly for different orbital degeneracies, pairing configurations, and lattice geometries beyond the simple hopping parameters. The quasiparticle hopping is also more strongly renormalized by correlation effects, which makes it difficult to measure in practice. It is for these reasons that we have chosen to treat $t$ as a tuning parameter and focus on the $T_{c}/J$ ratio that can be better compared with experiment.

It is therefore imperative to explore alternative avenues to enhance the ratio under ambient pressure. It has been noticed that three-layer cuprate superconductors have the highest $T_{c}$. One may therefore speculate that multi-layer may promote $T_{c}$. Indeed, the maximum $T_{c}$ increases from 97 K in the single-layer HgBa₂CuO_4+δ to 127 K in the two-layer HgBa₂CaCu₂O_6+δ and 135 K in the three-layer HgBa₂Ca₂Cu₃O_9+δ ⁷³. However, the ratio $T_{c}/J\approx 0.062$ seems to remain unchanged and the increase of $T_{c}$ seems to come purely from the increase of $J$ ²⁰. On the other hand, the maximum $T_{c}/J$ does increase from 0.021 in the single-layer Bi₂Sr_2-xLa_xCuO_6+δ to 0.058 in the three-layer Bi_2+xSr_2-xCa₂Cu₃O_10+δ in Bi-systems ²⁰, but the latter still lies within our proposed range, implying that increasing the number of layers from Bi2201 to Bi2223 only helps to tune the optimal conditions for maximizing $T_{c}/J$, while the constraint itself is not touched. We have also examined the effect of additional local interlayer hopping and find that it actually reduces the maximum $T_{c}/J$. Additionally, one may follow the studies of FeSe films ^{113; 114} and consider to improve $T_{c}$ by introducing phonons, but this seems empirically at most to provide an increase of around 40 K, given the limited characteristic phonon frequencies under ambient pressure ^{115; 116}. A larger spin interaction occurs for the Hund’s rule coupling inside an atom. However, it is not clear if intra-atomic inter-orbital pairing may support a high $T_{c}$ due to their very different orbital characters of the paired electrons.

Putting together, the known unconventional superconductor families, possibly except the infinite-layer nickelates, seem to have almost exhausted their potentials in reaching the highest $T_{c}$ allowed by their respective spin exchange interactions. As a result, room-temperature superconductivity at ambient pressure is unlikely to occur based on a single pairing mechanism within the current theoretical framework. This not only helps rule out some evidently wrong directions ^{117; 118}, but also points out the necessity of exploring alternative approaches to achieve room-temperature superconductors at ambient pressure ^{119; 79; 121; 122; 123; 77; 101; 1; 124; 125; 120; 126}. It encourages the possibility of incorporating different pairing mechanisms ^{130; 128; 127; 8; 129; 131}, including but not limited to magnetic, charge, orbital, or nematic fluctuations, excitons, bipolarons, etc, to improve the overall effective pairing interaction, for which FeSe films may be a good example ^{132; 133; 134}. Our derived ratios provide a tentative guide in future material exploration of novel high-temperature superconductors. Theoretically, by utilizing $J$ from newly developed methods ¹³⁵ and effective hopping $t$ from strongly correlated calculations ¹³⁶, an approximate estimate of the upper limit of $T_{c}$ may be predicted for the selection of promising candidates. Experimentally, estimating $J$ from RIXS, INS, or other state-of-the-art techniques in newly discovered materials may also help identify their potential in reaching the desired $T_{c}$. Last but not least, understanding unconventional superconductivity from a real-space, strong-coupling perspective may already provide an operational and more practical avenue for material design compared to the momentum-space, weak-coupling approach.

To derive the $T_{c}/J$ constraint, we extend the above model to the following variations:

(1) A two-layer model with intralayer nearest-neighbour pairing and interlayer hopping:

where the subscript $a=1,2$ represents the layer index, $\psi_{aij}=\frac{1}{\sqrt{2}}(d_{ai\downarrow}d_{aj\uparrow}-d_{ai\uparrow}d_{aj\downarrow})$, and $t_{p}$ denotes the local interlayer hopping.

(2) A two-layer model with an extra conduction layer motivated by the possible importance of apex oxygens in cuprates:

where $t_{p}$ denotes local ($i=j$) or nearest-neighbour ($j=i\pm\hat{x}$ or $i\pm\hat{y}$) interlayer hopping.

(3) A single-layer model with onsite pairing interaction as in the attractive Hubbard model:

where $\psi_{i}=d_{i\downarrow}d_{i\uparrow}$ and $J$ is given by the local attractive Hubbard interaction.

(4) A two-layer model with onsite pairing in one layer and an extra conduction layer:

where $\psi_{i}=d_{i\downarrow}d_{i\uparrow}$ and $t_{p}$ denotes local ($i=j$) or nearest-neighbour ($j=i\pm\hat{x}$ or $i\pm\hat{y}$) interlayer hopping.

(5) A two-layer model with interlayer pairing:

where $\psi_{12i}=\frac{1}{\sqrt{2}}(d_{1i\downarrow}d_{2i\uparrow}-d_{1i\uparrow}d_{2i\downarrow})$.

Models (1) and (2) are constructed to reflect the effects of interlayer hopping and apex oxygen in cuprate superconductors, models (3) and (4) apply for onsite pairing with local attractive interaction, and model (5) is motivated by the bilayer nickelate superconductor.