Abstract
The role of electronphonon interactions in ironbased superconductor is currently under debate with conflicting experimental reports on the isotope effect. To address this important issue, we employ the renormalizationgroup method to investigate the competition between electronelectron and electronphonon interactions in these materials. The renormalizationgroup analysis shows that the ground state is a phonondressed unconventional superconductor: the dominant electronic interactions account for pairing mechanism while electronphonon interactions are subdominant. Because of the phonon dressing, the isotope effect of the critical temperature can be normal or reversed, depending on whether the retarded intra or interband interactions are altered upon isotope substitutions. The connection between the anomalous isotope effect and the unconventional pairing symmetry is discussed at the end.
Introduction
Superconductivity^{1,2,3,4,5} is a novel phenomenon of zero electric resistance in some materials when cooled below the characteristic critical temperature T_{c}. The magic arises from electron pairing in superconductors such that the lowenergy excitations are described by an exotic quantum condensate without any dissipation. In conventional superconductors, such as aluminium, the interactions between electrons and the lattice vibrations generate effective attraction and lead to electron pair formation. In quantum language, these vibrations can be treated as particlelike excitations named phonons. It is generally believed that the electronphonon interactions explain the pairing mechanism for conventional superconductors.
On the contrary, the pairing mechanism of the unconventional superconductors, such as cuprates, seems to stem from the strong electronelectron interactions. Despite of intensive experimental and theoretical studies^{1,2} in the past decades, there are still plenty of unsettled controversies about these unconventional superconductors. One of the most important issues is the interplay between the electronelectron and the electronphonon interactions^{6,7,8,9,10,11,12,13,14,15,16,17,18}. The recently discovered ironbased superconductors^{3,4,5,19,20,21,22,23} provide a unique testing ground to address this issue^{24,25,26,27}. Gathered from theoretical and experimental investigations, the interaction strength in the ironbased superconductors is only weak to medium, rendering controlled theoretical understanding possible.
One of the checking points is the critical temperature of superconductivity upon isotope substitutions^{28,29,30,31}. According to the BardeenCooperSchrieffer theory for the conventional superconductors, the critical temperature T_{c} is related to the mass of the isotope element M,
where α is the exponent for the isotope effect. If the dominant interaction is electronphonon in nature, theoretical calculations give \(\alpha =1/2\). In the extreme opposite, if the pairing is completely driven by electronelectron interactions, the critical temperature should not change with isotope substitutions and the corresponding exponent is \(\alpha \approx 0\). In realistic superconductors, we expect the isotope exponent to be inbetween. Note that, in unconventional superconductors, the phononmediated interactions are insufficient to explain the pairing mechanism and it is of crucial importance to study the interplay between electronelectron and electronphonon interactions^{24,25,26,27}. For instance, even when the pairing mechanism is electronic origin, dispersions observed in angleresolved photoemission spectroscopy manifest distortions upon isotope substitutions^{13,14,15,16,17,18,27}.
The isotope effect observed in ironbased superconductor^{28,29,30,31,32,33,34} seems to tell a more complicated story. For instance, a strong isotope effect by iron substitution^{28} is found in SmFeAs(O, F) and (Ba, K)Fe_{2}As_{2}, almost as large as that in conventional superconductors. On the contrary, inverse isotope effect^{29} is spotted in (Ba, K)Fe_{2}As_{2} with different isotope substitutions. Later, it was proposed that the isotope substitutions may give rise to structural change^{35} and further complicate the story. On the theoretical side, Yanagisawa et al.^{36} proposed a multiband and multichannel model to explain the possibility of observing the inverse isotope effect. However, BussmannHolder and Keller^{37} commented that an inversion of the exponent cannot occur upon iron isotope substitutions. The controversies about the isotope effect of the ironbased superconductor are still on. And, it is of crucial importance to clarify the subtle role of the electronphonon interactions in ironbased superconductors.
Results
Instantaneous and retarded interactions
Motivated by the controversy, we investigate the competition between electronelectron and electronphonon interactions by the unbiased renormalizationgroup (RG) method. Due to the retarded nature of the phononmediated interactions, the energy dependence must be included. The minimal approach to include both simultaneous and retarded interactions can be accomplished by the stepshape approximation^{38,39,40,41} as shown in Fig. 1(a),
where g_{i} and \({\tilde{g}}_{i}\) represent (instantaneous) electronic interactions and (retarded) phononmediated ones. The energy scale for the retarded interactions is set by the Debye frequency \({\omega }_{D}\). Our RG analysis reveals that the pairing mechanism is dominated by the electronic interactions g_{i}. But, the retarded interactions \({\tilde{g}}_{i}\) also grow under RG transformation and become relevant in lowenergy limit. Inclusion of these subdominant interactions leads to anomalous isotope effect. The isotope exponent α can be extracted numerically from RG flows in weak coupling. It is quite remarkable that the sign of the exponent α sensitively depends on whether the inter and/or intraband interactions are altered by isotope substitutions.
Multiband model
To illustrate how the RG scheme works, we start with a fiveorbital tightbinding model for ironbased superconductors with generalized onsite interactions,
where \(a,b=1,2,\ldots ,5\) label the five dorbitals of Fe, \(1:{d}_{3{Z}^{2}{R}^{2}}\), \(2:{d}_{XZ}\), \(3:{d}_{YZ}\), \(4:{d}_{{X}^{2}{Y}^{2}}\), \(5:{d}_{XY}\), and α = ↑,↓ is the spin index. The kinetic matrix K_{ab} in the momentum space has been constructed in previous studies^{42}. The generalized onsite interactions consist of three parts: intraorbital U_{1}, interorbital U_{2} and Hund’s coupling J_{H}. Adopted from previous studies, we choose the values, \({U}_{1}=4\,{\rm{eV}}\), \({U}_{2}=2\,{\rm{eV}}\) and \({J}_{H}=0.7\,{\rm{eV}}\) for numerical studies here.
Fermiology is important in the multiband superconductors. The electron doping x is related to the band filling \(n=6+x\) (\(n=10\) for completely filled bands) here and the Fermi surface at \(x=0.1\) is illustrated in Fig. 1(b). There are five active bands: two hole pockets centered at (0, 0) and another hole pocket centered at (π, π) while two electron pockets located at (π, 0)and (0, π) points^{43}. To simplify the RG analysis, we sample each pocket with one pair of Fermi points (required by timereversal symmetry). This is equivalent to a fourleg ladder geometry with quantized momenta as shown in Fig. 1(b). In the lowenergy limit, the effective Hamiltonian^{44,45,46} is captured by five pairs of chiral fermions with different velocities. The RG equations for all couplings can be found in Methods.
Pairing mechanism
By integrating the two sets of RG equations numerically, we found all couplings are well described the scaling ansatz^{47},
where G_{i}, \({\tilde{G}}_{i}\) are nonuniversal constants and \({\gamma }_{{g}_{i}}\), \({\gamma }_{{\tilde{g}}_{i}}\) are RG exponents for simultaneous and retarded couplings. The divergent length scale l_{d}, associated with the pairing gap, is solely determined by electronic origin. The dominant pairing occur within band 1 and band 2 and the Cooper scatterings c_{11}, c_{22} c_{12} have maximum exponent \({\gamma }_{i}=1\). Other Cooper scatterings are subdominant with exponents close to 0.9, as shown in Fig. 2(a). Meanwhile, by Abelian bosonization^{45,46}, the signs of c_{ij} from numerics lead to signrevised (between electron and hole pockets) s_{±}wave pairing, agreeing with the previous functional RG study^{48}. Note that these exponents are rather robust within the doping range where the same Fermiology maintains. What about the phononmediated interactions? As clearly indicated in Fig. 2(b), the RG exponents for \({\tilde{c}}_{11}\), \({\tilde{c}}_{22}\) are roughly 0.6, much smaller than the dominant electronic interactions, showing the pairing mechanism is electronic origin. However, since the RG exponents are positive, the retarded interactions also grow under RG transformation. These subdominant phononmediated interactions can lead to anomalous isotope effect as explained in the following.
Twostep RG scheme
To achieve quantitative understanding in weak coupling, the rescaled Debye frequency must be taken into account carefully. Under RG transformations, \({\omega }_{D}\to {\omega }_{D}{e}^{l}\) as shown in Fig. 3. At the (logarithmic) length scale \({l}_{D}\equiv \,\mathrm{log}({{\rm{\Lambda }}}_{0}/{\omega }_{D})\), the difference between g_{i} and \({\tilde{g}}_{i}\) disappears. The Debye frequency \({\omega }_{D}\sim 30\,{\rm{meV}}\) in ironbased materials^{24} and the band width (thus Λ_{0}) is 3–4 eV, giving rise to l_{D} ~ 5. Note that the RG is truncated at the cutoff length scale l_{c} where the maximal coupling reaches order one. In weak coupling, it is clear that \({l}_{c} > {l}_{D}\) and thus the RG scheme must be divided into two steps. For \(l < {l}_{D}\), both sets of RG equations are employed. At \(l={l}_{D}\), the functional form for the retarded interactions is the same as the instantaneous one. Thus, one should add up both types of couplings \({g}_{i}({l}_{D})+{\tilde{g}}_{i}({l}_{D})\) and keep running RG by just the first set of equations. In physics terms, this means that the difference between simultaneous and retarded interactions vanishes before the pairing gaps open.
Extracting isotope exponent
Numerical results for the twostep RG indicate the same superconducting phase as described in previous paragraphs but the isotope exponent α can be extracted numerically. Under RG transformation, the critical temperature satisfies the scaling form, \({k}_{B}{T}_{c}\sim {\rm{\Delta }}\,[g(0)]={{\rm{\Delta }}}_{c}{e}^{{l}_{c}}\), where Δ_{c} is the pairing gap at the cutoff length scale. By varying the length scale l_{D}, the critical temperature changes, i.e.
Furthermore, from the definition of the isotope exponent, the standard scaling argument under RG transformation gives rise to the isotope exponent.
where \(d(\mathrm{log}\,M)=\,2d(\mathrm{log}\,{\omega }_{D})=2d{l}_{D}\), because \({\omega }_{D}\sim {M}^{1/2}\). The above formula for the isotope exponent α is the central result in this paper. For conventional superconductor, \({{\rm{\Delta }}}_{c}\sim {\omega }_{D}\) and the cutoff length scale is not sensitive to the Debye frequency (the second term vanishes). Thus, \(\alpha \approx 1/2\). On the other hand, for unconventional superconductors without relevant electronphonon interactions, \({{\rm{\Delta }}}_{c}\sim {{\rm{\Lambda }}}_{0}\) and the cutoff length scale is also not sensitive to the Debye frequency. It is clear that α = 0 in this case. But, what happens if the electronphonon interactions, though not dominant, are actually relevant under RG transformation? We shall elaborate the details in Discussion.
Discussion
To extract the isotope exponent, we study how the cutoff length scale l_{c} varies with different Debye frequencies due to isotope substitutions. In weak coupling, we found that g_{i} are much larger than \({\tilde{g}}_{i}\). Thus, Δ_{c} has very weak dependence on \({\omega }_{D}\) and the first term can be ignored. The contribution from the second term is shown in Fig. 4. We tried two different profiles for the retarded interactions. Include only intraband interactions, \({\tilde{c}}_{ii}(0)=\,0.3\,U\) first, where U is the strength of electronelectron interactions. The isotope exponent is positive (reading from the slope), \(\alpha \approx 0.1\), with very smooth variation. On the other hand, with only interband interactions, \({\tilde{c}}_{ij}(0)=\,0.14\,U\), the isotope exponent is negative and changes gradually from zero to \(\alpha \approx \,0.03\).
These anomalous isotope effects are closely related to the unconventional pairing symmetry. For the s_{±}wave pairing, \({c}_{ii} < 0\) but \({c}_{ij} > 0\) at the cutoff length scale. The phononmediated intraband interactions \({\tilde{c}}_{ii} < 0\) help to develop the pairing instability and thus lead to a positive isotope exponent. On the other hand, the interband ones \({\tilde{c}}_{ij} < 0\) have opposite sign with their simultaneous counterparts c_{ij}. In consequence, the pairing instability is suppressed and an inverses isotope effect is in order. The RG analysis presented here provides clear and natural connection between the anomalous isotope effect and the unconventional pairing symmetry.
Although the isotope exponent α can be extracted numerically in weak coupling, extending the quantitative description to intermediate coupling may not be easy. If the pairing gaps open before hitting the Debye energy scale, i.e. \({l}_{c} < {l}_{D}\), our numerical results show that l_{c} solely depend on electronic interactions and thus \(d{l}_{c}/d{l}_{D}=0\). The isotope exponent in this regime mainly arises from the first term. The pairing gap \({{\rm{\Delta }}}_{c}={{\rm{\Delta }}}_{c}({{\rm{\Lambda }}}_{0},{\omega }_{D}{e}^{l})\), depending on both the bandwidth and the rescaled Debye frequency, is now quite complicated. The RG analysis alone is not sufficient to obtain α in a quantitative fashion. However, we recently found that the effective Hamiltonian at the cutoff length scale is well captured by meanfield theory (not yet published). In principle, one can combine RG and meanfield approaches together to compute the isotope exponent in intermediate coupling more accurately.
In the end, we discuss the recent discovery of superconductivity in FeSe/STO systems^{49,50}. We emphasize that our current approach includes fermiology, electronelectron interactions, and electronphonon interactions within only the superconducting(SC) layers. One crucial assumption is the profile of the mediated electronphonon interactions can be captured by the step function. The RG scheme built upon this approximation works as explained in the manuscript. However, according to the recent literatures in FeSe/STO systems^{49,50,51,52,53,54,55,56}, to include the nonSC (SrTiO_{3}) layers we need to devise a new theoretical approach which is beyond our model at this point. The profile of the electronphonon interactions arisen from nonSC layers is probably not captured by the simple step function anymore. One needs to find out the interaction profile generated by the nonSC layers first so that one can devise the RG scheme accordingly. This is going to be an interesting and challenging topic to explore in the future.
Methods
RG equations
The interactions between these chiral fermions fall into two categories^{39}: Cooper scattering \({c}_{ij}^{l}\), \({c}_{ij}^{s}\) and forward scattering \({f}_{ij}^{l}\), \({f}_{ij}^{s}\). The retarded ones share the same classification, denoted with an extra tilde symbol. The RG equations for the simultaneous interactions are,
where \(\dot{g}=dg/dl\), where \(l=\,\mathrm{ln}({{\rm{\Lambda }}}_{0}/{\rm{\Lambda }})\) is the logarithm of the ratio between bare energy cutoff \({{\rm{\Lambda }}}_{0}\) and the running cutoff \({\rm{\Lambda }}\). The tensor \({\alpha }_{ij,k}=({v}_{i}+{v}_{k})\,({v}_{j}+{v}_{k})/[2{v}_{k}({v}_{i}+{v}_{j})]\) with v_{i} representing the Fermi velocities.
The second set of equations describes how the retarded interactions are renormalized,
Note that we separate the intraband and interband couplings for clarity, i.e. \(i\ne j\) in the above RG equations. In fact, the separation is necessary because we shall see later that interband and intraband couplings play different roles in the lowenergy limit. In addition, \({f}_{ii}=0\) and \({\tilde{f}}_{ii}=0\) to avoid double counting.
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Acknowledgements
We acknowledge supports from the National Science Council in Taiwan through grant MOST 1072112M005008MY3 and MOST 1062112M007011MY3. Financial supports and friendly environment provided by the National Center for Theoretical Sciences in Taiwan are also greatly appreciated.
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Both W.M.H. and H.H.L. contribute extensively to the work in all aspects. W.M.H. and H.H.L. prepare the manuscript together.
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Huang, W., Lin, H. Anomalous isotope effect in ironbased superconductors. Sci Rep 9, 5547 (2019). https://doi.org/10.1038/s4159801942041z
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Further reading

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