Conventional superconductivity in quasicrystals

TL;DR

This study investigates superconductivity in quasicrystals, revealing that despite their quasiperiodic structure, they tend to exhibit conventional BCS-like superconductivity, especially when the Fermi energy is not at a pseudogap.

Contribution

The paper provides the first detailed theoretical analysis of superconductivity in quasicrystals using the Bogoliubov-de Gennes approach, highlighting the inhomogeneous nature and size dependence of the order parameter.

Findings

Superconducting state is inhomogeneous but lacks superconductivity islands.

Order parameter depends on system size when Fermi energy is at a pseudogap.

Results align with experimental observations of superconductivity in Al-Zn-Mg quasicrystals.

Abstract

Motivated by a recent experimental observation of superconductivity in the Al-Zn-Mg quasicrystal, we study the low-temperature behavior of electrons moving in the quasiperiodic potential of the Ammann-Beenker tiling in the presence of a local attraction. We employ the Bogoliubov-de Gennes approach for approximants of different sizes and determine the local pairing amplitude $\Delta_{i}$ as well its spatial average, $\Delta_{0}$, the superconducting order parameter. Due to the lack of periodicity of the octagonal tiling, the resulting superconducting state is inhomogeneous, but we find no evidence of the superconductivity islands, as observed in disordered systems, with $\Delta_{i}\rightarrow0$ at $T_{c}$ for all sites. In the weak-coupling regime, we find that the superconducting order parameter depends appreciably on the approximant size only if the Fermi energy sits at a pseudogap in the noninteracting density of states, with $\Delta_{0}$ decreasing as the system size increases. These results are in line with the experimental observations for the Al-Zn-Mg quasicrystal, and they suggest that, despite their electronic structure, quasicrystals are prone to display conventional BCS-like superconductivity.

Equations

$${\mathcal{H}}_0$$

$$IP{R}_{\nu }$$

$${\lambda }_{\gamma \delta }=\frac{1}{N}\sum _{i,j}{\left({\mathbf{r}}_i-{\mathbf{r}}_j\right)}_{\gamma }{\left({\mathbf{r}}_i-{\mathbf{r}}_j\right)}_{\delta }{\left|P\left(i,j\right)\right|}^2.$$

$$\mathcal{H}={\mathcal{H}}_0-U\sum _i{n}_{i\uparrow }{n}_{i\downarrow },$$

$${\mathcal{H}}_{BdG}$$

$$+\sum _i\left({\Delta }_i{c}_{i\uparrow }^{†}{c}_{i\downarrow }^{†}+{\Delta }_i^{\star }{c}_{i\downarrow }{c}_{i\uparrow }\right).$$

$${\Delta }_i=-U⟨c_{i\downarrow }{c}_{i\uparrow }⟩,⟨n_i⟩=\sum _{\sigma }⟨c_{i\sigma }^{†}{c}_{i\sigma }⟩,$$

$$A\left(\mathbf{k},\omega \right)=\sum _{\nu }\delta \left(\omega -{\epsilon }_{\nu }\right){\left|{\psi }_{\nu }\left(\mathbf{k}\right)\right|}^2,$$

$$\tilde{\mathcal{H}}=\sum _{\nu ,\sigma }{\xi }_{\nu }{c}_{\nu \sigma }^{†}{c}_{\nu \sigma }-U\sum _{\nu ,\zeta }{M}_{\nu ,\zeta }{c}_{\nu \uparrow }^{†}{c}_{\bar{\nu }\downarrow }^{†}{c}_{\bar{\zeta }\downarrow }{c}_{\zeta \uparrow },$$

$$M_{\nu ,\zeta }=\sum _i{\left|{\psi }_{\nu }\left(i\right)\right|}^2{\left|{\psi }_{\zeta }\left(i\right)\right|}^2,$$

$${\Delta }_{\nu }$$

$$⟨n⟩$$

$${\Delta }_{\nu }$$

Full text

I Introduction

Quasicrystals display a non-periodic, yet ordered, arrangement of atoms.(Shechtman et al., 1984; Levine and Steinhardt, 1984) They contain a small set of local environments which reappear again and again, albeit not in a periodic fashion. Their structure is not random either, since the diffraction pattern shows sharp Bragg peaks, although their symmetry is noncrystallographic, with the $n$-fold symmetries ($n=5,8,10,\ldots$) stemming from the fact that these local environments occur with $n$ equiprobable orientations. Because of this arrangement of atoms, the Bloch theorem no longer holds and the electronic states of quasicrystals show a remarkably rich behavior,(Grimm and Schreiber, 2003; Jagannathan and Piéchon, 2007) which includes critical states,(Kohmoto et al., 1987; Tsunetsugu et al., 1991; Yuan et al., 2000; Tanese et al., 2014; Macé et al., 2017) confined states in the middle of the band,(Kohmoto and Sutherland, 1986; Arai et al., 1988; Rieth and Schreiber, 1995) pseudogap in the density of states,(Fujiwara, 1989; Fujiwara and Yokokawa, 1991; Ishikawa et al., 2017; Jazbec et al., 2014) and unconventional conduction properties.(Pierce et al., 1993; Trambly de Laissardière and Fujiwara, 1994; Timusk et al., 2013; de Laissardière and Mayou, 2014)

Given their unusual electronic properties, there are several works addressing the effects of electronic correlations in quasicrystals, especially investigating their magnetic properties both in localized and itinerant regimes.(Wessel et al., 2003; Vieira, 2005; Thiem and Chalker, 2015; Hartman et al., 2016; Koga and Tsunetsugu, 2017) Although many interesting properties arise due of the intricate real space arrangement of the lattice sites, some of the physical properties inside phases with long-range order are similar to those of periodic systems.(Luck, 1993) Therefore, the experimental observation of non-Fermi liquid behavior in the $\mbox{Au}_{51}\mbox{Al}_{34}\mbox{Yb}_{15}$ heavy-fermion quasicrystal(K. Deguchi et al., 2012) immediately prompted several theoretical studies.(Watanabe and Miyake, 2013; Shaginyan et al., 2013; Andrade et al., 2015; Takemura et al., 2015)

Superconductivity was observed in approximants,(Graebner and Chen, 1987; Deguchi et al., 2015) which are periodic rational approximations to the quasicrystal, shortly after the discovery of quasicrystals. However, only recently a convincing observation of bulk superconductivity in the Al-Zn-Mg quasicrystal was reported.(Kamiya et al., 2018) Reference Kamiya et al., 2018 finds that the critical temperature $T_{c}$ is very low, $T_{c}\simeq 0.05\mbox{ K}$, and that $T_{c}$ is suppressed as one goes from the approximants to the quasicrystal. Moreover, the authors show that the thermodynamic properties can be understood within the usual BCS weak-coupling scenario. Motivated by these experimental findings, in this paper we study the attractive Hubbard model in a bidimensional quasicrystal. We employ a Bogoliubov-de Gennes approach and our results provide a scenario which is consistent with the experimental observations.

Our paper is organized as follows. In Sec. II, we review the basic properties of the Ammann-Beenker tiling model and its electronic properties. In particular, we employ the Kohn’s localization tensor to probe the spatial extent of the electronic states. In Sec. III, we introduce the attractive Hubbard model and the inhomogeneous Bogoliubov-de Gennes (BdG) mean-field theory to obtain the properties of our model inside the superconducting phase. In Sec. IV, we then compare our results to experiments and contrast our findings with the known results for random systems, after which we conclude the paper. We also have two appendices. Appendix A discusses the spectral function of the noninteracting model, while Appendix B introduces a complementary approach to study the superconductivity, namely the pairing of exact eigenstates (PoEE).

II Tiling model and electronic properties

For simplicity, we consider models on quasiperiodic tilings to mimic the geometrical properties of a quasicrystal. We report results obtained for a $2D$ tiling, where it is easier to handle large system sizes numerically.(lcd, ) The $2D$ tiling we consider is the octagonal, or Ammann-Beenker, tiling.(Socolar, 1989) This tiling is composed of two types of decorated tiles: squares and $45^{o}$ rhombuses, which combine to create six distinct local environments with coordination numbers $z=3,\cdots,8$, Fig. 1(a). These square approximants are obtained by the standard method of projecting down from a higher dimensional cubic lattice,(Socolar, 1989; Levine and Steinhardt, 1987; Duneau et al., 1989; Benza and Sire, 1991) and we consider approximant of sizes $N=41$, $239$, $1393$, $8119$, and $47321$. Even though it is easy to convince oneself on the absence of translational invariance by examining the real-space arrangement of the lattice sites in Fig. 1(a), the eightfold rotational symmetry (present at many different scales) becomes evident as we go to momentum space and calculate the x-ray structure factor, Fig. 1(b). The lack of periodicity manifests itself in reciprocal space both by the absence of a Brillouin zone and the presence of several intense Bragg peaks. As we increase the approximant size, more and more spots appear in the structure factor until it becomes densely filled in the reciprocal space in the limit of the infinite quasicrystal. Another important property of quasicrystals is their self-similarity under inflation transformations. These are site-decimation operations on a subset of vertices of the tiling followed by an increase in the length scale and the reconnection of the surviving vertices. It globally preserves the quasiperiodic structure, see Fig. 1(a), and the infinite quasicrystal is invariant under such transformation.(Duneau et al., 1989; Jagannathan, 2000)

The presence of diffraction spots of widely differing intensities has important consequences on the electronic properties of the quasicrystal. In a periodic system, if the Fermi wave vector $\mathbf{k}_{F}$ satisfies $2\mathbf{k}_{F}=\mathbf{H}$, where $\mathbf{H}$ is a reciprocal lattice vector, a band gap is expected to emerge. In a quasicrystal, because the structure factor is densely filled, this condition is easily met and then we expect the brighter peaks to lead to strong scattering of conduction electrons, giving rise to spikes in the density of states (DOS) [see also Fig. 6(b)].(Smith and Ashcroft, 1987; Zijlstra and Janssen, 2000) The scattering due to the remaining peaks, while weaker, results in wave functions which show fluctuations at all length scales. The Fibonacci chain, a one-dimensional quasicrystal, provides an example of such wavefunctions,(Kohmoto et al., 1987) often referred to as critical,(Kohmoto et al., 1987; Tsunetsugu et al., 1991; Yuan et al., 2000; Grimm and Schreiber, 2003; Jagannathan and Piéchon, 2007; Macé et al., 2017) in analogy with those found at the Anderson metal-insulator transition.(Richardella et al., 2010; Rodriguez et al., 2010, 2011)

As a minimal model to describe the electronic properties of quasicrystals, we study a nearest-neighbor tight-binding Hamiltonian in the Ammann-Beenker tiling,

[TABLE]

where $c_{i\sigma}^{\dagger}\left(c_{i\sigma}\right)$ is the creation (annihilation) operator of an electron at site $i$ with spin $\sigma$ and $t$ is the hopping amplitude between sites $i$ and $j$. In the following, energies are measured in units of $t$. In our calculation, we consider open boundary conditions because: (i) it preserves the particle-hole symmetry of the tiling; and (ii) the finite size effects are comparable to those of periodic boundary conditions due to the quasiperiodic arrangement of the different local environments.(Hartman et al., 2016)

The resulting DOS $\rho\left(\omega\right)=\left(1/N\right)\sum_{\nu}\delta\left(\omega-\varepsilon_{\nu}\right)$, where $\varepsilon_{\nu}$ are the eigenenergies of $\mathcal{H}_{0}$ in Eq. (1), is shown in Fig. 1(c) (we replace each delta function by a Lorentzian of width $\gamma$). As anticipated, $\rho\left(\omega\right)$ displays a strong energy dependence with several spikes, which are largely independent of the broadening $\gamma$. The large peak at $\omega=0$ can be traced to families of strictly localized states, a consequence of the local topology of the octagonal tiling.(Grimm and Schreiber, 2003; Rieth and Schreiber, 1995) The integrated density of states $n\left(\mu\right)=\int_{-\infty}^{\mu}d\omega\rho\left(\omega\right)$, $\mu$ is the chemical potential, is shown in Fig. 1(d). Besides the discontinuity close to $\mu=0$, corresponding to the peak in $\rho\left(0\right)$, $n\left(\mu\right)$ also shows a kink at the filling $2/s^{2}\approx 0.34315$, where $s=1+\sqrt{2}$ is the silver ratio. This is analogous to the case of the Fibonacci chain, where plateaus in $n\left(\mu\right)$ appear at $2/g^{n}$, where $n$ is an integer and $g$ is the golden ratio.(Tanese et al., 2014) A plateau in $n\left(\mu\right)$ corresponds to a gap in the single-particle spectrum of Eq. 1. However, conversely to the one-dimensional case, the DOS in the Ammann-Beenker tiling has at most a pseudogap close to $\omega\approx-1.9t$, corresponding to the filling $2/s^{2}$, and thus we observe only a kink.

A pseudogap at the Fermi level assists in the stabilization of the quasiperiodic structure via the Hume–Rothery mechanism, and it is indeed predicted and observed in several quasicrystals.(Smith and Ashcroft, 1987; Fujiwara and Yokokawa, 1991; Trambly de Laissardière and Fujiwara, 1994; Stadnik et al., 1997; Rotenberg et al., ; Rogalev et al., 2015; Jazbec et al., 2014)

The unambiguous existence of a pseudogap is hindered due to the finite broadening $\gamma$ employed in the numerical calculation of $\rho\left(\omega\right)$. Therefore, we now probe the spatial extent of the wave function, especially close to the filling $2/s^{2}$. First, we compute the inverse participation ratio (IPR),

[TABLE]

where $\psi_{\nu}$ is an eigenstate of $\mathcal{H}_{0}$ with eigenenergy $\varepsilon_{\nu}$. The scaling of the IPR with the system size is related to the spatial structure of the single-particle electronic states. If we write $IPR_{\nu}\propto N^{-\beta}$, then $\beta=1$ for extended and $\beta=0$ for exponentially localized states. In a quasicrystal, we expect $0\leq\beta\leq 1$, due to the multifractal character of the eigenstates.(Grimm and Schreiber, 2003; Chhabra and Jensen, 1989; Richardella et al., 2010; Rodriguez et al., 2010) In Fig. 2 we calculate $IPR_{\nu}$ at different positions in the band. For most fillings, we obtain $\beta\approx 0.90$, a value similar to the one observed in the Penrose tiling.(Grimm and Schreiber, 2003) At the band center, we get $\beta\approx 1$, a value one expects for extended states (at half filling, we have flat bandlike structures coexisting with dispersive ones, see Appendix A). For $n=2/s^{2}$, we have $\beta=0.55\left(6\right)$. Although smaller than the values at the other fillings, this value does not indicate that this particular state is localized and it seems inconsistent with the presence of a pseudogap.

While the IPR is a very useful tool in the context of disordered systems, it may not be able to capture all the subtleties of quasicrystalline electronic states. Indeed, a recent study of the one-dimensional Fibonacci chain showed that the IPR is unable to capture the expected insulating behavior inside the band minigaps.(Varma et al., 2016) Following Ref. Varma et al., 2016, we then decided to study the scaling behavior of the Kohn’s localization tensor(Resta and Sorella, 1999; Resta, 2011; Souza et al., )

[TABLE]

Here $\mathbf{r}_{i}$ is the position of site $i$ inside the approximant, $\gamma,\,\delta$ correspond to the spatial directions $x$ and $y$, and $P\left(i,\,j\right)=\sum_{\nu}\psi_{\nu}\left(i\right)\psi_{\nu}^{\star}\left(j\right)$, with $\varepsilon_{\nu}\leq\mu$, is the one-particle density matrix for a Slater determinant. Therefore, the localization tensor takes into account all states up to the chemical potential and not only a single-particle state. Because time-reversal symmetry is preserved in the problem, the transverse terms vanish identically: $\lambda_{xy}=\lambda_{yx}=0$. Moreover, we have $\lambda_{xx}=\lambda_{yy}=\lambda$, so we drop the spatial subscripts henceforth. The scaling of length $\lambda$ with the approximant size then determines if the system is a metal or an insulator. In a metal, we expect $\lambda^{2}$ to diverge with $N$, whereas for an insulator we expect $\lambda^{2}$ to saturate to a constant.(Varma et al., 2016) If we write $\lambda^{-2}\propto N^{-\alpha}$, we then expect $0\leq\alpha\leq 1$. The results for the scaling of Kohn’s localization tensor are displayed in Fig. 2(b), where it is clear that its dependence with the band filling is indeed more pronounced as compared to the IPR. For arbitrary filling, the states have an extendedlike nature, particularly at the band center where we have $\alpha\approx 1$. For the filling $n=2/s^{2}$, however, $\lambda^{-2}$ is weakly size dependent, $\alpha\approx 0.1$, suggesting a localized-like nature for this state, consistent with the presence of a pseudogap.

Overall, we find that the Kohn’s localization tensor has a stronger dependence with the band filling and it is better suited to decide whether the electronic states are conducting or insulating.(Varma et al., 2016) We stress, however, that $\lambda$ is not simply related to the spatial extent of the single-particle eigenstates, and a more detailed characterization of the multifractal character of the eigenstates is not straightforward within this formalism.

III Superconductivity

After discussing the electronic properties of the octagonal tiling, we now move to the main topic of this paper which is the study of superconductivity. We describe an $s$-wave superconductor using the attractive Hubbard Hamiltonian,

[TABLE]

where $\mathcal{H}_{0}$ is given by Eq. (1), $U>0$ is the uniform on-site pairing attraction, and $n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$ is the number operator. This model naturally neglects the effects of Coulomb interaction, as well as the nontrivial phonon spectrum of quasicrystals.(Luck and Petritis, 1986; Los et al., 1993; Quilichini, 1997; de Boissieu, 2012; Brown et al., 2018) However, we feel that it is a useful exercise to understand the physics of this simple model, where we can contrast our results with a similar investigation in the Penrose tiling.(Sakai et al., 2017) Moreover, we will show that it provides a useful starting point to understand the recently reported quasicrystal superconductivity.(Kamiya et al., 2018)

We study the model in Eq. (4) for different values of the pairing attraction $U$ and filling $n$. Because the DOS in a quasicrystal is strongly energy dependent, Fig. 1(c), one could expect, in principle, strong filling dependence of the results. Nevertheless, as we discussed in the previous section, the behavior of the electronic states is qualitatively the same for all fillings, i.e., metalliclike, except at the special filling of $2/s^{2}$ where we observe an insulating behavior due to a pseudogap in the DOS.

To solve the Hamiltonian (4), we employ the BdG approach, following the works in Refs. Ghosal et al., 1998, 2001, to write the mean-field Hamiltonian:

[TABLE]

The local pairing amplitude $\Delta_{i}$ and the local density $n_{i}$ are determined via the self-consistent equations,

[TABLE]

where the thermal averages are taken considering the eigenstates and eigenenergies of Eq. (5), which we determine via a numerical Bogoliubov transformation. We also introduce an effective chemical potential to incorporate a site-depend Hartree shift: $\tilde{\mu}_{i}=\mu+U\left\langle n_{i}\right\rangle/2$, building thus the most general mean-field theory for an inhomogeneous $s$-wave superconductor.(Dobrosavljević et al., 2012) We remark that this mean-field theory keeps only the amplitude fluctuations of $\Delta_{i}$ and should only be valid at weak-coupling.

We solve the self-consistency Eqs. (6) on finite approximants with $N$ sites, open boundary conditions, and fixed electronic filling $n=\sum_{i}n_{i}/N$. We consider $U\geq 1.5t$ because smaller values of $U$ generate very large coherence lengths and are harder to simulate. To solve Eqs. (6), we start with an initial guess for the local density $n_{i}$ and the pairing amplitude $\Delta_{i}$, and we iterate the procedure until convergence is achieved on all sites. We then adjust the chemical potential $\mu$ to target the desired filling $n$. Notice, therefore, that we have two self-consistency loops, making the whole procedure quite demanding numerically. To complement this procedure, and to access larger system sizes, we also implement the method of PoEE(Ghosal et al., 1998, 2001) (see Appendix B).

We start by showing the spatial distribution of the local pairing amplitude $\Delta_{i}$ at $T=0$ in Figs. 3 (a) and 3 (b). For small values of the pairing attraction $U$, the spatial pattern of $\Delta_{i}$ roughly follows that of the local density of states,(Collins et al., 2017) see Figs. 1 (e) and 1 (f), and it is essentially determined by the local environment of a given site $i$. At low fillings, the sites with a larger coordination number $z$ show larger values of $\Delta_{i}$, whereas for $n\rightarrow 1$ sites with smaller $z$ are the ones with larger $\Delta_{i}$, Figs. 3 (c) and 3 (d). Importantly, we do not observe the formation of superconducting islands as in disordered superconductors.(Ghosal et al., 1998, 2001; Bouadim et al., 2011; Nandkishore et al., 2013; Potirniche et al., 2014; Dodaro and Kivelson, 2018) This is not unexpected because these islands occur in regions where the random disorder potential is unusually small, corresponding thus to rare regions,(Andrade et al., 2009; Vojta and Hoyos, ) a situation which cannot take place in the presence of a deterministic quasiperiodic potential obeying inflation rules. Because the distribution of $\Delta_{i}$ consists essentially of six delta peaks, each one associated to a local environment, and is neither broad nor shows weight at $\Delta_{i}\approx 0$, we also conclude that the system is far away from a possible quasiperiodicity-induced quantum phase transition.

In smaller approximants, $N\leq 1393$, we are able to solve the BdG solutions at finite temperatures, Figs. 3 (e) and 3 (f). Even though $\Delta_{i}$ is spatially inhomogeneous, we find that the superconducting phase transition takes place at all approximant sites at once within our numerical precision. In a translational invariant system, we naturally expect all $\Delta_{i}$’s to vanish concomitantly at $T_{c}$. For the octagonal tiling, we believe that its self-similarity under inflation transformations, see Fig. 1(a), forces all $\Delta_{i}$’s to vanish simultaneously at $T_{c}$. To see this, suppose we start with a subset of sites for which $\Delta_{i}=0$ inside the superconducting phase. Now we successively apply the inflation transformations (remember also that $\Delta_{i}$ is essentially determined by its local environment). Because these transformations leave the infinite system invariant due to their self-similarity, we would then be able to eventually move the entire system to the normal phase. This argument also highlights that the existence of rare regions is not possible, thus precluding both the existence of the superconducting islands at $T=0$, as discussed before, and the presence of a thermal Griffiths phase close to $T_{c}$.

To explore the role of the coherence length $\xi$ — as defined by the spatial decay of the sample averaged correlation function $\left\langle\Delta_{i}\Delta_{j}\right\rangle$ — we study the Fourier transform of both $\Delta_{i}$ and $n_{i}$, see Fig. 4. The Fourier transform of $\Delta_{i}$ shows the expected eightfold structure, as in Fig. 1(b), but there are several missing peaks which we link to the presence of a coherence length $\xi$ ($\xi$ is the largest, circa $10$ lattice spacing, at the pseudogap $n=2/s^{2}$). As we increase the local attraction $U$, the Cooper pairs become more and more local, resulting in the suppression of $\xi$ and in the observation of a densely filled $\Delta_{i}$ spectrum in reciprocal space. The local density $n_{i}$, on the other hand, is quite insensitive to $U$ as it varies on the scale of one lattice spacing, always following the lattice potential.

We now investigate the superconducting ground-state evolution as a function of the approximant size $N$. On one hand—based on previous studies considering the multifractal states observed at the Anderson metal-insulator transition—(Richardella et al., 2010; Rodriguez et al., 2010, 2011) one could naively expect an enhancement of the superconductivity(Feigel’man et al., 2007, 2010) as one moves toward the infinite quasicrystal since the electronic states become more and more critical. On the other hand, all experiments so far find a reduction, or even a complete suppression, of $T_{c}$ as one goes from the approximant to the quasicrystal.(Graebner and Chen, 1987; Deguchi et al., 2015; Kamiya et al., 2018) In Fig. 5, we show the superconducting order parameter $\Delta_{0}=\sum_{i}\Delta_{i}/N$ and $T_{c}$ as a function of the approximant size $N$. We compare the results for $\Delta_{0}$ using the full numerical solution of the BdG equations, Eqs.(6), and that coming from the PoEE, Eqs. (10) and (11) [the PoEE is cheaper numerically and allows us to go up to $N=47321$ and also to estimate $T_{c}$, see Eq. (12)]. We find that $\Delta_{0}$ and $T_{c}$ remain essentially constant for $N\geq 239$ at all fillings but $n=2/s^{2}$, and the results of both methods agree even quantitatively. This implies that the approximants are able to capture the behavior of the infinite quasicrystal, and that the nature of the electronic wavefunctions in the infinite quasicrystal does not affect its superconductivity. This somewhat disappointing result also shows that the expected analogy to disordered systems close to the Anderson metal-insulator transition does not hold (for all parameter sets we simulate, considering $U\geq 1.5t$, we find no enhancement superconductivity as we increase $N$). Our scaling results for a general filling in Fig. 5 also do not agree with the experimentally observed suppression of $T_{c}$ as $N$ increases. The only filling which captures the experimental trend is $n=2/s^{2}$, corresponding to the pseudogap in the DOS. Here, both $\Delta_{0}$ and $T_{c}$ are suppressed, at weak coupling, as we increase the approximant size, due to the pseudogap in the DOS. From Fig. 2(b), we see that the pseudogap gets more and more pronounced as $N$ increases, and for $N\rightarrow\infty$ we must have $U>U_{c}$, a critical coupling, for the system to display superconductivity. For $U=1.5t$, the PoEE approach suggests that $\Delta_{0}\rightarrow 0$, whereas the full BdG solution finds $\Delta_{0}>0$, albeit small. The difference comes from the fact that the BdG method modifies the eigenstates of the non-interacting Hamiltonian, and it shows that we are already above the critical coupling $U_{c}$. As $U$ increases, all fillings behave similarly and the results are essentially size independent. Therefore, the suppression of $T_{c}$ in a quasicrystal occurs only if the Fermi level sits at a pseudogap — a condition routinely met in real quasicrystals — and at weak coupling.

IV Discussion and connection to

experiments

Our results show that the physics observed at the mean-field level in randomly disordered superconductors is not present in quasicrystals. In particular, we do not observe the formation of superconducting islands due to the deterministic character of the lattice potential we consider,(Duneau et al., 1989) a conclusion which should be valid for a broad range of quasicrystals that can be similarly constructed via inflation or substitution rules. As was recently shown,(Dodaro and Kivelson, 2018) the existence of such islands circumvents Anderson’s theorem(Anderson, 1959; Abrikosov and Gor’kov, 1961) and generically enhances $T_{c}$. We also do not find an increase of $T_{c}$ due to the multifractal nature of the electronic state in the infinite quasicrystal, as is expected for disordered systems close to the Anderson metal-insulator transition.(Feigel’man et al., 2007, 2010) Taken together, these observations imply that the BdG solution for superconductivity in a quasicrystal essentially fulfills Anderson’s theorem, i.e., the electrons form pairs with time-reversed eigenstates, say $\nu\uparrow$ and $\bar{\nu}\downarrow$, of the non-interacting model. Thus the superconductivity is of the conventional weak-coupling BCS type, with both $T_{c}$ and $\Delta_{0}$ depending weakly on $N$ (this point is further supported by the excellent agreement between the results of the BdG and PoEE methods in Fig. 5). An interesting consequence of this observation is the fact that Anderson’s theorem also implies that $T_{c}$ does not depend on the wave functions of $\mathcal{H}_{0}$, but only on its spectrum, and thus $T_{c}$ should be mainly governed by the DOS in Fig. 1(c). Our scaling of the order parameter $\Delta_{0}$ in Fig. 5 illustrates this conclusion as the only distinct behavior is observed at the pseudogap and at weak coupling. Concerning the finite temperature critical properties of the model, the absence of rare regions, together with the fact that Luck’s criterion(Luck, 1993) holds for the octagonal tilling (as it does for most tilings constructed via inflation rules), implies that the mean-field BCS solution is expected to hold, as is observed experimentally.(Kamiya et al., 2018)

Although the filling $n=2/s^{2}$ looks like a finely tuned exception in our model, this point is actually very relevant experimentally, since in most quasicrystals the Fermi energy is located at a pseudogap.(Fujiwara, 1989; Fujiwara and Yokokawa, 1991; Ishikawa et al., 2017; Jazbec et al., 2014) Because of that, our results naturally account for the suppression of superconductivity as one goes from the approximant to the quasicrystal,(Graebner and Chen, 1987; Deguchi et al., 2015) and for the conventional superconductivity in the Al-Zn-Mg quasicrystal, as reported in Ref. Kamiya et al., 2018. One obvious implication of our findings is that a small amount of non-magnetic impurities should enhance $T_{c}$ in a quasicrystal, similarly to what is predicted for superconducting semimetals.(Nandkishore et al., 2013; Potirniche et al., 2014; Dodaro and Kivelson, 2018) A similar possibility to increase $T_{c}$ is to dope the system, moving the Fermi level away from the pseudogap.

We can also use our results to understand the absence of superconductivity in $\mbox{Au}_{51}\mbox{Al}_{34}\mbox{Yb}_{15}$(K. Deguchi et al., 2012) and other related heavy-fermion quasicrystals.(Deguchi et al., 2015) In this class of quasicrystals, a non-Fermi liquid behavior was reported without the tuning of an external parameter.(K. Deguchi et al., 2012) Interestingly, such electronic behavior is absent in the approximants. A plausible scenario to understand these observations is the presence of unscreened magnetic moments down to $T\rightarrow 0$ in the quasicrystals, while in the approximants the moments are always screened below a temperature $T^{\star}$.(Andrade et al., 2015) Therefore, while superconductivity is observed in the approximants of heavy-fermion quasicrystals,(Deguchi et al., 2015) the unscreened local moments in quasicrystals act as local pair-breaking defects and further suppress the superconductivity, making it unlikely for this phase to appear in an experimentally accessible temperature.

Overall, despite the fractal geometry of quasicrystals,(Kempkes et al., 2019) the observation of unconventional superconductivity in quasicrystals(Sakai et al., 2017) will probably require the same ingredients as in periodic metals, meaning appreciable electron-electron interaction, most likely in a material which does not involve $f$ electrons. Another interesting problem would be to understand the compounds with a strong electron-phonon coupling where the ubiquitous phonon spectrum of quasicrystals would come into play,(Brown et al., 2018) contrasting these findings with superconductivity in elastically strained crystals where the electronic structure is modulated in response to local lattice deformations.(Zhu et al., 2003)

V Conclusions

In this paper, we have studied the electronic properties and the $s$-wave superconductivity in the two-dimensional Ammann-Beenker tiling. For the electronic properties, we employed the Kohn’s localization tensor and the IPR to access the extent of the electronic states. As in one-dimensional examples,(Varma et al., 2016) we find that the localization tensor gives a more detailed account on the conduction properties of a quasiperiodic system, as shown, for instance, in the better description of the insulating behavior expected for the pseudogap at $n=2/s^{2}$.

To investigate the superconductivity, we considered both real-space BdG and the PoEE approaches to calculate the local pairing amplitude $\Delta_{i}$. We show that $\Delta_{i}$ is essentially determined by its local environment and that the formation of superconducting islands is absent due to the deterministic nature of the lattice potential we consider.(Duneau et al., 1989) Therefore, we find conventional BCS superconductivity in a quasicrystal, despite the nature of their noninteracting electronic states. The pairing mechanism is the one suggested by Anderson,(Anderson, 1959) with time-reversed eigenstates forming the Cooper pairs. In the weak-coupling limit, the superconductivity is suppressed at the pseudogap as we increase the approximant size. Our findings are in accordance with recent experimental observations.(Kamiya et al., 2018)

Since the physics of rare events, which has profound effects in random inhomogeneous systems,(Vojta, 2006) is absent in quasicrystals, one may expect their electronic and magnetic responses to display a more conventional behavior whenever long-range order is present.(Wessel et al., 2003; Jagannathan, 2004; Vieira, 2005; Thiem and Chalker, 2015; Hartman et al., 2016) Of course, the local response is still highly non-trivial due to the fractal geometry of quasicrystalline lattice,(K. Deguchi et al., 2012; Andrade et al., 2015) and future work on correlation effects on quasicrystals are certain to provide many more surprises, an avenue that nowadays can also be explored using different platforms such as cold atoms,(Viebahn et al., 2019) electronic systems with incommensurate order,(Flicker and van Wezel, 2015) or strongly correlated electronic systems at fractional filling.(Sagi and Nussinov, 2016)

Acknowledgements.

We acknowledge J. H. Garcia, J. A. Hoyos, G. F. Magno, P. B. Mendonça, and R. T. Scalettar for useful discussions. R.N.A was supported by the CAPES (Brazil) - Finance Code 001. E.C.A. was supported by CNPq (Brazil) Grant No. 302065/2016-4.

Appendix A Spectral function

To gain further insight on the electronic structure of the non-interacting model, we calculate the spectral function

[TABLE]

where $\psi_{\nu}\left(\boldsymbol{k}\right)$ is the $\nu$-th eigenstate of the non-interacting tight-binding Hamiltonian $\mathcal{H}_{0}$, projected onto the momentum basis, and $\varepsilon_{\nu}$ is its corresponding eigenenergy. To calculate $A\left(\boldsymbol{k},\omega\right)$ numerically, we represent the Dirac-delta function as a Lorentzian with a broadening $\gamma=0.01t$.

In Fig. 6(a), we show the constant energy surfaces for the filling $2/s^{2}$ and the eightfold rotation symmetry is evident.(Rotenberg et al., ; Rogalev et al., 2015) Because of the pseudogap, the Fermi surface-like contours are broken into pockets which are centered around the brightest x-ray spots displayed in Fig. 1(b). Since we have a dense set of Bragg peaks, there are several of these pockets and they intersect each other, making an immediate association to a pseudogap difficult in a finite approximant, where we superimpose our numerical broadening to the true physical broadening of the curves, coming from the fact that momentum is not a good quantum number (see the finite size scaling in Fig. 2).

The energy as a function of momentum, for a given path in reciprocal space, is displayed in Fig. 6(b). There, we see that close to the band edges the dispersionlike curves show a parabolic behavior. Therefore, a nearly-free-electron viewpoint appears to be a good starting point to understand the electronic properties of quasicrystals at these extreme fillings.(Smith and Ashcroft, 1987) As we move on toward the band center, the dispersive features become more and more blurred due to the presence of gap openings at the crossing of the many parabolic bandlike curves. The boundaries between these two regimes are roughly set by the location of the pseudogap, where seemly linearly dispersing features are present.(Timusk et al., 2013) Precisely at the band center, there is a flat bandlike structure, which can be directly linked to the huge peak in the DOS in Fig. 1(c). However, one also observes dispersive features, confirming that at half-filling the model given by Eq. (1) is metallic.

Appendix B Pairing of exact eigenstates (PoEE)

To complement the BdG results in the main text, we also consider the so-called pairing of exact eigenstates.(Ghosal et al., 1998, 2001) This approach is a generalization of Anderson’s original idea of pairing an exact eigenstate of an inhomogeneous system to its time-reversed pair.(Anderson, 1959) Since this formalism considers only the eigenstates and eigenenergies of Eq. (1), it allows us to investigate up to the $N=47321$ approximant. A full self-consistent solution of the BdG equations for this number of sites is computationally prohibitive within our exact diagonalization scheme, although it may be possible using, for example, the kernel polynomial method.(Weiße et al., 2006; Covaci et al., 2010)

We now briefly review the method. We start with the eigenstates $\psi_{\nu}$ of the non-interacting Hamiltonian $\mathcal{H}_{0}$ in Eq. (1). We then pair up electrons in time-reversed eigenstates, say $\nu\uparrow$ and $\bar{\nu}\downarrow$. The analogous BCS Hamiltonian in this basis is then given by(Ma and Lee, 1985)

[TABLE]

where the matrix $M_{\nu,\zeta}$ is given by

[TABLE]

and $\xi_{\nu}=\varepsilon_{\nu}-\tilde{\mu}$ is the energy of the non-interacting problem measured with respect to effective, Hartree-shifted, chemical potential. A mean-field treatment of Eq. (8) then leads to the following set of self-consistent equations:

[TABLE]

where $E_{\nu}=\sqrt{\xi_{\nu}^{2}+\Delta_{\nu}^{2}}$ and $T$ is the temperature. The first equation determines the pairing amplitude of each eigenstate $\nu$, whereas the second one fixes the effective chemical potential $\tilde{\mu}$ (notice here we are only able to fix the average electronic density in the system). Once we solve these equations, we can then obtain the real space pairing amplitudes at $T=0$,(Ghosal et al., 2001, 1998) and we have that $\left\langle\Delta_{i}\right\rangle=\left\langle\Delta_{\nu}\right\rangle=\Delta_{0}$. The scaling of the order parameter is shown in Fig. 5 and we see that the PoEE results nicely follow the full BdG solution, thus confirming Anderson’s pairing and pointing to the conventional nature of the superconductivity in quasicrystals.

If we now linearize Eq. (10), we get the following set of linear equations:

[TABLE]

where we defined the matrix $B_{\nu,\zeta}\left(T\right)=UM_{\nu,\zeta}\tanh\left(\left|\xi_{\zeta}\right|/2T\right)/\left|\xi_{\zeta}\right|$. $T_{c}$ is then given by temperature where the largest eigenvalue of $B_{\nu,\zeta}\left(T\right)$ becomes equal to $1$ with $\left\langle n\right\rangle$ fixed [although $B_{\nu,\zeta}\left(T\right)$ is non-symmetric, we checked that its eigenvalues are real]. The resulting $T_{c}$ are also shown in Fig. 5, and we can compare it with the BdG results for $N=1393$ in Figs. 3 (c) and 3 (d). For $n=0.25$, we have $T_{c}/t=0.011$ (BdG) and $0.008$ (PoEE) whereas for $n=1.00$ we have $T_{c}/t=0.137$ (BdG) and $0.120$ (PoEE), again showing a good agreement between the methods.