Quasiperiodic criticality and spin-triplet superconductivity in superconductor-antiferromagnet moire patterns

Quasiperiodicity has long been known to be a potential platform to explore exotic phenomena, realizing an intricate middle point between ordered solids and disordered matter. In particular, quasiperiodic structures are promising playgrounds to engineer critical wavefunctions, a powerful starting point to engineer exotic correlated states. Here we show that systems hosting a quasiperiodic modulation of antiferromagnetism and spin-singlet superconductivity, as realized by atomic chains in twisted van der Waals materials, host a localization-delocalization transition as a function of the coupling strength. Associated with this transition, we demonstrate the emergence of a robust quasiperiodic critical point for arbitrary incommensurate potentials, that appears for generic relative weights of the spin-singlet superconductivity and antiferromagnetism. We show that the inclusion of residual electronic interactions leads to an emergent spin-triplet superconducting state, that gets dramatically enhanced at the vicinity of the quasiperiodic critical point. Our results put forward quasiperiodicity as a powerful knob to engineer robust superconducting states, providing an alternative pathway towards artificially designed unconventional superconductors.


I. INTRODUCTION
Unconventional superconductivity [1,2] encompasses one of the most exotic states found in quantum materials. In particular, recent interest in topological superconductors hosting Majorana states has been boosted by their potential for topological quantum computing. [3,4] However, unconventional superconductivity and in particular spin-triplet superconductivity remains a highly elusive state in natural compounds. [5][6][7][8] Whereas several compounds have been identified as a potential candidates for spin-triplet superconductivity, [9][10][11][12][13][14] finding generic mechanisms for its engineering remain still an open challenge. [15,16] A highly successful strategy for engineering spin-triplet superconductors consists of focusing on materials with potential coexisting magnetic and superconducting orders. [15,16] This procedure has been heavily exploited for the engineering of Majorana bound states with a variety of platforms [17][18][19][20][21][22][23][24]. Most of these schemes have relied on engineering periodic structures with competing orders, while its study in non-periodic systems has remained relatively unexplored. [25,26] In stark contrast, the study of conventional superconductivity in disordered, [27][28][29] and quasiperiodic systems [30][31][32][33] has a long history, in particular, demonstrating potential critical advantages of non-periodicity for engineering robust superconducting states.
Quasiperiodic patterns display a never-repeating arrangement of elements [34,35], yet hosting long-range order. Due to the lack of conventional periodicity, standard band-structure arguments no longer hold, and their electronic structure exhibits a notably rich behavior [36], such as the presence of confined states [37][38][39], fractal dimensions [40][41][42], pseudogap in the density of states [43][44][45][46], and unconventional conduction properties [47][48][49][50]. More importantly, the incommensurate structure of quasicrystals has prominent effects on the electron eigen- (a) Engineered antiferromagnetic atomic chain on top of a superconducting twisted graphene bilayer. The twisted system shows a spatially modulated superconducting state (yellow), that coexists with the spatially modulated antiferromagnetism of the chain. Panel (b) shows the modulation of the superconducting and antiferromagnetic order parameters along the chain direction, associated to the Hamiltonian of Eq. 1.
states. Incommensurate potentials give rise to electronic wave functions extended, localized or critical [51][52][53][54], and ultimately can host topological states of matter fully associated to the quasiperiodicity. [55,56] Here we demonstrate that quasiperiodic patterns arising from a combination of spin-singlet superconductivity and antiferromagnetism provide a powerful platform to engineer spin-triplet superconductivity. In particular, we show that this antiferromagnetic-superconductor pattern hosts a localization-delocalization phase transi-tion, with an associated quasiperiodic critical point with multifractal wavefunctions. We further show that upon inclusion of residual interactions, a spin-triplet superconducting state emerges, that gets dramatically enhanced at the proximity of the localization-delocalization critical point. Our results show that magnetic-superconducting quasiperiodic patterns, as those found in atomic chains in twisted van der Waals materials, provide a new mechanism to engineer unconventional superconducting states by exploiting quasiperiodic criticality. Our manuscript is organized as follows. First, in Sec. II we introduce a realization of our model, and we show the emergence of a critical point in quasiperiodic superconductorantiferromagnet patterns, separating the extended and localized regime. In Sec. III we show that interactions give rise to a spin-triplet superconducting state, and analyze its dependence with respect to the details of the quasiperiodic modulation. In Sec. IV, we demonstrate the robustness of the interaction induced spin-triplet superconducting state with respect to perturbations in the quasiperiodic Hamiltonian. Finally, in Sec V we summarize our conclusions.

II. ANTIFERROMAGNET-SUPERCONDUCTOR QUASIPERIODIC CRITICALITY
The system that we will study combines a spatially modulated antiferromagnetism and superconductivity, as shown in Fig. 1ab. This type of spatially modulated parameters appears in generic twisted twodimensional materials that combine superconductivity and magnetism. [24] Here we will focus on a specific case in which the system is purely one dimensional. This situation can be realized by taking a twisted graphene multilayer in a superconducting state, [57][58][59] whose superfluid density follows the modulation of the moire pattern, and depositing an array of ad-atoms on top it [60][61][62][63][64][65][66][67] ( Fig. 1a) The ad-atoms will have a long-range antiferromagnetic order stemming from the graphene RKKY interaction, [68] leading to a one-dimensional antiferromagnetic state. [69][70][71] Both electronic orders will have a modulation following the moire pattern, effectively realizing to a one dimensional model with modulated antiferromagnetism and superconductivity. [72][73][74][75] We describe this system combining a modulated superconducting and antiferromagnetic exchange by the following effective Hamiltonian [76][77][78][79]: where c † i,s (c i,s ) denotes the fermionic creation (annihilation) operator for site n and spin s, and σ z is the spin Pauli matrix. The first term denotes the kinetic energy of the system, the second term denotes the spatially modulated antiferromagnetism, and the third term corresponds to the modulated superconductivity. The parameters ∆ and m are responsible for the strength of the modulation corresponding to the antiferromagnetism and superconductivity, respectively, and Ω is the wavelength of the modulation. The model of Eq. 1 assumes that the superconducting state will be stronger when the magnetism is weaker (Fig. 1b), as often happens for spin-singlet superconductivity. For convenience, we will parameterize the superconducting and antiferromagnetic strength as m = λ sin α and ∆ = λ cos α, so that the net strength of the quasiperiodic modulation can be defined by the parameter λ = √ m 2 + ∆ 2 . For irrational values of Ω, the model of Eq. 1 lacks translational symmetry and thus does not accept a description in terms of Bloch states. As a result, the eigenstates of this Hamiltonian are not guaranteed to be extended states, as the Hamiltonian is inherently non-periodic. Whereas random disorder creates localization at arbitrarily small coupling constants in one-dimension, [80] quasiperiodic patterns are known to give rise to a localization transition at finite coupling constant. [52,81] In particular, it is worth noting that for ∆ = 0, our model is mathematically equivalent to the Aubry-Andre-Harper model. [52] Therefore, in the limit of ∆ = 0, the previous model will have a localization transition at m = 2t, so that for m < 2t all the states will be extended and for m > 2t all the states will be localized. As we show below, the generalized model of Eq . 1 with ∆ = 0 shares many of the characteristics of the AAH model, in particular a critical transition at finite coupling constant.
We now address the localization-delocalization transition in the previous model. In order to determine the extended and localized nature of the states, we compute the inverse participation ratio (IPR) of each eigenstate Ψ n as where i runs over all the components of each eigenstate. For localized states whose wavefunction spans a certain number of sites L, the value of the IPR is a finite nonzero number. In stark contrast, for extended states, the value of the IPR scales as 1/L, becoming zero in the thermodynamic limit.
Let us now explore the model of Eq. 1, and in particular, analyze how the localization of the states evolve as we increase the strength of the quasiperiodic modulation λ. We show in Fig.2a the evolution of the IPR for the different eigenstates, as a function of the modulation strength λ for α = π/3, which corresponds to taking ∆/m ≈ 0.57. In particular, as shown in Fig.2abc all the states remain extended up to λ = 2t is reached, at which point all become localized. Note that the single in-gap modes that remain all the time localized correspond to topological edge states. [55,56] This can be systematically studied by looking at the average IPR of the states as a function of ∆ and m. This is shown in Fig. 2d, where it can be seen that a boundary with the functional form ∆ 2 + m 2 = λ 2 = 4t 2 separates the localized region from the extended region. At the previous phase boundary separating extended from localized states, wavefunctions with critical behavior emerge. The different nature of the extended, localized, and critical states can be easily observed by plotting individual wavefunctions. In particular, we show in Fig. 2efg the wavefunction closest to charge neutrality for an extended (Fig. 2e), critical (Fig.   FIG. 3. Interaction-induced spin-triplet superconducting order as a function of the position (a), and spatial profiles of the spin-singlet superconductivity and antiferromagnetism (b). It is observed that the spin-triplet component is maximal in the regions where the spin-singlet superconductivity and antiferromagnetism coexist.
2f) and localized (Fig. 2g) regime. Whereas the extended wavefunctions span over the whole system (Fig. 2e), localized wavefunctions are strongly localized in a few lattice sites (Fig. 2g). The critical wavefunction of Fig.  2f are characterized by multifractal revivals. [82][83][84][85][86][87][88] This will become especially important in the next section, as the multifractal behavior of the states will substantially increase the impact of interactions in the system.

III. INTERACTION-DRIVEN SPIN-TRIPLET SUPERCONDUCTIVITY
Let us now move on to consider the impact of interactions in the previous quasiperiodic system. In particular, we will show that the inclusion of interactions will lead to spin-triplet superconductivity, where the criticality driven by the quasiperiodic pattern can be used as a knob to enhance superconducting order parameter close to the critical point. Local interactions of the form H int ∼ n c † n,↑ c n,↑ c † n,↓ c n,↓ are already accounted for in the stagger antiferromagnet and superconducting terms of the Hamiltonian. Therefore, we will now consider the effect of residual non-local interactions, in particular nearest neighbor density-density interactions. For that sake, we will now include an interaction term in our Hamiltonian of the form where V controls the strength of the nearest-neighbor attractive interaction. We solve the previous interacting term using a mean-field approximation . We will focus in this last term, whose contribution to the mean-field Hamiltonian is of the form where by definition of spin-triplet ∆ s,s n,n+1 = −∆ s ,s n+1,n . As the spin-triplet component of the interaction induced superconducting state has several degrees of freedom, it is convenient to define an spatially dependent d-vector d n,n+1 that parameterizes the spin-triplet superconducting order as where σ are the spin-Pauli matrices. As a first step, it is interesting to look at the real-space distribution of the unconventional superconducting state. In particular we show in Fig. 3 the real-space distribution of the spin-triplet state defined as We observe that the spin-triplet density follows the quasiperiodic pattern, and that it becomes zero in regions only having antiferromagnetism or s-wave superconductivity (Fig. 3ab). Interestingly, we find that such spintriplet component is maximal right in the region where the s-wave superconductivity and antiferromagnetism coexist in the same footing (Fig. 3ab), highlighting the key interplay of magnetism and superconductivity for driving as spin-triplet superconducting state. Moreover, it is interesting to examine the specific type of spin-triplet state that the interactions promote. In particular, we find that the d n,n+1 is always locked to the same direction of the antiferromagnetism, which in terms of the superconducting order parameters is associated to an interaction induced spin-triplet ∆ ↑↓ n,n+1 order parameter for antiferromagnetism in the z-axis.
We now move on to examine the impact of the quasiperiodic criticality on the induced spin-triplet state. As anticipated above, the critical behavior of the wavefunctions is known to provide an effective mechanism for enhancing electronic instabilities. [89][90][91][92][93] To verify this, we now compute the selfconsistent spin-triplet order parameter of Eq. 6 as a function of the modulation strength λ, as shown in Fig. 4a. It is observed that, as the modulation strength increases, the induced spin triplet parameter grows, becoming maximal around the critical point and decreasing as the system goes deeper into the localized regime. The enhancement associated to the critical point can also be verified by computing the induced spintriplet order parameter as a function of the interaction strength V , as shown in Fig. 4b. It is seen that for all coupling constants, the spin-triplet state is stronger close to the critical point λ = 2t, than deep into the extended (λ = t) or localized (λ = 5t) regime.
We now move on to examine the impact of the two quasiperiodic modulations as parameterized by α. As we showed above, the extended-localized transition takes place for λ = 2t, and independently on the value of α. This means that a critical point appears independently on the relative strengths between ∆ and m, and it only depends on λ = √ ∆ 2 + m 2 . In stark contrast, the emergence of a spin-triplet component due to interactions turns out to be highly dependent on α, as shown in Fig. 4c. In particular, we observe that when the system is purely antiferromagnetic or purely superconducting (α = 0, π/2) the spin-triplet component generated is exactly zero. In comparison, the spin-triplet state is maximal for α = π/4, that corresponds to having an equal weight on the singlet superconducting and antiferromag- netic order parameters. This observation emphasizes the importance of the coexistence of antiferromagnetism and superconductivity for the emergence for the interaction induced spin-triplet state.
We finally consider the impact of the spatial modulation frequency Ω in the interaction induced spin-triplet state. As shown in Fig. 4d we observe that the enhancement at the critical point happens for generic values of the modulation frequency. In the limit of small Ω, the system is essentially formed by patches of superconductor and antiferromagnet, [76][77][78][79] having a typical length on the order l ∼ 1/Ω, and thus the interaction between the superconducting and antiferromagnetic state happens in a limited part of the system. In comparison, for Ω ≈ π/2 there is a quick oscialltion between the two orders, promoting a dense coexistence of antiferromagnetism and spin-singlet superconductivity in the system. We observe that the interaction induced spin-triplet component is especially strong in this regime (Fig. 4d), reflecting the key interplay between spin-singlet superconductivity and antiferromagnetism for driving the unconventional superconducting state.

IV. ROBUSTNESS TO PERTURBATIONS
In this section, we address the robustness of our phenomenology with respect to perturbations. In particular, we will focus on the impact of next to nearest-neighbor hopping and Anderson disorder. The next to nearestneighbor hopping breaks the bipartite nature of the lattice, whereas the Anderson disorder would drive the system to a localized state for all λ. In particular, we obtain that the critically enhanced spin-triplet superconducting state also happens with those additional perturbations, as elaborated below. Second neighbor perturbations are expected to appear in a realization of the previous model and break the original bipartite nature of the system. The previous term is included in our Hamiltonian by means of a perturbation of the form The results with this additional perturbation are shown in Fig. 5a, where we took t = 0.2t. It is observed that the enhancement of the interaction-induced spin-triplet state happens in the presence of this additional perturbation. It is worth to note that in the presence of second neighbor hoppings, the localization-delocalization transition becomes state dependent, and it will no longer happen at λ = 2t. Nevertheless, it is observed that the qualitative behavior remains analogous to the idealized case with t = 0. This is especially important for potential realizations of our model in twisted two-dimensional materials and cold atoms setups, as generically these systems present small additional contributions to the Hamiltonian such as a second neighbor hopping. Next, we consider the impact of random disorder in the system, included as an onsite Anderson perturbation.
where n is a random number between [−0.1, 0.1]t. First, it is interesting to note that the inclusion of an arbitrarily small amount of disorder would drive the extended states to a localized regime. As a result, in the presence of disorder the localization-delocalization transition as a function of λ is completely destroyed, as the state becomes localized for all λ. The disorder strength λ will define a minimal localization length for the system. As λ is ramped up, the system will go from a localized regime dominated by the disorder, to a regime in which the localization is dominated by the quasiperiodic potential. Although the critical point is washed out, the enhancement of the superconducting state will still be visible at this quasiperiodic-disorder localization crossover. This is shown in Fig. 5b, where it is seen that the spin-triplet enhancement close to the former critical point is still visible. This phenomenology shows that even in experimental setups that host small imperfections, the enhancement of an unconventional spin-triplet superconducting state can be observed.

V. CONCLUSION
To summarize, we have demonstrated that antiferromagnet-superconductor moire patterns show a critical point associated with a localization-delocalization transition. We showed that the quasiperiodic criticality happens for arbitrary ratios between the superconducting and antiferromagnetic order parameters, and that the critical point is universally located in a curve defined by the two order parameters. Upon inclusion of residual electronic interactions, we demonstrated the emergence of an unconventional spin-triplet state, whose d-vector is locked along the antiferromagnetic spin direction. We showed that the emergence of this unconventional superconducting state is finely related to the interplay between antiferromagnetism and superconductivity, having a spatially inhomogeneous superconducting order maximal when the two parent orders coexist. We finally showed that this phenomenology happens for generic quasiperiodic modulation frequencies and survives the presence of perturbations to the Hamiltonian. Ultimately, the phenomenology presented can be realized in twisted graphene superlattices with atomically engineered impurities, and generically on moire patterns between two-dimensional antiferromagnets and superconductors. Our results put forward antiferromagnetic-superconducting quasiperiodicity as a powerful knob to engineer robust superconducting states, providing a new route towards the design of artificial unconventional superconductors.