Progresses on topological phenomena, time-driven phase transitions, and unconventional superconductivity

The last decade has seen the condensed matter community focus on three primary fields that regard ultrafast material science, topological phase transitions and unconventional superconductivity. Fundamental discoveries in these fields could have a huge impact in technological applications that range from spintronics to quantum computation and quantum simulations.

Regarding ultrafast materials science, the counterintuitive experimental observation of light-induced superconductivity [1] has stimulated a lot of interest in the field and soon the interest has developed also towards the ultrafast magnetism. In fact, magnetic switching is the basis for numerous applications, such as the processing and storage of information on magnetic media. Magnets are typically polarized or switched by a magnetic field pulse, which exerts a torque on the spins to reverse them. But an attractive alternative is to use light pulses to modify the magnetic state of a material. Nowadays one can produce bursts of photons with a pulse width of less than one femtosecond $(1\ \text{fs}=10^{-15}\ \text{s})$ —far faster than the time in which a magnetic field pulse can be generated. There is, however, still much to learn about the ways short light pulses interact with the magnetic properties of a material. The field of ultrafast spin dynamics —or femtomagnetism— that is developing around this topic has still researchers working to tackle fundamental questions, like the generation and detection of spin waves [2].

Regarding topological phase transitions, the search for Majorana fermions has quickly become a primary goal in condensed matter community. The intense interest is related to their definition: a Majorana fermion is a fermion that is its own antiparticle. While sophisticated particle physics experiments are testing for Majorana character in neutrinos propagating in three dimensions [3], solid state physicists are more interested in lower-dimensional counterparts. The most interesting Majorana fermions that are predicted to appear in materials are zero-dimensional bound states, called Majorana bound states (MBSs). Once discovered, MBSs are supposed to exhibit exotic non-Abelian statistics when exchanged among each other. While of great fundamental interest, perhaps the biggest driving factor in the search is a well-regarded proposal for (topological) quantum computation, which uses this unique statistical property of the MBS to robustly process quantum information free from local sources of decoherence [4,5]. The seminal proposal of Fu and Kane predicts that if the surface of a three-dimensional topological insulator is proximity-coupled to an s-wave superconductor, then vortex lines in the superconductor will trap MBSs where the lines intersect the topological insulator surface [6]. This proposal requires two main ingredients: i) a topological insulator and ii) an s-wave superconductor that can effectively proximity-couple to the surface of the topological insulator. Despite the recent discovery of three- and two-dimensional topological insulators [7,8] finding a suitable topological insulator for these experiments is still a difficult task. The reason being that, as of yet, there are no topological insulator materials that are completely insulating in the bulk, despite intense experimental programs dedicated to this task. The effect of interactions, for example, has been recently afforded in a topological quantum point contact [9]. In general, beyond interaction effects, quasiparticle poisoning is the main obstacle towards the realization of Majorana-based quantum computation specially in hybrid superconducting heterostructures. Parafermions are a natural generalization of Majorana fermions and can encode topological qudits immune to quasiparticle poisoning and recent proposals have appeared, based on photonic platforms, as the once discussed afterward [10,11]. Other proposals for the search of Majorana fermions are instead based on novel platforms where proximity effects are ruled out by intrinsic superconductivity in combination with unconventional spin-orbit coupling, magnetism, and two-dimensional (2D) superconductivity, as the quasi-2D electron gases (q2DEG) at the interface between band insulating oxides, like LaAlO₃ (LAO) and SrTiO₃ (STO). These systems naturally possess most of the fundamental characteristics needed for the realization of a topological superconductor [12].

Regarding unconventional superconductors, the full understanding of these materials still eludes the physics community and represents one of the main concerns in condensed matter physics [13]. Most of the experimental results in this field have yet to be explained and remain an outstanding challenge for theory. In recent years, many new classes of superconductors, whose behavior deviates significantly from traditional superconductivity, have been discovered. The origin of the superconductivity of these materials is certainly different from the interactions that sustain conventional superconductivity. Realizing an accurate theory for each of these classes of superconductors has proved to be a serious challenge and will surely remain one of leading issue for future researches. Moreover, understanding the superconducting unconventional coupling could pave the way for further advances in high-temperature superconductivity. In light of this, in this perspective we present a roundup of the main families of unconventional superconductors, giving an insight on recent advances in this research area.

Due to the lack of space and the vastness of the bibliography existing nowadays in these fields, it is fair to point out that the list of works presented in this perspective is quite limited and the discussion is intentionally as non-mathematical as possible to simply offer an overview of topics that are still in evolution.

Identifying and understanding all the processes which prevent thermalization and decoherence in driven-dissipative quantum systems [14] is a unifying theme in condensed matter research [15]. This comes with the potential to realize exotic out-of-equilibrium quantum phases, both for the process of pushing fundamental research and for wider technological purposes. In ref. [16] a review of geometrical methods employed in the analysis of quantum phase transitions and non-equilibrium dissipative phase transitions was provided. In ultrafast materials science, the counterintuitive experimental observation of light-induced superconductivity [1] has stimulated a lot of interest in the field. Experimental demonstrations of phonon-induced changes in electronic ground states, i.e., evidence of strong vibrational excitations that can induce electronic phase transitions capable of triggering transient superconductivity and metallicity, were first reported in manganites [17], cuprates [18], and nickelates [19]. In these experiments intense laser pulses have been reported to induce superconducting-like features, such as an inverse-frequency divergence of the imaginary part of the optical conductivity and vanishing resistivity, well above the materials' equilibrium critical temperatures T_c . A number of theoretical studies have explored the effects of carefully tuned coherent driving on the prethermal dynamics of one- and two-dimensional bi-partite fermionic lattice models [20–22]. These studies are motivated by the opportunities arising from having dynamical time-dependent Hubbard parameters, which have been experimentally realised in contexts ranging from quantum simulators [23] to strongly correlated materials via electronic [24] as well as vibrational excitations [25]. Recently the interest has moved to explore also organic materials $k-$(BEDT-TTF)₂Cu[N(CN)₂]Br that can be modeled by a triangular Hubbard model where the effect of frustration and hopping anisotropy can lead to rich equilibrium phase diagrams containing unique states of matter. The interplay between equilibrium states and generic periodic driving manifests complex non-equilibrium behavior and have been recently discussed in ref. [26].

Very recently, selective excitation of elementary vibrations of the crystal lattice (phonons) by ultrashort pulses of light has also emerged as a novel, low-energy route to control magnetoelectric materials (see fig. 1). This has opened the route to ultrafast magnetic switching in which one excites spin dynamics via electronic absorption [27]. In ref. [2] light-induced magnetism has been investigated in dysprosium orthoferrite (DyFeO₃), a magnetoelectric material with one of the strongest interactions between spins and the crystal lattice [28]. It was shown that a sub-ps pulse of an intense mid-infrared electric field, tuned to resonance with a phonon mode, drives a coherent spin reorientation within a half-cycle of the spin precession, developing long-living weak ferromagnetic order. Light-induced magnetism emerges via a non-equilibrium metastable route, inaccessible via a thermodynamic transformation. Also magnetic van der Waals materials provide an ideal playground for exploring the fundamentals of low-dimensional magnetism and open new opportunities for ultrathin spin processing devices. Using ultrashort pulses of light, it has been demonstrated all-optical control of magnetic anisotropy in the two-dimensional van der Waals antiferromagnet NiPS₃. Tuning the photon energy in resonance with an orbital transition between crystal-field split levels of the nickel ions, the selective activation of a sub-THz 2D magnon mode has been demonstrated [29]. This result establishes pumping of orbital resonances as a universal route for manipulating magnetic order in low-dimensional (anti)ferromagnets. Both experiments in refs. [2,29] open the route to an alternative ability to encode spin information with minimal energy dissipation which is of great scientific relevance.

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The quantum spin Hall effect has been predicted in several systems [30] and was first realized in HgCdTe/HgTe quantum wells [8]. Later, this phase has been observed in other semiconducting material systems such as InAs/GaSb double quantum wells [31] and in monolayers of WTe₂ and bismuthene [32]. The main properties of this state are related to the helical nature of the edge states that emerges thanks to spin-momentum locking and are nowadays well established by numerous experiments. One of these is the observation of conductance quantization of two-spin-polarized edge channel ${G_0 = 2e^2/h}$, with e the electron charge and h the Planck's constant [30]. Additionally, non-local edge transport and spin polarization of the edge channels were demonstrated by suitable transport experiments [33]. Recently, a still open question has been targeted, namely, how helical edge states interact with each other when brought close. A topological quantum point contact has been employed to guide edge channels from opposite sides into a quasi–one-dimensional constriction, based on inverted HgTe quantum wells [9]. Apart from the expected quantization in integer steps of $2e^2/h$, an additional plateau at $e^2/h$ has been surprisingly found. This characteristic has been explained both by combining band structure calculations and repulsive electron-electron interaction effects captured within the Tomonaga-Luttinger liquid model. Investigations of this 0.5 anomaly give a hint on the importance of the underlying band structure. Especially, the difference between a Dirac point in the band gap and one buried in the valence band implies a scattering term, which opens a spin gap. Furthermore, the results could be important for the detection of MBSs since the identified mechanism might be related to the observation of the $4\pi$-periodic Josephson current in HgTe Josephson junctions in the absence of an explicit time-reversal symmetry breaking mechanism. In general, beyond interaction effects, quasiparticle poisoning is the main obstacle towards the realization of Majorana-based quantum computation specially in hybrid superconducting heterostructures [34]. Parafermions are a natural generalization of Majorana fermions and can encode topological qudits immune to quasiparticle poisoning. While parafermions are expected to emerge in superconducting fractional quantum Hall systems they are not yet attainable with current technology. To bypass this problem, a photonic quantum simulator has been employed to experimentally demonstrate the key components of parafermion-based universal quantum computation [10]. By manipulating the photonic states the Clifford operator Berry phases have been realized corresponding to parafermion braiding. The advantage that parafermion qudits have against non-topological ones when they are exposed to local noise was finally established. This photonic quantum simulation provides the first step towards a physically robust methodology for realizing topological quantum computation.

Another possibility to demonstrate quantum manipulations using MBS is to take advantage of their topological character. Recently how to design various non-standard types of Andreev bound-state (ABS) dispersions has been described, via a composite construction relying on MBSs [35]. The MBSs appear at the interface of a Josephson junction consisting of two topological superconductors (TSCs). Each TSC harbors multiple MBSs per edge by virtue of a chiral or unitary symmetry. Interestingly it has been found that, while the ABS dispersions are $2\pi$-periodic, they still contain multiple crossings which are protected by the conservation of fermion parity. A single junction with four interface MBSs and all MBS couplings fully controllable, or networks of such coupled junctions with partial coupling tunability, open the door for topological band structures with Weyl points or nodes in synthetic dimensions. The possible experimental demonstration of ABS engineering in these devices further promises to unveil new paths for the detection of MBSs and higher-dimensional chiral anomaly.

Beyond engineering topological states in nanostructures based on proximity superconducting phenomena, innovative technological platforms for the realization of topological quantum systems based on novels material have been put forward. An example are the q2DEGs at the interface between band insulating oxides, like LAO and STO. Being characterized by a unique combination of unconventional spin-orbit coupling, magnetism, and 2D superconductivity, these systems naturally possess most of the fundamental characteristics needed for the realization of a TSC. These properties can be widely tuned by electric field effect acting on the orbital splitting and occupation of the non-degenerate 3d_xy and 3d_xz 3d_yz bands. The topological state in oxide q2DEGs quasi–one-dimensional nanochannels could be therefore suitably controlled, leading to conceptual new methods for the realization of a topological quantum electronics based on the tuning of the orbital degrees of freedom [12]. Even if the road towards possible quantum applications is very long, both theoretical and experimental developments in the field provide promises in the right directions.

Unconventional superconductivity represents nowadays an active frontier of condensed matter physics. The variety of superconducting materials showing properties that cannot be described by the Bardeen, Cooper, and Schrieffer (BCS) approach is continuously growing. Indeed, unconventional superconductivity refers to superconductors that deviate from the BCS description, i.e., in which the Cooper pairs are not coupled together by phonon exchange but rather by exchange interactions of some other kind [36–38]. In unconventional superconductors, in addition to the one-dimensional global gauge symmetry U(1), other symmetries, e.g., time-reversal or reflection symmetry, are often broken at the onset of superconductivity [39]. Since the symmetry properties of the order parameter reflect the symmetry breaks affecting the Cooper coupling state, phase-sensitive analysis, such us Josephson tunneling measurements, can unravel the nature of the order parameter symmetry of these systems [39].

The first superconductor recognized as unconventional was CeCu₂Si₂, showing a critical temperature equal to $T_c=0.6\ \text{K}$, as reported by Steglich et al. in 1979 [40]. After that, different classes of unconventional superconducting materials have been identified, i.e., cuprates, iron-based, heavy fermion, non-centrosymmetric, organic, topological, and interfacial superconductors, although in many cases the physics mechanisms causing the superconductivity are far from being fully understood. Cognizant of the continuing vibrant developments in this field, here we present briefly the unconventionality of few of these emerging superconductors.

The discovery of unconventional superconductivity in stratified cuprates occurred first in 1986 in Ba (hole)-doped La₂CuO₄ [41], with $T_c = 35\ \text{K}$, and shortly after Wu et al. disclosed the superconductivity at $93\ \text{K}$ in what was later identified as YBa₂Cu₃O$_{7-\delta}$ [42]. The properties of this class of materials are driven by electronic interactions within the CuO₂ plane, which is a universal feature of all cuprates due to their anisotropic 2D structure that drastically weakens the out-of-plane interlayer coupling. Despite the search of high-temperature superconductivity led to the discovery of many cuprate superconductors, also with a T_c as high as $134\ \text{K}$ [43] (up to $T_c=164\ \text{K}$ under high pressure [44]), there is still a hot debate for understanding many fundamental issues of both electron- and hole-doped cuprates [45–47]. In fact, the cuprate superconductors exhibit unusual collective phenomena [48], including unconventional spin and charge density modulations [49–54], Fermi surface reconstructions, and a pseudogap in various physical observables [55,56], but investigations of the commonalities and differences of cuprates across the optimal doping level are continuously in progress [54,57,58].

Iron-based superconductors consist of different classes of iron materials, like pnictides, chalcogenides, intermetallics, and oxides [59]. First discovered by Kamihara et al. [60,61], they continuously trigger interest in novel physics and applications [62], including the emergence of peculiar aspects related to unusual charge and spin ordering. Iron-based superconductors may be even influenced by magnetism and magnetic fluctuations, since, unlike the cuprates, in this case magnetism can coexist with superconductivity [63–65]. In these superconductors correlations are orbital selective and the connection between Hund's metal and Mott physics is still a hot matter of discussion [66–70].

In trying to boost the critical temperature of these materials, experiments focused also on layering the compounds. In fact, in addition to ongoing discoveries on bulk iron-based superconductors, efforts have been devoted to study interfacial superconductivity involving monolayer of FeSe (bulk $T_c=8\ \text{K}$) on STO, showing a $T_c=109\ \text{K}$ in single layer FeSe on STO [71]. Other examples of interfacial superconductivity are reported in the literature [72,73], since electronic systems with unusual properties can be established at interfaces between complex oxides. This is the case, for instance, of the superconductivity in the electron gas formed at the interface between two insulating dielectric perovskite oxides, LAO and STO [74–79]. Despite considerable experimental and theoretical efforts, the microscopic details of superconductivity in these systems are still not fully comprehended. Since phase-sensitive measurements could unveil the symmetry of the superconducting order parameter, new paradigms for the creation of superconducting circuit elements based on LAO-STO interfaces were conceived, where local gates enable the in situ creation and control of Josephson devices [80–83].

It is also worth mentioning that the Josephson effect, brought up here as a tool for detecting unconventional superconductivity, can show itself a sort of "unconventionality". In essence, we are referring to the so-called anomalous Josephson effect, emerging when both time-reversal and inversion symmetries are broken, so that a finite phase shift $0 < \varphi_0 < \pi$ can be induced [84] and the current-phase relation becomes $I_J(\varphi)=I_c\sin(\varphi+\varphi_0)$, with φ being the phase difference between the wave functions describing the superconducting condensate in the two electrodes. These so-called $\varphi_0$-junctions [85] can generate a constant phase bias $\varphi= -\varphi_0$, in an open-circuit configuration, and an anomalous Josephson current $I_J=I_c\sin(\varphi_0)$, if enclosed in a superconducting loop. Recently, both experimental [86–89] and theoretical [90–92] works continue apace, also in the light of concrete applications in superconducting electronics and spintronics [93,94].

Still staying in the realm of 2D superconductivity, we conclude this section by citing the recent discovery of nonconventional superconductivity in the so-called magic angle graphene [95], from which a plethora of experimental and theoretical works turned out. Graphene was also efficiently inserted into Josephson setups [96,97], on the path of promising applications [98,99] and fresh theoretical investigations, including noise phenomena studies [100–104].

In conclusion, we have given an overview of three emerging fields of condensed matter physics which in recent years have attracted considerable attention. In particular, we have considered the recent challenging topics on time-dependent phase transitions, topological phenomena, and superconductivity, with the aim to foster the community towards new ideas, applications and technological advancements.