Strong Interactions and Correlations

  • Edoardo Baldini
Part of the Springer Theses book series (Springer Theses)


One of the most intriguing yet challenging fields of research in contemporary condensed matter physics is the investigation of many-body effects in strongly correlated quantum systems. This class of materials provides an excellent playground for discovering exotic phenomena involving charge, lattice, spin and orbital degrees of freedom and leading to extraordinarily varied chemical and physical properties. Understanding electronic correlations in prototypical systems like cuprates and manganites can pave the route to the potential design and engineering of novel materials with tailored functionalities.

One of the most intriguing yet challenging fields of research in contemporary condensed matter physics is the investigation of many-body effects in strongly correlated quantum systems. This class of materials provides an excellent playground for discovering exotic phenomena involving charge, lattice, spin and orbital degrees of freedom and leading to extraordinarily varied chemical and physical properties. Understanding electronic correlations in prototypical systems like cuprates and manganites can pave the route to the potential design and engineering of novel materials with tailored functionalities. As a consequence, the main goal of current research is to identify the underlying mechanisms shared by these solids, so that their emerging phenomena can be accounted for by new theoretical models. In this Chapter, we describe the current status of understanding of the paradigm behind these systems. In Sect. 1.1, we put strong emphasis on the role played by the excitation spectrum in reflecting the signature of the many-body interactions and correlations that simultaneously act in the solid. In this regard, we provide some prototypical examples by gradually increasing the degree of complexity from the weakly correlated Fermi liquids to the systems governed by an intermediate correlations regime, which is the most difficult to address on both the theoretical and experimental sides. In Sect. 1.2, we introduce the concept of collective excitations in many-body systems, and explain why their study can provide valuable insights in the behaviour of complex materials, and open up new avenues for technological developments. Finally, in Sect. 1.3, we give a perspective on the intriguing frontiers offered by the physics of collective excitations for addressing long-standing problems in condensed matter research and for reaching the control of the material functionalities at ultrafast timescales.

1.1 The Paradigm of Strong Interactions and Correlations

The search for a complete theoretical framework to capture the physics of strong interactions and correlations in complex materials is an old problem faced by condensed matter physics. It has deep roots in the history of physics soon after the formulation of the band theory by Bethe, Sommerfeld and Bloch [1, 2, 3]. Band theory provided an intuitive yet profound description of the quantum mechanical behaviour of electrons in solids, leading to explain the origin of the difference between metals and insulators. One of the cornerstones of this theory is the assumption of a free electron model in metals, in which the interactions among electrons are completely neglected. It is therefore natural to believe that the first considerations on the transport and photoconductivity properties of NiO created a huge surprise in the physics community [4]. While band theory was predicting NiO to behave as a metal, experimental data were pointing towards a heavily insulating scenario. Peierls and Mott readily recognized that the breakdown of band theory in the description of NiO had to be related somehow to the presence of an incomplete d electron shell in the transition metal ions [5], whose main role is to produce a non-negligible Coulomb interaction among the electrons. This observation opened the doors to new frontiers in condensed matter physics and led to coin the term “Mott insulator” to describe a solid violating band theory.

Since then, a number of systems has been found to contradict the expectations of band theory. The simplest examples of metals at low temperatures could be readily treated within the framework of Landau’s theory of Fermi liquids [6], where the role of electron-electron interactions was reduced to small corrections of susceptibilities of the free electron gas. Bardeen-Cooper-Schrieffer (BCS) theory of phonon-mediated superconductivity [7] and the subsequent extension by Eliashberg [8] unveiled the need of considering the electron-boson interaction for explaining the superconducting (SC) instability in Fermi liquids. It was soon recognized that the physics of more complex solids with incomplete d- or f- electron shells required the inclusion of both electron-boson interactions and electron-electron correlations, which could not be simply considered as weak perturbations of the free electron gas regime. Although this called for the search of more exotic microscopic mechanisms beyond band theory, for long time the task of extending the quantum theory of solids was just considered as a mere puzzle for theorists, without any possible implications for the field of technology. New enthusiasm towards this area of research followed the unexpected discovery of a variety of exotic electronic phenomena, such as high-temperature (high-\(\mathrm {T_C}\)) superconductivity [9] and colossal magnetoresistance [10], which hold huge promise for the development of future devices and applications. Indeed, such phenomena are thought to be governed by two distinctive and common features, that have been revealed by a number of spectroscopies: (i) The presence of strong interactions (strong electron-boson coupling), which gives rise to effects like polaronic transport, incipient charge- and spin-density wave (CDW, SDW) instabilities and renormalizations of the electronic structure [11]; (ii) A subtle interplay between low- and high-energy scales, whose origin lies in the strong electronic correlations spreading the material’s spectral-weight (SW) over a wide energy range [12].

Corollary to these features is the dual nature that electrons exhibit in these materials [13]. For example, in most situations, itinerant carriers originating from wavefunction hybridization are found to coexist with localized carriers generated by boson-mediated carrier dressing, electron-electron repulsion and intrinsic inhomogeneities. Unravelling the details of this duality would shed light on many of the unsolved enigmas of contemporary condensed matter physics.

As the main source of uncertainty lies in the electronic structure of many materials, experimental probes of the many-body electron motion and energetics have been widely applied [12, 14, 15]. The deep understanding of a solid’s excitation spectrum in response to an external perturbation provides a benchmark for tests of theoretical models and can lead to the comprehension of the elusive ground state (GS) properties of the material. It is therefore pivotal to clarify the effects that strong interactions and correlations produce on the electronic structure of a solid, which become apparent in the so-called one- and two-particle excitation spectra of the material. The one-particle excitation spectrum results from the removal or the addition of an electron in the solid and represents a signature of charged excitations that can be probed by photoemission and inverse-photoemission spectroscopies [14]. The two-particle excitation spectrum arises instead from the interaction between the solid and a low-energy light field (i.e. with a photon energy below the material ionization threshold), resulting in the creation of neutral excitations that are detectable by optical probes [12].
Fig. 1.1

Schematic diagram representing the impact of electron-boson interaction and electron-electron correlations on the one- and two-particle excitation spectrum of a solid. In the top panels, the momentum-resolved single-particle spectral function is shown for a a weakly correlated, b an interacting and c a correlated material. The two particle spectral function is represented in the three cases in Panels (d), (e) and (f), respectively. Panels (g), (h) and (i) display the frequency-dependent scattering rate retrieved in the three different cases.

The figure has been adapted from Ref. [12]

To introduce a qualitative picture of how the electron-boson interaction and the electron-electron correlation renormalize the electronic structure of a material, it is instructive to consider first the example of Fermi liquids. In the simplest picture, the Fermi liquid quasiparticles, obeying the Pauli exclusion principle, reside in a partially filled parabolic band (Fig. 1.1a). The electrodynamic response of the system can be described by the Drude model [16], in which the complex optical conductivity is expressed as
$$\begin{aligned} \sigma (\omega ) = \frac{\widetilde{n} e^2 \tau _D}{m_b}\frac{1}{1-i\omega \tau _D} = \frac{\sigma _{dc}}{1-i \omega \tau _D}, \end{aligned}$$
where e is the electronic charge, \(\widetilde{n}\) is the effective carrier density, and \(m_b\) is the band mass of the carriers (which typically differs from the free electron mass \(m_e\)), 1/\(\tau _D\) is the scattering rate, and \(\sigma _{dc}\) is the dc conductivity that can be probed by transport. From this description, it results that the electrodynamic response of the Fermi liquid is characterized by the emergence of a Lorentzian peak centered around zero frequency and related to intraband processes (red feature in Fig. 1.1d). The contribution of interband transitions (blue feature in Fig. 1.1d) is found to arise at higher energies and is typically well reproduced by band structure calculations. The electronic kinetic energy in a Fermi liquid is proportional to the SW associated with the intraband Drude contribution.

To describe the emergence of electron-boson coupling, a milestone is represented by Eliashberg theory of interactions [8]. The presence of a strong interaction with a bosonic collective mode at \(\Omega _0\) leads to a renormalization of the electronic energy-momentum dispersion near the Fermi level \(E_F\) (Fig. 1.1b). The spectral shape of the optical conductivity undergoes a marked change, revealing an absorption sideband at \(\omega >\Omega _0\) (Fig. 1.1e). This absorption sideband can take the form of a satellite progression or can merge into a broad incoherent background as the electron-boson coupling strength increases. In any case, the SW associated with this sideband is transferred from the coherent Drude peak, thus conserving the total SW of the system typically below 1 eV. Consequently, electron-boson interaction alone does not alter the kinetic energy with respect to the Fermi liquid scenario. The frequency-dependent scattering rate 1/\(\tau _D\)(\(\omega \)) shows a step in the proximity of \(\Omega _0\), which is the signature of an enhanced scattering probability at higher energies (Fig. 1.1h). The same concept of electron-boson interaction is also central to the physics of insulators, as excitons and high-energy interband transitions can display strong couplings to other collective modes, modifying their profile [17].

We conclude by discussing the case of strong electron-electron correlations. Once correlations are introduced, they manifest as a pronounced broadening of the energy-momentum dispersion relation far from from \(\mathrm {E_F}\) (Fig. 1.1c), thus suggesting that the concept of weakly damped Landau quasiparticles does not hold over the entire energy range. This leads to a remarkable reduction of the low-energy SW compared to band-structure expectations, suppressing the value of the kinetic energy. Importantly, the SW removed from the coherent Drude response is transferred to high energies of the order of the on-site Coulomb energy U and/or of interband transitions (violet feature in Fig. 1.1f). At the same time, large values of the scattering rate are reached and become evident in the two-particle excitation spectrum (Fig. 1.1i).

In the following sections, we focus on several examples that will be extensively treated in the framework of this Thesis. All these examples involve scenarios in which the electron-boson interaction and the electron-electron correlations act either separately or simultaneously to offer a variety of emergent phenomena in quantum many-body systems.

1.1.1 Electron-Phonon Interaction in the Strong Coupling Regime

By adopting an increasing degree of complexity in the many-body problem, the first type of interaction that can be considered is the one that establishes among electrons and phonons. Although this interaction is present to a certain extent in all solids, it is its magnitude and structure that lead to remarkable consequences on the behaviour of some materials. In general, the electron-phonon interaction describes how the energy of an electronic charge is affected by alterations of the atomic positions in the encompassing medium, and can be written in the form
$$\begin{aligned} V_{ep} = \frac{1}{\sqrt{\Omega }} \sum _\mathbf{k ,\mathbf q ,\eta } M_{\eta }(\mathbf k +\mathbf q , \mathbf k ) A_\mathbf{q ,\eta } c_\mathbf{k +\mathbf q ,\sigma }^{\dagger } c_\mathbf{k ,\sigma }, \end{aligned}$$
where \(A_\mathbf{q ,\eta } = a_\mathbf{q ,\eta } + a_{{\text{-q }},\eta }^{\dagger }\). The operators \(c_\mathbf{k +\mathbf q ,\sigma }^{\dagger }\) \(c_\mathbf{k ,\sigma }\) describe scattering of an electron with spin \(\sigma \) from the band state k to k+q, while destroying \(a_\mathbf{q ,\eta }\) or creating \(a_{{\text{-q }},\eta }^{\dagger }\) a phonon in band \(\eta \) with wave vector q. The main task for theory and experiment is to derive expressions and reveal the details of the matrix element \(M_{\eta }(\mathbf k +\mathbf q ,\mathbf k )\) regulating this process.

In the presence of a sizeable coupling, the atoms of the lattice undergo a local displacement around the electron during its propagation through the crystal. This, in turn, can have a tremendous impact on the behaviour of the system, producing instabilities of the electronic structure towards other phases of matter or strongly renormalizing the charge dynamics. In the following, we describe two remarkable manifestations of this interaction, namely the phonon-driven SC instability in conventional metals and the concept of polarons in the specific case of insulators. Here, only the salient features that are relevant to the topics of the following Chapters will be discussed. A more detailed description of the quantum many-body theory behind the electron-phonon interaction is provided in books such as Refs. [18, 19, 20].

The electron-phonon interaction has acquired renewed interest from the late 1950s, right after the formulation of the BCS theory of superconductivity [7]. According to the BCS theory, the SC transport in conventional metals is realized upon the formation of bound pairs of electrons (i.e. Cooper pairs) via the exchange of virtual phonons. This interaction is frequency dependent and becomes attractive for specific frequency regions. As a result, only the electrons within a Debye energy of the Fermi surface undergo pairing, thus becoming part of a many-electron eigenfunction describing the GS of the superconductor. The success of this theory lies in the correct description of the SC properties of weak superconductors, in which the magnitude of the electron-phonon interaction is rather small. Further extensions of this theory by Eliashberg [8] allowed to capture the properties of those materials that lie in the strong coupling regime, such as lead and tin [21]. The parameter discriminating the weak and the strong coupling regimes is represented by the electron-phonon mass enhancement factor \(\lambda \) [22]. The discovery of new classes of superconductors, such as cuprates [9] and pnictides [23], has shed light on the fragility of the BCS framework in providing a universal description of the SC state of matter. Despite this limitation, BCS theory still represents one of the most valuable successes of many-body physics and its influence persists even today in the formulation of novel theories of superconductivity.

In this regard, more recent examples of phonon-mediated superconductivity at \(\mathrm {T_C}>35\) K have paved the route to new applications of BCS/Eliashberg theories. The discovery of superconductivity at 39 K in the layered system MgB\(_2\) [24] and that at 203 K in H\(_2\)S under pressure [25] have come as huge surprises to the physics community. While the case of H\(_2\)S is still under intense debate at the time of writing to clarify the strength of the electron-phonon coupling, the lattice anharmonicity and the involvement of multiple bands in the SC properties [26, 27], the case of MgB\(_2\) has reached a general consensus concerning its main properties. It is nowadays well established that MgB\(_2\) represents a prototypical example of two-band superconductor [28, 29, 30, 31, 32, 33, 34, 35, 36, 37], in which two distinct electronic bands (the \(\sigma \) and \(\pi \) bands) crossing \(\mathrm {E_F}\) contribute to the SC properties by offering distinct coupling strengths with the phonon degrees of freedom [38, 39, 40, 41, 42]. In other words, MgB\(_2\) embodies a rare type of phonon-mediated two-band superconductor in the strong coupling regime and, as such, has been the perfect playground to go beyond the established Eliashberg theory by implementing its anisotropic version [28, 43]. The phonon of interest is the optical boron stretching branch along the \(\Gamma \)-A direction in the BZ, which gives rise to a pronounced peak around 70 meV in the Eliashberg function \(\alpha ^2\)F(\(\omega \)) and is characterized by a large degree of anharmonicity [40, 44]. Besides governing the SC properties of MgB\(_2\), this phonon mode is also expected to play a central role in the normal state properties of the material, offering an off-diagonal scattering channel that connects the two bands. This phonon-mediated interband scattering process is intimately related to the electron-phonon coupling mechanism and has remained elusive over time, despite 15 years of research on MgB\(_2\). As this mechanism reflects a dynamical manifestation of the electron-phonon coupling, it cannot be addressed directly via equilibrium spectroscopy, thus requiring the intrinsic time resolution of ultrafast spectroscopies to be unraveled. The first excursion of this Thesis towards the comprehension of the electron- phonon coupling mechanism involves the real-time observation of the phonon-mediated \(\sigma \)-\(\pi \) interband scattering in MgB\(_2\), to which Chap.  3 is devoted. To track the dynamics of this process, we rely on the renormalization of a high-energy plasmon mode that can be detected by measuring the material transient reflectivity [45].

Besides the BCS/Eliashberg theory of superconductivity, another key manifestation of the electron-phonon coupling in solids is the emergence of polarons. The motion of an electronic charge through the lattice generates a potential well that, after overcoming a threshold and becoming sufficiently deep, can even bind the carrier itself. In this situation, the carrier is said to “self-trap”, i.e. it is bound in the well that it induces and cannot escape without a significant movement of the ions. The unit comprising the self-trapped carrier and the associated pattern of displaced ions is typically known as “polaron”. The polaronic coupling results in a renormalization of the bare electron mass, velocity, and scattering processes, thus leading to a slow motion of the self-trapped carrier along the crystal. The polaron model is an old topic in condensed matter physics, dating back to the pioneering work of Landau and Pekar [46]. The subsequent developments by Fröhlich, Holstein and Feynman aimed at a classification of the polaron species in terms of large, small and intermediate, depending on the extension of the polaron wavefunction over the crystal lattice [47, 48, 49, 50]. The Fröhlich model relies on the coupling with (polar) optical phonon modes, whose parameters (like the longitudinal phonon quantum energy) can be obtained via infrared spectroscopy [47]. The Holstein model depicts instead a situation where the electron-phonon interaction is local [48, 49]. The interplay between the electron-phonon coupling and the presence of defects/impurities in the crystal lattice can also give rise to more complex scenarios of electron localization [51].

The polaron problem has been subsequently extended to more exotic cases in which an agile body interacts with a relatively deformable environment. In this regard, one of the frontiers is represented by the study of the polaronic coupling between an exciton and the surrounding atoms [17]. Nowadays, this still represents a hard task, since it implies to understand the combined effects of electron-hole correlations and phonon coupling. This problem is addressed in Chap.  4, where we discover strongly bound excitons in anatase TiO\(_2\) and set the basis for the detailed description of their interaction with the polar modes of the lattice. Another example of interest is offered by materials in which an electron occupies one of the empty degenerate \(e_g\) levels on a transition metal ion in an octahedral crystal field. The lattice reacts locally by undergoing a Jahn-Teller (JT) distortion, thus lowering the energy cost associated with the presence of the excess charge. The electron dressed by this JT displacement is known as JT polaron [20]. On the contrary, in lattices where a collective JT distortion is already at play, the presence of an excess carrier can lead to the local relaxation or even removal of the JT distortion. The latter situation is referred to as anti-JT polaron [52]. This concept led to the initial search and discovery of high-\(\mathrm {T_C}\) superconductivity [9], following the prediction that high-\(\mathrm {T_C}\) values could be found in JT-distorted materials due to the condensation of a bipolaronic charged Bose liquid. This subject will be discussed in Chaps.  5 and  7.

We remark that both the BCS-Eliashberg theory of superconductivity and the polaron model provide an elegant framework to describe the effects produced by the electron-phonon interaction but neglect the possible coexistence of strong electron-electron correlations. The interplay between the electron-phonon interaction and the electron-electron correlation, with disorder tipping the balance on a local scale, is expected to give rise to a variety of exotic phenomena that are hardly captured by current theoretical models. In this regard, estimating the electron-phonon coupling in a correlated material for specific lattice modes can contribute advancing our knowledge on the preferential phonons to which the electronic excitations are mainly coupled. In this Thesis, we make the first step to this purpose, by evaluating the electron-phonon matrix elements in La\(_2\)CuO\(_4\) for a specific mode. The extension of this approach to many lattice modes and across a material phase diagram can in turn provide valuable inputs for simplified theoretical models such as the Hubbard-Holstein model [53, 54, 55, 56, 57].

1.1.2 Interplay between Low- and High-Energy Scales in Cuprates

Elucidating the origin of high-\(\mathrm {T_C}\) superconductivity in cuprates is probably the most studied, yet unsolved, problem faced by condensed matter physics. The interest in addressing this long-standing open question is driven mostly by fundamental research rather than by applications. Cuprates have always represented a novel paradigm for broadening the frontiers of solid state physics, offering unique phenomena that are not shared simultaneously by other classes of materials: Unconventionalsuperconductivity [58, 59], stripe/nematic/CDW order formation [60, 61, 62, 63, 64, 65, 66], opening of a pseudogap (PG) [67, 68, 69, 70], precursor and critical phenomena [71, 72, 73, 74, 75], strange metal behaviour [76, 77], dichotomy of the charge carrier nature [78], possible quantum criticality [79, 80], Mott physics [59, 81, 82], electronic phase separation. The emergence of such a variety of electronic phases is mostly associated with the extreme sensitivity that cuprates manifest towards external parameters and stimuli. This sensitivity is reflected in the complexity of the cuprate phase diagram, which is shown in Fig. 1.2. This has been adapted from very recent works in literature at the time of writing, and it represents the state-of-the-art of the knowledge on cuprate physics [66, 83].
Fig. 1.2

Tentative phase diagram for the cuprates as a function of temperature and hole doping on the CuO\(_2\) planes. The temperature scale \(\mathrm {T_N}\) defines the Néel temperature for the AFM Mott insulating phase. The temperature scale \(\mathrm {T^*}\) describes the emergence of the PG, which competes with the SC phase. The latter settles below the temperature \(\mathrm {T_C}\) and extends from p \(=\) 0.05 to p \(=\) 0.32. Below \(\mathrm {T_{CO}}\), some form of short-range charge ordering arises as another competing phase and leads to the suprression of \(\mathrm {T_C}\). \(\mathrm {T_{ONS}}\) represents the temperature below which SC phase fluctuations can be detected. In the normal phase, cuprates are considered as strange metals to account for non-Fermi liquid behaviour. In the extreme overdoped region, Fermi liquid behaviour is retrieved.

The phase diagram has been adapted from a combination of those reported in Refs. [66, 83]

A possible explanation behind this exotic behaviour is that cuprates are characterized by an intermediate regime of electronic correlations, which connects the extreme Mott insulator and Fermi liquid scenarios [84, 85]. This imposes several limitations to theory, as both the independent particle picture and the fully correlated Mott regime fail miserably in describing the microscopic properties of these compounds. Moreover, an intermediate-to-strong electron-boson interaction is expected to play a substantial role in the renormalization of the charge dynamics, as in most transition metal oxides (TMOs) [11]. Among the prominent bosonic excitations that have been identified or proposed, we can mention phonons, magnons, Josephson plasmons, excitons and polarization waves, and each of these collective excitations can generate a field that substantially affects the transport and the interactions of the fermionic particles. As a result, it is this lying in the “world of the intermediate” that makes cuprate physics so elusive and puzzling to both theory and experiment. As an additional element, disorder modifies the balance among the different forces on a local scale and leads to an intrinsic degree of inhomogeneity even in high quality single crystals [86]. All these subtle microscopic features concur in making cuprates the ideal playground for testing theories of cooperation and/or competition among several order parameters at a macroscopic scale.

The current Paragraph has neither the ambition, nor the intention, of providing a rigorous description of the complex physics behind the cuprates. Its goal is twofold: (i) To briefly illustrate how the physics of the electron-boson interactions and the electron-electron correlations have given foundation to two distinct schools of thought in the past for understanding unconventional superconductivity in cuprates; (ii) To connect this long-standing debate to the framework of this Thesis.

In regard to the first task, it is crucial to recall the main conclusion hold by the BCS theory of superconductivity and by its extensions for describing the so-called “conventional pairing mechanism” [7, 8]. In this scenario, low-energy virtual phonons act as a glue between two electrons, providing a retarded attractive interaction between them and leading to the formation of Cooper pairs. In cuprates, the role of the retarded interactions have been widely considered after the discovery that the formation of Cooper pairs is still valid and lies behind the mechanism of superconductivity (which is a priori not obvious). This has led to the search for the glue lying at the origin of high-\(\mathrm {T_C}\) superconductivity. As this glue cannot be provided solely by lattice vibrations, other frameworks involving spin fluctuations/polarization, electric polarization or charge density have been put forward [87, 88, 89]. This picture resembles the conventional mechanism, in which the mediating virtual phonons are replaced by a different low-energy glue of bosonic nature. Experimental data of ARPES and optics showed the presence of a peak in the electron-boson coupling function at \({\sim }\)60 meV, which may be related to a spin resonance or to a phonon mode [90, 91, 92, 93]. Additional modelling of the optical data also provided evidence for the decreased contribution of the 60 meV peak for increasing doping and for the presence of a broader contribution extending to higher energies (300 meV) [94]. This observation may support the idea that the pairing glue involves some kind of electronic degrees of freedom, such as spin, charge or orbital fluctuations.

In an opposite scenario, the existence of the retarded interactions has been doubted and a completely different mechanism involving non-retarded interactions associated with high-energy electronic scales has been proposed [95, 96]. Here, the most important element is represented by the proximity of the SC state to the Mott phase. Since the significant timescale of the interactions taking part to the pairing is the inverse of the Mott-Hubbard energy U (which lies in the range of some eV), such interaction can therefore be considered instantaneous. This is well depicted by the famous cartoon reported by Anderson in Ref. [95] and shown in Fig. 1.3a, representing an elephant (Mott-Hubbard energy scale U), a mammoth (superexchange interaction J) and a little mouse (electron-phonon coupling \(\lambda \)) in a refrigerator. “We have a mammoth and an elephant in our refrigerator—do we care much if there is also a mouse?” Consistent with this high-energy picture, a variety of studies showed that the physics of cuprates is not governed by processes living at energies close to \(\mathrm {E_F}\) alone. The underlying “Mottness is at the origin of the SW transfer between high- and low energies detected by a number of studies [59, 97, 98, 99, 100, 101]. These peculiar effects have been found to set in around \(\mathrm {T_C}\) and involve surprisingly high energies, from the mid-infrared (MIR) to the visible range. Remarkably, transfers of SW from the interband spectral region are not expected to occur in the conventional BCS picture and imply that the Ferrell-Glover-Tinkham sum rule is not satisfied within the intraband region alone [59, 102].
Fig. 1.3

a Cartoon showing a mammoth, an elephant and a mouse inside a refrigerator. The elephant, the mammoth and the mouse embody the Mott-Hubbard energy U, the superexchange interaction J and the electron-phonon coupling \(\lambda \), respectively, thus setting a direct comparison among the magnitude of the three energy scales in cuprates. The image has been reprinted with permission from Ref. [95], credits of Joe Sutliff. b Cartoon representing the concept behind our ultrafast broadband optical spectroscopy experiment to unveil the importance of the different energy scales to the SC pairing mechanism in cuprates. A light pulse opens the refrigerator and the SC condensate is set out-of-equilibrium. The oscillations of the SC condensate will resonate on the energy scale/scales that is/are playing a role in the pairing mechanism

A different framework involving the high-energy scales, known as “the MIR scenario”, was put forward on theoretical basis [103, 104, 105]. In this picture, it was postulated that superconductivity in cuprates is driven by a saving of the inter-conduction electron Coulomb energy, and specifically in the part associated with long wavelengths (q \(\le \) 0.3 \(\AA ^{-1}\)) and MIR frequencies (0.6–1.8 eV). In other words, all or most of the SC condensation energy comes from saving the Coulomb energy in this particular wavevector- and energy regime. The decrease in Coulomb energy would reflect an enhanced screening caused by the Cooper pair formation, as the latter modifies the bare density response function of the system. From these argument, it is evident that this scenario results in a strong modification of the electrodynamic properties of the system [104], producing remarkable changes in the MIR loss function (i.e. in the region that mainly involves the contributions from the conduction electrons in the CuO\(_2\) planes and is characterized by a broad feature known as MIR peak). Experimentally, these changes can be detected as sizeable modifications in the loss function measured by optics at q \(=\) 0 [78, 106] or as large drops in the electron scattering cross-section probed by differential transmission Electron Energy Loss Spectroscopy (EELS) at finite q.

To settle the controversy behind the origin of superconductivity in cuprates and to elucidate possible synergies between different mechanisms, one needs a technique that can monitor both the electron-boson interaction and the high-energy scales. In the context of this Thesis, we make a first step towards this goal by using a nonlinear optical technique out-of-equilibrium, based on the pump-probe spectroscopy method (described in details in Chap.  2). This approach has two main strengths: (i) to disentangle the contributions of the different electron-boson scattering channels on the basis of their characteristic timescales; (ii) the possibility to set specific collective excitations out-of-equilibrium using a laser pump pulse and subsequently monitor their impact on other degrees of freedom using a delayed optical probe pulse. In Chap.  2, we will refer to the first observable as the “incoherent response” of pump-probe spectroscopy, and to the second observable as the “coherent response”. In the latter case, when the collective excitation is represented for example by an optical phonon, this technique can provide a very selective and quantitative estimate of the electron-phonon coupling matrix elements. When the collective excitation is represented by the pair-breaking Bogoliubov mode [107], this technique can obtain information on possible pairing actors in superconductivity, by studying how the perturbed condensate affects and couples to other elementary excitations [108, 109]. In the language of the cartoon of Fig. 1.3a, this is like opening the refrigerator using a light pulse and setting superconductivity out-of-equilibrium. Therefore, the oscillating SC condensate can be considered as a trampoline. By monitoring how the elephant and the mammoth react to the oscillations of the SC condensate, one can gain information on the pair-mediating or the pair-breaking mechanism (Fig. 1.3b). Using this technique in optimally doped (OP) La\(_{2-x}\)Sr\(_x\)CuO\(_4\), it has been recently shown that the Bogoliubov mode couples to a high-energy excitation lying on the Mott scale, i.e. the remnant of the CT excitation [108]. Due to the crucial involvement of collective modes in our approach, a substantial part of this Chapter is devoted to discuss the physics of collective excitations in solid matter. The main limitations imposed by our approach are: (i) the limited spectral range that can be monitored; (ii) the possibility to study collective excitations only in the long-wavelength limit, i.e. at q \(=\) 0. The first drawback can be overcome by extending the probing region to cover the THz, MIR or the ultraviolet (UV) spectral ranges [110, 111]. The second restriction will be possibly released by the future development of the x-ray/electron analogues of the technique, which are expected to give access to a wider q-range of the elementary excitations.

At the time of writing, the task is therefore to set the basis for these future developments to occur, while gaining novel insights into the physics of cuprates. In Chap.  5, we investigate the electron-phonon matrix elements in the undoped parent compound La\(_2\)CuO\(_4\), where superconductivity and density wave effects are quenched and the system is electronically governed by Mott physics. In Chap.  6, we also demonstrate that our nonequilibrium method becomes a sensitive probe to the onset of fluctuating phenomena, like precursor SC correlations. In this regard, we also show how the emergence of precursor SC effects renormalizes some features in the bosonic excitation spectrum of a cuprate. In other words, the modification of the bosonic spectrum encodes the onset of short-range SC correlations.

In conclusion, addressing the physics of cuprates still remains a formidable task after thirty years since their discovery. Despite the disagreement between the different theoretical frameworks, a general consensus has been reached on the experimental side, in that the elementary excitation spectrum hides the fundamental manifestation of the mechanism behind superconductivity. This calls for the implementation of novel experimental techniques that can reveal new details imprinted on the elementary excitation spectrum with unprecedented time (spectral) and spatial (momentum) resolution.

1.1.3 Cooperative Phenomena in Manganites

Another intriguing manifestation of the balance between interactions and correlations is the emergence of cooperative phenomena in a wide class of oxides. Cooperative mechanisms lie at the heart of exotic electronic instabilities governing the properties of numerous materials at the macroscopic scale. Prototypical systems for the study of these effects are the manganites, which, similarly to cuprates, are also controlled by Mott physics and by an extreme sensitivity to the external conditions.

Since the 1990s, the main open question in the physics of mixed-valence manganites has been represented by the microscopic explanation of their colossal magnetoresistance, i.e. the colossal reduction of the electric resistivity that doped manganites undergo upon application of a magnetic field [10]. However, it was soon realized that addressing this problem had to involve another change of paradigm in the field of strongly correlated quantum systems. Indeed, manganites represent a fertile ground for the coexistence and mutual coupling among different order parameters, which complicates the description of the phenomena at the origin of the colossal magnetoresistance. As a result, in the last years, the focus of much research has been shifted to understand at first the complex relationship among the different degrees of freedom in the undoped parent compounds.

In these Mn\(^{3+}\) solids, the electronic properties arise from the subtle balance among a number of relevant energy scales [112, 113]. The on-site effective Coulomb interaction and the charge-transfer (CT) energy, embodied respectively by the Mott-Hubbard energy scale U and \(\Delta \), govern the electronic behaviour of the system within the framework of intermediate-to-strong electronic correlations. The intra-atomic exchange term, represented by the Hund coupling, is also relatively strong and implies that all electron spins must be aligned within the \(d^4\) configuration. The electron-phonon interaction lies at the heart of the orbital ordering in the material through the cooperative JT distortion of the lattice and strongly renormalizes the charge transport via polaronic coupling. The antiferromagnetic (AFM) superexchange interaction plays a fundamental role in defining the spin ordering pattern of the system. The coexistence of all these energy scales reflects in the richness and complexity characterizing the spectra of the elementary excitations in the undoped manganites [114, 115]. Assigning all these excitations is highly desirable, because they could play a crucial role in multiband Hubbard models used to capture the properties of the unconventional states developed upon doping. Despite huge efforts, this task still remains a real challenge. Experimentally, features associated with distinct phenomena often overlap and give rise to overcrowded spectra, thus hindering their clear assignment [115]. Moreover, the evolution of the spectroscopic signatures as a function of external parameters, crucial for unravelling many-body effects at play, are often too complicated to be reproduced even with phenomenological models. This makes manganites very elusive and ambiguous materials even to the most advanced experimental probes. A prototypical example of this ambiguity regards the classification of these solids within the Zaanen-Sawatzky-Allen (ZSA) scheme [114, 115, 116, 117, 118]. Since manganites, similarly to cuprates, also lie in the “land of the intermediate”, their salient properties become hard to capture on the theory side with simplified model Hamiltonians or ab initio calculations.
Fig. 1.4

Schematic magnetic phase diagram of RMnO\(_3\) as a function of the R\(^{3+}\) ion size, adapted from Ref. [119]. The evolution of the A-type AFM ordering, the incommensurate spin ordering and the E-type AFM ordering temperatures are displayed using open circles, open triangles and open diamonds, respectively. The red area highlights the region where the commensurate spin order and ferroelectric property emerges

Despite imposing a limitation to a detailed understanding of the physics behind these materials, such an internal complexity opens up new frontiers for the observation of exotic phenomena. Indeed, it becomes clear that key to this goal is to modify the energy landscape and the balance among the different interactions by applying an internal or an external perturbation. The simplest degree of freedom that can be varied is represented by the rare-earth ion R in the formula RMnO\(_3\) of orthorhombic manganites. What can appear in first instance as an innocent substitution leads in reality to a dramatic rearrangement of the structural, electronic, orbital and magnetic properties of the solid. Despite preserving the insulating character of the manganite, rare-earth substitution causes a strong structural deviation from the ideal perovskite lattice and eventually leads to a structural phase transition from orthorhombic to hexagonal. This is shown in the phase diagram of Fig. 1.4, which has been adapted from Ref. [119]. More interestingly, this structural distortion has profound consequences on magnetism, activating new paths for the spin interaction among different Mn\(^{3+}\) ions. The AFM ordering changes from A-type to E-type across the transition, but a small portion of the phase diagram involves more interesting spin patterns in correspondence of TbMnO\(_3\) and DyMnO\(_3\). Indeed, when the emerging magnetic interactions retain a competing character, magnetic frustration precludes simple spin ordering to establish and the system reorganizes with exotic chiral spin patterns to save energy. Thus, these unconventional spin orderings are a beautiful result of the interplay between the physics of cooperation and competition among several degrees of freedom. It is in these conditions that novel unexpected and fascinating phenomena can arise, the most subtle of which is the development of a ferroelectric polarization inside the manganite, thus giving rise to multiferroicity. At first sight this effect is counter-intuitive, as ferroelectricity and magnetism are generally mutually exclusive or competing order parameters. In the manganites, instead, the ferroelectric polarization rises as a secondary order parameter which is induced by the presence of the spiral order via the inverse Dzyaloshinskii-Moriya interaction and it is further stabilized by specific lattice displacements [120, 121, 122]. As multiferroicity in the manganites is a manifestation of the concept of multiple coupled order parameters, novel hybrid magnon-phonon modes (known as “electromagnons”) are expected to appear in the excitation spectrum of the system [123]. Studying these collective modes is attracting much interest as a way to reveal all the sources and symmetry of the magnetoelectric coupling [124, 125, 126, 127].

From the previous discussion, it becomes evident that the central task in solving the paradigm of manganites is to establish the “cause-effect” relationship behind their phenomenology. To this aim, a powerful approach is to investigate this relationship directly in the time domain, in which the different elementary excitations of the solid appear with distinct timescales. This approach relies on the use of a tailored pump pulse to set the manganite in a nonequilibrium state, followed by a delayed pulse that probes the evolution of the system. In the simplest scheme, one can separate the contributions related to the different degrees of freedom (charge, lattice, spin, orbital) depending on their characteristic temporal response, which is already a remarkable result. This is the playground of single- or two-colour pump-probe spectroscopy, whose application on the manganites has provided a wealth of information on the electron-lattice/electron-spin interactions, the spin order melting or the generation of coherent bosonic modes as phonons and orbitons [128, 129, 130]. More advanced ultrafast methods allow to selectively monitor an observable associated with a specific degree of freedom or to gain spectrally-resolved insights into the nonequilibrium response of the system. In the manganites, the former framework has elucidated the complex interplay among different order parameters, following the selective melting of the charge, orbital or spin order in real time [131, 132]. In contrast, spectrally-resolved nonequilibrium studies have never been applied to manganites. Mapping the ultrafast renormalization of the elementary excitation spectrum is highly desirable, as it can (i) reveal the presence of dynamical processes manifesting only in a specific spectral region; (ii) unveil subtle shifts of the SW associated with many-body effects; (iii) find novel observables contributing to the assignment of ambiguous spectral features. These are the main goals behind the experiments presented in Chap.  7 of this Thesis on the multiferroic manganite TbMnO\(_3\). Using high-energy photons covering the broad spectral range of the intersite d-d transitions, we are able to reveal the temporal hierarchy of low-energy phenomena leading to the melting of the cycloid spin-order after photoexcitation. Moreover, by studying the behaviour of the coherent collective modes of the material, we also gain insights into one of the unsolved problems behind the physics of insulating manganites, i.e. the character of their fundamental gap.

All these arguments lead to conclude that the nonequilibrium approach is a very promising scheme for shedding light on some controversial aspects of the manganite physics. In this regard, our conclusions for manganites are very similar to the ones we have drawn for cuprates in the previous section. On the contrary, the main difference between the two classes of materials upon perturbation with an optical pump pulse lies in the fact that doped manganites are more prone than cuprates to exhibit interesting photoinduced phase transitions [128, 133, 134, 135]. Although this feature is simply related to the rich phase diagrams shown by the manganites as a function of doping, it still justifies the efforts to realize the light-induced control of phase transitions in an ultrafast fashion. A different framework will be presented at the end of this Chapter, as the resonant excitation of specific collective modes is paving the route for the observation of even more exotic phenomena in both classes of solids. Once again, this in turn confirms the importance of describing the role of collective excitations in many-body systems, which is the subject of the following Section.

1.2 The Role of Collective Excitations in Many-Body Systems

From the previous Sections, it emerges that the general challenge in condensed matter physics is to approach the problem of many particles in complex systems. In a few cases, the physics of a many-body system can be still treated within the single-particle approximation by considering each particle as not the original electron but as a fermionic quasiparticle that becomes dressed of many-body interactions. As we have already mentioned in Sect. 1.1, this is the cornerstone of Landau’s theory of Fermi liquids, which applies to the situation of weakly interacting particles in a metal [6]. The breakdown of the Fermi liquid approach naturally leads to novel scenarios, which typically involve the presence of strong interactions, strong correlations and lower dimensionality. In these exotic cases, a deep and complete comprehension of the system becomes a formidable task.

The approach that is commonly adopted is therefore to construct a minimal theoretical model that is able to capture the salient properties of a specific material. However, for unravelling the hidden details of a many-body system, a different strategy must be used, which involves predicting and understanding the excitation spectrum of the material. Indeed, elementary excitations result from the balance among several forces inside the solid and govern all the relevant physical properties, such as the transport mechanisms and the response of the system to external perturbations. In other words, elementary excitations represent the signature of the many-body system. Some of these elementary excitations can be still traced back to the quasiparticle concept and associated with an effective mass and a characteristic lifetime. However, there also exists another kind of fictitious particles, mostly of bosonic nature, which are known under the name of “collective modes”. These excitations do not center around individual particles, but involve a cooperative, wave-like motion of many particles in the system simultaneously. This motion is therefore governed by the global interaction among the constituent particles. Most collective excitations (e.g. phonons, plasmons, magnons...) are intrinsic to the solid, while others arise as a consequence of the interaction with an external field. For example, the strong coupling of the electromagnetic field with a collective excitation carrying an electric- or magnetic-dipole moment results in the emergence of “polaritons” [136], which are hybrid modes characterized by unique quantum optical properties [137, 138, 139]. While collective excitations are present to some extent in all solids, it is in the case of strongly correlated systems that collectivity gives rise to a rich variety of emergent phenomena and leads to profound transformations depending on the external conditions [140]. This aspect has been widely described in Sects. 1.1.2 and 1.1.3. Thus, investigating the physics of collective modes in complex solids can reveal invaluable information on the many-body problem in the presence of cooperative or competing interactions.

Besides embodying the paradigm of the many-body problem, collective modes hold a huge promise for new technological developments and applications, especially in relation to energy sciences, health sciences, data storage and signal processing. The dream is to engineer novel devices where collective modes can be efficiently generated, manipulated and detected, similarly to what occurs in electronics with the electrons and in photonics with the photons [141, 142, 143, 144].
Fig. 1.5

Phononics, example of a novel research field involving the use of collective excitations for technological purposes.

Credits of Joerg M. Harms, Max Planck Institute for the Structure and Dynamics of Matter, Hamburg

First steps towards this goal have been made in the field of plasmonics, which has now become a reality [145]. Plasmonics deals with the interaction between the electromagnetic field and peculiar transverse collective modes known as surface plasmons. The latter are coherent oscillations of delocalized electrons, which set at the interface between two materials where the real part of the dielectric function changes sign across the interface [146]. Surface plasmons possess remarkable properties, including strongly enhanced local fields, dramatic sensitivity to the modification of the local environment and the ability to localize energy to tiny volumes not restricted by the wavelength of the exciting light. For these reasons, the generation, manipulation and detection of surface plasmons is nowadays exploited in several applications, the most famous of which is the surface plasmon resonance for detecting nanometer changes in thickness, density fluctuations or molecular adsorption [147].

Following the successful example of plasmonics, new areas of research are quickly emerging to explore the potential of collective excitations. For example, excitonics is now viewed as an ideal route for mediating the flow of energy at the nanoscale in photovoltaic materials, since strongly localized excitons are relatively immune to longer-range structural defects and disorder [148]. Phononics (Fig. 1.5) is gradually establishing as a powerful tool for controlling the structure of materials, thus tailoring their functionalities [149]. Magnonics is a promising field that aims to use propagating spin waves, i.e. magnons, as information carriers in devices [150]. Finally, polaritonics is expected to bridge the gap between electronics and photonics (100 GHz–10 THz) or to go beyond photonics (>300 THz) for high-speed signal processing applications, by exploiting the coupling between the electromagnetic field and transverse polarization fields inside the solid (e.g. those from transverse optical phonons or from excitons) [151].
Fig. 1.6

Pictorial representation of emergent collective modes in condensed matter systems. a Orbital waves (orbitons) in orbitally-ordered materials, involving a periodic modulation in the shape of the electronic clouds (adapted from Ref. [152]). b Amplitude and phase modes (amplitudon and phason) in a CDW system. Pink lines denote the static CDW, grey lines depict the change occurring in the CDW. Blue dots are the ions and the arrows indicate their motion

Such a technological perspective for the physics of collective excitations naturally leads to an important question concerning the fate of this area of research. Are there other collective excitations that can provide a valuable platform for facing future challenges in technology? The answer to this question is deeply intertwined with the fundamental concept of spontaneous symmetry breaking in condensed matter [153]. Whenever the symmetry of a system spontaneously breaks, an order parameter establishes in the lowest-energy phase and collective modes naturally emerge as low-lying excitations. The higher is the complexity of the order parameter, the more exotic is the nature of the collective modes arising in the system. In this framework sits the research on strongly correlated quantum systems, as these materials are natural candidates to search for new collective modes that can be manipulated and controlled by external stimuli. A prototypical example is provided by the orbitally-ordered manganites, where emerging collective modes are orbital waves named orbitons (displayed in Fig. 1.6a), involving a periodic modulation in the shape of the electronic clouds [152]. CDW systems offer a valuable platform for exploring collective modes of hybrid electronic-structural character [154], like the amplitude mode of the CDW (amplitudon), involving periodic changes in the charge density amplitude throughout the lattice, or the phase mode (phason), which gives rise to a translation of the condensate (shown in Fig. 1.6b). Finally, more exotic collective modes of electronic origin emerge in the physics of superconductors. Prominent examples involve: the Higgs mode, a neutral excitation associated with the amplitude of the SC gap [155, 156]; the Anderson-Bogoliubov mode, which involves charge and phase fluctuations of the SC condensate [107]; the Leggett mode, consisting of dynamic oscillations of Cooper pairs between two distinct superfluids [157].

In the following sections, we put emphasis on the collective excitations that will play a fundamental role in the framework of this Thesis, namely plasmons, excitons and phonons.

1.2.1 Plasmons

Plasmons are quantized collective oscillations of the charge density that arise in a plasma, i.e. a system of mobile charged particles interacting with one another via Coulomb forces. In metals and highly-doped semiconductors, the plasma is represented by the electron density and plasmons emerge as collective excitations of the many-electron system, leading to a coherent motion of all particles with a common frequency and wave-vector [158]. As a result, these modes play a fundamental role in the electrodynamics of solids and their study provides valuable information on intrinsic many-body phenomena governing the physics of materials.

Depending on the character of the collective excitation, plasmons can be transverse or longitudinal modes. Prototypical examples of transverse plasmons are the surface plasmons in metallic thin films/nanoparticles (NPs) [146] or a particular type of Josephson plasmons in high-\(\mathrm {T_C}\) cuprate superconductors [159]. The transverse nature of these collective modes allows them to absorb light, thus becoming detectable in the absorption spectrum of a material. Examples of longitudinal plasmons are the conventional bulk plasmons of metals and doped semiconductors, which arise in correspondence of the zero-crossings of the dispersive part of the material dielectric function \(\epsilon _1\)(q, \(\omega \)). As the longitudinal nature of these waves prevents their direct excitation by photon absorption, their observation becomes possible only via photon- or electron inelastic scattering [160, 161]. In the last decades, these probes have established as a powerful tool for investigating the charge-density response function of metals, unveiling the details of the dielectric screening and the associated crystal potential, local-field, and exchange-correlation effects.

In the following, we focus our discussion only on the longitudinal plasma modes, since in Chap.  3 the renormalization of a high-energy longitudinal plasmon is monitored to track the low-energy carrier dynamics of the two-band superconductor MgB\(_2\).
Fig. 1.7

Schematic representation of a metallic slab sustaining longitudinal plasma oscillations. a At equilibrium, a perfect compensation of the positive ion and negative electron charges occurs along the whole metallic slab. b, c Collective displacements of the electron gas in one direction or the other are shown. This in turn creates positive and negative surface charge layers, depicted in blue and violet, respectively. In the presence of a collective electronic displacement, a restoring force sets in and gives rise to oscillations at a frequency \(\omega _p\)

The origin of the concept of longitudinal plasma oscillations lies in the semiclassical Drude model of the electron conduction in metals, in order to explain the cut-off response of the reflectivity above a certain frequency \(\omega _p\) [16, 162]. A striking implication of the free carrier model is that the dielectric constant changes from negative to positive as the frequency goes through \(\omega _p\). Within the Drude model, the plasma frequency of the free-electron gas takes the form
$$\begin{aligned} \omega _p = \sqrt{\frac{n e^2}{\epsilon _0 m_0}}, \end{aligned}$$
where n is the number of electrons per unit volume, e is the electron charge, \(\epsilon _0\) is the vacuum permittivity and \(m_0\) is the bare electron mass. From Eq. 1.3, it follows that very large values of n lead to a high plasma frequency, which can be even pushed to the UV spectral region. This implies that, in the absence of other absorption processes, the reflectivity drops abruptly above \(\omega _p\) and some of the light can be transmitted through the metal. In doped semiconductors, it is the presence of impurities that leads to a free-carrier plasma reflectivity edge, lying typically in the infrared [163]. In this case, Eq. 1.3 has to be modified to account for the carrier effective mass and the background polarizability of the bound electrons.

From the semiclassical framework of the Drude model, plasmons arise as oscillations of the whole displaced electron gas with respect to the fixed lattice of positive ions [16] (Fig. 1.7). These collective oscillations proceed with a frequency \(\omega _p\) and can be sustained only in the absence of dissipative phenomena. In reality, the inspection of the plasmon energy-momentum dispersion relation allows to highlight the presence of dissipative mechanisms. For small values of the wave vector q, the lack of single-particle excitations in the region of the plasma oscillations leads to a negligible damping; for larger q, the plasmon dispersion curve merges into the continuum of single-electron excitations at a wavevector q\(_c\) and for q > q\(_c\) the oscillations will be damped and decay into the single-particle continuum (Landau damping) [164]. A notable exception to this behaviour is offered by the c-axis plasmon mode of MgB\(_2\), which reappears periodically in higher BZs due to the strong coupling between the single-particle and collective excitation channels [45]. As already observed, this mode will be subject of investigation in Chap.  3, although in the q \(=\) 0 limit.

Finally, the renormalization of the bare plasma frequency as a function of temperature can offer valuable information on peculiar many-body processes occurring in a system. In single-band metals, the bare plasma frequency is expected to decrease for increasing temperature [165, 166, 167]. In multiband systems with electron- and hole-like bands and strong electron-boson coupling, the carrier density n in every band depends on the details of the electron-boson interaction. As a result, the temperature dependence of n (i.e. the size of the Fermi surface) is strictly related to the thermal activation of the interaction processes [168, 169]. An analogous scenario will be proposed also in Chap.  3 for explaining the anomalous behaviour of the a- and c-axis bare plasma frequencies of MgB\(_2\).

1.2.2 Excitons

Excitons represent collective excitations emerging in a non-metallic system. An ideal exciton can be considered as an electrically neutral quantum of electronic excitation energy, travelling in the periodic structure of a crystal. As such, it can be viewed as a bound electron-hole pair that involves the transport of energy, but not of a net charge. As the exciton arises just in the presence of both the electron and the hole, its existence is intimately related to the absorption of a light field at a given frequency. The collective nature of an exciton provides a clear manifestation during the absorption of light. Differently from single-particle optical transitions, in which only one single-particle state gets excited by the incident light field and all the other states do not participate (Fig. 1.8a), the excitonic transition is a “collective” one, which involves many states of the electronic band structure. A simple cartoon for this phenomenon is shown in Fig. 1.8b.
Fig. 1.8

a Schematic representation of single-particle optical transitions, in which only an individual particle is excited by the incident light and all the other states are silent; b Schematic representation of a “collective” excitation associated with an excitonic transition, in which many states participate. The arrow shading indicates the strengths of the different transitions. As a result, the excitation energy becomes smaller than the bandgap since the collective character of the response is energetically stable

Fig. 1.9

Schematic representation of the wavefunction for a a Wannier-Mott, b a CT and c a Frenkel exciton

A necessary condition for this many-state transition to occur and for the exciton to form is that the electron and hole group velocities are the same, i.e. that the gradients of the CB and the VB are identical in a specific portion of the BZ. As a consequence, while the excitation energy of a single-particle transition is just the difference between the initial and final single-particle states, the excitation energy of an excitonic transition is lower than the fundamental band gap. The stabilization energy with respect to the uncorrelated particle-hole limit is the fingerprint for the collective nature of the state created in the absorption process and is typically known as the “exciton binding energy” (\(\mathrm {E_{B}}\)). This quantity measures the strength of the electron-hole interaction giving rise to the peculiar bound state, in which the electron and hole are found to move together with the same group velocity.

From the discussion above, it follows that the rigorous determination of \(\mathrm {E_{B}}\) relies on the estimate of the fundamental energy gap for both charged (single-particle) and neutral (two-particle) excitations. The former gap, known as “electronic gap” (\(\mathrm {E_{el}}\)), can be measured by a combination of photoemission and inverse photoemission spectroscopy; the latter gap, the so-called “optical gap” (\(\mathrm {E_{opt}}\)), can be extracted by optical spectroscopy. It follows that \(\mathrm {E_{B}}\) can be simply defined as
$$\begin{aligned} \mathrm {E_B} = \mathrm {E_{el}} - \mathrm {E_{opt}}. \end{aligned}$$
Depending on the value of \(\mathrm {E_{B}}\) and the spread of the wavefunction, excitons in condensed matter can be classified under different types. Wannier-Mott excitons represent hydrogenic electron-hole pairs delocalized over several unit cells of the lattice and arise in semiconductors with high dielectric constant (Fig. 1.9a). The other extreme is represented by Frenkel excitons, which correspond to a correlated electron-hole pair localized on a single lattice site and mostly prevail in solid rare gases and molecular solids (Fig. 1.9c). Intermediate between these two regimes is a number of exotic excitations that are typically referred to as CT excitons. In the language of molecular physics, CT excitons form when the electron and the hole occupy adjacent molecules in a molecular structure of an organic semiconductor [170]; on the other hand, in the language of strong electron correlations, CT excitons embody the lowest-energy neutral excitation that is found in insulating TMOs, involving an electron in the transition metal 3d orbitals and a hole in the oxygen 2p orbitals [171, 172]. Irrespective of the origin, the concept of CT exciton is always related to a transfer of charge from one atomic site to another, thus spreading the wavefunction over several unit cells and retaining an intermediate character between the Wannier-Mott and Frenkel limits (Fig. 1.9b).

Studying these collective electronic excitations is of great importance, as it provides detailed information about the magnitude and spatial distribution of the electron-hole correlations. The natural technique to probe excitons is represented by optical spectroscopy at low temperature, as the excitons manifest as narrow peaks preceding the absorption continuum [163]. Thus, optical spectroscopy alone can reveal the nature of the lowest-energy charge excitations of an insulator, provided that the fundamental gap has a direct character. In the presence of a fundamental indirect gap, the determination of \(\mathrm {E_{B}}\) by optical spectroscopy alone is hindered and the situation becomes more complex. We will illustrate an example of such a scenario in Chap.  4. Additional information on the exciton energy-momentum dispersion, which offers a precise estimate of the exciton transport properties, are instead provided by resonant inelastic x-ray scattering (RIXS) and EELS [171, 172].

To describe the properties of the neutral excitations occurring upon photon absorption or scattering, in the last decade much progress has been made by ab initio theoretical calculations. One of the main achievements for theory has been the inclusion of the screened electronhole interaction involved in the formation of excitons. This interaction is described in the so-called BetheSalpeter equation (BSE), which solves for the neutral excitation energies as the poles of a two-particle Green’s function [173]. The solution of the BSE typically begins with a many-body electronic structure calculation to solve for the single-particle quasielectron and quasihole. The BSE formalism subsequently introduces an interaction term that mixes the two types of charged transitions. Despite being computationally demanding, this technique proved successful in reproducing the salient feature of the experimental optical spectra of standard semiconductors and TMOs [174].

Novel frontiers in fundamental exciton research involve the study of exciton physics in the presence of strong interactions and correlations in complex materials. Excitons are high-energy bosonic collective excitations and, as such, they are more prone to undergo severe renormalizations upon interaction with other particles in a material. In the past, a body of work focused on the renormalization processes that the exciton experience in simple semiconductors upon chemical- or photo-doping [175]. Several single-particle and many-body effects as Pauli blocking, exciton screening and carrier-induced bandgap renormalization (BGR) were identified experimentally and modelled theoretically. More remarkably, research in this area successfully identified the sources of optical nonlinearity occurring in bulk semiconductors and nanostructures, opening the doors to the use of excitonic materials for applications and technology [176]. Extending these concepts to the case of strongly interacting and correlated quantum systems is not trivial, as excitons are likely to interact with the phonon field or with other elementary excitations [17]. On the experimental side, one needs to reveal the fingerprints of such exotic interactions by performing extensive temperature studies or by applying state-of-the-art techniques. On the theory side, suitable extension of the BSE should be developed in order to include the effects of electron-phonon interaction or of a finite doping. In the framework of this Thesis, these concepts will be deeply explored in Chap.  4, in which we report on the experimental discovery and the complete theoretical modelling of bound excitonic quasiparticles in the strongly interacting \(d^0\) insulator anatase TiO\(_2\). In this regard, the application of advanced ultrafast spectroscopic techniques paves the route toward a selective and quantitative estimate of the exciton-phonon interaction, an aspect which has never been accessible via conventional steady-state methods.

1.2.3 Phonons

Collective excitations of the atoms in a solid are known as phonons. These modes play a fundamental role in the physics of a crystal, determining its thermodynamic, transport and electrodynamic properties. The description of phonons within the quantum theory of solids has been the subjects of numerous books [18, 19, 20] and it is out of the scope of this Thesis. In Sect. 1.1.1, we have already explored some remarkable manifestations of the electron-phonon interactions in the strong coupling regime. Here, in this section, we restrict our discussion to another aspect of primary importance for the development of the following Chapters, namely the impact that phonons have on the electrodynamic properties of materials. In this regard, it is useful to distinguish between two separate effects: the direct manifestation of phonon modes in the infrared or in the Raman spectra of a material [160] and the signature that phonons leave on the high-energy charge excitations of the solid (like the optical bands of polarons, excitons and uncorrelated particle-hole excitations) [17].

In the first scenario, insightful information on the phonon degrees of freedom at q \(=\) 0 can be retrieved either via direct absorption or via scattering of the electromagnetic radiation by a phonon mode in the crystal. Direct absorption relies on a dipole-allowed process and is achieved by shining a resonant infrared field on the crystal. As the electromagnetic waves are transverse, they can couple to the transverse optic phonon modes of the solid at q \(=\) 0, which thus become observable in infrared spectroscopy. In contrast, the spontaneous inelastic scattering of radiation by phonons is a nonlinear process (quadratic in the electric field) and is typically achieved using photons in the visible or near-infrared range, which interact with the polarizable electron density and the bonds in the solid. As a consequence, the system is maintained in a virtual energy state for a short period of time before releasing a new photon (which is inelastically scattered). The signature of the phonon mode is left on the scattered photon, which possesses lower (Stokes) or higher (anti-Stokes) energy than the incoming photon [160].

From the above arguments, a phonon is named infrared/Raman-active if its excitation involves the absorption/scattering of the incident electromagnetic radiation in the long-wavelength limit. The emergence of a phonon mode in the infrared or in the Raman spectrum of a material relies solely on the symmetry of the mode according to group theory considerations. Typically, when the crystal possesses inversion symmetry, Raman-active modes are silent in infrared spectroscopy and infrared-active modes are silent in spontaneous Raman scattering. This concept is deeply rooted in the field of molecular spectroscopy, as the rule of mutual exclusion states that no normal modes can be both infrared and Raman-active in a molecule possessing a centre of symmetry. However, upon breaking spatial inversion symmetry, the same mode can simultaneously appear or be silent in both spectroscopies.
Fig. 1.10

a Real part of the optical conductivity in SrTi\(_{1-x}\)Nb\(_x\)O\(_3\) samples at 300 and 7 K for different values of the doping x. An incoherent band, which is the signature of large polarons in the material, is found to emerge as a function of doping and shows a satellite-like structure at 7 K. Adapted from [177]. b Temperature-dependent spectra of the dielectric function \(\epsilon = \epsilon _1 + i\epsilon _2\) of SrTiO\(_3\), in which the bandgap energy is found to blueshift for increasing temperature. This anomalous behaviour is a manifestation of the electron-phonon coupling in the material.

Adapted from Ref. [178]

Another manifestation of phonons on the electrodynamics of a solid is represented by their effect on the high-energy excitations of the material. In the simple case of indirect bandgap semiconductors like Si or Ge, indirect absorption is only related to the mediation of phonons when impurities and defects are neglected [163]. In more complex scenarios of intermediate electron-phonon coupling, a series of sidebands due to coupling to several phonon modes are typically observed in correspondence of polaron absorption or interband particle-hole excitations. An example is offered in Fig. 1.10a, in which a large polaron band is shown to develop in the optical conductivity spectra of SrTi\(_{1-x}\)Nb\(_x\)O\(_3\) as a function of doping x and at two different temperatures (adapted from Ref. [177]). The sidebands are smeared out in the presence of strong electron-phonon coupling, leading to broad Gaussian lineshapes manifesting even at the lowest temperature [17].

More subtle effects of the electron-phonon coupling emerge when temperature dependences are performed. Both the phonon modes detected in infrared spectroscopy/Raman scattering and the high-energy excitations of the solid are expected to soften with increasing temperature. This renormalization is accounted for by the thermal expansion of the lattice and, in the case of the high-energy interband transitions, it is well described by the empirical Varshni model [179]. However, anomalous situations have also been observed [178, 180, 181], in which the energy of an excitation hardens as the temperature is increased. An example is given in Fig. 1.10b, in which the temperature-dependent dielectric function of SrTiO\(_3\) (adapted from Ref. [178]) is shown. An anomalous temperature dependence of the direct gap is found. The description of the microscopic processes behind these effects are a current topic of research and no conclusive explanation has been provided yet. In the framework of this Thesis, an anomalous temperature dependence has been revealed in the exciton spectrum of anatase TiO\(_2\) and associated with the effect of the electron-phonon coupling at finite temperature.

1.3 Novel Frontiers in the Physics of Collective Excitations

In the previous Sections, we underlined how the study of collective excitations can decisively shape our comprehension of quantum many-body phenomena in complex matter. Nowadays, this task is being accomplished by two distinct, yet intimately related, areas of research, namely atomic physics and solid state physics. On the atomic physics side, tremendous developments in the field of trapped ultracold atomic gases in optical lattices has provided a physical realisation of a variety of quantum models, both for fermionic and bosonic particles [182]. The use of these quantum simulators provides a wealth of information concerning the emergence and the dynamics of several collective modes, which is important for unravelling the underlying symmetries of a system. In this regard, strong emphasis is being given to the search for those collective modes that play a fundamental role in elementary particle physics [183]. On the solid state physics side, the use of advanced spectroscopies has contributed to the discovery and assignments of novel collective excitations in quantum materials, whose properties shed light on the presence of interactions and correlations at the microscopic scale.

As already described above, one of the main frontiers in the latter field involves the study of the collective excitations under nonequilibrium conditions using ultrafast spectroscopies. Mapping the temporal evolution of these modes is highly desirable, as it bears the signature of the low-energy phenomena occurring in the system during its path to reach the thermodynamic equilibrium. This framework represents the main subject of this Thesis and, as such, it is applied to a number of materials characterized by a gradual degree of internal complexity beyond conventional band theory. The fingerprint of the collective excitations of interest is revealed either in the frequency or in the time domain, which is a powerful tool for accessing the dynamics of collective excitations lying at very different energy scales. The extension of this scheme to retrieve momentum-resolved information represents the future step in this field of research.
Fig. 1.11

Pictorial representation of the ultrafast control on a strongly correlated system via resonant excitation of an infrared-active phonon mode. The atoms associated with the eigenvector of the excited mode are strongly displaced from their equilibrium positions. The excitation of this infrared-active phonon can trigger additional lattice modes via a nonlinear coupling. A new nonequilibrium structure is developed and this can favor the emergence of novel quantum phenomena.

Credits of Joerg M. Harms, Max Planck Institute for the Structure and Dynamics of Matter, Hamburg

A relatively new frontier in collective mode physics is strictly related to the concept of “control” under nonequilibrium conditions. This implies either the control of a specific collective excitation or the control of a material through a collective excitation. The former scenario holds special promise in magnonics and it relies on the use of coherent collective excitations in the time domain. These modes are coherent in the sense that two time-separated optical pulses can enhance or suppress the oscillation associated with the mode, depending on their mutual separation in time. As a result, a particular mode can be switched on and off by designing a suitably shaped train of laser pulses. In other words, this is a type of quantum control based on the interference between a collective mode and itself. This framework has been particularly studied in the case of phonons in band semiconductors and CDW materials [184, 185], but its extension to the characteristic modes of magnets and superconductors is attracting interest for engineering new protocols of coherent control that may have an impact on technology and applications. An alternative and very powerful approach is instead based on the control of a material properties through the excitation of a specific dipole-allowed (i.e. infrared-active) collective mode using high laser fluences (Fig. 1.11). As these modes lie typically in the THz or FIR spectral range, strong limitations in achieving their intense resonant excitation have been imposed by the scarcity of suitable laser sources. Recent progress in ultrafast laser physics and quantum electronics have opened the doors to the implementation of laser schemes that can selectively excite these infrared-active collective modes in the THz and FIR range. The impact of this method has been exceptional, particularly in the field of strongly correlated quantum systems. As these materials are very sensitive to external stimuli, tailored photoexcitation of a collective mode can produce a variety of fascinating, yet still quite unpredictable, effects. The most impressive results have been achieved by resonant pumping infrared-active phonon modes in high-\(\mathrm {T_C}\) cuprate superconductors and in manganites. In the cuprates, this has led to the establishment of a highly coherent state above \(\mathrm {T_C}\) that resembles the properties of the SC phase [186, 187]. In underdoped (UD) monolayer cuprates of the La\(_{2-x}\)Ba\(_x\)CuO\(_4\) family, the Josephson plasmon resonance has been shown to emerge above \(\mathrm {T_C}\) upon suppression of the stripe order competing with superconductivity [186]. More surprisingly, in UD bilayer cuprates of the YBa\(_2\)Cu\(_3\)O\(_{7-{\delta }}\) family, this nonequilibrium state has even been shown to persist up to room temperature (RT) and to involve a deformation of the crystal lattice which disrupts the competing CDW order [187, 188]. As a result, this ultrafast pathway is likely to stabilize superconductivity. In the manganites, pumping specific infrared-active modes has led to the creation of metastable electronic states or to the melting of the spin and orbital orders [149, 189, 190]. As the laser pump pulses used for these experiments are very intense and are believed to be absorbed selectively by the excited collective mode, the radiation-matter interaction is highly nonlinear and has paved the route to the development of the so-called “nonlinear phononics” scheme. The goal here is to reach the control of the lattice of a material within the framework of ionic Raman scattering [149, 191]. Motivated by these results, new advances in the field are exploring the resonant excitation of Josephson plasmons [192] and electromagnons [193], opening new intriguing perspectives for the future of this research area.

In conclusion, all these developments make the physics of collective excitations a fertile area of research, justifying the scope of their detailed analysis to gain insights into elusive low-energy phenomena in many-body systems. Before discussing the case studies of this Thesis, in the next Chapter we first introduce the spectroscopic methods aimed at the identification of the elementary excitations in condensed matter systems. We put special emphasis on the nonequilibrium techniques, as these will represent the main experimental tool for our investigation.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsMassachusetts Institute of TechnologyCambridgeUSA

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