At half-filling, the CuO₄ square-planar motif generic to all cuprates is insulating due to the strong on-site Coulomb repulsion that causes the single charge carrier within each unit cell to localize. Upon hole doping, the initial excess charge preferentially locates on the oxygen sites–giving rise to a charge-transfer insulator–which then couples antiferromagnetically with the localized charge on the copper site to create a band of Zhang-Rice singlets. With further doping, the charge-transfer gap collapses and long-range antiferromagnetic (AFM) order is destroyed, leading to a system of highly-frustrated short-range spin correlations and ultimately d-wave superconductivity. At the end of the superconducting (SC) dome, the charge initially localized in the Mott state becomes fully itinerant and a correlated Fermi liquid (FL) state with a large cylindrical Fermi surface containing the total carrier density emerges.

Despite a decades-long investigation, the above summary is almost the extent of our current understanding of the fate of the initially localized carrier as further holes are introduced into the CuO₂ plane. The starting point and endpoint are well established–it is the transition from localization to itinerancy as the number of doped holes (p) increases that is proving the biggest challenge. The fact that this transition incorporates the most enigmatic features of the cuprate problem; the normal state pseudogap, the strange metal and high temperature superconductivity itself makes steps to understand its evolution all the more pressing.

The aim of this Perspective is to highlight a number of intriguing correlations that shed new light, not only on the fate of the localized hole–as reflected in both single-particle probes like angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM) and collective responses like transport–but also on the impact of its enhanced mobility on the superconductivity and in particular, on the superfluid density. The totality of the transport behaviour across the cuprate phase diagram provides compelling evidence for the presence of two current-carrying fluids, one associated with coherent fermions, the other incoherent and possibly bosonic in nature. The relative occupation of these two entities and its evolution with doping suggests that the superfluid density is dominated by only one of these fluids. Finally, in order to motivate the development of a consistent theoretical description of this dichotomy, we highlight systems that exhibit such a two-fluid response before presenting results from a treatment of the Hubbard model on a 2 × 2 plaquette that captures a number of the essential elements seen in the cuprates. The overall picture that emerges is one that departs significantly from the standard BCS description.

Despite the diverse and complex chemistry of the cuprates, the evolution of their physical (i.e., transport, thermodynamic and spectroscopic) properties across the phase diagram is found to be strikingly similar for all well-studied families. This remarkable uniformity suggests that Fermiology plays only a minor role in defining the overall behaviour, though certain unique aspects, such as the Lifshitz transition in La_2−x Sr _x CuO₄ (LSCO) or the CuO chains in YBa₂Cu₃O_7−δ (Y123) or YBa₂Cu₄O₈ (Y124), need to be fully taken into account before a complete picture can emerge. In this experimental survey, we focus predominantly on the single-layered cuprates Tl₂Ba₂CuO_6+δ (Tl2201) and Bi_2+z−y Pb _y Sr_2−x−z La _x CuO_6+δ (Bi2201) on which the bulk of our recent high-field transport studies have been performed, though for completeness, we also include complementary data obtained on LSCO, Y123, Y124 and single-layered HgBa₂CuO_6+δ (Hg1201). Moreover, in order to aid subsequent discussions, we redefine the phase diagram in Figure 1A in terms of the superfluid density n _s (0), rather than T _c . In this way, the SC dome can be divided simply into two regimes–pseudogapped (PG) and strange metal (SM)—rather than the usual three (underdoped, optimally doped and overdoped).

The growth of n _s (0) with increasing p inside the PG regime–known as the Uemura relation–reflects the fact that the pseudogap itself is states-non-conserving, i.e. that states lost through the opening of the pseudogap (preferentially near the Brillouin zone edges) are not recovered below T _c [1]. The reduction in n _s (0) beyond p*, on the other hand, has commonly been attributed to pair breaking–the so-called dirty d-wave scenario–though recent studies on LSCO thin films have challenged this association [2, 3]. Robust correlations between n _s (0) and certain key transport coefficients–highlighted in Figure 1 and to be addressed in turn in the following sections–provide further clues as to the origin of this loss of condensate, with the nature of the carriers within the SM regime and proximity to the Mott state at half-filling playing pivotal roles.

The picture emerging from this survey of recent high-field studies on hole-doped cuprates is of a transport current comprising two electron fluids–one coherent and FL-like (albeit correlated), the other non-FL-like and of unknown origin–whose relative ratio evolves smoothly across the phase diagram. Before discussing peculiarities in cuprates from a theoretical perspective, it is worthwhile to start with a general remark on the coexistence of two types of electronic liquids for interacting electrons in crystals. FL theory, together with its microscopic justification [38–40] has long been considered the ‘default’ many-body theory of conducting condensed matter. Recently, the concept of a ‘non-particle’ state was developed, mostly within the language of holographic duality [41, 42], which is argued to be a more appropriate framework to describe cuprates and other strange metals [7]. In both cases, the normal state of the system is described as a single liquid, either as a FL or a non-FL. It is not obvious therefore whether the hypothesis of a two-liquid state is consistent with general ideas of contemporary quantum many-body theory. Being very far from a complete theoretical description of the experimental situation, we are nevertheless able to give a positive answer to this question. To this aim, we will first make a general remark and then give two examples where the coexistence of two liquids in the same many-body system is reliably established.

We proceed by discussing the emerging physics in terms of the single-particle electron Green’s function [38–40]. To avoid possible misunderstanding, one should emphasise that this formalism contains full information on the many-particle system. In particular, it is sufficient to reproduce rigorously all equilibrium [43, 44] and non-equilibrium [45] properties of interacting fermionic systems. The simplest analysis can be done in terms of single-particle density matrix (let us stress again that, despite its ‘single-particle’ name, it characterizes the whole many-particle system). One can derive the following general expression [46] for the accelerating action of a constant uniform electric field E: d j a d t = e 2 E b ∑ k , n m a b − 1 k , n ρ n k , (1)

where j is the electric current, t the time, e the electron charge and a, b are Cartesian indices, m a b − 1 k , n = ℏ − 2 ∂ 2 ∂ k a ∂ k b k n H ̂ 0 k n (2)

plays the role of inverse effective mass tensor, k n and ρ _n (k) are eigenfunctions and eigenvalues of the single-particle density matrix ρ ̂ dependent on the wave vector k, n labels different eigenstates and plays the role of a ‘band index’ in the many-body description of crystalline solids [46], H ̂ 0 is the single-particle part of the Hamiltonian (we assume that the interaction Hamiltonian depends on coordinates only), and the k-summation in Eq. 1 is taken over the Brillouin zone. Equations 1 and 2 can be considered as a real-time equivalent of the optical f-sum rule frequently used to analyse strongly correlated systems [47].

The single-particle density matrix is nothing but the equal-time single-particle Green’s function G ̂ , and the latter, for energies close enough to the Fermi surface, can be decomposed into QP (that is, pole) and non-QP (incoherent) parts [38]. These two contributions to the density matrix corresponds to two contributions to Eq. 1. Rigorously speaking, the QP and non-QP parts of ρ ̂ do not necessarily commute one with one another and thus do not share a common set of eigenfunctions, but at the level of the operator ρ ̂ , this decomposition is always possible and accurate.

Half-metallic ferromagnets [48] provide a neat example of a physical system in which two types of electron liquids coexist naturally and rigorously. Whereas for one spin projection, the electron subsystem is metallic and FL-like, for the other, non-QP states in the gap dominate close to the Fermi surface, their density of states being non-zero above the Fermi energy only for the minority-spin gap and below it for the majority-spin gap [48]. Importantly, their contribution to the occupation number ρ _n (k) is almost k-independent and thus these states are effectively currentless [46]. Nevertheless, they may contribute to photoemission and scanning probe spectra, tunneling, etc. [48], as confirmed experimentally [49, 50].

The case of half-metallic ferromagnets is useful as it provides us with a unique opportunity to prove the separation of electron liquids into QP and non-QP (or incoherent) components. Moreover, this separation is rigorous due to its connection to conserving quantities such as the total spin (we assume that spin-orbit effects are negligible). Unsurprisingly, the situation in cuprates is much more complicated however, and we need to find a closer analog, even if its consideration cannot be done with the same rigour. In this regard, the heavy fermion (or ‘Kondo lattice’) systems [51, 52] provide a useful bridge. In these systems, the 4f or 5f electrons are mostly atomic-like and form local magnetic moments but in some regime they become itinerant, contributing to the shape of the Fermi surface, charge transport etc. In some sense, these states can be considered as ‘Abrikosov pseudofermions’—usually a purely mathematical construction introduced as a way to represent local spin operators–that hybridize with the true conduction-band fermions and become real [53]. Therefore one needs to distinguish between large and small Fermi surface constructions with and without their contribution [52].

With this in mind, let us now return to the cuprate problem. The minimal model that is supposed to keep the essential elements of cuprate physics is the single-band t − t′ Hubbard model derived via mapping of density-functional computational results for various representatives of this family [54, 55]. The corresponding Hamiltonian has the form H = − ∑ i j t i j c i σ † c j σ + ∑ i U n i ↑ n i ↓ (3)

where i, j are sites belonging to the simple square lattice, t _ij is an effective hopping that is supposed to be non-zero only for nearest- (t) and next-nearest-neighbours (t′) and U is the on-site Coulomb interaction parameter. Operators c i σ † ( c i σ ) create (annihilate) fermions at site i with spin σ = ↑, ↓ and n i σ = c i σ † c i σ . The ratio t′/t is about 0.3 for most of the cuprates, with the exception of LSCO where it is closer to 0.15. Empirically, T _c grows with the value of t′/t [55]. The Hubbard parameter U is comparable with the bandwidth, so we deal with the most theoretically demanding case of intermediately strong correlations.

Arguably the simplest approach to correlated electrons is dynamical mean-field theory (DMFT) where the lattice problem is mapped onto a single-site effective impurity problem with self-consistent determined characteristics [56]. For cuprates, however, the minimal DMFT model should start with a 2 × 2 plaquette rather than a single site since the d-wave SC order parameter lives on bonds rather than on sites [57]. There have been an enormous number of cluster DMFT calculations, the corresponding references for which can be found in [58, 59]. Rather than discussing numbers, we focus here on an interesting observation [60]; that for the values of t′/t near 0.3 and a realistic value of U, the ground state of the minimal plaquette is degenerate, i.e. the ground states for two, three and four electrons on the plaquette coincide. As a result, the plaquette Green’s function has poles at energies close to zero which may be a physical realization of the soft hidden fermion mode suggested earlier phenomenologically [61]. This makes the system in some sense analogous to the Kondo lattice, and this degeneracy plays a role similar to spin degeneracy in the conventional Kondo problem. In particular, making the lattice from such resonant centers naturally explains the appearance of the pseudogap [58, 60].

In the context of the two-fluid concept discussed here, the most important theoretical observation was made in [59] at the numerically exact solution of the many-body problem for a larger, 4 × 4 size plaquette. It turns out that the binding energy of two holes per plaquette is very strongly dependent on the ratio t′/t and can reach values about 0.8t ≈ 0.3 eV which is more than an order of magnitude larger than T _c , as shown explicitly in Figure 2. This means that, at least, for some range of doping, holes build incoherent pairs already in the normal phase. Despite the holes being initially expected to form something similar to a charged Bose liquid, one can suppose that the interaction with the thermal bath makes this liquid rather more classical than of Bose-Einstein character [62].

Thus, in general one can propose the following phenomenological picture. The coherent or QP part of the fermionic spectrum is mostly responsible for electron transport in the normal state and completely responsible for its coherent manifestations such as quantum oscillations. The incoherent part, on the other hand, may be associated with the formation of two-hole bound states (one could call them bipolarons but we prefer to avoid any associations with phonons since they do not appear to be relevant in this simplified picture). At high enough temperatures, the Fermi liquid of coherent electron excitations acts as a dissipative environment for the ‘bipolarons’ preventing their Bose-Einstein condensation. The latter can nevertheless occur as the temperature decreases leading ultimately to superconductivity, while the coherent part remains largely untouched. While this picture provides a natural explanation for the presence of the two fluids, it remains to be explored how this incoherent part can play such a prominent role in the magnetic field response [19]. This will be the subject of future studies.