## Abstract

Central to the normal state of cuprate high-temperature superconductors is the collapse of the pseudo-gap, briefly reviewed here, at a critical point and the subsequent onset of the strange metal characterized by a resistivity that scales linearly with temperature. A possible clue to the resolution of this problem is the inter-relation between two facts: (i) a robust theory of *T*-linear resistivity resulting from quantum criticality requires an additional length scale outside the standard one-parameter scaling scenario and (ii) breaking the Landau correspondence between the Fermi gas and an interacting system with short-range repulsions requires non-fermionic degrees. We show that a low-energy theory of the Hubbard model that correctly incorporates dynamical spectral weight transfer has the extra degrees of freedom needed to describe this physics. The degrees of freedom that mix into the lower band as a result of dynamical spectral weight transfer are shown to either decouple beyond a critical doping, thereby signalling Mottness collapse, or unbind above a critical temperature, yielding strange metal behaviour characterized by *T*-linear resistivity.

## 1. Introduction

High-temperature superconductivity in the copper oxide ceramics remains an unsolved problem because we do not know what the propagating degrees of freedom are in the normal state. Consequently, we cannot say with any certainty what are the weakly interacting degrees of freedom that pair up to form the superconducting condensate. In low-temperature superconductivity in metals, the existence of a Fermi surface simplified the identification of the propagating degrees of freedom. As shown by Polchinski [1], Shankar [2] and others [3], all renormalizations arising from short-range repulsive interactions are towards the Fermi surface. As a result, such interactions can effectively be integrated out, leaving behind dressed electrons or quasi-particles as the propagating degrees of freedom.

Undoped, the cuprates are Mott insulators. Charge localization obtains in Mott systems, not because the band is full, and in fact it is not, but because strong local electron correlations dynamically generate a charge gap by splintering the half-filled band in two. Hence, unlike low-temperature superconductors that are metals in which the (short-range repulsive) interactions are irrelevant, Mott insulation obtains from strong coupling physics. Precisely how the strong correlations mediate the myriad of phases in the doped cuprates remains unsettled. Nonetheless, there are some experimental facts that are clear. For example, when Mott insulators are doped, a ‘gap’ still remains [4] in the normal state. Dubbed the pseudo-gap, as zero-energy states exist [5] at some momenta, in particular along the diagonal connecting (0,0) and (*π*,*π*), this phenomenon remains one of the most nettling problems in cuprate phenomenology, as many of the articles in this Theme Issue attest. A typical value for the maximum of the gap along the (*π*,0) direction in underdoped Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} (Bi2212) is approximately 45 meV at an estimated hole content of 0.1 [6]. This hole-content level was obtained, not by counting the number of dopant atoms, as in the case of La_{1−x}Sr_{x}CuO_{2} (LSCO), in which the number of doped holes is obtained by counting the number of strontium atoms, but through the empirical formula [7]
1.1which accurately describes the evolution of the superconducting transition temperature, T_{c}, for LSCO as a function of doping. While this formula is on a firm experimental footing for LSCO, it has been widely criticized in the context of cuprates such as YBa_{2}Cu_{3}O_{7−δ} (Y123) and Tl_{2}Ba_{2}CuO_{6+δ} (Tl-2201), in which it is the oxygen content that determines the doping level. For example, Tokura *et al*. [8] and Tutsch *et al.* [9] found optimal doping values of *x*=0.21 and *x*=0.24, respectively, for YBCO. Others [10–12] have reached similar conclusions. Hardy and collaborators [13] have investigated the validity of equation (1.1) for Y123. However, their work does not offer an independent check on the validity of equation (1.1) for Y123 because, in correlating the change in the *c*-axis lattice constant (see [13], caption to fig. 2) with the doping level, they used equation (1.1). Consequently, they must confirm that *p*_{opt}=0.16 as they report.

I have digressed here on the doping level in the cuprates because one of the key issues with the pseudo-gap is where precisely it terminates, as articulated in the review by Norman *et al*. [14]. If the superconducting regions of all the cuprates are artificially made to have optimal doping at a hole content of 0.16, then the pseudo-gap, as determined [6] by spectroscopic probes such as angle-resolved photoemission spectroscopy (ARPES), scanning tunnelling microscopy and Raman scattering, will terminate at the end of the dome. While not all probes [15,16] find that *T**(*x*) merges with the terminus of the superconducting dome, the problem is partially one of accurately determining the doping level in the cuprates. Luckily, Honma & Hor [17] have recently addressed the problem of how to determine the doping level unambiguously in the cuprates in which it is the oxygens that act as the dopants. They have advocated [17] that, because the room temperature thermopower data for *all* the cuprates collapse onto a single curve, that curve can be used to calibrate the doping level and hence map out the superconducting region as a function of doping in an unbiased fashion. I reprint here a figure (figure 1) from their paper which illustrates that the thermopower scale is in excellent agreement with the doping level in Y123 determined by three different experimental methods. With this scale, the optimal doping level for most of the cuprates, except LSCO, occurs at *p*_{opt}≈0.21–0.23 [17]. This is roughly the doping level at which the pseudo-gap closes [16], as determined by transport measurements [15]. In this article, our focus is primarily transport, and hence we consider a phase diagram in which the pseudo-gap crosses the superconducting dome as shown in figure 2. That the superconducting region is depicted with a dome should not be taken literally.

Several questions arise with the pseudo-gap. Is it simply a remnant of the Mott gap? Is it a precursor to superconductivity? Does it represent a new ordered state? Does it compete with superconductivity? Regardless of how one answers these questions, the phase diagram makes it abundantly clear that the correct theory of the pseudo-gap should, above a characteristic scale, *T**, explain the strange metal in which the resistivity scales linearly with temperature. It is indeed odd then that the plethora of models [24–29] proposed to explain pseudo-gap behaviour have had little to say about the strange metal as a robust phenomenon persisting to high temperatures. Part of the problem is that a series of associated phenomena, for example incipient diamagnetism [30] indicative of incoherent pairing [26,28,29], electronic inhomogeneity [27, 31–35], time-reversal symmetry breaking [36–39] and quantum oscillations [40] in the Hall conductivity, possibly associated with the emergence of closed electron (not hole) pockets in the first Brillouin zone, are all consistent with some type of order with no immediate connection with the strange metal. We review briefly some of the proposals for the pseudo-gap, identify the problems in constructing a theory of the strange metal and sketch a possible resolution of this problem based on our recent work involving the exact integration of the high-energy scale in the Hubbard model [41–44].

## 2. Pseudo-gap phenomena

The phenomena surrounding the pseudo-gap [4,5,45] in the cuprates used to be fairly simple. In zero magnetic field, lightly doped cuprates possess Fermi arcs [5] in the normal state. That is, the Fermi surface that is present in the overdoped, more conventional Fermi liquid regime is destroyed on underdoping, leaving behind only a Fermi arc. In actuality, the situation is much worse. That the arc does not represent a collection of well-defined quasi-particles has been clarified by Kanigel *et al.* [46], who showed that, in Bi_{2}Sr_{2}CaCu_{2}O_{8−δ}, the length of the Fermi arc shrinks to zero as *T*/*T** tends to zero. Consequently, the only remnant of the arc at *T*=0 is a quasi-particle in the vicinity of (*π*/2,*π*/2) and hence the consistency with nodal metal phenomenology [47]. Aside from ARPES [46], Raman data [48] also support the nodal/antinodal dichotomy in the underdoped regime.

Recently, however, new ingredients have been added to the pseudo-gap story in the underdoped regime, which, on the surface, are difficult to reconcile with Fermi arcs. At high magnetic fields, quantum oscillations, indicative of a closed Fermi surface, have been observed [16] in Y123 and Tl-2201 through measurements of the Hall resistivity, Shubnikov–de Haas effect and the magnetization in a de Haas–van Alphen experiment. In underdoped samples, the Hall coefficient is negative, indicating closed electron-like orbits. Hence, there is an obvious incompatibility, though some have advocated none occurs [16], in interpreting the quantum oscillation experiments as a zero-field property of the underdoped cuprates where Fermi arcs obtain. Further, the fields, roughly 50 T, at which the quantum oscillations are observed are insufficient to kill the large gap that exists at the antinodal regions. The recent proposal by Pereg-Barnea *et al.* [49], which shows that quantum oscillations can arise from Fermi arcs terminated by a pairing gap, is noteworthy since it points to a possible resolution of the two phenomena.

Also attracting much attention is the recent experimental evidence for nematic order [50,51]. Daou *et al.* [50] observed that the Nernst signal in Y123 exhibits a large in-plane anisotropy that increases with decreasing temperature. The onset temperature is roughly *T*/*T**≈0.8. While the Nernst signal [30] had been measured previously [30], interpreted as evidence for incoherent pair formation above *T*_{c} but not all the way to *T**, it had not been measured with temperature gradients parallel to the *a*- and *b*-axes separately. While nematic order [27] can give rise to an anisotropy in the Nernst signal, Y123, in the doping regime measured, is already anisotropic in the *a*–*b* plane as it is orthorhombic. Further, it has been known for quite some time [52] that the thermopower has different signs parallel and perpendicular to the chains in Y123. As the Nernst signal involves the thermopower, such a sign difference could drastically amplify any modest anisotropy that exists among the combination of transport coefficients that contribute to the Nernst signal. Consequently, further experiments are needed to disentangle the inherent *a*/*b*-axis asymmetry in Y123 from that arising from an electronic phase that spontaneously breaks rotational symmetry.

Theoretical proposals for the pseudo-gap fall into three groups: (i) Mott-related physics having nothing to do with order [24,53,54], (ii) precursor superconductivity [26,28,29,55], and (iii) ordered states that break some type of symmetry, be it fourfold rotational symmetry [27], translational symmetry [25,34] or time-reversal symmetry [37]. In fact, experimental data support [38,39,50,51] many of the ordered states proposed for the onset of the pseudo-gap. However, it is unclear how any of these proposals are related to the origin of the strange metal. Nonetheless, a common approach [56] to meld ordered states with *T*-linear resistivity is to invoke quantum criticality. However, in its simplest one-parameter form, quantum criticality fails [56] to yield *T*-linear resistivity unless the dynamical critical exponent is negative, thereby violating causality. This result follows from three simple assumptions: (i) the charges are critical, (ii) one-parameter scaling is valid, and (iii) the *U*(1) charge is conserved. These three assumptions yield immediately to a general scaling form for the conductivity
2.1where *Q* is the charge, *Σ*(0) is the conductivity at *ω*/*T*=0 and *z* is the dynamical exponent. As a result, quantum criticality in its present form yields *T*-linear resistivity (for *d*=3) only if the dynamical exponent satisfies the unphysical constraint *z*<0. The remedy here might be threefold: (i) some other yet-unknown phenomenon is responsible for *T*-linear resistivity, (ii) the charge carriers are non-critical, or (iii) the single-parameter scaling hypothesis must be relaxed.

Marginal Fermi liquid theory (MFL) [57] does, however, offer a phenomenological account of *T*-linear resistivity. The key posit [57] here is that at *T*=0, regardless of momentum, the single-particle scattering rate is proportional to |*ω*|. Because of the linear dependence of the scattering rate on frequency, MFL phenomenology yields immediately *T*-linear resistivity. However, at present, there is no microscopic derivation of this highly successful account. Recently, however, the gauge/gravity duality [58] has proved useful [59] in this context. The key claim of the gauge/gravity duality is that some interacting quantum theories at strong coupling in *d*-space–time dimensions are dual to gravity theories in *d*+1-dimensional asymptotically anti-de Sitter (AdS_{d+1}) space–time. Unlike the standard equivalence between partition functions for *d*-dimensional quantum and *d*+1-dimensional classical systems in which the extra dimension represents time, in the gauge/gravity duality, the extra dimension represents the renormalization group (RG) scale. That is, in the AdS_{d+1} construction, an infinite number of copies of the original quantum mechanical theory, each at a different RG scale, fill the extra dimension. Hence, the original strongly coupled theory lives entirely at the boundary of the AdS_{d+1} space. In this construction, finite temperature and finite density correspond to having a charged black hole in the bulk geometry. Since the gravity theory is purely classical, all questions surrounding the strong-coupling physics in the original problem can be obtained from solving a set of linear wave equations in the charged black-hole geometry. Using the AdS_{4} construction, Faulkner *et al*. [59] computed the single-particle electron spectral function and showed that a range of non-Fermi liquid self-energies can emerge, including that of MFL theory. This result is truly remarkable as it represents the only derivation, albeit not microscopic, of MFL theory. However, a key question remains. Faulkner *et al*. [59] worked entirely with the gravity theory, and hence the underlying quantum theory is not known. Nonetheless, some hints as to the nature of the underlying theory are contained in the gravity solution to MFL theory. A key feature of the quantum theory on the charged AdS_{4} geometry is that the IR or low-frequency analytical structure of the correlation functions for the charge degrees of freedom is determined entirely by the near horizon metric . This serves to illustrate that the degrees of freedom that emerge in the IR and govern the analytical structure of the theory have no correspondence with those in the original UV-charged AdS_{4} limit. Perhaps the same is true of the underlying microscopic quantum theory that ultimately describes MFL theory. This suggests that extracting MFL behaviour from the basic model for a doped Mott insulator might be tricky because the natural variables that would expose this behaviour are not the bare electrons. Within the Hubbard model, it is dynamical spectral weight transfer across the Mott gap that makes the construction of a low-energy theory difficult. Isolating the propagating degrees of freedom which make the low-energy physics weakly interacting amounts to choosing a set of variables that essentially gets rid of the dynamical mixing. Such variables should in principle hold the key to *T*-linear resistivity. It is precisely this problem that we now address.

## 3. Fermi liquid theory breakdown: dynamical spectral weight transfer

To isolate how dynamical spectral weight transfer leads to a breakdown of Fermi liquid theory, it suffices to focus on how the intensity of the band in which the chemical potential resides scales with doping. For hole doping, the relevant band is the lower Hubbard band (LHB). The intensity of a band is equal to the number of charges that can fit into that band. A problem arises for a Fermi liquid account whenever there is a mismatch between the intensity of a band and the number of electrons that can fill the band. Such is the case in the Hubbard model close to half-filling. In the atomic limit, everything is known exactly. There are two bands split by the on-site repulsion *U*. When *x* holes are introduced, the intensity of the LHB is 1+*x* and that of the upper Hubbard band (UHB) is 1−*x*. The weight in the LHB separates into two parts. Since each hole leaves an empty site that can be occupied by either a spin-up or a spin-down electron, the empty part of the LHB has weight 2*x* [60] and the occupied part an intensity of 1−*x*. Hence, we see that the weight of the UHB and the occupied part of the LHB are equal in the atomic limit. This follows from the simple fact that removal of a doubly occupied site also removes one state from the LHB as well. There are, however, *t*/*U* corrections to these intensities beyond the atomic limit. In 1967, Harris & Lange [61] showed that the intensity of the LHB,
3.1has *t*/*U* corrections [61] that are entirely positive. Here, *f*_{iσ} is related to the original bare fermion operators via a canonical transformation that brings the Hubbard model into block diagonal form in which the energy of each block is *nU*. In fact, all orders of perturbation theory [61,62] increase the intensity of the LHB beyond its atomic limit of 1+*x*. It is these dynamical corrections that *α* denotes. While the intensity of the LHB increases away from the atomic limit, the total number of ways of assigning electrons to the LHB still remains fixed at 1+*x*. That is, the number of electron states in the LHB is independent of the hopping. Consequently, there is a mismatch between the intensity of the LHB and the number of electrons this band can hold. Since *m*_{LHB}>1+*x*, additional non-fermionic degrees of freedom are needed to exhaust the phase space of the LHB. These degrees of freedom affect the physics at all energy scales in the LHB. This has a profound consequence. The conserved charge is still the electron filling, *n*_{e}, which is obtained by integrating the density of states in the LHB up to the chemical potential, appropriately defined. But this quantity now has two contributions,
3.2one coming from the fermionic low-energy degrees of freedom, *n*_{qp}, in the LHB and the non-fermionic part. As a result, the fermionic quasi-particles in the LHB and the bare electrons can no longer correspond one to one as *n*_{e}>*n*_{qp}. This constitutes a breakdown of Fermi liquid theory.

We propose that the chemical potential for the effective number of low-energy fermionic degrees of freedom can be determined by partitioning the spectrum in the LHB so that dynamical spectral weight transfer is essentially removed. In such a picture, the empty part of the spectrum per spin is equal to the weight removed from the occupied part of the LHB when a hole is created. Hence, we arrive at the assignments of the spectral weights in figure 3*b*, in which the doping level is renormalized by the dynamics, i.e. *x*′=*x*+*α*. In other words, the effective number of fermionic degrees of freedom in the LHB is less than the conserved charge. This result already follows from the fact that, if the intensity of the LHB exceeds 1+*x*, electrons alone cannot exhaust the total degrees of freedom at low energies. Hence, by equation (3.2), there are two contributions to the conserved charge, thereby implying that the number of fermionic quasi-particles is less than the conserved charge. In other words, the dynamical degrees of freedom, denoted by *α*, serve to supplement the effective phase space of a hole-doped system and *x*′=*x*+*α* now denotes the effective number of hole degrees of freedom per spin at low energy.

For sufficiently large doping levels, the UHB collapses and the standard weakly interacting picture emerges. At this point, it is no longer meaningful to expand around the atomic limit and the analysis leading to figure 3*b* fails. This failure arises because the non-interacting ground state is not adiabatically connected to the atomic limit. Rather, perturbation theory around the band limit should be performed. In this limit, the weight of the band in which the chemical potential resides is 2, completely independent of doping. Hence, there must be a critical point as a function of doping or interaction strength at which the intensity of the lower band jumps to 2. This constitutes a collapse of Mottness. We have advocated [44] that the doping level at which the UHB collapses coincides with the closing of the pseudo-gap. This is physically reasonable because, unless a gap exists, there is no real separation between the UHB and LHB. The simulations of Kyung *et al.* [64] on the Hubbard model also provide clear evidence that a strong correlation exists between the pseudo-gap and the separation of the low- and high- energy bands. Experimental evidence for this collapse has been reported recently by Peets *et al.* [65] from soft X-ray scattering on the oxygen K-edge. In such an experiment, an electron is promoted from the core 1s to an unoccupied level. The experimental observable is the fluorescence yield as a function of energy as electrons relax back to the valence states. Experimentally [65], the fluorescence yield is related to the empty part of the spectrum projected onto the oxygen p-orbitals. Consequently, within a one-band Hubbard model, the relevant quantity is
3.3As shown in figure 4, beyond a critical doping level, the slope of the oxygen K-edge intensity changes abruptly. Peets *et al.* [65] interpreted the slope change as evidence for a saturation and hence a deviation from what is expected in the one-band Hubbard model. However, given the error bars on the data and the additional point at *x*=0.34 of Chen *et al.* [66], one cannot rule out that the data are simply associated with a slope change around a doping level of *x*=0.22. To verify this assertion, we plotted (figure 4) a computation of *L* (solid line) by Liebsch [69] on the two-dimensional Hubbard model using a self-consistent cluster method. As is evident, the agreement with the solid curve and most of the data points is excellent. Hence, over the complete doping range of interest, the low-energy spectral weight in the cuprates is well described by the doping-induced states in the one-band Hubbard model.

To summarize, dynamical spectral weight has two profound effects in the normal state of a doped Mott insulator. As long as it is present, a low-energy theory with electrons alone cannot exhaust the total number of charge degrees of freedom in the LHB. This results in a mismatch between the number of fermionic quasi-particles and bare electrons, thereby leading to a breakdown of Fermi liquid theory. In addition, *L*/*n*_{h}>1 implies that the number of ways of adding particles at low energies exceeds the number of electrons that can be added to the LHB. Consequently, *L*/*n*_{h}>1 is an indication that some states at low energy are gapped to the addition of an electron. Within the experimental accuracy [65], there is a dramatic slope change in *L* in the doping range where the pseudo-gap closes, adding credence to the emphasis placed here on dynamical spectral weight transfer as the signature of the pseudo-gap. However, to be completely consistent with the opening of a pseudo-gap, dynamical spectral weight transfer should also be temperature dependent. We show here that this is the case. Namely, the dynamical correction turns on only below a characteristic temperature set by *T**. This is an indication that some type of bound-state formation underlies *α*≠0. Above *T**, the new degrees of freedom are unbound and create the *T*-linear resistivity. Since this mechanism is capable of generating *T*-linear resistivity beyond *T** and a pseudo-gap below, we arrive at a consistent theory of two of the most elusive characteristics of the normal state of the cuprates.

## 4. Low-energy theory of Mottness

In the parameter space relevant to the cuprates, the Hubbard model represents strongly coupled physics. In this regime, no continuum limit exists and, as a consequence, no firm statement can be made as to the precise nature of the charge carriers. Ultimately, it is only correct in this regime to focus on the current, which can be given an interpretation in terms of single particles if the propagating degrees of freedom can be identified. Such an identification should be possible once the high-energy scale is correctly integrated out. A successful low-energy reduction of the Hubbard model should reveal the new degree of freedom that causes the intensity of the lower band to exceed 1+*x*. As this degree of freedom arises from the mixing with the doubly occupied sites (or the lack of rigidity of the Hubbard bands), it should have charge 2*e*. Charge 2*e* objects do not contribute to the fluorescence yield of the single-particle spectrum unless they combine with something else to produce a charge *e* entity. The proposed mechanism for the pseudo-gap is ‘doublon–holon’ bound-state formation mediated by a collective IR charge 2*e* mode. Below a characteristic temperature, such bound states are stable. Quite generally, such bound states are expected to mediate the Mott insulating state as well. Consider half-filling. Even in the half-filled state, the dynamical mixing that leads to a non-zero *α* is still present. That is, double occupancy is present even in the ground state of a half-filled band. Since a doubly occupied site must result in the simultaneous creation of an empty site, the insulating state persists only if the doubly occupied and empty site are bound, as has been proposed previously [70–72]. Otherwise, hole conduction obtains.

Such binding should emerge from the correct low-energy theory of a half-filled Hubbard band. We have shown previously [41–44,53,73] how the high-energy degrees of freedom can be integrated out exactly at any filling. The key idea is [41,43,44,73] to extend the Hilbert space by introducing a new fermionic field that creates excitations on the *U* scale without identifying such physics with double occupancy. At half-filling, the Lagrangian [43] simplifies to
4.1
4.2
This theory contains two bosonic fields with charge 2*e*(*φ*^{†}) and . These bosonic modes are collective degrees of freedom, not made out of the elemental excitations, which represent dynamical mixing with *U*-scale physics, namely the contribution of double holes (−2*e*) and double occupancy (2*e*) to any state of the system. Here *s* and are constants with units of energy, all operators in equation (4.1) have the same site index, repeated indices are summed over both the site index and frequency, *ω*, creates a fermion on site *i* with spin *σ*,
4.3
and the dispersion is given by , where **k** and **p** are the centre of mass and relative momenta of the fermion pair. The coefficients
4.4
play a special role in this theory as they account for the turn-on of the spectral weight. At the level of a Lagrangian, the vanishing of the coefficient of a quadratic term defines the dispersion of the associated particle. All the terms that are naively quadratic (equation (4.1)), possess constant coefficients and hence we reach the conclusion that there are no propagating bosons or electrons. What equation (4.1) makes plain is that the turn-on of the spectral weight in a Mott insulator cannot be formulated in terms of bare electrons, at least in a low-energy theory. This is consistent with the emerging view [74–76] that, in a Mott insulator, the single-particle Green function vanishes along a locus of points in momentum space. Physically, a vanishing of the single-particle electron Green function implies that the electrons are not the propagating degrees of freedom. Precisely what the propagating degrees of freedom are is determined by the dispersing modes in the low-energy Lagrangian. Consider the second line of the Lagrangian, equation (4.2). Appearing here are two interaction terms, which describe composite excitations, whose coefficients can vanish. Figure 5*b* shows explicitly that the vanishing of *γ* and leads to spectral weight that is strongly peaked at two distinct energies, ±*U*/2. Each state in momentum space has spectral weight at these two energies. The width of the bands is 8*t*. The particles that give rise to the turn-on of the spectral weight are composite excitations or the bound states of the bosonic and fermionic degrees of freedom determined by the interaction terms *φ*^{†}*cc* and . In terms of the variables appearing in the Hubbard model, we make the heuristic association of the composite excitations with bound states of double occupancy and holes, as has been postulated previously [70], to be the ultimate source of the gap in a Mott insulator. This association is entirely heuristic because the variables in the original UV variables are not straightforwardly related to the degrees of freedom in the IR. The gap (Mott gap) in the spectrum for the composite excitations obtains for *U*>8*t* as each band is centred at ±*U*/2 with a width of 8*t*. Figure 5 demonstrates that the transition to the Mott insulating state found here proceeds by a discontinuous vanishing of the spectral weight at the chemical potential to zero. However, a continuous evolution of the Mott gap is seen in numerical calculations [77] in finite-dimensional lattices but not in the [78] solution.

In terms of the bare electrons, the overlap with the composite excitations determines the Mott gap. To determine the overlap, it is tempting to complete the square on the *φ*^{†}*cc* term bringing it into a quadratic form, *Ψ*^{†}*Ψ*, with *Ψ*=*Aφ*+*Bcc*. This would lead to composite excitations having charge 2*e*, a vanishing of the overlap and hence no electron spectral density of any kind. However, the actual excitations that underlie the operator *φ*^{†}*cc* correspond to a linear combination of charge *e* objects, *c*^{†} and *φ*^{†}*c*. In terms of the UV variables, the latter can be thought of as a doubly occupied site bound to a hole. At half-filling [42,44], the exact representation of the electron creation
4.5
is indeed a sum of two composite excitations, the first having to do with spin fluctuations (*b*^{†}*c*) and the other with high-energy physics, *φ*^{†}*c* and , i.e. excitations in the UHB and LHB, respectively. We can think of the overlap
4.6
in terms of the physical process of passing an electron through a Mott insulator. The overlap will involve that between the bare electron with the low-energy excitations of equation (4.5), , and the overlap with the propagating degrees of freedom, with *P*_{Ψ}, the propagator for the composite excitations. Because of the dependence on the bosonic fields in equation (4.5), *O* retains destructive interference between states above and below the chemical potential. Such destructive interference between excitations across the chemical potential leads to a vanishing of the spectral weight at low energies [60]. Consequently, the turn-on of the *electron* spectral weight cannot be viewed simply as a sum of the spectral weight for the composite excitations. As a result of the destructive interference, the gap in the electron spectrum will always exceed that for the composite excitations. Hence, establishing (figure 5) that the composite excitations display a gap is a sufficient condition for the existence of a charge gap in the electron spectrum. A simple calculation [43] (figure 5*c*) of the electron spectral function at *U*=8*t* confirms this basic principle that a gap in the propagating degrees of freedom guarantees that the electron spectrum is gapped. Further, figure 5 confirms that the electron spectral function involves interference across the Mott scale. Consequently, although the composite excitations are sharp, corresponding to poles in a propagator as in equation (4.2), the electrons are not.

### (a) Binding–unbinding transition

The bound states that mediate the Mott gap also survive in the doped state. In fact, through dynamical spectral weight transfer, such bound states are transferred down in energy from the UHB. That some type of bound state must exist in the doped state of a Mott insulator can be inferred from the work of Gor’kov & Teitel’baum [79]. They observed remarkably that the charge carrier concentration, *n*_{Hall}, extracted from the inverse of the Hall coefficient in La_{2−x}Sr_{x}CuO_{4} (LSCO) obeys an empirical formula,
4.7
appropriate for a two-component or two-fluid system. One of the components is independent of temperature, *n*_{0}(*x*) (*x* the doping level), while the other is strongly temperature dependent, . The key observation here is that the temperature dependence in *n*_{Hall} is carried entirely within Δ(*x*,*T*), which defines a characteristic activation energy scale for the system, the pseudo-gap scale. Hence, a charge density that obeys equation (4.7) is symptomatic of a gap in the spectrum. Equation (4.7) offers direct confirmation for a renormalization of the doping level purely from dynamical effects. Our key contention here is that *α* arises from the second term in equation (4.7) and is mediated by the binding of to a hole.

We computed the Hall coefficient [80] in this theory by first obtaining the spectral function. As the computation of the spectral function is instructive in uncloaking the new propagating degrees of freedom, we recount it here. Noting that the conserved charge is given by
4.8
the spectral function for just the fermionic component at low energies *c*_{iσ} will necessarily have an integrated weight less than 1−*x* [81]. To illustrate this, we treat *φ*_{i} initially as a spatially independent field, owing to the lack of any gradient terms of *φ*_{i} in the action. Note that the integrated weight of the UHB is less than 1−*x* follows strictly from the form of the conserved charge not from any approximation scheme that is used to compute the spectral function. The electron Green function is then written as a path integral over the *φ* fields as
4.9
where the effective Lagrangian *L* is expressed in a diagonalized form
4.10
where the *γ*_{kσ} are the Boguliubov quasi-particles and are given by
4.11
and
4.12
where , , *E*_{0}=(−2*μ*+*s*^{2}/*U*)*φ***φ*, *E*_{k}=−*g*_{t}*tα*_{k}−*μ*, , and the gap is proportional to *s*, Δ_{k}=*sφ**(1−(2*t*/*U*)*α*_{k}), and hence vanishes when *φ* is absent and *g*_{t}=2*δ*/(1+*δ*), *δ*=1−*n*. The *g*_{t} term originates from the correlated hopping term, . The *γ*_{kσ}’s play the role of the fundamental low-energy degrees of freedom in a doped Mott insulator. That is, they are the natural propagating charge degrees of freedom. Note that they depend in a complicated way on the *φ*_{i} field and consequently are heavily mixed with the doubly occupied sector. Upon integrating out the fermions, we obtain
4.13
where
4.14
is the exact Green function corresponding to the Lagrangian, equation (4.10), which has a two-branch structure, corresponding to the bare electrons and the coupled holon–doublon state, respectively. The role of the *φ* field, which determines the weight of the second branch, is vital to our understanding of the properties of Mott systems, as was demonstrated previously [42,44]. It is trivial to see that, in the limit of vanishing *s* (no *φ* field), the *γ*_{kσ}’s reduce to the bare electron operators *c*_{k} and the first term in equation (4.14) vanishes. The two-fluid nature of the response stems from this fact of the theory. Namely, the first term contributes only when *φ*≠0 and the second when *φ*=0, as depicted in the spectral function in figure 6*a*. These contributions correspond to the dynamical and static components of the spectral weight, respectively, and appear as two distinct branches in the electron spectral function. Since the second branch corresponds to a bound degree of freedom, a gap opens at the chemical potential. As we demonstrate in figure 6*c*, the total weight of both branches that composes the total number of fermionic degrees of freedom is less than the conserved charge 1−*x*. This is dictated by the fact that conserved charge at low energies (equation (4.8)), consists of a fermionic as well as a bosonic component, thereby directly supporting the partitioning of the spectral weight in figure 3*b*.

We obtained the Green function *G*(**k**,*ω*) by numerical integration and the subsequent spectral function, *A*(**k**,*ω*). We computed the Hall coefficient from the spectral function using
4.15
where
4.16
and
4.17
in which *σ*_{xx} and *σ*_{xy} are the diagonal and off-diagonal components of the conductivity tensor, respectively, *f*(*ω*) is the Fermi distribution function and *B* is the normal component of the external magnetic field. The effective charge carrier density *n*_{Hall} is then obtained using the relation *R*_{H}=−1/(*n*_{Hall}*e*). Figure 7 shows a set of plots of *n*_{Hall} as a function of the inverse temperature, each corresponding to a different value of hole-doping, *x*, in the underdoped regime (*x* ranging from 0.05 to 0.20). The plots fit remarkably well to an exponentially decaying form. In other words, the computed charge carrier density within the charge 2*e* boson theory of a doped Mott insulator agrees well with the form given in equation (4.7) proposed by Gor’kov & Teitel’baum [79]. The inset shows the temperature-independent part of the charge density as a function of *x*. This quantity exceeds the nominal doping level [82]. Such a deviation in the nominal doping level is also supported by ARPES. Shown in figure 8 is a plot of the measured Fermi surface volume, *x*_{FS}, in LSCO as a function of the nominal doping level *x*. As is clear, *x*_{FS} deviates from linearity precisely where our calculated value of *n*_{0}(*x*) does. In fact, the agreement between the inset in figures 7 and 8 is striking. Nonetheless, in previous publications [82], this deviation was attributed to a band dispersion effect. The clear corroboration of this effect with our strong-coupling calculation suggests that the ultimate source of the deviation from linearity is Mottness. In a hole-doped Mott insulator, the Hall coefficient must change sign [83] before the particle-hole symmetric condition for the atomic limit, namely when 2*x*=1−*x* or equivalently before *x*=1/3 [84,85].

The ‘binding energy’, Δ(*x*), was extracted for each doping and plotted in figure 9 using equation (4.7). Shown here also are the values for the experimentally determined pseudo-gap energy for LSCO [86–88]. The magnitude of Δ(*x*) falls with increasing hole doping as is seen experimentally and hence is consistent with its interpretation, even quantitatively, as a measure of the pseudo-gap temperature *T**. A rough estimate of *T**,
4.18
may be obtained from Δ(*x*), by equating the number of doped carriers *x* with that of the activated ones , as proposed by Gor’kov & Teitel’baum [79]. Figure 10 shows a plot of *T** as a function of *x*. This result is in qualitative agreement with the experimentally obtained estimates of *T** [45,90,91].

### (b) T-linear resistivity

The physical picture for the charge 2*e* boson calculation is now clear. Below a characteristic temperature, the boson is bound to a hole and produces charge *e* states. This leads to a non-zero value for *α*. Above *T**, the bound states break up. The simplest way of understanding why the charge 2*e* boson must be bound at low energies, aside from the fact that it has no bare dynamics, is that, once the high-energy sector is integrated out exactly, the Hilbert space shrinks back to the Fock space of the Hubbard model.

The mechanism for *T*-linear resistivity is simple within this model. Once the binding energy of the boson vanishes, bosons are free to scatter off the electrons. The absence of a kinetic energy term for the bosons implies that their dynamics are classical. The resistivity of electrons scattering off classical bosons is well known to scale linearly with temperature above the energy to create the boson as depicted in figure 11. Hence, this mechanism is robust and should persist to high temperatures. Consequently, the charge 2*e* boson reduction of the Hubbard model offers a resolution of the pseudo-gap and the transition to the strange metal regime of the cuprates. Assuming the pseudo-gap coincides with optimal doping (Mottness collapse), the resistivity should acquire its maximum value here. Consequently, the resistivity should scale as a function of 1/|*E*_{B}−*μ*|, where *E*_{B} is the energy of the bound state and *μ* the chemical potential. Such a functional form should explain the experimentally observed [92] broad region over which *T*-linear resistivity is observed and why the region is roughly symmetrical with respect to doping. The corrections to *T*-linear resistivity are determined by the radiative corrections to the boson mass
4.19
with
4.20
Hence, the bare boson dynamics become non-trivial at *O*(*t*^{3}/*U*^{2}) in perturbation theory. This implies that the corrections to *T*-linear resistivity are subleading in powers of *t*/*U*. One of the outstanding problems as of this writing is the precise nature of the unbinding transition of the charge 2*e* boson. Is the transition sharp? Whether it is or not, there should be subleading terms (in powers of *t*/*U*) to the conductivity that will provide corrections to the *T*-linear term as observed experimentally [92].

Further, the mechanism for the strange metal is consistent with the scaling analysis leading to equation (2.1). Namely, *T*-linear resistivity requires an additional energy scale absent from a single-parameter scaling analysis. In the exact low-energy theory, a charge 2*e* boson emerges as a new degree of freedom. While it is bound in the pseudo-gap regime, its unbinding beyond a critical temperature or doping provides the added degree of freedom to generate the anomalous temperature dependence for the resistivity. A further experimental prediction of this work then is that the strange metal regime should be populated with charge 2*e* excitations, without the usual diamagnetic signal. Shot noise measurements are ideally suited for testing this prediction.

## 5. Final remarks

We have focused here on explaining three inter-related facts. (F1) Breaking the Landau correspondence between the Fermi gas and the interacting system requires new degrees of freedom. (F2) The connection between *T*-linear resistivity and quantum criticality is only possible if an additional degree of freedom outside the one-parameter paradigm is present as equation (2.1) makes plain. (F3) The total weight of the low-energy band exceeds the number of ways electrons can be assigned to this band, thereby requiring new low-energy degrees of freedom. The common ingredient among F1–F3 is a new degree of freedom distinct from the electrons. Explicitly carrying out the Wilsonian programme for the Hubbard model by integrating out the high-energy sector produces the missing degree of freedom that is capable of explaining all of these facts. Since it is the mixing with the doubly occupied sector that enhances the intensity of the LHB beyond 1+*x*, the new degree of freedom at low energies must have charge 2*e* and hence must be bosonic. This has been explicitly demonstrated by the exact procedure to integrate out the high-energy sector. At low energies and temperatures, the charge 2*e* boson mediates new charge *e* states in the LHB. Such bound states already exist in the half-filled system and constitute the propagating degrees of freedom that describe the UHB and LHB. Hence, the pseudo-gap emerges in the charge 2*e* theory as a remnant of the Mott gap. Heuristically, the bound states and hence the gap originate from a doubly occupied site bound to a hole.

This work makes a series of experimental predictions. First, below the *T** line *L*/*n*_{h}>1, whereas above it *L*/*n*_{h}=1. This information can be extracted from temperature-dependent experiments of the kind leading to figure 4. Second, any experimental probe that couples to the low-energy excitations should be interpreted in terms of *x*′, not the bare hole number *x*. These include measurements of the (i) optical conductivity, (ii) superfluid density, and (iii) ‘Fermi surface’ volumes extracted from quantum oscillation experiments (typically Hall measurements). The last is particularly germane because the Fermi surface volumes extracted experimentally [40,93] for YBCO are not consistent with any integer multiple of the physically doped holes.

That the charge 2*e* boson mediates local bound states is *a priori* expected as it lacks a bare kinetic term. The bound states that form account for the pseudo-gap and their break-up leads to a resistivity that quite generally scales linearly with temperature. As long as the UHB is present, the programme carried out here is sufficient to describe the physics of a doped Mott insulator. Once the UHB collapses, traditional Fermi liquid descriptions obtain. As a separation between the UHB and LHB is only meaningful if there is a gap in the spectrum, the pseudo-gap lies at the heart of Mottness. Precisely the role such bound states play in the superconducting state remains open.

## Acknowledgements

I thank R. Leigh, T.-P. Choy, S. Hong, S. Chakraborty, D. Galanakis and S. Chakraborty for extensive collaborations that led to the work summarized here. I also thank Frank Krüger, Lance Cooper and P. Abbamontte for extensive discussions on Mottness. I also acknowledge financial support from the NSF DMR-0940992 and the Center for Emergent Superconductivity, a DoE Energy Frontier Research Center, award number DE-AC0298CH1088.

## Footnotes

One contribution of 7 to a Theme Issue ‘Normal state of the copper oxide high-temperature superconductors’.

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