1 Introduction

Gauge approach to the ”pseudogap” phenomenology

of the spectral weight in high cuprates

P A Marchetti and M Gambaccini

Dipartimento di Fisica e Astronomia, U. di Padova and INFN, I-35131 Padova, Italy

We assume the -- model to describe the planes of hole-doped cuprates and we adapt the spin-charge gauge approach, previously developed for the - model, to describe the holes in terms of a spinless fermion carrying the charge (holon) and a neutral boson carrying spin (spinon), coupled by a slave-particle gauge field. In this framework we consider the effects of a finite density of incoherent holon pairs in the normal state. Below a crossover temperature, identified as the experimental ”upper pseudogap”, the scattering of the ”quanta” of the phase of the holon-pair field against holons reproduces the phenomenology of nodal Fermi arcs coexisting with gap in the antinodal region. We thus obtain a microscopic derivation of the main features of the hole spectra due to pseudogap. This result is obtained through a holon Green function which follows naturally from the formalism and analytically interpolates between a Fermi liquid-like and a -wave superconductor behaviour as the coherence length of the holon pair order parameter increases. By inserting the gauge coupling with the spinon we construct explicitly the hole Green function and calculate its spectral weight and the corresponding density of states. So we prove that the formation of holon pairs induces a depletion of states on the hole Fermi surface. We compare our results with ARPES and tunneling experimental data. In our approach the hole preserves a finite Fermi surface until the superconducting transition, where it reduces to four nodes. Therefore we propose that the gap seen in the normal phase of cuprates is due to the thermal broadening of the SC-like peaks masking the Fermi-liquid peak in the spectral weight. The Fermi arcs then correspond to the region of the Fermi surface where the Fermi-liquid peak is unmasked.

PACS numbers: 71.10.Hf, 11.15.-q, 74.72.-h, 74.72.Kf

## 1 Introduction

The phenomenon of ”pseudogap” in hole-doped cuprates appears rather complex, exhibiting different features in various regions of the doping-temperature phase diagram and even the word ”pseudogap” is used with different meanings by different authors. One of the most spectacular features associated with pseudogap phenomenology is the appearance revealed by ARPES of a truncated Fermi surface consisting of Fermi arcs. There have been several theoretical explanations of this phenomenon, many of them critically analyzed in [1], but no consensus has been reached, since none of these proposals explains all the characteristic features of the experimental data.

In this paper we develop an ”explanation” of the Fermi arcs within a generalization of a gauge approach to superconductivity in cuprates recently proposed [2], comparing the results with ARPES and tunneling data. To understand our proposal it is useful first to sketch the pairing mechanism, eventually leading to superconductivity, for underdoped cuprates presented in [2] within a gauge approach to the - model: as we dope a vortex-like quantum distortion of the AF background is generated around the empty sites (described in terms of fermionic spinless holons) with opposite chirality for cores on the two Néel sublattices. The spin excitations (bosonic spin- 1/2 spinons) are gapless without doping, corresponding to long-range AF order, but above a critical doping density they acquire a finite gap due to scattering against the spin vortices and the long-range anti-ferromagnetic order is converted to a short-range order. Due to the no-double occupation constraint, decomposing the hole into holon and spinon generates a local gauge symmetry inducing in turn a gauge attraction between holon and spinon binding them into a physical hole.

The primary pairing force for the charge carriers is an attraction due to chirality between spin vortices with cores on two different Néel sublattices, inducing an attraction between holons in their cores. As a consequence of this attraction at a crossover temperature, denoted , a finite density of incoherent holon pairs are formed. In [2] it was claimed that this phenomenon produces a reduction of the hole spectral weight on the Fermi surface (FS) and it was proposed to identify this temperature with the experimentally observed ”upper pseudogap” , where the in-plane resistivity deviates from the linear behaviour. At a lower crossover temperature, denoted , also a finite density of incoherent spinon pairs appears, giving rise to a gas of incoherent preformed hole pairs through holon-spinon gauge attraction .

Finally, at a even lower temperature, , the hole pairs become coherent and superconductivity emerges. This approach exhibits yet another crossover temperature [3], , intersecting in the doping-temperature phase diagram. Such crossover is not directly related to superconductivity. It corresponds to a change in the holon dispersion. It is characterized by the emergence of a ”small” holon Fermi surface around the momenta (), with complete suppression of the (coherent) spectral weight for holes in the antinodal region and partial suppression outside of the magnetic Brillouin zone (MBZ).

This crossover appears only in bipartite lattices. Below the effect of short-range AF fluctuations becomes stronger and the transport physics of the corresponding normal state region is dominated by the interplay between the short-range anti-ferromagnetism of spinons and the thermal diffusion induced by the gauge fluctuations triggered by the Reizer [4] momentum. This interplay produces in turn the metal-insulator crossover [3]. We identify in experimental data with the inflection point of in-plane resistivity and the broad peak in the specific heat coefficient [5]. The region above in the doping()-temperature() phase diagram will be called the “strange metal phase” (SM), the one below will be called the “pseudogap phase” (PG). Actually there is no agreement between the experimentalists on the existence of two crossovers (in our approach and ) associated to the pseudogap phenomenology and the same is true for most of the theoretical approaches. For clarity in Fig. 1 we present a schematic behaviour of the crossovers and in the phase diagram.

In this paper we extend the pairing mechanism for holes developed for the PG of the - model to the normal phase of the -- model showing explicitely that and are indeed distinct.

It turns out that a finite density of incoherent holon pairs induces an angle dependent reduction of the (physical) hole spectral weight on the FS, starting from the antinodal regions, thus producing a microscopic derivation of the phenomenology of Fermi arcs coexisting with gap in the antinodal region. Notice that this effect on the holes occurs even if the pairing is only among unphysical holons, the spinons being still unpaired, so that there is not yet a gas of preformed hole-pairs.

We show that, when a gas of incoherent holon pairs is present, an energy scale separating low energy modes with a Fermi liquid (FL) behaviour from high energy modes with a -wave superconducting behaviour results naturally and self consistently. This energy scale will be identified with the the inverse correlation length (”mass”) of the quanta of the phase of the holon pairs field. The scattering of these excitations against holons produces in the holon Green function lowering a gradual reduction of the spectral weight on the FS at small frequency as we move away from the diagonals of the Brillouin zone. Simultaneously at larger frequencies we have the formation and increase of two peaks of intensity precursors of the excitations in the superconducting (SC) phase. The value of in fact decreases with , in an extended range approximately linearly, and this decrease drives the system towards the superconducting phase occurring when, possibly discontinously (as experimentally suggested e.g. by [6]), jumps to 0, in correspondence with the condensation of the hole pairs. Decreasing , the well defined FL quasi-particles appearing at high gradually lose their coherence in favor of SC-like excitations. The latter gain spectral weight and become increasingly well defined excitations. In the SC phase, all the modes are above and the holon system is a -wave superconductor, in particular the holon spectral weight at the FS is reduced to zero except on four nodes.

The physical hole is obtained as a holon-spinon resonance produced by the gauge attraction and it inherits the above holon features, but with a strongly enhanced scattering rate, due to the spinon contribution.

The behaviour of the spectral weight derived from the above sketched mechanism appears consistent with many ARPES data and it is able to explain the evolution of the density of states as derived in tunneling experiments. This mechanism of spectral weight suppression exhibits the fingerprint of the presence of the slave-particle gauge field, because the smooth interpolation between FL and SC behaviour is actually due to the interaction of the phase of the holon pairs with the gauge field. Without such interaction the SC-like peaks are strongly suppressed outside of the SC phase, disagreeing with experiments.

Although many phenomenological features in our approach are similar to those of the approach proposed phenomenologically in [7] and partially justified microscopically in [8], at odds with those approaches in ours the hole for (i.e except in the SC phase) has always a FS without gap. This is consistent with the fact that the ”pseudogap” region can be reached (at least in our approach) from a FL behaviour without crossing a phase transition, only crossovers being involved in the process, which appears to agree with experiments. We propose therefore that the gap seen in the normal phase of cuprates is due to the thermal broadening of the SC-like peaks, the Fermi arcs corresponding to the region of the Fermi surface where the Fermi-liquid peak is unmasked from that broadening. A further complete suppression of the spectral weight in the antinodal region occurs in the PG, as previously discussed. Readers only interested in comparison of final theoretical results with experiments can skip the next sections and go directly to section 8, where we recapitulate the key points of our approach before comparing with experimental data. The effects of the reduction of the spectral weight on transport properties will be discussed in a companion paper [9] where we prove, in particular, a deviation with negative curvature from linearity in of in-plane resistivity.

## 2 The spin-charge gauge approach to the t-t′-J model

The approach in its original version [3] assumed the - model as model Hamiltonian for the low-energy physics of the CuO planes of the cuprates. Here we add a negative next-nearest-neighbor hopping term (with ) which is known necessary to fit the FS of cuprates. The scheme of the approach, however, remains unchanged and we sketch it here for reader’s convenience and to set up the notation, emphasizing only the new features.

We decompose the hole operator at site of the model as , where is a spinless fermionic holon, carrying charge, while is a spin 1/2 bosonic spinon, carrying spin and obeying the constraint . The redundant degree of freedom arising from this decomposition is cured by an emergent slave-particle gauge field, , minimally coupled to holon and spinon with the same charge. With the choice of statistics adopted the no-double occupation constraint is automatically satisfied because of the Pauli principle for the holon. [ For simplicity in this paper we use the same symbols to denote the field operators in the hamiltonian formalism and the corresponding fields in the path-integral lagrangian formalism. ] One then uses the possibility offered in 2D (and 1D) to add a “statistical” spin flux () to and a “statistical” charge flux () to “compensating” each other so that the product is still a fermion. The introduction of these fluxes in the lagrangian formalism is materialized through Chern-Simons gauge fields. We then optimize their choice in an improved mean-field approximation (MFA). A key step of MFA is to find a reference spinon configuration with respect to which expand the fluctuations that will be described by a new staggered spinon field, . This reference configuration is found optimizing the free energy of holons in the presence of a fixed holon-dependent spinon configuration. One can show as in the 1D [10] and 2D [11] - model that the optimization involves a spin-flip associated to every holon jump between different Néel sublattices. Furthermore, if we neglect the spinon fluctuation in the spin flux, the effect of the optimal spin flux is to attach a spin-vortex to the holon, with opposite chirality on the two Néel sublattices. More precisely with this approximation

 Φs(x)≈−∑lh∗lhl(−1)|l|2arg(→x−→l). (2.1)

The gradient of can be seen as the potential of a vortex. These vortices take into account the long-range quantum distortion of the AF background caused by the insertion of a dopant hole, as first discussed in [12], the rigidity holding up them being provided precisely by the AF background.

Neglecting the holon fluctuations in , the optimal charge flux in the 2D - model was argued to provide a staggered flux per plaquette for small enough doping and temperature. This anomalous behaviour near half-filling was justified on the basis of a rigorous result by Lieb [13] and a numerical simulation, proving that in a square 2D lattice at half-filling the optimizing flux for the free energy of fermions is indeed translationally invariant and per plaquette. The contribution from the reference spinon configuration in the optimization can be shown to trivialize the optimal flux at high enough doping or temperature, in agreement with diamagnetic inequality. Although Lieb’s proof does not extend to the - model, one can still rely on the results of [14] which show that for elementary circuits, triangles and squares, the optimal flux at half-filling is for triangles and for squares. Assuming that the above rule for fluxes matches with the staggered flux per square plaquette of the underlying model one founds that along the diagonal links of the - model the flux should be trivial, this implying that parity and time-reversal are still preserved. On the basis of the above considerations we extend the conjecture made for the - model as

Assumption The optimal charge flux in the -- model at sufficiently small doping and temperature is given as follows: let be a site in the even Nèel sublattice and be its n.n. sites in the 1 and 2 directions respectively and its n.n.n. sites. Then

 Φh(i±^α)−Φh(i)=(−1)α(±iπ/4),Φh(i±^β)−Φh(i)=0. (2.2)

Also for the -- model the contribution from the reference spinon configuration in the optimization can be shown to trivialize the optimal flux at high enough doping or temperature. Extending a conjecture made for the - model we propose that the crossover between the two behaviours discussed above corresponds to the crossover between the “pseudogap phase”(PG) and the “strange metal phase”(SM) in the cuprates. Neglecting the fluctuations of the gauge field which can be reinserted by Peierls minimal substitution, the leading terms of the Hamiltonian can then be written as:

 H≃∑n.n.(−t)AMijh∗ihjei(Φhi−Φhj)+J(1−h∗ihi−h∗jhj)(1−|AMij|2) +Jh∗ih∗jhjhi|RVB|2ij+∑n.n.n.<>t′AMijh∗ihj. (2.3)

where , with (resp.) if is in the even (resp. odd) Nèel sublattice, and it is a kind of Affleck-Marston spinon parameter [15] and is an RVB spinon singlet order parameter. The AM/RVB dichotomy in (2) is due to the spin-flip in the optimization procedure described above . Then we use the following MFA for holons and spinons: in the first term in (2) we take , while in the second term we replace the hole density by its average and in the normal state we neglect the third term because of being higher order in doping (). A long-wavelength treatment of the second term in (2) leads to a spinon (CP) non-linear sigma model with an additional term coming from the spin flux,

 J(1−2δ)(∇Φs)2z∗z, (2.4)

where with the 2D lattice laplacian. In a quenched treatment of spin vortices one finds for the average , producing a mass gap for the spinon, consistent with AF correlation length at small derived from the neutron experiments [16].

In the parameter region to be compared with the PG of the cuprates the charge flux per plaquette converts the spinless holons into Dirac fermions with small Fermi surfaces centered at : , where is the (average) holon Fermi momentum and the Fermi energy. The holon dispersion is defined in the Magnetic Brillouin Zone. In the parameter region to be compared with the SM of the cuprates, where the optimal charge flux per plaquette is , we recover a standard tight-binding “large” FS ( ) for the holons, centered at (). Let us briefly comment how these FS arise. In the model the density of the Gutzwiller projected holes is proportional to the doping , the holes corresponding in coordinate space to the empty sites. Substituting the representation of the hole fields in terms of holon and spinon in the lagrangian in the presence of the reference spinon configuration identified by the optimization sketched above, for the holons dressed by the charge flux in the SM one finds the 0-energy level at the position of the FS of the tight-binding unprojected model. This result is compatible with the holon density if 2 holons can occupy the same momenta and the ”vacuum” of the model is set at half-filled holon band, namely the holons relevant physically for the projected holes are those corresponding to the deviation from half-filling. Noticing [3] that the (fermionic) holon fields dressed by the charge flux describe semions (with statistics intermediate between fermions and bosons), the first property is guaranteed if we apply to them the Haldane-Wu statistics [17] for semions (as in the 1D model [10]) allowing precisely double-occupation in momentum space for spinless semions. We argue that this MF treatment of holons, although not strictly equivalent to the Gutzwiller projection for holes, nevertheless it is still reasonable for small holon energies, close to the FS . In the same approximation in the PG one finds for the holons dressed by the charge flux the 0-energy level at the position of the small FS quoted above, the ”vacuum” for holons then corresponds to the filling of the lower branch of the Dirac double cones with vertices at generated by the charge -flux background, yielding a result characteristic of a 2D doped Mott insulator.

Holons and spinons are coupled by the emergent gauge field yielding overdamped resonances with strongly -dependent life-time. This dependence originates from the fluctuations of the transverse mode of the gauge field, dominated by the contribution of the gapless holons. Their Fermi surface produces an anomalous skin effect, with momentum scale

 Q≈(Tk2F)1/3, (2.5)

the Reizer momentum [4]. One can take into account approximately an external frequency replacing by in the life-time and in the Reizer momentum.

Let us now turn to holon pairing. Spin vortices centered on holons on the two Néel sublattices have opposite vorticity and this produces an attraction, previously neglected in the MFA; this is the origin of the attractive force between holons. Physically it is due to the quantum distortion of the AF background caused by the holes. We include this effect in MFA by introducing also the term coming from the average of in (2.4). We get the contribution:

 J(1−2δ)∑i,j(−1)|i|+|j|Δ−1(i−j)h∗ihih∗jhj, (2.6)

where is the 2D lattice laplacian. In the static approximation for holons (2.6) describes a 2D Coulomb gas with coupling constant , with , where is a UV cutoff, and charges depending on the Néel sublattice. For 2D Coulomb gases with the above parameters a pairing develops for a temperature , which turns out to appear in the SM. As it occurs for non-weakly coupled attractive Fermi systems, below there is a temperature at which the pairs condense, and it will turn out to be the superconducting transition temperature , see [2]; in-between there is a crossover, , due to spinons, but it will not be discussed here. This omission affects only the lower temperature region for the hole Green function discussed in this paper; in principle the approach is able to derive this correction, but it would highly complicate the calculations without affecting most of the main features. We defer this issue to a future publication.

## 3 Low energy Hamiltonian for holons

In this section we discuss in detail the pairing among holons in the SM region of the model in the BCS approximation, adapting the framework developed in [2], in turn inspired by [18]. A brief comment on the modifications needed for the PG region are added at the end. In the next section we incorporate in this framework the fluctuations of the phase of the holon-pairs order parameter.

We start treating the kinetic hamiltonian

 Hh0=−t∑⟨ij⟩(h∗ihj+h.c.)−t′∑⟨⟨ij⟩⟩(h∗ihj+h.c.)−μ∑ih∗ihi. (3.1)

in the two sublattices (even and odd ) scheme, defining and for , as this will be useful in the later treatment in presence of holon pairing that distinguishes the two Néel sublattices.

The two fields defined within the magnetic Brillouin zone (MBZ) and diagonalizing are

 ψ±(→k)=a→k±eik1b→k (3.2)

where MBZ and the Fourier transforms and are periodic outside the MBZ. The energy eigenvalues are , where and .

Both fields have pieces of FS within the MBZ (see Fig. 2, panel (a) ). has four hole-like (FS increases as doping increases) arcs centered at , in the middle of the sides of the boundary of the MBZ, while has electron-like (FS reduces as doping increases) Fermi arcs near , at its vertices. In terms of the fields defined in the MBZ the holon field, defined in the whole BZ, turns out to be

 Missing dimension or its units for \hskip (3.3)

with () chosen to keep the argument of inside the MBZ (where it is defined).

Now we exploit the two reciprocal primitive vectors to translate the 3rd and 4th quadrants respectively in order to obtain a upper rectangular zone , equivalent to the MBZ, as done in the PG (Fig. 2 panel (b)). Notice that each translation exchanges the index of the fields, because of the minus sign due to the exponential factor in eq. (3.2), and of the eigenvalues because .

In the presence of holon pairing the field extended by periodicity, restricted to the right quadrant, , and to the the left quadrant, , in is denoted by . It has a good continuum limit because it has a closed hole-like Fermi surface centered at in and at in , as discussed in a slightly different framework in [19].

Even if mostly neglected in what follows, the field extended by periodicity, and then denoted by , has a small closed electron-like Fermi surface around the point , see Fig. 2 panel (b). It is however convenient to restrict also to and to treat it consistently with , these restrictions are denoted by . We now discuss the holon attraction arising from (2.6). This interaction is able to distinguish between the two sublattices, therefore, when incoherent holon pairs appear the description in the MBZ is appropriate. However the pieces of the original FS of the Nearly Free Electron description will deform to reach orthogonally the boundary of the MBZ as a consequence of the interaction between the previously defined segments of Fermi surface inside and outside the MBZ, and the hole-like and electron-like FS discussed above split-off, the fields having a closed FS. In the BCS treatment adopted in the following the deformation of the FS is concentrated near the boundary of the MBZ, where hole-like and electron-like FS are closer. Finally we point out the relevance in our discussion of the negative next-nearest-neighbor hopping term which bends the FS of holons and allows BCS pairing of quasi-particles having a good continuum limit in the SM region. In the simple - model (where ), the FS in the SM does not cross the boundaries of the MBZ and our approach to the phase fluctuations of the holon pair order parameter presumably is possible only in the PG region where closed Fermi surfaces centered at arise. Therefore we guess that in the pure - model the formation of holon pairs only appears close to and not at the naively deduced , which instead can give a rough estimate of the crossover in the full -- model.

The structure of FS for holons discussed above will be qualitatively inherited by physical holes, via gauge coupling to spinons discussed later. Such structure bares some resemblance with that appearing in the spin-density wave approaches [19,20], but in our case its origin is the holon-holon pairing interaction distinguishing the two sublattices, not directly the standard AF interaction, although since the holon pairing originates from the term it is still of AF origin. Coexisting hole- and electron-like FS at intermediate dopings appear also in large-U treatments of the 2D Hubbard model in terms of an effective --- model [21]. In both cases going from low to high doping level, one first find small pockets like in our PG, then coexistence, like in our region between and and finally a large FS, as above in the SM.

The attractive interaction between holons in different sublattices, hence with opposite vortex chirality, of eq. (2.6) is treated at large scales in analogy with the treatment in the PG for the - model. Since not all vortices form pairs, a finite screening effect persists and the gas of vortices still have a finite correlation length [22], which we denote by , where is the average Fermi momenta of the corresponding closed FS. We keep track of the screening effect by replacing in the long wavelength limit in eq. (2.6) by an effective potential . Hence we approximate the interaction hamiltonian at large scales as:

 HhI=−∑i,jV(→i−→j)a∗ib∗jbjai≈−~J∑→p,→qVeff(→p−→q)a∗→pb∗−→pb−→qa→q (3.4)

We perform the translation discussed above, add right and left labels () to distinguish the two hole-like FS in and we measure momenta from and respectively. We neglect the interaction between and sectors, except for the effect of deformation near the MBZ boundary discussed above and taken into account only phenomenologically, and we adopt the BCS approximation for the and sectors defining the order parameter

 Δhα,→k=~J∑→qVeff(→k−→q)⟨bα,−→qaα,→q⟩. (3.5)

For the fields the Hamiltonian kernel appears in block-diagonal form:

 ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−μ−t′α,→k−tα,→k00−Δhα,→k0−μ−t′α,→k+tα,→kΔhα,→k00Δh∗α,→kμ+t′α,→k−tα,→k0−Δh∗α,→k00μ+t′α,→k+tα,→k⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ (3.6)

with

 tα,→k=t→k+→Qα=−2t[sinkx−(−1)αsinky],t′α,→k=t′→k+→Qα=−4t′(−1)αsinkxsinky (3.7)

provided is odd in ; the symbols are omitted here and often in the following, when their presence can be obviously understood. The choice of the order parameter with lowest angular momentum and hence energy has then -wave symmetry. We assume that this order parameter has -wave symmetry for and -wave symmetry for , so that

 Δhα,→k=Δhα(|→k|)γα(→k),γα(→k)=sinkx+(−1)αsinky (3.8)

with assumed constant, and denoted , near the FS in the BCS approximation. From eq.(3.5) we get Then we obtain for the field a -wave symmetry gluing and sectors when the full BZ is restored (as we see from its definition, the order parameter changes its sign when the 3rd and 4th quadrants are translated). This -wave symmetry in generates also an -wave pairing for the electron-like Fermi surface of , as discussed in a similar situation in [19].

The energy spectrum of the quasi-particles described by has the BCS form where

 ϵα,±,→k=√(±tα,→k−t′α,→k−μ)2+|Δα,→k|2. (3.9)

Let us briely comment on the PG, where

 tα,→k=−2t√(sinkx)2+(sinky)2. (3.10)

measuring the momenta from () in the () sectors of . In the PG the analog of the -field in has no FS, but there is a strong matrix effect due to the Dirac structure of the holon field, eventually leading for the hole to an angle-dependent wave-function renormalization constant, in , even in absence of holon pairing [3]. For the holon pairing the main differences between the SM and the PG are the shape of the FS and the higher value of the (average) Fermi momentum of the former w.r.t. the latter (), with an induced difference for the modulus of the holon-pair field. The Fermi velocity instead is roughly the same.

## 4 Integration of high energy modes

In this section we turn to the path-integral formalism and derive the low energy effective action which describes the pairing process in the normal state as we lower the temperature until the superconducting transition is reached.

We start from the spinon sector. The relevant action is the non-linear -model action with a mass term discussed in section 2. This allows us to safely integrate out spinon degrees of freedom obtaining an effective action for the slave particle gauge field . By gauge invariance the leading order is a Maxwell-like action:

 Sseff(Aμ)=13πms∫[0,β]×R2d3xF2μν. (4.1)

For the holon sector, in the BCS approximation discussed in section 3, the holon is gapless only at the 4 nodal points of . However in a large-scale gauge-invariant treatment whereas one can keep constant the modulus of the order parameter near the FS as in BCS, we must include its spatially dependent phase, which we denote by . This is done setting:

 Δhα(→x,x0)=Δheiϕhα(→x,x0), (4.2)

(A precise procedure to go from the lattice to the continuum phase field is discussed in [23].) The effects of on holons is non-trivial, as first suggested in a different setting in [24], and will be discussed here in detail. We generalize the BCS interaction between the holon pairs of section 3 and the FL quasi-particles by allowing phase fluctuations of . The holon action for near the FS in terms of Matsubara frequencies then reads:

 Sh,Δ(Ψα,Δhα)=∑α,k[ik0−E(→k)+μ]Ψ∗α,kΨα,k (4.3) −∑α,k,p12{Δhα(k+p)[γα(→k)−γα(→p)]Ψ∗α,kΨ∗α,p+c.c.}

where we put . Linearizing the dispersion we set near the FS, with an angle parametrizing the FS. We notice that in the case of constant order parameter , Eq. (4.3) reproduces the standard BCS coupling discussed in section 3.

One could then reinsert in the kinetic term the gauge field by Peierls subsitution and naively one would integrate out the holons, again considering the leading term in . However this is not correct since, as will be shown in section 5, above the phase fluctuations have short-range decay, with an energy scale that we denote by and therefore only the holon modes with higher energy can be safely integrated out. In this way no singular terms due to the integration of gapless excitations arise and both dynamics and interactions of the order parameter will be approximately local in space. A similar approach was proposed in [25], but directly for the hole and using different additional approximations. We assume that the high-frequency integrated modes are superconducting modes and this assumption will be proved to be self consistent later. The result of integration of these high-energy modes in the presence of the gauge field can be deduced from the Anderson-Higgs mechanism: the gap equation (3.5) for each has a degenerate manifold of solutions for arbitrary phase , therefore the energy should depend only on gradients of . Then by gauge invariance of the holon-gauge system we obtain in the continuum limit (we take and summation over repeated indices is understood henceforth):

 SΔeff(ϕhα,A)∼∑αcμ(Δhα)2∫d3x(∂μϕhα−2Aμ+2πnμα)2(x) (4.4)

where are positive functions and the integer-valued vector currents allow to be self-consistently an angle function taking into account the presence of vortices in the phase of the holon-pair field. Summation on is understood in the partition function.

The formalism of integer currents can be made precise with a lattice regularization which we use in the following. One should finally add the action describing the coupling of the low-energy modes of with and derived from (4.3) with UV cutoff .

## 5 Propagator of the holon-pair field

In this section we argue on the basis of a self-consistent argument that the Euclidean correlation function of the phase of the holon-pair field because of its coupling to a gapless gauge field for has a purely exponential decay:

 GΔ(→x,x0)≡⟨ei[ϕh(→x,x0)−ϕh(→0,0)]⟩≃e−mϕ√v2ϕx20+→x2. (5.1)

This behaviour was assumed in [24] in a different setting; here we explain the origin of this behaviour in our approach. (We neglect here for simplicity the index , as it turns out that is the same for both values of .) The relation (5.1) defines the holon-pair phase coherence length . can be thought as the mass of the phase of the holon pairs field and as the mean distance between phase vortices.

The important point of Eq. (5.1) is that besides its exponential decay for large it does not show any singularity for small (in particular it is a constant at ) as it would occur for a true massive field which exhibits Ornstein-Zernike decay with power in 3D. It is the above stressed feature that eventually is responsible for the peaks in the spectral weight precursors of SC, yielding an ”effective” gap in the antinodal region for the hole, a mark of pseudogap phenomenology.

We proceed as follows: we first calculate the assuming as action the one obtained in section 4, integrating the high-energy modes of spinons and holons and we show that it is self-consistent with the assumed superconducting-like behaviour of the high-energy modes of the holons. Then in section 6 we calculate perturbatively the effect of with such propagator on the low-energy holon modes and we show that the derived formula for the holon propagator if extended to high energy is consistent with the superconducting behaviour previously assumed.

To prove Eq. (5.1) in the setting described above we evaluate the correlation in the Coulomb gauge ( indicates spatial components and ) with a lattice regularization. A gauge-fixing is necessary in view of Elitzur theorem [26] that states that without gauge-fixing all non-gauge-invariant correlators vanish. Due to the presence of the integer vector currents in (4.4) one cannot evaluate it perturbatively. Therefore we use a construction due to Dirac [27] to circumvent the problem: We introduce the gauge invariant field

 eiϕE(x)=ei[ϕh(x)+2∑yExμ(y)Aμ(y)] (5.2)

where . Here is the classical (lattice) 2D electric field generated by a unit charge at , hence satisfying the 2D Gauss law and denotes the -dimensional Kronecker delta. Since Eq. (5.2) reduces to in the Coulomb gauge, we can evaluate the correlation estimating the expectation value without gauge fixing, because it is gauge-invariant. Notice that the ”electric” field has no temporal component and it is different from zero only in the temporal plane (see Fig. 3).

Taking into account the periodicity of the phase, i.e. the presence of vortices, we apply the Poisson summation formula to the lattice regularization of (4.4) rewriting

 ∑nexp(−SΔeff(ϕh,A,n))=∑Jexp(−SΔeff(ϕh,A,J)) (5.3)

where are integer vector currents and

 SΔeff(ϕh,A,J)=∑x[JμJμ2cμ+iJμ(∂μϕh−2Aμ)](x). (5.4)

Then we have:

where denotes the standard measure for lattice fields. Defining a current of charge 2 supported on a path from to (i.e. ), we can write

 ϕh(x)−ϕh(0)=∑zJμx,0(z)∂μϕh(z) (5.6)

and performing the functional integration in , appearing linearly in the action, we obtain the constraint in the numerator of eq. (5.5) and the constraint in the denominator. The final step is to integrate out the slave particle gauge field . The result at zero temperature is:

 GΔ(x)=∑{Jμ:∂μJμ=2(δ3x−δ30)}e−∑zJμJμ2cμe−3πms2∑z,w[Jμ(z)+2Exμ(z)−2E0μ(z)]Δ−13(z−w)[Jμ(w)+2Exμ(w)−2E0μ(w)]∑{Jμ:∂μJμ=0}e−∑zJμJμ2cμe−3πms2∑z,wJμ(z)Δ−13(z−w)Jμ(w)

where is the inverse of the 3-dimensional lattice Laplacian (at finite we just replace it by its finite temperature version and restrict the time axis to with periodic boundary conditions on the fields). The sum in the numerator is on currents starting from the point and reaching the point and on closed currents, while only closed currents are present in the denominator. Hence the denominator can be interpreted as the partition function of a gas of current loops interacting via the 3D (lattice) Coulomb potential. In the numerator the open current has endpoints where the electric current spread out in fixed time planes, as described by . The currents can be interpreted as the Euclidean worldlines of charge 2 particles. Currents supported on loops correspond to worldlines of virtual particle-antiparticle pairs, the open current corresponds to the worldline of a particle created at one end of the line and annihilated at the other one. A similar problem has been dealt with in [28] and the result can be summarized as follows: for sufficiently small the closed currents in the leading approximation just weakly renormalize the coefficient of the Maxwell action for , so one can approximate (5) setting the denominator to 1 and retaining in the sum of the numerator only a single fluctuating open current with weighting factor

 e−∑zJμJμ2cμe−3πms2∑z,w[Jμ(z)+2Exμ(z)−2E0μ(z)]Δ−13(z−w)[Jμ(w)+2Exμ(w)−2E0μ(w)]. (5.8)

The factor produces an exponential decay in . The power decay of the correlation function is then decided by how strongly the current can fluctuate. Gaussian fluctuations would produce a power of , however in our case the Coulomb potential in 3 (Euclidean) or less dimensions is confining. Every fluctuation away from a current of minimal length can be described as the addition of a closed loop to the minimal current. Because of confinement such loop produces an exponential factor decaying with its area and thus strongly suppressing every deviation from a current of minimal length. This forces the fluctuations of the current to be non-gaussian, lying within a thin tube surrounding the shortest straight path (see Fig. 3). An approximate evaluation of Eq. (5.8) is given in the Appendix and it reproduces Eq. (5.1) with a depending on through (contained in ). The role of the gauge field was crucial in obtaining this result. Without it the open current in (5) would have gaussian fluctuations and the correlation would exhibit the standard Ornstein-Zernike decay of a free massive field.

In Fig. 4 we show the typical temperature behaviour of and obtained with the approximations described in Appendix .

## 6 Holon self energy and self-consistency

In this section we extend the discussion of the previous one about holon pairing considering the scattering of holon quasi-particles against the fluctuations of the phase of the holon-pairs field. This section was inspired by [24], however here we give an explicit analytical formula interpolating from the Fermi-liquid and the superconducting behaviour of holons which differs from the one proposed there and in particular our treatment of inclusion of dissipation is completely different from that adopted there. We exploit the action in Eq. (4.3) and Eq. (5.1) to evaluate, in the continuum limit, the zero temperature self energy of holon quasi-particles.

What follows holds both in the PG region and in the SM region since the FS shape is irrelevant; for concreteness we discuss the SM case. The difference between the two regions lies in the value of the involved parameters, essentially . The only condition we need is parity invariance which implies that the quasi-particle excitation energy verifies .

We consider the holon right + field , the left is similar, with . Also the field can be treated along similar lines with obvious changes. In the limit the self energy reads

 Σ(ω,→k)=|Δh|2γR(→k)22πmϕddm2ϕI(k), (6.1)
 I(k)=∫d3q(2π)31q2+m2ϕ1i(q0−k0)−E(→q−→k). (6.2)

where the dependence of has been neglected for and we used . Considering quasi-particles on a shell of thickness about the we can linearize the quasi-particle dispersion and assuming we have , where , obtaining:

 I(k)≈14π√k20+δk2+m2ϕ−mϕik0+δk (6.3)

Inserting Eq. (6.3) into Eq. (6.1) and replacing by we obtain for the self energy

 Σ(ω,→k)=|Δh|2γR(→k)21iω+E(→k)⎡⎢ ⎢⎣1−mϕ√ω2+E(→k)2+m2ϕ⎤⎥ ⎥⎦ (6.4)

where and is the angle between the vector and the nodal direction in the right sector. The structure of the self-energy (6.4) is reminiscent of the electron Green functions of one-dimensional models with dynamically generated mass, when the charge and spin velocity coincide [50].

The quasi-particle Green’s function of the holon is given by

 G(ω,→k) = [iω−E(→k)−Σ(ω,→k)]−1 (6.5) = 1{1+|Δh|2γR(→k)21ω2+E(→k)2[1−mϕ√ω2+E(→k)2+m2ϕ]}[iω−E(→k)].

This quite complicated expression is our key result. We notice that at zero frequency, , the only pole of the Green’s function is located at , that is at the FS. This means that there is always a FS except exactly at , which occurs only in the superconducting phase. For the existence of the FS the second term in (6.4) is crucial, without it a gap would appear. The absence of the gap is consistent with the fact that in spite of a non-vanishing no phase transition has occurred.

To understand the meaning of we analyze the two limits of low () and high () frequency, which coincide with those in [24].

For , we expand the self energy in powers of up to the second order and the Green’s function becomes

 G(ω,→k)≃ZΔ(→k)iω−E(→k),ZΔ(→k)=11+|Δh|22m2ϕγR(→k)2. (6.6)

Thus for low frequencies the effect of holon pairing appears through the wave function renormalization weight . The system behaves like a FL with unchanged Fermi velocity and FS of the Nearly Free Electron approximation, but with a strongly direction dependent weight that heavily suppresses (if ) quasi-particles in antinodal directions reducing the effective FS. For , we expand the self energy in powers and we get the Green’s function

 G(ω,→k)≃−iω+E(→k)ω2+E(→k)2+|Δh|2γR(→k)2 (6.7)

which is the quasi-particle Green’s function of a -wave superconductor. Thus for high frequencies the holon system behaves like a -wave superconductor in the MBZ, the -wave being obtained gluing the and behaviours in and , respectively.

As soon as , the FS and the FL holon quasi-particle peak of weight appear for low frequencies and the system behaves like a metal. When , quasi-particles begin to scatter strongly with the quanta of the phase of the order parameter changing significantly both quasi-particle dispersion and coherence. FL quasi-particles modes are suppressed in favor of -wave SC modes that become much more coherent and relevant. This means that decreasing the value of one drives the behaviour of the holon system from metallic towards superconducting. Therefore the energy scale separates FL from SC modes, consistently matching with the cutoff in section 4. In fact, fermionic modes of energy higher than were previously assumed to be SC-like and had been integrated out to give dynamics to the holon-pair field, see eq. (4.4). In this way one can justify the assumption on high energy modes made in sections 4 and 5 to compute the propagator of the phase field, because the obtained self-energy is consistent with the assumption made.

Performing the analytic continuation in Eq. (6.5) we obtain the retarded Green’s function

 GR(ω,→k)=ω+iη+E(→k)(ω+iη)2−E(→k)2−|Δh|2γR(→k)2[1−mϕ√E(→k)2+m2ϕ−(ω+iη)2] (6.8)

and its imaginary part is proportional to the spectral weight of the holon.

Since there are FL modes, the scattering rates for holons, , is dominated by Reizer singularity and can be computed as in [29]. Combining (6.5) with Ioffe-Larkin rule one can show that the reduction of the spectral wight reducing temperature yields a deviation from below of the -linear behaviour of in-plane resistivity, typical of the SM , as discussed in [30].

Finally we should remember that the continuum fields whose spectral weight has been discussed above should eventually be converted again to the original holon field using (3.3), obtaining in in this approximation

 ⟨h∗h⟩(ω,→k)=(⟨Ψ∗+,RΨ+,R⟩(ω,→k)+⟨Ψ∗+,LΨ+,L⟩(ω,→k))χ(→k∈MBZ) +⟨Ψ∗−Ψ−⟩(ω,→k)χ(→k∉MBZ), (6.9)

where denotes the characteristic function. As a consequence of (6) the spectral weight of the holon coming from the fields does not have peaks outside the MBZ, i.e. it only exhibits Fermi arcs along the original FS, except near the MBZ boundary where the FS of the NFE approximation is distorted. The same is true inside the MBZ for the field, whose spectral weight therefore exhibits peaks only outside the MBZ. This suppression of the spectral weight in the outer ”shadow” FS is consistent due to the destruction, produced by the fluctuations of the phase field , of the long-range order appearing in the BCS approximation for the holon pairing. A similar suppression in the spin-density-wave approach was found in [31], but there was due to fluctuating local anti-ferromagnetic order. In this respect physically more similar to our approach is the phenomenological model of [32], where the suppression of the outer part of the FS is also due to pairing. However, there mathematically this is realized through the presence of a zero in the electron Green function, as in the cluster DMFT treatment of the 2D Hubbard model [33], which has no direct counterpart in our holon Green function. As a result, in those approaches the spectral gap in the antinodal region is asymmetric w.r.t. the Fermi energy in contrast with the symmetric behaviour found in our approach below in the SM.

Assuming the Green’s function in eq. (6.8) we plot in Fig. 5 panel (a) the holon spectral weight at different points of the FS of in the SM with fixed small . The spectral weight is symmetric in frequencies, and exhibits three maxima, the FL-like peak at and two symmetric SC-like peaks roughly at

 ωscp∼±√|Δh|2γR(→k)2+m2ϕ. (6.10)

Approximately (al least for small ) as long as , the FL peak is higher than the SC peaks which become negligible for , while approximately for , the SC peaks are higher than the FL one. The size of the area under the peaks determines the main behaviour of the system.

Even if small, the FL peak is always present unless strictly vanishes and in this case the spectral weight at the FS does not vanish only because of . A key point is the direction dependence of the condition on the maxima, due to