#### Abstract

We consider a two-dimensional fermion system on a square lattice described by a mean-field Hamiltonian involving the singlet id-density wave (DDW) order, assumed to correspond to the pseudo-gap (PG) state, favored by the electronic repulsion and the coexisting -wave superconductivity (DSC) driven by an assumed attractive interaction within the BCS framework. Whereas the single-particle excitation spectrum of the pure DDW state consists of the fermionic particles and holes over the reasonably conducting background, the coexisting states corresponds to Bogoliubov quasi-particles in the background of the delocalized Cooper pairs in the momentum space. We find that the two gaps in the single-particle excitation spectrum corresponding to PG and DSC, respectively, are distinct and do not merge into one “quadrature” gap if the *nesting property *of the normal state dispersion is absent. We show that the PG and DSC are representing two competing orders as the former brings about a depletion of the spectral weight available for pairing in the anti-nodal region of momentum space where the superconducting gap is supposed to be the largest. This indicates that the PG state perhaps could not be linked to a preformed pairing scenario. We also show the depletion of the spectral weight below at energies larger than the gap amplitude. This is an important hallmark of the strong coupling superconductivity.

#### 1. Introduction

A general consensus among the condensed matter physics community regarding the existence of the pseudogap (PG) phase in underdoped high superconductors has emerged after nearly a decade and half of the intensive theoretical and experimental studies [1–27]. However, regarding the origin of the PG and its relation with superconductivity (SC), there are divergent views. The interpretations run from descriptions where the PG is regarded as a superconducting precursor state involving incoherent electron-electron pairings above [1–7] with particle-hole symmetry of the SC state preserved to others where the PG, distinct from SC, corresponds to an ordered state with particle-hole asymmetry and both the phases compete [8–12]. In the former description, the preformed pairs appear at relatively high temperatures K compared to and one views as a “crossover” temperature, rather than a sharp phase transition. The origin of these preformed pairs is not fully known. They are supposed to arise from the attractive interaction which drives the superconductivity [21].

Our view regarding the origin of the PG is, however, centered around the simple paradigm that PG corresponds to -density wave ordering [9–11] (or its more complex variant, namely, density wave ordering [22, 23, 28–30]), at the antiferromagnetic wave vector . Starting with a two-dimensional fermion system on a square lattice described by a mean-field Hamiltonian involving the singlet -density wave (DDW) order ( ()) at the wave vector , we intend to show that, for a transition to the PG state, upon lowering the temperature from to at a fixed doping level (10%) on the underdoped side (see Figure 1), the entropy difference between the PG and normal paramagnetic states is negative. As an ordered state is expected to have lower entropy, there is nothing unusual about it. There appears to be no discontinuity in entropy at during this passage. There is no discontinuity in the electronic specific heat either. This is due to the fact that (high-energy) electronic excitations, responsible for the contributions towards the entropy and the specific heat, along the antinodal directions of momentum space where the pseudogap are maximal become scattered with shorter lifetimes (compared to hole excitations) and therefore unable to show any striking feature in at . The outcome of the calculation of in-plane quasiparticle thermal conductivity in both the phases via the Boltzmann equation in the relaxation-time approximation shows that there is a mild discontinuity in at for is found to be higher than for . This is an unusual feature. We needed to solve the pseudogap equation and the equation to determine the chemical potential () consistently in order to calculate . We pinned close to van Hove singularities (vHSs) of the normal state dispersion (involving the first, the second, and third neighbor hoppings) in an effort to do so. We have rendered all quantities, such as , dimensionless expressing , and energy gaps, in units of the first neighbor hopping. The origin of the above-mentioned discontinuity in is found to rest on the fact that the PG phase has nodes in the gap function resulting in the longer-lived excitation of single-fermion states with negligible energies even down to very low temperature. These low-lying excitations have significant contribution in . In fact, the entire bunch of energy states corresponding to the Fermi arc are the contributors. We find that the in-plane (dimensionless) longitudinal electrical conductivity in the PG phase is significantly greater than 1. As regards the Hall conductivities (electrical () and thermal ()), we found them nearly constant and close to unity for all the temperatures considered between and . Thus, for the in-plane electrical conductivities, the Hall angle . These results show reasonably conducting character of the PG state as has been reported by Levchenko et al. [13]. These authors have found the in-plane electrical and thermal conductivities are metal-like, while the -axis resistivity and the Hall number are insulator-like in the pseudogap phase of the cuprates.

As regards -wave superconductivity (DSC), we model the effective two-particle pairing interaction in the singlet pairing channel by suitable function of the form , where is the coupling strength (model parameter). We assume implicitly that this unconventional superconductivity is initiated by the strongly coupled bosonic modes, such as those corresponding to the electron spin fluctuations (proximity to an antiferromagnetic phase raises the possibility of spin-fluctuation-mediated pairing), leading to singlet pairing and concomitant kinetic energy reduction of the nodal quasiparticles at the pairing temperature which ultimately generates sufficiently high “quantum pressure” due to the temperature reduction to make the entire system undergo further lowering of kinetic energy and the free energy at . Whereas the single-particle excitation spectrum of the pure DDW state consists of the fermionic particles and holes over the insulating background, the coexisting (CXS) DDW and DSC state corresponds to Bogoliubov quasiparticles (or “bogolons”) in the background of the delocalized Cooper pairs in the momentum space. We find that the single-particle excitation spectrum , where , , and . The symbol stands for the normal state dispersion and for the chemical potential of the fermion number. In our scheme, as already mentioned, the Fermi level is pinned at the Van Hove singularity of the dispersion involving the first, the second, and the third neighbor hoppings plus a constant term. All energies are expressed in units of the first neighbor hopping. With all these paraphernalia, there are only two energy gaps and corresponding to PG and DSC, respectively, and two distinct quasiparticle dynamics in our formulation of the problem. The second neighbor hopping in the dispersion, which is known to be important for cuprates [26] and frustrates the kinetic energy of electrons, leads to Fermi surface sheets being not connected by (nonnesting property). One may notice that the two gaps in the excitation spectrum are distinct and do not merge into one “quadrature” gap () if the nesting property, , of the dispersion is absent; for the nested situation we do obtain such a merger yielding . As we shall see below (9), the nonnesting property of the dispersion is one of the important requirements for the onset of pure DDW ordering. To explain a little more, we note that the nesting is a meaningful phenomenon for an interacting system when we have a Fermi liquid (FL) description for the system. For interacting systems, all many-body effects are lumped into the self-energy () part which is generally -independent. The Re changes the quasiparticle dispersion away from the one corresponding to the noninteracting case whereas Im gives the quasiparticle lifetime. The simplest example is the one corresponding to the repulsive Hubbard model () on a two-dimensional tight-binding square (bipartite) lattice. The kinetic energy connects only one sublattice to the other. The single-particle eigenstates for the noninteracting case have energies here. When fermions are poured into such a band, if initially the Fermi surface is circular that is, free-electron-like, at half-filling becomes a perfect square. The Fermi surface (FS) is nested with only at half-filling; close to half-filling the nesting is approximate. The density of states (DOS) displays nesting singularity. Upon inclusion of , say, at the Hartree-Fock level, we obtain fermionic quasiparticles with reasonable FL description. In the 2D system under consideration in this communication, the FS nested with in the pure DDW state also presents a particularly striking though untenable situation with single semimetallic band. In order to present a suitable description of cuprates, we thus have to turn our attention to nonnested dispersion (NND). However, the kinetic energy and DOS do not display the same behavior as in the 2D Hubbard model with this type of dispersion due to the second neighbor hopping. As we shall see below, the specific heat shows anomalous temperature dependence, a typical non-Fermi liquid (NFL) feature, with the onset of DDW ordering. We thus note that nonnested dispersion and NFL behavior are perhaps as deeply connected as the nesting and the Fermi liquid behavior are for 2D systems.

The particle-hole asymmetry in the single-particle excitation spectrum (SPES) of the pure DDW state with NND is also reflected in the coexisting DDW and DSC states (see Figure 4(c)) though the latter is characterized by the Bogoluibov quasiparticle bands—a prominent fingerprint of superconductivity. These results are qualitatively the same as those obtained by Hashimoto and his coworkers [24]. The particle-hole asymmetry in the CXS is an indication of the interplay of the two orderings. Obviously enough, the second neighbor hopping is partly responsible for this. We shall show that the pseudogap and high temperature superconductivity are representing two competing orders as the former brings about a depletion of the spectral weight (SW()) available for pairing in the antinodal region of momentum space where the superconducting gap is supposed to be the largest. This indicates that the PG state could not be linked to a preformed pairing scenario. Furthermore, there is depletion of the spectral weight below at energies larger than the energy gap. We show this result analytically for the coexistent states calculating SW() within the BCS framework for a two-dimensional fermion system on a square lattice starting with a Hamiltonian corresponding to the -density wave (DDW) order plus the superconducting pairing . This is a prominent spectroscopic evidence for the strong coupling superconductivity observed by Kaminski et al. [31].

The paper is organized as follows. In Section 2, we discuss the particle-hole asymmetry aspect of SPES in the coexistent DDW and DSC states. In Section 3, we derive expressions for the thermodynamic potential and entropy and exploit the latter for the estimation of the pseudogap transition temperature . The electronic specific heat is shown to display anomalous temperature dependence. We also calculate the quasiparticle thermal conductivity in the normal and pseudogap phases via the Boltzmann equation in the relaxation-time approximation. The observed mild discontinuity in at indicates that the passage of the system from the normal to the PG state is a nonsharp thermal phase transition. The optical conductivity or the spectral weight occupies the centre-stage [32–40] in determining whether the pseudogap and high temperature superconductivity are representing two competing or cooperating orders. In Section 4, we discuss this issue in detail and conclude that these are competing orders. Besides, the calculation/plot of the integral where the subscript “” stands for the conduction band, is the density of states (DOSs) or spectral density (SD), is the second derivative with respect to of dispersion , and is the momentum distribution function at a temperature , as a function of energy in units of the first neighbor hopping is obtained as a decreasing function for energy larger than that corresponding to the SC gap amplitude. This is an important hallmark of the strong coupling superconductivity. The paper ends in Section 5 with the concluding remarks.

#### 2. Bogoluibov Quasiparticle Bands

In the second quantized notation, the Hamiltonian to deal with the -density wave (DDW) order at the antiferromagnetic wave vector plus the -wave superconductivity can be expressed as where and . The time-reversal invariance of the normal state requires that the dispersion . The function involves the near neighbor hopping terms to be specified later (see (10)). The gap function . The conical brackets stand for the thermal average calculated with the Hamiltonian in (2). This step ensures the self-consistency.

The imaginary -wave order parameter describing the PG state breaks the time-reversal symmetry of the normal state. The time-reversal operator transforms the order parameter to its complex conjugate: . If the time-reversal symmetry is preserved, and are identical to within a common spatially independent phase. If, however, the time-reversal symmetry is broken, the two states are distinct albeit with the same free energy. We shall see that the equation corresponding to this gap in the pure DDW case is , where the function being the difference of two Fermi functions and is positive, the interaction has to be of the form . The nonzero DSC order parameter or the gap , on the other hand, requires an appropriate attractive interactions , where is the coupling strength. The pure DSC gap is given by which is the usual BCS form. If the pairing interaction is imagined to be a “probe” applied to the Fermi system in the PG state, then the gap function (where ) is perhaps a “response” that the system displays. Naturally, the structure of the “probe” in momentum space will have tremendous influence on the “response.” For example, the usual electron-phonon (e-ph) type pairing interaction leads to a fully gapped state-a “conventional” BCS superconductor. The electron-bosonic mode (e-bm) interaction or a combination of electron-electron (e-e) and e-bm interactions, on the other hand, are expected to produce gaps with nodes and antinodes (or, more generally, Fermi surface (FS) pockets of the “unconventional” superconductors) and these are interpreted as the manifestation of the non--wave symmetry of the order parameter. For a conventional “e-ph pairing interaction” which is structureless in momentum space such a solution of the gap equation would never be possible. Thus, it is natural to surmise, as we have done above, that a combination of e-e and/or e-bm interactions will lead to a -wave gap . For the quantities , given in the form of , the amplitude for the two orderings is to be obtained solving a set of self-consistent equations to be specified below.

At this point, we introduce few thermal averages determined by , namely, , , , and . Here, is the time-ordering operator which arranges other operators from right to left in the ascending order of imaginary time . The first step of the scheme involves the calculation of (imaginary) time evolution of the operators where, in units such that , . We obtain, for example, and so on. Here, and the argument part has been dropped in writing the operators and their derivative. As the next step, upon using (3), we find that the equations of motion of these averages are given by

The final step is the calculation of the Fourier coefficients: (where the Matsubara frequencies are with ) of these temperature Green’s functions. Here, . We refrain from writing explicitly the equations to determine these coefficients as this is a trivial exercise in view of (4). Upon solving the equations, we obtain , , , and , where , and . The denominator of the Fourier coefficients yields the single particle excitation spectrum. We find that this is a quadratic in :

It may be noted that, in the pure -wave case to be investigated below, will get replaced by . Correspondingly, the single-particle excitation spectra will be given by , and . The quasiparticle excitations in cuprates (where we have a pure -wave SC order together with a coexistent pseudogap right up to 0 K) are thus demonstrably Bogoliubov quasiparticles in the SC phase. It may be noted that the result obtained here is different from the one surmised by Leblanc et al. [41], within the ansatz for the RVB state proposed by Yang et al. [42, 43], to explain the angle-resolved photoemission spectroscopy (ARPES) data published by Hashimoto and his collaborators [24]. The conjectured energy of the gapped excitations in the superconducting state is , with Bogoliubov amplitudes and which are applied to the pseudogapped bands indexed by and given as . The energies and , where is a third nearest neighbor tight-binding dispersion and is that for first nearest neighbor which for defines the antiferromagnetic Brillouin zone boundary. Obviously enough, the difference between our result and that proposed by LeBlanc et al. [41] lies in the nonmerger of the two gaps and in the excitation spectrum of hole-doped cuprates into one “quadrature gap” in the former. Despite this, as we shall see below, we could capture qualitatively some key aspects (see Figure 4) of the results obtained by Hashimoto and his coworkers [24].

At this point, we note that many theorists and experimentalists [22, 23, 28–30] subscribe to the view that the pseudogap is of variety (chiral DDW (CDDW) order at the wave vector ). In such a situation, , where the real and the imaginary parts are given by , and . The quantities in (5) would now be given by where here is equal to . The single particle excitation spectrum is given by , , and . Unlike the pure DDW case, for the CDDW case when the dispersion is perfectly nested. Thus, and the Fourier coefficient , where , , , , , and . The sum of the coherence factors , but the factors are complex unlike the usual Bogoluibov picture. The outcome suggests that the investigation on the possibility of state requires a deeper analysis. We shall, therefore, presently focus our attention on the imperfect nesting and the pure DDW scenario.

It may be seen that, in the pure DDW case, and which may be written as

The first result in (9), upon applying Luttinger’s theorem [44], leads to the equation to determine the chemical potential , where is the hole-doping level, and is the number of unit cells in the -space. The second result in (9) leads to the DDW gap equation . Quite obviously, the difference of the Fermi functions within the square brackets is positive and therefore the interaction needs to be repulsive for this equation to be meaningful. The quasiparticle coherence factors are given by the expressions , . Here , , , and . The index is equal to () with corresponding to the upper branch and corresponding to the lower branch . The single-particle spectral function in the spin- channel is given by , where is the retarded Green’s function given by and . Upon using the result , where represents a Cauchy’s principal value, the spectral function in the DDW phase is given by a sum of functions at the quasiparticle energies: . These results are the same as those reported by Chakravarty et al. [9–11]. In particular, if the dispersion is nested, we obtain Bogoluibov-like dispersion , and , where , but the coherence factors and . This situation being inadmissible as it effectively corresponds to a single semimetallic band, we need to specify the normal-state dispersion at this stage. It is well known [45] that near a van Hove singularity (vHS) the fermion density of states diverges, so that even arbitrarily weak interactions can produce large effects. When the Fermi level reaches these points, a variety of response functions diverge. As already stated, we have a two-dimensional fermionic system with a square lattice. Suppose we have a tight-binding dispersion of the form where, for the hole-doped materials, (for the electron-doped materials ), and, in all cases, . For example, typical values are eV, , and . Upon ignoring the third neighbor hopping term above, we find that the dispersion typically has two inequivalent saddle points at and in the first Brillouin zone. Upon assuming that for fillings such that the Fermi curve lies close to the singularities, the majority of states participating in the pairing formation will come from regions in the vicinity of these saddle points. This is the key strategy we adopt below to plot the single-particle excitation spectrum and calculate all quantities of interest, such as the the thermodynamic and transport properties. As we shall show below, the anomalous temperature dependence of the electronic specific heat (a typical non-Fermi liquid (NFL) behavior) due to the onset of exotic DDW ordering has its origin in the nonnesting property of the dispersion in (10).

The -summation in the equations for and in the preceeding paragraph may be replaced by the integration: , where the Fermi energy density of states (DOSs) should be determined by the inclusion of the disorder potential ideally (see Section 3.2). The quantity here is obtained from the spectral function given below (9) by a sum of functions at the quasiparticle energies. We simply replace the functions by Lorentzians with an assumed intrinsic lifetime broadening . A 2D plot of as a function of , in the pseudogap phase (doping level 9.94%), is shown in Figure 2. The numerical values, in the units of the first neighbor hopping , are , the PG gap amplitude , and the SC gap amplitude . The hopping parameters are , and . The DDW ordering leads to pining of the Fermi level close to, but not precisely at, the vHs. The plot shows a cusp at . For comparison purpose, we have plotted the square lattice tight band DOS in the Hubbard model as well which clearly shows vHS. There is logarithmic singularity at the centre (saddle point singularity) and the step-like discontinuities at the band-edges [46].

In the absence of the DDW gap, with the modulation vector set at and , we have , , and . This yields , and , where and the coherence factors are given by . Upon using the expression for the Fourier coefficient the chemical potential , according to the Luttinger rule [34], is given by the equation , where is the doping level and is the number of -vectors in the first Brillouin zone. The Fourier coefficient leads to the weak coupling BCS gap equation for the singlet pairing: , where . We, thus, notice that the Matsubara propagators obtained in our general analysis, where the DDW and DSC orderings have been assumed to be coexisting, are able to yield the already known results [9–11] albeit with slightly different expression for the excitation spectrum.

For the coexistent DDW and DSC state, from (5) and (6) we find that
where , and . We find that the coherence factors and satisfy the sum rule . The single-particle spectral function in the DDW + DSC state, as before, is given by . Similarly, the spectral function is given by . We find that, when the nesting property is satisfied**, **, , , and . In this situation, the sum = . As we shall see below, the sum will be required for the derivation of an expression for the thermodynamic potential following the Kadanoff-Baym approach [47, 48]. The remaining Fourier coefficients and which correspond to the DDW gap and the DSC gap, respectively, are given by

Upon replacing and , respectively, by and (and ), as in the pure -wave case, the two Fourier coefficients and lead to the DDW () and DSC gap () equations while leads to the equation for the chemical potential. With , in view of the definition and the first equation in (12), we obtain where . In the zero-temperature limit when nesting of the Fermi surface is near perfect, as in the square lattice with nearest-neighbor hopping and a small second neighbor hopping ( small compared to unity), (13) may be written as . With the model interaction , where is the repulsive coupling strength (), this equation assumes a simple form . For the gap , on the other hand, we obtain where . In the zero-temperature case, for a nonnested dispersion, (14) reduces to

The near-nested dispersion yields which is similar to the weak coupling BCS gap equation. With an appropriate attractive interaction , where is the coupling strength, we find that the equation assumes a simple form . With the two gap equations combined, we obtain

We also obtain as the sum is not zero. Since has its origin in the coulombic interaction [9–11], the condition obtained hints at the possibility of noninvolvement of the electron-phonon type interaction in the hole-doped cuprates. Upon modeling the functional dependence of the pairing interactions and order parameter as , , and squared order parameter , where and belong to the first Brillouin zone (BZ), we may write the gap equation (16) as where the dimensionless quantity and . In writing energy integrations in the gap equation, we have assumed an arbitrary energy cut-off “” less than the Fermi energy. This facilitates integration over a length larger than . The quantity with dimensions ,which will be assumed to be a constant (see Figure 16), is the density of energy states. In the weak coupling limit , for the special situation , we obtain . This is reminiscent of the corresponding result of the BCS theory for the “conventional superconductors.” For the unconventional superconductor under consideration here, we adopt the simple strategy of assigning numerical values to and performing the integration in (17), by discretizing the integral, to obtain . The 2D graph in Figure 3 displays the outcome. We find that the quantity is an increasing function of . Though the quantitative aspect of the result may be an artefact of the model adopted, qualitatively the exercise underscores the fact that reasonable strong-coupling solution for the order parameter amplitude for a superconductor of -wave variety is available within the BCS framework. Now at all temperatures above K, there is a finite possibility of finding electrons in the nonsuperconducting state. At finite temperature, the occupation of the excited one-electron state obeys the Fermi statistics with the Fermi distribution . Equation (17) is, therefore, replaced by where . We assume that which makes sense if the DDW and DSC are noncooperating/competing orders (see Section 4). In that case, one may regard that the factor of 2 multiplying the Fermi function appears because either one of the states or may be occupied. One can derive the equation for the critical temperature readily from above:

**(a)**

**(b)**

**(c)**