A Theoretical Approach to Pseudogap and Superconducting Transitions in Hole-Doped Cuprates

We consider a two-dimensional fermion system on a square lattice described by a mean-�eld Hamiltonian involving the singlet iddensity wave (DDW) order, assumed to correspond to the pseudo-gap (PG) state, favored by the electronic repulsion and the coexisting dd-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 themomentum space.We �nd 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. is 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 TTcc at energies larger than the gap amplitude. is is an important hallmark of the strong coupling superconductivity.


Introduction
A general consensus among the condensed matter physics community regarding the existence of the pseudogap (PG) phase in underdoped high   superconductors has emerged aer nearly a decade and half of the intensive theoretical and experimental studies .However, regarding the origin of the PG and its relation with superconductivity (SC), there are divergent views.e interpretations run from descriptions where the PG is regarded as a superconducting precursor state involving incoherent electron-electron pairings above   [1][2][3][4][5][6][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][9][10][11][12].In the former description, the preformed pairs appear at relatively high temperatures  * ∼ 200 K compared to   and one views  * as a "crossover" temperature, rather than a sharp phase transition.e origin of these preformed pairs is not fully known.ey 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][10][11] (or its more complex variant, namely,    density wave ordering [22,23,[28][29][30]), at the antiferromagnetic wave vector    .Starting with a two-dimensional fermion system on a square lattice described by a mean-�eld Hamiltonian involving the singlet -density wave (DDW) order (    0 2 (cos     cos   )) at the wave vector    , we intend to show that, for a transition to the PG state, upon lowering the temperature from    * to    * at a �xed doping level (∼10%) on the underdoped side (see Figure 1), the entropy difference    PG   Normal  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.ere appears to be no discontinuity in entropy at  * during this passage.ere is no discontinuity in the electronic speci�c heat  el either.is   F 1: A generic phase diagram of the hole-doped cuprates in the doping level ()-temperature () plane.e antiferromagnetic phase corresponds to the blackened region in the far le.While the lightly red region to its right is the pseudogap (PG) state, the near-semicircular red region corresponds to the superconducting (SC) phase.e curves  * ( and   (, respectively, correspond to the PG transition (from the paramagnetic insulating phase) and the SC transition.e vertical, thick green line within the SC phase corresponds to the region investigated which is a part of the superconducting DDW region.
is due to the fact that (high-energy) electronic excitations, responsible for the contributions towards the entropy and the speci�c 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  el at  * .e 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   (   * / * → 0  is found to be higher than   for  → 0 + .is 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 �rst, 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 �rst neighbor hopping.e 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 singlefermion states with negligible energies even down to very low temperature.ese low-lying excitations have signi�cant contribution in   .In fact, the entire bunch of energy states corresponding to the Fermi arc are the contributors.We �nd that the in-plane (dimensionless) longitudinal electrical conductivity in the PG phase   is signi�cantly 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  * .us, for the in-plane electrical conductivities, the Hall angle tan   ≡   /  ≪ 1. ese results show reasonably conducting character of the PG state as has been reported by Levchenko et al. [13].ese 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 | 1 |(cos     cos   (cos  ′    cos  ′  , where  1 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 �uctuations (proximity to an antiferromagnetic phase raises the possibility of spin-�uctuation-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 �nd that the single-particle excitation spectrum   (      ± ( 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 �rst, the second, and the third neighbor hoppings plus a constant term.All energies are expressed in units of the �rst neighbor hopping.With all these paraphernalia, there are only two energy gaps   and  (sc)    corresponding to PG and DSC, respectively, and two distinct quasiparticle dynamics in our formulation of the problem.e 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 ( eff () 2  [ 2   +  (sc)2  ]) if the nesting property,    − + , of the dispersion is absent; for the nested situation we do obtain such a merger yielding  ()    ()     ( 2   +  eff () 2 ) 1/2 .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 (Σ) part which is generally -independent.e Re Σ changes the quasiparticle dispersion away from the one corresponding to the noninteracting case whereas Im Σ gives the quasiparticle lifetime.e simplest example is the one corresponding to the repulsive Hubbard model (  0) on a two-dimensional tight-binding square (bipartite) lattice.e kinetic energy connects only one sublattice to the other.e single-particle eigenstates for the noninteracting case have energies    {−2 1 (cos(  ) + cos(  )) here.
When fermions are poured into such a band, if initially the Fermi surface is circular that is, free-electron-like, at half-�lling becomes a perfect square.e Fermi surface (FS) is nested with   ( ) only at half-�lling; close to half-�lling the nesting is approximate.e density of states (DOS) displays nesting singularity.Upon inclusion of   0, 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 speci�c heat  el 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.e particle-hole asymmetry in the single-particle excitation spectrum (SPES) of the pure DDW state with NND is also re�ected in the coexisting DDW and DSC states (see Figure 4(c)) though the latter is characterized by the Bogoluibov quasiparticle bands�a prominent �ngerprint of superconductivity.ese results are qualitatively the same as those obtained by Hashimoto and his coworkers [24].e 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.is 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  (sc)   ∑  ′  (  ′ )⟨ − ′ −   ′  ⟩. is is a prominent spectroscopic evidence for the strong coupling superconductivity observed by Kaminski et al. [31].
e 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  * .e electronic speci�c 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.e 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.e optical conductivity or the spectral weight occupies the centre-stage [32][33][34][35][36][37][38][39][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 �rst neighbor hopping is obtained as a decreasing function for energy larger than that corresponding to the SC gap amplitude.is is an important hallmark of the strong coupling superconductivity.e paper ends in Section 5 with the concluding remarks.

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 ) is perhaps a "response" that the system displays.Naturally, the structure of the "probe" in momentum space will have tremendous in�uence on the "response." For example, the usual electron-phonon (e-ph) type pairing interaction leads to a fully gapped state-a "conventional" BCS superconductor.e 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.us, 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 Δ sc  .For the quantities Δ   Δ sc  , given in the form of Δ 0 cos   −cos   , the amplitude Δ 0  for the two orderings is to be obtained solving a set of self-consistent equations to be speci�ed below.
At this point, we introduce few thermal averages determined by , namely,      −⟨   †  0}⟩, Γ     −⟨ † −−  †  0}⟩,  ′     −⟨ +  †  0}⟩, and Γ ′     −⟨ † −−−  †  0}⟩.Here,  is the time-ordering operator which arranges other operators from right to le in the ascending order of imaginary time .e �rst step of the scheme involves the calculation of (imaginary) time evolution of the operators    where, in units such that ℏ  1,       0 −.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 �nd that the equations of motion of these averages are given by It may be noted that, in the pure -wave case to be investigated below, |  | 2 will get replaced by Δ 2  .Correspondingly, the single-particle excitation spectra will be given by   (    ) =  ()  .e 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].e conjectured energy of the gapped excitations in the superconducting state is   sc =  {(  ) 2  Δ 2 sc }, with Bogoliubov amplitudes  2  = (    sc )2 and  2  = (−    sc )2 which are applied to the pseudogapped bands indexed by  =  and given as   =     { 2  2  Δ 2 PG }. e energies   = (  −  0  )2 and  2 = (   0  )2, where   is a third nearest neighbor tightbinding dispersion and  0  is that for �rst nearest neighbor which for  0  = 0 de�nes 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 Δ  and Δ (sc)   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][29][30] subscribe to the view that the pseudogap is of    variety (chiral DDW (CDDW) order at the wave vector  = ( )).In such a situation,   = (−   Δ  ), where the real and the imaginary parts (   Δ  ) are given by   = − 0 sin(  ) sin(  ), and Δ  = (Δ (PG) 0 ())(s    − s   ).e quantities (Δ Δ   Δ 2  Δ 3  Δ 4 ) in ( 5) would now be given by where  eff () 2 here is equal to which may be written as e �rst 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.e 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.e quasiparticle coherence factors  2      [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 twodimensional fermionic system with a square lattice.Suppose we have a tight-binding dispersion of the form where, for the hole-doped materials,  2 >  (for the electrondoped materials  2 < ), and, in all cases,  2 <  1 /2.For example, typical values are  1 ∼ .2 eV,  2 / 1  ∼ .2, and  3 / 1  ∼ .1.Upon ignoring the third neighbor hopping term above, we �nd that the dispersion typically has two inequivalent saddle points at   and   in the �rst Brillouin zone.Upon assuming that for �llings 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.is 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 speci�c 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) 2. e numerical values, in the units of the �rst neighbor hopping  1 , are ( 1 ) = −0.0189, the PG gap amplitude (Δ (pg) 0  1 ) = 0.0871, and the SC gap amplitude (Δ (sc)  0  1 ) = 0. e hopping parameters are ( 2  1 ) = 0.3925, and ( 3  1 ) = 0.0005.e DDW ordering leads to pining of the Fermi level close to, but not precisely at, the vHs.e plot shows a cusp at  = 0.For comparison purpose, we have plotted the square lattice tight band DOS in the Hubbard model as well which clearly shows vHS.ere 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 (0, 0) and ) and the coherence factors are given by  (±)   = (12)1 ± (   έ())].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 �rst Brillouin zone.e Fourier coefficient Γ  (,   ) leads to the weak coupling BCS gap equation for the singlet pairing: , where  = (  ) −1 .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][10][11] albeit with slightly different expression for the excitation spectrum.
e above-mentioned exercise leads to the graphical representations of  ()    in the pure DDW state, ± έ() in the pure DSC state, and ± () () in the coexistent DDW and DSC states.In Figure 4(a), we have shown the plot of  ()    along the antinodal cut (− )-(0 )-( ), while in Figures 4(b) and 4(c) we have shown the plot of ± έ() and ± () (), respectively.In order to highlight their special features, namely, the particle-hole asymmetry (symmetry) for  ()    and ± ()  (± έ()), we note that the optical and electronic phenomena in solids arise from the behavior of electrons and holes (unoccupied states in a �lled electron sea).Electron-hole symmetry can oen be invoked as a simplifying description, which states that electrons with energy above the Fermi sea behave the same as holes below the Fermi energy.In semiconductors, for example, electronhole symmetry is generally absent because the energy-band structure of the conduction band differs from the valence band.In Figure 4(a), we �nd easy to notice that particlehole asymmetry as the upper band is parabolic while the lower band is characterized by dip at (0 ) and the socalled back-bending momentum at (−0.45 ) and (0.45 ).A plot of the upper and the lower bands ± έ() in Figure 4(b) indicates perfect electron-hole symmetry as both the upper and lower bands are parabolic.In Figure 4 ).We notice that (i) in the presence of superconducting gap the elementary excitations are the Bogoluibov quasiparticles which mix electron and hole states, and (ii) as a manifestation of interplay of DDW and DSC the band (− ()   ) no more peak at   (0 ) but rather at the back-bending momentum position as shown due to the presence of a particle-hole asymmetric pseudogap.It must be noted that the Bogoluibov quasiparticle band features, agreeable with -wave BCS theory, observed in the superconducting state here have been reported time and again in the ARPES experiments [14,24,[50][51][52].
e particle-hole asymmetry of  ()  and ± ()  could also be visualized through the corresponding contour plots in the �rst Brillouin zone (B�) shown in Figure 5.In Figures 5(a) and 5(b), respectively, we have shown plots of  ()   and  ()  .e band-maxima in  ()   occur at the points (± ±) and (0 0) whereas those in  ()   occur at the points (±/2 ±/2).Since the electrons with energy above the Fermi sea do not behave the same as the holes below the Fermi energy, we have the particle-hole asymmetry in the pure DDW state.In Figures 5(c) and 5(d), respectively, we have shown (− ()   ) and (− ()   ).Since the bands  ()  above the Fermi energy are the re�ected ones of those below the Fermi energy, that is, (− ()  ), we have not shown  ()   and  ()  here.In fact, the band-maxima in  ()   occur at the points (± ±) and (0 0) whereas those in  ()   occur at the points (±/2 ±/2).On the other hand, the band-maxima in (− ()   ) occur at the antinodal points (± 0) and (0 ±) while the band-minima in (− ()   ) occur at the nodal points (±/2 ±/2).Since the Bogoluibov quasiparticles here with energy above the Fermi sea do not behave the same as those below the Fermi energy, we have clear particle-hole asymmetry in the DDW + DSC state as well. may be written down using (11).In fact, this has already been done below (11).Since in the pure -wave case Δ †(sc  = Δ (sc  , we need not calculate   (   as this is given by   (  .In the pure DDW state, we �nd that the thermodynamic potential is given by the expression

Thermodynamics and Transport Properties in PG Phase
Using this expression, we shall calculate the entropy () and the electronic speci�c heat ( el ).However, this requires the calculation of the Fermi energy density of states (DOSs) which we wish to discuss �rst before we calculate (  el .

Elastic Scattering by Impurities and Quasiparticle Lifetime.
We now wish to calculate the lifetimes of the electron and hole quasiparticle below and show that the former is short-lived compared to the latter.We also wish to link this important fact with the entropy, the electronic speci�c heat and the thermal conductivity.In order to calculate the quasiparticle lifetime (QPLT), we need to consider the effect of elastic scattering by impurities.is involves calculation of self-energy Σ(   , involving the momentum and the Matsubara frequencies   .e self-energy alters the singleparticle excitation spectrum in a fundamental way.A few diagrams contributing to the self-energy are shown in Figure 6.e unsmooth lines carry momentum but no energy as the scattering is assumed to be elastic.e total momentum entering each impurity vertex, depicted by a dark circle, is zero.We assume that impurities are alike, distributed randomly, and contribute a potential term ( =    =   ( −   , where ( −    is the potential due to a single impurity at   =     for a given  and  =   .e term ( −    is expanded in a Fourier series: ( −    =   (   ( −   ].We �rst consider only the contribution of Figure 6(a).Assuming the scattering by impurities weak, we may write it as (q 2 ) (−q 1 − q 2 ) + + + = F 6: A few diagrams contributing to the self-energy.e jagged lines carry momentum but no energy.e total momentum entering each impurity vertex, depicted by a dark circle, is zero.�e have assumed that impurities are alike, and distributed randomly.�hereas the �rst two �gures correspond to one impurity vertex, the remaining two correspond to the product of four impurity potentials with nonzero averages.ese are the cases where two impurities each give rise to two potentials.us, the remaining two �gures involve the interference of the scattering by more than one impurity.�e have assumed low concentration of impurities, and therefore the �gures yield smaller contributions compared to those corresponding to the �rst two and the other diagrams of the same class involving only one impurity vertex.
′ ) by a screened exponential falloff of the form to consider the effect of the in-plane impurities, where  1 characterizes the range of the impurity potential.e limit      ′ , which corresponds to a point-like isotropic scattering potential characterizing the in-plane impurities, will only be considered here for simplicity.For low concentration of impurities and weak disorder potential  1  1  0  0  2 ≪ 1), we have ί1  ≪ 1. Upon using the Dyson's equation, under this condition, the full propagator may be written as We have calculated formal expressions of the propagator  (Full (    and reciprocal QPLT (/ ί(   above with the inclusion of impurity scattering.e corresponding retarded Green's function  ( ( , in units such that ℏ  , is given by  ( (   ∫ +∞ −∞ (/2 (− (Full (  where in the upper half-plane  (Full (    is given by (28) At this stage, assuming low concentration of impurities, one may include the contributions of all such diagrams which involve only one impurity vertex.is gives the equation to determine the total self-energy (   : (         ( (Full (−   Γ(    , where the Lippmann-Schwinger equation to determine Γ(     is Γ(      (− +   ′ ( ′ −  (Full ( −  ′    Γ(  ′    .is is the -martix approximation.Upon using the optical theorem for the -matrix [53] .us, the effect of the inclusion of contribution of all the above-mentioned diagrams, which involve only one impurity vertex, is to replace the Born approximation for scattering by the exact scattering cross-section for a single impurity, that is, ί−  → Γ −  . Since    ) and ) are known, one can determine Γ −1  in terms of ).In the limit    −  ′ , the disorder potential )   0 , and, therefore, we obtain Γ   )   0 /1 −  0    )).In view of    ) given above (27), we �nd that Im Γ   )  − 0  0  2 /1 +  2 0  2  0  2 ).From the equation   Γ    )) = −   1 /2   Γ  ), we consequently �nd that Γ −1  , in the �rst approximation, is given by Γ (30) and (35).e reciprocal QPLT for the electron and the hole quasiparticles are plotted in Figure 7 as a function of Γ −1 0 (measure of disorder potential) for Σ  / 1 = −1.2.e former is assumed to be centered at ± 0) and 0 ±) whereas the latter at ±/2 ±/2) of the �rst Brillouin zone.We �nd that the former is short-lived compared to the latter for very weak to moderately weak disorder potential.We note that, even though Γ  is found to be -independent in the �rst approximation, the term ±2 −1 1  1/2  sin  /2) in (30) will ensure that 1/ Γ sin  /2)} are momentum-dependent.In Figure 8, we have shown the contour plots of the Fermi energy DOS or spectral density given by (35) on the Brillouin zone (BZ) at 9.94% hole doping where Σ  / 1 = −1.2) rede�nes the chemical potential.Whereas for Figure 8 e dimensionless entropy per unit cell is given by  =  2 /).For the pseudogapped (PG) phase    * ) and the normal phase    * ), the entropy expressions are  PG = 2 ∫ ) Fermi ) PG ) and   = 2 ∫ ) Fermi )  ), respectively, where  (30) for Σ  / 1 = −1.2.e former is assumed to be centered at ± 0) and 0 ±) whereas the latter at ±/2 ±/2) of the �rst Brillouin zone.One notices that the former is short-lived compared to the latter for very weak to moderately weak disorder potential.
el = −/)), for the pseudo-gapped (PG) phase    * ), is We have ignored the temperature dependence of the chemical potential above.For the -summation purpose, we shall �rst divide the BZ into �nite number of rectangular patches.We shall next determine the numerical values corresponding to each of these patches of the momentum-dependent density (e.g., for  el the momentum dependent density is ) and sum these values.We generate these values through the surface plots using "Matlab." With these inputs, we now embark on a calculation of the entropy and the speci�c heat.e entropy difference Δ =  PG −  Normal ) between the PG and normal paramagnetic states is negative (see Table 1).As an ordered state is expected to have lower entropy, there is nothing unusual about it.
e physical quantities, such as speci�c heat  el , electrical resistivity , and magnetic susceptibility , are expected to show anomalous temperature dependences due to the onset of exotic DDW ordering which, as we have seen, has its origin in the nonnesting property of the dispersion.In order to examine this aspect, we have calculated the speci�c heat, using (38), as a function of decreasing temperature in the PG phase starting from  *  200 � at a �xed underdoping level  9.94%.e dimensionless electronic speci�c heat displays a non-Fermi liquid behavior with decreasing temperature in T 1: e values of the pseudo-gap (PG) state entropy  PG given by (36) (column (ii)) and the normal state entropy   given by (37) (column (iv)) at the temperatures indicated at the doping level 9.94%.We also have noted the values of  el calculated using (38).We identify the pseudo-gap temperature  * (∼200 K) as the one at which the entropy difference Δ =  PG −   becomes zero.e entropy is almost a smoothly increasing function of temperature.ere appears to be a small discontinuity (not clearly discernable) in entropy at  * during the passage from the PG state to normal state.We �nd that there is no discontinuity in the electronic speci�c heat  el either.We �nd that, for (− * )/ * → 0 ± ,  el ≈ 1.7100×10 −4    * , that is, there is no discontinuity in  el at the PG temperature  * .Within a mean �eld theoretic framework, we observe that even slightly shorter lifetime of electron quasiparticles for a weak disorder potential ∼0.1 (see Figure 7) in PG phase leads to a severe non-Fermi liquid (NF�) behavior in speci�c heat which is absent in the normal phase.

Transport Properties.
We now consider the simplest description of quasiparticle transport [54] in the DDW state in terms of the Boltzmann equation in the relaxationtime approximation.e elastic impurity scattering with the inclusion of the disorder potential leads to a momentumdependent relaxation time as seen in the Section 3.2.In our calculations and graphical representations in Section 2, however, we assumed an intrinsic lifetime broadening (/ 1 ) ∼ 0.1 independent of momentum.We make the same assumption here in the �rst approximation and consider a momentum-and energy-independent relaxation time  and its dimensionless counterpart  0 =  1 /.is is seemingly quite adequate for elastic impurity scattering for a weak disorder potential ∼0.1 as the lifetimes of electron quasiparticles are not strikingly different from that of hole quasiparticles (see Figure 7).e electrical and thermal conductivities may then be expressed as /  ), at the doping level 9.94% in the PG phase, is found to be of ∼ 6.7 × 10 −4 (indicating that (dimensionless)   ≫ 1 in the PG phase) albeit with slight variation.e variation may be due to the total omission of the interband contributions in (39).
precisely for this reason the thermal conductivity   shows discontinuity in the passage from the PG to normal phase; in PG phase the dominant contributions to   come through the states corresponding to Fermi arc including the nodal quasiparticles (see Figure 11(a)) whereas in the normal phase the main contributions are from the portion of the arc close to the antinodal region (± 0) (see Figure 11(b)).As shown in Figure 11(c), the main contributions to   as well come through the states corresponding to Fermi arc.Furthermore, we also �nd that as we move to the SC phase from the PG phase along the line indicated in Figure 1, there is considerable enhancement in   as expected.
e underlying assumption in writing down the expressions for the electrical and the thermal Hall conductivities (     ) in ( 39) is that we are in the linear response regime.e invocation of the Landau level (LL) quantization is not necessary in this regime as the magnetic �eld () may be assumed to be much smaller than 1 Tesla.Besides, the LL quantization is expected to be relevant for a large and ultraclean sample.e normal state dispersion in (10) (40).Likewise, the reason for the appearance of  in the third term of (40) could be explained.Taking lattice constant  3. Å, we �nd that   2.1 × 10 −4  where  should be in Tesla.For   1 Tesla,   1. erefore, a nonzero   1 is not expected to affect the linear dependence of (     ) on , as could be apparently inferred from (26), in any signi�cant way.We indeed �nd the linear dependence of (     ) on  in the PG phase at a given temperature for the doping level ∼10% (see Figure 12).Now as shown in Figure 13, where we have plotted the electrical and the thermal Hall conductivity densities in the �rst B� at   127 K,   .4%, and   0.0 Tesla, the main contributions to (     ) arise from the density of states corresponding to those portions of the Fermi arc which are not part of the nodal region inhabited by quasiparticles of longer lifetime; there is no contribution from the antinodal regions centered at [(± 0) (0 ±)] either.ese �ndings are consistent as long as   1 and the temperature is lower than  * .erefore, these are very much supportive of the experimental �ndings of Doiron-Leyraud et al. [55][56][57] who have detected quantum oscillations in the electrical resistance of underdoped YBCO establishing the existence of a well-de�ned Fermi surface (FS) with Fermi pockets in the antinodal region when the superconductivity is suppressed by a strong magnetic �eld.Furthermore, Riggs et al. [58] have observed the oscillations in the speci�c heat of YBCO-Ortho II samples (in the presence of a magnetic �eld   4 T), the same type of YBCO samples investigated by Doiron-Leyraud and coworkers [55][56][57], con�rming the �ndings regarding Fermi pockets.
As shown in Table 2, we could �nd reasonably conducting character of the PG state for the near-nested Fermi surface.We obtain nearly constant small Hall (electrical) conductivity for all the temperatures considered close to   and those below it in the underdoped region.e longitudinal conductivity   varies slowly, while the Hall angle tan   ≡   /   1. if a problem so requires.For example, the screened plasma frequency   , approximated with the plasma edge in the re�ectivity (), may be introduced in place of ώ while dealing with the problem of metals.We, however, refrain from doing so as we are not dealing with a similar system.e longitudinal optical conductivity   () due to an applied �eld   , in the linear response theory, is given by the frequency-dependent current   ()    ()  ().e quantity   () may also be expressed in terms the vector potential   () of the applied �eld as   ()    () +   ()  (), where   ()  ()  ∞ 0   [  ()   (0)] is the current-current correlation function, and  is the number of -vectors in the Brillouin zone.Here,   () is the current operator in the interaction representation:   ()  ( 0 )  (− 0 ), where  0 is the Hamiltonian with   0. e conical brackets ⋯ stand for an equilibrium average de�ned with the Hamiltonian  0 .For simplicity, one may assume the electron self-energy to be -independent or the vector potential to be positionindependent.It can then be shown that the vertex corrections in the current-current correlation function vanish for   0, due to the odd parity of the current operator.As in [32,40], the equilibrium average of the current operator   ()  −  (), where   ( −1 ∑ ′    ()  ()),   () is the second derivative with respect to   of dispersion (), and   () is the momentum distribution function.e prime symbol in the sum above is to emphasize that only the conduction band contributions need to be taken into account.In view of this result, we obtain   ()  (− +   ())  ().Now as noted above,   ()    ()  ().e vector potential   () is related to the �eld component   () by   ()  ( + 0 + )  ().
is eventually yields SW()   −1 ∑ ′    ()  ().We note that the elastic scattering by impurities, as discussed in Section 3.2, has not been taken into account in this derivation.We shall assume below, instead, an intrinsic lifetime broadening ( 1 ) ∼ 0.1 independent of momentum.
As the outline of the derivation above shows, the spectral weight SW() is obtained from the real part of the optical conductivity (OC).We reemphasize that we have taken into account only the carrier contribution from the conduction band to OC and assumed the direction  to be the direction of the current �ow when induced by an electromagnetic �eld in the same direction.
We slightly modify (43) replacing the -summation above by  + − (  )2  + − ((  )2)( ), where ( ), given below (11), is the density of states (DOSs) or spectral function (SF) obtained using the Matsubara propagators with poles at ± ()  .e Fermi energy DOS corresponds to   0. e reason for not using (35) giving the DOS with impurity scattering is the fact that in deriving (43) the scattering aspect has not been taken into account.To obtain the SF and SW(), the coupled gap equations for   and  (sc)  together with   1 )  .15.e hopping parameters are ( 2  1 )  .39125 and ( 3  1 )  .5.e intrinsic lifetime broadening ( 1 ) has been assumed to be 0.1.the equation to determine the chemical potential () of the fermion number have been solved in a consistent manner with the pinning of the van Hove singularity (vHS) close to .
We consider the quantity ()   + − (  )/ 2  + − (  )2   ( )  ()  , where the subscript "" stands for the conduction band.In Figures 14(a) and 14(b) above, we have contour-plotted the density   ( )  ()  in the SC phase and the PG phase, respectively, whereas in Figure 14(c) we have plotted the quantity () as a function of  for    K <   ∼ 1 K.We �nd that there is considerable depletion of the spectral weight density available for pairing in the antinodal region of momentum space in the PG phase compared to the SC phase as could be inferred from the shrinkage of the hot region in Figure 14(b) in contrast with Figure 14(a).To obtain the data for Figure 14(c), -integration is necessary.For the integration purpose, we �rst divide the �� into �nite number of rectangular patches.We next determine the numerical values corresponding to each of these patches of the momentum-dependent density and sum these values.e sum is then divided by the number of patches.We have generated these values through a surface plot.e gap amplitude at 60 K is (Δ ()  sc  1 ) ∼ .1.e integral () is a decreasing function for energy larger than that corresponding to the gap amplitude.is is supportive of the fact that DSC of the hole-doped cuprates corresponds to the strong coupling superconductivity.

Concluding Remarks
In this communication, we have assumed that the SC order parameter is such that it is linked with an attractive interaction (  ′ )  −| .We have assumed that a combination of e-e and e-bm interactions will lead to a -wave gap Δ (SC)  .It must be clari�ed that our e�ort to obtain the solution of the gap equation has not been ambitious enough to aim at attempting the settlement of the long debated issue [59] whether the pairing interactions are of purely electronic and (or) electron-bosonic mode origin.In fact, assuming that these agents jointly or in a mutually exclusive manner provide us the requisite pairing interactions, we bypass this vital issue.
e angle-resolved photoemission spectroscopy (ARP ES), which is a valuable tool in the toolkit of experimentalists for studying the PG phase, shows Fermi arcs [15][16][17][18][19][20], below the characteristic temperature  * , centered along the zone diagonal instead of the expected Fermi surface (FS).ough a general consensus exists about the existence of the Fermi arcs, there is again a debate on their main characteristics which revolves around the following issues.Some authors [15][16][17] report that the Fermi arcs are linked to the preformed pair scenario.Near the antinode there exists a single gap, nearly independent of temperature, and in the superconducting state the gap follows the expected -wave behavior along the FS.In contrast, other group of authors [18][19][20]24] claim that arcs are associated to a pairing which is distinct from that corresponding to SC. ese theoretical and experimental studies carried out so far have been able to keep the interest alive for solving this puzzle.In this communication, as already mentioned, our focal point is the latter view.While the singleparticle excitation spectrum of the DDW state involves the usual fermionic particle () and hole (ℎ) pairing/mixing in the singlet channel with an associated energy scale of roughly

F 2 :
e 2D plot of the spectral function (  ∫ ((   ∫ +  (  / ∫ +  (  /(  as a function of , with a cusp at   0. e negative  states by and large correspond to 's close to Γ-point whereas positive  states are close to (± ±.For the comparison purpose, we have also plotted the square lattice tight band DOS in the Hubbard model which clearly shows vHS at   0. amplitude/energy cut-off Dimensionless coupling strength ((  )) F 3: e 2D plot of ( 0 /ℏ  ) as a function of the dimensionless coupling strength (  ) and second-degree polynomial �t.

F 4 :
Momentum component along the anti-nodal cut Dispersion in DDW-SC phase in units of first (a) A plot of the upper and the lower bands  (  in the pure DDW state (doping level 9.94%) along the antinodal cut (− -(0 -( .e back-bending (or saturation) momentum of the dispersion  (  and the Fermi momentum (   are indicated.e numerical values, in the units of the �rst neighbor hopping  1 , are ( 1  = −0.0183, the PG gap amplitude (Δ (pg 0  1  = 0.0610, and the SC gap amplitude (Δ (sc 0  1  = 0. e hopping parameters are ( 2  1  = 0.3925 and ( 3  1  = 0.0005.e DDW ordering leads to pining of the Fermi level close to, but not precisely at, the Van Hove singularity.(b) A plot of the upper and the lower bands ± έ( in the pure SC state along the antinodal cut (− -(0 -(  indicating perfect electron-hole symmetry (both the upper and lower bands are parabolic).e numerical value of the SC gap amplitude (Δ (sc 0  1  = 0.0150.e hopping parameters values and ( 1  are the same as above.(c) A plot of the upper bands + (  and the lower bands − (  in the DDW-DSC state along the antinodal cut (− -(0 -( .e back-bending (or saturation) momentum of the dispersion − (  is indicated by double-headed arrows.is shoulder-type feature also exists in the experimental data of Hashimoto et al. [24].Evidently, there are four quasiparticle bands (shown as a function of momentum ( along the antinodal cut) with two positioned at negative energy and two at positive energy for the Fermi energy taken as zero.e parameter values used are ( 1  = −0.0018, the PG gap amplitude (Δ (pg 0  1  = 0.024, and the SC gap amplitude (Δ (sc 0  1  = 0.015.e hopping parameters are ( 2  1  = 0.3913 and ( 3  1  = 0.0005.

F 5 :
(i) e contour plots of the upper band  (  (a) and the lower band  (  (b) in the pure DDW state (doping level 9.94%) on the �rst Brillouin zone (B�).e wave vector component (   is plotted along the horizontal direction while (   is plotted along the vertical direction.e scale of the plot in (a) is from 0 to 8 whereas in (b) it is from 0 to −2. e band-maxima in (a) occur at the points (± ± and (0 0.e band-maxima in (b) occur at the nodal points (± ±.(ii) e contour plots of the band (− (   (c) and the band (− (   (b) in the DDW + DSC state (doping level 9.94%) on the �rst Brillouin zone (B�).e scale of the plot in (c) is from 0 to −8 while in (d) it is from 0 to −2. e band-maxima in (c) occur at the antinodal points (± 0 and (0 ±.e band-minima in (d) occur at the nodal points (± ±.e nodal quasiparticles correspond to the cone-like structure.

1 0 1 0
, for Figure 8(b) it is 0.30.e usual Fermi arc (Figure 8(a)) for very low value of Γ = 0.05 gets nearly washed out (Figure 8(b)) for higher value Γ −= 0.3 since, as seen in Figure 7, the increase in Γ −makes the lifetime of nodal excitations shorter.us, the disorder seems to have signi�cant in�uence on the Fermi energy DOS.

F 8 :
e contour plots of the Fermi energy DOS on the Brillouin zone (BZ) at 9.94% hole doping for Σ  / 1 = −1.2 which rede�nes the chemical potential.e scale of the plots is from −0.2 to 0.6.In (a) Γ −1 0 = 0.05 whereas in (b) it is 0.30.e usual Fermi arc scenario in (a) gets nearly washed out in (b) due to increase in the disorder potential.

F 16 :
(a) e plot of the spectral function in the pseudogap phase (doping level 9.94%) at    (or, the Fermi energy density of states) on the �rst �rillouin zone (��) for ( 1 )  −.189, (Δ (pg)   1 )  .77, and (Δ(sc)    1 )  .e hopping parameters are ( 2  1 )  .3925 and ( 3  1 )  .5.(b) e plot of the spectral function in the superconducting phase (doping level 9.94%) at    (or, the Fermi energy density of states) on the �rst �rillouin zone (��) for ( 1 )  −.183, (Δ (pg)   1 )  .24, and (Δ(sc) 2 2. Unlike the pure DDW case, for the CDDW case      sc  when the dispersion is perfectly nested., but the factors are complex unlike the usual Bogoluibov picture.eoutcome suggests that the investigation on the possibility of CDDW + DSC 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,   {  +     +  +  −  2  } × {  −     −  +  −  2  } and          −  +    −      −  +  −  2 −      −  +  − 2 ′ /2{    ′ / −  ′ +  + } and      1/{       | − + −        | + + }.Upon using the result  ±  +  −1  [ −1 ±1/], 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:    2[ 2   −     +  2   −    ].ese 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  †   ∑  ′    ′  †  ′ /2  ′  tanh  ′ /2, where      −1 , but the coherence factors  2   1 and  2   .is 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 +  + /2,       −  + /2, and    [    2 + |  | 2 ] 1/2 .e index  is equal to (±1) with   +1 corresponding to the upper branch  and   −1 corresponding to the lower branch .e single-particle spectral function in the spin- channel is given by      − −1  Im     , where      is the retarded Green's function given by        ∞ −∞ Fermi , where the Fermi energy density of states (DOSs)  Fermi  should be determined by the inclusion of the disorder potential ideally (see Section 3.2).e quantity  Fermi  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 / 1  ∼ .1.A 2D plot of + −   /2  + −   /2  as a function of , in the pseudogap phase (doping level 9.94%), is shown in Figure . e remaining Fourier coefficients  ′  (,   ) and Γ  (,   ) which correspond to the DDW gap and the DSC gap, respectively, are given by    and Γ  (,    lead to the DDW (Δ  ) and DSC gap (Δ (sc  ) equations while   (,    leads to the equation for the chemical potential.With  ′  (,   , in view of the de�nition  †  ≡ − ∑  ′ , (,  ′ ⟨ †     −  eff ( 2 }.In the zero-temperature limit when nesting of the Fermi surface is near perfect, as in the square lattice with nearestneighbor hopping and a small second neighbor hopping (( 2 / 1  small compared to unity), (13) may be written as where  1 is the coupling strength, we �nd that the equation assumes a simple form 1 ≈ | 1 | ∑  [(cos   −cos    2 /{ 2   )(cos     cos   ) 2 } 1/2 ].With the two gap equations combined, we obtain −1 which is similar to the weak coupling BCS gap equation.With an appropriate attractive interaction (,  ′  = −| 1 |(cos    − cos   (cos  ′   − cos  ′  , 4.    where the angle  spans the half-Fermi arc as measured from the nodal direction.It may be relevant to note that the anomalous pairing [47,48]ng the Kadanoff-Baym approach[47,48], one can write a relation between the thermodynamic potential and the spectral functions  (  , corresponding to the Matsubara propagators    (   ,    (   , and so forth as where the spectral functions   (  ,   (    ,   (  , and   (   are given by   (   =            =  −         =−      (     =              =  −           =−      (   =   Γ         =  −Γ   (      =−      (   =            =  −         =−   . , and so on.Since we already have calculated    (    and Γ   (   , we shall be able to calculate the spectral functions   (   and   (  .To calculate   (    , we use the result (2)nt to de�ne a thermodynamic potential Ω( in terms of the Hamiltonian (   where  is a variable and  is given by(2).One can write Ω( − Ω(0  ((  , where Ω(0 is an integration constant and the angular brackets ⋯  denote thermodynamic average calculated with (.e system under consideration corresponds to Ω(  1.×    (      (     Δ (sc    (    Δ †(sc    (    (22) =     ′ ( −  ′  (1 1  ∼ 1.65 eV which gives   = (1 1  −1 ∼ .6 (eV) −1 We, thus, obtain (    ≈ −  (  (  , and +  + }.e electronic excitations in cuprates are thus demonstrably particle-hole mixed quasiparticles in the pseudogap phase with �nite lifetime for the states of de�nite momentum due to the impurity scattering.Using the integral representation of ( above, it is not difficult to show that  (   ′    In order to determine the Fermi energy density of states (DOS)  Fermi (, we shall put    in(34).
Kelvin) at the doping level 9.94%.e discontinuity in   is clearly visible at  * ∼ 200 K. e dimensionless Lorentz number ( For the �rst neighbor hopping, say, corresponding to the sites   ( 0) and   ( ), the quantity    T 2: e values of the pseudo-gap (PG) state electrical hall conductivity (  ), longitudinal conductivity (  ), and e Hall angle (  ) for   0.0 Tesla.We �nd tan   ≡   /  ∼ 0.01 in the PG phase.   , where   (2 2 /ℎ) is the Peierls phase factor.Similarly, for the second neighbor hopping, say, corresponding to the sites   ( 0) and   (2 ), the quantity    −(/ 0 ) ∫    /2.ese explain the reason behind the appearance of  and /2, respectively, in the �rst and the second terms of where the vector potential  may be assumed to be in the Landau gauge:   ( 0 −  0 ).e hopping amplitudes   , corresponding to the sites  and , now may assume the form [  exp(  )], where    (/ 0 ) ∫     dL and  0  (ℎ/2).