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

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 𝑑-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 𝑇_𝑐 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 𝑇_𝑐 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 𝑇_𝑐 [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 𝑇^∗∼200 K compared to 𝑇_𝑐 and one views 𝑇^∗ 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 𝑖𝑑-density wave ordering [9–11] (or its more complex variant, namely, 𝑑+𝑖𝑑 density wave ordering [22, 23, 28–30]), at the antiferromagnetic wave vector 𝐐=(𝜋,𝜋). Starting with a two-dimensional fermion system on a square lattice described by a mean-field Hamiltonian involving the singlet 𝑖𝑑-density wave (DDW) order (𝑖Δ_𝑘=(𝑖Δ₀(𝑇)/2) (cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)) at the wave vector 𝐐=(𝜋,𝜋), we intend to show that, for a transition to the PG state, upon lowering the temperature from 𝑇>𝑇^∗ to 𝑇<𝑇^∗ at a fixed doping level (∼10%) on the underdoped side (see Figure 1), the entropy difference Δ𝑆(=𝑆_PG−𝑆_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. There appears to be no discontinuity in entropy at 𝑇^∗ during this passage. There is no discontinuity in the electronic specific heat 𝐶_el 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 𝐶_el at 𝑇^∗. The outcome of the calculation of in-plane quasiparticle thermal conductivity 𝛼_𝑥𝑥 in both the phases via the Boltzmann equation in the relaxation-time approximation shows that there is a mild discontinuity in 𝛼_𝑥𝑥 at 𝑇^∗∶𝛼_𝑥𝑥 for 𝑡=(𝑇−𝑇^∗)/𝑇^∗→0₋ is found to be higher than 𝛼_𝑥𝑥 for 𝑡→0₊. This is an unusual feature. We needed to solve the pseudogap equation and the equation to determine the chemical potential (𝜇) consistently in order to calculate 𝛼_𝑥𝑥. We pinned 𝜇 close to van Hove singularities (vHSs) of the normal state dispersion 𝜀_𝐤 (involving the first, the second, and third neighbor hoppings) in an effort to do so. We have rendered all quantities, such as 𝛼_𝑥𝑥, dimensionless expressing 𝜀_𝐤, and energy gaps, in units of the first neighbor hopping. The origin of the above-mentioned discontinuity in 𝛼_𝑥𝑥 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 𝛼_𝑥𝑥. 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 𝜎_𝑥𝑥 is significantly greater than 1. As regards the Hall conductivities (electrical (𝜎_𝑥𝑦) and thermal (𝛼_𝑥𝑦)), we found them nearly constant and close to unity for all the temperatures considered between 𝑇_𝑐 and 𝑇^∗. Thus, for the in-plane electrical conductivities, the Hall angle tan𝜃_𝐻≡𝜎_𝑥𝑦/𝜎_𝑥𝑥≪1. 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 𝑐-axis resistivity and the Hall number are insulator-like in the pseudogap phase of the cuprates.

Figure 1 

A generic phase diagram of the hole-doped cuprates in the doping level (𝑥)-temperature (𝑇) plane. The antiferromagnetic phase corresponds to the blackened region in the far left. While the lightly red region to its right is the pseudogap (PG) state, the near-semicircular red region corresponds to the superconducting (SC) phase. The curves 𝑇∗(𝑥) and 𝑇𝑐(𝑥), respectively, correspond to the PG transition (from the paramagnetic insulating phase) and the SC transition. The vertical, thick green line within the SC phase corresponds to the region investigated which is a part of the superconducting DDW region.

As regards 𝑑-wave superconductivity (DSC), we model the effective two-particle pairing interaction 𝑉(𝐤,𝐤^′) in the singlet pairing channel by suitable function of the form −|𝑉₁|(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)(cos𝑘^′_𝑥𝑎−cos𝑘^′_𝑦𝑎), where 𝑉₁ 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 𝑇_𝑐 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 𝑇<𝑇_𝑐. 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 𝐸_𝐤(𝑇<𝑇_𝑐)=±𝐸^(𝑈,𝐿)(𝐤),𝐸_𝐤^(𝑈,𝐿)2=(1/2)[(𝜉²_𝐤+𝜉²_𝐤+𝐐+2𝐷_eff(𝐤)²)±𝑝(𝐤)], where 𝑝(𝐤)=[(𝜉²_𝐤−𝜉²_𝐤+𝐐)²+4Δ²_𝐤(𝜉_𝐤+𝜉_𝐤+𝐐)²]^1/2, 𝐷_eff(𝐤)²=[Δ²_𝐤+Δ^(sc_𝐤⁾²], and 𝜉_𝐤=𝜀_𝐤−𝜇. The symbol 𝜀_𝐤 stands for the normal state dispersion and 𝜇 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 𝜀_𝐤 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 𝑖Δ_𝐤 and Δ^(sc)_𝐤 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 𝐐=(𝜋,𝜋) (nonnesting property). One may notice that the two gaps in the excitation spectrum are distinct and do not merge into one “quadrature” gap (𝐷_eff(𝐤)²=[Δ²_𝐤+Δ^(sc_𝐤⁾²]) if the nesting property, 𝜀_𝐤=−𝜀_𝐤+𝐐, of the dispersion is absent; for the nested situation we do obtain such a merger yielding 𝐸_𝐤^(𝑈)=𝐸_𝐤^(𝐿)=(𝜉²_𝐤+𝐷_eff(𝐤)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 𝐤-independent. The Re Σ changes the quasiparticle dispersion away from the one corresponding to the noninteracting case whereas Im Σ gives the quasiparticle lifetime. The simplest example is the one corresponding to the repulsive Hubbard model (𝑈>0) 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 𝜀_𝐤={−2𝑡₁(cos(𝑘_𝑥𝑎)+cos(𝑘_𝑦𝑎))} 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 𝐐=(±𝜋,±𝜋) only at half-filling; close to half-filling the nesting is approximate. The density of states (DOS) displays nesting singularity. Upon inclusion of 𝑈>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 𝐐=(±𝜋,±𝜋) 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 𝐶_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.

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(𝑇)) 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 𝑇_𝑐 at energies larger than the energy gap. We show this result analytically for the coexistent states calculating SW(𝑇) within the BCS framework for a two-dimensional fermion system on a square lattice starting with a Hamiltonian 𝐻 corresponding to the 𝑖𝑑-density wave (DDW) order plus the superconducting pairing Δ^(sc)_𝑘=∑_𝑘′,𝜎𝑉(𝐤,𝐤^′)⟨𝑑_{−𝑘′,−𝜎}𝑑_𝑘′,𝜎⟩. 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 𝑇^∗. The electronic specific heat is shown to display anomalous temperature dependence. We also calculate the quasiparticle thermal conductivity 𝛼_𝑥𝑥 in the normal and pseudogap phases via the Boltzmann equation in the relaxation-time approximation. The observed mild discontinuity in 𝛼_𝑥𝑥 at 𝑇^∗ 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 𝐹(𝜔)=^+𝜋_−𝜋𝑑𝑘_𝑥𝑎2𝜋^+𝜋_−𝜋𝑑𝑘_𝑦𝑎2𝜋_𝛼𝐴(𝐤,𝜔)𝜀_𝐤^(𝑥𝑥)𝑛_𝑘,𝛼(𝑇), (1) where the subscript “𝛼” stands for the conduction band, 𝐴(𝐤,𝜔) is the density of states (DOSs) or spectral density (SD), 𝜀_𝐤^(𝑥𝑥) is the second derivative with respect to 𝑘_𝑥 of dispersion 𝜀_𝐤, and 𝑛_𝑘,𝛼(𝑇) is the momentum distribution function at a temperature 𝑇<𝑇_𝑐, as a function of energy (𝜔) 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 𝑖𝑑-density wave (DDW) order at the antiferromagnetic wave vector 𝐐=(𝜋,𝜋) plus the 𝑑-wave superconductivity can be expressed as 𝐻=_𝑘,𝜎𝜀_𝐤𝑑^†_𝑘,𝜎𝑑_𝑘,𝜎+_𝑘,𝜎𝜀_𝐤+𝐐𝑑^†_{𝑘+𝑄,𝜎}𝑑_{𝑘+𝑄,𝜎}+_𝑘,𝜎𝐷_𝑘𝑑^†_𝑘,𝜎𝑑_{𝑘+𝑄,𝜎}+_𝑘,𝜎𝐷^†_𝑘𝑑^†_{𝑘+𝑄,𝜎}𝑑_𝑘,𝜎+_𝑘,𝜎Δ^†(sc)_𝐤𝑑_{−𝑘,−𝜎}𝑑_𝑘,𝜎+_𝑘,𝜎Δ^(sc)_𝐤𝑑^†_𝑘,𝜎𝑑^†_{−𝑘,−𝜎}−𝜇_𝑘,𝜎𝑑^†_𝑘,𝜎𝑑_𝑘,𝜎+𝑑^†_{𝑘+𝑄,𝜎}𝑑_{𝑘+𝑄,𝜎}, (2) where 𝐷_𝐤=(𝑖Δ_𝐤)≡−∑_𝑘′,𝜎𝑈(𝐤,𝐤^′)⟨𝑑^†_{𝐤′+𝐐,𝜎}𝑑_𝐤′,𝜎⟩ and 𝐷^†_𝐤=(−𝑖Δ_𝐤)≡−∑_𝑘′,𝜎𝑈(𝐤,𝐤^′)⟨𝑑^†_𝐤′,𝜎𝑑_{𝐤′+𝐐,𝜎}⟩. The time-reversal invariance of the normal state requires that the dispersion 𝜀_𝐤=𝜀_−𝐤. The function 𝜀_𝐤 involves the near neighbor hopping terms to be specified later (see (10)). The gap function Δ^†_𝐤^(sc)≡∑_𝑘′,𝜎𝑉(𝐤,𝐤^′)⟨𝑑^†_𝑘,𝜎𝑑^†_{−𝑘,−𝜎}⟩=−Δ^†(_𝐤+𝐐^sc). The conical brackets stand for the thermal average calculated with the Hamiltonian in (2). This step ensures the self-consistency.

The imaginary 𝑑-wave order parameter 𝐷_𝐤 describing the PG state breaks the time-reversal symmetry of the normal state. The time-reversal operator 𝑂^‵ transforms the order parameter to its complex conjugate: 𝑂^‵𝐷_𝐤=𝐷^†_𝐤. If the time-reversal symmetry is preserved, 𝐷_𝐤 and 𝐷^†_𝐤 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 𝐷^†_𝐤≈∑_𝐤′𝑈(𝐤,𝐤^′)𝐷^†_𝐤′𝑓(𝐤^′,𝑇)×[(𝜉²_𝐤′+|𝐷_𝐤′|²)]⁻¹, where the function 𝑓(𝐤^′,𝑇) being the difference of two Fermi functions and is positive, the interaction 𝑈(𝐤,𝐤^′) has to be of the form 𝑈(𝐤,𝐤^′)=|𝑈₁|(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)(cos𝑘^′_𝑥𝑎−cos𝑘^′_𝑦𝑎). The nonzero DSC order parameter or the gap Δ^(sc)_𝑘, on the other hand, requires an appropriate attractive interactions 𝑉(𝐤,𝐤^′)=−|𝑉₁|(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)(cos𝑘^′_𝑥𝑎−cos𝑘^′_𝑦𝑎), where 𝑉₁ is the coupling strength. The pure DSC gap is given by Δ^(sc)_𝐤=−(1/2)∑_𝐤′𝑉(𝐤,𝐤^′)Δ^(sc)_𝐤′×[(𝜉²_𝐤′+Δ^(sc_𝐤⁾²_′)]⁻¹ which is the usual BCS form. If the pairing interaction 𝑉(𝐤,𝐤^′) is imagined to be a “probe” applied to the Fermi system in the PG state, then the gap function Δ^(sc)_𝐤 (where Δ^(sc)_𝐤=Δ^†_𝐤^(sc)) 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-𝑠-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 𝑑-wave gap Δ^(sc)_𝑘. For the quantities (Δ_𝐤,Δ^(sc)_𝐤), given in the form of Δ₀(𝑇)(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎), the amplitude Δ₀(𝑇) 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 𝐻, namely, 𝐺_𝜎(𝐤,𝜏)=−⟨𝑇{𝑑_𝐤,𝜎(𝜏)𝑑^†_𝐤,𝜎(0)}⟩, Γ_𝜎(𝐤,𝜏)=−⟨𝑇{𝑑^†_{−𝐤,−𝜎}(𝜏)𝑑^†_𝐤,𝜎(0)}⟩, 𝐺^′_𝜎(𝐤,𝜏)=−⟨𝑇{𝑑_{𝐤+𝐐,𝜎}(𝜏)𝑑^†_𝐤,𝜎(0)}⟩, and Γ^′_𝜎(𝐤,𝜏)=−⟨𝑇{𝑑^†_{−𝐤−𝐐,−𝜎}(𝜏)𝑑^†_𝐤,𝜎(0)}⟩. Here, 𝑇 is the time-ordering operator which arranges other operators from right to left in the ascending order of imaginary time 𝜏. The first step of the scheme involves the calculation of (imaginary) time evolution of the operators 𝑑_𝐤,𝜎(𝜏) where, in units such that ℏ=1, 𝑑_𝐤,𝜎(𝜏)=exp(𝐻𝜏)𝑑_𝐤,𝜎(0)exp(−𝐻𝜏). We obtain, for example, 𝜕𝑑_𝐤,𝜎=−𝜀_𝐤−𝜇𝑑_𝐤,𝜎−Δ^(sc)_𝐤𝑑^†_{−𝐤,−𝜎}−𝐷_𝐤𝑑_{𝐤+𝐐,𝜎},𝜕𝑑^†_{−𝐤,−𝜎}=𝜀_𝐤−𝜇𝑑^†_{−𝐤,−𝜎}−Δ^†_𝐤^(sc)𝑑_𝐤,𝜎+𝐷^†_𝐤𝑑^†_{−𝐤−𝐐,−𝜎}, (3) and so on. Here, 𝜕≡(𝜕/𝜕𝜏) and the argument part has been dropped in writing the operators 𝑑_𝑘,𝜎(𝜏) and their derivative. As the next step, upon using (3), we find that the equations of motion of these averages are given by 𝜕𝐺_𝜎(𝐤,𝜏)=−𝜀_𝐤−𝜇𝐺_𝜎(𝐤,𝜏)−Δ^(sc)_𝐤Γ_𝜎(𝐤,𝜏)−𝐷_𝑘𝐺^′_𝜎(𝐤,𝜏)−𝛿(𝜏),𝜕Γ_𝜎(𝐤,𝜏)=𝜀_𝐤−𝜇Γ_𝜎(𝐤,𝜏)−Δ^†_𝐤^(sc)𝐺_𝜎(𝑘,𝜏)+𝐷^†_𝐤Γ^′_𝜎(𝐤,𝜏),𝜕𝐺^′_𝜎(𝐤,𝜏)=−𝜀_𝐤+𝐐−𝜇𝐺^′_𝜎(𝐤,𝜏)−𝐷^†_𝑘𝐺_𝜎(𝐤,𝜏)−Δ^(sc)_𝐤+𝐐Γ^′_𝜎(𝑘,𝜏),𝜕Γ^′_𝜎(𝐤,𝜏)=𝜀_𝐤+𝐐−𝜇Γ^′_𝜎(𝐤,𝜏)+𝐷_𝐤Γ_𝜎(𝐤,𝜏)−Δ^†(_𝐤+𝐐^sc)𝐺^′_𝜎(𝐤,𝜏). (4)

The final step is the calculation of the Fourier coefficients: 𝐺^𝜆_𝜎(𝐤,𝜔_𝑛)=∫^𝛽₀𝑑𝜏𝑒^{𝑖𝜔𝑛𝜏}𝐺^𝜆_𝜎(𝐤,𝜏) (where the Matsubara frequencies are 𝜔_𝑛=[(2𝑛+1)𝜋/𝛽] with 𝑛=0,±1,±2,…) of these temperature Green’s functions. Here, 𝛽=(𝑘_𝐵𝑇)⁻¹. 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 𝐺_𝜎(𝐤,𝜔_𝑛)=(Δ₁/Δ), Γ_𝜎(𝐤,𝜔_𝑛)=(Δ₂/Δ), 𝐺^′_𝜎(𝐤,𝜔_𝑛)=(Δ₃/Δ), and Γ^′_𝜎(𝐤,𝜔_𝑛)=(Δ₄/Δ), where Δ₁=𝑖𝜔_𝑛+𝜉_𝐤𝑖𝜔_𝑛²−𝜉²_𝐤+𝐐−Δ^(sc_𝐤⁾²−Δ²_𝐤𝑖𝜔_𝑛−𝜉_𝐤+𝐐,Δ₂=𝑖𝜔_𝑛²−𝜉²_𝐤+𝐐−𝐷_eff(𝐤)²Δ^(sc)_𝐤,Δ₃=𝐷^†_𝑘𝑖𝜔_𝑛+𝜉_𝑘𝑖𝜔_𝑛+𝜉_𝑘+𝑄−𝐷_eff(𝐤)²,Δ₄=𝐷^†_𝑘Δ^(sc)_𝐤𝜉_𝐤+𝜉_𝐤+𝐐=,Δ𝑖𝜔_𝑛²−𝜉²_𝐤𝑖𝜔_𝑛²−𝜉²_𝐤+𝐐−𝑖𝜔_𝑛+𝜉_𝐤𝑖𝜔_𝑛+𝜉_𝐤+𝐐+𝑖𝜔_𝑛−𝜉_𝐤𝑖𝜔_𝑛−𝜉_𝐤+𝐐Δ²_𝐤−Δ^(sc_𝐤⁾²𝑖𝜔_𝑛²−𝜉²_𝐤+𝑖𝜔_𝑛²−𝜉²_𝐤+𝐐+Δ^(sc_𝐤⁾⁴+Δ⁴_𝐤+2Δ²_𝐤Δ^(sc_𝐤⁾², (5)𝜉_𝐤=𝜀_𝐤−𝜇, and 𝐷²_eff(𝐤)=(Δ²_𝐤+Δ^(sc_𝐤⁾²). The denominator Δ of the Fourier coefficients yields the single particle excitation spectrum. We find that this is a quadratic in (𝑖𝜔_𝑛)²: Δ=𝑖𝜔_𝑛⁴−𝑖𝜔_𝑛²𝜉²_𝐤+𝜉²_𝐤+𝐐+2𝐷_eff(𝐤)²+𝜉²_𝐤𝜉²_𝐤+𝐐−2𝜉_𝐤𝜉_𝐤+𝐐Δ²_𝐤+Δ^(sc_𝐤⁾²𝜉²_𝐤+𝜉²_𝐤+𝐐+Δ^(sc_𝐤⁾⁴+Δ⁴_𝐤+2Δ²_𝐤Δ^(sc_𝐤⁾². (6)

It may be noted that, in the pure 𝑑-wave case to be investigated below, |𝐷_𝐤|² will get replaced by Δ²_𝐤. Correspondingly, the single-particle excitation spectra will be given by 𝐸_𝐤(𝑇<𝑇_𝑐)=±𝐸^(𝑈,𝐿)(𝐤),𝐸_𝐤^(𝑈,𝐿)2=(1/2)[(𝜉²_𝐤+𝜉²_𝐤+𝐐+2𝐷_eff(𝐤)²)±𝑝(𝐤)], and 𝑝(𝐤)=[(𝜉²_𝐤−𝜉²_𝐤+𝐐)²+4Δ²_𝐤(𝜉_𝐤+𝜉_𝐤+𝐐)²]^1/2. The quasiparticle excitations in cuprates (where we have a pure 𝑑-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 𝐸_𝐤(𝑇<𝑇_𝑐)=±𝐸^(𝑈,𝐿)(𝐤) 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 𝐸^𝜎_sc={(𝐸^𝜎)²+Δ²_sc}, with Bogoliubov amplitudes 𝑢²_𝜎=(1+𝐸^𝜎/𝐸^𝜎_sc)/2 and 𝑣²_𝜎=(1−𝐸^𝜎/𝐸^𝜎_sc)/2 which are applied to the pseudogapped bands indexed by 𝜎=± and given as 𝐸^±=𝜀₁±{𝜀²₂+Δ²_PG}. The energies 𝜀₁=(𝜖_𝐤−𝜖⁰_𝐤)/2 and 𝜀₂=(𝜖_𝐤+𝜖⁰_𝐤)/2, where 𝜖_𝐤 is a third nearest neighbor tight-binding dispersion and 𝜖⁰_𝐤 is that for first nearest neighbor which for 𝜖⁰_𝐤=0 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 Δ_𝐤 and Δ^(sc)_𝐤 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 𝑑+𝑖𝑑 variety (chiral DDW (CDDW) order at the wave vector 𝐐=(±𝜋,±𝜋)). In such a situation, 𝐷_𝐤=(−𝜒_𝐤+𝑖Δ_𝐤), where the real and the imaginary parts (𝜒_𝐤,Δ_𝐤) are given by 𝜒_𝐤=−𝜒₀sin(𝑘_𝑥𝑎)sin(𝑘_𝑦𝑎), and Δ_𝐤=(Δ^(PG)₀(𝑇))(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎). The quantities (Δ,Δ₁,Δ₂,Δ₃,Δ₄) in (5) would now be given by Δ=𝑖𝜔_𝑛⁴−𝑖𝜔_𝑛²𝜉²_𝐤+𝜉²_𝐤+𝐐+2𝐷_eff(𝐤)²+𝜉²_𝐤𝜉²_𝐤+𝐐−2𝜉_𝐤𝜉_𝐤+𝐐𝐷_𝐤²+Δ^(sc_𝐤⁾²𝜉²_𝐤+𝜉²_𝐤+𝐐+Δ^(sc_𝐤⁾⁴+𝐷_𝐤⁴−𝐷^†_𝐤²Δ^(sc_𝐤⁾²−𝐷_𝐤²Δ^(sc_𝐤⁾²,Δ₁=𝑖𝜔_𝑛+𝜉_𝐤𝑖𝜔_𝑛²−𝜉²_𝐤+𝐐−Δ^(sc_𝐤⁾²−𝐷_𝐤²𝑖𝜔_𝑛−𝜉_𝐤+𝐐,Δ₂=Δ^(sc)_𝐤𝑖𝜔_𝑛²−𝜉²_𝐤+𝐐−Δ^(sc_𝐤⁾²+𝐷^†_𝐤²,Δ₃=𝐷^†_𝐤𝑖𝜔_𝑛+𝜉_𝐤𝑖𝜔_𝑛+𝜉_𝐤+𝐐−𝐷_𝐤−Δ^(sc_𝐤⁾²+𝐷^†_𝐤²,Δ₄=𝐷^†_𝐤Δ^(sc)_𝐤𝑖𝜔_𝑛+𝜉_𝐤+𝐷_𝐤Δ^(sc)_𝐤𝑖𝜔_𝑛−𝜉_𝐤+𝐐, (7) where 𝐷_eff(𝐤)² here is equal to (|𝐷_𝐤|²+Δ^(sc_𝐤⁾²). The single particle excitation spectrum 𝐸_𝐤(𝑇<𝑇_𝑐) is given by 𝐸_𝐤(𝑇<𝑇_𝑐)=±𝐸^(𝑈,𝐿)(𝐤), 𝐸_𝐤^(𝑈,𝐿)2=(1/2)[(𝜉²_𝐤+𝜉²_𝐤+𝐐+2𝐷_eff(𝐤)²)±𝑝(𝐤)], and 𝑝(𝐤)=[(𝜉²_𝐤−𝜉²_𝐤+𝐐)²+4|𝐷_𝐤|²(𝜉_𝐤+𝜉_𝐤+𝐐)²+4{(𝐷^†_𝐤)²+(𝐷_𝐤)²−2|𝐷_𝐤|²}Δ^(sc_𝐤⁾²]^1/2. Unlike the pure DDW case, for the CDDW case 𝑝(𝐤)=4𝑖Δ_𝐤Δ^(sc)_𝐤 when the dispersion is perfectly nested. Thus, 𝐸_𝐤^(𝑈,𝐿)=±[𝜉²_𝐤+𝐷_eff(𝐤)²±2𝑖Δ_𝐤Δ^(sc)_𝐤]^1/2 and the Fourier coefficient 𝐺_𝜎(𝐤,𝜔_𝑛)=∑_𝑗=±1{𝑈_𝐤^2(𝑗)(𝑖𝜔_𝑛+𝑗(𝜀_𝐤́+𝑖𝜄́_𝐤⁻¹))⁻¹+𝑉_𝐤^2(𝑗)(𝑖𝜔_𝑛+𝑗(𝜀_𝐤́−𝑖𝜄́_𝐤⁻¹))⁻¹}, where 𝜀_𝐤́=𝑅_𝐤^1/2cos(𝜃_𝐤/2), 𝜄́_𝐤⁻¹=𝑅_𝐤^1/2sin(𝜃_𝐤/2), 𝑅_𝐤=[(𝜉²_𝐤+𝐷_eff(𝐤)²)²+4Δ²_𝐤Δ^(sc_𝐤⁾²]^1/2, tan²(𝜃_𝐤)=(4Δ²_𝐤Δ^(sc_𝐤⁾²)/(𝜉²_𝐤+𝐷_eff(𝐤)²)², 𝑈_𝐤^2(𝑗)=(1/4)[1−(𝑗𝜉_𝐤/𝑅_𝐤^1/2exp(𝑖𝜃_𝐤/2))], and 𝑉_𝐤^2(𝑗)=(1/4)[1−(𝑗𝜉_𝐤/𝑅_𝐤^1/2exp(𝑖𝜃_𝐤/2))]. The sum of the coherence factors ∑_𝑗=±1{𝑈_𝐤^2(𝑗)+𝑉_𝐤^2(𝑗)}=1, but the factors are complex unlike the usual Bogoluibov picture. The outcome 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, Δ={(𝑖𝜔_𝑛+𝜉_𝐤)(𝑖𝜔_𝑛+𝜉_𝐤+𝐐)−Δ²_𝐤}×{(𝑖𝜔_𝑛−𝜉_𝐤)(𝑖𝜔_𝑛−𝜉_𝐤+𝐐)−Δ²_𝐤} and 𝐺_𝜎𝐤,𝜔_𝑛=𝑖𝜔_𝑛−𝜉_𝐤+𝐐𝑖𝜔_𝑛−𝜉_𝐤𝑖𝜔_𝑛−𝜉_𝐤+𝐐−Δ²_𝐤,𝐺^′_𝜎𝐤,𝜔_𝑛=𝐷^†_𝐤𝑖𝜔_𝑛−𝜉_𝐤𝑖𝜔_𝑛−𝜉_𝐤+𝐐−Δ²_𝐤, (8) which may be written as 𝐺_𝜎𝐤,𝜔_𝑛=𝑢²_𝐤𝑖𝜔_𝑛−𝜀_𝑘^(𝑈)−1+𝑣²_𝐤𝑖𝜔_𝑛−𝜀_𝑘^(𝐿)−1,𝐺^′_𝜎𝐤,𝜔_𝑛=𝐷^†_𝐤2𝑤_𝑘𝑖𝜔_𝑛−𝜀_𝑘^(𝑈)−1−𝑖𝜔_𝑛−𝜀_𝑘^(𝐿)−1. (9)

The first result in (9), upon applying Luttinger’s theorem [44], leads to the equation to determine the chemical potential 𝜇∶(1+𝑝)=(𝑁_𝑠/2)⁻¹∑_𝐤′[𝑢²_𝐤′×(exp(𝛽𝜀_𝐤^(𝑈)_′)+1)⁻¹+𝑣²_𝐤′×(exp(𝛽𝜀_𝐤^(𝐿)_′)+1)⁻¹], where 𝑝 is the hole-doping level, and 𝑁_𝑠 is the number of unit cells in the 𝐤-space. The second result in (9) leads to the DDW gap equation 𝐷^†_𝐤=∑_𝐤′,𝜎𝑈(𝐤,𝐤^′)(𝐷^†_𝐤′/2𝑤_𝐤′)[(exp(𝛽𝜀_𝐤^(𝐿)_′)+1)⁻¹−(exp(𝛽𝜀_𝐤^(𝑈)_′)+1)⁻¹]. Quite obviously, the difference of the Fermi functions within the square brackets is positive and therefore the interaction 𝑈(𝐤,𝐤^′) needs to be repulsive for this equation to be meaningful. The quasiparticle coherence factors (𝑢²_𝐤,𝑣²_𝐤) are given by the expressions 𝑢²_𝐤=(1/2)[1+(𝜀^𝐿_𝐤/𝑤_𝐤)], 𝑣²_𝐤=(1/2)[1−(𝜀^𝐿_𝐤/𝑤_𝐤)]. Here 𝜀_𝐤^(𝑈,𝐿)=[𝜀^𝑈_𝐤+𝑗𝑤_𝐤], 𝜀^𝑈_𝐤=(𝜉_𝐤+𝜉_𝐤+𝐐)/2, 𝜀^𝐿_𝐤=(𝜉_𝐤−𝜉_𝐤+𝐐)/2, and 𝑤_𝐤=[(𝜀^𝐿_𝐤)²+|𝐷_𝐤|²]^1/2. The index 𝑗 is equal to (±1) with 𝑗=+1 corresponding to the upper branch (𝑈) and 𝑗=−1 corresponding to the lower branch (𝐿). The single-particle spectral function in the spin-𝜎 channel is given by 𝐴_𝜎(𝐤,𝜔)=(−𝜋⁻¹)Im𝐺_𝜎^(𝑅)(𝐤,𝜔), where 𝐺_𝜎^(𝑅)(𝐤,𝜔) is the retarded Green’s function given by 𝐺^𝑅_𝜎(𝑘,𝜔)=∫^∞_−∞(𝑑𝜔^′/2𝜋){𝜁_𝜎(𝑘,𝜔^′)/(𝜔−𝜔^′+𝑖0⁺)} and 𝜁_𝜎(𝑘,𝜔)=(1/𝑖){𝐺_𝜎(𝑘,𝑧=𝑖𝜔_𝑛)|_{𝑧=𝜔−𝑖0+}−𝐺_𝜎(𝑘,𝑧=𝑖𝜔_𝑛)|_{𝑧=𝜔+𝑖0+}}. Upon using the result (𝑥±𝑖0⁺)⁻¹=[𝑃(𝑥⁻¹)±(1/𝑖)𝜋𝛿(𝑥)], where 𝑃 represents a Cauchy’s principal value, the spectral function 𝐴(𝐤,𝜔) in the DDW phase is given by a sum of 𝛿 functions at the quasiparticle energies: 𝐴(𝐤,𝜔)=2𝜋[𝑢²_𝐤𝛿(𝜔−𝜀_𝐤^(𝑈))+𝑣²_𝐤𝛿(𝜔−𝜀_𝐤^(𝐿))]. 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 𝜀_𝐤^(𝑈,𝐿)=𝑗𝑤_𝐤, and 𝐷^†_𝐤=∑_𝐤′,𝜎𝑈(𝐤,𝐤^′)(𝐷^†_𝐤′/2𝑤_𝐤′)tanh(𝛽𝑤_𝐤′/2), where 𝛽=(𝑘_𝐵𝑇)⁻¹, but the coherence factors 𝑢²_𝐤=1 and 𝑣²_𝐤=0. 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 𝜀_𝐤=−2𝑡₁cos𝑘_𝑥𝑎+cos𝑘_𝑦𝑎+4𝑡₂cos𝑘_𝑥𝑎×cos𝑘_𝑦𝑎−2𝑡₃cos2𝑘_𝑥𝑎+cos2𝑘_𝑦𝑎+4𝑡₂, (10) where, for the hole-doped materials, 𝑡₂>0 (for the electron-doped materials 𝑡₂<0), and, in all cases, 𝑡₂<(𝑡₁/2). For example, typical values are 𝑡₁∼0.2 eV, (𝑡₂/𝑡₁)∼0.24, and (𝑡₃/𝑡₁)∼0.01. Upon ignoring the third neighbor hopping term above, we find that the dispersion typically has two inequivalent saddle points at (𝜋,0) and (0,𝜋) 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 𝐤-summation in the equations for 𝐷^†_𝐤 and 𝜇 in the preceeding paragraph may be replaced by the integration: ∫𝑑(𝐤𝑎)𝜌_Fermi(𝐤)→∫^+𝜋_−𝜋𝑑(𝑘_𝑥𝑎)/2𝜋∫^+𝜋_−𝜋(𝑑(𝑘_𝑦𝑎)/2𝜋𝜌_Fermi(𝐤), where the Fermi energy density of states (DOSs) 𝜌_Fermi(𝐤) should be determined by the inclusion of the disorder potential ideally (see Section  3.2). The quantity 𝜌_Fermi(𝐤) here is obtained from the spectral function 𝐴(𝐤,𝜔) given below (9) by a sum of 𝛿 functions at the quasiparticle energies. We simply replace the 𝛿 functions by Lorentzians with an assumed intrinsic lifetime broadening (𝛾/𝑡₁)∼0.1. A 2D plot of 𝐴(𝜔)=∫𝑑(𝐤𝑎)𝐴(𝐤,𝜔)=∫^+𝜋_−𝜋𝑑(𝑘_𝑥𝑎)/2𝜋∫^+𝜋_−𝜋𝑑(𝑘_𝑦𝑎)/2𝜋𝐴(𝐤,𝜔) as a function of 𝜔, in the pseudogap phase (doping level 9.94%), is shown in Figure 2. The numerical values, in the units of the first neighbor hopping 𝑡₁, are (𝜇/𝑡₁)=−0.0189, the PG gap amplitude (Δ^(pg)₀/𝑡₁)=0.0871, and the SC gap amplitude (Δ^(sc)₀/𝑡₁)=0. The hopping parameters are (𝑡₂/𝑡₁)=0.3925, and (𝑡₃/𝑡₁)=0.0005. The DDW ordering leads to pining of the Fermi level close to, but not precisely at, the vHs. The plot shows a cusp at 𝜔=0. 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].

Figure 2 

The 2D plot of the spectral function 𝐴(𝜔)=∫𝑑(𝐤𝑎)𝐴(𝐤,𝜔)=∫+𝜋−𝜋𝑑(𝑘𝑥𝑎)/2𝜋∫+𝜋−𝜋𝑑(𝑘𝑦𝑎)/2𝜋𝐴(𝐤,𝜔) as a function of 𝜔, with a cusp at 𝜔=0. The negative 𝜔 states by and large correspond to 𝐤’s close to Γ-point whereas positive 𝜔 states are close to (±𝜋,±𝜋). For the comparison purpose, we have also plotted the square lattice tight band DOS in the Hubbard model which clearly shows vHS at 𝜔=0.

In the absence of the DDW gap, with the modulation vector 𝐐 set at (0,0) and |𝐷_𝐤|=0, we have Δ=((𝑖𝜔_𝑛)²−𝜀²́(𝐤))((𝑖𝜔_𝑛)²−𝜉²_𝐤), Δ₁=(𝑖𝜔_𝑛+𝜉_𝐤)((𝑖𝜔_𝑛)²−𝜉²_𝐤), and Δ₂=((𝑖𝜔_𝑛)²−𝜉²_𝐤)Δ^(sc)_𝐤. This yields 𝐺_𝜎(𝐤,𝜔_𝑛)=𝑎_𝐤⁽⁻⁾(𝑖𝜔_𝑛+𝜀(́𝐤))⁻¹+𝑎_𝐤⁽⁺⁾(𝑖𝜔_𝑛−𝜀(́𝐤))⁻¹, and Γ_𝜎(𝐤,𝜔_𝑛)=−(Δ^(sc)_𝐤/2𝜀(́𝐤))[(𝑖𝜔_𝑛+𝜀(́𝐤))⁻¹−(𝑖𝜔_𝑛−𝜀(́𝐤))⁻¹], where 𝜀(́𝐤)=(𝜉²_𝐤+|Δ^(sc)_𝐤|²) and the coherence factors are given by 𝑎_𝐤^(±)=(1/2)[1±(𝜉_𝐤/𝜀(́𝐤))]. Upon using the expression for the Fourier coefficient 𝐺_𝜎(𝐤,𝜔_𝑛) the chemical potential 𝜇, according to the Luttinger rule [34], is given by the equation (1+𝑝)=(𝑁_𝑠/2)⁻¹∑_𝑘[(1−(𝜉_𝐤/𝜀(́𝐤))×(exp(−𝛽𝜀(́𝐤))+1)⁻¹+(1+(𝜉_𝐤/𝜀(́𝐤^′))×(exp(𝛽𝜀(́𝐤^′))+1)⁻¹], where 𝑝 is the doping level and 𝑁_𝑠 is the number of 𝐤-vectors in the first Brillouin zone. The Fourier coefficient Γ_𝜎(𝐤,𝜔_𝑛) leads to the weak coupling BCS gap equation for the singlet pairing: Δ^(sc)_𝐤=−∑_𝐤′𝑉(𝐤,𝐤^′)(Δ^(sc)_𝐤′/2𝜀(́𝐤^′))tanh(𝛽𝜀(́𝐤^′)/2), where 𝛽=(𝑘_𝐵𝑇)⁻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 [9–11] albeit with slightly different expression for the excitation spectrum.

For the coexistent DDW and DSC state, from (5) and (6) we find that 𝐺_𝜎𝐤,𝜔_𝑛=𝑢_𝐤²⁽⁺⁾𝑖𝜔_𝑛+𝐸_𝐤^(𝑈)−1+𝑢_𝐤²⁽⁻⁾𝑖𝜔_𝑛−𝐸_𝐤^(𝑈)−1+𝑣_𝐤²⁽⁺⁾𝑖𝜔_𝑛+𝐸_𝐤^(𝐿)−1+𝑣_𝐤²⁽⁻⁾𝑖𝜔_𝑛−𝐸_𝐤^(𝐿)−1,𝑢_𝐤²⁽⁺⁾=12𝜉1−^𝐤𝐸_𝐤^(𝑈)×𝐸_𝐤^(𝑈)2−𝜉^′2_𝐤+𝐐𝑞(𝐤)−𝜉1+^𝐤+𝐐𝐸_𝐤^(𝑈)×Δ²_𝐤𝑞(𝐤),𝑢_𝐤²⁽⁻⁾=12𝜉1+^𝐤𝐸_𝐤^(𝑈)×𝐸_𝐤^(𝑈)2−𝜉^′2_𝐤+𝐐𝑞(𝐤)−𝜉1−^𝐤+𝐐𝐸_𝐤^(𝑈)×Δ²_𝐤𝑞(𝐤),𝑣_𝐤²⁽⁺⁾=−12𝜉1−^𝐤𝐸_𝐤^(𝐿)×𝐸_𝐤^(𝐿)2−𝜉^′2_𝐤+𝐐𝑞(𝐤)−𝜉1+^𝐤+𝐐𝐸_𝐤^(𝐿)×Δ²_𝐤𝑞(𝐤),𝑣_𝐤²⁽⁻⁾=−12𝜉1+^𝐤𝐸_𝐤^(𝐿)×𝐸_𝐤^(𝐿)2−𝜉^′2_𝐤+𝐐𝑞(𝐤)−𝜉1−^𝐤+𝐐𝐸_𝐤^(𝐿)×Δ²_𝐤𝑞(𝐤), (11) where 𝜉^′2_𝐤+𝐐=(𝜉²_𝐤+𝐐+Δ^(sc_𝐤⁾²), and 𝑞(𝐤)=(𝐸_𝐤^(𝑈)2−𝐸_𝐤^(𝐿)2). We find that the coherence factors 𝑢_𝐤^2(±) and 𝑣_𝐤^2(±) satisfy the sum rule (𝑢_𝐤²⁽⁺⁾+𝑢_𝐤²⁽⁻⁾+𝑣_𝐤²⁽⁺⁾+𝑣_𝐤²⁽⁻⁾)=1. The single-particle spectral function in the DDW + DSC state, as before, is given by 𝐴(𝐤,𝜔)=2𝜋[𝑢_𝐤²⁽⁺⁾𝛿(𝜔+𝐸_𝐤^(𝑈))+𝑢_𝐤²⁽⁻⁾𝛿(𝜔−𝐸_𝐤^(𝑈))+𝑣_𝐤²⁽⁺⁾𝛿(𝜔+𝐸_𝐤^(𝐿))+𝑣_𝐤²⁽⁻⁾𝛿(𝜔−𝐸_𝐤^(𝐿))]. Similarly, the spectral function 𝐴(𝐤+𝐐,𝜔) is given by 𝐴(𝐤+𝐐,𝜔)=2𝜋[𝑢²⁽⁺⁾_𝐤+𝐐𝛿(𝜔+𝐸_𝐤^(𝑈))+𝑢²⁽⁻⁾_𝐤+𝐐𝛿(𝜔−𝐸_𝐤^(𝑈))+𝑣²⁽⁺⁾_𝐤+𝐐𝛿(𝜔+𝐸_𝐤^(𝐿))+𝑣²⁽⁻⁾_𝐤+𝐐𝛿(𝜔−𝐸_𝐤^(𝐿))]. We find that, when the nesting property is satisfied, 𝑢²⁽⁺⁾_𝐤+𝐐=𝑢_𝐤²⁽⁻⁾, 𝑢²⁽⁻⁾_𝐤+𝐐=𝑢_𝐤²⁽⁺⁾, 𝑣²⁽⁺⁾_𝐤+𝐐=𝑣_𝐤²⁽⁻⁾, and 𝑣²⁽⁻⁾_𝐤+𝐐=𝑣_𝐤²⁽⁺⁾. In this situation, the sum (𝐴(𝐤,𝜔)+𝐴(𝐤+𝐐,𝜔)) = 2𝜋[(𝑢_𝐤²⁽⁺⁾+𝑢_𝐤²⁽⁻⁾)(𝛿(𝜔+𝐸_𝐤^(𝑈))+𝛿(𝜔−𝐸_𝐤^(𝑈)))+(𝑣_𝐤²⁽⁺⁾+𝑣_𝐤²⁽⁻⁾)(𝛿(𝜔+𝐸_𝐤^(𝐿))+𝛿(𝜔−𝐸_𝐤^(𝐿)))]. As we shall see below, the sum (𝐴(𝐤,𝜔)+𝐴(𝐤+𝐐,𝜔)) will be required for the derivation of an expression for the thermodynamic potential following the Kadanoff-Baym approach [47, 48]. The remaining Fourier coefficients 𝐺^′_𝜎(𝑘,𝜔_𝑛) and Γ_𝜎(𝐤,𝜔_𝑛) which correspond to the DDW gap and the DSC gap, respectively, are given by 𝐺^′_𝜎𝑘,𝜔_𝑛𝐷=−^†𝐤2𝐸_𝐤^(𝑈)×𝐸_𝐤^(𝑈)−𝜉_𝐤×𝐸_𝐤^(𝑈)−𝜉_𝐤+𝐐−𝐷_eff(𝐤)²𝑞(𝐤)×𝑖𝜔_𝑛+𝐸_𝐤^(𝑈)−1+𝐷^†_𝐤2𝐸_𝐤^(𝑈)×𝐸_𝐤^(𝑈)+𝜉_𝐤×𝐸_𝐤^(𝑈)+𝜉_𝐤+𝐐−𝐷_eff(𝐤)²𝑞(𝐤)×𝑖𝜔_𝑛−𝐸_𝐤^(𝑈)−1+𝐷^†_𝐤2𝐸_𝐤^(𝐿)×𝐸_𝐤^(𝐿)−𝜉_𝐤×𝐸_𝐤^(𝐿)−𝜉_𝐤+𝐐−𝐷_eff(𝐤)²𝑞(𝐤)×𝑖𝜔_𝑛+𝐸_𝐤^(𝐿)−1−𝐷^†_𝐤2𝐸_𝐤^(𝐿)×𝐸_𝐤^(𝐿)+𝜉_𝐤×𝐸_𝐤^(𝐿)+𝜉_𝐤+𝐐−𝐷_eff(𝐤)²𝑞(𝐤)×𝑖𝜔_𝑛−𝐸_𝐤^(𝐿)−1,Γ_𝜎𝐤,𝜔_𝑛Δ=−^(sc)𝐤2𝐸_𝐤^(𝑈)𝐸_𝐤^(𝑈)2−𝜉²_𝐤+𝐐−𝐷_eff(𝐤)²𝐸_𝐤^(𝑈)2−𝐸_𝐤^(𝐿)2×𝑖𝜔_𝑛+𝐸_𝐤^(𝑈)−1+Δ^(sc)_𝐤2𝐸_𝐤^(𝑈)×𝐸_𝐤^(𝑈)2−𝜉²_𝐤+𝐐−𝐷_eff(𝐤)²𝐸_𝐤^(𝑈)2−𝐸_𝐤^(𝐿)2𝑖𝜔_𝑛−𝐸_𝐤^(𝑈)−1+Δ^(sc)_𝐤2𝐸_𝐤^(𝐿)𝐸_𝐤^(𝐿)2−𝜉²_𝐤+𝐐−𝐷_eff(𝐤)²𝐸_𝐤^(𝑈)2−𝐸_𝐤^(𝐿)2×𝑖𝜔_𝑛+𝐸_𝐤^(𝐿)−1−Δ^(sc)_𝐤2𝐸_𝐤^(𝐿)×𝐸_𝐤^(𝐿)2−𝜉²_𝐤+𝐐−𝐷_eff(𝐤)²𝐸_𝐤^(𝑈)2−𝐸_𝐤^(𝐿)2𝑖𝜔_𝑛−𝐸_𝐤^(𝐿)−1. (12)

Upon replacing 𝐷_𝐤 and 𝐷^†_𝐤, respectively, by (𝑖Δ_𝐤) and (−𝑖Δ_𝐤) (and Δ^(sc)_𝐤=Δ^†_𝐤^(sc)), as in the pure 𝑑-wave case, the two Fourier coefficients 𝐺^′_𝜎(𝑘,𝜔_𝑛) and Γ_𝜎(𝐤,𝜔_𝑛) lead to the DDW (Δ_𝐤) and DSC gap (Δ^(sc)_𝐤) equations while 𝐺_𝜎(𝐤,𝜔_𝑛) leads to the equation for the chemical potential. With 𝐺^′_𝜎(𝑘,𝜔_𝑛), in view of the definition 𝐷^†_𝐤≡−∑_𝑘′,𝜎𝑈(𝐤,𝐤^′)⟨𝑑^†_𝐤′,𝜎𝑑_{𝐤′+𝐐,𝜎}⟩ and the first equation in (12), we obtain 𝐷^†_𝐤′=_𝐤,𝜎𝑈𝐤^′𝐷,𝐤^†𝐤2𝑞(𝐤)𝐸_𝐤^(𝑈)×𝑆^(𝑈)_𝐤−exp−𝛽𝐸_𝐤^(𝑈)+1⁻¹−𝑆^(𝑈)_𝐤+exp𝛽𝐸_𝐤^(𝑈)+1⁻¹−𝐷^†_𝐤2𝑞(𝐤)𝐸_𝐤^(𝐿)×𝑆^(𝐿)_𝐤−exp−𝛽𝐸_𝐤^(𝐿)+1⁻¹−𝑆^(𝐿)_𝐤+exp𝛽𝐸_𝐤^(𝐿)+1⁻¹, (13) where 𝑆^{(𝑗=𝑈,𝐿)}_𝐤±≡{(𝐸_𝐤^(𝑗)±𝜉_𝐤)×(𝐸_𝐤^(𝑗)±𝜉_𝐤+𝐐)−𝐷_eff(𝐤)²}. 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 ((𝑡₂/𝑡₁) small compared to unity), (13) may be written as 𝐷^†_𝐤′≈(1/2)∑_𝐤,𝜎𝑈(𝐤^′,𝐤){𝐷^†_𝐤/(𝜉²_𝐤+𝐷_eff(𝐤)²)^1/2}. With the model interaction 𝑈(𝐤^′,𝐤)=𝑈₁(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)(cos𝑘^′_𝑥𝑎−cos𝑘^′_𝑦𝑎), where 𝑈₁ is the repulsive coupling strength (𝑈₁>0), this equation assumes a simple form 1≈𝑈₁∑_𝐤[(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)²/{𝜉²_𝐤+(Δ^(PG)₀(0)²+Δ^(sc)₀(0)²)(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)²}^1/2]. For the gap Δ^(sc)_𝐤, on the other hand, we obtain Δ^(sc)_𝐤=−_𝐤′,𝜎𝑉𝐤,𝐤^′Δ^(sc)_𝐤′2𝑞𝐤^′𝐸_𝐤^(𝑈)_′×𝐸_𝐤^(𝑈)2_′−𝑎𝐤^′tanh𝛽𝐸_𝐤^(𝑈)_′2+_𝐤′,𝜎𝑉𝐤,𝐤^′Δ^(sc)_𝐤′2𝑞𝐤^′𝐸_𝐤^(𝐿)_′×𝐸_𝐤^(𝐿)2_′−𝑎𝐤^′tanh𝛽𝐸_𝐤^(𝐿)_′2, (14) where 𝑎(𝐤)≡(𝜉²_𝐤+𝐐+𝐷_eff(𝐤)²). In the zero-temperature case, for a nonnested dispersion, (14) reduces to Δ^(sc)_𝐤1=−2_𝐤′,𝜎𝑉𝐤,𝐤^′Δ^(sc)_𝐤′𝐸_𝐤^(𝑈)_′+𝐸_𝐤^(𝐿)_′×𝑎𝐤1+^′𝐸_𝐤^(𝑈)_′𝐸_𝐤^(𝐿)_′. (15)

The near-nested dispersion yields Δ^(sc)_𝐤≈−(1/2)∑_𝐤′,𝜎𝑉(𝐤,𝐤^′)Δ^(sc)_𝐤′×[√(𝜉²_𝐤′+𝐷_eff(𝐤^′)²)]⁻¹ which is similar to the weak coupling BCS gap equation. With an appropriate attractive interaction 𝑉(𝐤,𝐤^′)=−|𝑉₁|(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)(cos𝑘^′_𝑥𝑎−cos𝑘^′_𝑦𝑎), where 𝑉₁ is the coupling strength, we find that the equation assumes a simple form 1≈|𝑉₁|∑_𝐤[(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)²/{𝜉²_𝐤+(Δ^(PG)₀(0)²+Δ^(sc)₀(0)²)(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)²}^1/2]. With the two gap equations combined, we obtain 11≈2𝑈₁+𝑉₁×_𝐤cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎²×𝜉²_𝐤+Δ^(PG)₀(0)²+Δ^(sc)₀(0)²×cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎^2−1/2. (16)

We also obtain 𝑈₁≈|𝑉₁| as the sum ∑_𝐤[(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)²×{𝜉²_𝐤+(Δ^(PG)₀(0)²+Δ^(sc)₀(0)²)×(cos(𝑘_𝑥𝑎)−cos(𝑘_𝑦𝑎))²}^−1/2] is not zero. Since 𝑈₁ has its origin in the coulombic interaction [9–11], the condition 𝑈₁≈|𝑉₁| 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 𝑈(𝐤,𝐤^′)→𝑈₁cos(2𝜑)cos(2𝜑^′), 𝑉(𝐤,𝐤^′)→−|𝑉₁|cos(2𝜑)cos(2𝜑^′), and squared order parameter Δ_𝐤(𝑇=0)²=(Δ^(PG)₀(𝑇=0)²+Δ^(sc)₀(𝑇=0)²)(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)²→Δ₀(0)²cos²(2𝜑), where 𝜑≡arc(tan(𝑘_𝑦/𝑘_𝑥)) and (𝑘_𝑥,𝑘_𝑦) belong to the first Brillouin zone (BZ), we may write the gap equation (16) as 1≈𝑔𝑘_𝐹0^2𝜋𝑑𝜑×2𝜋₀₊₁𝑑𝜉cos²2𝜑𝜉²+Δ₀(0)/ħ𝜔_𝑐²cos²𝑘2𝜑=𝑔_𝐹0^2𝜋𝑑𝜑2𝜋cos²2𝜑×logΔ₀(0)/ħ𝜔_𝑐²cos²2𝜑+1+1Δ₀(0)/ħ𝜔_𝑐,×cos2𝜑 (17) where the dimensionless quantity 𝑔(𝑘_𝐹)=(𝐷/2)(𝑈₁+|𝑉₁|) and Δ₀(𝑇=0)=(Δ^(PG)₀(𝑇=0)²+Δ^(sc)₀(𝑇=0)²)^1/2. In writing energy integrations in the gap equation, we have assumed an arbitrary energy cut-off “ħ𝜔_𝑐” less than the Fermi energy. This facilitates integration over a length larger than 𝑘_𝐹⁻¹. The quantity 𝐷 with dimensions (Energy)⁻¹,which will be assumed to be a constant (see Figure 16), is the density of energy states. In the weak coupling limit 𝑔(𝑘_𝐹)≪1, for the special situation 𝜑→0, we obtain (Δ₀(0)/ħ𝜔_𝑐)≈2exp(−1/𝑔(𝑘_𝐹)). 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 (Δ₀(0)/ħ𝜔_𝑐) and performing the integration in (17), by discretizing the integral, to obtain 𝑔(𝑘_𝐹). The 2D graph in Figure 3 displays the outcome. We find that the quantity (Δ₀(0)/ħ𝜔_𝑐) is an increasing function of 𝑔(𝑘_𝐹). 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 Δ₀(𝑇=0) for a superconductor of 𝑑-wave variety is available within the BCS framework. Now at all temperatures above 𝑇=0 K, there is a finite possibility of finding electrons in the nonsuperconducting state. At finite temperature, the occupation of the excited one-electron state 𝐸_𝑘=(𝜉²_𝐤+Δ_𝐤(𝑇)²) obeys the Fermi statistics with the Fermi distribution (exp𝛽𝐸_𝑘+1)⁻¹. Equation (17) is, therefore, replaced by 1≈𝑔𝑘_𝐹0^2𝜋𝑑𝜑×2𝜋₀₊₁𝑑𝜉cos²2𝜑√𝜉²+Δ₀/ħ𝜔_𝑐²cos²×Δ2𝜑1−2𝑓𝑇,𝜉,⁰ħ𝜔_𝑐,𝜑, (18) where 𝑓(𝑇,𝜉,(Δ₀/ħ𝜔_𝑐),𝜑)=(exp{𝛽ħ𝜔_𝑐((𝜉²+(Δ₀/ħ𝜔_𝑐)²cos²(2𝜑)))}+1)⁻¹. We assume that Δ₀(𝑇=0)²=(Δ^(PG)₀(𝑇=0)²+Δ^(sc)₀(𝑇=0)²)≈Δ^(sc)₀(𝑇=0)² 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 𝐤 or −𝐤 may be occupied. One can derive the equation for the critical temperature 𝑇_𝑐 readily from above: 2=𝑔𝑘_𝐹0⁺¹1𝑑𝜉𝜉tanh𝜉ħ𝜔_𝑐2𝑘_𝐵𝑇_𝑐. (19)

Figure 3 

The 2D plot of (Δ0/ħ𝜔𝑐) as a function of the dimensionless coupling strength 𝑔(𝑘𝐹) and second-degree polynomial fit.

(a)

(b)

(c)

Figure 4 

(a) A plot of the upper and the lower bands 𝜀𝑘(𝑈,𝐿) in the pure DDW state (doping level 9.94%) along the antinodal cut (−𝜋,𝜋)-(0,𝜋)-(𝜋,𝜋). The back-bending (or saturation) momentum of the dispersion 𝜀𝑘(𝐿) and the Fermi momentum (𝑘𝐹) are indicated. The numerical values, in the units of the first neighbor hopping 𝑡1, are (𝜇/𝑡1)=−0.0183, the PG gap amplitude (Δ(pg)0/𝑡1)=0.0610, and the SC gap amplitude (Δ(sc)0/𝑡1)=0. The hopping parameters are (𝑡2/𝑡1)=0.3925 and (𝑡3/𝑡1)=0.0005. The 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 ±𝜀(́𝐤) in the pure SC state along the antinodal cut (−𝜋,𝜋)-(0,𝜋)-(𝜋,𝜋) indicating perfect electron-hole symmetry (both the upper and lower bands are parabolic). The numerical value of the SC gap amplitude (Δ(sc)0/𝑡1)=0.0150. The hopping parameters values and (𝜇/𝑡1) are the same as above. (c) A plot of the upper bands +𝐸𝑘(𝑈,𝐿) and the lower bands −𝐸𝑘(𝑈,𝐿) in the DDW-DSC state along the antinodal cut (−𝜋,𝜋)-(0,𝜋)-(𝜋,𝜋). The back-bending (or saturation) momentum of the dispersion −𝐸𝑘(𝐿) is indicated by double-headed arrows. This 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 (𝑘) along the antinodal cut) with two positioned at negative energy and two at positive energy for the Fermi energy taken as zero. The parameter values used are (𝜇/𝑡1)=−0.0018, the PG gap amplitude (Δ(pg)0/𝑡1)=0.024, and the SC gap amplitude (Δ(sc)0/𝑡1)=0.015. The hopping parameters are (𝑡2/𝑡1)=0.3913 and (𝑡3/𝑡1)=0.0005.

Equation (19) immediately gives 𝑘_𝐵𝑇_𝑐=0.57ħ𝜔_𝑐exp(−1/𝑔(𝑘_𝐹)). With the aid of data available (for 𝑔(𝑘_𝐹)=1.0 (strong coupling) we have (Δ₀/ħ𝜔_𝑐)=0.5) from Figure 3, we then obtain the ratio (2Δ₀/𝑘_𝐵𝑇_𝑐)≈4.8. This value is quite different from the experimental value [19, 49], where 𝑇_𝑐 is approximately related both to the gap (G) and to the extension of Fermi arcs by 2G (4𝛼/𝜋)=4.3𝑘_𝐵𝑇_𝑐 where the angle 𝛼 spans the half-Fermi arc as measured from the nodal direction. It may be relevant to note that the anomalous pairing 𝑑^†_𝐤,𝜎𝑑^†_{−𝐤−𝐐,−𝜎}=−𝐷^†_𝑘Δ^(sc)_𝐤𝜉_𝐤+𝜉_𝐤+𝐐2𝑞(𝐤)𝐸_𝐤^(𝑈)×tanh𝛽𝐸_𝐤^(𝑈)2+−𝐷^†_𝑘Δ^(sc)_𝐤𝜉_𝐤+𝜉_𝐤+𝐐2𝑞(𝐤)𝐸_𝐤^(𝐿)×tanh𝛽𝐸_𝐤^(𝐿)2 (20) is required to be nonzero for the validity of the results, such as (5) and (6) and the four Bogoluibov bands 𝐸_𝐤(𝑇<𝑇_𝑐)=±𝐸^(𝑈,𝐿)(𝐤), obtained above. This is possible only when the nesting is absent, that is, (𝜉_𝐤+𝜉_𝐤+𝐐)≠0. Thus, the plot in Figure 3 and the result (2Δ₀/𝑘_𝐵𝑇_𝑐)≈4.8 obtained are only qualitatively accurate. This is why we have solved the coupled equations (13) and (14) numerically assuming the specific form for the interactions 𝑈(𝐤,𝐤^′) and 𝑉(𝐤,𝐤^′) given above for the nonnested dispersion together with the equation to determine the chemical potential (𝜇) obtainable from (11). The equation, according to the Luttinger rule, is given by =𝑁1+𝑝^𝑠2⁻¹_𝑘′𝑢_𝐤⁽⁺⁾_′²×exp−𝛽𝐸_𝐤^(𝑈)_′+1⁻¹+𝑢_𝐤⁽⁻⁾_′²×exp𝛽𝐸_𝐤^(𝑈)_′+1⁻¹+𝑣_𝐤⁽⁺⁾_′²×exp−𝛽𝐸_𝐤^(𝐿)_′+1⁻¹+𝑣_𝐤⁽⁻⁾_′²×exp𝛽𝐸_𝐤^(𝐿)_′+1⁻¹, (21) where 𝑝 is the hole-doping level, and 𝑁_𝑠 is the number of unit cells in the 𝐤-space. The 𝐤-summations in these equations, as before, will be replaced by the integration ∫𝑑(𝐤𝑎)𝜌_Fermi(𝐤)→∫^+𝜋_−𝜋𝑑(𝑘_𝑥𝑎)/2𝜋∫^+𝜋_−𝜋𝑑(𝑘_𝑦𝑎)/2𝜋𝜌_Fermi(𝐤), where 𝜌_Fermi(𝐤) stands for the Fermi energy density of states (DOSs). For example, for the doping level 9.94% and temperatures 𝑇=60 K and 20 K, respectively, solving these equations simultaneously, we have found (𝜇/𝑡₁=−0.0018, Δ^(PG)₀/𝑡₁=0.0240, Δ^(sc)₀/𝑡₁=0.0150), and (𝜇/𝑡₁=−0.0009, Δ^(PG)₀/𝑡₁=0.0170, Δ^(sc)₀/𝑡₁=0.0294) for ((𝑡₂/𝑡₁)≈0.4, (𝑡₃/𝑡₁)=0.0005).

The above-mentioned exercise leads to the graphical representations of 𝜀_𝑘^(𝑈,𝐿) in the pure DDW state, ±𝜀(́𝐤) in the pure DSC state, and ±𝐸^(𝑈,𝐿)(𝐤) in the coexistent DDW and DSC states. In Figure 4(a), we have shown the plot of 𝜀_𝑘^(𝑈,𝐿) along the antinodal cut (−𝜋,𝜋)-(0,𝜋)-(𝜋,𝜋), while in Figures 4(b) and 4(c) we have shown the plot of ±𝜀(́𝐤) and ±𝐸^(𝑈,𝐿)(𝐤), respectively. In order to highlight their special features, namely, the particle-hole asymmetry (symmetry) for 𝜀_𝐤^(𝑈,𝐿) and ±𝜀_𝐤^(𝑈,𝐿) (±𝜀(́𝐤)), we note that the optical and electronic phenomena in solids arise from the behavior of electrons and holes (unoccupied states in a filled electron sea). Electron-hole symmetry can often 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, electron-hole symmetry is generally absent because the energy-band structure of the conduction band differs from the valence band. In Figure 4(a), we find easy to notice that particle-hole asymmetry as the upper band is parabolic while the lower band is characterized by dip at (0,𝜋) and the so-called back-bending momentum at (−0.45,𝜋) and (0.45,𝜋). A plot of the upper and the lower bands ±𝜀(́𝐤) in Figure 4(b) indicates perfect electron-hole symmetry as both the upper and lower bands are parabolic. In Figure 4(c), we have shown a plot of the upper bands 𝐸_𝑘^(𝑈,𝐿) and the lower bands −𝐸_𝑘^(𝑈,𝐿) in the DDW-DSC state once again along the antinodal cut (−𝜋,𝜋)-(0,𝜋)-(𝜋,𝜋). The shoulder-type feature of the dispersion −𝐸_𝑘^(𝐿), indicated by double-headed arrows, also exists in the experimental data of Hashimoto et al. [24]. The bands (−𝐸_𝑘^(𝑈,𝐿)) below the Fermi energy are the reflected ones of those above the Fermi energy (𝐸_𝑘^(𝑈,𝐿)). 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 (−𝐸_𝑘^(𝐿)) no more peak at 𝑘_𝐹(0,𝜋) 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 𝑑-wave BCS theory, observed in the superconducting state here have been reported time and again in the ARPES experiments [14, 24, 50–52].

The particle-hole asymmetry of 𝜀_𝐤^(𝑈,𝐿) and ±𝐸_𝐤^(𝑈,𝐿) could also be visualized through the corresponding contour plots in the first Brillouin zone (BZ) shown in Figure 5. In Figures 5(a) and 5(b), respectively, we have shown plots of 𝜀_𝐤^(𝑈) and 𝜀_𝐤^(𝐿). The band-maxima in 𝜀_𝐤^(𝑈) occur at the points (±𝜋,±𝜋) and (0,0) whereas those in 𝜀_𝐤^(𝐿) occur at the points (±𝜋/2,±𝜋/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 (−𝐸_𝐤^(𝑈)) and (−𝐸_𝐤^(𝐿)). Since the bands 𝐸_𝐤^(𝑈,𝐿) above the Fermi energy are the reflected ones of those below the Fermi energy, that is, (−𝐸_𝐤^(𝑈,𝐿)), we have not shown 𝐸_𝐤^(𝑈) and 𝐸_𝐤^(𝐿) here. In fact, the band-maxima in 𝐸_𝐤^(𝑈) occur at the points (±𝜋,±𝜋) and (0,0) whereas those in 𝐸_𝐤^(𝐿) occur at the points (±𝜋/2,±𝜋/2). On the other hand, the band-maxima in (−𝐸_𝐤^(𝑈)) occur at the antinodal points (±𝜋,0) and (0,±𝜋) while the band-minima in (−𝐸_𝐤^(𝐿)) occur at the nodal points (±𝜋/2,±𝜋/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.

(a)

(b)

(c)

(d)

Figure 5 

(i) The contour plots of the upper band 𝜀𝑘(𝑈) (a) and the lower band 𝜀𝑘(𝐿) (b) in the pure DDW state (doping level 9.94%) on the first Brillouin zone (BZ). The wave vector component (𝑘𝑥𝑎) is plotted along the horizontal direction while (𝑘𝑦𝑎) is plotted along the vertical direction. The scale of the plot in (a) is from 0 to 8 whereas in (b) it is from 0 to −2. The band-maxima in (a) occur at the points (±𝜋,±𝜋) and (0,0). The band-maxima in (b) occur at the nodal points (±𝜋/2,±𝜋/2). (ii) The contour plots of the band (−𝐸𝐤(𝑈)) (c) and the band (−𝐸𝐤(𝐿)) (b) in the DDW + DSC state (doping level 9.94%) on the first Brillouin zone (BZ). The scale of the plot in (c) is from 0 to −8 while in (d) it is from 0 to −2. The band-maxima in (c) occur at the antinodal points (±𝜋,0) and (0,±𝜋). The band-minima in (d) occur at the nodal points (±𝜋/2,±𝜋/2). The nodal quasiparticles correspond to the cone-like structure.

3. Thermodynamics and Transport Properties in PG Phase

3.1. Thermodynamic Potential

We wish to obtain now expressions for the entropy and the specific heat in closed form for the DDW + SC state. For this purpose, it is convenient to define a thermodynamic potential Ω(𝜆) in terms of the Hamiltonian 𝐻(𝜆)=𝜆𝐻 where 𝜆 is a variable and 𝐻 is given by (2). One can write Ω (𝜆)−Ω(0)=∫(𝑑𝜆/𝜆)⟨𝐻(𝜆)⟩_𝜆, where Ω(0) is an integration constant and the angular brackets ⟨⋯⟩_𝜆 denote thermodynamic average calculated with 𝐻(𝜆). The system under consideration corresponds to Ω(𝜆=1). Following the Kadanoff-Baym approach [47, 48], one can write a relation between the thermodynamic potential and the spectral functions 𝐴_𝜆(𝑘,𝜎,𝜔), corresponding to the Matsubara propagators 𝐺^𝜆_𝜎(𝐤,𝜔_𝑛), 𝐺^𝜆_𝜎(𝐤+𝐐,𝜔_𝑛), and so forth as Ω(𝜆)(0)=−Ω𝑑𝜆𝜆_𝑘,𝜎^+∞_−∞𝑑𝜔𝑒2𝜋^𝛽𝜔+1⁻¹×𝜔𝐴_𝜆(𝐤,𝜎,𝜔)+𝐴_𝜆(𝐤+𝐐,𝜎,𝜔)+Δ^(sc)_𝐤𝐵_𝜆(𝐤,𝜎,𝜔)+Δ^†(sc)_𝐤𝐶_𝜆(𝐤,𝜎,𝜔), (22) where the spectral functions 𝐴_𝜆(𝐤,𝜎,𝜔), 𝐴_𝜆(𝐤+𝐐,𝜎,𝜔), 𝐵_𝜆(𝐤,𝜎,𝜔), and 𝐶_𝜆(𝐤,𝜎,𝜔) are given by 𝐴_𝜆(𝐤,𝜎,𝜔)=𝑖𝐺^𝜆_𝜎𝐤,𝜔_{𝑛𝑖𝜔𝑛=𝜔+𝑖0+}−𝐺^𝜆_𝜎𝐤,𝜔_{𝑛𝑖𝜔𝑛=𝜔−𝑖0+},𝐴_𝜆(𝐤+𝐐,𝜎,𝜔)=𝑖𝐺^𝜆_𝜎𝐤+𝐐,𝜔_{𝑛𝑖𝜔𝑛=𝜔+𝑖0+}−𝐺^𝜆_𝜎𝐤+𝐐,𝜔_{𝑛𝑖𝜔𝑛=𝜔−𝑖0+},𝐵_𝜆(𝐤,𝜎,𝜔)=𝑖Γ^𝜆_𝜎𝐤,𝜔_{𝑛𝑖𝜔𝑛=𝜔+𝑖0+}−Γ^𝜆_𝜎(𝐤,𝜔_𝑛)_{𝑖𝜔𝑛=𝜔−𝑖0+},𝐶_𝜆(𝐤,𝜎,𝜔)=𝑖𝐹^𝜆_𝜎𝐤,𝜔_{𝑛𝑖𝜔𝑛=𝜔+𝑖0+}−𝐹^𝜆_𝜎𝐤,𝜔_{𝑛𝑖𝜔𝑛=𝜔−𝑖0+}. (23)

Here 𝐺^𝜆_𝜎(𝐤,𝜏)=−⟨𝑇{𝑑_𝐤,𝜎(𝜏)𝑑^†_𝐤,𝜎(0)}⟩_𝜆, Γ^𝜆_𝜎(𝐤,𝜏)=−⟨𝑇{𝑑^†_{−𝐤,−𝜎}(𝜏)𝑑^†_𝐤,𝜎(0)}⟩_𝜆, 𝐹^𝜆_𝜎(𝐤,𝜏)=−⟨𝑇{𝑑_𝐤,𝜎(𝜏)𝑑_{−𝐤,−𝜎}(0)}⟩_𝜆, and 𝐺^𝜆_𝜎(𝐤+𝐐,𝜏)=−⟨𝑇{𝑑_{𝑘+𝑄,𝜎}(𝜏)𝑑^†_{𝑘+𝑄,𝜎}(0)}⟩_𝜆. Besides, 𝐺^𝜆_𝜎(𝐤+𝐐,𝜔_𝑛)=∫^𝛽₀𝑑𝜏𝑒^{𝑖𝜔𝑛𝜏}𝐺^𝜆_𝜎(𝐤+𝐐,𝜏), 𝐹^𝜆_𝜎(𝑘,𝜔_𝑛)=∫^𝛽₀𝑑𝜏𝑒^{𝑖𝜔𝑛𝜏}𝐹^𝜆_𝜎(𝑘,𝜏), and so on. Since we already have calculated 𝐺^𝜆_𝜎(𝐤,𝜔_𝑛) and Γ^𝜆_𝜎(𝐤,𝜔_𝑛), we shall be able to calculate the spectral functions 𝐴_𝜆(𝐤,𝜎,𝜔) and 𝐵_𝜆(𝐤,𝜎,𝜔). To calculate 𝐴_𝜆(𝐤+𝐐,𝜎,𝜔), we use the result 𝐺_𝜎𝐤+𝐐,𝜔_𝑛=𝑢²⁽⁺⁾_𝐤+𝐐𝑖𝜔_𝑛+𝐸_𝐤^(𝑈)−1+𝑢²⁽⁻⁾_𝐤+𝐐×𝑖𝜔_𝑛−𝐸_𝐤^(𝑈)−1+𝑣²⁽⁺⁾_𝐤+𝐐𝑖𝜔_𝑛+𝐸_𝐤^(𝐿)−1+𝑣²⁽⁻⁾_𝐤+𝐐𝑖𝜔_𝑛−𝐸_𝐤^(𝐿)−1, (24) where the coherence factors (𝑢^2(±)_𝐤+𝐐,𝑣^2(±)_𝐤+𝐐) may be written down using (11). In fact, this has already been done below (11). Since in the pure 𝑑-wave case Δ^†(sc)_𝐤=Δ^(sc)_𝐤, we need not calculate 𝐶_𝜆(𝐤,𝜎,𝜔) as this is given by 𝐵_𝜆(𝐤,𝜎,𝜔). In the pure DDW state, we find that the thermodynamic potential is given by the expression Ω𝑇,𝜇=Ω₀−2𝛽⁻¹𝑑(𝐤𝑎)𝜌_Fermi(𝐤)_{𝑗(=𝑈,𝐿)}lncosh𝛽𝜀_𝐤^(𝑗)2,Ω₀=𝑑(𝐤𝑎)𝜌_Fermi(𝐤)_{𝑗(=𝑈,𝐿)}𝜀_𝐤^(𝑗). (25)

Using this expression, we shall calculate the entropy (𝑆) and the electronic specific heat (𝐶_el). However, this requires the calculation of the Fermi energy density of states (DOSs) which we wish to discuss first before we calculate (𝑆,𝐶_el).

3.2. 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 specific heat and the thermal conductivity. In order to calculate the quasiparticle lifetime (QPLT), we need to consider the effect of elastic scattering by impurities. This involves calculation of self-energy Σ(𝐤,𝜔_𝑛), involving the momentum and the Matsubara frequencies 𝜔_𝑛. The self-energy alters the single-particle excitation spectrum in a fundamental way. A few diagrams contributing to the self-energy are shown in Figure 6. The unsmooth lines carry momentum but no energy as the scattering is assumed to be elastic. The 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 𝑉(𝐫=𝐑+𝑧𝐤)=∑_𝑖𝑢(𝐫−𝐫_𝑖), where 𝑢(𝐫−𝐫_𝑖) is the potential due to a single impurity at 𝐫_𝐢=𝐑_𝐢+𝑧𝐤 for a given 𝑧 and 𝐑=𝑥𝐢+𝑦𝐣. The term 𝑢(𝐫−𝐫_𝑖) is expanded in a Fourier series: 𝑢(𝐫−𝐫_𝑖)=∑_𝐪,𝑖𝑢(|𝐪|)exp[𝑖𝐪⋅(𝐫−𝐫_𝑖)]. We first consider only the contribution of Figure 6(a). Assuming the scattering by impurities weak, we may write it as Σ⁽¹⁾(𝐤,𝜔_𝑛)=𝑁_𝑗∑_𝐤′|𝑢(𝐤−𝐤^′)|²𝐺(𝐤^′,𝜔_𝑛)=Σ₀⁽¹⁾(𝐤,𝜔_𝑛)+Σ_𝑒 where Σ₀⁽¹⁾𝐤,𝜔_𝑛=−𝑁_𝑗𝐤′𝑢𝐤−𝐤^′2𝑖𝜔_𝑛×^+∞_−∞𝑑𝜀𝜌(𝜀)×𝑢²_𝐤′𝜔²_𝑛+𝜀_𝐤^(𝑈)2_′⁻¹+𝑣²_𝐤′𝜔²_𝑛+𝜀_𝐤^(𝐿)2_′⁻¹, (26)𝑁_𝑗 is the impurity concentration, 𝑢(𝐤−𝐤^′) characterizes the momentum-dependent impurity potential, and Σ_𝑒<0 is the part of the first-order contribution which redefines the chemical potential (see the first equation in (29) below). To evaluate the integrals in (26), such as ∫^+∞_−∞𝑑𝜀𝜌(𝜀)(𝜔²_𝑛+𝜀_𝐤^(𝑗)2)⁻¹, we assume 𝜌(𝜀)=𝜌₀𝛿(𝜀−𝜀_𝐤^(𝑗)) where we take a broad bandwidth, say, (10𝑡₁)∼1.65 eV which gives 𝜌₀=(10𝑡₁)⁻¹∼0.6 (eV)⁻¹ We, thus, obtain 𝐺(𝐤,𝜔_𝑛)≈−𝜌₀(𝑖𝜔_𝑛)(𝜋/|𝜔_𝑛|), and Σ₀⁽¹⁾𝐤,𝜔_𝑛=−𝑁_𝑗𝜌₀𝑖𝜔_𝑛𝑘′𝑢𝐤−𝐤^′2𝜋𝜔_𝑛. (27)

Figure 6 

A few diagrams contributing to the self-energy. The jagged lines carry momentum but no energy. The total momentum entering each impurity vertex, depicted by a dark circle, is zero. We have assumed that impurities are alike, and distributed randomly. Whereas the first two figures correspond to one impurity vertex, the remaining two correspond to the product of four impurity potentials with nonzero averages. These are the cases where two impurities each give rise to two potentials. Thus, the remaining two figures involve the interference of the scattering by more than one impurity. We have assumed low concentration of impurities, and therefore the figures yield smaller contributions compared to those corresponding to the first two and the other diagrams of the same class involving only one impurity vertex.

We may write the right-hand side of (27) as [−𝑖𝜔_𝑛𝑡₁/(2|𝜔_𝑛|𝜄_𝐤́)], where 𝜄́_𝐤⁻¹≡(1/𝑡₁𝜏_𝐤)=2𝜋𝑁_𝑗𝑡₁⁻¹𝜌₀∑_𝑘′|𝑢(𝐤−𝐤^′)|². Note that 𝜄_𝐤́, which corresponds to the quasiparticle lifetime (QPLT), is expressed as a dimension-less quantity. We model 𝑢(|𝐤−𝐤^′|) by a screened exponential falloff of the form 𝑢(|𝐤−𝐤^′|)=[|𝑢₀|²𝜅²/{|𝐤−𝐤^′|²+𝜅²}]^1/2 to consider the effect of the in-plane impurities, where 𝜅⁻¹ characterizes the range of the impurity potential. The limit 𝜅≫|𝐤−𝐤^′|, 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 (𝑡₁⁻¹𝜌₀|𝑢₀|²≪1), we have 𝜄́_𝐤⁻¹≪1. Upon using the Dyson’s equation, under this condition, the full propagator may be written as 𝐺^(Full)𝐤,𝜔_𝑛≈𝑢²⁽¹⁾_𝑟,𝐤𝑖𝜔_𝑛−𝜀́^(𝑗=+1)_𝑟,𝐤𝑡+𝑖¹4𝜄́_𝐤^{(𝑗=+1)−1}+𝑢²⁽²⁾_𝑟,𝐤𝑖𝜔_𝑛−𝜀́^(𝑗=−1)_𝑟,𝐤𝑡+𝑖¹4𝜄́_𝐤^{(𝑗=−1)−1}, (28) where 𝜀́^(𝑗=±1)_𝑟,𝐤=12𝜉^′_𝐤+𝑗𝑅_𝐤^1/2𝜃cos^𝐤2,𝜉^′_𝐤=𝜀_𝐤−𝜇+Σ_𝑒𝑅_𝐤=𝐸₁(𝐤)⁴+𝐸₂(𝐤)^41/2,tan𝜃_𝐤𝐸=𝜋−²(𝐤)²𝐸₁(𝐤)²,𝐸₁(𝐤)²=𝜉_𝐤+𝐐+2𝑤_𝐤²−𝜉²_𝐤+𝜉^′_𝐤²−𝑡²₁𝜄́_𝐤⁻²4−4Σ_𝑒𝑢²_𝐤𝜀_𝐤^(𝐿)+𝑣²_𝐤𝜀_𝐤^(𝑈),𝐸₂(𝐤)²=𝜉^′_𝐤𝑡₁𝜄_𝐤́−𝑡₁𝜄́_𝐤⁻¹2𝑢²_𝐤𝜀_𝐤^(𝐿)+𝑣²_𝐤𝜀_𝐤^(𝑈),𝑢²⁽¹⁾_𝑟,𝐤=𝜀́^(𝑗=+1)_𝑟,𝐤−𝑢²_𝐤𝜀_𝐤^(𝐿)+𝑣²_𝐤𝜀_𝐤^(𝑈)×𝜀́^(𝑗=+1)_𝑟,𝐤−𝜀́^(𝑗=−1)_𝑟,𝐤⁻¹,𝑢²⁽²⁾_𝑟,𝐤=𝑢²_𝐤𝜀_𝐤^(𝐿)+𝑣²_𝐤𝜀_𝐤^(𝑈)−𝜀́^(𝑗=−1)_𝑟,𝐤×𝜀́^(𝑗=+1)_𝑟,𝐤−𝜀́^(𝑗=−1)_𝑟,𝐤⁻¹. (29)

The QPLT for 𝑗=±1 are given by 1𝜄́_𝐤^(𝑗=±1)=𝜄́_𝐤⁻¹+𝑗2𝑡₁⁻¹𝑅_𝐤^1/2𝜃sin^𝐤2. (30)

We have calculated formal expressions of the propagator 𝐺^(Full)(𝐤,𝜔_𝑛) and reciprocal QPLT (1/𝜄́_𝐤^(𝑗=±1)) above with the inclusion of impurity scattering. The corresponding retarded Green’s function 𝐺^(𝑅)(𝐤,𝑡), in units such that ħ=1, is given by 𝐺^(𝑅)(𝐤,𝑡)=∫^+∞_−∞(𝑑𝜔/2𝜋)exp(−𝑖𝜔𝑡)𝐺^(Full)(𝐤,𝜔) where in the upper half-plane 𝐺^(Full)(𝐤,𝜔_𝑛) is given by (28) while in the lower half-plane 𝐺^(Full)𝐤,𝜔_𝑛≈𝑢²⁽¹⁾_𝑟,𝐤𝑖𝜔_𝑛−𝜀́^(𝑗=+1)_𝑟,𝐤1−𝑖4𝛽𝜄́_𝐤^{(𝑗=+1)−1}+𝑢²⁽²⁾_𝑟,𝐤𝑖𝜔_𝑛−𝜀́^(𝑗=−1)_𝑟,𝐤1−𝑖4𝛽𝜄́_𝐤^{(𝑗=−1)−1}. (31)

Thus, 𝐺^(𝑅)(𝐤,𝜔^′)=∫^+∞_−∞𝑑𝑡exp(𝑖𝜔^′𝑡)𝐺^(𝑅)(𝐤,𝑡) is given by (31) with 𝜔 real. We obtain 𝐺^(𝑅)(𝐤,𝑡)=_{()𝜈=1,2,𝑗=±1}𝑢^2(𝜈)_𝑟,𝐤𝑖exp−𝑖𝜀́^(𝑗)_𝑟,𝐤𝑡𝑡−4𝛽𝜄́_𝐤^(𝑗)𝜃(𝑡), (32) where for 𝜈=1 and 𝜈=2, respectively, are the values of the index 𝜈 which have correspondence with the index 𝑗 equal to +1 and −1. The unit step function 𝜃(𝑡)=∫^+∞_−∞(𝑖𝑑𝜔/2𝜋){exp(−𝑖𝜔𝑡)/(𝜔+𝑖0⁺)}. The electronic excitations in cuprates are thus demonstrably particle-hole mixed quasiparticles in the pseudogap phase with finite lifetime for the states of definite momentum due to the impurity scattering. Using the integral representation of 𝜃(𝑡) above, it is not difficult to show that 𝐺^(𝑅)𝑘,𝜔^′=^+∞_−∞𝑑𝑡exp𝑖𝜔^′𝑡𝐺^(𝑅)(𝑘,𝑡)=_{()𝜈=1,2;𝑗=±1}𝑢^2(𝜈)_𝑟,𝐤𝜔^′−𝜀́^(𝑗)_𝑟,𝐤+𝑖𝑡₁4𝜄́_𝐤^(𝑗)×𝜔^′−𝜀́^(𝑗)_𝑟,𝐤²+𝑡₁4𝜄́_𝐤^(𝑗)2−1. (33)

The dimensionless density of states 𝜌(𝐤,𝜔)≡(−1/2𝜋²𝜌₀)Im𝐺^(𝑅)(𝐤,𝜔), consequently, may be written as 𝜌(𝐤,𝜔)=∑_{𝜈=1,2;𝑗=(±1)}𝜌^(𝑗,𝜈)(𝐤,𝜔), where 𝜌^(𝑗,𝜈)(𝐤,𝜔)=𝑍_𝐤^(𝜈)𝛾_𝐤^(𝑗)×𝜔−𝜀́^(𝑗)_𝑟,𝐤²+𝛾_𝐤^(𝑗)2−1, (34)𝑍_𝐤^(𝜈)=(1/2𝜋²𝜌₀)𝑢^2(𝜈)_𝑟,𝐤, and 𝛾_𝐤^(𝑗)=(𝑡₁/4𝜄́_𝐤^(𝑗)) (the level broa-dening factors). In order to determine the Fermi energy density of states (DOS) 𝜌_Fermi(𝐤), we shall put 𝜔=0 in (34). It is clear that 𝜌_Fermi(𝐤)=_{()𝜈=1,2;𝑗=±1}𝑍_𝐤^(𝜈)𝛾_𝐤^(𝑗)×𝜀́^(𝑗)2_𝑟,𝐤+𝛾_𝐤^(𝑗)2−1. (35)

At this stage, assuming low concentration of impurities, one may include the contributions of all such diagrams which involve only one impurity vertex. This gives the equation to determine the total self-energy Σ(𝐤,𝜔_𝑛): Σ(𝐤,𝜔_𝑛)=𝑁_𝑗∑_𝐪𝑢(𝐪)𝐺^(Full)(𝐤−𝐪,𝜔_𝑛)Γ(𝐤,𝐪,𝜔_𝑛), where the Lippmann-Schwinger equation to determine Γ(𝐤,𝐪,𝜔_𝑛) is Γ(𝐤,𝐪,𝜔_𝑛)=𝑢(−𝐪)+∑_𝐪′𝑢(𝐪^′−𝐪)𝐺^(Full)(𝐤−𝐪^′,𝜔_𝑛)Γ(𝐤,𝐪^′,𝜔_𝑛). This is the 𝑡-martix approximation. Upon using the optical theorem for the 𝑡-matrix [53], one may write Σ(𝐤,𝜔_𝑛)=𝑖ImΓ(𝐤,𝐤,𝜔_𝑛)=−𝑖𝜔_𝑛𝑡₁/(2|𝜔_𝑛|Γ_𝐤́) where Γ́_𝐤⁻¹=2𝜋𝑁_𝑗𝑡₁⁻¹𝜌₀∑_𝐤′|Γ(𝐤,𝐤^′)|². Thus, 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_𝐤⁻¹→Γ́_𝐤⁻¹.

Since 𝐺(𝐤,𝜔_𝑛) and 𝑢(𝐪) are known, one can determine Γ́_𝐤⁻¹ in terms of 𝑢(𝐤). In the limit 𝜅≫|𝐤−𝐤^′|, the disorder potential 𝑢(|𝐪|)≈|𝑢₀|, and, therefore, we obtain Γ(𝐤,𝜔_𝑛)≈|𝑢₀|/(1−|𝑢₀|𝐺(𝐤,𝜔_𝑛)). In view of 𝐺(𝐤,𝜔_𝑛) given above (27), we find that Im Γ(𝐤,𝜔_𝑛)≈−𝜌₀𝜋|𝑢₀|²/(1+𝜌²₀𝜋²|𝑢₀|²). From the equation (𝑖ImΓ(𝐤,𝐤,𝜔_𝑛))=−𝑖𝜔_𝑛𝑡₁/(2|𝜔_𝑛|Γ_𝐤́), we consequently find that Γ́_𝐤⁻¹, in the first approximation, is given by Γ́₀⁻¹=[2𝜋𝑁_𝑗𝑡₁⁻¹𝜌₀|𝑢₀|²/(1+𝜌²₀𝜋²|𝑢₀|²)]. We now replace 𝜄́_𝐤⁻¹ by Γ́₀⁻¹ in (30) and (35). The reciprocal QPLT for the electron and the hole quasiparticles are plotted in Figure 7 as a function of Γ́₀⁻¹ (measure of disorder potential) for Σ_𝑒/𝑡₁=−1.2. The former is assumed to be centered at (±𝜋,0) and (0,±𝜋) whereas the latter at (±𝜋/2,±𝜋/2) of the first Brillouin zone. We find that the former is short-lived compared to the latter for very weak to moderately weak disorder potential. We note that, even though Γ_𝐤́ is found to be 𝐤-independent in the first approximation, the term ±2𝑡₁⁻¹𝑅_𝐤^1/2sin(𝜃_𝐤/2) in (30) will ensure that 1/Γ́_𝐤^(𝑗=±1)={Γ́₀⁻¹+𝑗2𝑡₁⁻¹𝑅_𝐤^1/2sin(𝜃_𝐤/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_𝑒/𝑡₁(=−1.2) redefines the chemical potential. Whereas for Figure 8(a) the disorder potential measure Γ́₀⁻¹=0.05, for Figure 8(b) it is 0.30. The usual Fermi arc (Figure 8(a)) for very low value of Γ́₀⁻¹(Γ́₀⁻¹≪1)=0.05 gets nearly washed out (Figure 8(b)) for higher value Γ́₀⁻¹=0.3 since, as seen in Figure 7, the increase in Γ́₀⁻¹ makes the lifetime of nodal excitations shorter. Thus, the disorder seems to have significant influence on the Fermi energy DOS.

Figure 7 

A plot of the electron and hole reciprocal QPLT as a function of weak disorder potential using (30) for Σ𝑒/𝑡1=−1.2. The former is assumed to be centered at (±𝜋,0) and (0,±𝜋) whereas the latter at (±𝜋/2,±𝜋/2) of the first Brillouin zone. One notices that the former is short-lived compared to the latter for very weak to moderately weak disorder potential.

(a)

(b)

Figure 8 

The contour plots of the Fermi energy DOS on the Brillouin zone (BZ) at 9.94% hole doping for Σ𝑒/𝑡1=−1.2 which redefines the chemical potential. The scale of the plots is from −0.2 to 0.6. In (a) Γ́0−1=0.05 whereas in (b) it is 0.30. The usual Fermi arc scenario in (a) gets nearly washed out in (b) due to increase in the disorder potential.

3.3. Entropy and Specific Heat

The dimensionless entropy per unit cell is given by 𝑆=𝛽²(𝜕Ω/𝜕𝛽). For the pseudogapped (PG) phase (𝑇<𝑇^∗) and the normal phase (𝑇>𝑇^∗), the entropy expressions are 𝑆_PG=2∫𝑑(𝐤𝑎)𝜌_Fermi(𝐤)𝑠_PG(𝐤) and 𝑆_𝑁=2∫𝑑(𝐤𝑎)𝜌_Fermi(𝐤)𝑠_𝑁(𝐤), respectively, where 𝑠_PG(𝐤)=_{𝑗()=𝑈,𝐿}lnexp−𝛽𝜀_𝐤^(𝑗)++1𝛽𝜀_𝐤^(𝑗)+𝛽²𝜕𝜀_𝐤^(𝑗)×𝜕𝛽exp𝛽𝜀_𝐤^(𝑗)+1⁻¹,𝑠_𝑁(𝐤)=_𝑗=1,2lnexp−𝛽𝜀_𝑗(𝐤)++1𝛽𝜀_𝑗(𝐤)+𝛽²𝜕𝜀_𝑗(𝐤)×𝜕𝛽exp𝛽𝜀_𝑗(𝐤)+1⁻¹, (36)(37)𝜀₁(𝐤)=𝜀_𝐤, and 𝜀₂(𝐤)=𝜀_𝐤+𝐐. We calculate the value of the entropy difference between the DDW state and the normal state (Δ𝑆=𝑆_PG(𝑇<𝑇^∗)−𝑆_𝑁(𝑇>𝑇^∗)). The specific heat (𝐶_el=−𝛽(𝜕𝑆/𝜕𝛽)), for the pseudo-gapped (PG) phase (𝑇<𝑇^∗), is 𝐶_el≈2𝑘_𝐵𝑑(𝐤𝑎)𝜌_Fermi(𝐤)_{𝑗()=𝑈,𝐿}𝛽𝜀_𝐤^(𝑗)2×exp𝛽𝜀_𝐤^(𝑗)×exp𝛽𝜀_𝐤^(𝑗)+1⁻². (38)

We have ignored the temperature dependence of the chemical potential above. For the 𝐤-summation purpose, we shall first divide the BZ into finite 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 𝐶_el the momentum dependent density is 𝜌_Fermi(𝐤)∑_{𝑗(=𝑈,𝐿)}(𝛽𝜀_𝐤^(𝑗))²×exp(𝛽𝜀_𝐤^(𝑗))×(exp(𝛽𝜀_𝐤^(𝑗))+1)⁻²) 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 specific heat. The entropy difference Δ𝑆(=𝑆_PG−𝑆_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.

Table 1 

The values of the pseudo-gap (PG) state entropy 𝑆_PG given by (36) (column (ii)) and the normal state entropy 𝑆_𝑁 given by (37) (column (iv)) at the temperatures indicated at the doping level 9.94%. We also have noted the values of 𝐶_el calculated using (38). We identify the pseudo-gap temperature 𝑇^∗ (~200 K) as the one at which the entropy difference Δ𝑆=𝑆_PG−𝑆_𝑁 becomes zero. The entropy is almost a smoothly increasing function of temperature. There appears to be a small discontinuity (not clearly discernable) in entropy at 𝑇^∗ during the passage from the PG state to normal state. We find that there is no discontinuity in the electronic specific heat 𝐶_el either.

The physical quantities, such as specific heat 𝐶_el, electrical resistivity 𝜌, and magnetic susceptibility 𝜒, 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 specific heat, using (38), as a function of decreasing temperature in the PG phase starting from 𝑇^∗≈200 K at a fixed underdoping level ≈9.94%. The dimensionless electronic specific heat displays a non-Fermi liquid behavior with decreasing temperature in Figure 9(b): 𝐶_el≈3.1218−0.9001[(𝑘_𝐵𝑇/𝑡₁)log(𝑘_𝐵𝑇/𝑡₁)]+0.0713[(𝑘_𝐵𝑇/𝑡₁)log(𝑘_𝐵𝑇/𝑡₁)]². The plot above shows that the second-degree polynomial fit (in [(𝑘_𝐵𝑇/𝑡₁)log(𝑘_𝐵𝑇/𝑡₁)]) approximately takes care of the anomalous behavior in 𝐶_el. In the normal phase, 𝐶_el is linear in 𝑇 as shown in Figure 9(c). The Sommerfield constant 𝛾 is found to be 1.7143×10⁻⁴𝑘_𝐵. We find that, for (𝑇−𝑇^∗)/𝑇^∗→0_±, 𝐶_el≈1.7100×10⁻⁴𝑘_𝐵𝑇^∗, that is, there is no discontinuity in 𝐶_el at the PG temperature 𝑇^∗. Within a mean field 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 (NFL) behavior in specific heat which is absent in the normal phase.

(a)

(b)

(c)

Figure 9 

(a) A plot of the entropy in the DDW and normal phases as a function of temperature. We identify the pseudogap temperature 𝑇∗ (~200 K) as the one at which the entropy difference Δ𝑆=𝑆PG−𝑆𝑁 becomes zero. (b) A plot of the dimensionless electronic specific heat in the PG phase as a function of [(𝑘𝐵𝑇/𝑡1)log(𝑘𝐵𝑇/𝑡1)] and a second-degree polynomial fit. (c) A plot of the specific heat (𝐶el) in the normal phase as a function of 𝑇. We have the usual linear dependence of 𝐶el on 𝑇 in the normal phase.

3.4. Transport Properties

We now consider the simplest description of quasiparticle transport [54] in the DDW state in terms of the Boltzmann equation in the relaxation-time approximation. The elastic impurity scattering with the inclusion of the disorder potential leads to a momentum-dependent 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 (𝛾/𝑡₁)∼0.1 independent of momentum. We make the same assumption here in the first approximation and consider a momentum- and energy-independent relaxation time 𝜏 and its dimensionless counterpart 𝜏₀=𝑡₁𝜏/ħ. This 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). The electrical and thermal conductivities may then be expressed as 𝜎_𝑥𝑥=−2𝑒²ℎ𝜏_{0()𝐤,𝑗=𝑈,𝐿}𝜕𝑓_𝐤^(𝑗)𝜕𝜀_𝐤^(𝑗)𝑣^(𝑗)_𝑘,𝑥²,𝜎_𝑥𝑦=−𝐷𝑒²ℎ𝜏²_{0()𝐤,𝑗=𝑈,𝐿}𝜕𝑓_𝐤^(𝑗)𝜕𝜀_𝐤^(𝑗)𝑣^(𝑗)_𝑘,𝑥×𝑣^(𝑗)_𝑘,𝑦𝜕𝑣^(𝑗)_𝑘,𝑦𝜕𝑘_𝑥𝑎−𝑣^(𝑗)_𝑘,𝑥𝜕𝑣^(𝑗)_𝑘,𝑦𝜕𝑘_𝑦𝑎,𝛼_𝑥𝑥=2𝑒²ℎ𝜏₀𝑒𝑇_{()𝐤,𝑗=𝑈,𝐿}𝜀_𝐤^(𝑗)𝜕𝑓_𝐤^(𝑗)𝜕𝜀_𝐤^(𝑗)𝑣^(𝑗)_𝑘,𝑥²,𝛼_𝑥𝑦=𝐷𝑒²𝜏ℎ𝑒𝑇²⁰_{()𝐤,𝑗=𝑈,𝐿}𝜀_𝐤^(𝑗)𝜕𝑓_𝐤^(𝑗)𝜕𝜀_𝐤^(𝑗)𝑣^(𝑗)_𝑘,𝑥×𝑣^(𝑗)_𝑘,𝑦𝜕𝑣^(𝑗)_𝑘,𝑦𝜕𝑘_𝑥𝑎−𝑣^(𝑗)_𝑘,𝑥𝜕𝑣^(𝑗)_𝑘,𝑦𝜕𝑘_𝑦𝑎, (39) where the dimensionless quantity 𝐷≡4𝜋(𝑎/𝑙_𝐵)²(𝑎/𝐿)², 𝑙_𝐵≡(ħ/𝑒𝐵)^1/2 is the magnetic length, ∑_𝑗 runs over 𝑗=(𝑈,𝐿), and 𝜀_𝐤^(𝑗), 𝑓_𝐤^(𝑗), 𝑣^(𝑗)_𝑘,𝛼(𝛼=𝑥,𝑦)=(𝜕𝜀_𝐤^(𝑗)/𝜕𝑘_𝛼𝑎) are the corresponding energies, Fermi functions and velocity components, respectively. The energies 𝜀_𝐤^(𝑗) are expressed in units of the first neighbor hopping 𝑡₁. Equation (39) shows that, in the presence of multiple quasiparticle bands, the elements of the conductivity tensors are simply a superposition of the contributions from the individual bands. The reason for the interband contributions not showing up in (39) is that the spectral functions, appearing in the general quantum Boltzmann equations [54], are replaced by 𝛿 functions at the quasiparticle energies. The spectral densities do appear as bunch of 𝛿 functions as can be seen in the discussions below in (9) in Section 2. We have artificially introduced an intrinsic lifetime broadening though to make these 𝛿 functions appear Lorentzians. Figure 10 shows a clear discontinuity in 𝛼_𝑥𝑥 at 𝑇^∗∼200 K. The dimensionless Lorentz number 𝐿≡𝛼_𝑥𝑥/𝑇𝜎_𝑥𝑥(∼6.7×10⁻⁴) is nearly a constant in the PG phase. The reason for the slight nonconstancy may be ascribed to the total omission of the interband contributions alluded to above. We see that 𝐿≪1 and 𝛼_𝑥𝑥∼1, and 𝜎_𝑥𝑥≫1 (see Table 2) inside the DDW phase. This shows somewhat metallic character of the DDW phase. It indicates that, though the DDW phase is characterized by the gap maxima in the antinodal regions and the zero gap in the nodal regions of the BZ, the electron quasiparticles which inhabit the former region must have a shorter lifetime compared to those inhabiting the latter region (cf. Figure 7). In fact, precisely for this reason the thermal conductivity 𝛼_𝑥𝑥 shows discontinuity in the passage from the PG to normal phase; in PG phase the dominant contributions to 𝛼_𝑥𝑥 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 (±𝜋,0) (see Figure 11(b)). As shown in Figure 11(c), the main contributions to 𝜎_𝑥𝑥 as well come through the states corresponding to Fermi arc. Furthermore, we also find that as we move to the SC phase from the PG phase along the line indicated in Figure 1, there is considerable enhancement in 𝜎_𝑥𝑥 as expected.

Table 2 

The values of the pseudo-gap (PG) state electrical hall conductivity (𝜎_𝑥𝑦), longitudinal conductivity (𝜎_𝑥𝑥), and The Hall angle (𝜃_𝐻) for 𝐵=0.08 Tesla. We find tan𝜃_𝐻≡𝜎_𝑥𝑦/𝜎_𝑥𝑥 ~ 0.01 in the PG phase.

Figure 10 

A plot of the values of the pseudogap (PG) state quasiparticle (dimensionless) thermal conductivity (𝛼PG𝑥𝑥) and the normal state thermal conductivity (𝛼𝑁𝑥𝑥) as a function of temperature (𝑇 in Kelvin) at the doping level 9.94%. The discontinuity in 𝛼𝑥𝑥 is clearly visible at 𝑇∗∼200 K. The dimensionless Lorentz number (𝐿=𝛼𝑥𝑥/𝑇𝜎𝑥𝑥), at the doping level 9.94% in the PG phase, is found to be of ∼6.7×10−4 (indicating that (dimensionless) 𝜎𝑥𝑥≫1 in the PG phase) albeit with slight variation. The variation may be due to the total omission of the interband contributions in (39).

(a)

(b)

(c)

Figure 11 

The contour plots of (a) the pseudogap (PG) state quasiparticle (dimensionless) thermal conductivity (𝛼PG𝑥𝑥) density, (b) the normal state thermal conductivity (𝛼𝑁𝑥𝑥) density, and (c) the electrical conductivity (𝜎PG𝑥𝑥) density on the first BZ at 120 K. The relative scale of the plots in (a) and (b) is from −3 to +2 while in (c) is from 0 to 0.8. The dominant contribution to (𝛼PG𝑥𝑥,𝜎PG𝑥𝑥) come from the states corresponding to the Fermi arc. There is, however, no contribution from the antinodal quasiparticles. In the normal phase, the main contributions to 𝛼𝑥𝑥 are from the portion of the arc close to the antinodal region (±𝜋,0).

The underlying assumption in writing down the expressions for the electrical and the thermal Hall conductivities (𝜎_𝑥𝑦,𝛼_𝑥𝑦) in (39) is that we are in the linear response regime. The invocation of the Landau level (LL) quantization is not necessary in this regime as the magnetic field (𝐵) 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. The normal state dispersion in (10) now appears as 𝜀_𝐤(𝐵)=−2𝑡₁cos𝑘_𝑥𝑎+cos𝑘_𝑦𝑎+𝜑+4𝑡₂cos𝑘_𝑥𝑎cos𝑘_𝑦𝜑𝑎+2−2𝑡₃cos2𝑘_𝑥𝑎+cos2𝑘_𝑦𝑎+𝜑+4𝑡₂, (40) where the vector potential 𝐀 may be assumed to be in the Landau gauge: 𝐀=(^0−𝐵_𝑥⁰). The hopping amplitudes 𝑡_𝑖𝑗, corresponding to the sites 𝐢 and 𝐣, now may assume the form [𝑡_𝑖𝑗exp(𝑎_𝑖𝑗)], where 𝑎_𝑖𝑗=(𝜋/Φ₀)∫^𝐢_𝐣𝐀⋅dL and Φ₀=(ℎ/2𝑒). For the first neighbor hopping, say, corresponding to the sites 𝐢=(𝑎,0) and 𝐣=(𝑎,𝑎), the quantity 𝑎_𝑖𝑗=−(𝜋/Φ₀)∫^(𝑎,0)_(𝑎,𝑎)𝐵𝑥𝑑𝑦=𝜑, where 𝜑=(2𝜋𝑒𝐵𝑎²/ℎ) is the Peierls phase factor. Similarly, for the second neighbor hopping, say, corresponding to the sites 𝐢=(𝑎,0) and 𝐣=(2𝑎,𝑎), the quantity 𝑎_𝑖𝑗=−(𝜋/Φ₀)∫^(𝑎,0)_(2𝑎,𝑎)𝐵𝑦𝑑𝑦=𝜑/2. These explain the reason behind the appearance of 𝜑 and 𝜑/2, respectively, in the first and the second terms of (40). Likewise, the reason for the appearance of 𝜑 in the third term of (40) could be explained. Taking lattice constant =3.8 Å, we find that 𝜑=2.1998×10⁻⁴𝐵 where 𝐵 should be in Tesla. For 𝐵≪1 Tesla, 𝜑≪1. Therefore, a nonzero 𝜑≪1 is not expected to affect the linear dependence of (𝜎_𝑥𝑦,𝛼_𝑥𝑦) on 𝐵, as could be apparently inferred from (26), in any significant way. We indeed find the linear dependence of (𝜎_𝑥𝑦,𝛼_𝑥𝑦) on 𝐵 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 first BZ at 𝑇=127 K, 𝑝=9.94%, and 𝐵=0.08 Tesla, the main contributions to (𝜎_𝑥𝑦,𝛼_𝑥𝑦) 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 [(±𝜋,0),(0,±𝜋)] either. These findings are consistent as long as 𝐵≪1 and the temperature is lower than 𝑇^∗. Therefore, these are very much supportive of the experimental findings of Doiron-Leyraud et al. [55–57] who have detected quantum oscillations in the electrical resistance of underdoped YBCO establishing the existence of a well-defined Fermi surface (FS) with Fermi pockets in the antinodal region when the superconductivity is suppressed by a strong magnetic field. Furthermore, Riggs et al. [58] have observed the oscillations in the specific heat of YBCO-Ortho II samples (in the presence of a magnetic field 𝐵=45T), the same type of YBCO samples investigated by Doiron-Leyraud and coworkers [55–57], confirming the findings regarding Fermi pockets.

Figure 12 

A 2D plot showing the linear dependence of 𝜎𝑥𝑦 on 𝐵 in the PG phase at the temperature 120 K for the doping level ~10%.

(a)

(b)

Figure 13 

The contour plots of (a) the electrical Hall conductivity density (𝜎(PG)𝑥𝑦(𝐤,𝑇,𝐵)). (b) The thermal Hall conductivity density (𝛼(PG)𝑥𝑦(𝐤,𝑇,𝐵)) on the first Brillouin zone in the pseudogap phase at 𝑇=127 K, 𝑝=9.94%, and 𝐵=0.08 Tesla. The scale of the plot in (a) is from −0.06 to 0.06 while that in (b) is from −3 to 3 in arbitrary unit. The main contributions to (𝜎𝑥𝑦,𝛼𝑥𝑦) arise from the density of states corresponding to those portions of the Fermi arc which are not part of the nodal region; there is no contribution from the antinodal region either.

As shown in Table 2, we could find 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 𝑇_𝑐 and those below it in the underdoped region. The longitudinal conductivity 𝜎_𝑥𝑥 varies slowly, while the Hall angle tan𝜃_𝐻≡𝜎_𝑥𝑦/𝜎_𝑥𝑥≪1.

4. Spectral Weight

We now focus our attention on the optical spectral weight [32–40] SW(𝑇)=∫^∞₀Re𝜎_𝑥𝑥(𝜔,𝑇)(𝑑𝜔/𝜋)∝⟨−𝐻_kin⟩, where Re𝜎_𝑥𝑥(𝜔,𝑇) is the real part of the optical conductivity, assuming the direction 𝑥 to be the direction of the current flow when induced by an electromagnetic field in the same direction. The quantity SW(𝑇) provides valuable information on the evolution of the electronic dynamics with temperature [35]. The issue of the constancy and the temperature dependence of the total optical sum above is discussed at length in [34–37]. One may refer to the works of Baeriswyl et al. [36, 37], in particular, to examine the validity of the relation for SW(𝑇) above for discrete lattice models at finite temperature. We shall see, for our single-band system of cuprates, that the optical sum is dependent on the temperature when the full temperature dependence of the single-particle distribution due to the tight-binding band dispersion with the van Hove singularity and the unconventional orderings (DDW and DSC) are consistently accounted for. Our treatment produces a similar temperature dependence in this sum as seen in experiments reported nearly a decade ago [38, 39]. It may be noted that one sometimes introduces a suitable upper cutoff frequency 𝜔́ in place of ∞ in the optical sum above if a problem so requires. For example, the screened plasma frequency 𝜔_𝑝, approximated with the plasma edge in the reflectivity 𝑅(𝜔), 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.

The longitudinal optical conductivity 𝜎_𝑥𝑥(𝜔) due to an applied field 𝐸_𝑥, in the linear response theory, is given by the frequency-dependent current 𝑗_𝑥(𝜔)=𝜎_𝑥𝑥(𝜔)𝐸_𝑥(𝜔). The quantity 𝑗_𝑥(𝜔) may also be expressed in terms the vector potential 𝐴_𝑥(𝜔) of the applied field as 𝑗_𝑥(𝜔)=⟨𝐽_𝑥(𝜔)⟩+Π_{𝑗𝑥,𝑗𝑥}(𝜔)𝐴_𝑥(𝜔), where Π_{𝑗𝑥,𝑗𝑥}(𝜔)=(𝑖/𝑁)∫^∞₀𝑑𝑡𝑒^𝑖𝜔𝑡⟨[𝐽_𝑥(𝑡),𝐽_𝑥(0)]⟩ is the current-current correlation function, and 𝑁 is the number of 𝐤-vectors in the Brillouin zone. Here, 𝐽_𝑥(𝑡) is the current operator in the interaction representation: 𝐽_𝑥(𝑡)=exp(𝑖𝐻₀𝑡)𝐽_𝑥exp(−𝑖𝐻₀𝑡), where 𝐻₀ is the Hamiltonian with 𝐀=0. The conical brackets ⟨⋯⟩ stand for an equilibrium average defined with the Hamiltonian 𝐻₀. For simplicity, one may assume the electron self-energy to be 𝐪-independent or the vector potential to be position-independent. It can then be shown that the vertex corrections in the current-current correlation function vanish for 𝐪→0, due to the odd parity of the current operator. As in [32, 40], the equilibrium average of the current operator ⟨𝐽_𝑥(𝜔)⟩=−𝑆𝐴_𝑥(𝜔), where 𝑆=(𝑁⁻¹∑^′_𝑘𝜀_𝑥𝑥(𝐤)𝑛_𝑘(𝑇)), 𝜀_𝑥𝑥(𝐤) is the second derivative with respect to 𝑘_𝑥 of dispersion 𝜀(𝐤), and 𝑛_𝑘(𝑇) is the momentum distribution function. The 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 𝑗_𝑥(𝜔)=(−𝑆+Π_{𝑗𝑥,𝑗𝑥}(𝜔))𝐴_𝑥(𝜔). Now as noted above, 𝑗_𝑥(𝜔)=𝜎_𝑥𝑥(𝜔)𝐸_𝑥(𝜔). The vector potential 𝐴_𝑥(𝜔) is related to the field component 𝐸_𝑥(𝜔) by 𝐸_𝑥(𝜔)=𝑖(𝜔+𝑖0⁺)𝐴_𝑥(𝜔). It follows that [32, 40] 𝜎_𝑥𝑥(𝜔)=[−𝑆+Π_{𝑗𝑥,𝑗𝑥}(𝜔)]/𝑖(𝜔+𝑖0⁺) where the usual spectral form for Π_{𝑗𝑥,𝑗𝑥}(𝜔) is Π_{𝑗𝑥,𝑗𝑥}=∫𝑞𝑥,𝜔𝑑𝜔^′/𝜋𝜒_{𝑗𝑥,𝑗𝑥}𝑞_𝑥,𝜔^′𝜔^′−𝜔−𝑖0⁺. (41)

Here, the Fourier space current-current correlation function 𝜒_{𝑗𝑥,𝑗𝑥}(𝑞_𝑥,𝜔) is related to the charge correlation function 𝜒_𝜌,𝜌,(𝑞_𝑥,𝜔) in the form 𝜒_{𝑗𝑥,𝑗𝑥}(𝑞_𝑥,𝜔)=(𝜔²/𝑞²_𝑥)𝜒_𝜌,𝜌(𝑞_𝑥,𝜔) through the equation of continuity (𝜕𝜌(𝐫,𝑡)/𝜕𝑡)+∇⋅𝐣(𝐫,𝑡)=0. One may then write 𝑞_𝑥⁻²∫(𝜔𝑑𝜔/𝜋)𝜒_𝜌,𝜌(𝑞_𝑥,𝜔)=∫(𝑑𝜔/𝜋𝜔)𝜒_{𝑗𝑥,𝑗𝑥}(𝑞_𝑥,𝜔). The integral in the right-hand side of this equation may be shown equal to 𝑆 as the corresponding left-hand side is 𝑞_𝑥⁻²𝜔𝑑𝜔𝜋𝜒_𝜌,𝜌𝑞_𝑥,𝜔=𝑞_𝑥⁻²𝑖𝜕𝜕𝑡𝑑𝜔()2𝜋exp−𝑖𝜔𝑡2𝜒_𝜌,𝜌𝑞_𝑥=𝑁,𝜔_1/2𝑞_𝑥⁻²𝑖𝜕𝜌()(𝜕𝑡𝐪,𝑡,𝜌−𝐪,0)_𝑡=0=𝑁^1/2𝑞_𝑥⁻²𝜌(𝐪),𝐻(𝑡),𝜌(−𝐪,0)_𝑡=0, (42) and the double commutator, in the long wavelength limit, is found to be approximately equal to 𝑆. We substitute, for 𝑆, the outcome of this exercise, namely, ∫(𝑑𝜔/𝜋𝜔)𝜒_{𝑗𝑥,𝑗𝑥}(𝑞_𝑥,𝜔)=𝑆, in the result 𝜎_𝑥𝑥(𝜔)=[−𝑆+Π_{𝑗𝑥,𝑗𝑥}(𝜔)]/𝑖(𝜔+𝑖0⁺) above to obtain 𝜎_𝑥𝑥(𝜔)=∫(𝑑𝜔/𝜋𝑖𝜔)𝜒_{𝑗𝑥,𝑗𝑥}(𝐪,𝜔)/(𝜔+𝑖0⁺). This eventually yields SW(𝑇)=𝑁⁻¹∑^′_𝑘𝜀_𝑥𝑥(𝐤)𝑛_𝑘(𝑇). 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 (𝛾/𝑡₁)∼0.1 independent of momentum.

As the outline of the derivation above shows, the spectral weight SW(𝑇) 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 𝑥 to be the direction of the current flow when induced by an electromagnetic field in the same direction. For ℏ=1, we are, thus, led to SW(𝑇)=𝜋𝑒²𝑁_𝑠^′_𝑘𝜀_𝑥𝑥(𝐤)𝑛_𝑘(𝑇). (43)

We slightly modify (43) replacing the 𝐤-summation above by ∫^+𝜋_−𝜋𝑑(𝑘_𝑥𝑎)/2𝜋∫^+𝜋_−𝜋(𝑑(𝑘_𝑦𝑎)/2𝜋)𝐴(𝐤,𝜔), where 𝐴(𝐤,𝜔), given below (11), is the density of states (DOSs) or spectral function (SF) obtained using the Matsubara propagators with poles at ±𝐸_𝐤^(𝑈,𝐿). The Fermi energy DOS corresponds to 𝜔=0. The 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(𝑇), the coupled gap equations for 𝑖Δ_𝑘 and Δ^(sc)_𝑘 together with the equation to determine the chemical potential (𝜇) of the fermion number have been solved in a consistent manner with the pinning of the van Hove singularity (vHS) close to 𝜇.

We consider the quantity 𝐹(𝜔)=∫^+𝜋_−𝜋𝑑(𝑘_𝑥𝑎)/2𝜋∫^+𝜋_−𝜋𝑑(𝑘_𝑦𝑎)/2𝜋∑_𝛼𝐴(𝐤,𝜔)𝜀_𝑥𝑥(𝐤)𝑛_𝑘,𝛼, where the subscript “𝛼” stands for the conduction band. In Figures 14(a) and 14(b) above, we have contour-plotted the density ∑_𝛼𝐴(𝐤,𝜔)𝜀_𝑥𝑥(𝐤)𝑛_𝑘,𝛼 in the SC phase and the PG phase, respectively, whereas in Figure 14(c) we have plotted the quantity 𝐹(𝜔) as a function of 𝜔 for 𝑇=60 K < 𝑇_𝑐∼100 K. We find 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),  𝐤-integration is necessary. For the integration purpose, we first divide the BZ into finite 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. The sum is then divided by the number of patches. We have generated these values through a surface plot. The gap amplitude at 60 K is (Δ⁽⁰⁾_sc/𝑡₁)∼0.01. The integral 𝐹(𝜔) is a decreasing function for energy larger than that corresponding to the gap amplitude. This is supportive of the fact that DSC of the hole-doped cuprates corresponds to the strong coupling superconductivity.

(a)

(b)

(c)

Figure 14 

(a) The contour plot of the density ∑𝛼𝐴(𝐤,𝜔)𝜀𝑥𝑥(𝐤)𝑛𝑘,𝛼, where the subscript “𝛼” stands for the conduction band, on the first BZ for the SC phase at 60 K. The gap amplitude (Δ(0)sc/𝑡1)∼0.01. (b) The contour plot of the density ∑𝛼𝐴(𝐤,𝜔)𝜀𝑥𝑥(𝐤)𝑛𝑘,𝛼 on the first BZ for the PG phase at 120 K. The gap amplitude (Δ(0)PG/𝑡1)∼0.07. 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. (c) A 2D plot of 𝐹(𝜔) at 𝑇=60 K as a function of energy in units of the first neighbor hopping. The integral 𝐹(𝜔) is a decreasing function for energy larger than that corresponding to the gap amplitude.

5. Concluding Remarks

In this communication, we have assumed that the SC order parameter is such that it is linked with an attractive interaction 𝑉(𝐤,𝐤^′)=−|𝑉₁|(cos𝑘_𝑥𝑎−cos𝑘_𝑦𝑎)(cos𝑘^′_𝑥𝑎−cos𝑘^′_𝑦𝑎), where 𝑉₁ is the coupling strength, that produces the DSC pairing given approximately by Δ^(SC)_𝐤=−∑_𝐤′𝑉(𝐤,𝐤^′)Δ^(SC)_𝐤′×[𝜀²_𝐤′+Δ²_𝐤′+Δ^(SC_𝐤⁾²_′]⁻¹. We have assumed that a combination of e-e and e-bm interactions will lead to a 𝑑-wave gap Δ^(SC)_𝑘. It must be clarified that our effort 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.

The angle-resolved photoemission spectroscopy (ARP ES), which is a valuable tool in the toolkit of experimentalists for studying the PG phase, shows Fermi arcs [15–20], below the characteristic temperature 𝑇^∗, centered along the zone diagonal instead of the expected Fermi surface (FS). Though 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 [15–17] 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 𝑑-wave behavior along the FS. In contrast, other group of authors [18–20, 24] claim that arcs are associated to a pairing which is distinct from that corresponding to SC. These 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 single-particle excitation spectrum of the DDW state involves the usual fermionic particle (𝑝) and hole (ℎ) pairing/mixing in the singlet channel with an associated energy scale of roughly 20 meV, the coexisting and competing DDW and DSC states correspond to the well-known Bogoliubov quasiparticles with an associated energy scale of nearly 10 meV. Most importantly, from our viewpoint, the passage to the PG state from the normal state, upon cooling at a fixed underdoping level, corresponds to a nonsharp thermal transition rather than a smooth crossover as is suggested by the authors who advocate the preformed pairing scenario. The PG phase, as we have shown in Section  3.3, is characterized by the non-Fermi liquid (NFL) behavior. This NFL behavior persists in the superconducting DDW phase.

We have shown above (see Figure 15) a plot of the dimensionless electronic specific heat (𝐶_el) in the SC and PG phases as a function of [(𝑘_𝐵𝑇/𝑡₁)log(𝑘_𝐵𝑇/𝑡₁)] and a second-degree polynomial fit. The details of this part of the investigation will be reported elsewhere. The anomalous temperature dependence of 𝐶_el in Figure 15 is indicative of the non-Fermi liquid (NFL) behavior persisting even in the DDW + DSC phase. The discontinuity in 𝐶_el occurs roughly at 𝑇≈100 K due to the passage from the PG phase to SC phase at the fixed doping level 9.94%. This allows us to infer that the critical temperature (𝑇_𝑐) of the superconducting transition for a generic hole-doped cuprate is approximately 100 K.

Figure 15 

A plot of the dimensionless electronic specific heat (𝐶el) in the DSC and DDW phases as a function of [(𝑘𝐵𝑇/𝑡1)log(𝑘𝐵𝑇/𝑡1)] and a second-degree polynomial fit (𝐶el≈ −14.5830 + 3.9355[(𝑘𝐵𝑇/𝑡1)log(𝑘𝐵𝑇/𝑡1)]−0.0136[(𝑘𝐵𝑇/𝑡1)log(𝑘𝐵𝑇/𝑡1)]2). The anomalous temperature dependence of 𝐶el is indicative of the non-Fermi liquid (NFL) behavior.

(a)

(b)

Figure 16 

(a) The plot of the spectral function in the pseudogap phase (doping level 9.94%) at 𝜔=0 (or, the Fermi energy density of states) on the first Brillouin zone (BZ) for (𝜇/𝑡1)=−0.0189, (Δ(pg)0/𝑡1)=0.0707, and (Δ(sc)0/𝑡1)=0. The hopping parameters are (𝑡2/𝑡1)=0.3925 and (𝑡3/𝑡1)=0.0005. (b) The plot of the spectral function in the superconducting phase (doping level 9.94%) at 𝜔=0 (or, the Fermi energy density of states) on the first Brillouin zone (BZ) for (𝜇/𝑡1)=−0.00183, (Δ(pg)0/𝑡1)=0.024, and (Δ(sc)0/𝑡1)=0.015. The hopping parameters are (𝑡2/𝑡1)=0.39125 and (𝑡3/𝑡1)=0.0005. The intrinsic lifetime broadening (𝛾/𝑡1) has been assumed to be 0.1.

We have mentioned below (11) that the spectral function 𝐴(𝐤,𝜔) in the DDW + DSC phase is given by a sum of 𝛿 functions at the quasiparticle energies: 𝐴(𝐤,𝜔)=2𝜋_𝑗=±1𝑢_𝐤^2(𝑗)𝛿𝜔+𝑗𝐸_𝐤^(𝑈)+𝑣_𝐤^2(𝑗)𝛿𝜔+𝑗𝐸_𝐤^(𝐿). (44)

We simply replace the 𝛿 functions here by Lorentzians with an assumed intrinsic lifetime broadening (𝛾/𝑡₁)∼0.1 as was done in Section 2. From the expression for spectral density, putting put 𝜔=0, we then obtain 𝜌_Fermi(𝐤)=12𝜋𝜌_0(±)𝑗=𝑢_𝐤^2(𝑗)𝛾𝑡₁×𝐸_𝐤^(𝑈)2+𝛾𝑡₁²⁻¹+𝑣_𝐤^2(𝑗)𝛾𝑡₁×𝐸_𝐤^(𝐿)2+𝛾𝑡₁²⁻¹. (45)

This leads to the usual Fermi arc picture on the first Brillouin zone in the PG phase and nearly a constant in the SC phase. In Figure 16, we have depicted the plots of the spectral density in the PG and SC phases assuming the above-mentioned value of the intrinsic lifetime broadening. The plot in Figure 16(b) validates the approximation of the Fermi energy density of states (DOS) by a constant we made while seeking solutions of the gap equations analytically in the SC phase in (17).

The authors, who link PG to preformed pair scenario, advocate the existence and relevance of superconducting fluctuations (SCF) with short correlation lengths in the underdoped cuprates for transport properties. In particular, the survival of a large Nernst signal [3, 4] up to temperatures much larger than the superconducting transition temperature 𝑇_𝑐 in the pseudogap region has been interpreted as the evidence for such SCFs. The fluctuations are supposed to correspond to vortex-like phase fluctuations with a Beresinkii-Kosterlitz-Thouless (BKT) character due to the quasi-two-dimensional (2D) nature of the system [4]. In our mean field approach neither there is such superconducting-fluctuations nor there are vortices in the normal state, as in the preformed pair scenario, supporting a Nernst signal. In a future work, our aim would be to look into this aspect.