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Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2010, Article ID 928419, 11 pagesdoi:10.1155/2010/928419

Research Article

Investigation of a CDDW Hamiltonian to Explore Possibility ofMagneto-Quantum Oscillations in Electronic Specific Heat ofHole-Doped Cuprates

Partha Goswami1 and Manju Rani1, 2

1 D.B.College, University of Delhi, Kalkaji, New Delhi 110019, India2 Department of Electronic Science, University of Delhi South Campus, New Delhi 110021, India

Correspondence should be addressed to Partha Goswami, physics [email protected]

Received 3 February 2010; Revised 31 July 2010; Accepted 25 August 2010

Academic Editor: P. Guptasarma

Copyright © 2010 P. Goswami and M. Rani. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We investigate a chiral d-density wave (CDDW) mean field model Hamiltonian in the momentum space suitable for thehole-doped cuprates, such as YBCO, in the pseudogap phase to obtain the Fermi surface (FS) topologies, including the anisotropyparameter(ε) and the elastic scattering by disorder potential (|v0|). For ε = 0, with the chemical potential μ = −0.27 eV for 10%doping level and |v0| ≥ |t| (where |t| = 0.25 eV is the first neighbor hopping), at zero/non-zero magnetic field (B), the FS on thefirst Brillouin zone is found to correspond to electron pockets around antinodal regions and barely visible patches around nodalregions. For ε /= 0, we find Pomeranchuk distortion of FS. We next relate our findings regarding FS to the magneto-quantumoscillations in the electronic specific heat. Since the nodal quasiparticle energy values for B = 0 are found to be greater than μ for|v0| ≥ |t|, the origin of the oscillations for nonzero B corresponds to the Fermi pockets around antinodal regions. The oscillationsare shown to take place in the weak disorder regime (|v0| = 0.25 eV) only.

1. Introduction

In the recent past, extensive theoretical [1–9] and experi-mental efforts [10–20] have been carried out to understandthe pseudogap (PG) phenomenon in the normal state ofthe underdoped cuprates. These enterprises have led to sub-stantial understanding of the Fermi surface (FS) topologiesof the material. The noteworthy aspect of the PG state ofthe hole-doped cuprates, such as YBCO, is that it could becharacterized with a variety of coexisting/competing orders,as have been reported by previous workers [1–11, 21]. Thetheorists [1–9] among them have put forward explanationsof the (experimentally) observed facts/anomalies [11–20, 22]in various physical properties, which carry the signature ofthe complexities in PG state, with high degree of success. Inthis communication, starting with a chiral d-density wave(CDDW) [1, 2] mean field Hamiltonian Hd+id in momentumspace involving suitable dispersion with hopping integrals

(ti j) and incorporating the effect of elastic scattering bydisorder potential (|v0|) in the Fourier coefficient of single-particle Green’s function (defined by Hd+id), we show thatthe reasonably well-developed Fermi pockets in the antinodalregions and barely visible (hole-like) patches in the nodalregions of FS, at zero/nonzero magnetic field, can beobtained from the energy eigenmodes of the matrix in Hd+id

for |v0| ≥ |t| and |t′/t| = 0.4 where t(=0.25 eV) and t′,respectively, are the first neighbor and the second neighborhopping integrals. It must be mentioned that the appearanceof the pockets depends crucially on the ratio |t′/t| (seeFigure 2(e)). The angle resolved photo-emission spectroscpic(ARPES) studies [23–28] (including the vacuum ultraviolet(VUV) laser-based ARPES [29]) of cuprates, where theexperimental observations roughly correspond to the so-called “maximal intensity surface” explained in [25], havenot shown the evidence of the existence of the Fermipockets so far. Doiron-Leyraud et al. [13–15], however, have

2 Advances in Condensed Matter Physics

detected quantum oscillations in the electrical resistanceof underdoped YBCO establishing the existence of a well-defined FS with Fermi pockets in the antinodal region whenthe superconductivity is suppressed by a magnetic field.Furthermore, Riggs et al. [12] have observed the oscillationsin the specific heat of YBCO-Ortho II samples (in thepresence of a magnetic field B = 45 T), the same typeYBCO samples were investigated by Doiron-Leyraud et al.[13–15], confirming the findings regarding Fermi pockets.The aims of the present theoretical investigation are (i)to show the existence of Fermi pockets in the antinodalregions and (ii) to explore the possibility of approximate(1/B)-oscillations in the specific heat in order to verify thefinding of Riggs et al. [12]. While we succeed in showingthe existence of the Fermi pockets (see Figures 2 and 3) forthe weak (|v0| = 0.25 eV) as well as the strong disorder(|v0| = 1 eV) regimes, the quantum oscillations are shownto be possible for B ≥ 17 T in the weak disorder regimeonly. Our reason for the identification of the PG state withthe CDDW state rather than the well-known [3, 4] d-densitywave (DDW) state is that the CDDW ordering offers atheoretical explanation [5] of the nonzero polar Kerr effectobserved recently in YBCO by Xia et al. [11]. Also, since theexperimental signature of nematic order has been observedrecently in cuprates in neutron scattering experiments [10],it is felt that the hopping anisotropy [30], which captures theelectronic nematicity at the mean-field level, ought to be apart of the ongoing investigation to be dealt with in future.

The paper is organized as follows. In Section 2 wepresent the mean field Hamiltonian Hd+id in momentumspace involving in-plane hopping anisotropy. In Section 3we discuss the elastic scattering by impurities and relate itto the issue of the Fermi pockets as this occupies the centrestage [31] in the magneto-quantum oscillation context. InSection 4 we discuss the issue of the magneto-quantumoscillations in electronic specific heat in detail. The paperends in Section 5 with the concluding remarks.

2. Model Hamiltonian

For a magnetic field applied in z-direction, we consider anormal state tight-binding energy dispersion involving t, t′,and t′′ which are, respectively, the hopping elements betweenthe nearest, next nearest (NN), and NNN neighbours:

εk,N (B) = −2(tx cos(kxa) + ty cos

(kya + ϕ

))

+ 4t′ cos(kxa) cos(kya +

ϕ

2

)

− 2(t′′x cos 2kxa + t′′y cos 2

(kya + ϕ

))

+ �

(N +

(12

))ωc.

(1)

Here the Landau level(LL) index N = 0, 1, . . . and “a” isthe lattice constant (of YBCO). The vector potential A isassumed to be in the Landau gauge: A = (0 − Bx 0). Thequantity ωc = eB/m∗ is the cyclotron frequency where m∗

is the effective mass of the electrons. The Zeeman splitting

has not been taken into account. Following Hackl and Vojta[30], we have introduced a hopping anisotropy parameter ε,such that the hopping elements obey tx,y = (1 ± ε/2)t andt′′x,y = (1± ε/2)t′′. For ε /= 0, the lattice rotation symmetry isspontaneously broken. In the numerical calculations belowwe take t as an energy unit, where t = 0.25 eV, t′ = 0.4t,and t′′ = 0.0444t. These values are the same as in [3].Furthermore, we shall consider the value of the chemicalpotential (μ) of the fermion number to be (−0.27 eV)for 10% hole doping [3]. In the presence of a vectorpotential A, the hopping amplitudes ti j , corresponding tothe sites i and j, assume the form [ti j exp(ai j)], where ai j =(π/Φ0)

∫ ij A.dl and Φ0 = (h/2e). For the first neighbor

hopping, say, corresponding to the sites i = (a, 0) and

j = (a, a), the quantity ai j = −(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 tothe sites i = (a, 0) and j = (2a, a), the quantity ai j =−(π/Φ0)

∫ (a,0)(2a,a) Bydy = ϕ/2. These explain the reason

behind the appearance of ϕ and ϕ/2, respectively, in the firstand the second terms of (1). Likewise, the reason for theappearance of ϕ in the third term of (1) could be explained.

In the second quantized notation, the Hamiltonian (withindex j = (1, 2) below corresponding to two layers of YBCO)for the chiral d + id density-wave state, together with theanisotropy in the hopping parameters, in the presence ofmagnetic field (B) can be expressed as

Hd+id(B) =∑

k,N ,σ ,i=1,2

Φ(i)†k,σ EN (k,B)Φ(i)

k,σ , (2)

where Φ(i)†k,σ = (d†(1)

k,σ d†(1)k+Q,σ d

(2)†k,σ d†(2)

k+Q,σ) and EN (k,B)=[εUk,N (B)I4×4 + ζk(B).α]. Here α = (α1 α2 α3 α4) with

αi =(σi 00 σi

)(i = 1, 2, 3), α4 =

(0 I2×2

I2×2 0

), (3)

I4×4 and I2×2, respectively, are the 4×4 and 2×2 unit matrices,σi are the Pauli matrices, and ζk(B) = (−χk −�k ε

Lk (B) tk)—

a four-component vector. Here the chiral order parameter

[1], Dk exp(iθk), is given by Dk = (χ2k + �2

k)1/2

and cotθk =(−χk/�k) with

χk = −(χ0

2

)sin(kxa) sin

(kya

),

�k =(�0(T)

2

)(cos kxa− cos kya

).

(4)

We consider the simplest form of the modulation vector,Q = (±π,±π). The quantity tkis momentum conservingtunneling matrix element which for the tetragonal structureof cuprates is given by tk = (t0/4)(cos kxa−−cos kya)2. Asin [3], we take t0 = 0.032t. The reader may note that in (2)we have disregarded the LL mixing completely. The energyeigenvalues of EN (k,B) are

E( j,ν)N (k,B) =

[εUk, N (B) + jwk,N (B) + νtk

], (5)

Advances in Condensed Matter Physics 3

where εUk, N (B) = (εk, N (B) + εk +Q,N (B))/2, εLk, N (B) =(εk,N (B) − εk+Q,N (B))/2 and wk,N (B) = [(εLk, N (B))

2+ D2

k]1/2.

Here j is equal to (±1) with j = +1 corresponding to theupper branch (U) and j = −1 to the lower branch (L);for a given j, ν = ±1. For LL index N = 0, 1, we shall

have twofold splitting of E( j,ν)N (k,B) for a given ( j, ν). With

these eigenvalues, we obtain the noninteracting Matsubarapropagator

GN (k,ωn) =∑

ν = ±1

{V (U ,ν)2k,N

(iωn − E(U ,ν)

N (k,B))−1

+V (L,ν)2k,N

(iωn − E(L,ν)

N (k,B))−1

}.

(6)

The quasiparticle coherence factors (V (U ,ν)2k,N , V (L,ν)2

k,N ) aregiven by the expressions

V (U ,ν)2k,N =

(14

)[1 +

(εLk,N

wk,N

)],

V (L,ν)2k,N =

(14

)[1−

(εLk,N

wk,N

)].

(7)

The magnetic field dependence of these factors arisesthrough εLk,N (B). At hole doping level ∼10%, we find that thepseudogap (PG) temperature T∗ ∼155 K. We have assumedthe value �0(T < T∗) = 0.0825 eV = 0.3300 t inthe vicinity of T∗ and (χ0/�0(T < T∗))2 = 0.0025. Weshall now consider the effect of the elastic scattering byimpurities on the Fermi surface topology and search forthe Fermi pockets. This is an important issue as withoutthese pockets the Onsager relation [31] does not allow oneto investigate magneto-quantum oscillations. The impuritypotential/disorder with finite range not only has drasticeffects on the Fermi surface (FS) topology but also will beseen to affect the density of states at Fermi energy, relevantfor the calculation of physical properties, in the followingsections.

3. Elastic Scattering by Impurities andFermi Energy Density of States

The effect of elastic scattering by impurities involves thecalculation of self-energy Σ(k,ωn), in terms of the momen-tum and the Matsubara frequencies ωn, which alters thesingle-particle excitation spectrum in a fundamental way. Afew diagrams contributing to the self-energy are shown inFigure 1. The wiggly lines carry momentum but no energy asthe scattering is assumed to be elastic. The total momentumentering each impurity vertex, depicted by a slim ellipse,is zero. We assume that impurities are alike, distributedrandomly, and contributea potential term V(r = R + zk) =

∑i u(r − − ri), where u(r − ri ) is the potential due to a

single impurity at ri = Ri + zk for a given z and R =xi + yj. The potential term u(r− ri) is expanded in a Fourierseries V(r) = ∑

q,i V(|q|) exp[iq · (r − − ri )]. We firstconsider only the contribution of Figure 1(a). Assuming thescattering by impurities weak, as in [32], we may write it as

Σ(1)(k,ωn) = NjΣk′|V(k− k′)|2GN (k

′,ωn) = ∑(1)

0 (k,ωn) +Σe where

Σ(1)0 (k,ωn) = −Nj

∑

k′, ν = ±1

∣∣∣V(

k− k′)∣∣∣2

(iωn)∫ +∞

−∞dερ(ε)

×[V (U ,ν)2k,N

(ωn

2 + E(U ,ν)N (k,B)2

)−1

+V (L,ν)2k,N

(ω2n + E(L,ν)

N (k,B)2)−1

],

(8)

Nj is the impurity concentration, V(k − k′) characterizes

the momentum-dependent impurity potential, and Σe isthe part of the first-order contribution which can beshown to be independent of k and ωn for k close toFermi momentum. To evaluate the integrals in (8), such

as∫ +∞−∞ dερ(ε)(ω2

n + E( j,ν)N (k,B)2)

−1, we assume that ρ(ε) =

ρ0δ(ε − E( j,ν)N (k,B)), where we take a broad band-width, say,

(10 t) ∼ 2.5 eV which gives ρ0 ∼ 0.4 (eV)−1. We thus obtain

GN (k,ωn) ≈ −ρ0(iωn)(

π

|ωn|)

, (9a)

Σ(1)0 (k,ωn) = −Njρ0(iωn)

∑

k′

∣∣V(k− k′)∣∣2(

π

|ωn|). (9b)

We may write the right-hand side of (9b) as

[−iωn/(2|ωn|τk)], where (1/τk) = 2πNjρ0Σk′ |V(k− k′)|2.

Note that τk , which corresponds to quasiparticle lifetime(QPLT), is expressed in reciprocal energy units. Upon usingDyson’s equation, the full propagator may be written as

G(Full)N (k, ωn) ≈∑ν=±1{D/E}, where

D =[V (U ,ν)2k,N

(iωn − E(L,ν)

N

)+ V (L,ν)2

k,N

(iωn − E(U ,ν)

N

)],

E = (iωn)2 − (iωn)(E(U ,ν)N + E(L,ν)

N −(

i

2τk

)+ Σe

)

+{E(U ,ν)N E(L,ν)

N +(−(

i

2τk

)+ Σe

)

×(V (U ,ν)2k,N E(L,ν)

N + V (L,ν)2k,N E(U ,ν)

N

)}.

(10)


We have dropped the argument part from the single-particleexcitation spectra above for convenience. The roots of theequation E = 0 are

iωn =⎧⎨⎩

(E(U ,ν)N + E(L,ν)

N − (i/2τk) + Σe

)

2

⎫⎬⎭

±⎛⎝R(ν)

k,N1/2

2

⎞⎠⎛⎝cos

⎛⎝θ(ν)

k,N

2

⎞⎠− i sin

⎛⎝θ(ν)

k,N

2

⎞⎠⎞⎠,

R(ν)k,N =

[E(ν)2

1 + E(ν)22

]1/2, tan

(θ(ν)k,N

)= E(ν)

2

E(ν)1

,

E(ν)1 =

(E(U ,ν)N − E(L,ν)

N

)2+ Σe

2

−(

14τ2

k

)+ 2

(E(U ,ν)N + E(L,ν)

N

)Σe

− 4Σe

(V (U ,ν)2k,N E(L,ν)

N + V (L,ν)2k,N E(U ,ν)

N

),

E(ν)2 =

⎧⎨⎩

(E(U ,ν)N + E(L,ν)

N + Σe

)

τk

⎫⎬⎭

−(

2τk

)(V (U ,ν)2k,N E(L,ν)

N + V (L,ν)2k,N E(U ,ν)

N

).

(11)

It follows that the denominator E of GFullN (k, ωn) may

be written as the product of two factors (iωn − (α(+)ν,N +

iβ(+)ν,N ))(iωn − (α(−)

ν,N + iβ(−)ν,N )), where

α( j=±)ν,N =

⎧⎨⎩

(E(U ,ν)N + E(L,ν)

N + Σe

)

2

⎫⎬⎭±

⎛⎝R(ν)

k,N1/2

2

⎞⎠ cos

⎛⎝θ(ν)

k,N

2

⎞⎠,

β( j=±)ν,N = −

⎧⎨⎩(

14τk

)±⎛⎝R(ν)

k,N1/2

2

⎞⎠ sin

⎛⎝θ(ν)

k,N

2

⎞⎠⎫⎬⎭.

(12)

In view of (12), GFullN (k,ωn) ≈∑ν=±1{D/E}may be written

as

G(Full)N (k,ωn) =

∑

ν=±1

V (+,ν)2ren,k,N

⎡⎣iωn − ε(+,ν)

r N + i

⎛⎝ 1

4τ(+,ν)k,N

⎞⎠⎤⎦−1

+ V (−,ν)2ren,k,N

⎡⎣iωn − ε(−,ν)

r N + i

⎛⎝ 1

4τ(−,ν)k,α,N

⎞⎠⎤⎦−1

,

(13)

where the superscript j = (±),

ε( j,ν)r N = α

( j)ν,N ,

(1/τ

( j,ν)k,N

)={

(1/τk)±(R(ν)k,N

1/2)

sin(θ(ν)k,N/2

)},

(14)

V( j,ν)2ren,k,N =

(14

)(1± δk,N

), δk,N =

⎛⎝δ(1)

k,N

δ(2)k,N

⎞⎠,

δ(1)k,α,N =

[(α(+)

ν,N + iβ(+)ν,N

)+(α(−)

ν,N + iβ(−)ν,N

)

−2(V (+,ν)2k,N E(−,ν)

N + V (−,ν)2k,N E(+,ν)

N

)],

δ(2)k,N =

[(α(+)

ν,N + iβ(+)ν,N

)−(α(−)

ν,N + iβ(−)ν,N

)].

(15)

The renormalized Bogoliubov coherence factors V( j,ν)2ren,k,N turn

out to be complex quantities.We have calculated explicitly the propagators GFull

N (k,ωn)above with the inclusion of impurity scattering. The corres-

ponding retarded Green function G(R)N (k,t), in units such that

�=1, is given by G(R)N (k,t)=∫ +∞

−∞ (dω/2π)exp(−iωt)G(Full)N (k,ω),

where in the upper and lower half-plane, respectively,

G(Full)N (k,ωn)

=∑

j=(±),ν=±1

V( j,ν)2ren,k,N

⎡⎣iωn − ε

( j,ν)r N +

⎛⎝ i

4τ( j,ν)k,N

⎞⎠⎤⎦−1

,(16)

G(Full)N (k,ωn)

=∑

j=(±),ν=±1

V( j,ν)2ren,k,N

⎡⎣iωn − ε

( j,ν)r N −

⎛⎝ i

4τ( j,ν)k,N

⎞⎠⎤⎦−1

.(17)

Thus G(R)N (k,ω′) = ∫ +∞

−∞ dt exp(iω′t)G(R)N (k, t) is given by (16)

with ω real. We obtain

G(R)N (k, t)

=∑

j=(±),ν=±1

V( j,ν)2ren,k,Ni exp

⎛⎝−i ε( j,ν)

r N (k)t −⎛⎝ t

4τ( j,ν)k,N

⎞⎠⎞⎠θ(t),

(18)

where the unit step function θ(t)= ∫ +∞−∞ (idω/2π){exp(−iωt)/

(ω + i0+)}. The electronic excitations in cuprates are thusdemonstrably Bogoliubov quasiparticles in the pseudogapphase with finite lifetime for the states of definite momentum


due to the impurity scattering. Using the integral representa-tion of θ(t) above, it is not difficult to show that

G(R)N (k,ω′) =

∫ +∞

−∞dt exp(iω′t)G(R)

N (k, t)

=∑

j=(±),ν=±1

V( j,ν)2ren,k,N

⎡⎣ω′ − ε

( j,ν)r N +

⎛⎝ i

4τ( j,ν)k,N

⎞⎠⎤⎦

×⎡⎢⎣(ω′ − ε

( j,ν)r N

)2+

⎛⎝ 1

4τ( j,ν)k,N

⎞⎠

2⎤⎥⎦−1

.

(19)

As the renormalized Bogoliubov coherence factors V( j,ν)2ren,k,N

are found to be complex, the dimensionless density of states

ρN (k,ω) ≡ (−1/2π2ρ0) ImG(R)N (k,ω) comprises of two parts:

ρN (k,ω) = ρ′N (k,ω) + ρ′′N (k,ω), where

ρ′N (k,ω) =(

12π2ρ0

) ∑

j=(±),ν=±1

(ReV

( j,ν)2ren,k,N

)γ

( j,ν)k,N

×[(

ω − ε( j,ν)r N (k)

)2+ γ

( j,ν)k,N

2]−1

,

(20)

ρ′′α,N (k,ω) =(− 1πρ0

) ∑

j=(±),ν=±1

(ImV

( j,ν)2ren,k,N

)

×(ω − ε

( j,ν)r N (k)

)

×[(

ω − ε( j,ν)r N (k)

)2+ γ

( j,ν)k,N

2]−1

,

(21)

ReV( j,ν)2ren,k,N =

(14

)(1± Re δk,α,N

),

ImV( j,ν)2ren,k,N =

(14

)(1± Im δk,α,N

),

(22)

and γ( j,ν)k,N = τ

( j,ν)−1k,N /4 (the level-broadening factors). In

order to determine the Fermi energy density of states (DOS)ρN ,Fermi(k), since we shall put ω = μ in (20) and (21), it is

clear that ρN ,Fermi(k) =∑ j=(±),ν=±1 ρ( j,ν)N ,Fermi(k), where

ρ( j,ν)N ,Fermi(k) =

(1

2π2ρ0

)(ReV

( j,ν)2ren,k,N

)γ

( j,ν)k,N

×[(

μ− ε( j,ν)r N (k)

)2+ γ

( j,ν)k,N

2]−1

.

(23)

Equation (21) does not contribute here as the branches of

the Fermi surface are given by (ε( j,ν)r N (k)− μ) = 0. However,

for ω /=μ, definitely this equation will contribute towards theDOS. The chemical potential μ, according to the Luttingerrule, is given by the equation

(1 + p

) =∫d(ka)

∑

j,ν,N

ρ( j,ν)N ,Fermi(k)

×(

exp(β(ε

( j,ν)r N (k,B)− μ

))+ 1

)−1,

(24)

where p is the doping level,∫d(ka) → ∫ +π

−π (d(kxa)/2π∫ +π−π (d(kya)/2π, and β = (kBT)−1. It must be mentioned

that ρN ,Fermi(k,B = 0) roughly corresponds to the so-called“maximal intensity surface” [24, 25] of the ARPES studies,provided the momentum dependence of the level broadeningfactors is ignored.

We now model V(|k − k′ |) by a screened exponential

falloff of the form V(|k−k′ |) = [|v0|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 limitκ |k − k

′ |, which corresponds to a point-like isotropicscattering potential characterizing the in-plane impurities,will only be considered here for simplicity. At this stage,assuming low concentration of impurities, one may includethe contributions of all such diagrams which involve onlyone impurity vertex. This gives the equation to determine thetotal self-energy ΣN (k,ωn):

ΣN (k,ωn) = Nj

∑q

V(

q)GN

(k− q,ωn

)ΓN(

k, q,ωn), (25)

where the Lippmann-Schwinger equation to determineΓN (k, q,ωn) is

ΓN(

k, q,ωn) = V

(−q)

+∑

q′V(

q′ − q

)GN

(k− q

′,ωn

)

× ΓN(

k, q′,ωn

).

(26)

This is the t-matrix approximation.Upon using the opticaltheorem for the t-matrix [32], one may write

ΣN (k,ωn) = i ImΓN (k, k,ωn) = − iωn(2|ωn|Γk,N

) , (27)

where Γ−1k,N = 2πNjρ0

∑k′ |ΓN (k, k

′)|2. Thus the effect of

the inclusion of contribution of all the above mentioneddiagrams is to replace the Born approximation for scatteringby the exact scattering cross-section for a single impurity,that is, τ−1

k → Γ−1k . Since GN (k,ωn) and V(q) are known,

using (24), (25), and (26), one can determine Γ−1k,N in terms

of V(k). In the limit κ |k − k′ |, the disorder potential

V(|q|) ≈ |v0|, and, therefore, from the latter we obtain

ΓN (k,ωn) ≈ |v0|(1− |v0|Gα,N (k,ωn)

) . (28)

In view of (9a), we find that

ImΓN (k,ωn) ≈ −ρ0 π |v0|2(1 + ρ2

0π2|v0|2) . (29)

From (27) we now find that Γ−1k,N , in the first approximation,

is given by [2ρ0π|v0|2/(1 + ρ20π

2|v0|2)]. We take a moderatedisorder potential |v0| = 0.25 eV. This gives γk,N = Γ−1

k,N/4 ≈0.0357 eV. We next take a stronger disorder potential |v0| =1 eV. This gives γk,N = Γ−1

k,N/4 ≈ 0.2436 eV. We note


that, even though Γk,N is found to be k-independent in thefirst approximation, the term ±4R1/2

k,N sin(θk,N/2) in (15) will

ensure that τ( j,ν)k,N are momentum dependent and different

for the upper and lower branches. It may be mentioned in

passing that the total DOS ρ( j,ν)N ,Fermi has been obtained by the

summation of the density ρ( j,ν)N ,Fermi(k) over the BZ and found

to assume positive values although, as in Figures 2 and 3,

ρ( j,ν)N ,Fermi(k) becomes negative on some specific points of the

BZ. For the k-summation purpose, we first divide the BZinto finite number of rectangular patches. We next determinethe numerical values corresponding to each of these patches

of the momentum-dependent density ρ( j,ν)N ,Fermi(k) and sum

these values. We have generated these values through thesurface plots above. With the inputs obtained in this section,we embark on a calculation of the specific heat in the nextsection.

4. Magneto-Quantum Oscillations inSpecific Heat

Following the Kadanoff-Baym approach [33], the thermody-namic potential may be given by the expression

Ω(T ,B,μ

)

= Ω0(B)− 2(βNs

)−1 ×∑

j,k,ν,N

lncosh

⎛⎝β

(ε

( j,ν)r N (k,B)− μ

)

2

⎞⎠,

(30)

where Ω0(B) = Ns−1 ∑

j,k,ν,N ((ε( j,ν)r N (k,B) − μ) and β =

(kBT)−1. The dimensionless entropy per unit cell is givenby S = β2(∂Ω/∂β) while the electronic specific heat, for thepseudogapped (PG) phase (T < T∗), is Cel = −β(∂S/∂β). Weobtain

Cel ≈ 2kBN−1s

∑

k, j,ν,N

(β(ε

( j,ν)r N (k,B)−−μ

))2

× exp(β(ε

( j,ν)r N (k,B)− μ

))

×(

exp(β(ε

( j,ν)r N (k,B)− μ

))+ 1

)−2.

(31)

We have ignored the temperature dependence of the chemi-cal potential above. Now strictly speaking, for the magneticfield-dependent phenomena, whenever we need to replacethe sum over physical momenta k by an integral over a regionof k-space, the Berry-phase-corrected result is to be used.It is known [1, 2] that for some isolated points in k-space,the correction is much larger than those corresponding tothe other points for a magnetic field B even of order 1 Tesla.In what follows, we, however, ignore this correction. Uponusing (23) in (31) we find that the electronic specific heat

at a given doping level is given by Cel ≈ γ(B)T , where thespecific heat coefficient for B /= 0 may be expressed as

γ(B) ≈(k2B

π2

) ∑

j,ν,N

∫d(ka)Q1(B, k)×Q2(B, k),

Q1(B, k) =⎛⎝γ

( j,ν)k,N

ρ0

⎞⎠×

[(μ− ε

( j,ν)r N (k)

)2+ γ

( j,ν)k,N

2]−1

,

Q2(B, k) =(

ReV( j,ν)2ren,k,N

)× β ×

(β(ε

( j,ν)r N (k,B)− μ

))2

× exp(β(ε

( j,ν)r N (k,B)− μ

))

×(

exp(β(ε

( j,ν)r N (k,B)− μ

))+ 1

)−2.

(32)

The quantity Q2(B, k) is given by the expression

Q2(B, k) =∫ +∞

−∞dxI

( j,ν)k,N (x,B)

{x2ex

( ex + 1)2

}, (33)

where

I( j,ν)k,N (x,B) = (�ωc)

−1(

ReV( j,ν)2ren,k,N

)

× δ

((x

β�ωc

)− (�ωc)

−1(ε

( j,ν)r N (k, B)− μ

)).

(34)

The delta functions can be expanded in cosine Fourier series:δ(x−a) = (2π)−1 +π−1

∑∞m=1 cos[m(x−a)]. Upon doing so,

we find that the nonoscillatory part of the specific heat, withthe linear dependence on T, is

Cnonoscl ∼(

k2BT

3π�ωc

)∑

j,ν,N

∫d(ka)Q1(B, k)×

(ReV

( j,ν)2ren,k,N

).

(35)

The oscillatory part may be expressed as

Coscll =(k2BT

π3�ωc

) ∑

j,ν,N

∫d(ka)Q1(B, k)×

(ReV

( j,ν)2ren,k,N

)

×∫ +∞

−∞dx

∞∑

m=1

cos

(mx

β�ωc

){x2ex

( ex + 1)2

}

× cos{m(�ωc)

−1(ε

( j,ν)r N (k, B)− μ

)}.

(36)

Equation (36) prima facie indicates that the origin of theapproximate (1/B)-oscillations in the electronic specific heatof the underdoped YBCO is the upper and the lowerbranches of the excitation spectrum. The appearance of theDingle factors in (36), due to the scattering by impurities,is ensured by the Lorentzian Q1(B, k). In this Lorentzian,for the moderate disorder potential |v0| = 0.25 eV, the levelbroadening factor (LBF) is 0.0357 eV while, for the strongerdisorder potential |v0| = 1 eV, the same is 0.2436 eV.


∑(k,ωn)

G0(k,ωn)

G0(k,ωn)G0(k,ωn) 〈G0(k,ωn)〉 G0(k− q,ωn)

G0(k− q1,ωn) G0(k− q1 − q2,ωn)

+

+ +

=

(a)

(b)

(c) (d)

G(k,ωn)

G(k,ωn)

v(q)

v(q)

v(−q) = v∗(q)

v(q1)

v(q2) v∗(q1 + q2)

Figure 1: A few diagrams contributing to the self-energy. The wiggly lines carry momentum but no energy. The total momentum enteringeach impurity vertex, depicted by a slim ellipse, is zero. We have assumed that impurities are alike and distributed randomly. WhereasFigures 1(a) and 1(b) correspond to one impurity vertex, Figures 1(c) and 1(d) correspond to a product of four impurity potentials withnonzero averages. These are the cases where two impurities each gives rise to two potentials. Thus the figures involve the interferenceof the scattering by more than one impurity. We have assumed low concentration of impurities and therefore these figures yield smallercontributions compared to those corresponding to Figures 1(a), 1(b), and the other diagrams of the same class involving only one impurityvertex.

In an effort to explain the possible quantum oscillationsin the specific heat, for the anisotropy parameter ε = 0 with

LL index N = 0 and 1, we introduce the two quantities ˜Ean,N

and ˜En,N , where

ε(±,ν)r N (k,B) |k=[(±π,0),(0,±π)] ≡ ˜Ean,N (B),

ε(±,ν)r N (k,B) |k=(±π/2,±π/2) ≡ ˜En,N (B) ,

(37)

corresponding to the anti-nodal and the nodal excitationsrespectively, and calculate these quantities for the disorderpotential |v0| = 0.25 eV and 1 eV. Each of the quantities˜Ean,N (B) and ˜En,N (B) will have 8 values corresponding tothe superscripts (±, ν) and the subscript N. Since we take

N = 0 and 1, the 4 × 4 = 16 possible values are displayedin Table 1 for B = 0; likewise, 8 × 4 = 32 possiblevalues are in Table 2 for B = 45 T. From Tables 1 and2 (see also table legends below) we clearly notice that asthe magnetic field (B) is increased, provided one is in theweak disorder regime (|v0| = 0.25 eV), the Landau statesmove to a higher energy, ultimately rising above the Fermilevel. They are thereby emptied, and the excess fermionsfind a place in the next lower Landau level (LL). Duringthe crossing of an LL, its occupation by fermions is haltedand then reduced. The specific heat consequently decreasesslightly. As the excess fermions get accommodated in the nextlower LL, the specific heat rises again. Thus the oscillations inthe fermion density in the vicinity of Fermi energy manifestthemselves as oscillations in the specific heat. These events


−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3

kxa

kya

(a)

−3−3−2 −2

−1 −10

01

12

23

3

3210

−1−2−3−4−5−6

kxakya

Ferm

ien

ergy

DO

Sat

10%

dopi

ng

leve

l

(b)

−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3

kxa

kya

(c)

−3−3−2 −2

−1 −10

01

12

23

3

3210−1−2−3−4

kxaky a

Ferm

ien

ergy

DO

Sat

10%

dopi

ng

leve

l

(d)

−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3

kxa

kya

(e)

Figure 2: The contour/3D plots of the Fermi energy DOS on the Brillouin zone (BZ) at 10% hole doping, |t′/t| = 0.4 (except (e)) andB = 0. (a) The anisotropy parameter ε = 0 and the level broadening factor γ = 0.0357 eV. (b) ε = 0 and γ = 0.2436 eV. (c) ε = 0.05 andγ = 0.0357 eV. (d) ε = 0.05 and γ = 0.2436 eV. (e) ε = 0.05,γ = 0.0357 eV, and |t′/t| = 0.5. The scale of the plots in Figures 2(a), 2(c)and 2(e) is from −0.2 to 1.2. In the weak disorder regime we obtain prominent electron pockets and barely visible hole-like patches. In thestrong disorder regime the scenario does not change radically. We have assumed Σe = 0.05 eV above.The comparison of figures 2(a) with2(c) (and figures 2(b) with 2(d)) indicates the slight change (Pomeranchuk distortion) in the topology of the Fermi surface due to ε /= 0;electron pockets around (0,±π) widen slightly compared to those around (±π, 0). (e) The figure indicates that the higher value of the ratio|t′/t|may lead to the disappearance of the electron pockets.


−3−3−2 −2

−1 −10 0

11

22

33

kxakya

1

0.5

0

−0.5Ferm

ien

ergy

DO

Sat

10%

dopi

ng

leve

l

(a)

−3−3−2 −2

−1 −10 0

11

22

33

kxakya

3210

−1−2−3

4

Ferm

ien

ergy

DO

Sat

10%

dopi

ng

leve

l

(b)

Figure 3: The 3D plots of the Fermi energy DOS on the Brillouinzone (BZ) at 10% hole doping and B = 45 Tesla. (a) The anisotropyparameter ε = 0 and the level broadening factor γ = 0.0357 eV. (b)ε = 0 and γ = 0.2436 eV. The features for B = 45 T are by and largethe same as in Figure 2.

take place for 17 T ≤ B ≤ 53 T (see the table legends). Thesimilar oscillations in the electrical conductivity (Shubnikov-de Haas oscillations (SdHOs)) are already reported [13–16].As we notice above, in YBCO, these oscillations have theirorigin at the electron pockets in the Fermi surface locatedaround the anti-nodal points provided one is in the weakdisorder regime (|v0| = 0.25 eV). For the strong disorder

potential (|v0| = 1 eV), the energy values ˜Ean,N (B = 0) inTable 1, less than μ = −0.27 eV, are too far below μ to be ableto contribute to the oscillation process up to a reasonablyhigh magnetic field B of order 50 T. The basic reason, as weinfer, behind this magneto-oscillation phenomenon is theLandau quantization in the presence of a strong magneticfield (B > 17 T) (In weak magnetic fields (B < 1 T),i.e. in the linear response regime, one can neglect Landauquantization, since the existence of Landau levels (LLs) insmall magnetic fields requires extremely clean and, possiblyvery large samples) [31] in a reasonably clean sample. Thepoint to be noted here is that one cannot afford to have thetotal absence of disorder, for in that case the level broadening

factors γ will be zero in (23) for ρ( j,ν)N ,Fermi(k). Thus, in any

investigation on the quantum oscillations, including a recentnoteworthy theoretical work [34] involving the Fermi arcpicture, the inclusion of moderate disorder seems to benecessary.

Table 1: The values of the anti-nodal and the nodal excitationsfor B = 0 for the disorder potential |v0| = 0.25 eV and 1 eV. All

the values of ˜En,N (B = 0) are above the value of μ = −0.27 eV.Therefore, the origin of the specific heat oscillations shown in(36) would not correspond to the nodal patches around k =(±π/2,±π/2) shown in Figure 2.

|v0| = 0.25 eV |v0| = 1 eV˜Ean,N (B = 0)

in eV

˜En,N (B = 0)in eV

˜Ean,N (B = 0)in eV

˜En,N (B = 0)in eV

−0.2578 0.1639 −0.0857 0.3389

−0.5130∗ 0.0249 −0.6871∗ −0.1501

−0.2724∗∗ 0.1639 −0.0982 0.3389

−0.5324∗ 0.0249 −0.7066∗ −0.1501∗ The energy values are less than μ = −0.27 eV.∗∗The energy value is the closest to μ = −0.27 eV.

Table 2: The values of the anti-nodal and the nodal excitations forB = 45 T for the disorder potential |v0| = 0.25 eV and 1 eV. Thefirst four rows correspond to LL index N = 0 while the next four

to N = 1. All the values of ˜En,N (B = 45 T) are above the value ofμ = −0.27 eV.

|v0| = 0.25 eV |v0| = 1 eV˜Ean,N (B =45 T)

in eV

˜En,N (B =45 T)in eV

˜Ean,N (B =45 T)in eV

˜En,N (B=45 T)in eV

−0.2578 0.1695 −0.0837 0.3418

−0.5098∗ 0.0245 −0.6839∗ −0.1478

−0.2703∗∗ 0.1695 −0.0962 0.3418

−0.5293∗ 0.0245 −0.7034∗ −0.1478

−0.2538 0.1766 −0.0797 0.3472

−0.5035∗ 0.0278 −0.6775∗ −0.1429

−0.2662∗∗∗ 0.1766 −0.0921 0.3472

−0.5230∗ 0.0278 −0.6971∗ −0.1429∗ The energy values are less than μ = − 0.27 eV.∗∗This is the energy value (corresponding to LL index N = 0) whichincreases to −0.2700 eV at B = 53 T.∗∗∗This is the energy value (corresponding to N = 1) which attains thevalue of μ = −0.2700 eV at B = 17 T.

5. Concluding Remarks

The well-known theoretical developments, such as thedynamical mean field theory (DMFT) [7, 8, 35], may requirein future a revisit of the problem of the quantum oscillationswith a new perspective. Particularly, the development ofDMFT and its cluster extensions provides new path toinvestigate strongly correlated systems; the DMFT study ofsuperconductivity near the Mott transition establishes theremarkable coexistence of a superconducting gap, stemmingfrom the anomalous self-energy, with a pseudogap stemmingfrom the normal self-energy. This theory also leads to thegeneration of the Fermi arc behavior of the spectral function[7, 8].


We conclude that the key input in our analysis is the Lan-dau level split dispersion in (1); the chirality aspect perhapsplays a minor role as these oscillations are also possible inthe pure d-density wave state [4]. We note that though theinclusion of the elastic scattering by impurities has led to aclearer understanding, of the Fermi surface topology in thepresence of a magnetic field at the semiphenomenologicallevel, the further examination of the single-particle excitationspectrum of the system in a fully self-consistent approxi-mation framework is necessary to impart a comprehensivemicroscopic basis to the findings presented. Finally, we hopethat our results, namely. the one relating to the reconstructedFermi surface in the weak as well as the strong disorderregimes and the other to the electronic specific heat anomaly,will persuade researchers to look for them in the hole-dopedcuprates. It must be added that the experimental observationof the latter is quite a difficult proposition, for the domi-nant phononic contribution is expected to overshadow theanomaly in the heat capacity measurements [36].

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