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arXiv:0811.1056v4 [cond-mat.supr-con] 11 Sep 2009

Resolution of two apparent paradoxes concerning quantum oscillations in underdoped

high-Tcsuperconductors

Xun Jia,1Pallab Goswami,2and Sudip Chakravarty1

1Department of Physics and Astronomy, University of California Los Angeles

Los Angeles, CA 90095-1547

2Department of Physics and Astronomy, Rice University, TX 777005

(Dated: September 11, 2009)

Recent quantum oscillation experiments in underdoped high temperature superconductors seem

to imply two paradoxes. The first paradox concerns the apparent non-existence of the signature of

the electron pockets in angle resolved photoemission spectroscopy (ARPES). The second paradox is

a clear signature of a small electron pocket in quantum oscillation experiments, but no evidence as

yet of the corresponding hole pockets of approximately double the frequency of the electron pocket.

This hole pockets should be present if the Fermi surface reconstruction is due to a commensurate

density wave, assuming that Luttinger sum rule relating the area of the pockets and the total number

of charge carriers holds. Here we provide possible resolutions of these apparent paradoxes from the

commensurate d-density wave theory. To address the first paradox we have computed the ARPES

spectral function subject to correlated disorder, natural to a class of experiments relevant to the

materials studied in quantum oscillations. The intensity of the spectral function is significantly re-

duced for the electron pockets for an intermediate range of disorder correlation length, and typically

less than half the hole pocket is visible, mimicking Fermi arcs. Next we show from an exact transfer

matrix calculation of the Shubnikov-de Haas oscillation that the usual disorder affects the electron

pocket more significantly than the hole pocket. However, when, in addition, the scattering from

vortices in the mixed state is included, it wipes out the frequency corresponding to the hole pocket.

Thus, if we are correct, it will be necessary to do measurements at higher magnetic fields and even

higher quality samples to recover the hole pocket frequency.

I.INTRODUCTION

High temperature superconductors have been ad-

dressed from a remarkable number of vantage points.

Nonetheless many of the basic questions still remain un-

resolved, and the notion of broken symmetries arising

from a Fermi liquid has been traditionally discarded in

favor of many exotic ideas. Here we revisit the Fermi

liquid concept and a particular broken symmetry in re-

sponse to a class of recent quantum oscillation exper-

iments.1,2,3,4,5,6We and others7,8,9,10,11,12,13have had

some success in this respect. If these theories and ex-

periments are correct, one will have to radically alter our

twenty year-old view of these superconductors.14The to-

tality of phenomenology cannot of course be explained

without serious Fermi liquid corrections. But as long as

the quasiparticle residue is finite, we hope that the low

energy properties can be understood from our perspec-

tive.

Two paradoxes have arisen in the context of the quan-

tum oscillation measurements. The first is the contrast

between Fermi arcs observed in angle resolved photoe-

mission (ARPES) experiments15on one hand and recent

quantum oscillation experiments suggesting small Fermi

pockets in underdoped YBa2Cu3O6+δ(YBCO),1,2,3,4,5,6

on the other. This is particularly clear in recent ARPES

experiments where an effort was made to examine YBCO

with similar doping as in the quantum oscillation mea-

surements.16

The second paradox is the non-existence of any evi-

dence of the hole pockets in quantum oscillation mea-

surements. Within the density wave scenario of wave

vector Q = (π,π) (the lattice spacing set to unity), there

should be two dominant frequencies in quantum oscilla-

tions. One corresponding to the electron pocket at about

500T and the other corresponding to the hole pocket at

around 900 T. These are of course rough numbers cor-

responding to approximately 10% doping, assuming that

the Luttinger sum rule is satisfied in the mixed state,

that is, the quantum oscillations reflect the normal state

even if the measurements may lie within the mixed state.

The concept of a broken symmetry is very powerful

because deep inside a phase a physically correct effec-

tive Hamiltonian can address many important questions,

whereas our inability to reliably predict properties of

even a single band Hubbard model, while widely pursued,

has been a limiting factor. This is not an empty exer-

cise, if new phenomena can be predicted or striking facts

can be explained with some degree of simplicity. That

broken symmetry both dictates and protects the nature

of elementary excitations, determining the properties of

matter, is important to emphasize.

The suggested form of order, the d-density wave

(DDW),17explains numerous properties of these super-

conductors, including the concomitant suppression of the

superfluid density18, Hall number,19and more recently

the large enhancement of Nernst effect in the pseudogap

state,20in addition to the existence of a single-particle

gap of dx2−y2 form above Tc. There are also theoretical

reasons why DDW is a possibility. It competes favorably

with other ordering tendencies in variational studies of

extended Hubbard models with nearest-neighbor repul-

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sion and pair-hopping terms.21,22It is also realized in

a class of two-leg ladder models with nearest-neighbor

repulsion.23However, whatever form the correct Hamil-

tonian takes, we know that it must favor d-wave super-

conductivity (DSC). Such a Hamiltonian will almost cer-

tainly favor DDW order as well, in light of the abundance

of local Hamiltonians which do not discriminate between

DSC and DDW order. In fact, two carefully designed dif-

ficult polarized neutron scattering experiments have pro-

vided tantalizing direct evidence of DDW order,24,25al-

though other experiments have claimed otherwise.26,27,28

A natural enemy of the pristine properties of matter

is disorder that is unavoidable in complex systems such

as high temperature superconductors. The role of dis-

order was emphasized in the original proposal of DDW

order as a relevant competing order in the phase dia-

gram of high-Tc superconductors,17although our views

of disorder have greatly evolved during the intervening

years. It is this DDW order combined with disorder that

would be the focus of the present manuscript in resolving

the paradoxes stated above. The disorder considered are

of two different types: (1) scattering due to impurities

and defects and (2) scattering from vortices in the mixed

states.

We consider two kinds of intrinsic disoredr: (a) Gaus-

sian white noise and (b) correlated disorder with a finite

correlation length. In the momentum space the scatter-

ing rate for correlated disorder will decay as exp(−q2l2

where q is the momentum transfer between the initial and

the final states, lD being the correlation length. There-

fore, because of its smaller size, the states correspond-

ing to the electron pockets are scattered more than on

the hole pockets. This is an interpretation of the phe-

nomenon and is based on intrapocket scattering. An al-

ternative interpretation involves the density of states on

the Fermi surface of the electron pockets. In contrast, for

white noise, scattering is independent of momentum and

affects both pockets similarly. Disorder naturally has a

strong effect on ARPES spectral function, which is sensi-

tive to the coherence factors that are analogs of Wannier

functions. In Shubnikov-de Haas oscillations it is only

the averaged effect of disorder that enters by determin-

ing the effective lifetime on the Fermi surface. Therefore,

the role of disorder is quite different, as we shall explicitly

see.

Correlated disorder is also experimentally relevant.

Unlike quantum oscillation experiments which probes

bulk properties, ARPES is inherently a surface probe.

In the relevant case of YBCO, as cleaved surfaces show

that CuO and BaO terminations give different contribu-

tions to the total photoemission intensity, with a hole

doping nh = 30%, almost irrespective of the nominal

bulk doping. This self-doping was controlled by evapo-

rating potassium in situ on the cleaved surface, so as to

reduce the hole content down to the value of underdoped

bulk YBCO (δ ≈ 0.5),16the doping level for which many

quantum oscillation experiments are carried out. The

potassium overlayer is likely to produce an effective cor-

D),

related disorder in the CuO plane.

To explain the second paradox we shall adapt an anal-

ysis of Stephen29to include a normal state that exhibits

DDW order. We shall see that the relativistic charac-

ter of the nodal fermions of the hole pocket, as opposed

to the nonrelativistic nature of the charge carriers of

the electron pocket provides a possible explanation. If

we denote the Dingle factor of the electron pocket by

De = e−π/ωcτv, the Dingle factor of the hole pocket is

Dh≈ D4.4

of the missing frequency from the hole pocket.

ωcis the cyclotron frequency corresponding to the elec-

tron pocket and 1/τv is the scattering rate of the elec-

trons from the vortices in the mixed state. The analysis

of Stephen also leads to a tiny shift of the relative fre-

quency of the quantum oscillations in the mixed state,

of the order of 10−6. Thus, there is enough leeway that

even a very large error in this estimate will not affect our

conclusions.

The organization of the manuscript is as follows. In

Sec. II we compute the ARPES spectral function and

show that disorder can destroy the evidence of electron

pockets. Section III is devoted to an exact transfer ma-

trix computation of Shubnikov-de Haas oscillations and

the effect of disorder on it. Section IV contains a discus-

sion of scattering of quasiparticles of the putative normal

state from the vortices in the mixed state. In Sec. V we

briefly summarize the salient features of our work.

e. This huge suppression may be the resolution

Here,

II. SPECTRAL FUNCTION

A.Hamiltonian

The Hamiltonian of commensurate DDW order in

terms of the fermion creation and destruction operators,

c†

kand ck, in the momentum space is

H1=

?

k∈RBZ

?

ǫkc†

kck+ ǫk+Qc†

k+Qck+Q

?

+

?

k∈RBZ

(iWkc†

kck+Q+ h.c.),

(1)

where the ordering wave vector Q = (π,π), and ǫkis the

single particle spectra. The lattice constant is set to be

unity for simplicity. The reduced Brillouin zone (RBZ)

is bounded by ky± kx= ±π. We define ǫkby

ǫk= − 2t(coskx+ cosky) + 4t′coskxcosky

− 2t′′(cos2kx+ cos2ky) (2)

and the DDW gap by

Wk=W0

2(coskx− cosky). (3)

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B.Disorder

Potential disorder in real space with a finite correlation

length lDis modeled by

V (r) =

gV

2πl2

D

?

dx e

−|r−x|2

2l2

D G(x), (4)

where the disorder averages are ?G(x)?

?G(x)G(y)? = δ(x − y); the disorder intensity is set by

gV. This disorder Hamiltonian in the momentum space

is then

= 0 and

H2=

?

k1,k2∈BZ

V (k1,k2)c†

k1ck2+ h.c.,(5)

where the matrix elements are

V (k,k + q) =gV

2πe−

q2l2

2 u(q),

D

(6)

and u(q) is

u(q) =

1

2π

?

dy G(y)e−iq·y, (7)

satisfying the conditions of ?u(q)? = 0 and ?u(q)u(q′)? =

δ(q + q′). In practice, we generate u(q) directly with

the desired statistical properties and then compute the

matrix elements in Eq. (5).

C.Computation of ARPES spectral function

Once the full Hamiltonian H = H1+ H2is generated,

it is diagonalized by the transformation ck=?

where γl is the annihilation operator of quasiparticles

with energy El. The coefficients Pk,land energy Elare

obtained through an exact numerical diagonalization pro-

cedure. Finally, the ARPES spectral function A(k,ω) at

a temperature T is given by:

lPk,lγl,

A(k,ω) = 2π

?

l

|Pkl|2nlδ(ω − El), (8)

where nl = 1/[1 + exp((El− µ)/kBT)] is the fermion

occupation number. Note that the numerical implemen-

tation of Eq. (8) requires an approximation of the delta

function by, for example, a Lorentzian distribution.

We discretize the BZ with a mesh of size 80 × 80, and

diagonalize the corresponding Hamiltonian. The param-

eters we choose for YBCO at 10% doping are: t = 0.3 eV,

t′= 0.3t, t′′= t′/9.0, and W0= 0.0825 eV, same as be-

fore.7The chemical potential µ is set to be −0.2627 eV.

These parameters yield a hole doping of nh∼ 10%. The

temperature T = 10 K is chosen, where a typical ARPES

experiment is performed. For gV/(2π) = 0.1t, the quasi-

particle life time for lD= 0 is of the order τ ∼ 10−12s

from Fermi’s golden rule, which is a reasonable value.3,6

Note that the band width is 8t. The scattering rate for

FIG. 1:

related disorder corresponding to lD = 4.

parameters are stated in the text.

The spectral function A(k,ω) at ω = µ with cor-

The remaining

correlated disorder for a finite lD ∼ 4 is considerably

smaller, as can be seen from Eq. 6. The final spectral

function is obtained by averaging over 20 - 50 disorder

configurations until no difference is detected upon fur-

ther averaging. A typical result with disorder correlation

length lD= 4, in units of the lattice constant, is plotted

in Fig. 1. The electron pockets are barely visible, resem-

bling experimental observations. From Fermi’s golden

rule, the scattering rate is proportional to the square

of the matrix element between the initial and the final

states, V (k,k + q) ∼ exp(−q2l2

q is the momentum transfer in the scattering process. On

average, the scattering rate is therefore proportional to

exp(−q2

In particular, qtis roughly the size of a pocket. Because

electron pockets are much smaller than hole pockets, the

scattering rate is greater for electron pockets. The hole

pockets centered at (±π/2,±π/2) have vanishingly small

spectral function on the back side due to the coherence

factors and appear as Fermi arcs instead,30whose lengths

are further reduced by disorder.

We also demonstrate the dependence of the spectral

function A(k,ω = µ) on lDin Fig. 2. The disorder corre-

lation lengths are lD= 0,2,8,16 for panels (a) through

(d), respectively. There are three distinct regimes de-

pending on lD. For small lD, the electron and hole

pockets will be almost equally scattered. As a conse-

quence, the spectral function is smeared out everywhere;

see Fig. 2(a). For intermediate values of lD, for example

lD = 2 in Fig. 2(b) and lD = 4 in Fig. 1, scattering is

more prominent for electron pockets, resulting in a pic-

ture consisting of only four Fermi arcs. Finally, as the

correlation length lDincreases further, Fig. 2(c) and (d),

the electron pockets reappear. Indeed, though more dis-

order scattering occurs on the electron pockets, the spa-

tial variation of disorder, hence the net effect of disorder,

D/2) (see Eq. (6)), where

tl2

D/2), where qtis a typical momentum transfer.

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FIG. 2: The spectral function A(k,ω = µ) for gV/(2π) =

0.1t. The correlation lengths are lD = 0,2,8,16 from panels

(a) through (d), respectively. The remaining parameters are

given in the text.

becomes weaker.

?

?

?

?

??

?

FIG. 3:

A(k,ω) at two k points in the Brillouin zone, as indicated

in the insert.

The energy dependence of the spectral function

To characterize the energy dependence of the spectral

function, we compute A(k,ω) as a function of ω at two k

points in the Brillouin zone. One of them is at the inter-

section ΓY line with the inner side of the Fermi surface,

the other is situated on the electron pocket along the di-

rection ΓM; see Fig. 3. The disorder correlation length

was chosen to be lD = 4. Although there are peaks at

ω ∼ µ for both, the peak corresponding to the electron

pocket is significantly suppressed by disorder; the sec-

ond peak at ω − µ ∼ −0.2eV is clearly an artifact of our

simple theory and such high energy states would surely

decay once correlation effects are taken into account by

the creation of particle-hole pairs.

In Fig. 4, A(k,ω) is plotted as a function of both k and

ω. The horizontal axis is along the path Γ → Y → M →

Γ in the Brillouin zone. The spectral function is negli-

gibly small outside the reduced Brillouin zone bounded

by kx±ky= ±π,30and consequently there are no peaks

in the central region of Fig. 4. Close to k = (π,0) and

ω = µ, A(k,ω) has very small intensity due to long range

correlated disorder, consistent with our previous observa-

tion that the electron pockets are most likely unobserv-

able.

??

FIG. 4: (Color online)The gray scale plot of the spectral func-

tion A(k,ω) as a function of both k and ω. Red dashed curves

indicate the quasi-particle dispersion relation. A horizontal

dashed line shows the chemical potential µ. The remaining

parameters are given in the text.

III. SHUBNIKOV-DE HAAS OSCILLATIONS

A.The transfer matrix method

Let us now consider the effect of disorder on

Shubnikov-de Haas (SdH) oscillations of the conductivity,

σxx. The tight-binding Hamiltonian on a square lattice

in a sample of dimension N×M, with the lattice constant

set to unity, is

H =

?

i

ǫic†

ici+

?

i,j

ti,jeiai,jc†

icj+ h.c., (9)

where ciis the fermionic annihilation operator at the site

i. The spin degrees of freedom are omitted for simplicity.

The hopping amplitude ti,j vanishes except for nearest

and next nearest neighbors. To include two-fold commen-

surate DDW order, the nearest neighbor hopping ampli-

tudes are chosen to be

ti,i+ˆ x= −t +iW0

ti,i+ˆ y= −t −iW0

4

(−1)(n+m),

4

(−1)(n+m),

(10)

where (n,m) are a pair of integers labeling a site: i =

nˆ x + mˆ y, and W0 is the DDW gap; for the next near-

est hopping ti,j = t′. The on-site impurity energy ǫi is

defined by

ǫi=V0

Z

?

r

Gre

−|r−i|2

2l2

D ,(11)

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which is analogous to Eq. (4).

disorder, we set the disorder averages ?Gr? = 0 and

?GrGr′? = δr,r′, and Z =?

factor. V0parameterizes the disorder intensity. Note that

Eq.(11) reduces to ǫi= V0Giin the limit lD→ 0, and ǫi

becomes uncorrelated random variables. A constant per-

pendicular magnetic field B is included via the Peierls

phase factor ai,j=2πe

h

is the vector potential in the Landau gauge. We note

that a perpendicular magnetic field even as large as 60 T

has little effect on DDW order.31

In this section we choose t = 0.29 eV, t′= 0.1 eV,

and W0= 0.065 eV. The chemical potential is set to be

µ = −0.28 eV. Note that these parameters are slightly

different from those in the previous section, although the

hole doping is again ∼ 10%. We have left out the third

nearest neighbor hopping, which greatly complicates the

transfer matrix calculation without offering any particu-

lar insight. The disorder intensity V0 = 0.4t leads to a

quasi-particle life time of the order of ∼ 10−12s in the

limit of lD= 0. The magnetic field ranges from B = 20 T

to B = 75 T, representative of the quantum oscillation

experiments. The only relevant length scale here is the

magnetic length lB =

??/eB, which for B = 20 T is

∼ 15a, a being lattice constant equal to 3.85˚ A.

Now consider a quasi-1D system, N ≫ M, with

a periodic boundary condition along y-direction.

Ψn = (ψn,1,ψn,2,...,ψn,M)Tbe the amplitudes on the

slice n for an eigenstate with a given energy E, then the

amplitudes on three successive slices satisfy the relation

To model correlated

re−|r|2/2l2

Dis a normalization

?i

jA · dl, where A = (0,−Bx,0)

Let

?Ψn+1

Ψn

?

=

?T−1

n(E − Hn) −T−1

1

nTn−1

0

??

Ψn

Ψn−1

?

(12)

,

where Hn is the Hamiltonian within the slice n, and

the matrix Tn corresponds to the hopping between the

slices n and n + 1. Tn is tridiagonal, as electrons can

hop from a site on slice n to three sites on the slice

n + 1. All postive Lyapunov exponents of the transfer

matrix,32γ1> γ2> ... > γM, are computed by iterating

Eq. (12) and performing orthonormalization regularly.

The convergence of this algorithm is guaranteed by the

well known Oseledec theorem.33For the above parame-

ters a transverse dimension corresponding to M = 40 is

sufficient. Equation (12) was iterated 105to 106times

until the relative errors of less than 1% of all the Lya-

punov exponents were achieved.

B. Computation of σxx

The conductivity σxxat zero temperature is obtained

from the Landauer formula:34,35,36

σxx(B) =e2

h

M

?

i=1

1

cosh2(Mγi). (13)

The SdH oscillations of σxx at zero temperature for

?

?

?

?

FIG. 5: Fourier transform of SdH oscillations at zero temper-

ature for lD = 0 in arbitrary units. Insert shows oscillations

as a function of the inverse magnetic field. Note the presence

of higher harmonics. The parameters are described in the

text.

lD= 0 is shown in the insert of Fig. 5. A third order poly-

nomial was used to subtract the background. The Fourier

transform is shown in the main panel of Fig. 5 . Clearly,

there are two main oscillation frequencies F1= 490±30T

and F2= 915 ± 40T, corresponding to electron and hole

pockets respectively, although a few harmonics are also

visible. Though the first peak at F1agrees with experi-

mental observations, the second peak at F2has not yet

been observed.

For correlated disorder, the Fourier transform of σxxis

plotted in Fig. 6. In all the cases, two main frequencies of

F1∼ 500T and F2∼ 900T are prominent. As for A(k,ω),

an increase of lDincreases the amplitudes of F1and F2,

because the effective disorder becomes weaker with the

increased correlation length. Also the higher harmonics

are more visible. There is, however, a sharp distinction

between the dependence of the two physical quantities,

which becomes clear when we consider the white noise

case: A(k,ω) is completely smeared out in the momen-

tum space because white noise scatters between all pos-

sible wave vectors; see Fig. 2 (a). The coherence factors

are of crucial importance for the spectral function. In

contrast, the SdH oscillations are damped by the Dingle

factor, which is parametrized by a single lifetime and dis-

order enters in an averaged sense. This striking contrast

is clear in Fig. 5. The surprise is that impurity scatter-

ing affects the electron pocket more than the hole pocket,

which is remarkably robust even for white noise. There

must be a piece of physics missing, if we are to explain

why the hole pocket is not observed in SdH. This miss-

ing physics we argue is the vortex scattering rate which

affects the two pockets very differently.

IV.VORTEX SCATTERING RATE IN THE

MIXED STATE

We have shown previously9that the scattering rate of

the DDW quasiparticles, corresponding to the electron