1. Symmetry, Disorder and the Josephson E ect
in d-Wave Superconductors
Fabian Wenger
Institute of Theoretical Physics
Chalmers University of Technology
and Goteborg University
412 96 Goteborg
Sweden
1995
AKADEMISK AVHANDLING
for avlaggande av teknisk doktorsexamen i teoretisk fysik.
Examinator och huvudhandledare: Prof. Stellan Ostlund
co-advisor: Prof. Alexander A. Nersesyan, Tbilisi, Georgia
Fakultetsopponent: Prof. Eduardo Fradkin, Urbana-Champaign, USA
Avhandlingen forsvaras vid en o entlig disputation den 2:a juni 1995 kl. 10.15
i sal FB, Origohuset, vid Chalmers tekniska hogskola, Goteborg.
Avhandlingen forsvaras pa engelska.
5. Abstract
This thesis focuses on aspects of symmetry, disorder and the Josephson e ect in
d-wave superconductors which are generically characterized by sign changes of the
gap function k in certain directions with respect to the underlying crystal lattice.
A basic feature for the pure d-wave superconductors is the linear quasiparticle
density of states at low energies: n(E) / E.
We consider the in uence of disorder on a 2D d-wave superconductor. The cen-
tral result of our solution is that the exponent for the density of states decreases
under the in uence of weak non-magnetic disorder n(E) / E , and 0 < < 1
and we calculate for several types of non-magnetic disorder. We use a com-
bination of numerical, perturbative, self-consistent and eld theoretical methods
in our calculation. The perturbative approach reveals the presence of strong ver-
tex corrections i.e. a partial summation of Feynman diagrams is insu cient. The
non-perturbative eld theoretical approach gives asymptotically exact result for
the density of states and we nd that the property n(0) = 0 for non-magnetic
disorder is a consequence of a continuous symmetry in the e ective model which
remains unbroken.
We study the Josephson e ect in d-wave superconductors within a tunneling
Hamiltonian approach. We use a systematic expansion of the tunneling matrix el-
ement to investigate the e ects of its anisotropy on the IcRn product in bicrystal
junctions. The IcRn product is strongly varying as function of crystal orienta-
tion at constant relative misorientation angle. The comparison of the theoretical
results with experiments on YBa2Cu3O7 bicrystal Josephson junctions gives
qualitative agreement for gap functions of generalized s- and dx2 y2-wave symme-
try showing that the combination of anisotropy in the superconducting gap and
in the tunneling matrix is required to explain the experimental results.
We investigate the generic weak-coupling BCS states in a singlet paired trans-
lationally invariant tetragonal superconductor. We classify the possible order pa-
rameters and nd a possibility of a nodeless (dx2 y2 + i dxy) - pairing in the
presence of short-range antiferromagnetic interactions.
We discuss the relevance of our results to the order parameter symmetry in the
high-Tc cuprate superconductors.
Key-words: d-wave, superconductor, symmetry, disorder, Josephson e ect,
density of states, high-Tc, cuprate, order parameter.
5
7. This thesis consists of an introductory text that contains original research
results in Secs. 4.2 and 5.1. Appended are ve publications that are listed below
and will be referred to in the text as Paper I to V.
I : F. Wenger, S. Ostlund:
d-wave Pairing in Tetragonal Superconductors",
Phys. Rev. B 47, 5977 (1993).
II : A.A. Nersesyan, A.M. Tsvelik, F. Wenger:
Disorder E ects in Two-Dimensional d-wave Superconductors",
Phys. Rev. Lett. 72, 2628 (1994).
III: A.A. Nersesyan, A.M. Tsvelik, F. Wenger:
Disorder E ects in Two-Dimensional Fermi Systems with Conical
Spectrum: Exact Results for the Density of States",
Nucl. Phys. B 438 FS], 561 (1995).
IV: F. Wenger:
Strong Vertex Corrections from Weak Disorder in 2D d-wave
Superconductors",
Z. Phys. B, (in press).
V : Z.G. Ivanov, E.A. Stepantsov, T. Claeson, F. Wenger, S.Y. Lin, N. Khare,
P. Chaudhari:
Highly Anisotropic Supercurrent Transport in YBa2Cu3O7 Bicrystal
Josephson Junctions",
(submitted to Phys. Rev. Lett.).
7
9. Acknowledgement
I want to express my sincere gratitude to all the people who made this thesis
possible; my thesis advisor Stellan Ostlund for his guidance during the years,
providing an exciting research environment and for inspiring many of the projects
in this thesis; my co-advisor Alexander Nersesyan, who has been a wonderful
teacher and friend, for his countless e orts to share with me his insights and ideas
on our research and theoretical physics in general, the results of our collaboration
on disorder e ects are a major part of this thesis; Zdravko Ivanov who introduced
me to his exciting experimental results, which has given me a chance to apply
theoretical concepts to a concrete experiment and to experience the excitement
that followed my attempts to understand this new e ect.
I would like to thank all the present and former members of the solid state
theory group: Anders Eriksson, Sebastian Eggert, Torbjorn Einarsson, Per Frojdh,
Henrik Johannesson, Ann Mattsson, Mimmi Ockermann, Stefan Rommer and
Yvonne Steen for providing a stimulating working atmosphere. Especially I want
to thank Mimmi, Shura, Stellan and Zdravko for proofreading the introduction.
A very special thank goes to Jan-Olov Branander, Ivana Kawikova, Andreas
Klinkmuller, Lars Osterlund and Prayoon Songsiriritthigul for their friendship
and nonprofessional activities.
Finally I want to thank Annika for her love, companionship and endurance.
Fabian Wenger
Goteborg
May 1995
9
13. 1. Introduction
Superconductivity is one of the most fascinating phenomena of nature. It is a
dramatic manifestation of quantum mechanics which we can observe by our own
senses when a magnet is levitated by a superconductor due to the ability of the
superconductor to repel magnetic elds.
The discovery of superconductivity in 1911 by H. Kammerlingh Onnes 1] was
characterized as a transition of matter to a state with no measurable resistance
i.e. the induced electric current carried by the electrons in a superconducting ring
does not decrease. This phenomenon only occurs below the critical temperature
Tc which is dependent on the type of material we use. The search for new mate-
rials displaying higher Tc's has therefore been a decisive factor to develop useful
applications of superconductivity. This proved to be a nontrivial task but in 1986
J. Georg Bednorz and Karl Alex Muller 2] kicked o high Tc research by discov-
ering materials with a critical temperature above 30 K and within few months
after publication of their results Maw-Kuen Wu and his collaborators 3] reached
a Tc of 93 K. The new materials which made those values possible became known
as the cuprate superconductors. Though those temperatures are still rather low
compared to room temperature of about 300 K, current Tc's 4, 5] are consider-
ably above the liquefying temperature of nitrogen of 77 K which can be achieved
relatively simple (recent updates on the applications of high Tc devices can be
found in 6] and 7]) 1.
At the time of discovery of superconductivity quantum mechanics was still in
its infancy and even after the foundations were laid in the 1920's, a microscopic
theory of superconductivity was not found until 1957, when Bardeen, Cooper and
Schrie er (BCS) 9] discovered that a superconductor could be described by a
quantum mechanical wavefunction that incorporated electron pairing as the basic
ingredient.
Certain aspects of their theory can be seen by considering the structure of the
wave function of a single pair of electrons 10], when we consider the amplitude
to nd the two electrons of the pair at a xed relative position. This amplitude
is a complex number and according to the basic rules of quantum mechanics the
1As an aside I can recommend the book by Hazen 8] for an entertaining and interesting
glimpse of the rst year of high Tc research with emphasis on the discovery of the 93 K
YBa2Cu3O7 superconductor.
13
14. 1. Introduction
square of its modulus gives the probability to nd the two electrons at a xed
relative position in space. For most pre-1986 superconductors it is accepted that
this amplitude is isotropic and can be chosen real and positive in the absence of
any external electromagnetic elds, i.e. it does not depend on the orientation with
respect to the crystal lattice. This type of symmetry is called s-wave.
In this thesiswe are going to investigatea di erentcase such that the amplitude,
still real in the absence of elds, changes sign in certain directions. The pairing
amplitude in these special directions vanishes. A particular choice of such a sym-
metry for the pairing amplitude is the dx2 y2-wave symmetry as shown to the left
in Fig. 1.1. The amplitude in a given direction is proportional to the distance from
the center of the four-leaved clover to the boundary times the indicated sign. The
orientation is xed with respect to the underlying crystal lattice. One peculiar
aspect of this type of symmetry is that despite the fact that we assume that the
crystal lattice is equivalent in the x and y directions the pairing amplitudes di er
in sign. This is called a spontaneously broken (discrete) symmetry.
As a measure of the pairing correlation BCS introduced the gap function (r)
where r is a chosen relative position between lattice points. As a quantum me-
chanical amplitude it is allowed to be a complex function. It should therefore not
be confused with a positive real excitation energy in general.
But in the particular case of an isotropic s-wave superconductor this thinking
is correct as can be seen on the bottom right of Fig. 1.1. The full curve is the
density of quasiparticle excitations in the superconductor. The gap function is in
this case just a positive constant and no quasiparticle excitations exist below
.
For the dx2 y2-wave superconductor however the gap function varies continu-
ously as we rotate the direction r from the minimum value to the maximum
+ . There still is a strongly enhanced density of quasiparticle excitations at
but states are found at all energies. At the lowest energies the density of states
is a linear function of energy n(E) / E. This is a direct consequence of the sign
change of the gap function.
During the last three years there has been a lively debate whether the gap
function in cuprate superconductors could be of dx2 y2-wave symmetry. Espe-
cially possible new aspects of the Josephson e ect 11] have started an intense
experimental and theoretical activity. I recommend the interested reader to con-
sult the recent review article by Beasley 12] and Scalapino 13]. Those authors as
well as the present one agree that we don't know for sure yet. But we have good
reasons to believe that there is a pronounced anisotropy of the gap function.
I hope that the theoretical aspects of dx2 y2-wave superconductors studied in
this thesis will be helpful in resolving the symmetryissue in the cuprates. Though
this in itself does not generate the solution to all problems of high Tc, to identify
the correct symmetry of a problem has often proven to be the key to a deeper
understanding.
14
15. E E
n(E)n(E)
∆ ∆
n nF F
Figure 1.1: The comparison of dx2 y2-wave and isotropic s-wave superconductors.
The wavefunction for electron pairs changes sign in the dx2 y2-wave
case, hence in certain directions there is a zero amplitude for pairing
correlations. In the isotropic case the pairing correlations are approxi-
matelyequalinall directions.Underneath wesee thedi erentdensities
of states n(E) for the two cases. The dx2 y2-wave superconductor has
a linearly increasing n(E) at low energies and a peak at . The dotted
line is our prediction if we include the e ect of weak non-magnetic dis-
order. The density of states follows a sublinear power law n(E) / E
with 0 < < 1 at low energies. In contrast the s-wave superconductor
possesses no electronic excitations below the gap .
15
17. 2. Superconductivity and BCS
Theory
The history of superconductivity goes back to the discovery by Kammerlingh
Onnes 1] in 1911 that mercury shows a sudden drop in resistivity below 4K.
Within experimental errors it seemed to be a perfect conductor. In 1933 Meiss-
ner and Ochsenfeld 14] found another astonishing property which is now the
fundamental experimental hallmark of superconductors, namely their perfect dia-
magnetism so that magnetic elds cannot penetrate into the bulk up to a certain
critical eld.
A truly microscopic theory of the phenomena was discovered by Bardeen,
Cooper and Schrie er 9] in 1957 and is known since then as the BCS theory.
It was extremely successful in explaining experiments and led even to new dis-
coveries such as the quantization of magnetic ux inside a superconducting ring
15, 16] and the Josephson e ect 17].
The materials which show superconducting low-temperature properties were
typically poor" metals in their normal state so that the free electron model had
to be modi ed. The superconducting current densities suggested that only a small
percentage of the valence electrons actually contributed to the phenomena. The
most likely candidates to change their properties are those close to the Fermi
surface, since they have arbitrarily small excitation energies in the free electron
model.
Leon Cooper 10] investigated a toy model of such a situation where he showed
that an arbitrarily small nonlocal attraction of two electrons which can only oc-
cupy states above an inert Fermi sea always form a bound state with respect to
the Fermi level. Such an e ective attraction could be found when electrons inter-
acted through the exchange of phonons and in the vicinity of the Fermi surface
it could even dominate the repulsive Coulomb interactions between electrons.
BCS then developed the idea that the true many-electron ground state should
be a coherent superposition of such Cooper pairs of electrons. There are many
excellent textbooks on the subject 18, 19, 20]. I will stick to a rather short
exposition of the main points. The interacting electron system can be described
17
18. 2. Superconductivity and BCS Theory
by the following general Hamiltonian (N is the number of unit cells)
H =
X
k;
kcy
k ck
1
2N
X
k;l;q; ; 0
Vk;l;qcy
k+q cy
l 0cl+q 0ck (2:1)
The rst term contains the single electron energy k which includes the chemical
potential for the electrons. This term is diagonal in momentum space and at
zero temperature the electrons simply ll up all states with negative energy, and
we have a sharp drop from one to zero in the occupation number at the Fermi
surface. The energy of the complete system is simply the sum over all negative
single electron energies.
The second term scatters electrons in and out of the Fermi sea (see Fig.2.1)
and if Vk;l;q is positive (attractive) the energy can be lowered by including con-
gurations with states occupied above the Fermi sea.
-k
k -l
l
k σ
σ σ
σ
l+q ’ ’l
k+q
Figure 2.1: Scattering processes due to the interaction between electrons for a gen-
eral potential (above) and the processes that cause the superconducting
transition (below).
A crucial step was to nd an Ansatz for the wavefunction which could describe
these pair correlations and still could be handled analytically. The interaction
conserves the total momentum of the pair and for the ground state we expect it
to be zero i.e. k and k. The total spin can be 1 or 0 corresponding to triplet
or singlet pairing and both have to be investigated. But for superconductivity
singlet pairing seems to be the relevant case so that the electrons have opposite
spin (k;") and ( k;#).
18
19. Ifthe wavefunctionis splitin two parts corresponding to unoccupiedor occupied
pair state cy
k;"cy
k;# BCS assumed that there is a xed phase between these two
parts so that they can interfere. This led them to consider wavefunctions of the
form
BCS =
Y
k
(uk + vkcy
k;"cy
k;#)j0i (2:2)
with jukj2 + jvkj2 = 1 and ukv k = u kvk.
At rst sight this wavefunction looks very peculiar. It doesn't contain a de nite
number of electrons and evaluating the energy we get indeed non-zero matrix
elements between states with di erent number of electrons. It can be shown that
these uctuations are indeed negligible in the thermodynamic limit and that the
electron density is well-de ned.
To nd the ground state we can minimize the energy EBCS by a variational
procedure. The expectation value of the single electron term in the BCS state is
2
X
k
kjvkj2 (2:3)
The interaction leads to three types of terms
1
N
X
k
X
l
Vk;l;0jvlj2
!
jvkj2 (2.4)
1
N
X
k
X
q
Vk;k;qjvk+qj2
!
jvkj2 (2.5)
1
N
X
k
X
l
Vk; l;l kul vl
!
ukvk + h:c: (2.6)
The rst two terms can be included in a rede nition of the single electron energy.
The third term however describes the scattering from one pair state to another.
It is this term which gives superconductivity and which xes the pair amplitudes
in such a way that the pairing amplitude for a certain k value depends on the
average of the other pairing amplitudes.
This average is contained in the gap function
k = 1
N
X
l
Vk; l;l kul vl (2:7)
Now we can write the total energy as a quadratic form in k-space
EBCS =
X
k
2 0
kjvkj2 + kukvk + kukvk
=
X
k
0
k(jvkj2 jukj2 + 1) + kukvk + kukvk (2.8)
19
20. 2. Superconductivity and BCS Theory
Minimizing the resulting 2 2 matrices termwise we nd that we can choose uk
real and
uk =
0
k + Ekq
( 0
k + Ek)2 + j kj2
; vk = kq
( 0
k + Ek)2 + j kj2
(2:9)
where Ek =
q
0
k
2 + j kj2. We get immediately that ukvk = k=2Ek. We can
insert this expression in the de nition of the gap function and get a non-linear
integral equation determining k self-consistently
k = 1
N
X
l
Vk; l;l k
l
2El
(2:10)
The trivial solution k = 0 at all k values gives simply the non-interacting
electron gas while a non-trivial solution is a possible superconducting ground
state. Usually only a discrete set of such solutions can be found making the form
of the BCS wavefunction stable against small perturbations.
There is a very neat way to get the same results in a self-consistent mean-
eld theory in
D
cy
k;"cy
k;#
E
= ukvk which neglects uctuations to higher than linear
order in
D
cy
k;"cy
k;#
E
cy
k;"cy
k;# 21, 22]. Within this mean- eld theory the low-lying
excitations can be found by a canonical transformation to a new set of Fermionic
states, called Bogoliubons, which are of the form
k;" = ukck" vkcy
k# (2.11)
k;# = vkcy
k"+ ukc k# (2.12)
In the limitof in nite volume the approximate Hamiltonian only contains a single
particle term in the Bogoliubons
H = EBCS +
X
k;
Ek
y
k; k; + ::: (2:13)
The energy of the Bogoliubons is given by Ek =
q
0
k
2 + j kj2. At low tempera-
tures the Bogoliubons are the main thermal excitations in a superconductor.
The BCS theory was subsequently developed to include lifetime e ects of the
quasiparticles in the normal state and retardation of the electron-phonon inter-
action (see for example ch.7 in Schrie er's book 18]) but it still relies on the
fundamental assumption that there is a phase-coherence between empty and oc-
cupied pair states of electrons i.e. an order parameter of the form
D
cy
k;"cy
k;#
E
.
Many superconductors were shown to t in the BCS scheme and the pairing
theory was successfully applied to liquid 3He and neutron stars.
On the other hand experimentalists developed new superconducting materials
with new peculiar properties. A milestone in this search was the discovery of the
cuprates which we will describe in the next chapter.
20
21. 3. The Cuprate Superconductors
Previous to 1986 most experimentalists and theorists were convinced that the
upper limit in Tc for any superconductor would be around 30 K.
In that year Karl Alex Muller and Georg Bednorz 2] from the IBM research
laboratory in Ruschlikon started a scienti c avalanche by discovering a new class
of materials which are now called cuprate superconductors or high-Tc compounds.
The great attention is due to the fact that the critical temperature Tc in these
compounds was eventually by 1993 raised up to 135 K in HgBa2Ca2Cu3O8+ at
atmospheric pressure 4] and in the same compound Tc can be increased up to
164 K at 30 GPa pressure 5].
This is more than ve times higher than in any conventional superconductor!
Below we give a list of the most prominent cuprates together with their transition
temperature.
Table 3.1: Some well-known cuprate superconductors with their critical temper-
ature Tc below which the materials are superconducting:
chemical formula short names Tc
La1:85Sr0:15CuO4 214 38K
Bi2Sr2CaCu2O8+ 2212, BiSCCO 80-91K
YBa2Cu3O7 123, YBCO 92K
Tl2Ba2Ca2Cu3O10 2223, TlBCCO 120-125K
HgBa2Ca2Cu3O8+ 1223, HgBCCO 135 K (164 K at 30 GPa)
The question arises which new features permit this high transition temper-
atures. The basic superconducting properties are the same as in conventional
superconductors e.g. zero resistance and perfect diamagnetism. The ux quanti-
zation in units of 0 = h=(2e) inside a superconducting ring 23] suggests that
the basic building blocks are once more paired quasiparticles of total charge 2e.
Even more astonishing is the fact that in an open ring that is bridged with a
conventional superconductor the same ux quantum is observed 24] so that we
expect a coupling between high-Tc and low-Tc pairs. If the nature of supercon-
ductivity would be totally di erent we wouldn't expect such a coupling. However
this statement has to be taken with a grain of salt since boundary processes can
21
22. 3. The Cuprate Superconductors
charge reservoirs
Cu-O planes
Figure 3.1: Schematic building blocks of the cuprate superconductors.
Cu : O :
Figure 3.2: The strong bonds between Cu and O atoms that cause the planar antifer-
romagnetic correlations
potentially couple order parameters with di erent symmetries.We will investigate
such a possibility in the chapter on the Josephson e ect.
The superconducting materials are derived from parent compounds without
superconducting properties. They are antiferromagnetic insulators as shown by
neutron scattering 25]. The typical structure for high-Tc compounds consists
of 2D layers of CuO2 which are separated by charge reservoir layers of varying
chemicalcontent between the di erent cuprate superconductors (see Fig.3.1). The
strong antiferromagnetic correlations in the CuO2 planes emerge from chemical
bonds between the copper 3dx2 y2 orbitals and neighbouring oxygen 2px and 2py
orbitals 26]. See Fig. 3.2 for a sketch of the bonding orbitals.
The CuO2 planes become increasingly conducting upon removing charge by
(hole) doping and at su cient doping levelsthere is a transition from a conducting
state into a superconductor below Tc. See Fig. 3.3 for a typical phase diagram.
The electronic structure of the undoped and doped CuO2 plane is sketched in
22
24. 3. The Cuprate Superconductors
U
∆CT
E
Cu 3d
O 2p
Cu 3d
U
∆CT
E
Cu 3d
O 2p
Cu 3d
Cu 3d
O 2p
Figure 3.4: Two di erent scenarios for the evolving electronic structure in the doped
CuO2 plane. On the left is the undoped insulating parent compound, in the
middle the moderately doped and on the right the strongly doped cuprate.
The upper row shows the presence of mid-gap states and the lower row the
scenario without mid-gap states.
Fig. 3.4. The undoped plane has two Cu 3d bands which correspond to singly
occupied states on Cu which is lled completely and a band which corresponds to
doubly occupied Cu which is completely empty. The splitting of those two bands
is governed by strong the Coulomb repulsion energy U on Cu. The O 2p band
falls within these two bands and is also completely lled. The gap between the
O 2p and the unoccupied Cu 3d band is the charge transfer gap which is in the
range of 1.5-2 eV 27]. The chemical potential lies in the charge transfer gap.
What happens upon doping is a matter of debate. One scenario is that the
chemical potential does not shift. However doping induces states in the mid-gap
region which are formed by a rearrangement of both conduction and valence band
states of the insulators and in the strongly doped (overdoped) region a single band
of mixed character remains 27]. Another scenario is that the chemical potential
drops immediately into the O 2p band removing progressively states from those
bands and at the same time the electronic structure of this band changes rapidly
creating a large Fermi surface and extended saddle points in the vicinity of the
Fermi energy 28] .
The nature of conductance in these materials seems also to be rather exotic.
While the undoped compound is a correlation induced insulator it is possible
to create a non-zero carrier density by doping. The carrier density is correlated
approximately linearly with the hole concentration in the CuO2 plane 27]. The
24
25. resistivity shows a pronounced anisotropy c= ab 40 100 of the value perpen-
dicular to the planes divided by the in-plane resistivity. This anisotropy increases
by lowering the temperature pointing to a strongly two-dimensional character of
the electronic states at low temperatures. At room temperature a typical in-plane
resistivity is of the order of 10 6 m while a typical metal has a value of 10 8
m. The value for the cuprates is rather typical for 3d transition metal oxides.
The in-plane resistivity falls however rapidly and approximately linear with de-
creasing temperature across several hundred Kelvin for the cuprates. If the hole
doping is less than a few percent the antiferromagnetic properties set in at TN,
the Neel temperature. For roughly 5 % - 25% hole doping no antiferromagnetic
long range order is present. This is the doping range where the cuprates have a
superconducting phase transition at Tc and the resistivity falls to zero together
with a perfect diamagnetic behaviour at su ciently small elds.
The question arises what is special about the cuprates. A comparison with other
layered transition metal oxides reveals the following unique properties 27]:
a large superexchange coupling J 0:1eV / t3
pd=( CT ( CT + U)),
a large on-site Coulomb energy U,
the smallest non-zero spin quantum number S = 1=2 on Cu,
two-dimensionality.
The large J comes from the small CT and the large overlap of Cu 3d and O
2p states. The combination of these four factors is somewhat complementary
because they combine to a high antiferromagnetic ordering temperature TN for
the undoped compound but the last two lead to strong quantum and thermal
uctuations of the magnetic correlations. Therefore a small percentage of doping
can destroy long range magnetic order and this ease of metalization" is quite
exceptional.
One interesting possibility for comparison is the recently studied compound
Sr2RuO4 29] of the same layered perovskite structure as the cuprate La2CuO4,
however Cu is substituted by Ru. The resistive behaviour is very similar to a
cuprate but below 100 K the c-axis resistivity also drops linearly with decreasing
temperature at an approximately constant anisotropy ratio until at Tc = 0:93 K
the undoped compound becomes superconducting. It remains to be seen whether
Tc can be enhanced in this compound by doping and which the important di er-
ences compared to the cuprates are.
During the past three years it has also become increasingly evident that not
only the normal state is peculiar but also the superconducting state. The main
impact came from photoemission and microwave absorption measurements. The
angular resolved photoemission experiments by Wells et al. 30] showed a strongly
anisotropic superconducting gap in Bi2Sr2CaCu2O8+ with a large gap along the
25
26. 3. The Cuprate Superconductors
Cu-O bond direction of about 20 meV and a nearly vanishing gap in the Cu-Cu
directions. This variation of the magnitude of the gap function is consistent with
a dx2 y2-wave gap. Recent measurements by Ding et al. 31] show evidence that
the direction of a zero gap might be not exactly at the dx2 y2-wave locations.
Microwave experiments by Hardy et al. 32] show a linear temperature depen-
dence of the magnetic penetration depth in YBa2Cu3O6:95. This was interpreted
as evidence of nodes i.e. planes in 3D momentum space where the gap function
changes sign. We can understand this interpretation if we study the linear re-
sponse of a superconductor in a weak magnetic eld, see e.g. 20] ch.25. In the
isothermal limit i.e. at zero frequency of the eld letting the wavevector go to
zero the London penetration depth is proportional to the inverse square root of
the density of superconducting electrons ns
L / 1=pns (3:1)
with ns /
X
k
k2 (f0( k) f0(Ek)) (3:2)
where f0 is the derivative of the Fermi-Dirac distribution, k the excitation energy
in the normal state and Ek the excitation energy in the superconducting state. At
low T,f0 is an approximate" delta function of width T.Therefore the rst term is
approximately constant but the second term is varying linearly with temperature
due to the linear density of states in the vicinity of a node. Hence we have
L / 1=
q
c1 c2T / 1 + c2
2c1
T ; since c1 c2: (3:3)
with a positive proportionality constant and hence L increases linearly with
temperature. Recently it has been shown by Mao et al. 33] that this linear tem-
perature dependence is also present in the c-direction penetration depth though
the absolute magnitude is increased roughly by a factor 10. In the same compound
the e ect of impurity doping by Ni and Zn has been measured 34]. The former is
most likely a magnetic impurity and the latter a non-magnetic impurity. Both of
them substitute for Cu in the CuO2 plane. However it is Zn which has a strongly
pair breaking e ect and 0.31 % of Zn is enough to make YBa2Cu3O6:95 gapless"
(in a isotropic superconductor this e ect is caused by magnetic impurities,see e.g.
35]), i.e. ns decreases with the square of the temperature at low temperatures,
not linearly anymore. Therefore L / T2. On the other hand up to 0.75 % Ni
has no such e ect. This was interpreted by Hirschfeld et al. as the e ect of strong
scattering in a dx2 y2-wave superconductor in the unitary limit 36].
Probably the most surprising experiments in the superconducting state are
those which test the directiondependence of the Josephson e ect. We will however
postpone their discussion to the last chapter and go instead through the theory for
the dx2 y2-wave superconductor. Whether a dx2 y2-wave order parameter really is
present in any cuprate superconductor is at the moment of writing still an open
26
27. question, but we will see that there's a fair amount of evidence in support of
dx2 y2-wave .
27
29. 4. Models for d-Wave Pairing
4.1 A short glance at group theory
As a rst step I want to explain some aspects of the group theoretical classi cation
for the pairing symmetries.
In quantum mechanics we often consider problems in an isotropic space so that
the Hamiltonian of the system is invariant under an arbitrary rotation R of our
coordinate system H(x) = H(Rx) i.e. all space directions are the same. So we can
generate new eigenstates by writing them in rotated coordinates H(x) (Rx) =
H(Rx) (Rx) = E (Rx).
In the Hilbert space of wavefunctions such a transformation can be described by
a unitary operator UR : (x) 7! (Rx) which doesn't mix invariant subspaces of
the Hamiltonian i.e. the transformed state (Rx) has the same eigenvalues as the
original state (x). In a subspace that consists of states with equal eigenvalue the
UR can be represented as matrices. Such a set of matrices UR forms a representa-
tion of the group of rotations. If we multiplytwo such matrices we get the matrix
that corresponds to the composition of the two matrices: UR1UR2 = UR1R2, fur-
thermore UR 1 = U 1
R . If the matrices are in addition unitary we call it a unitary
representation.
It is obviously important to nd all such symmetry representations for a given
Hamiltonian to split the Hilbert space into relevant pieces which can be diagonal-
ized separately and to nd the whole body of relevant eigenstates with the same
energy eigenvalue.
In a crystal we no longer get the same Hamiltonian back for arbitrary rotations.
There is only a nite set of allowed symmetrytransformations for a certain kind of
lattice. The possible unitary representation for these nite groups are well-known
and any possible eigenstate has to t into one of these reps.
The lattice symmetry forms a subgroup of arbitrary rotations and re ections
and the irreducible representations can therefore be labeled by the angular mo-
mentum of the simplest basis function. We list the possible symmetries of scalar
functions transforming according to a irreducible unitary rep under the symmetry
group D4 of the square lattice in Fig. 4.1.
In the cuprates the lattice symmetryis tetragonal, written symbolically D4h, or
orthorhombic due to small distortions depending primarily on the charge reser-
29
30. 4. Models for d-Wave Pairing
s d d g,l=4x
2
- y
2
xy
Figure 4.1: Symmetry of the scalar functions which transformaccording to a irreducible
unitary rep of D4. The labels underneath give the lowest allowed angular
momentum.
voirs which undergo changes by varying doping and temperature 37]. But since
there are superconducting phases found with both symmetries we restrict our-
selves to the case of tetragonal symmetry where the CuO2 planes form square
lattices.
In BCS theory the quasiparticle spectrum is of the form Ek =
q
2
k + j kj2.
The single particle energy is invariant under lattice symmetry transformations g
i.e. k = gk . Under the assumption that the spectrum Ek is invariant under the
lattice symmetries Ek = Egk we have j kj = j gkj, i.e. the complex gap function
k transforms according to a unitary representation. If we restrict ourselves to
singlet pairing we also have k = k.
4.2 Weak coupling models for d-wave pairing
In low-temperature superconductors many materials are consistent with an iso-
tropic gap with s-wave symmetry so why should it change now? The basic reason
to choose d-wave pairing is because experimentally 38, 27] there is strong on-site
repulsion and nearest-neighbour antiferromagnetic exchange. Though this means
in principle that we have to use a strong coupling model since the energy scale of
the interactions is comparable to the bandwidth we will take an easier route and
analyze the problem with the mean- eld treatment presented in the rst chapter.
It naturally will lead us to d-wave pairing.
Our rst basic assumption is that the important electronic states can be de-
scribed by a single band. We include in the single-particle energy k nearest- and
next-nearest-neighbour hopping as well as a constant term to shift the lling fac-
tor to appropriate values. The interaction consists of a repulsive Hubbard term,
antiferromagnetic nearest- and next-nearest-neighbour exchange i.e.
H =
X
k;
( 2t(coskx + cosky) 4t0 coskx cosky ) nk
30
31. 4.2 Weak coupling models for d-wave pairing
+1
2
X
r
U nr;"nr;# + 1
2
X
r;r0
(n:n:)
J Sr Sr0 + 1
2
X
r;r0
(n:n:n:)
K Sr Sr0 (4.1)
where Sr = 1
2cy
r; ~ cr; and ~ is the vector of Pauli matrices.
The energy of a BCS-state is then given by
EBCS = 2
X
k
kjvkj2 1
N
X
k;l
Wk;ljvkj2jvlj2 + 1
2N
X
k;l
Vk;lukvkul vl (4:2)
where N is the number of unit cells and
k = 2t(coskx + cosky) 4t0 coskx cosky
Wk;l = U + 1
2J(coskx coslx + cosky cosly) + K coskx cosky coslx cosly
Vk;l = 2U 3J(coskx coslx + cosky cosly)
6K(coskx cosky coslx cosly + sinkx sinky sinlx sinly) (4.3)
We introduce the renormalized single particle energy
0
k = k
1
N
X
l
Wk;ljvlj2 (4:4)
which leads to new values for the hopping energies t;t0 and for . According to
the BCS formulas (2.9) we get jvkj2 = 1
2(1
0k
Ek
) and ukvk = k=2Ek. Noting
that the e ective potential Vk;l is separable" i.e. each term in the sum is a
product of factors depending on k respectively l only, the gap equation allows
for three possible forms of a real gap function that belong to the representation
s;dx2 y2;dxy:
s
k = s
(000) + 1
2
s
(100)(coskx + cosky)
dx2 y2
k = 1
2
dx2 y2
(100) (coskx cosky) (4.5)
dxy
k = dxy
(110)sinkx sinky (4.6)
(4.7)
The subscript for the expansion parameters on the r.h.s. expresses that the
corresponding parameter of the gap comes from the interaction with a lattice
neighbour at relative position (xyz), see paper I for explanations. This subscript
will be dropped from now on. We get a self-consistency equation for each param-
eter in this expansion but which are the ones that lower the energy? Neglecting
the renormalization of the single-particle energy we note from Eq.(4.2) that only
31
32. 4. Models for d-Wave Pairing
the s-wave solution gives a non-zero contribution to the on-site repulsion i.e. the
term proportional to U since ukvk = k=2Ek and the integral over this quantity
is exactly zero for symmetries other than s-wave.
A similar symmetry analysis shows that the attractive nearest-neighbour ex-
change couples to both s and dx2 y2 and the attractive next-nearest-neighbour
exchange couples to s and dxy. So the best choice in the case U J K > 0
is a dx2 y2 gap function. K could eventually lead to an admixture with a relative
phase of =2 i.e. k = dx2 y2
k + i dxy
k .
The self-consistency or gap equations in this case will read
1
J = 3
8N
X
k
(coskx cosky)2
2Ek
tanh( Ek=2) (4.8)
1
K = 3
N
X
k
(sinkx sinky)2
2Ek
tanh( Ek=2) (4.9)
where = 1
kBT (at nite temperatures a factor tanh( Ek=2) is multiplied to
the integrand of the gap equation). These gap equations have to be solved at a
constant electron density. Numerically there is however a very weak dependence
of the electron density on the gap values since the bandwidth is much larger than
the maximal gap values. In Fig. 4.2 we plot J for a pure dx2 y2-paired state with
gap values at zero temperature that are appropriate for cuprate superconductors
with the following values for the hopping terms: t = 440 meV, t0 = 60 meV 39].
Together with = 330 meV this gives a lling factor of 0.78 of the band i.e.
22% hole doping in the normal state. Since there is a node of the gap i.e. points
with Ek = 0, an arbitrary small J gives rise to a non-vanishing gap.
In Fig. 4.3 J and K are plotted for a gap of symmetry dx2 y2 but with a small
imaginary admixture of dxy (again we have T = 0). The quasi-particle spectrum
is now nodeless and therefore we need nite values of J and K to sustain a gap.
In both cases the absolute values of J and K are considerably above results
from rst principle calculations 39] possibly due to the mean- eld treatment.
In the case of Bi2Sr2CaCu2O8+ tunneling measurements 40] give dx2 y2 30
meV. Solving the implicit relation for the gap at constant J numerically we nd
the typical form of the temperature dependent gap (see Fig. 4.4) with a value of
2 (T = 0)=kBTc = 4:3 somewhat higher than the BCS-value of 3.52 but not high
enough to yield the correct Tc which would require a gap ratio of about 7. This
may be once more a de ciency of the mean- eld approach.
Plotted on a dimensionless scale the gap falls almost BCS-like but at interme-
diate values somewhat faster (see Fig. 4.5).
The phase diagram for our model at zero temperature is shown in Fig. 4.6. If
we vary the antiferromagnetic couplings the gap equations allow at su ciently
low temperatures for non-trivial solutions of the indicated form. It is conceivable
32
33. 4.2 Weak coupling models for d-wave pairing
Figure 4.2: Strength of the nearest-neighbour antiferromagnetic exchange interaction
as a function of a pure dx2 y2-gap function.
Figure 4.3: Strength of the nearest- and next-nearest-neighbour antiferromagnetic ex-
change as a function of a mixed gap function of the form
dx2 y2
k +i dxy
k .
that the most likely pairing alternative is dx2 y2 since we usually expect J K.
We note also that the mixed phase is only existent in a fairly small parameter
region. The mixed phase generically exhibits a double transition, i.e. the real and
imaginary part of the gap function vanish at di erent temperatures. Only for J
and K on the dashed line the two contributions vanish at the same temperature.
The fact that we have superconducting phases at arbitrarily small parameter
values is due to the node structure of the gap. That parameter region is however
numerically di cult and the lines are extrapolated for J < 150 meV.
The phase boundaries are obtained by setting the gap in the corresponding
gap equations (4.8) and (4.9) to zero (e.g. dx2 y2 = 0) and evaluating the corre-
sponding coupling constants J,K at T = 0 using the value of the non-vanishing
33
34. 4. Models for d-Wave Pairing
Figure 4.4: The temperature dependence of a pure dx2 y2-gap.
Figure 4.5: The temperature dependence of a pure dx2 y2-gap on a dimensionless scale
compared with the famous BCS curve.
gap contribution (e.g. dxy 6= 0) to parameterize the curve. The dashed line is
obtained by setting the gap identically zero and varying the critical temperature
in the gap equations (4.8) and (4.9).
Genericfor d-wave pairing is the occurrence of zeroes in the quasiparticle energy
at the points on the Fermi surface where kx = ky since the gap function has
to be odd by re ections about this axis. Therefore the low-temperature physics
should be changed from what is known from conventional s-wave superconductors
where thereis a completedepletionof quasiparticlesbelow the constant gap .For
examplethespeci cheat fromthe Bogoliubons willcontributea termproportional
to T2 instead of being exponentially small.
In the case of the cuprates band structure calculations 41] and angle-resolved
photoemission 42] show that the shape of the Fermi surface is well described by
the single band with a nearest- and a next-nearest-neighbour hopping term.There
34
35. 4.2 Weak coupling models for d-wave pairing
Figure 4.6: Phase diagram of the superconducting states for our model given by the
nearest- and next-nearest-neighbour antiferromagnetic couplings J and K
at T = 0. Below the broken line in the mixed phase dxy +idx2 y2, the part
of the gap function with dxy-symmetry vanishes at a lower temperature
than the dx2 y2-part and opposite above the broken line.
are however extended saddle points close to the zone boundary 43] which seems
to be a strong coupling e ect 44].
The lineardensityof states at low energiesshould manifestitselfinan a bplane
tunneling conductivity for break junctions that is proportional to the square
of the voltage in contrast to conventional superconductors. This characteristic
feature was indeed observed by Forro et al. 40].
An important issue is the fact that we look at a quasiparticle pairing with
rather strong interactions. This leads to serious problems since we should start
from a variational ground state that already incorporates the e ect of these in-
teractions. For strong on-site repulsion one should therefore include the single
occupancy constraint i.e. the ground state must not contain doubly occupied
sites. The question is whether a re ned treatment of the interaction will change
the symmetry properties of the order parameter.
Before dealing with some of the strong coupling approaches I want to mention
35
36. 4. Models for d-Wave Pairing
another model which predicts a dx2 y2-wave order parameter for the cuprates.
Pines and coworkers 45, 46, 47] considered a weak coupling model where the
physics of the CuO2 planes is described by an antiferromagnetic Fermi liquid"
starting from non-interacting quasiparticles which have the excitation spectrumof
the tight-binding model k = 2t(coskx+cosky) 4t0 coskx cosky. The interaction
is described by quasiparticle-paramagnon scattering. The quasiparticles scatter
against a magnetic background characterized by a spin susceptibility which ts
NMR experiments,
(q;!) = Q
1 + 2(q Q)2 i!=!SF
(4:10)
where Q is the static spin susceptibility at Q = ( ; ), the antiferromagnetic
correlation length and !SF the paramagnon energy. We see that the static sus-
ceptibility ! = 0 is peaked at the antiferromagnetic wave-vector Q and that the
imaginary part Im( (q;! ! 0)) ! at all q. This scattering was shown to give
the linear in T in-plane resistivity ab in the normal state 46].
Solving the Eliashberg equations 48] the exchange of antiferromagnetic para-
magnons between the electron quasiparticles leads to a phase transition into a
superconducting state with dx2 y2 symmetry. It is however unclear how the phe-
nomenological spin susceptibility can be justi ed within a microscopic description
of the interactions 49].
4.3 Strong coupling models for d-wave pairing
As we have seen earlier the electronic structure of the CuO2 plane involves a large
energy scale from Coulomb interaction on the Cu sites compared to the band
widths. A natural choice for a Hamiltonian would be to start from a two band
model e.g.
H =
X
i;
ddy
i di +
X
l;
ppy
l pl (4.11)
X
hi;li;
Vildy
i pl + h:c: (4.12)
+U
X
i
dy
i"di"dy
i#di# (4.13)
with obvious notations and taking only into account the overlap between Cu and
O and a strongly repelling Hubbard term on Cu. It has been shown by Zhang
and Rice 26] that this model reduces for realistic values at low doping to a single
band t J model
Ht J = t P0
X
<r;r0>;
cy
r cr0 + h:c: P0 + J
X
<r;r0>
Sr Sr0
1
4nrnr0 (4:14)
36
37. 4.3 Strong coupling models for d-wave pairing
where P0 is a Gutzwiller projection operator eliminating states with doubly oc-
cupied sites. This model is the strong coupling limit of a single band Hubbard
model (t Ueff)
HHubbard = t
X
r;r0;
n:n:
cy
r cr0 + Ueff
X
r
nr;"nr;# (4:15)
Note that Ueff CT 6= U, the e ective Hamiltonian has not the bare Coulomb
repulsion on Cu but a reduced value. The physical explanation is that the hole
which resides on O hybridize with Cu to form a local singlet around each Cu
atom. This singlet then moves through the lattice of Cu ions in a similar way as
a hole in the single band e ective Hamiltonian.
It has been argued however that it is essential to keep the two band structure
intact 50] since it could play a key role for the normal state of the cuprates.
A lot of insight into the t J model has come from numerical diagonalization
of small lattices. For an extensive review see Dagotto's review article 51]. The
pairing susceptibility at small doping seems indeed to be favourable in the dx2 y2-
wave channel. This might indicate that the pairing symmetry remains intact in
going from weak to strong coupling. A slave boson approach to the t J model
also has found evidence of dx2 y2-wave pairing in the appropriate range of doping
52].
Daniel Rokhsar 53] has proposed an analogue to Cooper's problem 10] to
determine the correct symmetry of the order parameter. He considers the motion
of a singlet pair of holes in a spin liquid characterized by local spin correlations.
The hole singlet is formed on nearest- and next-nearest-neighbour sites which
overlap due to e ective hopping processes which can be expressed in terms of the
electron hopping processes and the spin correlations.
In a doped disordered antiferromagnet with only local Neel like correlations and
no broken symmetries the pair ground state has dx2 y2 symmetry for t0=t > 0:11
(t0 is the diagonal next-nearest-neighbour hopping). In a doped chiral spin liquid
of anyons 54], the new feature is the introduction of a non-zero triple product
hSi (Sj Sk)i 0:07 which gives rise to an imaginary part of the e ective hop-
ping terms. The pair wave state is then a complex d-wave state of the symmetry
dx2 y2 +idxy. The latter phase has also been advocated by Laughlin who used an
anyon approach to the t J model 55].
I want to conclude this chapter with some words of caution. Though I described
here models which potentially predict a dx2 y2-wave order parameter we should
be aware that all these models contain essential simpli cations of the electronic
structure of the cuprates. A priori we do not know which simpli cations are
allowed in order to catch the essential physics of the problem. Therefore I do not
regard any of those models to give a de nite proof for dx2 y2-wave pairing in the
cuprates but they demonstrate that this is certainly a viable option.
37
39. 5. Disordered d-Wave
Superconductors
Why do we study the e ects of disorder on dx2 y2-wave superconductors ? This
question is of course related to the question how we can possibly identify a dx2 y2-
wave superconductor in real experiments. One of the characteristic features is the
existence of power laws in di erent physical quantities, for example the depen-
dence of the break junction conductivity on the square of the applied voltage
coming from the linear density of states as noted by Zhou and Schulz 56] and
independently in paper I. The question is therefore whether disorder is capable
to change such power laws. Indeed we will nd that non-magnetic disorder can
change the exponent so that their experimental value could serve to characterize
the sample properties with regard to disorder.
Since the dx2 y2-wave symmetryhas an intrinsicallyanisotropic gap function we
would expect that even non-magnetic impurities potentially reduce the Tc in an
analogous way as magnetic magnetic impurities in isotropic s-wave superconduc-
tors since they are both potentially pair breaking. The latter case is for example
explained in Rickayzen's book 35], chapter 8.5. The magnetic impurities reduce
both the gap and Tc and enhance the density of states below the gap leading even
to a nite density of states at the Fermi level with a still non-vanishing gap. This
is possible due to the strong damping (decay) of the quasiparticle states.
We expectthat the reduction of the gap and Tc also are present for a dx2 y2-wave
superconductor. However since the gap belongs to a certain symmetry represen-
tation we expect that this fact is robust for su ciently weak disorder i.e. there is
a nite range of impurity concentrations where the symmetry of the gap remains
d-wave. Within this range we can study the residual e ects of the impurities on
an energy scale T T(imp)
c . This systems still has the manifest dx2 y2-wave sym-
metry of the pairing state but as we will show the spectrum of the quasiparticles
is a ected at low energies.
As our starting point we will therefore use already the renormalized value of
the dx2 y2-wave gap. This is probably a strength of the analysis because otherwise
we would have to nd a suitable gap equation which we solve in the presence of
disorder. This would limit us to a particular model and as we argued previously
there is no general agreement on what is supposed to be the correct gap equation
39
40. 5. Disordered d-Wave Superconductors
0.0 5.0 10.0 15.0
E
0.0
100.0
200.0
N(E)
Figure 5.1: The piecewise constant curve shows the integrated density of states
N(E) for a 16 16 lattice in momentum space in the vicinity of a
node. Progressively deviating curves are for increasing range V;V ]
of the disorder potential. The plotted curves have V = 0, V = 2,
V = 4, V = 6 and V = 8. The energy is normalized such that one
momentum lattice spacing times the Fermi velocity is equal to one.
for the cuprates.
Our case di ers from the strong non-magnetic impurity scattering in a dx2 y2-
wave superconductor which has been studied extensively rst in the context of
heavy fermion superconductors 57, 58] and later in the cuprates 59]. This is in
essence a single-site approach taking however into account all repeated scattering
on the same impurity. At su ciently strong impurity strength a virtual bound
state 60] is created at the impurity which is believed to give rise to an impurity
band at nite concentrations.
5.1 Numerical simulation
To gain some insight we try to solve the problemnumerically rst. We use a 16 16
square lattice in momentum space which simulates the states in the vicinity of a
single node and determine the spectrum numerically with and without disorder.
The results for the integrated density of states
N(E) =
Z E
F
n(E)dE (5:1)
are shown in Fig. 5.1.
40
41. 5.1 Numerical simulation
For the pure system we see that the discrete nature of the lattice clearly shows
up in a piecewise constant N(E) but the overall shape follows rather well a
parabola except at the lowest energies where the lattice is not dense enough
and at the highest energies because the cut-o is not sharp in energy due to the
square boundary of the lattice. Therefore we expect that any scaling behaviour
typical for the conical spectrum would show up in the intermediate energy range.
The non-magnetic impurity potential was randomly distributed in real space
in the interval V;V ]. N(E) exhibits the following features:
The disorder causes both N(E) and its derivative n(E) to increase at low
energies with the opposite trend at high energies. The e ect gets stronger
as V is increased.
N(E) becomes much smoother as a function of E so the lattice e ects are
not as pronounced.
The bandwidth is broadened.
Particle-hole symmetry is obeyed on average i.e. within numerical accuracy
there is a one to one correspondence between positive and negative energy
states with E (not shown explicitly in Fig. 5.1).
We used here a speci c realization of the disorder potential without adjusting
the chemical potential but the lattice is su ciently large that this shift is within
numerical accuracy zero hence we have particle-hole symmetry. By plotting the
data on a doubly logarithmic scale excluding the lowest 20 and the highest 70
states we nd indeed a very nice scaling behaviour of N(E) / E1+ , see Fig.
5.2, across more than half a decade. That's the best we can expect for our small
lattice. The estimated variance of the data as determined from linear regression
is on the order of 10 3 or smaller.
In Fig. 5.3 we plot the corresponding power for the density of states n(E) / E
as a function of V 2= 2, where corresponds to the cut-o which gives the range
of energies that are well approximated by a conical spectrum i.e. the maximumof
the dx2 y2-wave gap. For our lattice = 10. The plot indicates a linear decrease
of with increasing V 2= 2. Observe that we get for the pure case = 0:95 and
not 1 due to our nite lattice size.
We nd that n(E) / E0:18 for V 2= 2 = 0:64. Interestingly this power law
has indeed been observed in the temperature dependence of the zero voltage
conductivity of a Pb/YBa2Cu3O7 junctions with a normal lead electrode and a
superconducting YBa2Cu3O7 electrode by Valles et al. 61] and they found that
G(V = 0;T) / E0:18. If we assume that this conductivity comes from thermally
excited quasiparticles this would indicate a disorder potential in YBa2Cu3O7
roughly on the order of the energy gap.
41
42. 5. Disordered d-Wave Superconductors
0.2 0.4 0.6 0.8
Log(E)
1.2
1.4
1.6
1.8
2.0
2.2
Log(N(E))
Figure 5.2: log10-log10 plot of the integrated density of states. Decreasing slope
means an increase in the range of the impurity potential. The slope
decreases linearly with the range of the disorder potential.
0.0 0.2 0.4 0.6 0.8 1.0
V
2
/ ∆
2
0.0
0.2
0.4
0.6
0.8
1.0
α
Figure 5.3: The exponent in the power law for the density of states n(E) / E
as a function of V 2= 2.
42
43. 5.2 Perturbation theory
This numerical analysis should give us con dence that the problem of disorder
is potentially relevant for experiments and that it is worth to pursue with more
advanced methods. As a second step we use conventional perturbation theory to
gain more insight into the solution.
It will nally emergethat the appearance of a non-trivial power law is analogous
to an e ect in the theory of electrons in 1+1 dimensions with short range inter-
actions where correlation functions can have non-trivial power laws depending on
the interaction strength (see e.g. Ref. 62, 63]). For our case it is the strength of
the disorder potential V 2 which will play a similar role.
5.2 Perturbation theory
Paper IV deals exclusively with the perturbative approach to the problem. In
this introduction we concentrate on the emerging picture rather than on the
calculational details.
The perturbation theory allows to express the inverse Green's function matrix
in terms of a self-energy which can be expanded in a small parameter
G 1(k;!n) = G 1(k;!n) (k;!n); (5.2)
= i~!n ~k 3 + ~k 1 : (5.3)
The bar on the left hand side stands for an average over di erent impurityrealiza-
tions hence we makethe assumption that the Green's function has a self-averaging
property for macroscopic samples. !n denotes a Matsubara frequency.
The self-energy expansion (k;!n) can be illustrated in terms of Feynman
diagrams shown in Fig. 5.4. If we x the external momentum to be at a node
and only take into account scattering with small momentum transfer we nd the
following frequency renormalization
~!n=!n = 1 + g log j!nj
!
+ (g2 1
2g2)log2
j!nj
!
: (5.4)
The second term comes from the rst order diagram with a single closed impu-
rity line and the third term from the uncrossed second order diagram with two
impurity lines and the crossed second order diagram. We see that the Feynman
diagrams provide an expansion in glog( =j!nj) which is our small parameter. The
contributions from the crossed diagram are of the same order as the uncrossed one
and hence no Migdal's theorem 64] can be formulated for this problem. To calcu-
late the density of states we have to make the analytic continuation i!n ! E+i
and take the limit of small values of E. This means that we have to sum up the
leading contributions to in nite order. This is however a formidable task but we
can hope to get a qualitative insight already at second order.
43
44. 5. Disordered d-Wave Superconductors
p
Σ =(k, )nω
p q p
+
+ ...
++
p
Σ =(k, )nω Σ ( )ωn
+ Σ ( )ωn
(1)
(2)
uc
+ Σ (k, )ωn
(2)
+
+ ...
cr
qp+q-k
Figure 5.4: The standard diagrammaticexpansion of the self-energy (k;!n) for a
weak static random potential. The full line stands for the propagation
of a quasiparticle with a certain momentum and each broken semi-
circle corresponds to the Born scattering at the same impurity center.
The frequency !n is conserved and therefore omitted.
44
45. 5.2 Perturbation theory
In order to understand the nature of disorder in this problem let us have a look
at the rst order self energy.The correct formwhich allows analytical continuation
is
(1)(!n) = i!n g 1
2 log
2
(i!n)2 (5.5)
) (1)(E + i ) !0! E g 1
2
"
log
2
E2 + i
#
: (5.6)
We see that the disorder indeed induces an imaginary part of the self energy for
real frequency (energy) E which corresponds to the decay of Bloch states in a
disordered media but at su ciently low energies E the real part dominates which
corresponds to a renormalized energy. We can understand this by the fact that
if we decrease the range of energies in which we allow scattering processes to
occur the corresponding momentum range gets arbitrarily well de ned. This is in
contrast to the case when we have an extended Fermi surface and the momentum
range therefore would converge to a continuum of allowed momenta.
Hence we expect that the dominant e ect of disorder in a dx2 y2-wave su-
perconductor is a renormalized energy which changes the density of states as a
consequence.
Apart from the terms considered above we also have to take into account the
momentum dependence of the self-energy. The rst contribution comes from the
crossed second order diagram. These terms renormalize k and k. They are
however less divergent for small scattering strength containing at most a single
logarithm / g2 log =Ek.
We see that the important point here is the existence of a zero-dimensional
Fermi surface" in a 2D momentumspace which is typical of the conical spectrum
of the dx2 y2-wavesuperconductor. The third componentof momentumis assumed
not to change the dispersion strongly so that we can always neglect it completely.
If this is not the case the disorder e ects di er radically 65, 66]. The problem
is simpli ed by the fact that the diagrams with crossing impurity lines are not
divergent, and one can easily sum the remaining diagrammatic series. Weak non-
magneticdisorder 65] givesriseto a nitedensityof states at zeroenergy,n(0) 6= 0
in this case.
A 2D conical spectrum is not unique to the dx2 y2-wave superconductors, it
also occurs in zero-gap (degenerate) semiconductors 67], and in heterojunctions
where the contact is made between semiconductors with inverted symmetry of
bands 68]. It can be realized in two-dimensional graphite sheets 69], as well
as for lattice electrons in a strong magnetic eld, the most well known example
being a tight-binding model of Fermions with 1/2 of a magnetic ux quantum
per plaquette ( ux phase) 70]. An interesting generalization of this model has
recentlybeen considered in the context of the quantumHalle ect 71]. The conical
spectrum is also a property of hypothetical orbital antiferromagnets and spin
45
46. 5. Disordered d-Wave Superconductors
n(E)
Ε
Λ
a)
c)
b)
Figure 5.5: Qualitative picture of the density of states n(E) in second order per-
turbation theory including the e ects from the rst crossed diagram.
a) Non-magnetic disorder is characterized by an in nite slope at zero
energy but n(0) = 0. b) A random magnetic eld in the z-direction
gives rise to a non-zero n(0). c) A randomly uctuating imaginary
isotropic gap i s gives a nite range of values where n(0) = 0. The
dotted line is the pure density of states. is the cut-o .
nematic states 72, 73, 74].
Nevertheless these terms have important consequences. To illustrate this we
have varied the nature of the disorder potential. We consider the perturbative
approach for the following types of quasiclassical disorder potentials with only
small momentum transfers:
a) a non-magnetic random potential
b) a magnetic eld in the z-direction with a random magnitude and
c) a pairing state with symmetry dx2 y2 +idxy with a small random second com-
ponent.
The results for the density of states are shown in Fig. 5.5 for the separate cases.
The non-magnetic disorder is dominated by the terms which renormalize the
frequency only but in the other two cases the contributions of the momentum
dependence cannot be neglected. The in nite derivative of the density of states
is in agreement with the sublinear power law observed numerically in the last
section.
46
47. 5.3 Field theoretical methods
5.3 Field theoretical methods
For the present problem which involves disorder for a conical spectrum the eld
theory approach was pioneered by Fisher and Fradkin 70, 67] when they con-
sidered the most general type of disorder potential present in a conical spectrum
containing all the cases a), b) and c) listed above. Their conclusion was that the
disorder always creates a nite density of states at zero energy n(0) > 0. This can
be understood by the fact that part of the disorder potential describes a random
chemical potential which gives on average a non-zero occupation of positive en-
ergy excitation excitations of the conical spectrum and hence an extended Fermi
surface on average.
It is shown in paper III that a) is an exactly marginal case. The non-magnetic
quasiclassical disorder will therefore give rise to a line of xed points labeled by
the disorder strength and continuously varying power laws for e.g. the density of
states n(E) / E which is sublinear < 1. Taking into account processes with
large momentumtransfer the multi-node structure gives rise to an additional non-
Abelian symmetry and leads to a universal power law for the density of states.
The case b) of a magnetic eld in the z-direction with a random magnitude is
shown to be a relevant perturbation destroying criticality and a nite density of
states at zero energy n(0) > 0 is the result. The case c) of a random imaginary
gap component with di erent symmetry is irrelevant and does not change the low
energy physics.
Hence we encounter the most interesting low-energy physics in case a) the
non-magnetic disorder where the density of states remains zero at zero energy.
How come? Though a slowly varying potential could be interpreted as a chemical
potential for the normal electrons this is no longer true in the dx2 y2-wave state
due to the special dispersion relation for the quasiparticle excitations.
Ek !
q
( k V (x))2 + 2
k (5:7)
in the case of a slowly varying potential and we see that at every location x we
still have a conical spectrum with zero chemical potential and hence n(0) = 0
everywhere, the cone is just shifted in the local" momentum space. Since k and
k can be used as local coordinates in momentum space the spectrum is of the
form
E =
q
(k1 A(x))2 + k22 (5:8)
which is the spectrum of 2+1D Dirac Fermions (e = c = h = 1) in the presence of
a special anisotropic random gauge eld. We see that the random non-magnetic
potential is translated into the random gauge eld and not a random chemical
potential of the excitations, i.e. Dirac Fermions. The situation is illustrated in
Fig. 5.6.
47
48. 5. Disordered d-Wave Superconductors
E
k
k1
2
A(x)
Figure 5.6: The e ect of a slowly varying non-magnetic random potential on the
quasiparticle spectrum of a dx2 y2-wave superconductor is the same
as an anisotropic random gauge eld on 2+1D Dirac Fermions. The
cone shifts to a new position in momentumspace depending on A(x).
Adding a magnetic eld in the z-direction with a slowly varying random mag-
nitude would result in a di erent spectrum
E =
q
k21 + k22 + B(x) (5:9)
which locally shifts the cones upwards or downwards and hence breaks particle-
hole symmetry i.e. it is in fact a random chemical potential for the conical spec-
trum. This important fact has been pointed out already by Fisher and Fradkin
70].
Adding an imaginary part of the gap function with dxy or s-wave symmetry
with a slowly varying random magnitude would result in a random mass term in
the spectrum
E =
q
k21 + k22 + m2(x) jm(x)j (5:10)
and it therefore has the tendency to create a gap in the spectrum.
In the subsequent sections we will demonstrate how this problem of 2+1D Dirac
Fermions with quenched disorder can be simpli ed if we deal with single particle
properties when only one single frequency (energy) is involved in the propagators
or correlation functions. Due to the STATIC nature of disorder the timedirection
can be eliminated from the problem and will enter simply as a parameter of a
2D theory. This 2D problem will be mapped by the replica trick to a special
48
49. 5.3 Field theoretical methods
limit of 1+1D interacting Fermion systems. Those systems will be solved non-
perturbatively by Abelian and non-Abelian Bosonization. The conical spectrum
is essential for this construction to work!
5.3.1 The Fermionic path integral
The material in this subsection follows partly the review article by Shankar 75]
which provides an excellent introduction to the renormalization group techniques
which I assume that the reader is familiar with.
Usually what we want to calculate are correlation functions of the form
hOi = hexp( H)Oi
hexp( H)i (5:11)
where the denominator is the partition function Z = hexp( H)i of the prob-
lem and h i denotes the trace taken over the Fock space. The problem would be
simpli ed if we could calculate these properties in a basis of eigenstates of the
Fermionic eld operators or y. Due to the exclusion principle such an eigen-
state cannot be constructed within the algebra of complex numbers. It is however
possible if we introduce the algebra of Grassmann numbers. Let's look at a single
Fermionic degree of freedom for which the Hilbert space consists only of j0i and
j1i, denoting the vacuum state or the occupied state respectively. We de ne now
a coherent Fermionic state by
j i = j0i j1i; (5:12)
and independently thereof D
= h0j h1j : (5:13)
In the case of many Fermionic degrees of freedom we attach a complete set of
quantum numbers as indices to the Grassmannians . So what are the rules for
computation with Grassmannians: all Grassmannian numbers anticommute with
each other and with all Fermionicoperators. They also anticommutewith bra and
ket vectors if they contain an odd number of Fermionic states and they commute
otherwise. With this set of rules it can easily be veri ed that
j i = j i and
D
y =
D
(5:14)
Due to the anticommutation with themselves only linear terms in Grassmannians
are non-zero therefore any function of Grassmannians is determined by the rst
two terms of the Taylor expansion in the corresponding .
At the next step we introduce the integral" over Grassmannians which is in
principle a bad concept since no measure of distance is introduced so we regard
the following rules just as formal de nitions
Z
d = 1 and
Z
d = 0 (5:15)
49
50. 5. Disordered d-Wave Superconductors
The di erential d is also a Grassmannian and anticommutes therefore with .
Using these rules we nd for example
Z
exp( M )
nY
i=1
d id i = det(M); (5:16)
where M is an n n matrix and denotes a n-component vector of Grassman-
nians. A resolution of the identity matrix is
=
Z
j i
D
exp( )d d (5:17)
and an expression for the trace of a bosonic" operator, i.e. every term in the
operator contains an even number of fermionic operators is
Tr(O) =
Z D
Oj iexp( )d d (5:18)
This enables us now to nd a path integral representation for the partition func-
tion. We assume that the Hamiltonian is written in a normal ordered form i.e. the
vacuum expectation value has been subtracted from it (H = : H :). It is found
by rewriting
exp( H) = limN!1
1 NH
!N
(5:19)
and inserting a resolution of the identity between each factor (1 H) where
= =N and letting ! 0 in the end. The result is
Z =
Z
N= 1
NY
i=1
exp i+1 i
H( i+1; i)
! !
d id i; (5.20)
=
Z
( )= (0)
exp
Z
0
( ) @
@ ( ) H( ( ); ( ))d
!
h
d d
i
:(5.21)
In the last line the limit ! 0 for the time" slices has been taken. d d ] =Q
d d for all fermionic degrees of freedom. We have substituted here each
fermionic operator in the Hamiltonian with the corresponding Grassmannian and
the index i of the Grassmannian labels the imaginarytimesliceto which it belongs
(we have a complete set of Grassmannians for every resolution of the identity),
and in the last line a symbolic continuum limit has been taken so that the Grass-
mannians depend on a continuous parameter . How should we understand the
derivative of the Grassmannians with respect to ? Let's introduce the Fourier
transform of the Grassmannians with respect to which is well-de ned being a
linear transformation:
( ) = 1 X
n
exp(i!n ) (!n); (5.22)
( ) = 1 X
n
exp( i!n ) (!n): (5.23)
50
51. 5.3 Field theoretical methods
The sum goes over the fermionic Matsubara frequencies !n = (2n + 1) = con-
sistent with the antiperiodic boundary conditions of the path integral. Inserting
these Fourier transforms the partial derivative @=@ is well-de ned giving a
factor i!n. The result is
Z =
Z
exp
X
n
(!n)i!n (!n) H( ; )
!
h
d d
i
: (5:24)
Naturally we have to insert the Fourier expansion in the Hamiltonian as well and
derivative terms are understood in the same way.
Correlation functions can now be computed as
D
( 0) ( 00)
E
=
R
( 0) ( 00)exp
R
0 ( ( ) @
@ ( ) H( ( ); ( )))d
h
d d
i
Z (5.25)
Now if we write out explicitlyall other quantum numbers (e.g. r) for the fermionic
states the expression reads
D
( 0;x) ( 00;y)
E
=
R
( 0;x) ( 00;y)exp(S)
h
d ( ;r)d ( ;r)
i
Z (5:26)
with
S =
Z
0
d
Z
dr
(
( ;r) @
@ ( ;r) H( ; )
)
: (5:27)
S is the action for the fermionic path integral.
5.3.2 An excursion to Dirac Fermions and back
Let us now insert our speci c form of the Hamiltonian. Let's consider only a single
node at rst. After linearizing and transforming back to real space the dx2 y2-wave
BCS Hamiltonian in Nambu-Gor'kov notation ( y
k = (cy
k";c k#) ) transforms to
HBCS =
Z
d2r y(r)( ick@1 1 ic?@2 3) (r) (5:28)
where ck and c? denote the gradient of the gradient of the gap function and the
single particle energy at the node respectively. The terms linear in momentum
have transformed to derivatives in the corresponding direction. The quasiclassical
part of the impurity potential transforms to
Himp =
Z
d2rV (r) y(r) 3 (r): (5:29)
V (r) is a Gaussian random eld with with zero average such that V (r)V (r0) =
(r r0).
51
52. 5. Disordered d-Wave Superconductors
Let's see what kind of action we get from the sum of HBCS + Himp if we use
the Fourier expansion in Matsubara frequencies
S =
X
n
Z
d2r (!n;r)i!n ( ;r) (5.30)
(!n;r)( ick@1 1 ic?(@2 V (x)=c?) 3) (!n;r) (5.31)
This means that the partition function factorizes into a product for each !n:
Z =
Y
n
Zn (5:32)
where Sn is given as a termin the action S with a xed Matsubara frequency !n. If
we calculate a correlation function for Fermionswhich all have the same frequency
the parts with di erent frequencies cancel both in numerator and denominator
such that the dimensionality of the problem is reduced by one and !n is only a
parameter of the theory. The crucial point is that we have free non-interacting
particles and the impurity potential is static and therefore the frequency is a con-
served quantity. In the diagrammatic language this would mean that the electrons
propagate with a given energy which is conserved in the presence of disorder.
Now we are ready to compare with massive Dirac Fermions in 2 Euclidean
dimensions. For details I refer the interested reader to the book by Zinn-Justin
76]. The action is given by
SDirac =
Z
d2r (r) (@ + iA ) + m] (r); (5:33)
Remember that the name Euclidean means that a continuation to imaginary time
of the Minkowski action. The key observation is now that this action is the same
as in our problem if we rst make a =2 rotation in the spinor space around the
1 axis such that 3 ! 2 and with the identi cations
m = !n; 1 = 1; 2 = 2; (5:34)
and a rescaling of the coordinates
r1 ! ckr1; r2 ! c?r2: (5:35)
Furthermore we have
(A1;A2) = (0;V (x)=c?); (5:36)
so that the disorder potential can be identi ed as a random gauge eld of the
Dirac electrons. The actions then di er only by a global factor i. This merely
adds a prefactor to the Green's function and can easily be taken into account
We can identify r1 as the the imaginary time direction which was a real space
direction of the original Fermions and r2 as the space direction of the Dirac
52
53. 5.3 Field theoretical methods
Fermions. The mass of the Dirac Fermions is given by the Matsubara frequency of
the original Fermions. BEWARE!!! Correlation functions of the original Fermions
have to be replaced according to the rules above and MUST NOT be substituted
with the corresponding physical quantities of the Dirac Fermions! As an example
we write out the density of states for the original Fermions of the dx2 y2-wave
superconductor:
n(E) = 1Im
D
(!n;r1;r2) (!n;r1;r2)
E
i!n!E+i
; (5.37)
= 1
c?ck
Re
D
( ;r) ( ;r)
E
i!n!E+i
: (5.38)
So what we calculate is in fact the mass renormalization" for the 2D Dirac
Fermions in an anisotropic random gauge" eld for small" given mass. We will
see below that in fact the density of states corresponds to an order parameter 70]
and therefore a zero density of states n(E) = 0 is connected to the criticality of
the disordered system. Therefore symmetry arguments known from the theory of
second order phase transition will play a key role to establish the zero density of
states in our case.
5.3.3 The replica trick
Now we want to address the question how to handle the quenched random disor-
der potential. The word quenched means that there is no relaxation process for
random variables, they do not equilibrate with the other degrees of freedom. We
have to deal with one given realization of disorder. There is a way to make such
a model tractable and that is to assume self-averaging. It assumes that physical
quantities like the density of states can be calculated by splitting up the systems
in many smaller parts and calculate the density of states in each of them and
taking an average over all these smaller samples instead. So this amounts in rst
calculating for a given realization of the impurity potential and then averaging
over the di erent realizations of disorder i.e.
hOi =
Z
DA P A] hOiA (5:39)
In the integral there is still the dynamicalaverage to be taken at xed A and this is
an inconvenient object. We use the replica trick 77] to get rid of the denominator
in the correlation function at the expense of introducing an additional integer
index for the Fermions and taking an analytic continuation of the total number
of these indices to zero. So we can for example rewrite the Green's function at
given A as follows:
D E
=
Z
DA P A]
R
exp(S A]) d d ]
Z A] ; (5.40)
53
54. 5. Disordered d-Wave Superconductors
= limr!0
Z
DA P A]
Z
exp(S A]) d d ]Zr 1 (5.41)
= limr!0
Z
DA P A]
Z
1 1 exp(
rX
a=1
S a; a;A]) d ad a]; (5.42)
= limr!0
1
r
Z
b b exp(Sr a; a]) d ad a]: (5.43)
where we assumed replica symmetry and that it is possible to do the functional
integration over A so that we arrive at an e ective disorder free action Sr a; a].
Now the central point is that if we can do the functional integration over A
we can get rid of disorder at the beginning and consider instead a system with
translational invariance. The price we pay is that the nal result is not described
by a limit of such systems which may or may not remind us of anything we are
familiar with.
Since the disorder potential enters the original action linearly we can indeed
perform the functional integration in the case of a Gaussian eld with
P A] = exp 1=(2g)
Z
d2rA(r)2 (5:44)
by completing the square
R
DAexp
h
1=(2g)
R
d2rfA(r)2 + 2g iA(r) a(r) 2 a(r) (g a(r) 2 a(r))2g
R
d2r g=2 ( a(r) 2 a(r))2
i
= const exp
h R
d2r g=2 ( a(r) 2 a(r))2
i
:
(5.45)
It is clear that the choice of the Gaussian eld is almost compulsory since only
in this case is the functional integration a simple and well-de ned operation for
e.g. square integrable functions A. We arrive now at the following disorder-free
e ective action:
Sr =
Z
d2r
h
a( @ + m) a + g=2( a 2 a)2i
: (5:46)
We see that we now have electronswith an interactioninstead of disorder. Since all
the replicated Fermions had the same disorder potential they are correlated with
each other after integrating out the disorder whichis expressed by the interactions.
The interactionis local betweenelectrons on the same site.Note also that repeated
indices are summed in the formula above.
5.3.4 The e ective model of interacting electrons
Now we can use this e ectiveaction to go from2D to a 1+1D Hamiltoniansystem.
This is possible since we can quantize the model in such a way to bring it to the
form
Seff = @ H( ; ) (5:47)
54
55. 5.3 Field theoretical methods
and quantize the model i.e. replace the Grassmannians with fermionic operators.
This is only possible since the kinetic energy term is linear in momentumand not
second order as for a non-relativistic electron spectrum. We choose r1 to represent
and r2 the spatial direction r. Since r1 = is unbounded we are in the zero-
temperature limit of the 1+1D path integral formulation. We quantize according
to 1 ! y and ! which means that the density of states of the original
Fermions now becomes
n(E) = 1
c?ck
Re
D
( ;r) ( ;r)
E
i!n!E+i
(5.48)
= 1
c?ck
Re
D
y( ;r) 1 ( ;r)
E
i!n!E+i
: (5.49)
From the e ective action we can read o the Hamiltonian
H =
Z
dr
h
y
a( i@r 3 + !n 1) a g=2(: y
a 3 a :)2i
(5:50)
The normal ordering comes from the fact that the original Hamiltonian had to
be normal ordered in the path integral formulation!
5.3.5 Abelian Bosonization
The method of Bosonization has a long history, the interested reader can nd a
lot of information in a reprint volume by Stone 78] or in Fradkin's book 79]. On
a formal level we can say that it expresses a coincidence of correlation functions
of exponentials of free Boson elds with correlation functions of free Fermions in
1+1 dimensions. The power of the method lies in the fact that it is applicable
to interacting Fermions as well. In recent years there have been many e orts
to generalize the method to higher dimensions (see e.g. 80, 81, 82]) to treat
interacting electron systems like the cuprates. Another recent development is the
application to single impurityproblems (see e.g. 83] and references cited therein).
Let us try to nd a physical motivation for the Bosonization approach. A key
idea is to consider the density uctuations above a non-interacting ground state
i.e. an operator (q) = P
r exp(iqr)cy
rcr = P
k cy
k+qck acting on the Fermi sea
j i. Linearizing the dispersion relation around the Fermi surface and keeping
only states within an energy from the Fermi surface we create in fact another
eigenstate with de nite energy of the free Hamiltonian. The energy of the new
state is given by vF(q mod 2kF ).
This is a unique property of a one-dimensionalsystemsince even for an isotropic
Fermisurface in higher dimensionsthe gradient of the single-particleenergy points
in all possible directions so that the energy for a particle-hole excitation cy
k+qck
even for small momentum transfer q can have a continuous range of values de-
pending on k. Therefore (q) does not create an eigenstate in higher dimensions.
55
56. 5. Disordered d-Wave Superconductors
In the one-dimensional case however we can hope to describe the physics purely
in terms of the density uctuations. What do we win by doing so? The answer is
that it will be helpful if we add a short range interaction to the system which can
be naturally written in terms of density uctuations e.g.
X
k;k0;q
Vk;k0;qcy
k+qckcy
k0 qck0 V (q) ( q) : (5:51)
We see that if we have a slowly varying dependence of the interaction potential V
on momentum transfer q which means short-ranged interaction in real space the
interaction term can be approximated by a product of two density uctuations.
We will demonstrate how our e ective model can be treated by Abelian Boso-
nization. The mainlineof ideas follows closelythe lecturenotes by A.A.Nersesyan
84]. We start considering only the rst term in the Hamiltonian which describes
massless Dirac Fermions in 1+1D:
H =
Z
dr y
a( i@r 3) a (5:52)
We see that it is diagonal in spinor space = ( L; R) where the two components
stand for left- and right-movers respectively. This gets transparent if we use the
Fourier transformed elds in which the Hamiltonian reads:
H =
Z
dx H(x) =
X
k
k y
R(k) R(k) y
L(k) L(k)]: (5:53)
We will in the following use the Heisenberg representation of the elds. The so-
lutions are conveniently written in light-cone coordinates:
x = x0 x1; @ @=@ = 1
2(@0 @1): (5:54)
Using this notation we get
R(x;t) = R(x ) = 1p
L
X
k
e ikx
R(k); (5.55)
L(x;t) = L(x+) = 1p
L
X
k
eikx+
L(k); (5.56)
Now we should remember the normal ordering of the original Hamiltonian which
amounts to subtracting the vacuum contribution of the corresponding operators
(the Heisenberg representation did not change this since the eld operators which
we used are eigenoperators" changing only by a phase at any instance t). These
contribution are often singular and therefore we need a regularization scheme for
calculating them. In our case this is provided by the operator product expansion.
It consists of splitting operators that are de ned on the same point in space-time
56
57. 5.3 Field theoretical methods
to sit at a distance apart and letting ! 0 at the end. Furthermore we will
need to damp out contributions coming from high k values lying in the high-
energy region where the relativistic spectrum also might be a poor approximation
of our physical problem.
Let us demonstrate how this works for a concrete example. We calculate the
vacuum average for the propagator of right-movers
D
y
R(x ) R(y )
E
L!1! i
2
1
x y + i (5.57)
Now from the normal ordering prescription : AB := AB hABi we nd the
following operator product expansion for the propagator:
y
R(x ) R(x + ) j !0 = : y
R(x ) R(x + ) : + i
4 (5.58)
= : y
R(x ) R(x ) : + i
4 + O( ); (5.59)
where we used in the last step that in a normal ordered product the singular
vacuum contribution have been subtracted such that they allow for a well-behaved
Taylor expansion. We get the same result for the left-movers taking x+ instead.
Now we take a rst step towards Bosonization which is called the Sugawara
construction 85] where we express the massless Dirac Hamiltonian which is bi-
linear in Fermi operators with a product of bilinears instead to achieve in the end
the incorporation of the interaction term. So we are going to study the following
current operators:
JR(x ) = : y
R(x ) R(x ) : (5.60)
JL(x+) = : y
L(x+) L(x+) : (5.61)
Calculating the commutators we get
h
JR(x );JR(y )
i
= i
2
0(x y ) (5.62)
h
JL(x+);JL(y+)
i
= i
2
0(x+ y+); (5.63)
while JL and JR commute. In order to derive this result, remember that we have
to point split the operators and use the operator product expansion and the fact
that vacuum averages commute with anything since they are complex numbers.
Now we use Wick's theoremto calculate the square of the currents. The product
of normal ordered operators is equal to the normal ordered product plus a sum of
the normal order of all contractions except those inside the factors (contraction
means a vacuum expectation value of the corresponding operators). So we nd
J(x) JR(x ):
57
58. 5. Disordered d-Wave Superconductors
J2(x) = lim!0 J(x )J(x+ ) (5.64)
= + i
2 : y^@ : 1
(4 )2 + O( ): (5.65)
So we get
: J2
R : = i
2 : y
R
^@ R :; (5.66)
: J2
L : = i
2 : y
L ^@+ L : : (5.67)
and hence
H = (: J2
R : + : J2
L :) (5:68)
Now that we have expressed the Hamiltonianin terms of currents we are able to
make the crucial observation that it can as well be described with a corresponding
bosonic theory. We identify
JR(x ) = 1p @ 'R(x ); JL(x ) = 1p @+'L(x+); (5:69)
and we de ne
' 'R + 'L (5:70)
We de ne the conjugate elds to ' as
= @0' (5:71)
and from the Kac-Moody relations for the currents we derive that the bosonic
elds ' and indeed ful ll the canonical commutation relations
'(x); (x0)] = i (x x0): (5:72)
and hencethey reallycan be calledbosonic scalar elds.The corresponding Hamil-
tonian is simply the one for a free scalar eld
H =
Z
dx : J2
R : + : J2
L : ] (5.73)
) HBose =
Z
dx : (@ ')2 : + : (@+')2 : ]
= 1
2
Z
dx : 2 : + : (@x')2 : ]: (5.74)
For our replicated Fermi elds we introduce one Gaussian eld for each replica
index separately ' ! 'a. We want to emphasize that in the corresponding ex-
pression for the currents the replica indices are xed and no changes of the alge-
braic relations of the currents are introduced therefore, the currents with di erent
replica index simply commute.
58
59. 5.3 Field theoretical methods
The most crucial part is the incorporation of the interaction term coming orig-
inally from the disorder. Rewritten in terms of the bosonic eld we get
Hint =
Z
dx g
2 (: JL;a JR;a :)2; (5.75)
) Hint; Bose =
Z
dx g
2
X
a;b
a b: (5.76)
This part can be incorporated into the Hamiltonian by a orthogonal transforma-
tion of the elds and a subsequent rescaling to restore the canonical commutation
relations. In these transformed elds we recover once more just free bosonic elds
H = 1
2
Z
dx : P2 : + : (@x )2 : ] (5:77)
We have been able to write the massless part of the Dirac Hamiltonian plus the
interaction as the theory for a free bosonic eld! The crucial observation is that
this eld theory has scale invariance i.e. correlation functions have power law
decays which characterize systems at a second order phase transition.
The correlation function which we are interested in is
D
y 1
E
. The Bosonized
version of this operator is given by P
a : cos(
p
4 'a) : 79]. It is a relevant
perturbation which breaks scale invariance. In our problem the transformed eld
is the correct free eld so we express the mass term as
D
y
1
E
=
X
a
: cos(
p
4
X
b
Oab b) : (5:78)
the critical dimension for this operator is
r =
rX
b=1
V 2
ab = 1 1
r +
2
r (5:79)
In the replica limit we just get the exponent at r = 0 which is
= 1 g
2 (5:80)
Since we have a scale invariant system at !n = 0 which is broken at nite values
we can estimate the response which gives us the density of states for the original
problem at small !n by a scaling argument
!n
Z
d2r
r 1: (5:81)
Then one nds:
j!j 1
2 (5:82)
59
60. 5. Disordered d-Wave Superconductors
and hence we get for the density of states indeed the power law
n(E) / E(1 g
2 )=(1+ g
2 ) (5:83)
which indeed is what we promised and which is in excellent agreement with the
numerical simulation if we identify g=(2 ) = V 2= 2. So what have we won in
comparison to the perturbation theory.
We have obtained an exact description of the disorder system at !n = 0 and
the density of states at low energies is indeed the response of this system to a
SMALL perturbation.
In comparison in the perturbative expansion for the self energy in the previous
section we had to extrapolate the Taylor expansion to large values of glog( =!n)
which would in principle require a summation of the whole diagrammatic series.
This is indeed achieved to leading order by the non-perturbative Bosonization
approach. So we can say that we expand now around the proper ( x) point.
Furthermore the perturbative results can indeed be recovered
(i!n)
i!n
= 1 ~!n
!n
(5.84)
~!n= 1
= 1 j!nj
!(1 )=(2 )
(5.85)
= g
2 log( =j!nj) 1
2
g
2 log( =j!nj)
2
+ ::: (5.86)
Observe the factor 1=2 in the second term which is in agreement with the pertur-
bative result after inclusion of the crossed diagram.
5.3.6 Non-Abelian Bosonization
The dx2 y2-wave superconductor has four nodes and hence we have to deal with
another index for our Dirac Fermions which characterizes to which node they
belong. The disorder potential can scatter electrons not only in the vicinity of a
node but the spatially short-range part induces large momentumscattering which
mixes electrons from di erent nodes. We will consider a case where we have N
nodes and discuss at the end the particular application which we have in mind.
Looking back at the preceding section we can see that Bosonization achieved to
express the dynamics of the system in terms of the symmetry group of the elec-
trons, i.e. we could formulate an equivalent model on the algebra of the generator
of the U(1) symmetry group which is simply a scalar eld . The corresponding
action was the model of a free bosonic eld. The U(1) symmetry comes from the
conserved particle number.
We will see now that the same is possible for a more complicated symmetry
group of the Fermions containing a non-Abelian symmetry group and once more
60
61. 5.3 Field theoretical methods
we will nd that we can describe the dynamics in terms of the underlying sym-
metry group and the action will in this case be a Wess-Zumino-Witten model.
We start with free massless Fermions on a group
G = U(1) SU(r) SU(N) (5:87)
The Hamiltonian is given by
H =
Z
dr i n @ n (5:88)
where = 1;2;:::;N is the node index, while n = 1;2;:::;r is the replica index.
We introduce now three type of currents which transform according to the fun-
damental representation of the corresponding Lie algebra (chirality is not written
out explicitly):
U(1) Abelian currents, being replica and node singlets:
J = : y
n n : (5:89)
SU(N) currents being replica singlets:
Ja = : y
nTa
n : (5:90)
SU(r) currents being node singlets:
Ji = : y
nGi
nm m : (5:91)
Ta and Gi are elements of the generating algebra su(N) and su(r) respectively.
Here we sum over all repeated indices so that we only have one U(1) current in
contrast to the previous section where we had r currents with U(1) symmetry.
What are the commutation relations of these new currents? The U(1) current
is simple and gives
J(x);J(y)] = iNr
2
0(x y); (5:92)
The algebra of the non-Abelian current is more interesting. For the SU(N) cur-
rents we get
Ja(x);Jb(y)] = ifabcJc(x) (x y) + ir
4 ab
0(x y) (5.93)
where fabc are the structure constants of the algebra su(N). And analogously for
the SU(r) currents
Ji(x);Jj(y)] = igijkJk(x) (x y) + iN
4 ij
0(x y): (5:94)
61
62. 5. Disordered d-Wave Superconductors
where gabc are the structure constants of the algebra su(r).
Now we can once more go through the Sugawara construction for the non-
Abelian currents i.e. we have to calculate the squared currents. The wonderful
result is that the emerging normal ordered four Fermion terms of both SU(r) and
SU(N) currents are of opposite sign but otherwise identical and we are once more
able to rewrite the Hamiltonian density in terms of bilinears of currents such that
it splits into three di erent sectors
H = HU(1) + HSU(N) + HSU(r)
= 2
N + r( : Ja
RJa
R : + : Ja
LJa
L : )
+ 2
N + r( : Ji
RJi
R : + : Ji
LJi
L : )
+Nr( : JRJR : + : JLJL : ): (5.95)
In the case of a very short ranged impurity potential all scattering processes be-
tween the nodes and in the vicinity of the same node are characterized by a single
coupling constant g and if we neglect the anisotropy in the Fermi velocities the ef-
fective interaction after replication and integrating out disorder has a particularly
simple form
Hint = 4g JR JL (5:96)
where the boldfaced letters denotes SU(N) currents and we have dropped terms
which only renormalize the velocities. This model has been solved exactly by
AlexeiTsvelik 86] by Bethe's Ansatz and the result is that the current interaction
which only a ects the SU(N) sector of the theory creates a gap which remains
nite even in the replica limit r ! 0. The low lying excitations are node singlets
and they are described by the rest of the Hamiltonian
Heff = HU(1) + HSU(r) (5:97)
Due to the additive form of the Hamiltonian we can treat the terms separately.
HU(1) can be Bosonized according to the rules in the previous section. Dealing
with HSU(r) requires an additional step. The Euclidean action of this Sugawara
Hamiltonian in which the currents satisfy a level N Kac-Moody algebra has cor-
relation functions of currents which are identical to the correlation functions for
the Wess-Zumino-Witten action 87]
W(g) = 1
16
Z
d2x Tr((@ g 1)(@ g)) (5.98)
+ i
24
Z
B
d3X" Tr(g 1(@ g)g 1(@ g)g 1(@ g)) (5.99)
where g 2 SU(r) and we identify
JL = iN
2
g@ g 1 ; JR = iN
2
g@+g 1 ; (5:100)
62
63. 5.3 Field theoretical methods
The integration of the second termis within a 3D sphere whose boundary coincides
with the original compacti ed 2D space.
This last term in the Wess-Zumino-Witten action is a topological term which
is quantized i.e. assuming that it is single valued it can only be equal to 2 iNm
where m is an integer. This term makes the theory conformally invariant and the
scaling dimensions of any operator written in terms of currents can be calculated.
Though HSU(N) seems to be eliminated from the spectrum it enters the scaling
exponents via the level of the Kac-Moody algebras. The quantity of interest which
we have is a mass bilinear
y
L a R b exp(i
q
4 =Nr ) gab; g 2 SU(r) (5:101)
The scaling dimension of g equals
g = r2 1
r(N + r): (5:102)
So
dim
h
TrSU(N)( y
1 )
i
= 1
Nr + r2 1
r(N + r) = Nr + 1
N(N + r) (5:103)
We see that, despite the fact that the dimensions of the U(1) phase exponential
and the eld ^g separately diverge in the replica limit,the dimension of the physical
quantity, the mass bilinear, remains nite at r ! 0:
limr!0dim
h
TrSU(N)( y
1 )
i
= 1
N2 (5:104)
Following the same scaling argument as in the preceding section we get
n(E) / E =(2 ) (5.105)
= E1=(2N2 1) (5.106)
So if we only take into account backscattering N = 2 we have the exponent 1=7
and if we have all scattering processes in the dx2 y2-wave superconductor then
N = 4 and the exponent is reduced to 1=31.
In paper III it is shown that the inclusion of anisotropy in the Fermi veloci-
ties plays an essential role in the last case only i.e. when we cannot get rid of
the anisotropy by a global rescaling of coordinates. The e ect is to reduce to
1=(N2( + 1)), where is the ratio of the Fermi velocities ck=c?.
We can say that the experimental result by Valles et al. 61] is closest to the
case N = 2 which is probably an intermediateregime since it is cut o by the Tc of
lead, so in principleit would be veryinterestingto extend the experimentsto lower
temperature by using di erent counterelectrodes. The most obvious suggestion is
to make a perfect tunnel junction of HTS-I-HTS structure and measure the power
63
64. 5. Disordered d-Wave Superconductors
law decay of the quasiparticle branch of the I-V curve which is twice the exponent
for the densityof states. Varying the degree of disorder by doping or using di erent
preparation techniques and a large number of samples could reveal whether the
power law scales with the impurity concentration or not. In the latter case this
would be a strong hint of a short range disorder leading to universal power laws.
The former case would suggest that the impurity potential is slowly varying and
well screened and could therefore be used to check the quality of the samples.
Such an e ect could possibly be detected even in a single piece of HTS due to the
intrinsic Josephson e ect in the c-axis where at least BISCCO seems to behave
like a stack of weakly coupled superconductors 88].
5.3.7 A note on symmetry
The fact that the density of states is zero at zero energy n(0) = 0 for a non-
magnetic random potential is in fact very robust since we can relate it to a
symmetry argument. The disorder free replicated action has a continuous global
symmetry under which it remains invariant the 5 symmetry.The central point is
that the e ective action even in the most general case with 4 nodes and unequal
values of c? and ck contains only two Pauli (or ) matrices i.e.
i ; i: 1,2 (5:107)
Then we have an additional anticommutingmatrix (the third Pauli matrix) which
is 5 = i 1 2. We de ne now the continuous 5 symmetry on the Grassmannian
eld as the transformation
! exp(i 5) (5.108)
! exp(i 5) (5.109)
It is easy to verify by using the anticommutation relations that
exp(i 5) i = i exp( i 5) ; i: 1,2 (5:110)
and therefore i ; i: 1,2 is indeed invariant under the continuous 5 symmetry.
However the mass" term which gives the density of states is within this notation
written as
!n (5:111)
and hence it does not remain invariant under the 5 symmetry. We have already
shown that the action without this term corresponds to a critical point and the
Matsubara mass" term destroys criticality being a relevant perturbation. Hence
a nite vacuum expectation value of the mass term in the limit of !n ! 0 corre-
sponds to a nite order parameter breaking the 5 symmetry in the ground state.
But as is known from the Mermin-Wagner theorem 89], a continuous symmetry
64
