Proceedings TH2002 Supplement (2003) 207 – 217
c Birkhäuser Verlag, Basel, 2003
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Proceedings TH2002
Disorder and Quantum Criticality in d-Wave
Superconductors
Julia S. Meyer, Igor V. Gornyi and Alexander Altland
Abstract.
To explain experimentally observed quasiparticle relaxation rates in the cuprate
superconductors, Sachdev and collaborators recently proposed that the d-wave system may be close to a quantum critical point towards a superconducting state with
an additional order parameter component of is- or id-wave symmetry. In the proximity of the transition, dynamical order parameter fluctuations modify the system
properties. Here, we explore the influence of disorder on this interacting fluctuation
regime. Our main finding is that unlike in the clean system, significant qualitative
differences between is and id exist: In the is-case due to the disorder-enhanced lowenergy density of states, a secondary superconducting transition (with a strongly
disorder-dependent Tc ) takes place. By contrast, in the id-case essential features of
the clean scenario are left intact by the disorder.
Quasiparticle relaxation rates in the cuprate superconductors are accessible
by high resolution angle-resolved photoemission (ARPES) experiments. Recent
experimental results [1] indicate a quasiparticle relaxation rate κ that is linear in
temperature (or excitation energy) over a wide range of temperatures both above
and below Tc . Clearly such a strong damping mechanism lies beyond the scope of
orthodox BCS-type formulations in terms of a non-interacting quasiparticle (QP)
system. Notably, the standard BCS picture predicts the much stronger scaling
κ ∼ ||3 [2, 3] within the superconducting phase due to the nodal structure, whereas
above the transition one would expect the conventional Fermi liquid result κ ∼ 2
to hold.
A possible explanation for this anomalously strong damping, put forward by
Sachdev and collaborators [4], is the close proximity of the system to some quantum critical point towards formation of a phase with an additional stable order
parameter component. Although the transition point may not lie within the physical parameter regime, its proximity will affect the system due to dynamical order
parameter fluctuations. Studying different possibilities allowed by the symmetries
of the system, Sachdev and collaborators [4] found that the most promising candidates are additional superconducting order parameter components of is- or idxy symmetry, breaking time-reversal invariance. Thus, they proposed a dynamical
ˆ 1 (x, t),
ˆ
ˆ 0 (x) + i∆
generalization of the BCS order parameter field, viz. ∆(x,
t) = ∆
ˆ 0 represents the background dx2 −y2 order parameter and i∆
ˆ 1 ≡ i∆
ˆ s/dxy
where ∆
is a time-dependent component of either s- or dxy -wave symmetry. The effect of
ˆ 1 (t) leads to the
coupling the relativistic quasiparticles to the collective mode ∆
following low-energy phenomenology of the system [4, 5]:
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J.S. Meyer, I.V. Gornyi, A. Altland
Proceedings TH2002
(i) No qualitative differences between s- and dxy -scattering exist.
(ii) Scattering off the time-dependent order parameter fluctuations enhances the
quasiparticle relaxation to the linear rate κ ∼ || that is observed experimentally.
(iii) Furthermore, the coupling of the quasiparticles to the dynamical order paˆ 1 leads to the formation of long-ranged collective flucrameter component ∆
tuations. These fluctuations should [4] show up in (neutron or Raman) scattering experiments.
(iv) However, the Dirac spectrum of the nodal quasiparticles prevents the formation of a secondary superconductor transition. I.e., if sufficiently weak, the
ˆ 1 does not open a quasiparticle gap.
attractive interaction mediated by ∆
How does disorder affect these findings? Since the Dirac spectrum of the
nodal quasiparticles is an essential ingredient in obtaining the above features, one
might expect that disorder has a drastic effect because it profoundly changes the
low-energy spectrum of the quasiparticles. However, at least in the id-case this
turns out not to be the case. We find that [6]:
ˆ 1 (t).
(i’) The system becomes sensitive to the symmetry of ∆
(ii’) Even in the presence of disorder, the coupling of quasiparticles to a dynamical
order parameter of id-symmetry induces an inelastic relaxation rate κin ∼
||. In addition, the disorder introduces a structureless elastic background
contribution κel ∼ τ −1 , where τ denotes the impurity scattering time. At low
frequencies, || < τ −1 , a logarithmically singular third contribution κsing ∼
ln(τ ||) to the relaxation rate κ = κin + κel + κsing arises due to the formation
of long-ranged diffusive excitations.
(iii’) Furthermore, the id-order parameter continues to exhibit fluctuation behavior similar to that found in Refs. [4, 5].
(iv’) By contrast, in the is-case, the interplay of impurity scattering and dynamical fluctuations of the collective mode drives the system into a secondary
superconductor transition: at the critical temperature T̃c ∼ ω0 exp(−r̃/ν0 ),
the system enters a fully gapped d + is superconducting state. Here ω0 =
min[ωD , 1/τ ], where ωD is a phenomenological parameter corresponding to
the maximum (Debye) frequency of the order parameter fluctuations, 1/r̃
denotes the amplitude of the order parameter fluctuations (i.e., r̃ is proportional to the distance from the quantum critical point), and ν0 the residual
low-energy density of states (DoS). Since ν0 is induced by impurity scattering, ν0 ∝ 1/τ , the transition temperature depends exponentially on the
disorder strength which can be deliberately changed by doping. Thus, a
disorder-dependent transition into a fully gapped state at low temperatures
would be a test criterion for the presence of dynamical is-order parameter
fluctuations.
Vol. 4, 2003
Disorder and Quantum Criticality in d-Wave Superconductors
(−π,π)
209
(π,π)
ky
(−π,−π)
kx
(π,−π)
Figure 1: Nodal structure in the Brillouin zone of the d-wave superconductor. The
low-energy quasiparticles at the four nodes a = 1, 1̄, 2, 2̄ have a Dirac spectrum.
Before introducing disorder, let us outline the physics of the clean system [4, 5]
which serves as a basis for discussing the effects of disorder. The d-wave system
possesses four low-energy nodes a = 1, 1̄, 2, 2̄ in the Brillouin zone (see Fig. 1). The
quasiparticles at these nodes have an anisotropic Dirac spectrum with velocities
vF (Fermi velocity) and vF γ −1 , where γ ≡ t/|∆0 | and t is the tight binding energy.
Denoting the two-component quasiparticle field by ψ a , the Dirac action of the
unperturbed quasiparticles reads
S[ψ] = d3 x ψ a† ∂τ + ivF (sa1 γ −1 σ1ph ∂1 + sa2 σ2ph ∂2 ) ψ a ,
(1)
where x = (τ, x)T is a three-component space-time argument, saj are sign factors
depending on the node index, and σjph Pauli matrices in particle/hole space [7].
The quasiparticle fields are coupled to the order parameter fluctuations φ through
the scattering vertex
(2)
Sc [φ, ψ] = λ d3 x ψ a† Γa σ3ph ψ a φ,
where λ is a coupling constant, and Γa a symmetry factor distinguishing between
the two possible order parameter components: Γa = 1 (Γa = (−)a ) for the case of
is (id) symmetry. Finally, the time-dependent amplitude of the order parameter
φ(x) is controlled by a φ4 -type action,
u 1
(∂τ φ)2 + c2 (∂x φ)2 + rφ2 + φ4 ,
S[φ] = d3 x
(3)
2
4!
where c, r, and u are phenomenological constants, and r = 0 marks the position
of the quantum critical point into the dx2 −y2 + is/dxy phase. Note that the model
described by the composite action S[φ, ψ] = S[ψ] + Sc[φ, ψ] + S[φ] is known in the
high-energy literature as the Yukawa-Higgs model.
To understand the consequences of this coupling mechanism, let us integrate
over the quasiparticle fields which leads to an effective φ-action Seff [φ] = S[φ] +
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J.S. Meyer, I.V. Gornyi, A. Altland
Proceedings TH2002
δS[φ]. Here, δS[φ] = − lnexp(−Sc [φ, ψ]) and . . . =
D(ψ † , ψ)
†
exp(−S[ψ , ψ])(. . .) is the functional average over the quasiparticle action. Using
an RPA type expansion (cf. Fig. 4), δS[φ] can be approximated by
δS[φ] =
λ2
2
d3 x d3 x φ(x)χ(x, x )φ(x ),
where χ(x, x ) ≡ tr Γa σ3ph G aa (x, x )Γa σ3ph G a a (x , x)
(4)
is the Cooper pair propagator. ‘tr’ stands for a trace over particle/hole indices,
and G aa (x, x ) = ψ a (x)ψ †a (x ) is the Gorkov Green function. In the clean case,
G becomes diagonal in momentum and nodal space, and one finds (after proper
rescaling of momenta)
2
(5)
δS[φ] = − d3 k |φk |2 C1 − C2 (ωm
+ k2 )1/2 ,
where d3 k ≡ T ωm d2 k denotes a summation over the three-momentum k ≡
(ωm , k)T , and ωm is a bosonic Matsubara frequency. The constant C1 merely shifts
the parameter r (where we assume that C1 < r). The absence of a secondary
instability (which would require C1 to be singular in the limit T → 0) can be
traced back
ω to the linearity of the low-energy DoS considering the rough estimate
C1 ∼ λ2 T 0 d ν()/. Here, the important effect comes form the non-analytic
2 1/2 2
operator ∼ (k2 + ωm
) φ which is responsible for the enhanced quasiparticle
relaxation rate [4, 5].
From the structure of χ it is obvious that the result does not depend on the
ˆ 1 (t). The difference between an s- and a d-scattering amplitude,
symmetry of ∆
d
respectively, is their nodal structure: while ∆s1 is constant, ∆1xy changes sign from
node to node. However, the pair susceptibility operator – and accordingly the
induced action δS – is quadratic in the scattering vertex and, therefore, oblivious
to this difference.
Before analyzing the influence of disorder quantitatively, let us highlight the
main elements. Note that even though the disorder is pair breaking, as long as
∆0 is sufficiently large, we can neglect this effect. Disorder scattering leads to a
smearing of spectral structures over scales ∼ τ −1 . In particular, it washes out the
low-energy cusp of the Dirac spectrum as the quasiparticle Green functions acquire
a self energy, G −1 (in ) → G −1 (in ζ) + iκel sgn(n ). Here the real constants κel =
(2τ )−1 and ζ can be obtained from self-consistent Born or T -matrix perturbation
theory [8], depending on the microscopic realization of the disorder. Relatedly, τ −1
is the crossover scale from ballistic (|| > τ −1 ) to diffusive (|| < τ −1 ) quasiparticle
dynamics.
Furthermore, in the diffusive low-energy regime, corrections to the relaxation
time (as well as other observables) are generated by mechanisms of quantum interference (similar to the weak localization corrections known from the physics of
Vol. 4, 2003
Disorder and Quantum Criticality in d-Wave Superconductors
211
normal metals). In the non-interacting case, these interference modes can be conveniently described by a field integral formulation [9] wherein the diffusion modes
D, rather than the microscopic quasiparticle propagators, are basic degrees of freedom. The extension of this formalism to the presence of interactions proceeds along
the lines of the general construction scheme of field theories of interacting disordered fermion systems [10]. The quasiparticle Dirac action Eq. (1) is generalized
to the presence of disorder by including a random potential V̂ describing disorder
in the potential and/or order parameter component. Taking the composite action S[φ, ψ] as a starting point, one finds that (after disorder averaging, HubbardStratoinovich tarnsformation, and integration over the fermion degrees of freedom)
the joined influence of order parameter fluctuations and diffusion
modes is de
scribed by an action S[R, φ] = S[φ]+ S̃[R, φ]. Here S̃[R, φ] = − a tr ln Ga−1 [R, φ],
and
i
−1
RΞ + iω̂σ3cc + λφ̂Γa
ha
(6)
Ga [R, φ] = 2τ
i
−1
h†a
+ iω̂σ3cc − λφ̂Γa
2τ ΞR
plays the role of the generalized Gorkov Green function. In (6), the matrix structure
is in Nambu (particle-hole) space, and ha defines the particle-hole component of
the (Dirac) quasiparticle Hamiltonian of node a. Further, ω̂ = diag (ωn ) is a vector
of Matsubara frequencies, and σ3cc is a Pauli matrix acting in ‘charge conjugation
space’, a construction required to consistently describe the behavior of the theory
under time reversal [11]. The fixed matrix Ξ is given as Ξ ≡ sgn(ω̂) ⊗ σ3cc , and the
cc
cc
structure of the order parameter field in cc-space as φ̂ = φE11
+ φT E22
. Finally,
rr ,αα
T
cc −1 cc
R = {Rnn } ∈ Sp(2N ) is a symplectic matrix field, R = σ2 R σ2 , whose
fluctuations describe the diffusively propagating soft modes mentioned above.
(n, n are frequency indices, r, r replica indices, and the indices α, α = 1, 2 act in
charge conjugation space.)
Eq. (6) describes the joint effect of the formation of diffusion modes, the
interaction, and the kinematics of the Dirac operator. For sufficiently low temperatures, T τ −1 , microscopic details of the Dirac dynamics become inessential,
and an expansion in slow fluctuations of the fields R and φ produces the effective
action
S[R, φ] = S[φ] + S[R] + Sc [R, φ],
(7)
where S[φ] is given by Eq. (3) and
πν0
S[R] =
d2 x tr D∂R∂R−1 − 2|ω̂|(R + R−1 ) ,
8
2
a
2
Sc [R, φ] = − d x i (
Γ ) c1 tr RΞφ̂ + c2 tr (RΞφ̂)
.
a
Here, S[R] is the action of the non-interacting disordered superconductor [7], where
ν0 is the induced low-energy DoS, D = (γ + γ −1 )/(π 2 ν0 ) the diffusion constant,
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J.S. Meyer, I.V. Gornyi, A. Altland
Proceedings TH2002
|ω̂| = {|ωn |}, and the trace extends over replica, charge conjugation and frequency
space (while the nodal (and Nambu) structure has already been traced out, see
below). Furthermore,
c1 and c2 are real positive constants, and the symmetry
factor as = a Γa distinguishes between the is (as = 4) and the idxy (as = 0)
case.
Focusing on the points (i’)-(iv’) summarized above, we next ask what can be
learned from the formal representations (6) and (7) of the disordered problem. The
structure of S[R] indicates that fluctuations of the field Rnn are impeded by finite
frequencies ωn . Indeed, a perturbative evaluation of the non-interacting action
(see Ref. [7] and below) shows that small fluctuations of R around the SC(B/T)A
saddle point R = 1 lead to logarithmic corrections ∼ g −1 ln(τ (|ωn | + |ωn |), where
g ∼ Dν0 1 is the dimensionless conductance, to practically all observables
coupling to the quasiparticle sector of the theory. Note that this observation entails
the existence of a crossover scale ω ∗ ∼ exp(−g), separating a perturbative high
energy regime from a more complex low energy regime. The logarithmic corrections
are similar to the ‘weak localization’ (WL) corrections of normal metals, inasmuch
as they form as a result of multiple traversal of the same scattering path in the
disordered environment. However, in marked contrast to the situation in normal
metals, quasiparticles of (Matsubara) energy ωn > 0 in disordered superconductors
can interfere with both, quasiparticles (ωn > 0) and quasi-holes (ωn < 0). In
particular, they can self-interfere as shown in Fig. 2.
Figure 2: Feynman path describing the self-interference of a quasiparticle.
Fortunately, most of the questions brought up above can be addressed without
explicit reference to the ultra-low energy regime |ωn | < ω ∗ . Focusing on the regime
of intermediate energies, ω ∗ < |ωn | < τ −1 , we next explore further aspects of the
interplay between interaction and disorder. As we are not yet in a fluctuation
dominated regime, the action (7) can be expanded in small fluctuations around
the constant configuration R = 1. Setting R = exp(W ), where W is a generator
of the symplectic group, a quadratic expansion in W leads to
1 2
Ξφ̂
Ξφ̂
(0)
(0)
λ
tr Ga
G
,
(8)
δS0 [φ] =
−φ̂Ξ ph a
−φ̂Ξ ph
2
a
πν0
−1
S[W, φ] = −
d2 x tr Wnn Dnn
Wn n
8
Vol. 4, 2003
Disorder and Quantum Criticality in d-Wave Superconductors
+ ias c1
d2 x tr Ξφ̂W − c2
(1)
Sc
213
d2 x tr (Ξφ̂)2 W 2 +(Ξφ̂W )2 , (9)
(2)
Sc
−1
2
where Ga = Ga [R = 1, φ = 0] (see Eq. (6)). Furthermore, Dnn
(q) = Dq + |ωn | +
|ωn | is the inverse of the diffusion mode operator. Eq. (8) describes the linear
coupling of φ to a field propagating through the diffusion mode. The meaning of
the two interaction operators is illustrated in Fig. 3. Heuristically, the absence of
(1)
a small momentum transfer operator, Sc in the idxy case follows from the fact
that the diffusion mode does not have structure in node space (Multiple impurity
scattering equilibrates the quasiparticle amplitude homogeneously over all four
nodes.) while the idxy scattering vertex changes sign from node to node. The
coupling of a diffusion mode to an isolated vertex, therefore, vanishes, and we are
(2)
(1)
left with the bilinear coupling Sc . As we will see below, it is Sc that entails the
most significant modifications as compared to the clean case.
(0)
ψ
(O1 ) 2 :
φ
O2 :
ψ
Figure 3: The scattering vertex (left) yields two different interaction terms in the
presence of disorder: due to the nodal structure in the id-case, however, the term
O1 vanishes.
To quantitatively explore the interplay of disorder and the collective mode
in the present context [12, 13, 14], let us – as for the clean case – evaluate the
φφ-propagator in an RPA like approximation (cf. Fig. 4). Here this amounts to
considering the configurational average of the susceptibility operator χ(x, x )dis .
In a ladder approximation, stabilized by the parameter γ 1, δS[φ] is given as
2
1 δS[φ] = δS0 [φ] + Sc(1) W .
2
(10)
Thus, in the is-case, the quasiparticle/order parameter vertex is coupled to the diffusion mode Dnn (q) given above. Taking into account this mode in the calculation
of the polarization operator yields
is
δS [φ] = − d3 k |φk |2 [C1 (T ) + . . . ] ,
(11)
where the ellipses stand for operators involving derivatives. In addition to a T independent shift δr (< r), the constant term C1 (T ) = λ2 ν0 ln(ω0 /T ) + δr now
214
J.S. Meyer, I.V. Gornyi, A. Altland
Proceedings TH2002
contains a logarithmic low-temperature divergence which is characteristic for swave superconductors. The low-T singularity of C1 implies that at the critical
temperature
(12)
T̃c ∼ ω0 exp(−r̃/ν0 ),
where r̃ = r/λ2 , the φ-action becomes unstable, and a BCS transition into a phase
with finite expectation value of the order parameter (φ = 0) and gapped quasiparticle states occurs. The difference to the conventional BCS transition is that
in the d-wave system the low-energy DoS ν0 is generated by impurity scattering
and, therefore, the critical temperature is exponentially sensitive to the disorder
concentration.
-1
=
-1
id xy
is
Figure 4: Effective interaction in the id and the is case, respectively. While in
the id case the polarization loop is infrared-finite, in the is case vertex renormalizations render the polarization loop infrared-divergent and, thus, lead to a
superconducting instability.
The physics of the id-case is markedly different because here the susceptibility
operator is not affected by vertex corrections due to the absence of the interaction
(1)
operator Sc (i.e., as = 0). Since disorder scattering distributes the quasiparticle
amplitude randomly over all nodes, the two nodal summations over the symmetry
factors Γa = (−)a at the vertices decouple and annihilate all contributions to
the susceptibility operator, safe for the ‘bare bubble’ without vertex corrections.
ˆ dxy and,
Heuristically, the quasiparticle only sees the vanishing nodal average of ∆
therefore, multiple scattering contributions to the pair propagator are absent.
Evaluating the susceptibility for the self-energy decorated Green functions
(i.e., the term δS0 [φ]), we find
(13)
δS id [φ] = − d3 k |φk |2 C1 − C2 |ωm | + Dk 2 .
Due to the absence of vertex corrections, the constant C1 ∼ λ2 ωD f (ωD τ ζ) (where
f (x) decreases from 1 at x 1 to x at x → 0) is now again finite at T → 0.
[Here the absence of an instability (even though ν0 finite) is due to the fact that
ω
the Greens functions are ‘massive’, i.e., C1 ∼ λ2 T 0 d ν()/κ.] Furthermore, a
non-analytic contribution C2 |ωm |φ2 , where C2 ∼ λ2 , exists despite the smearing
of the low-energy spectrum by the imaginary part of the (elastic) self energy. Thus,
the action still contains an operator of engineering dimension 1 and, therefore, the
fluctuation behavior of the collective modes is similar to that in the clean case.
Vol. 4, 2003
Disorder and Quantum Criticality in d-Wave Superconductors
215
Evaluation of the quasiparticle relaxation rate shows that, indeed, in the vicinity
of the instability (r C2 ||) κin ∼ || as in the absence of disorder. On the other
hand, for r C2 || we obtain Fermi liquid behavior κin ∼ 2 .
On top of the inelastic contribution, disorder leads to elastic relaxation. We
next analyze the functional representation of the elastic quasiparticle relaxation
rate,
1
Re (R + R−1 )11,11
,
(14)
κel () ∼
nn
2τ
iωn →+i0
in the disorder dominated low temperature regime < τ −1 . To a zeroth approximation the relaxation rate is, expectedly, set by the inverse scattering time, κel τ −1 .
Expanding R to second order in the generators W and computing the functional expectation value . . . wrt the composite action (8) (corresponding to an
expansion in the small parameter g −1 ), the soft modes D couple to the self-energy
operator through the diagram shown in Fig. 5. Similar to the weak localization
corrections to the conductance of normal metals, these processes lead to a negative logarithmic correction κsing () ∼ g −1 ln(τ ||) to the self energy. As mentioned above, the singularity of the interference correction implies that below the
crossover scale ω ∗ nonperturbative effects become important – leading to both,
Anderson localization of the quasiparticle states and a linear vanishing of their
DoS, ν() ∼ ν0 ||/ω ∗ [7]. [In the is-case, ω ∗ and T̃c depend on fundamentally
different input parameters (ω ∗ < T̃c or ω ∗ > T̃c , both are conceivable). Therefore the BCS instability may interfere with the formation of strong localization
phenomena [15].]
interacting mode:
=
+
Figure 5: Quantum interference correction to the self energy (left) and renormalization of the interference process by interactions (right).
In the presence of a dynamical id order parameter the bare diffusion mode
has to be replaced by the ‘interacting’ mode shown in Fig. 5, obtained by including
(2)
the interaction operator Sc . However, the quantitative summation over these selfinteraction processes only causes an inessential renormalization of the prefactor,
g −1 → g −1 (1 + F ), where F 1 is an interaction-dependent constant. Similar
logarithmic corrections, weakly renormalized by quasiparticle interactions, affect
the (spin-) conductance and the DoS [15]. Our findings conform with the results
of Ref. [14] which considered the case of globally broken time-reversal invariance
(i.e., symmetry class C in the notation of Ref. [16], while the present problem
belongs to symmetry class CI): in Ref. [14] it was shown within an RG analysis
that interactions do not significantly modify the low-energy disorder-generated
interference phenomena [17].
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J.S. Meyer, I.V. Gornyi, A. Altland
Proceedings TH2002
Summarizing, we find that disorder has essentially different effects on critical
fluctuations of an is- or id-order parameter component in two-dimensional cuprate
superconductors. In the is-case, a secondary BCS transition with a critical temperature that sensitively depends on the impurity concentration is induced. By
contrast, in the id-case, the addition of disorder to the system leaves much of the
phenomenology – in particular the characteristic singularity κin ∼ || – derived
in Refs. [4, 5] intact. Scattering times comparable to the energies (O(20K)) estimated [4] as characteristic for the dynamical order parameter fluctuations are
achieved even by moderate impurity concentrations [18]. Therefore, we believe
that the above findings may be made visible experimentally at sufficiently low
temperatures. Finally, we discussed the (essentially disorder-generated) singular
contribution to the relaxation rate appearing at low energies.
Acknowledgments It is a pleasure to acknowledge valuable discussions with D.V.
Khveshchenko, A.W.W. Ludwig, A.D. Mirlin, J. Paaske, S. Sachdev, M. Vojta,
and A.G. Yashenkin.
IVG was supported by the Schwerpunktprogramm “Quanten-Hall-Systeme”,
the SFB195 der Deutschen Forschungsgemeinschaft, and by the RFBR.
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[*] Also at A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia.
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Disorder and Quantum Criticality in d-Wave Superconductors
217
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[17] Note that close to the transition (r = 0), more pronounced interactiongenerated phenomena [15] in both, is and id, cases may appear.
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Julia S. Meyera
Igor V. Gornyib,∗
Alexander Altlandc
a
University of Minnesota
Department of Physics
116 Church St. SE
Minneapolis MN55455, USA
b
Institut für Nanotechnologie
Forschungszentrum Karlsruhe
76021 Karlsruhe, Germany;
∗
Institut für Theorie der Kondensierten Materie
Universität Karlsruhe
76128 Karlsruhe, Germany
c
Institut für Theoretische Physik
Universität zu Köln
Zülpicher Str. 77
50937 Köln, Germany