Kondo-lattice screening in a d-wave superconductor
Kondo-lattice screening in a d-wave superconductor
Daniel E. Sheehy1,2 and Jörg Schmalian2
Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803
Department of Physics and Astronomy, Iowa State University and Ames Laboratory, Ames IA 50011
(Dated: November 28, 2018)
We show that local moment screening in a Kondo lattice with d-wave superconducting conduction
electrons is qualitatively different from the corresponding single Kondo impurity case. Despite the
conduction-electron pseudogap, Kondo-lattice screening is stable if the gap amplitude obeys ∆ <√
TKD, in contrast to the single impurity condition ∆ < TK (where TK is the Kondo temperature
for ∆ = 0 and D is the bandwidth). Our theory explains the heavy electron behavior in the d-wave
superconductor Nd2−xCexCuO4.
I. INTRODUCTION
The physical properties of heavy-fermion metals are
commonly attributed to the Kondo effect, which causes
the hybridization of local 4-f and 5-f electrons with itin-
erant conduction electrons. The Kondo effect for a single
magnetic ion in a metallic host is well understood1. In
contrast, the physics of the Kondo lattice, with one mag-
netic ion per crystallographic unit cell, is among the most
challenging problems in correlated electron systems. At
the heart of this problem is the need for a deeper un-
derstanding of the stability of collective Kondo screen-
ing. Examples are the stability with respect to com-
peting ordered states (relevant in the context of quan-
tum criticality2) or low conduction electron concentra-
tion (as discussed in the so-called exhaustion problem3).
In these cases, Kondo screening of the lattice is believed
to be more fragile in comparison to the single-impurity
case. In this paper, we analyze the Kondo lattice in a
host with a d-wave conduction electron pseudogap4. We
demonstrate that Kondo lattice screening is then sig-
nificantly more robust than single impurity screening.
The unexpected stabilization of the state with screened
moments is a consequence of the coherency of the hy-
bridized heavy Fermi liquid, i.e. it is a unique lattice ef-
fect. We believe that our results are of relevance for the
observed large low temperature heat capacity and sus-
ceptibility of Nd2−xCexCuO4, an electron-doped cuprate
superconductor5.
The stability of single-impurity Kondo screening has
been investigated by modifying the properties of the con-
duction electrons. Most notably, beginning with the work
of Withoff and Fradkin (WF)6, the suppression of the
single-impurity Kondo effect by the presence of d-wave
superconducting order has been studied. A variety of an-
alytic and numeric tools have been used to investigate the
single impurity Kondo screening in a system with conduc-
tion electron density of states (DOS) ρ (ω) ∝ |ω|r, with
variable exponent r (see Refs. 6,7,8,9,10,11,12). Here,
r = 1 corresponds to the case of a d-wave superconduc-
tor, i.e. is the impurity version of the problem discussed
in this paper. For r ≪ 1 the perturbative renormaliza-
tion group of the ordinary13 Kondo problem (r = 0), can
be generalized6. While the Kondo coupling J is marginal,
0.05 0.1 0.15 0.2
0.05 0.1 0.15 0.2
local
moment
screened
Kondo lattice
screened impurity
FIG. 1: The solid line is the critical pairing strength
∆c for T → 0 [Eq. (33)] separating the Kondo screened
(shaded) and local moment regimes in the Kondo-lattice
model Eq. (4). Following well-known results6,7 (see also Ap-
pendix A), the single-impurity Kondo effect is only stable for
∆ . D exp(−2D/J) ∼ TK (dashed).
a fixed point value J∗ = r/ρ0 emerges for finite but small
r. Here, ρ0 is the DOS for ω = D with bandwidth D.
Kondo screening only occurs for J∗ and the transition
from the unscreened doublet state to a screened singlet
ground state is characterized by critical fluctuations in
time.
Numerical renormalization group (NRG) calculations
demonstrated the existence of a such an impurity quan-
tum critical point even if r is not small but also
revealed that the perturbative renormalization group
breaks down, failing to correctly describe this critical
point9. For r = 1, Vojta and Fritz demonstrated that
the universal properties of the critical point can be un-
derstood using an infinite-U Anderson model where the
level crossing of the doublet and singlet ground states
is modified by a marginally irrelevant hybridization be-
tween those states10,11. NRG calculations further demon-
strate that the non-universal value for the Kondo cou-
pling at the critical point is still given by J∗ ≃ r/ρ0,
even if r is not small8. This result applies to the case of
broken particle-hole symmetry, relevant for our compari-
son with the Kondo lattice. In the case of perfect particle
http://arxiv.org/abs/0704.1815v3
hole symmetry it holds that8 J∗ → ∞ for r ≥ 1/2.
The result J∗ ≃ r/ρ0 may also be obtained from a large
N mean field theory6, which otherwise fails to properly
describe the critical behavior of the transition, in partic-
ular if r is not small. The result for J∗ as the transition
between the screened and unscreened states relies on the
assumption that the DOS behaves as ρ (ω) ∝ |ω|r all the
way to the bandwidth. However, in a superconductor
with nodes we expect that ρ (ω) ≃ ρ0 is essentially con-
stant for |ω| > ∆, with gap amplitude ∆, altering the
predicted location of the transition between the screened
and unscreened states. To see this, we note that, for en-
ergies above ∆, the approximately constant DOS implies
the RG flow will be governed by the standard metallic
Kondo result1,13 with r = 0, renormalizing the Kondo
coupling to J̃ = J/ (1− Jρ0 lnD/∆) with the effective
bandwidth ∆ (see Ref. 9). Then, we can use the above
result in the renormalized system, obtaining that Kondo
screening occurs for J̃ρ0 & r which is easily shown to be
equivalent to the condition ∆ . ∆∗ with
∆∗ = e
1/rTK, (1)
where
TK = D exp
, (2)
is the Kondo temperature of the system in the absence
of pseudogap (which we are using here to clarify the typ-
ical energy scale for ∆∗). Setting r = 1 to establish the
implication of Eq. (1) for a d-wave superconductor, we
see that, due to the d-wave pseudogap in the density of
states, the conduction electrons can only screen the im-
purity moment if their gap amplitude is smaller than a
critical value of order the corresponding Kondo temper-
ature TK for constant density of states. In particular,
for ∆ large compared to the (often rather small) energy
scale TK, the local moment is unscreened, demonstrating
the sensitivity of the single impurity Kondo effect with
respect to the low energy behavior of the host.
Given the complexity of the behavior for a single im-
purity in a conduction electron host with pseudogap, it
seems hopeless to study the Kondo lattice. We will show
below that this must not be the case and that, moreover,
Kondo screening is stable far beyond the single-impurity
result Eq. (1), as illustrated in Fig. 1 (the dashed line
in this plot is Eq. (1) with ρ0 = 1/2D). To do this,
we utilize a the large-N mean field theory of the Kondo
lattice to demonstrate that the transition between the
screened and unscreened case is discontinuous. Thus, at
least within this approach, no critical fluctuations oc-
cur (in contrast to the single-impurity case discussed
above). More importantly, our large-N analysis also finds
that the stability regime of the Kondo screened lattice
is much larger than that of the single impurity. Thus,
the screened heavy-electron state is more robust and the
local-moment phase only emerges if the conduction elec-
tron d-wave gap amplitude obeys
∆ > ∆c ≃
TKD ≫ TK, (3)
0.02 0.04 0.06 0.08 0.1 0.12 0.14
local
moment
screened
Kondo lattice
FIG. 2: (Color online) The solid line is a plot of the Kondo
temperature TK(∆), above which V = 0 (and Kondo screen-
ing is destroyed), normalized to its value at ∆ = 0 [Eq. (14)],
as a function of the d-wave pairing amplitude ∆, for the
case of J = 0.3D and µ = −0.1D. With these parameters,
TK(0) = 0.0014D, and ∆c, the point where TK(∆) reaches
zero, is 0.14D [given by Eq. (33)] The dashed line indicates a
spinodal, along which the term proportional to V 2 in the free
energy vanishes. At very small ∆ < 2.7 × 10−4D, where the
transition is continuous, the dashed line coincides with the
solid line.
with D the conduction electron bandwidth. Below, we
shall derive a more detailed expression for ∆c; in Eq. (3)
we are simply emphasizing that ∆c is large compared to
TK [and, hence, Eq. (1)].
In addition, we find that for ∆ < ∆c, the renormalized
mass only weakly depends on ∆, except for the region
close to ∆c. We give a detailed explanation for this en-
hanced stability of Kondo lattice screening, demonstrat-
ing that it is a direct result of the opening of a hybridiza-
tion gap in the heavy Fermi liquid state. Since the re-
sult was obtained using a large-N mean field theory we
stress that such an approach is not expected to properly
describe the detailed nature close to the transition. It
should, however, give a correct order of magnitude result
for the location of the transition.
To understand the resilience of Kondo-lattice screen-
ing, recall that, in the absence of d-wave pairing, it is
well known that the lattice Kondo effect (and concomi-
tant heavy-fermion behavior) is due a hybridization of
the conduction band with an f -fermion band that rep-
resents excitations of the lattice of spins. A hybridized
Fermi liquid emerges from this interaction. We shall see
that, due to the coherency of the Fermi liquid state, the
resulting hybridized heavy fermions are only marginally
affected by the onset of conduction-electron pairing. This
weak proximity effect, with a small d-wave gap ampli-
tude ∆f ≃ ∆TK/D for the heavy fermions, allows the
Kondo effect in a lattice system to proceed via f -electron-
dominated heavy-fermion states that screen the local mo-
ments, with such screening persisting up to much larger
values of the d-wave pairing amplitude than implied by
the single impurity result6,7, as depicted in Fig. 1 (which
applies at low T ). A typical finite-T phase diagram is
shown in Fig. 2.
Our theory directly applies to the electron-doped
cuprate Nd2−xCexCuO4, possessing both d-wave
superconductivity14,15 with Tc ≃ 20K and heavy
fermion behavior below5 TK ∼ 2 − 3K. The latter
is exhibited in a large linear heat capacity coefficient
γ ≃ 4J/(mol×K2) together with a large low-frequency
susceptibility χ with Wilson ratio R ≃ 1.6. The lowest
crystal field state of Nd3+ is a Kramers doublet, well
separated from higher crystal field levels16, supporting
Kondo lattice behavior of the Nd-spins. The supercon-
ducting Cu-O-states play the role of the conduction
electrons. Previous theoretical work on Nd2−xCexCuO4
discussed the role of conduction electron correlations17.
Careful investigations show that the single ion Kondo
temperature slightly increases in systems with elec-
tronic correlations18,19, an effect essentially caused by
the increase in the electronic density of states of the
conduction electrons. However, the fact that these con-
duction electrons are gapped has not been considered,
even though the Kondo temperature is significantly
smaller than the d-wave gap amplitude ∆ ≃ 3.7meV
(See Ref. 20). We argue that Kondo screening in
Nd2−xCexCuO4 with TK ≪ ∆ can only be understood
in terms of the mechanism discussed here.
We add for completeness that an alternative sce-
nario for the large low temperature heat capacity of
Nd2−xCexCuO4 is based on very low lying spin wave
excitations21. While such a scenario cannot account for
a finite value of C (T ) /T as T → 0, it is consistent
with the shift in the overall position of the Nd-crystal
field states upon doping. However, an analysis of the
spin wave contribution of the Nd-spins shows that for
realistic parameters C (T ) /T vanishes rapidly below the
Schottky anomaly22, in contrast to experiments. Thus
we believe that the large heat capacity and susceptibility
of Nd2−xCexCuO4 at low temperatures originates from
Kondo screening of the Nd-spins.
Despite its relevance for the d-wave superconductor
Nd2−xCexCuO4, we stress that our theory does not ap-
ply to heavy electron d-wave superconductors, such as
CeCoIn5 (see Ref. 23), in which the d-wave gap is not
a property of the conduction electron host, but a more
subtle aspect of the heavy electron state itself. The latter
gives rise to a heat capacity jump at the superconducing
transition ∆C (Tc) that is comparable to γTc, while in
our theory ∆C (Tc) ≪ γTc holds.
II. MODEL
The principal aim of this paper is to study the screen-
ing of local moments in a d-wave superconductor. Thus,
we consider the Kondo lattice Hamiltonian, possessing lo-
cal spins (Si) coupled to conduction electrons (ckα) that
are subject to a pairing interaction:
kαckα +
i,α,β
Si · c†iασαβciβ + Upair. (4)
Here, J is the exchange interaction between conduction
electrons and local spins and ξk = ǫk − µ with ǫk the
conduction-electron energy and µ the chemical potential.
The pairing term
Upair = −
Ukk′c
c−k′↓ck′↑, (5)
is characterized by the attractive interaction between
conduction electrons Ukk′ . We shall assume the latter
stabilizes d-wave pairing with a gap ∆k = ∆cos 2θ with
θ the angle around the conduction-electron Fermi surface.
We are particularly interested in the low-temperature
strong-coupling phase of this model, which can be studied
by extending the conduction-electron and local-moment
spin symmetry to SU(N) and focusing on the large-N
limit24. In case of the single Kondo impurity, the large-
N approach is not able to reproduce the critical behavior
at the transition from a screened to an unscreeened state.
However, it does correctly determine the location of the
transition, i.e. the non-universal value for the strength of
the Kondo coupling where the transition from screened
to unscreened impurity takes place8. Since the location
of the transition and not the detailed nature of the tran-
sition is the primary focus of this paper, a mean field
theory is still useful.
Although the physical case corresponds to N = 2, the
large-N limit yields a valid description of the heavy Fermi
liquid Kondo-screened phase25. We thus write the spins
in terms of auxiliary f fermions as Si · σαβ → f †iαfiβ −
δαβ/2, subject to the constraint
iαfiα = N/2. (6)
To implement the large-N limit, we rescale the ex-
change coupling via J/2 → J/N and the conduction-
electron interaction as Uk,k′ → s−1Uk,k′ [where N ≡
(2s + 1)]. The utility of the large-N limit is that the
(mean-field) stationary-phase approximation to H is be-
lieved to be exact at large N . Performing this mean field
decoupling of H yields
k,m=−s
kmckm + V
kmckm + h.c.
kmfkm
k,m=1/2
c−k−mckm + h.c.
+ E0, (7)
with E0 a constant in the energy that is defined below.
The pairing gap, ∆k, and the hybridization between con-
duction and f -electrons, V , result from the mean field de-
coupling of the pairing and Kondo interactions, respec-
tively. The hybridization V (that we took to be real)
measures the degree of Kondo screening (and can be di-
rectly measured experimentally26) and λ is the Lagrange
multiplier that implements the above constraint, playing
the role of the f -electron level. The free energy F of this
single-particle problem can now be calculated, and has
the form:
F (V, λ,∆k) =
∆k∆k′U
k,α=±
(ξk + λ)−
Ekα − T ln
1 + e−βEkα
where T = β−1 is the temperature. The first three terms
are the explicit expressions for E0 in Eq. (7), and Ek± is
Ek± =
+ λ2 + 2V 2 + ξ2
Sk, (9)
Sk = (∆
− λ2)2 + 4V 2
(ξk + λ)
2 +∆2
describing the bands of our d-wave paired heavy-fermion
system.
The phase behavior of this Kondo lattice system for
given values of T , J and µ is determined by finding points
at which F is stationary with respect to the variational
parameters V , λ, and ∆k. For simplicity, henceforth we
take ∆k as given (and having d-wave symmetry as noted
above) with the goal of studying the effect of nonzero
pairing on the formation of the heavy-fermion metal char-
acterized by V and λ that satisfy the stationarity condi-
tions
= 0, (10a)
= 0, (10b)
with the second equation enforcing the constraint,
Eq. (6). We shall furthermore restrict attention to µ < 0
(i.e., a less than half-filled conduction band).
Before we proceed we point out that the magnitude of
the pairing gap near the unpaired heavy-fermion Fermi
surface (located at ξ = V 2/λ) is remarkably small. Tay-
lor expanding Ek− near this point, we find
Ek− ≃
ξ − V 2/λ− λ∆2
, (11)
giving a heavy-fermion gap ∆fk = (λ/V )
∆k [with am-
plitude ∆f = ∆(λ/V )
]. We show below that (λ/V )
≪ 1 such that ∆fk ≪ ∆k. In Fig. 3, we plot the
lower heavy-fermion band for the unpaired case ∆k = 0
(dashed line) along with ±Ek− for the case of finite ∆k
(solid lines) in the vicinity of the unpaired heavy-fermion
Fermi surface, showing the small heavy-fermion gap ∆fk.
Thus, we find a weak proximity effect in which the heavy-
fermion quasiparticles, which are predominantly of f -
character, are only weakly affected by the presence of
d-wave pairing in the conduction electron band.
0.6 0.8 1.0 1.2 1.4
FIG. 3: The dashed line is the lower heavy-fermion band
(crossing zero at the heavy-fermion Fermi surface) for the
unpaired (∆ = 0) case and the solid lines are ±Ek− for ∆k =
0.1D, showing a small f-electron gap ∆fk ≃ .014D.
-1 -0.5 0.5 1
FIG. 4: Plot of the energy bands E+(ξ) (top curve) and
(ξ) (bottom curve), defined in Eq. (13), in the heavy
Fermi liquid state (for ∆ = 0), for the case V = 0.2D and
λ = 0.04D, that has a heavy-fermion Fermi surface near
ξ = D and an experimentally-measurable hybridization gap26
(the minimum value of E+ − E−, i.e., the direct gap) equal
to 2V ∼
TKD. Note, however, the indirect gap is λ ∼ TK.
III. KONDO LATTICE SCREENING
A. Normal conduction electrons
A useful starting point for our analysis is to recall
the well-known27 unpaired (∆ = 0) limit of our model.
By minimizing the correpsonding free energy [simply the
∆ = 0 limit of Eq. (8)], one obtains, at low temperatures,
that the Kondo screening of the local moments is repre-
sented by the nontrivial stationary point of F at V = V0
and λ = λ0 = V
0 /D, with
D + µ
, (12)
Here we have taken the conduction electron density of
states to be a constant, ρ0 = (2D)
−1, with 2D the
bandwidth. The resulting phase is a metal accommo-
dating both the conduction and f -electrons with a large
density of states ∝ λ0−1 near the Fermi surface at
ǫk ≃ µ + V 20 /λ0, revealing its heavy-fermion character.
In Fig. 4, we plot the energy bands
E± (ξk) =
ξk + λ±
(ξk − λ)2 + 4V 2
, (13)
of this heavy Fermi liquid in the low-T limit.
With increasing T , the stationary V and λ decrease
monotonically, vanishing at the Kondo temperature
D2 − µ2 exp
, (14)
D − µ
D + µ
λ0. (15)
Here, the second line is meant to emphasize that TK is
of the same order as the T = 0 value of the f -fermion
chemical potential λ0, and therefore TK ≪ V0, i.e., TK
is small compared to the zero-temperature hybridization
energy V0.
It is well established that the phase transition-like be-
havior of V at TK is in fact a crossover onceN is finite
1,24.
Nevertheless, the large-N approach yields the correct or-
der of magnitude estimate for TK and provides a very use-
ful description of the strong coupling heavy-Fermi liquid
regime, including the emergence of a hybridization gap
in the energy spectrum.
B. d-wave paired conduction electrons
Next, we analyze the theory in the presence of d-wave
pairing with gap amplitude ∆. Thus, we imagine contin-
uously turning on the d-wave pairing amplitude ∆, and
study the stability of the Kondo-screened heavy-Fermi
liquid state characterized by the low-T hybridization V0,
Eq. (12). As we discussed in Sec. I, in the case of a single
Kondo impurity, it is well known that Kondo screening is
qualitatively different in the case of d-wave pairing, and
the single impurity is only screened by the conduction
electrons if the Kondo coupling exceeds a critical value
1 + lnD/∆
. (16)
For J < J∗, the impurity is unscreened. This result for
J∗ can equivalently be expressed in terms of a critical
pairing strength ∆∗, beyond which Kondo screening is
destroyed for a given J :
∆∗ = D exp
, (17)
[equivalent to Eq. (1) for r = 1], which is proportional
to the Kondo temperature TK. This result, implying
0.02 0.04 0.06 0.08
FIG. 5: (Color online) Main: Mean-field Kondo parameter V
as a function of the d-wave pairing amplitude ∆, for exchange
coupling J = 0.30D and chemical potential µ = −0.1D, ac-
cording to the approximate formula Eq. (31) (solid line) and
via a direct minimization of Eq. (8) at T = 10−4D (points),
the latter exhibiting a first-order transition near ∆ = 0.086D.
that a d-wave superconductor can only screen a local
spin if the pairing strength is much smaller than TK,
can also be derived within the mean-field approach to
the Kondo problem, as shown in Appendix A (see also
Ref. 7). Within this approach, a continuous transition to
the unscreened phase (where V 2 → 0 continuously) takes
place at ∆ ≃ ∆∗.
Thus, calculations for the single impurity case indi-
cate that Kondo screening is rather sensitive to a d-wave
pairing gap. The question we wish to address is, how
does d-wave pairing affect Kondo screening in the lattice
case? In fact, we will see that the results are quite differ-
ent in the Kondo lattice case, such that Kondo screening
persists beyond the point ∆∗. To show this, we have nu-
merically studied the ∆-dependence of the saddle point
of the free energy Eq. (8), showing that, at low temper-
atures, V only vanishes, in a discontinuous manner, at
much larger values of ∆, as shown in Fig. 5 (solid dots)
for the case of J = 0.30D, µ = −0.1D and T = 10−4D
(i.e., T/TK ≃ .069). In Fig. 2, we plot the phase diagram
as a function of T and ∆, for the same values of J and
µ, with the solid line denoting the line of discontinuous
transitions.
The dashed line in Fig. 2 denotes the spinodal Ts of the
free energy F at which the quadratic coefficient of Eq. (8)
crosses zero. The significance of Ts is that, if the Kondo-
to-local moment transition were continuous (as it is for
∆ = 0), this would denote phase boundary; the T → 0
limit of this quantity coincides with the single-impurity
critical pairing Eq. (17). An explicit formula for Ts can
be easily obtained by finding the quadratic coefficient of
Eq. (8):
tanhEk/2Ts(∆)
, (18)
with Ek ≡
, and where we set λ = 0 [which
must occur at a continuous transition where V → 0, as
can be seen by analyzing Eq. (10b)]. As seen in Fig. 2,
the spinodal temperature is generally much smaller than
the true transition temperature; however, for very small
∆ → 0, Ts(∆) coincides with the actual transition (which
becomes continuous), as noted in the figure caption.
Our next task is to understand these results within
an approximate analytic analysis of Eq. (8); before do-
ing so, we stress again that the discontinuous transition
from a screened to an unscreened state as function of T
becomes a rapid crossover for finite N . The large N the-
ory is, however, expected to correctly determine where
this crossover takes place.
1. Low-T limit
According to the numerical data (points) plotted in
Fig. 5, the hybridization V is smoothly suppressed with
increasing pairing strength ∆ before undergoing a discon-
tinuous jump to V = 0. To understand, analytically, the
∆-dependence of V at low-T , we shall analyze the T = 0
limit of F , i.e., the ground-state energy E. The essen-
tial question concerns the stability of the Kondo-screened
state with respect to a d-wave pairing gap, characterized
by the following ∆-dependent hybridization
V (∆) = V0
∆2typ
, (19)
with ∆typ an energy scale, to be derived, that gives the
typical value of ∆ for which the heavy-fermion state is
affected by d−wave pairing.
To show that Eq. (19) correctly describes the smooth
suppression of the hybrization with increasing ∆, and to
obtain the scale ∆typ, we now consider the dimensionless
quantity
χ∆ ≡ −
, (20)
that characterizes the change of the ground state en-
ergy with respect to the pairing gap. Separating the
amplitude of the gap from its momentum dependence,
i.e. writing ∆k = ∆φk, we obtain from the Hellmann-
Feynman theorem that:
χ∆ = −
. (21)
For ∆ → 0 this yields
Gcc (k,iω)Gcc (−k,−iω) . (22)
Here, Gcc (k,iω) is the conduction electron propagator.
As expected, χ∆ is the particle-particle correlator of the
conduction electrons. Thus, for T = 0 the particle-
particle response will be singular. This is the well known
Cooper instability. For V = 0 we obtain for example
χ∆ (V = 0) =
D2 − µ2
, (23)
where we used ∆ as a lower cut off to control the Cooper
logarithm. Below we will see that, except for extremely
small values of ∆, the corresponding Cooper logarithm
is overshadowed by another logarithmic term that does
not have its origin in states close to the Fermi surface,
but rather results from states with typical energy V ≃√
In order to evaluate χ∆ in the heavy Fermi liquid state,
we start from:
Gcc (k,ω) =
ω − E+ (ξk)
ω − E− (ξk)
, (24)
where E± is given in Eq. (13) and the coherence factors
of the hybridized Fermi liquid are:
1− ξk − λ√
(ξk − λ)2 + 4V 2
ξk − λ√
(ξk − λ)2 + 4V 2
 . (25)
Inserting Gcc (k,ω) into the above expression for χ∆
yields
∫ D−µ
4v2u2θ (E−)
E+ + E−
We used that E+ > 0 is always fulfilled, as we consider a
less than half filled conduction band.
Considering first the limit λ = 0, it holds E− (ξ) < 0
and the last term in the above integral disappears. The
remaining terms simplify to
χ∆ (λ = 0) =
∫ D−µ
ξ2 + 4V 2
D2 − µ2
. (27)
Even for λ nonzero, this is the dominant contribution to
χ∆ in the relevant limit λ ≪ V ≪ D. To demonstrate
this we analyze Eq. (26) for nonzero λ, but assuming
λ ≪ V as is indeed the case for small ∆. The calculation
is lengthy but straightforward. It follows:
D2 − µ2
D |µ|
. (28)
The last term is the Cooper logarithm, but now in the
heavy fermion state. The prefactor λ/D ≃ TK/D is a
result of the small weight of the conduction electrons on
the Fermi surface (i.e. where ξ ≃ V 2/λ) as well as the
reduced velocity close to the heavy electron Fermi sur-
face. Specifically it holds u2
ξ ≃ V 2/λ
≃ λ2/V 2 as well
as E−
ξ ≃ V 2/λ
ξ − V
Thus, except for extremely small gap values where
∆2 < D2
)−D/TK
, χ∆ is dominated by the λ = 0
result, Eq. (27), and the Cooper logarithm plays no role
in our analysis. The logarithm in Eq. (27) is not origi-
nating from the heavy electron Fermi surface (i.e. it is
not from ξ ≃ r
). Instead, it has its origin in the inte-
gration over states where E− < 0. The important term
in Eq. (26) is peaked for ξ ≃ 0 i.e. where
E± (ξ ≃ 0) = ±V and is large as long as |ξ| . V . For
ξ ≃ 0 holds v
≃ − u
. This peak at ξ ≃ 0 has
its origin in the competition between two effects. Usu-
ally, u or v are large when E± ≃ ξ. The only regime
where u or v are still sizable while E± remain small is
close to the bare conduction electron Fermi surface at
|ξ| ≃ V (the position of the level repulsion between the
two hybridizing bands). Thus, the logarithm is caused
by states that are close to the bare conduction electron
Fermi surface. Although these states have the strongest
response to a pairing gap, they don’t have much to do
with the heavy fermion character of the system. It is in-
teresting that this heavy fermion pairing response is the
same even in case of a Kondo insulator where λ = 0 and
the Fermi level is in the middle of the hybridization gap.
The purpose of the preceding analysis was to derive
an accurate expression for the ground-state energy E at
small ∆. Using Eq. (20) gives:
E = E(∆ = 0)− χ∆ρ0∆2, (29)
which, using Eq. (27) and considering the leading order
in λ ≪ V and ∆ ≪ V , safely neglecting the last term
of Eq. (28) according to the argument of the previous
paragraph, and dropping overall constants, yields
+V 2ρ0 ln
D + µ
− ρ0∆
D2 − µ2
. (30)
Using Eq. (10), the stationary value of the hybridization
(to leading order in ∆2) is then obtained via minimization
with respect to V and λ. This yields
V (∆) ≃ V0 −
, (31)
with the stationary value of λ = 2ρ0V
2, which estab-
lishes Eq. (19). A smooth suppression of the Kondo
hybridization from the ∆ = 0 value V0 [Eq. (12)] oc-
curs with increasing d-wave pairing amplitude ∆ at low
T . This result thus implies that the conduction electron
gap only causes a significant reduction of V and λ for
∆ ≃ ∆typ ∝
In Fig. 5 we compare V (∆) of Eq. (31) (solid line)
with the numerical result (solid dots). As long as V
0.0001 0.0002 0.0003 0.0004 0.0005
C/γ0T
FIG. 6: Plot of the low-temperature specific heat coefficient
= − ∂
, for the case of λ = 10−2D, V = 10−1D, and
µ = −0.1D, for the metallic case (∆ = 0, dashed line) and
the case of nonzero d-wave pairing (∆ = 0.1D, solid line).
This shows that, even with nonzero ∆, the specific heat coef-
ficient will appear to saturate at a large value at low T (thus
exhibiting signatures of a heavy fermion metal), before van-
ishing at asymptotically low T ≪ ∆f (= ∆(λ/V )2 = 10−4D)
Each curve is normalized to the T = 0 value for the metallic
case, γ0 ≃ 23π
2/λ2.
stays finite, the simple relation Eq. (31) gives an ex-
cellent description of the heavy electron state. Above
the small f -electron gap ∆f , these values of V and λ
yield a large heat capacity coefficient (taking N = 2)
γ ≃ 2
π2ρ0V
2/λ2 and susceptibility χ ≃ 2ρ0V 2/λ2, re-
flecting the heavy-fermion character of this Kondo-lattice
system even in the presence of a d-wave pairing gap. Ac-
cording to our theory, this standard heavy-fermion be-
havior (as observed experimentally5 in Nd2−xCexCuO4)
will be observed for temperatures that are large com-
pared to the f -electron gap ∆f . However, for very small
T ≪ ∆f , the temperature dependence of the heat capac-
ity changes (due to the d-wave character of the f -fermion
gap), behaving as C = AT 2/∆ with a large prefactor
A ≃ (D/TK)2. This leads to a sudden drop in the heat
capacity coefficient at low T , as depicted in Fig. 6.
The surprising robustness of the Kondo screening with
respect to d-wave pairing is rooted in the weak proximity
effect of the f -levels and the coherency as caused by the
formation of the hybridization gap. Generally, a pairing
gap affects states with energy ∆k from the Fermi en-
ergy. However, low energy states that are within TK of
the Fermi energy are predominantly of f -electron charac-
ter (a fact that follows from our large-N theory but also
from the much more general Fermi liquid description of
the Kondo lattice28) and are protected by the weak prox-
imity. These states only sense a gap ∆fk ≪ ∆k and can
readily participate in local-moment screening.
Furthermore, the opening of the hybridization gap co-
herently pushes conduction electrons to energies ≃ V
from the Fermi energy. Only for ∆ ≃ V ≃
TKD will
the conduction electrons ability to screen the local mo-
ments be affected by d-wave pairing. This situation is
very different from the single impurity Kondo problem
where conduction electron states come arbitrarily close
to the Fermi energy.
2. First-order transition
The result Eq. (31) of the preceding subsection strictly
applies for ∆ → 0, although as seen in Fig. 5, in practice
it agrees quite well with the numerical minimization of
the free energy until the first-order transition. To under-
stand the way in which V is destroyed with increasing ∆,
we must consider the V → 0 limit of the free energy.
We start with the ground-state energy. Expanding E
[the T → 0 limit of Eq. (8)] to leading order in V and
zeroth order in λ (valid for V → 0), we find (dropping
overall constants)
≃ −4ρ0V 2 ln
V 3, (32)
where we defined the quantity ∆c
∆c = 4
D2 − µ2 exp
, (33)
at which the minimum value of V in Eq. (32) vanishes
continuously, with the formula for V (∆) given by
V (∆) ≃ 1
, (34)
near the transition. According to Eq. (33), the equilib-
rium hybridization V vanishes (along with the destruc-
tion of Kondo screening) for pairing amplitude ∆c ∼√
TKD, of the same order of magnitude as the T = 0
hybridization V0, as expected [and advertised above in
Eq. (3)].
Equation (33) strictly applies only at T = 0, appar-
ently yielding a continuous transition at which V → 0
for ∆ → ∆c. What about T 6= 0? We find that, for
small but nonzero T , Eq. (33) approximately yields the
correct location of the transition, but that the nature
of the transition changes from continuous to first-order.
Thus, for ∆ near ∆c, there is a discontinuous jump to
the local-moment phase that is best obtained numeri-
cally, as shown above in Figs. 5 and 2. However, we can
get an approximate analytic understanding of this first-
order transition by examining the low-T limit. Since ex-
citations are gapped, at low T the free energy FK of the
Kondo-screened (V 6= 0) phase is well-approximated by
inserting the stationary solution Eq. (34) into Eq. (32):
2 ln3
, (35)
for FK at ∆ → ∆c. The discontinuous Kondo-to-local
moment transition occurs when the Kondo free energy
Eq. (35) is equal to the local-moment free energy. For
the latter we set V = λ = 0 in Eq. (8), obtaining (recall
ρ0(D + µ)
2 − 1
D2 − µ2
−T ln 2− T
1 + e−βEk
, (36)
where we dropped an overall constant depending on the
conduction-band interaction.
The term proportional to T in Eq. (36) comes from the
fact that Ek− = 0 for V = λ = 0, and corresponds to
the entropy of the local moments. At low T , the gapped
nature of the d-wave quasiparticles implies the last term
in Eq. (36) can be neglected (although the nodal quasi-
particles give a subdominant power-law contribution). In
deriving the Kondo free energy FK, Eq. (35), we dropped
overall constant terms; re-establishing these to allow a
comparison to FLM , and setting FLM = FK, we find
2 ln3
= T ln 2, (37)
that can be solved for temperature to find the transition
temperature TK for the first-order Kondo screened-to-
local moment phase transition:
TK(∆) =
6 ln 2
, (38)
that is valid for ∆ → ∆c, providing an accurate ap-
proximation to the numerically-determined TK curve in
Fig. 2 (solid line) in the low temperature regime (i.e.,
near ∆c = 0.14D in Fig. 2).
Equation (38) yields the temperature at which, within
mean-field theory, the screened Kondo lattice is destroyed
by the presence of nonzero d-wave pairing; thus, as long
as ∆ < TK(∆), heavy-fermion behavior is compatible
with d-wave pairing in our model. The essential feature
of this result is that TK(∆) is only marginally reduced
from the ∆ = 0 Kondo temperature Eq. (2), establishing
the stability of this state. In comparison, according to ex-
pectations based on a single-impurity analysis, one would
expect the Kondo temperature to follow the dashed line
in Fig. 2.
Away from this approximate result valid at large N ,
the RKKY interaction between moments is expected to
lower the local-moment free energy, altering the predicted
location of the phase boundary. Then, even for T =
0, a level crossing between the screened and unscreened
ground states occurs for a finite V . Still, as long as the
∆ = 0 heavy fermion state is robust, it will remain stable
at low T for ∆ small compared to ∆c, as summarized in
Figs. 1 and 2.
IV. CONCLUSIONS
We have shown that a lattice of Kondo spins coupled to
an itinerant conduction band experiences robust Kondo
screening even in the presence of d-wave pairing among
the conduction electrons. The heavy electron state is pro-
tected by the large hybridization energy V ≫ TK. The
d-wave gap in the conduction band induces a relatively
weak gap at the heavy-fermion Fermi surface, allowing
Kondo screening and heavy-fermion behavior to persist.
Our results demonstrate the importance of Kondo-lattice
coherency, manifested by the hybridization gap, which is
absent in case of dilute Kondo impurities. As pointed
out in detail, the origin for the unexpected robustness of
the screened heavy electron state is the coherency of the
Fermi liquid state. With the opening of a hybridization
gap, conduction electron states are pushed to energies
of order
TKD away from the Fermi energy. Whether
or not these conduction electrons open up a d-wave gap
is therefore of minor importance for the stability of the
heavy electron state.
Our conclusions are based on a large-N mean field the-
ory. In case of a single impurity, numerical renormaliza-
tion group calculations demonstrated that such a mean
field approach fails to reproduce the correct critical be-
havior where the transition between screened and un-
screened impurity takes place. However the mean field
theory yields the correct value for the strength of the
Kondo coupling at the transition. In our paper we are
not concerned with the detailed nature in the near vicin-
ity of the transition. Our focus is solely the location of
the boundary between the heavy Fermi liquid and un-
screened local moment phase, and we do expect that a
mean field theory gives the correct result. One possibility
to test the results of this paper is a combination of dy-
namical mean field theory and numerical renormalization
group for the pseudogap Kondo lattice problem.
In case where Kondo screening is inefficient and ∆ >√
TKD, i.e., the “local moment” phase of Figs. 1 and 2,
the ground state of the moments will likely be magneti-
cally ordered. This can have interesting implications for
the superconducting state. Examples are reentrance into
a normal phase (similar to ErRh4B4, see Ref. 29) or a
modified vortex lattice in the low temperature magnetic
phase. In our theory we ignored these effects. This is no
problem as long as the superconducting gap amplitude
∆ is small compared to
TKD and the Kondo lattice is
well screened. Thus, the region of stability of the Kondo
screened state will not be significantly affected by includ-
ing the magnetic coupling between the f -electrons. Only
the nature of the transition and, of course, the physics
of the unscreened state will depend on it. Finally, our
theory offers an explanation for the heavy fermion state
in Nd2−xCexCuO4, where ∆ ≫ TK.
Acknowledgments — We are grateful for useful discus-
sions with A. Rosch and M. Vojta. This research was
supported by the Ames Laboratory, operated for the U.S.
Department of Energy by Iowa State University under
Contract No. DE-AC02-07CH11358. DES was also sup-
ported at the KITP under NSF grant PHY05-51164.
APPENDIX A: SINGLE IMPURITY CASE
For a single Kondo impurity a critical value J∗ for
the coupling between conduction electron and impurity
spin emerges, separating Kondo-screened from local mo-
ment behavior for a single spin impurity in a d-wave su-
perconductor, see Eq. (16). As discussed in the main
text, this is equivalent to a critical pairing Eq. (17)
above which Kondo screening does not occur. The re-
sult was obtained in careful numerical renormalization
group calculations8,9. In the present section, we demon-
strate that the same result also follows from a simple
large-N mean field approach. It is important to stress
that this approach fails to describe the detailed critical
behavior. However, here we are only concerned with the
approximate value of the non-universal quantity J∗. In-
deed, mean field theory is expected to give a reasonable
value for the location of the transition.
Our starting point is the model Hamiltonian
kmckm ++
m,m′,k,k′
f †mfmc
kmck′m′
Ukk′c
. (A1)
with the corresponding mean-field action S = Sf + Sb +
Sint with (introducing the Lagrange multiplier λ and hy-
bridization V as usual, and making the BCS mean-field
approximation for the conduction fermions):
km(∂τ + ǫk)ckm + f
m(∂τ + λ)fm
V †V − λNq0
, (A2)
Sint =
f †mckmV + V
†ckmfm
m=1/2
−k−mckm + c
−k−m∆k
, (A3)
where the λ integral implements the constraint Nq0 =∑
mfm, with q0 = 1/2. Here, we have taken the large
N limit, with N = 2J + 1.
The mean-field approximation having been made, it is
now straightforward to trace over the fermionic degrees
of freedom to yield
N |V |2
− λNq0 −
(iω − λ− Γ1(iω))(iω + λ+ Γ1(−iω))− Γ2(iω)Γ̄2(iω)
, (A4)
for the free energy contribution due to a single impurity
in a d-wave superconductor. Here, we dropped an overall
constant due to the conduction fermions only, as well as
the quadratic term in ∆k (which of course determines the
equilibrium value of ∆k; here, as in the main text, we’re
interested in the impact of a given ∆k on the degree of
Kondo screening), and defined the functions
Γ1(iω) = |V |2
iω + ǫk
(iω)2 − E2
, (A5)
Γ2(iω) = V
(iω)2 − E2
, (A6)
Γ̄2(iω) = (V
(iω)2 − E2
. (A7)
At this point we note that, for a d-wave superconduc-
tor, Γ2 = Γ̄2 = 0 due to the sign change of the d-wave
order parameter. The self-energy Γ1(iω) is nonzero and
essentially measures the density of states (DOS) ρd(ω) of
the d-wave superconductor. In fact, one can show that
the corresponding retarded function Γ1R(ω) satisfies
Γ1R(ω) = |V |2
ρd(z)
ω + iδ − z
, (A8)
with δ = 0+, so that the imaginary part Γ′′1R(ω) =
−π|V |2ρd(ω) measures the DOS. Writing Γ1R(ω) ≡
|V |2G(ω), we have for the free energy
N |V |2
− λNq0 (A9)
nF(z) tan
( −|V |2G′′(z)
z − λ− |V |2G′(z)
and for the stationarity conditions, Eq. (10),
nF(z)G
′′(z)(z − λ)
(z − λ− |V |2G′(z))2 + |V |4(G′′(z))2
,(A10)
q0 = −
nF(z)|V |2G′′(z)
(z − λ− |V |2G′(z))2 + |V |4(G′′(z))2
,(A11)
which can be evaluated numerically to determine V and
λ as a function of T and ∆.
The Kondo temperature TK is defined by the temper-
ature at which V 2 → 0 continuously; at such a point, the
constraint Eq. (A11) requires λ → 0. Here, we are in-
terested in finding the pairing ∆ at which TK → 0; thus,
this is obtained by setting T = V = λ = 0 in Eq. (A10):
−πρd(z)
, (A12)
= −ρ0 log
D + µ
+ ρ0, (A13)
where, for simplicity, in the final line we approximated
ρd(z) to be given by
ρd(ω) ≃
ρ0|ω|/∆, for |ω| < ∆,
ρ0, for |ω| > ∆,
(A14)
that captures the essential features (except for the nar-
row peak near ω = ∆) of the true DOS of a d-wave
superconductor, with ρ0 the (assumed constant) DOS of
the underlying conduction band.
The solution to Eq. (A13) is:
∆∗ = (D + µ) exp
, (A15)
showing a destruction of the Kondo effect for ∆ → ∆∗, as
V → 0 continuously, thus separating the Kondo-screened
(for ∆ < ∆∗) from the local moment (for ∆ > ∆∗)
phases.
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