Theory of magnetic excitons in the heavy-fermion superconductor $UPd_{2}Al_{3}$

0.2 MB·English

by  Jun Chang

Theory of magnetic excitons in the heavy-fermion superconductor UPd Al
2 3
Jun Chang1, I. Eremin1,2, P. Thalmeier3, and P. Fulde1
1 Max-Planck Institut fu¨r Physik komplexer Systeme, D-01187 Dresden, Germany
2 Institute fu¨r Mathematische und Theoretische Physik,
Technische Universit¨at Carolo-Wilhelmina zu Braunschweig, 38106 Braunschweig, Germany
7 3 Max-Planck Institut fu¨r Chemische Physik fester Stoffe, D-01187 Dresden, Germany
0
(Dated: February 6, 2008)
0
2 We analyze the influence of unconventional superconductivity on the magnetic excitations in
n the heavy fermion compound UPd2Al3. We show that it leads to the formation of a bound state
a at energies well below 2∆0 at the antiferromagnetic wave vector Q=(0,0,π/c). Its signature is
J a resonance peak in the spectrum of magnetic excitations in good agreement with results from
inelastic neutron scattering. Furthermore we investigate the influence of antiferromagnetic order
8
on the formation of the resonance peak. We find that its intensity is enhanced due to intraband
1
transitionsinducedbythereconstructionofFermisurfacesheets. Wedeterminethedispersionofthe

resonancepeaknearQandshowthatitisdominatedbythemagneticexcitondispersionassociated
]
n withlocalmoments. WedemonstratebyamicroscopiccalculationthatUPd2Al3isanotherexample
o inwhichtheunconventionalnatureofthesuperconductingorderparametercanbeprobedbymeans
c of inelastic neutron scattering and determined unambiguously.
-
r
p
u I. INTRODUCTION It is important to note that the resonant spin excita-
s tionsinsuperconductingcupratescanbeseenasadirect
.
at Therelationshipbetweenunconventionalsuperconduc- ccoonnsdeuqcuteinngceorodfetrhpeardaxm2−eyt2e-rw.aNvaemsyemly,mtheteryresoofntahnecesuppeeark-
m
tivityandmagnetisminheavy-fermionsystemsandtran-
occurs only if the order parameter changes sign in the
d- sitionmetaloxidesisoneofthemostinterestingresearch first Brillouin zone (BZ).6 Thus, INS can be considered
areas in condensed matter physics. In both cases it is
n as a bulk probe for the unconventional nature of super-
o widely believed that the magnetic degrees of freedom conductivity in these compounds. Therefore it is impor-
c play an essential role in the formation of superconduc- tant to search for such an effect in other unconventional
[ tivity. Furthermore, unconventional superconductivity
superconductors as well. In this paper we analyze the
yields strong feedback on the magnetic spin excitations
2 consequences of the unconventional pairing on the mag-
v in these systems below the superconducting transition netic excitations in UPd Al . We show that in addition
2 3
0 temperature Tc. One example is the famous so-called to the magnetic exciton dispersion present in the nor-
5 resonancepeakobservedinhigh-T cupratesbymeansof
c mal state, unconventionalsuperconductivity induces the
5 inelastic neutron scattering (INS)1 whose nature is still
formation of a bound state below T with an associated
9 actively debated.2 Remarkably, it has been found that c
resonance peak in the magnetic spectrum at the antifer-
0
6 INS reveals the formation of a new magnetic mode in romagnetic (AF) wave vector Q=(0,0,π/c) where c is a
the superconducting state of the uranium based heavy-
0 latticeconstantalongthecrystallographicz axis. Itsfre-
/ fermioncompoundUPd2Al3withTc=1.8K.3Itssharply quency is well below 2∆ and in good agreement with
t 0
a peakedintensity,itstemperaturedependenceandtheen- experimental data. We show that similar to cuprates
m ergy position well below 2∆ (with ∆ being the maxi-
0 0 the resonance peak in UPd Al is a consequence of an
2 3
mumofthe superconductinggap)stronglyresemblesthe
- unconventional superconducting order parameter which
d resonance peak seen in high-T cuprates. This is par-
c changes sign at regions of the Fermi surface connected
n ticularly remarkable, since the origin of superconduc- by the antiferromagnetic wavevector Q. We analyze the
o
tivity in cuprates and UPd Al seems to be different.
c 2 3 influence of antiferromagnetism on the formation of the
While frequently discussed scenarios in cuprates are a
: resonance peak and surprisingly find that its intensity is
v spin-fluctuationmediatedCooperpairingortheelectron-
enhanceddue to the reconstructionofthe Fermisurface.
i
X phonon interaction, in UPd2Al3 a magnetic-exciton me- We find that the dispersion of the resonance peak away
r diated pairing model has been proposed5 basedon avail- from Q is controlled by the momentum dependence of
a able experiments. The latter model is built on the dual
excitations of the localized magnetic moment (magnetic
nature of the 5f electrons. It consists of localized 5f2
exciton).
crystallineelectric field(CEF)states whichdisperse into
amagneticexcitonbandduetointersiteinteractionsand
aconductionelectronband4 formedbyitinerant5f elec- The resonance peak in UPd Al has also been stud-
2 3
trons with enhanced hybridization. The model success- ied by Bernhoeft et al.7 within a phenomenological two
fully explains the formationof unconventionalsupercon- component spin susceptibility model. However, in our
ductivity in this compound5 based on the virtual ex- microscopic calculations we show that antiferromagnetic
changeofthe magneticexcitonsbetweenitinerantquasi- order plays a crucial role in the formation of the reso-
particles. nance peak below T .
c
2
II. THE HAMILTONIAN commonly done.
Eq.(1)givesrisetofermionicandbosonicself-energies
FollowingpreviousconsiderationbyMcHaleet al.5 we and is particularly relevant for electron-hole states sepa-
use the low-energy Hamiltonian describing the interac- ratedbytheantiferromagneticwavevectorQ=(0,0,π/c).
tion of the itinerant f electrons and magnetic excitons Previously it has been shown that this interaction ex-
originating from localized 5f2 crystalline electric field plains superconductivity in UPd2Al3.5 We define the
(CEF) states: electron and magnetic exciton Green’s functions as fol-
lows:
1
H = ξ c c + ω α α +
0 p †pσ pσ q †q q 2 Gσσ′(p,iωm) = − Tτcpσ(τ)c†pσ′(0) ,
pσ q (cid:18) (cid:19) FT
Xg X D E
− N c†pασαzβcp+q,βλq αq+α†−q , (1) D(q,ivn) = − Tτaq(τ)a†−q(0) FT, (2)
Xp,q (cid:16) (cid:17) D E
where λ2q = ∆ωCqEF, and ∆CEF = 6 meV is the energy pwsheeurdeoaspqi(nτ)s=uscαeqp(tτib)i+lityα.†−qT(τh)eabnadreDmisagesnseetnictiaelxlycittohne
gap of the 5f2 electrons between the ground and first Green’s function is given by D (q,iν )=−∆CEF 1 .
excitedstatesinthecrystallineelectricfield. Thedisper- 0 n 2 νn2+ωq2
Duetotheinteractionofthemagneticexcitonswithcon-
sionof the magnetic excitons is approximatelydescribed
ducting electrons, the feedback effect on the former re-
by ω(q ) = ω [1+β cos(cq )] with 0 < β ≃ 1, where g
z ex z sults in
is the coupling constant between the itinerant electrons
and the localized magnetic moments. We adopt param- D0 2ωq
D = =− , (3)
eter values ωex = 5.5 meV and β = 0.72. Note, here we 1−D0Π0 ω2−ωq2 +2ωqΠ0
follow Ref. 5 in assuming that only the σ component
z
oftheconductionelectronscanexcitemagneticexcitons. where the magnetic exciton self-energy is given by
Therefore the spin-space isotropy is broken in a maxi-
mal (Ising) way. As a result the usual classification of Π (q,iω )=g2∆CEFχ (q,iω ) . (4)
Cooper pairs into spin-triplet and spin-singlet states is 0 n ω 0 n
q
notvalidandthe notation equalandoppositespinpair-
ingstatesshouldbebetterusedinstead. However,wewill Here, the spin susceptibility of the conduction electrons
still speak of singlet and triplet Cooper-pairing states as in the superconducting state is
1
χ0(q,iωn)=− G(k,iωm)G†(k+q,iωm+iωn)+F(k,iωm)F†(k+q,iωm+iωn) , (5)
β
iωXm,k(cid:2) (cid:3)
where bare Green’s functions of superconducting electrons are G(k,iω )=− iωm+ξk , F(k,iω )= ∆k . A
m ωm2+ξk2+∆2k m ωm2+ξk2+∆2k
straightforwardevaluation of the sum over the Matsubara frequencies gives (at T =0 K)
1 1 ξ ξ +∆ ∆
Imχ (q,ω)= d3k 1− k+q k k k+q δ(ω−E −E ) . (6)
0 4(2π)3 E E k+q k
Z (cid:18) k+q k (cid:19)
The Fermi surface of the conducting electrons is almost with cos(ck ) momentum dependence and spin-triplet
z
likeacylinderwithweakdispersionalongthez direction. states (S = 0) 1 (|↑↓i+|↓↑i) with sin(ck ) have the
z √2 z
Neglectingtheanisotropyofdispersionintheplaneyields highest(degenerate)superconducting transitiontemper-
ξk = ǫk⊥ +ǫkz −µ = ǫ w2+ǫ cos(ckz)−µ (ǫ ≪ ǫ , ature. Thus, in the following we will consider the two
⊥ k k ⊥
w = k /k ≤ 1) and E = ξ2 +∆2. Here, we ap- superconducting order parameters, ∆s = ∆ cos(ck )
0 k k k k 0 z
proxim⊥atethe hexagonalunit cell by a circle with radius and ∆t = ∆ sin(ck ) as the most relevant ones in this
p k 0 z
k chosen so that the hexagon and the circle have the model.
0
same area. Furthermore, we assume a parabolic disper-
sionintheplane. DuetotheIsing-typeanisotropyofthe
Letusnowdiscusstheconsequencesofthebehaviorof
interaction between conduction electrons and magnetic
Imχ for the magnetic exciton dispersion which follows
excitons it has been previously found5 that both pure 0
paramagnetic, i.e., spin-singlet states 1 (|↑↓i−|↓↑i) from Eq. (3). The dispersion of the magnetic exciton in
√2 the presence of coupling to the conduction electrons is
3

given by
2.2
atIlnowthferenqouωrem2n=aclieωpsq2adr−eatm2ergam2g∆nineCetEidcFisRnteaotχue0,r(Rqc,aeωsχe)0biys.athceoncustrav(n7a)t- χω(Q,) (states/eV)000..0182 (a) tsriinpglelett χω (Q,) (states/eV)02.0 (b) tsriinpglelett
ture of the Fermi surface along the kz direction. At the Im0.04 Re
same time Imχ ∝ −iγω, where γ is a Landau damp-
0
ing constant. Thus, the bare magnetic exciton acquires 0.00 1.8
0.0 0.5 1.0ω1.5 2.0 2.5 3.0 0.0 0.5 1.0ω1.5 2.0 2.5 3.0
a linewidth and renormalizes slightly by a certain con- (meV) (meV)
stant which changes the original position of ω down-
q FIG.1: (Coloronline)The(a)imaginaryand(b)realpartsof
wards. In the superconducting state the renormalization
the longitudinal component of the conducting electrons spin
isstronglydependentonthegapsymmetry. Forexample,
susceptibility χ0(Q,ω) as a function of frequency for singlet
intheconventionals-wavestateReχ0islessthanitsnor- (reddashedcurve)andtriplet(blacksolidcurve)Cooperpair-
mal state value. First, the s-wave superconducting gap ing. The curves for the normal state (not shown) almost co-
gives a negative contribution to χ0 as follows from the incide with those for the spin singlet superconducting state.
coherence factor in Eq. (6), and second it yields the spin Hereand in thefollowing we useT =0.15 K,∆0 =0.5 meV,
excitation gap structure in Imχ0 as follows from the δ ǫk = 25 meV and a damping parameter 35 µeV for the nu-
function. Therefore,the feedbackofthe conductionelec- merical calculations.
tronsonthemagneticexcitonbecomesweakerandyields
ashift ofthe magneticexcitondispersiontowardshigher
frequencies in the superconducting state with respect to Furthermore, below Ωtcrr the Reχ0 is increased with re-
its normal state value. specttoitsnormalstatevalueforω 6=0asshowninFig.
At the same time, the unconventionalcharacterof the 1(b).
superconducting gap entails immediate consequences for A frequency dependence of the Reχ (Q,ω) can yield
0
Eqs. (6) and (7). Namely, at the Fermi surface (i.e., for more than one solution of Eq. (7). In order to demon-
ξk =0)onefinds∆sk+Q =−∆sk aswellas∆tkr+Q =−∆tkr stratehowthosesolutionscanbefoundaboveandbelow
where Q is the antiferromagnetic wave vector. Thus, T we illustrate in Fig. 2 the possible characteristic be-
c
in both cases the anomalous coherence factor equals havior of ω2 −2g2∆ Reχ (q,ω). Here, we assume
q CEF 0
2 for all k momenta which is in strong contrast to
z
the usual s-wave symmetry of the superconducting

gap. Simultaneously, the δ function in Eq. (6) starts 4
to contribute for Ωcr(kz,qz) = Min(Ek+ˆzqz +Ek) = ω
ǫ2[cos(ck +cq )−cos(ck )]2+(|∆ |+|∆ |)2 3 m
z z z kz kz+qz ω
wqhikch is determined by the value of the gap, ∆k and 2 q
the quasiparticle dispersion along k axis, ǫ . Note
z 1
that Ωscr(kz,qz = π/c) = 2|cos(ckz)| ∆20+|| ǫ2 and ω ωω2 ∆ χ ω
q k 0 r ωq2-2g2 CEFRe 0(q, )
Ωtcrr(kz,qz = π/c) = 2 ∆20+ ǫ2−∆20 cos2(ckz) for -1 q2
r k
thesingletandtripletsupercondu(cid:16)ctingga(cid:17)p,respectively.
-2
It is interesting to note that for the singlet case there 0 ω /12∆ 2
is no gap in Imχ and therefore, it is nearly the same 0
0
as in the normal state6 [see Fig. 1(a)]. Then Reχ0 is FIG. 2: (Color online) Illustration of thesolutions of Eq. (7)
a constant at low frequencies. Therefore the magnetic at wave vector q = Q. In the normal state there is only one
excitondispersionwillbe simplyshiftedinproportionto crossing point between the ω2 curve (solid) with the ωq line
the total value of Reχ exactly as in the normal state. (dotted)yieldingthefrequencyofthemagneticexciton. Note
0
Correspondingly, no strong feedback on the magnetic that ωq may slightly differ from the bare exciton dispersion
exciton due to superconductivity takes place. due to Reχ0(Q,ω) = const in the normal state. In the su-
perconducting state due to the strong frequency dependence
However, one sees that in the case of the triplet order
pΩatrra=me|t2e∆r I|m(ǫχ≫0 w∆ill)b.eTgahpenpedduaettloeaastcuopmbtoinveadlueeffsecotf opfoiRntesχo0f(qω,2ωw)it(hanωdq2/o−r2Rge2∆ΠC0)EFoRneeχfi0n(dqs,ωse)v.eTrahleilnotweressetcptionlge
ofcrthe anom0alokus coh0erence factor and the δ function, a (ωr < 2∆0) occurs at very small damping (Imχ0 is zero or
small) resulting in a resonancelike peak in ImD(q,ω). The
discontinuousjumpinImχ0 occursataboutω =2∆0for secondcrossingpointisnotvisibleinImDduetoalargepeak
the triplet order parameter. Via Kramers-Kronig trans- in Imχ0 or strong damping around 2∆0. The third crossing
formation the discontinuous jump in Imχ0 yields a log- point, ωm occurs at energies larger than 2∆0 and represents
arithmic singularity in Reχ . Note, the logarithmic sin- the feedback effect of superconductivity on the magnetic ex-
0
gularity in Fig. 1.(b) is suppressed by a weak damping. citon.
4
that in normal state the Reχ is almost frequency in- where m denotes the value of the effective antiferro-
0
dependent and magnetic exciton’s peak position shifts magnetic staggered field. This term leads to a splitting
slightly in superconducting state with respect to its nor- of the quasiparticle energy dispersion into two bands.11
mal state value. Due to the gap structure in supercon- In particular, the Hamiltonian (8) can be easily diag-
ducting state, depending on the value of g, a new pole onalized by a unitary transformation12 and the result-
mzearoy oatcctuhreaset efrneeqrugeienscileess,stthheanto2t∆al0.ImIfDIm, iχ.e0.,isthsme asplleocr- ing energy dispersions are Ek± = (ε±k)2+∆2k with
tral function of magnetic excitations, shows a resonance ε±k =[ǫ+k± (ǫ−k)2+m2]. Here,wehaqveintroducedǫ+k =
pTehaiksawgrheiecshwoecllcuwristhoenxlypeirnimtehnetasluIpNeSrcdoantdau3c.tMinogresotavteer., 12(ξk+ξk+qQ) and ǫ−k = 21(ξk−ξk+Q). The AF Fermi
at higher energies one finds in addition two more poles surfaceofthetwobandsε±k consistsoftwodisjointcylin-
ders in the reduced AF BZ |p | ≤ π (see Ref. 10). In a
in Eq. (7). The latter yields an additional structure in z c
pureAFstate,therealpartofthesusceptibilityχ atlow
ImD which is a renormalized magnetic exciton with fi- 0
energiesisdeterminedbytheintrabandprocessesandcan
nite damping. This typicalbehavior ofthe susceptibility
f(ε±) f(ε± )
can be found in Fig. 3. be approximated by χ0(q,ω) ∝ kAk,q ε±k −ε±k++ωq
So far we have ignored the coexistence of antiferro- k+q− k
where A is the AF coherence factor. At wave vec-
k,q P
magnetism and superconductivity in UPd Al for the
2 3 tor Q for ω = 0 the susceptibility is proportional to the
conductionelectrons.8 Antiferromagneticorderresultsin
density of states and decreases rapidly to zero as one in-
UPd Al due to the interaction of neighboring uranium
2 3 creases frequency. This is a consequence of the equality
ions. This leads to a dispersion for exciton state and
eventually to an antiferromagnetic instability and a new ε±k = ε±k+Q. Therefore the renormalization of the mag-
netic excitons due to conduction electrons can be safely
groundstate.9 TheunitcellisdoubledandtheBrillouine
ignored.
zone is correspondingly reduced. The new dispersion of
the conducting electrons enters the expression for the
spin susceptibility. Then the solutions of Eq. (7) must Most importantly, in the superconducting state coex-
be redetermined. This is done in the following. isting with AF, the imaginary part of the spin suscep-
The total Hamiltonian is tibility of the conduction electrons including intraband
andinterbandscatteringisgivenforT=0K,q=Q,and
H =H0+m σc†p+Qσcpσ (8) ω >0 by
p
X
Imχ (Q,ω)=
0
k 18δ ω−2Ek+ 1− (cid:0)ε+kE(cid:1)2k+−2∆2k! ǫ2kz2m+2m2!+ k 41δ ω−Ek+−Ek− (cid:18)1− ε+kEε−kk+E−k−∆2k(cid:19) ǫ2kz2+ǫ2kzm2!
X (cid:0) (cid:1) X (cid:0) (cid:1)
+ k 81δ ω−2Ek− 1− (cid:0)(cid:0)ε−kE(cid:1)2(cid:1)k−−2∆2k! ǫ2kz2m+2m2! . (9)
X (cid:0) (cid:1)
(cid:0) (cid:1)
Herethe firstandthe thirdtermsdescribe theintraband have only one-half the original period, i.e., ε±k = ε±k+Q.
quasiparticlepaircreationwhilethesecondtermrefersto ThereforeallpartsoftheFermisurfacecanbeconnected
the corresponding interband process. Note that Eq. (9) by the antiferromagnetic wave vector Q and simultane-
containstwotypesofcoherencefactors,i.e.,duetosuper- ouslyhaveasignchange∆ =−∆ ofthegapexcept
k+Q k
conductivity and antiferromagnetic order, respectively. at the nodal points. As a result, Imχ (Q,ω) is non-zero
0
As usual, the low-energybehavior of Imχ is dominated for ω > 0 due to the contribution of the nodal states
0
by the intraband contributions. We also assume that both in the singlet and the triplet Cooper-pairing cases.
the presence of antiferromagnetism does not change the WithincreasingfrequencyImχ increasesuptoenergies
0
qualitativebehaviorofthe superconductinggap,i.e.,the of about 2∆ and then decreases [see Fig. 3(a)]. The
0
positionofthelinenodeandthecorrespondingchangeof functionaldependenceofImχ atlowfrequenciesresem-
0
sign of the superconducting order parameter remain the bles the behaviorof the density of states except that the
same althoughsome higher harmonicsmay appear.11 As structure occurs at around 2∆ . Correspondingly, the
0
inEq.(6),thesuperconductingcoherencefactorsequal2 real part of χ (Q,ω = 0) is the same as in a pure AF
0
for k momenta close to the Fermi surface. At the same state. However, away from ω = 0 it does not drop as in
z
time,thereconstructedconductionbandsintheAFstate thepureAFstatebutincreasesquadraticallyuptoabout
5
0.5
12 10
eV)4 (a) Im χχ0(Q,ωω) eV)8 (b) Im D(Q,ωω)
s/ Re 0(Q, ) m Re D(Q, )
ate es/ c)
ω,) (Qst2 ω ) (stat4 πq (/z 1 5
χ(0 D(,Q0
0
-4
0.0 0.5 1.0ω (1.5 2.0 2.5 3.0 0.0 0.5 1.0ω (1.5 2.0 2.5 3.0 1.5 0
meV) meV) 0 1 ω (meV) 2 3
FIG. 3: (Color online) Results for χ0(Q,ω) and D(Q,ω)
in the coexisting AF and singlet superconducting state. FIG. 4: (Color online) Contour plot of theimaginary part of
(a) Calculated real and imaginary parts of the longitudi- thetotalpseudospinsusceptibilityasfunctionoffrequency,ω
nal component of the conduction electron spin susceptibil- and qz momentum. One clearly observes two distinct peaks
ity, χ0(Q,ω). (b) The real and imaginary part of the at Q. The one at low energies represents the resonance peak
total pseudospin susceptibility D(Q,ω). Here m = 50 induced by thefeedback of superconductivity and the one at
meV, g = 10 meV. We use the gap function ∆k ∝ higherω istherenormalizedmagneticexciton. AwayfromQ
∆0 cos(ckz)− 15cos(3ckz)+ 310cos(5ckz) ( Ref. 7 ). both peaks disperse upward in energy following thebehavior
of the normal state magnetic exciton.
(cid:2) (cid:3)
2∆ due to the structure of Imχ induced by the super-
0 0
conducting gap. Only then does Reχ drop to small the absolute magnitude arises from the different densi-
0
values. Altogether Reχ (Q,ω) increases in the super- ties of states at those regions of the Fermi surface where
0
conducting state for ω <2∆ due to the unconventional the maximum of the singlet and the triplet gaps occur.
0
nature of the superconducting order parameter. How- Altogether this does not change the functional form of
ever, the pure resonance (bound state) in ImD is not Imχ . We mention that thermal conductivity results
0
realized due to finite damping. An additional pole in in a rotating field14 are compatible with both cos(ck )
z
ImD still exists at frequencies smaller than 2∆ due to andsin(ck )orderparameterswhiletheobservedKnight
0 z
a strong increase of Reχ at small frequencies as shown shift15 seems to favor the former. Interestingly, we also
0
in Fig. 3(b). At higher frequencies it becomes small and foundthatarecentlyproposedsuperconductinggapwith
thus another pole appears corresponding to the broad- cos(2ck ) symmetry16 does not lead to the formation of
z
ened original magnetic exciton. Thus, ImD has a two- low-energyspin excitations aroundwave vector Q in the
pole structure as shown in Fig. 3(b). Note, in order to superconducting gap. The reason is that its momentum
increase the intensity of the low-energy pole in ImD we dependence yields no change of the sign of the super-
include the contribution of the higher harmonics to the conducting order parameter, ∆ = ∆ , and thus no
k k+Q
gapfunction. As already mentioned, they are due to the constructive contribution can result from the anomalous
presence of AF order11. coherence factor.
Finally we discuss the dispersionofthe magnetic exci- Inconclusion,wehaveinvestigatedtheeffectsofsuper-
tations awayfromQ along the qz direction. In Fig. 4 we conductivity on the magnetic excitations in the uncon-
showthecalculatedmomentumandfrequencydependen- ventionalsuperconductorUPd Al . Inparticular,dueto
2 3
ciesofImD(qz,ω). Itisclearthatassoonasqz 6=π/cthe the change in signof the superconducting order parame-
originalmagneticexcitonhasastrongupwarddispersion ter the conduction electron susceptibility is enhanced in
in the normal state. Therefore, effects connected with thesuperconductingstatewhichyieldsanadditionalpole
renormalization induced by superconductivity will also (bound state) in the total susceptibility. We further an-
be shifted towards higher energies. The pole induced by alyzed the role played by antiferromagnetism and found
superconductivity shows dispersion similar to the mag- that its presence increasesthe spectralweightofthe res-
netic exciton [see Fig. 4]. For both singlet and triplet onance due to the doubling of the unit cell. However,
order parameters our results are in fair agreement with the resonance peak in the AF phase becomes a virtual
recentINS data.13 Namely, in the superconducting state bound state due to finite damping. Finally we point out
one finds two distinct energy dispersions, one being the that UPd Al is another known example where the un-
2 3
resonancelikefeaturewithhighintensityinsidethesuper- conventionalnatureofthesuperconductingorderparam-
conducinggapandthesecond,thatoflocalizedmagnetic eter yields a structure in the magnetic susceptibility as
excitons renormalized by the conducting electrons. An- inlayeredhigh-T cuprates. Thereforeitcanberegarded
c
other interesting point worth noting is that due to the as a model system of unconventional superconductivity
doubling of the unit cell and the equality ε±k = ε±k+Q, studied by inelastic neutron scattering.
the effect of the sin(ck ) and cos(ck ) gaps leads to a The authors acknowledge helpful discussions with G.
z z
very similar behavior for Imχ . The slight difference in Zwicknagl, J.-P. Ismer, and A. Klopper.
0
6
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