Wave function for odd frequency superconductors

0.35 MB·

by  Hari P. Dahal

Wave function for odd frequency superconductors.
Hari P. Dahal,1 E. Abrahams,2 D. Mozyrsky,1 Y. Tanaka,3 and A. V. Balatsky1,4
1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
2Serin Physics Laboratory, Rutgers University, P.O. Box 849, Piscataway, NJ 08855
3Dept. of Applied Physics, Nagoya University, Chikusa-ku, Nagoya 464-8603, JAPAN
4Center for Integrated Nanotechnology, Los Alamos National Laboratory, Los Alamos, New Mexico 87545∗
(Dated: 1/15/09)
9 Werevisitthequestionofnatureofodd-frequencysuperconductors,firstproposedbyBerezinskii
0 in 1974.1 We start with the notion that order parameter of odd-frequency superconductors can be
0 thought of as a time derivative of the odd-time pairing operator. It leads to the notion of the
2 composite boson condensate.2 To elucidate the nature of broken symmetry state in odd-frequency
superconductors, we consider a wave function that properly captures the coherent condensate of
n
composite charge 2e bosons in an odd-frequency superconductor. We consider the Hamiltonian
a
J whichdescribestheequal-timecompositebosoncondensationasproposedearlierinPhys.Rev.B52,
1271(1995). WeproposeaBCS-likewavefunctionthatdescribesacompositecondensatecomprised
5
ofaspin-0Cooperpairandaspin-1magnonexcitation. Wederivethequasiparticledispersion,the
1
self-consistentequationfortheorderparameterandthedensityofstates. Weshowthatthecoherent

wave function approach recovers all theknown proposerties of odd-frequency superconductors: the
]
n quasi-particleexcitationsaregaplessandthesuperconductingtransitionrequiresacriticalcoupling.
o
c PACSnumbers:
-
r
p
u I. INTRODUCTION ity and/or near the quantum critical point in CeCu2Si2
s and CeRhIn5.9 In addition, hydrated NaxCoO2 is sug-
. gestedtosupportans-wavetripletodd-frequencygap10.
t The discussion about possible symmetry types of
a Veryrecently,Kalaset al.11 havearguedthat the boson-
m superconducting order parameter ∆(k,τ) (τ denotes
fermion cold atom mixture exhibits s-wave triplet odd-
imaginary time) has drawn significant research interest.
- frequency pairing above some critical coupling at which
d Theconventionalsinglet(triplet)superconductorfollows
the mixture phase separates.
n PT∆(k,τ) = ∆(k,τ) ([PT∆(k,τ) = −∆(k,τ)]) un-
o der parity P and time T transformations. The singlet Motivated by the growing interest and possibilities
c
(PT =1)andtriplet(PT =−1)conditionscanbesatis- of odd-frequency pairing, here we address the missing
[
fied by either taking P = T = 1 and P = −1,T = 1 part of the odd-frequency superconductivity discussion:

1 for an even-in-frequency gap, or P = T = −1 and what is the wave function of the odd-frequency super-
v P =1,T =−1 for odd-frequency pairing. conductors? One might wonder how one can even ask
3
2 Although mainstream discussions of superconudctiv- this question given that superconducting correlations of
3 ity are for even-frequency pairing, there is a growing in- two fermion operators in odd-frequency superconductor
2 terest in understanding odd-frequency pairing. The dis- do not have an equal time expectation value? We as-
. cussion of unconventional pairing (P = 1,T = −1) was sume (pretty safe assumption in fact) that any state, in-
1
0 initiated by Berezenskii1 to explain the superfluid phase cluding odd-frequency superconductor, does has a many
9 of 3He. Although his proposal of triplet odd-frequency body wave function that captures superconducting cor-
0 pairing could not explain the superfluid phase of 3He, it relations. Any state of matter has an associated wave
: certainly motivated a search of other possibilities of the function |ψi that captures the amplitude distribution of
v
i pairing symmetries. Balatsky and Abrahams3 later ex- the particles forming this state. Hence we are asking ex-
X tendedtheconceptofodd-frequencypairingtothesinglet actlythisquestionaboutthemanybodywavefunctionof
r superconductor (P =T =−1). the odd frequency superconductors. Our wave function
a builds upon along discussion? onthe possible orderpa-
Although the realization of the odd-frequency pair-
rameterandequal timecompositeoperatorsthatcapture
ing in current systems is still under debate, several re-
superconductingcorrelationsofodd-frequencysupercon-
ports consider this possibility in a a number of systems.
ductors in equal time domain.
Odd-frequency pairing in the Kondo lattice has been in-
vestigated to study superconductivity in heavy-fermion We propose a BCS-like pairing wave function for
compounds,5 The proximity effects in a superconductor- an odd-frequency superconductor, and study its conse-
ferromagnet structure6, a normal-metal/superconductor quencesfortheenergydispersion,superconductingorder
junction7 anddiffusive normalmetal/unconventionalsu- parameter, and density of states. The wave function,
perconductor interface8 have been attributed to odd- which describes a condensate of a spin-0 Cooper pair
frequency pairing. The p-wave singlet odd-frequency and a spin-1 magnon excitation, is consistent with the
pairing is argued to be a viable pairing in the coexis- Hamiltoniansuggestedearlierin2tostudyodd-frequency
tence region of antiferromagnetism and superconductiv- superconductivity. We minimize this Hamiltonian with
2
respect to the proposed wave function and derive an ex- II. HAMILTONIAN AND WAVE FUNCTION
pressionforthe quasiparticledispersion,aselfconsistent
gap equation and the density of states. We find that a) When the idea of the odd-frequency pairing was first
the quasi-particle dispersion is gapless, b) the gap equa- formulated for the singlet superconductor, an effective
tion has non zero solution only for a critical value of the spin-independent interaction mediated by phonon was
coupling, c) the density of states is finite even for an en- considered.4 It was realized that this kind of interaction
ergylessthanthegapenergy,andd)thedensityofstates wasunphysicalforthe singletpairing.4 Theproblemwas
is reduced at the gap edge compared to that of the BCS solved by considering spin dependent electron-electron
case. interactions. Odd-frequencypairingposedanotherprob-
Before introducing the wave function and getting into lemrelatedtotheselectionoftheorderparameter. Inthe
the details of the minimization of the Hamiltonian, we BCS case the order parameter is generated from the ex-
would like to show that PT = 1 can be obtained by pectation value, F(r,t;r′,t′ →t)=hψ(r,t)ψ(r′,t)i. But
takingP =T =−1inS =0singletcase. Anysupercon- fortheodd-frequencysuperconductortheequal-timegap
ducting order with translational invariance, equilibrium vanishes since the gapis odd infrequency. This problem
andbrokenU(1)symmetrywouldresultinananomalous was solved by taking dF(r,t;r′,t′)/dt|t→t′ as the equal-
(Gor’kov) Green’s function time order parameter.2
A Hamiltonian having a spin dependent electron-
F (τ,k)=hT c (τ)c (0)i, (1)
αβ τ α,k β,−k electroninteractionwasintroducedbyAbrahamsetal.2.
where α,β are spin indices. We assume that the transi- Using the equationof motionthey derivedan expression
tion occurs only in a well defined representation. Thus, for dF(r,t;r′,t′)/dt|t→t′. It was shown that the equal-
for S =0 singlet pairing, we may define time condensate for odd-frequency pairing is the expec-
tation value of the product of a pair operator and a spin
F(τ,k)=ǫ F (τ,k), (2)
αβ αβ
excitation operator. In what follows, we adopt this ap-
and for S =1 triplet pairing, proach, but for an odd frequency s-wave m = 1 triplet
phase. We rewrite the Hamiltonian from Ref. 2 in the
F~(τ,k)=(iσ~σ) F (τ,k). (3)
αβ αβ following form,
We now show the properties of F(τ,k) under P and T
transformations. For S =b0 from Eq. (2), H = ǫk↑c†k↑ck↑+ ǫk↓c†k↓ck↓+ ωqSq+Sq−
Xk Xk Xq
F(k,τ)=ǫ [θ hc (τ)c (0)i−θ hc (0)c (τ)i],
αβ τ α,k β,−k −τ β,−k α,k (4) + Vklqpc†k+q2↑c†−k+q2↓Sq+c−l+p2↓cl+p2↑Sp−, (8)
where θ is the Heaviside theta function. kXlqp
τ
We apply PT to this F:
whereǫ referstothekineticenergyofthe↑↓electrons
k↑↓
F(−k,−τ) measured from the Fermi energy, ωq is the magnon ki-
neticenergy,andV isanattractiveinteractionwhich
=ǫ [θ hc (−τ)c (0)i−θ hc (0)c (−τ)i] kl,qp
µν −τ µ,−k ν,k τ ν,k µ,−k mediates the condensation. c† and c creates and an-
=ǫ [θ hc (0)c (τ)i−θ hc (τ)c (0)i], kσ kσ
µν −τ µ,−k ν,k τ ν,k µ,−k nihilates electrons at the state kσ. S± describe magnon
(5)
excitations. Using this Hamiltonian, we propose a BCS-
where in the last line we have used the fact that like wave function and study the superconducting state.
hTA(−τ)B(0)i = hTA(0)B(τ)i which agrees with the The proposed wave function is written as
cyclicity of the trace,
|ψi= (u +v c† c† S+)|0i, (9)
hA(−τ)B(0)i = Tr(e−HτAeHτB)= Ykq kq kq k+q2↑ −k+q2↓ q
Tr(AeHτBe−Hτ) = hA(0)B(τ)i. (6)
where |0i represents the vacuum for both the electrons
Going back to Eq. 5, we permute µ↔ν, and the spin bosons. This wave function describes the
superposition of the wave functions having two paired
F(−k,−τ)
electronswithk+q and−k+q momentumandcarrying
=ǫ [Θ hc (τ)c (0)i−Θ hc (0)c (τ)i] (7) 2 2
µν τ µ,k ν,−k −τ ν,−k µ,k oppositespinsandcondensedalongwithspinexcitations
=F(k,τ). (S+). v (u ) represent the amplitude of the occupa-
q kq kq
tion (or unoccupation) of these electron pairs with the
All these properties of the Gor’kov function will be re-
spin excitation.
flectedinthebehaviorofthegapfunctionaswell. There-
There are key properties that explain this particular
fore,thegapfunctioningeneralisevenonlyundersimul-
choiceofvariationalfunction: i)|ψiisacoherentstateof
taneous transformation: k → −k (P) and τ → −τ (T).
We recall that PT = 1 is not only satisfied by P = +1, composite bosons (c†k+q2↑c†−k+q2↓Sq+)) that carry charge
T =+1butalsobyP =−1andT =−1. Theformerde- 2e; ii) this wave function describes a coherent state that
scribestheBCSs-wave(even-frequency)pairingwhereas has broken U(1) symmetry associated with supercon-
the latter describes odd-frequency pairing. ductingcondensate,ascanbeexplicitly verifiedbyusing
3
c → exp(iφ)c ; iii) Composite boson that condenses is We proceedby defining the twoquantities ∆ andE that
k k
not a simple Cooper pair2 butcontainstwofermions and will turn out to be the gap parameter and the energy of
a spin-1 boson; iv) composite boson field has finite ex- a composite excitation.
pectation value in this state
1
∆ =− V sin2θ hS−S+i hS−S+i ,
hψ|c†k+q2↑c†−k+q2↓Sq+|ψi=ukqvkq (10) kq 2Xlp klqp lpq qq p
(16a)
and therefore |ψi is a mean field wave function for the
comThpeosnitoermcoanlidzeantisoanteo.f the wave function is given by, Ekq = r(ǫk+q2 +2 ǫk−q2 + ω2qhS−S+iq)2+∆2kq
hψ|ψi= (|ukq|2+|vkq|2hS−S+iq)=1, (11) = (ǫ + q2 + ωqhS−S+i )2+∆2 . (16b)
Ykq r k 8m 2 q kq
which implies that |u |2 + |v |2hS−S+i = 1 for all Then
kq kq q
k,q.
∆
To make a next step we need to find the expectation sin2θkq =2ukqvkq hS−S+iq = kq, (17a)
valueoftheHamiltonian(Eq.8)withrespecttothewave q Ekq
function (Eq. 9) and minimize it. Then we will proceed and
to derive the quasi-particle dispersion, density of states,
and the self-consistentequationfor the order parameter. cos2θ =v2 hS−S+i −u2 = −ξkq, (17b)
kq kq q kq E
kq
III. TOTAL ENERGY AND ITS MINIMIZATION where we have introduced the abbreviation
ξ = ǫk+q2 +ǫk−q2 + ωqhS−S+i . (18)
The calculation of each term in Eq. 8 is shown in kq q
2 2
Appendix. Using Eqs. A2, B2, C2, D1, the total energy
can be written as SolvingthenormalizationconditionandEq.17,wecan
show that,
E = (ǫk+q2 +ǫk−q2 +ωqhS−S+iq)|vkq|2hS−S+iq 1 ξ
Xkq u2 = (1+ kq ), (19a)
kq 2 E
+ V v∗ u v u∗ hS−S+i hS−S+i . (12) kq
klqp kq kq lp lp q p
kXlqp
1 ξ
Following the BCS method, we choose ukq,vkq such vk2q = 2hS−S+i (1− Ekq ) (19b)
that they satisfy the normalization condition so that q kq
ukq = sinθkq and vkq = cosθkq/ hS−S+iq. Then the BCS limit canbe recoveredat any stage of this analy-
expression for the energy reads p sis if we assume that spin correlators are factorized and
havea peak atq=0. This limit correspondsto the con-
E = cos2θkq(ǫk+q2 +ǫk−q2 +ωqhS−S+iq)+ densation of spin field hS−S+iq = hS−iqhS+iqδq,0. In
Xkq this limit additional summation over q drops out and
1 we recover standard BCS logarithm in selfconsistency
V sin2θ sin2θ hS−S+i hS−S+i .
klqp kq lp q p
4kXlqp q q equation Eq.(16a), along with other features of BCS so-
lution. This limit corresponds to the factorizitation of
(13)
composite boson into product hψ|c† c† S+|ψi→
k+q2↑ −k+q2↓ q
Theminimizationoftheenergywithrespecttoθkq gives hψ|c† c† |ψihψ|S+|ψiδ .
k↑ −k↓ q q,0
∂E
∂θ =−sin2θkq(ǫk+q2 +ǫk−q2 +ωqhS−S+iq)+
kq IV. ENERGY SPECTRUM
V cos2θ sin2θ hS−S+i hS−S+i =0,
klqp kq lp q p
Xlp q q Unlike the BCS case, Ekq is not a single-particle ex-
(14) citation energy. Therefore, we shall derive an expression
fortheenergyrequiredtoexciteanelectronfromthe su-
which can be rewritten as perconducting groundstate. The excited state for an up
spin is given by,
V sin2θ hS−S+i hS−S+i
lp klqp lp q p
tan2θ = .
kq P ǫk+q2 +ǫk−pq2 +ωqhS−Sp+iq (15) ψ↑ =[q,Yk6=k′(ukq+vkqb†kq)]c†k′+q2↑|0i, (20)
e
4
where we have defined the composite creation operator gapless. A still further increase in q results in a finite
c
b†kq = c†k+q2↑c†−k+q2↓Sq+. We calculate the expectation DthOeSspaetctEra=l w0e.igFhotraqtct≥hekgFaptheedrgeei.s no enhancement of
value of the Hamiltonian Eq. (8) with respect to the ex-
The calculation of the DOS for ω = ω also shows
cited state wave function Eq. (20). The details are in q 0
the similar density of state as discussed above for both
Appendix E. The expectation value can be expressed as
q =0.25k and q ≥k .
c F c F
∆2
hψ↑|H|ψ↑i=hψ|H|ψi+ǫk′+q2+Ekk′′qq−2ξk′qvk2′qhS−S+iq. VI. SUPERCONDUCTING GAP VS COUPLING
e e (21)
CONSTANT
Using Eq. (19b), we can rewrite the above equation as,
Theself-consistentgapequation(Eq.16a)canbewrit-
∆E↑ =ǫk′+q2 −ξk′q+Ek′q, (22) ten as,
where ∆E =hψ |H|ψ i−hψ|H|ψi is the excitation en- V ∆
aersg∆yEof↑t=he↑ku′p·qsep/↑i2nme∗lee−↑ct(rωonqs/.2)∆hSE−↑Sc+ainqa+lsEokb′qe.wDriottineng ∆kq = 2 Xlp EllppqhS−S+iqqhS−S+ip, (24)
the same for the down spin excited state ψ↓, we find where we have taken
∆E↓ =−k′·q/2m∗−(ωq/2)hS−S+iq+Ek′q.
e −V : |ǫ |≤0.2µ
V = k (25)
klpq 0 : |ǫ |>0.2µ
n k
V. DENSITY OF STATES
Then ∆ = ∆ = ∆ . The use of a more compli-
kq lp
cated interaction potential with a momentum depen-
The density of states (DOS) as a function of energy,
dence. would bring additional calculational complica-
N(E), is defined as,
tions, which would not change the nature of the results.
Wedenotep2/8m∗+(ω /2)hS−S+i byf(p). Wefirst
k·q ω p p
N (E)= δ[E−(± − qhS−S+i +E )], perform the energy integral in Eq. 24 as follows,
± 2m∗ 2 q kq
Xkq
V hS−S+i
(23) 1 = p
2 (ǫ +f(p))2+∆2
Xlp p
wWheerneu±mecroicrarellsypocnadlcsutloatuepthaenddednoswitnysopfinsstaretesspefcotrivtewlyo. = g p2dpp ~ωc hS−S+ipdǫ
cases of the magnon dispersion: 1) ωq =q2/2M, and 2) Z Z0 (ǫp+f(p))2+∆2
ωq = ω0.. We set hS−S+iq = 1.0, and M = 10m∗. The p ǫ (p)+ ∆2+ǫ (p)2
DOS for case 1) is shown in Fig. 1. = g p2dphS−S+i log c c ,
InFig1aweshowtheDOSasafunctionofenergyand Z p f(p)+p∆2+f(p)2
orderparameter. Wehavesetamagnonmomentumcut- p (26)
off, q = 0.25k . We see that the DOS can be non-zero
c F
forenergieslessthanthesuperconductinggapparameter; where N(0) is the DOS in the normal state at the
hence the DOS is gapless. The maximum of the density Fermienergy,gisthedimensionlesscouplingN(0)V/2π2,
ofstateisalwaysatthegapedge,butitishighlyreduced ǫc(p) = ~ωc +f(p) and ~ωc = 0.2µ. If we assume spin
at the gap edge compared to the BCS case. At E = 0, correlatortohaveasharppeakδq,0 we recoverBCSself-
the DOS can be non-zero for small ∆. The calculation consistency equation from this equation.
for a smaller q (not shown in the figure) shows that the In the BCS case the gap equation is 1 =
c
gap becomes more prominent in the DOS and spectral N(0)V log[(~ωc + ∆2+~ωc2)/∆]. There is a solution
weightis transferredto the gapedge,similar to the BCS for ∆ for an arbitrpary small value of N(0)V due to log-
case. Hence q → 0 reproduces the BCS results. In Fig. arithmic divergence of the integral. In our case, in the
c
1b-dwe have shownthe plane cut of Fig.1a for different presenceofthe magnon,the denominatorwillhavesome
values of ∆. For ∆=0.1 (Fig. 1b) we see that the DOS nonzero value because of the non-zero magnon energy.
is non-zero for 0.05<E <∆. For ∆=0.04 (Fig. 1c) we Thentherighthandsidecanbemadeequalto1onlyfor
see that the gap is completely closedand the excitations some critical value of g, as can be seen in the numerical
will be gapless. The effect is even bigger for ∆=0.02. evaluation discussed below.
We also calculated the DOS using case 2): ω =
q
q2/2M for q ≥ k (the Fermi momentum) for a fixed
c F
value of ∆ = 0.1. The result is shown in Fig. 2. In A. Case 1), ωp=p2/2M
this figure we can see that the DOS almost closes the
gap when q = k . As we increase q , the gap closes We solveEq.26numerically for∆ asa function ofthe
c F c
completely. Then the quasiparticle excitations become couplingstrengthg. Wesetm∗/M =0.1andhS−S+i =
p
5
N(E) a) 1.2
1.5
1
1.2
0.8 bbbb)))) ∆∆∆∆====0000....1111
0.6
0.8 0.4 1
0.2 E)
0 N(
0.4
0.5
0.3
0 0.2
0
0.02 0∆.04 0.06 0.08 0.1 0 0.1 E 0 0 0.05 0.1 0.15 0.2 0.25 0.3
E
1.5 1.5
ccc))) ∆∆∆===000...000444 dd)) ∆∆==00..0022
1 1
E) E)
N( N(
0.5 0.5
0 0
0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3
E E
FIG. 1: The density of states (DOS) as a function of energy and superconducting order parameter in an odd-frequency
superconductor. The DOS is normalized with respect to the DOS of the normal state. All the energies are normalized with
Fermi energy of the system. The result in the upper panel is presented for the magnon momentum cutoff qc = 0.25. In this
figurewe show that thedensity of state is finiteeven for energy less than thegap energy. The maximum of theDOSis at the
gap edge but the DOS is highly reduced compared to the BCS case. For smaller gap energy the DOS is completely gapless.
Oncethe gap is closed the DOSstarts to pile up at E =0 for smaller values of ∆.
0.16
1
0.12
0.75
0.3
N(E) 0.5 =0) 0.2 ∆ 0.08
E
N( 0.1
0.04
0.25 0
0 0.5 1 1.5 2
q /k
c F
0
0 0 0.05 0.1 0.15 0.2 0.25 0.3
0.02 0.04 0.06 0.08 0.1 0.12 0.14
E N(0)V
FIG. 2: The DOS at fixed ∆ = 0.1 as a function of energy FIG. 3: The order parameter is numerically calculated for
for the magnon momentum cutoff qc ≥ kF. The values of qc ωq = q2/2M. The magnon momentum cutoff is given by
are0.9,1.1,1.4,1.7timeskF. Asweincreaseqc thegapinthe qc = BkF, where B = 0.12,0.1,0.08,0.06, top to bottom.
density of states gradually closes up. For bigger qc the DOS The order parameter is non zero only for critical value of
piles up at E =0. thecouplingg=N(0)V/2π2. ForagivenvalueofN(0)V the
larger∆correspondstothelargermagnonmomentumcutoff.
B. Case 2), ωp=ω0
The gap equation is again given by Eq. 26 but now
1.0. The cutoff for the magnon momentum is given by
qc = BkF, where B varies between 0.12 to 0.06 in equal f(p)=p2/8m∗−ω0hS−S+ip/2. (27)
stepsof0.02. TheresultisshowninFig.3. Inthisfigure
we can see that a nonzero order parameter requires a We solveEq.26numerically for∆ asa function ofthe
critical coupling. coupling strength g. We fix the cutoff for the magnon
6
momentum to be 0.1kF. The result for various ω0 =Cµ VII. MEISSNER EFFECT
where C = 0,0.02,0.04,0.06,0.08 is shown in Fig. 4.
Again, the superconducting transition requires a critical
coupling. The Meissner effect is one of the defining properties
of a superconductor. The Meissner effect has been de-
0.2 rivedforthecompositeodd-frequencysuperconductorby
Abrahams et al.2. Here, we summarize the derivation
0.16 given in that reference
0.12 A superconductor shows the Meissner effect when the
∆ paramagnetic electrodynamic response is less than the
0.08 diamagnetic response. The dc response is given by,
0.04
0 j (q)=−Q (q)A (q),
0 0.1 0.2 0.3 0.4 i ij j
Ne2 (28)
N(0)V Q (q)=δ +Qp(q),
ij ij m ij
FIG. 4: The order parameter is calculated for ωq =ω0. The
magnonmomentumcutoffisqc =0.1kF. Theresult isshown
for ω0 =Cµwhere C =0,0.02,0.04,0.06,0.8, top to bottom.
The order parameter is non-zero only for coupling exceeding where A(q) is the Fourier transform of vector potential
a critical value. For a given value of N(0)V the bigger ∆ A(r), N is the electron density, and m is their mass.
corresponds to the smaller valueof ω0. Qp(q) is given by,
ij
−e2 β
Qp(q)= k k′ dτhTc†(k ,τ)c (k ,τ)c†(k′ ,0)c (k′ ,0)i, (29)
ij 4m2 i jZ γ + γ − δ − δ +
Xγδ Xkk′ −β
where k =k±q/2. Qp can be evaluated near the crit- current correlation function for the Meissner effect are
±
ical temperature T by perturbation in the order param- used. The analytical expression for q →0 is
c
eter ∆. The relevant Feynman diagrams of the current-
e2T2∆2
Qp(q)−Qn(q)= k k′[G2(k,ω)G2(k′,ω′)−2G3(k,ω)G(k′,ω′)]D(k+k′,ω+ω′), (30)
ij ij m2 i j
ωωX′,kk′
where,G(k,ω)andD(k,ω)aretheelectronandmagnon Qp−Qn doesnotchange,thusapositive superfluidden-
propagators. The condition for the Meissner effect is sitywithavaluebetweenzeroandthe BCSvalue. Thus,
given by Qp −Qn > 0, which signifies the positive su- it is shown that the composite odd-frequency supercon-
perfluid density in the superconductor. ductors exhibit the Meissner effect.
Situations with several models of the magnon propa-
gators are discussed. If the magnon propagator is mo-
mentum independent, there is no contribution to Q (q) VIII. CONCLUSION
ij
since the momentum summands are odd functions. So
a momentum-dependent magnon propagator is used to In this paper we propose a BCS-like wave func-
discuss the Meissner effect. In the case of a static, spa- tion for the s-wave triplet odd frequency superconduc-
tiallyuniformmagnonpropagatorhavingfactorizedform tor. Our alternative approach to the odd-frequency su-
given by, D(q,ν) = −δ δ , the Meissner effect is found perconductivity is based on the earlier discussion on
q ν
(Qp −Qn > 0). For spread-out δ-functions, the sign of composite bosons2. We present the wave function for
7
the odd frequency superconductor |ψi = (u + order to get the superconducting transition in the odd
kq kq
v c† c† S+)|0i,,Eq.(9),thatexpliciQtlycontains frequency superconductor, which we have also shown in
kq k+q2↑ −k+q2↓ q thiswork. ii)wealsoderivethedispersionrelationforthe
onlytheequaltimeoperatorsandhencedoesnotinvolve
quasiparticles. We determine the density of states of the
frequencyortimedomain. Thewavefunctiondescribesa
excitations. The density of states is very different from
condensateofaCooperpairofspinS =0andamagnon
thatoftheBCScase. Thegaplessnatureofquasiparticle
ofspinS =1. |ψidoesdescribeacoherentstatethathas
excitations we find is also in agreement with earlier pre-
nonzero expectation value for the composite boson op-
dictions. The calculation of the density of states shows
erator, it captures the charge 2e condensate that breaks
thatitisalwayshigheratthegapedgebutitsmagnitude
gaugesymmetryandcorrespondstothesuperconducting
ishighlyreducedcomparedtotheBCScase. Forarange
state. Naturally, since this |ψi describes odd-frequency
ofparameters,unlike the BCScase,the DOSis finite for
superconductor,spatialparityPofthis condensateisre-
energies less than the gap energy and at E =0 it can be
versed compared to the even frequency pairing opera-
non-zero, hence odd-frequency supercoductor is gapless.
tors that corresponds to BCS condensate. Specifically,
WealsoargueshowtheBCSresultisrecoveredbytaking
for the case we considered of spin triple S = 1 odd fre-
the magnon operator to condense and momentum cutoff
quency condensate the spatial parity of the composite
q =0.
boson hc†(r)c†(r)S+(r)i is P =+1 and hence this order c
↑ ↓ Present discussion would be useful for the equal time
parameterdoesposessallthequantumnumbersinherent
formulation of the odd-frequency superconducting state
to the odd frequency S =1 superconductor.
and physical observables related to condensate. It also
Wepresentasimplifiedmodelthatcapturestheimpor-
would be useful in elucidating the nature of condensate
tantfeatures of the strongcoupling theory developedfor
in odd-frequency supercondutors.
the odd-frequencysuperconductorsandourresultsagree
with the predictions of earlier studies: i) we show that Work at Los Alamos was supported by US DOE
the superconductivityrequiresacriticalcoupling. Itwas through LDRD and BES. We also acknowledge hospi-
argued earlier1,3 that a critical coupling is necessary in tality of KITP at UC Santa Barbara.
∗ [email protected],http://theory.lanl.gov APPENDIX A: KINETIC ENERGY OF UP SPIN
1 V. L. Berezinskii, JETP Lett. 20, 287 (1974) ELECTRONS
2 ElihuAbrahams,AlexanderBalatsky,D.J.Scalapinoand
J. R. Schrieffer, Phys. Rev.B 52, 1271 (1995); A. V. Bal-
It is convenient to rewrite ǫ c† c as,
atsky and J. Bonca, Phys. Rev.B 48, 7445, (1993), k k↑ k↑ k↑
3 AlexanderBalatsky,andElihuAbrahams,Phys.RevB45 P
13125 (1992) 1 1
4 ElihuAbrahams,AlexanderBalatsky,J.R.Schrieffer,and (N ) ǫk↑c†k↑ck↑ = N ǫk+q2↑c†k+q2↑ck+q2↑.
Philip B. Allen, Phys.Rev.B 47 513 (1993) Xq Xk Xkq
5 P.Coleman,E.Miranda,andA.Tsvelik,Phys.Rev.B49 (A1)
8955 (1994) This is a trivial identity since we can shift k → k+ q
6 F.S.Bergeret,A.F.Volkov,andK.B.Efetov,Phys.Rev. and get the same result. 2
Lett. 86 4096 (2001)
7 Y.Tanaka,Y.Tanuma,and A.A.Golubov, Phys.Rev.B We denote the (mn) component of the wave function
8 7Y6.T0a5n4a5k2a2,(a2n0d07A).A.Golubov,Phys.Rev.Lett.98037003 pase,ct|aψtmionniv=alu(uemonf t+hevmkinnce†mti+cn2en↑ce†−rgmy+on2f↓tShn†e)|u0pi.spTinheeleexc--
9 (Y2u0k0i7)Fuseya, Hiroshi Kohno, and Kazumasa Miyake, J. trons KE↑ = N1 kqhψ∗|ǫk+q2↑c†k+q2↑ck+q2↑|ψi is given
by, P
Phys. Soc. Jpn. 72 2914 (2003)
10 M.D.Johannes,I.I.Mazin, D.J.Singh,andD.A.Papa-
constantopoulos, Phys.Rev.Lett. 93 097005 (2004)
11 Ryan M. Kalas, Alexander V. Balatsky, and Dmitry 1
Mozyrsky, Phys.Rev.B 78, 4513 (2008). KE↑ = N hψk∗′q′|ǫk+q2↑c†k+q2↑ck+q2↑|ψkqi
Xkq
(A2)
1
= N ǫk+q2|vkq|2hS−S+iq.
Xkq
Here we use the normalization condition that
hψm∗ ′6=k′,q′|ψm6=k,qi=δmm′δqq′.
Then the kinetic energy of the up spin electrons is
KE↑ = kqǫk+q2|vkq|2hS−S+iq.
P
8
APPENDIX B: KINETIC ENERGY OF DOWN APPENDIX E: ENERGY OF EXCITED STATES
SPIN ELECTRONS
The wave function of an excited state is,
Using the same argument as discussed in appendix A,
we rewrite, kǫk↓c†k↓ck↓ as, ψ = (ukq+vkqb†kq) c†k′+q2↑|0i. (E1)
P e (cid:2)q,Yk6=k′ (cid:3)
1 1
(N Xq )Xk ǫk↓c†k↓ck↓ = N Xkq ǫ−k+q2↓c†−k+q2↓ck+q2↓. Uthseinkgintehteicpernoecregdyuroef tohfetuhpe AsppinpeenldecixtroAn,sw(Ke Eca↑l)cuwlaitthe
(B1) respect to the excited state wave function:
g
Then the expectation value of the kinetic
1
energy of the down spin electrons KE↓ = KE↑ = N ǫk+q2|vkq|2hS−S+iq+ǫk′+q2 (E2)
N1 kqhψ∗|ǫ−k+q2↓c†−k+q2↓ck+q2↓|ψi is given by g q,Xk6=k′
P where the restriction on k in the summation is inherited
1 from the restriction imposed on the excited state wave
KE↓ = N Xkqhψk∗′q′|ǫ−k+q2↓c†−k+q2↓c−k+q2↓|ψkqi function. ǫk′+q2 is due to the creation operator c†k′+q2↑
(B2) whichcreatesanupspinelectronhavingunitprobability
= N1 Xkq ǫ−k+q2|vkq|2hS−S+iq. roefworcitceupthateioEnq.inE2thine tshtaetfeololofwminogmfeonrtmu,m k′ + q2. We
1
Then the kinetic energy of the down spin electrons is KE↑ = N ǫk+q2|vkq|2hS−S+iq+ǫk′+q2 −ǫk′+q2|vk′q|2.
KE↓ = N1 kqǫk−q2|vkq|2hS−S+iq. g Xkq
P (E3)
Proceeding similarly, we show that the kinetic energy
APPENDIX C: MAGNON ENERGY
of the down spin electrons can be written as,
1
KETmhe=expeqchtψa∗t|iωonqSvq+aSluq−e|ψofitchaenmbeagrenwonritkteinnetaisc energy KE↓ = N Xkq ǫk−q2|vkq|2hS−S+iq−ǫk′−q2|vk′q|2.
g
P (E4)
1
KEm =(N ) hψk∗′q′|ωqSq+Sq−|ψkqi, (C1) The kinetic energy of the magnon takes the following
Xk Xq form,
which gives, 1
KEm = N ωq|vkq|2hS−S+i2q−ωq|vk′q|2hS−S+i2q.
1 Xkq
KEm = ωq|vkq|2hS−S+S−S+iq (C2) g (E5)
N
Xkq
The interaction energy can be written as
1
APPENDIX D: INTERACTION ENERGY EI = N Vklqpvlpu∗lpvk∗qukqhS−S+iqhS−S+ip
kXlqp
e
In the calculation of the expectation value of the in- −2 Vk′lqpvlpu∗lpvk∗′quk′qhS−S+iqhS−S+ip.
teractionenergy,EI, it is easyto seethat the productof Xlp
only two states, kq and lp give non-zero contribution to (E6)
the interactionterm. All the otherstates arenormalized
to unity. Then, From Eq. 16a, we can show that
EI = N1 hψ∗|Vklqpb†kqblp|ψi ∆k′q =−Xlp Vk′lpqvlpu∗lpqhS−S+iqhS−S+ip. (E7)
kXlpq
1 We use this relation in the right hand side of Eq. E6.
= N Vklpqhψk∗qψl∗p|b†kqblp|ψlpψkqi (D1) The second term now gives +2vk∗′quk′q hS−S+iq∆k′q,
1 kXlqp which, using Eq. 17a gives ∆2k′q/Ek′q. Tphen,
= V v∗ u v u∗ hS−S+i hS−S+i
N kXlqp klqp kq kq lp lp q p E = 1 V v u∗ v∗ u hS−S+i hS−S+i + ∆2k′q.
I N kXlqp klqp lp lp kq kq q p Ek′q
where b†kq =c†k+q2↑c†−k+q2↓Sq+ e (E8)
9
Combining Eqs. (E3,E4,E5,E8), we get the result that
we will use in DOS calculation,
∆2
k′q
hψ|H|ψi−hψ|H|ψi=ǫk′+q2 + Ek′q (E9)
−e(ǫk′e+q2 +ǫk′−q2 +ωqhS−S+iq)|vk′q|2hS−S+iq