A Lorentz Invariant Pairing Mechanism: Relativistic Cooper Pairs

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by  A. Bermudez

A Lorentz Invariant Pairing Mechanism: Relativistic Cooper Pairs
A. Bermudez1 and M.A. Martin-Delgado1
1Departamento de F´ısica Te´orica I, Universidad Complutense, 28040. Madrid, Spain.
WestudyaLorentzinvariantpairingmechanismthatariseswhentworelativisticspin-1/2fermions
aresubjectedtoaDiracstringcoupling. Intheweakcouplingregime,wefindremarkableanalogies
between this relativistic bound system and the well known superconducting Cooper pair. As the
coupling strength is raised, quenched phonons become unfrozen and dynamically contribute to the
8 gluing mechanism, which translates into novelfeatures of thisrelativistic superconductingpair.
0
0 PACSnumbers: 71.10.Li,71.38-k,03.65.Pm,75.30.Ds
2

n I. INTRODUCTION HamiltonianinEq.(1)totwo-fermionsystemsispossible
a
[7, 8, 9, 10], which in the center of mass reference frame
J
0 A large class of superconducting materials can be ac- reads as follows
1 cuuproantetlwyodmesacrjoibrecdonbtyritbhuetiBonCsS. Fthiresotr,yF[r1¨o]l,icwhhsihchowisedbahsoewd H3D = √c2(α1−α2)(p−imωβ12r)+mc2(β1+β2), (2)
] thecouplingbetweenelectronsandcrystalphononsleads
con ttoronasn[2eff].ecItnisvpeiraetdtrbaycttivheisirnetseurlatc,tCioonopbeertwdeisecnustsheedehleocw- wI4herβe,αa1nd=βα12⊗=Iβ4, αβ2 =repIr4es⊗enαt,thβe1g=enβera⊗lizI4a,tiβon2 o=f
⊗ ⊗
- any attractive interaction can bind a couple of electrons the Dirac matrices in the two-body Hilbert space. Here
pr which lay around a filled Fermi sea [3]. This bound sys- p:=(p1 p2)/√2, and r:=(r1 r2)/√2 stand for the
− −
u tem,knownasaCooperpair,isresponsibleforseveralin- relative momentum and position operators.
s triguing properties displayed by superconductors, which In this work, we study the binding properties of this
.
t can be described as a many-body coherent state where two-body relativistic Hamiltonian, and discuss under
a
m electrons above the Fermi surface are bounded in pairs. which circumstances an analogy to Cooper pairs can be
The oversimplified picture developed by Cooper cap- performed. Phonons in this relativistic system are dy-
d- tures the essence of the underlying physical phenomena namical and always provide a pairing mechanism, as we
n occurring in superconducting solids. In the same spirit, shall see. Thus, there is no need to invoke a many-
o westudyasimplemodeloftworelativisticfermionswith body effect through the Pauli principle as in the origi-
c an effective attractive interaction. In order to maintain nal Cooper pair scenario. In fact, there are real mate-
[
the similarities with the Cooper problem, we must fulfill rials which deviate from standard BCS theory. In BCS,
1 the two following requirements: phonons are quenchedand their effect appears as a pair-
v Phonon gluing mechanism: Ina relativisticscenario, ing energy scale, but they are not explicit in the Hamil-
5 the simplest phonon-like coupling is modeled by a Dirac tonian. There is an extension of the BCS theory that
1 string coupling mechanism where the vibrations of the accounts for the effects of dynamical phonons, known as
6
string describe the lattice phonons (see fig. 1 left). This theMigdal-Eliashbergtheory[11,12,13]. Ourrelativistic
1
. interactionis introducedbyanon-minimalcouplingpro- fermionic pairing mechanism is thus closer to this latter
1 cedure in the free Dirac equation treatment.
0
We shall restrict ourselves to a two-dimensional sys-
8
∂ Ψ tem, where an exact solution is derived and several in-
0 i~ | i = cα(p imωβr)+mc2β Ψ , (1)
: ∂t − | i teresting properties can be neatly discussed. Two spa-
v (cid:0) (cid:1)
i where Ψ stands for the Dirac 4-component spinor, p
X | i
represents the momentum operator, and c the speed of
ar light. Here β = diag(I2, I2),αj = off-diag(σj,σj) are
−
theDiracmatricesinthestandardrepresentationwithσ
j
astheusualPaulimatrices[4]. ThisDiracstringcoupling
p p imωβr was introduced in [5, 6] as a relativis-
→ −
tic extension of the harmonic oscillator, usually coined
as the Dirac oscillator, where ω represents the oscilla-
tor’s frequency. In our picture, this frequency effectively
describes the lattice vibrations and its coupling to the
fermionic degrees of freedom.
Two-fermion binding: Regardless of the coupling
strength, we shall show that such an effective string FIG. 1: (left) Electron-phonon interaction in thelanguage of
coupling binds relativistic fermions in pairs (see fig. 1 Feynmandiagrams. (right)Effectiveelectron-electronattrac-
right). A Lorentz invariant extension of the Dirac string tiveinteraction duetotheexchange of a lattice phonon.
2
tial dimensions are also natural for other superconduct- phonon Fock states
ing materials like the cuprates [14]. In this case, the
1
Dσir,aαc m=aσtri,cβes=reσdu,caentdotthheereulsautaivliPstaicul1i-bmoadtyricsetastαexΨ= |nr,nli:= √n !n !(a†r)nr(a†l)nl|vaci, (6)
x y y z r l
| i
can be described by a 2-component spinor. The 2-body
where n ,n = 0,1... specify the number of right- and
relativisticHamiltonianintwodimensionscanbewritten r l
left-handed phonons coupling the two-fermion system.
as follows
One immediately observes that the Hilbert space can
c
H2D =√c2(σx⊗I2−I2⊗σx)(px−imωσz ⊗σzx) be∞nrd,invl=id0eHdnirnnl,awsheerrieeseaocfhisnuvbasrpiaancte csaunbsbpeacdeesscHribe=d
+√2(σy ⊗I2−I2⊗σy)(py −imωσz ⊗σzy) (3) bsLpyanHnnerdnlby:= Hn′r,nlLHn′′r,nl. These subspaces are
+mc2(σz I2+I2 σz).
⊗ ⊗
Hn′r,nl =span{|+i|nr,nli},
II. ENERGY SPECTRUM AND EIGENSTATES Hn′′r,nl =span{|↑↑i|nr,nl+1i,|−i|nr,nli,|↓↓i|nr+1,nli},
(7)
In two dimensions, chiral creation-annihilation opera- where the states := ( )/√2 , and + :=
tors which carry dual aspects of a left- or right-handed |−i | ↑↓i−| ↓↑i | i
( + )/√2 are maximally entangled unpolarized
symmetry are defined as follows | ↑↓i | ↓↑i
Bell states. In particular, describes a zero-energy
Hn′r,nl
ar := √12(ax−iay), a†r := √12(a†x+ia†y), (4) msuabisnpinacgesuEb+s,pnar,cnels= 0. ThcaenHbaemeixltporneisasned(5a)sifnolltohwesre-
al := √12(ax+iay), a†l := √12(a†x−ia†y), Hn′′r,nl
1 i ζ(n +1) 0
where a†x,ax,a†y,ay, are the usual creation- Hnrnl =∆i ζ(n +1) −p 0 l i ζ(n +1),
annihilation operators of the harmonic oscillator 2D l r
−
ato†i t=he √g1r2o(cid:16)u∆n1˜dris−tait∆e~˜pwi(cid:17)id,tha.ndU∆s˜ing=thpes~e/mopωeraistorresl,atthede  p 0 ipζ(nr +1) p −1 (8)
wherethe2-bodyinteractioncouplesthreedifferentlevels
relativistic Hamiltonian in Eq. (3) takes a simpler and
and can be exactly diagonalized. Using Cardano-Vietta
amenable form
solutiontothirdorderpolynomials,weobtainthefollow-
ing energies
∆ g∗a†l ga†l 0
 ga 0 0 g a 
H2D =g∗0all g0a g0a g∗a∆rr , (5) E1∆nrnl :=r4[1+ζ(nr3+nl+2)]cosΘ,
 †r ∗ †r − 
where ∆ := 2mc2 stands for the system rest mass, E2nrnl := 4[1+ζ(nr+nl+2)]cos Θ+ 2π , (9)
g := imc2√2ζ is a coupling parameter, and ζ := ∆ r 3 (cid:18) 3 (cid:19)
~ω/mc2 controlsthe strengthoftheeffectiveinteraction. E3nrnl := 4[1+ζ(nr+nl+2)]cos Θ+ 4π ,
Considering the two-body spinorial basis , , ∆ r 3 (cid:18) 3 (cid:19)
{| ↑↑i | ↑↓i
, ,wecanunderstandtheDiracstringcoupling
|↓↑i |↓↓i}
as a four-level system depicted in fig. 2. where
1 27(n n )ζ
l r
Θ:= arccos − . (10)
|↑↑i 3 (cid:20)2[3(1+ζ(n +n +2))]3/2(cid:21)
r l
These eigenstates are represented for different values of
|↑↓i |↓↑i the coupling strength ζ in Fig. 3, where the chiral quan-
tum numbers have been set to n =n +1.
r l
In this figure we observe two different regimes:
Weak Coupling regime ζ 1: In this case, the low
≪
energy properties can be accurately described by a two-
|↓↓i
level system. This feature will turn out to be crucial for
FIG.2: Fermionicspin-fliptransitionsduetotheDiracstring the analogies of the system to a non-relativistic Cooper
coupling and mediated by the creation-annihilation of chiral pair discussed in section III.
phonons. Strong Coupling regime ζ 1: Inthis case,the four
≫
levelsbecomeessentialinordertodescribethelowenergy
We now proceed to describe the energy spectrum of excitations. Consequently,thedescriptionbecomesmore
the2-bodyinteractingrelativisticsystem,intermsofthe involved but also gives a richer structure that may show
3
certain novel properties with respect to non-relativistic Since these expressions must satisfy the antisymmetric
Cooper pairs that are described in section IV. condition in Eq.(13), the number of chiral quanta are
Once the eigenvalues have been obtained, we may de- constrained as follows
rive the corresponding eigenstates, which we list below
|E+,nr,nli:=|1+,nr,nli, ||EE+j,,nnrr,,nnllii⇒⇒nnrr++nnll ==22kk+1 ::kk==00,,11,,22...... (15)
|Ej,nr,nli:= Ω [αj|−i|nr,nli+iβj|↑↑,nr,nl+1i (11) Due to the indistinguishability of the relativistic
j
+iδj ,nr+1,nl ], fermions, the eigenstates |E+,nr,nli must contain an odd
|↓↓ i numberofchiralquanta,whereas E arerestricted
| j,nr,nli
where we have defined the following parameters to even number of chiral quanta.
α :=∆2 E2 , We have thus derived a complete solution of the rel-
j − jnrn2l ativistic Dirac equation for two bodies interacting via a
β :=∆(∆+E ) ζ(n +1), Dirac string coupling. Therefore, this 2-fermion system
j jnrnl l
p (12) belongs to the small class of exactly solvable few-body
δ :=∆(∆ E ) ζ(n +1),
j − jnrnl r relativistic systems. In sections III and IV we show that
p
Ω := α2+β2+δ2. thisrelativisticinteractiondoesindeedleadtotheforma-
j q j j j tionofboundpairs,bothintheweakandstrongcoupling
Heretheindexesj =1,2,3correspondtothethreediffer- regimes. Furthermore,wepresentadetailedstudyofthe
enteigenvalues(9). Letusmentionthatthetotalangular similar properties that the relativistic bound pair shares
momentum J :=S +L is conserved. Thus, the eigen- with the well-known non-relativistic Cooper pair. As we
z z z
states (11) have well-defined angular momenta, namely, will see, there are profound analogies in the weak cou-
~(n n ). Finally, we must consider the consequences pling regime, whereas novel properties are found in the
r l
offer−mionindistinguishability. TheSymmetrizationpos- strong coupling limit.
tulate states that a system of identical fermions must be
described in terms of antisymmetrical states, which es-
tablishes the following constraint III. WEAK COUPLING REGIME
P21 Ψ(1,2) = Ψ(1,2) , (13)
| i −| i The standarddescriptionof Cooperpairsin supercon-
where P21 stands for the permutation operator that ducting solids is usually performed in a weak coupling
swaps the fermion labels 1 2. Considering the eigen- regime, where a slightly phonon-mediated attractive in-
↔
statesinEq.(11)underthepermutationoperator,weob- teractionbindselectronwhichlayclosetotheFermisur-
tain the following expressions face. Weshallconsiderthatthetwo-bodyHamiltonianin
Eq.(3) effectively describes the gluing mechanism above
P21|E+,nr,nli=(−1)nr+nl|E+,nr,nli, (14) the Fermi sea, and therefore a weak coupling regime is
P21|Ej,nr,nli=(−1)nr+nl+1|Ej,nr,nli. obtained when ζ ≪1.
In this weak coupling regime, we have seen in Fig. 3
that the low-lying excitations can be entirely described
byatwo-levelsystem. Thissituationisschematicallyde-
8
scribed fig. 4, where we see how spin-polarizedlevels be-
E1
6 comedecoupledfromthoseresponsibleofthe low-energy
properties. In this situation, we can obtain an effective
4
Hamiltonianforthelowenergysector,byadiabaticelim-
2 ination.
E/mc2 0 E3 Let us consider an arbitrary state |Ψ(t)i∈Hnrnl
E+
ψ(t) =c (t) ,n ,n +1 +c (t) ,n ,n
−2 | i ↑↑ |↑↑ r l i ↑↓ |↑↓ r li
+c (t) ,n ,n +c (t) ,n +1,n ,
r l r l
−4 ↓↑ |↓↑ i ↓↓ |↓↓ i
(16)
−6
E2 whose dynamical evolution, described by the Dirac
−8 Hamiltonian (5), can be represented as
0 0.5 1 1.5 2
1/ζ
FtarehpIpeGprrde.osi3are:acnhcDtstsehtapreeiwnnstegdraeocknnocaguetpctolorifauntcpghtlieisvnttergwecrnooe-ggufteiphmrl.mienNigζoon≫ζteen1te,h1rwag.tyheasrspeea1cs/tζ1ru/→ζm→0wiw∞the i~ddtcccc↓↓↑↑↓↑↓↑((((tttt)))) =gg∆∗0aall gg∗00aa†r†l gg∗00aa†l†r g−g∗0a∆arr cccc↓↓↑↑↓↑↓↑((((tttt))))(17.)
≪
4
In the weak coupling limit, the transitions to the spin- can take on two different channels via the consecutive
polarized ,n ,n +1 , ,n +1,n upper and creation-annihilation of right- or left-handed phonons.
r l r l
{| ↑↑ i | ↓↓ i}
lower levels can be considered negligible. Therefore, the This process can be understood as an instance of a su-
level population does not evolve under the action of the perexchange coupling between the spins
two-body interaction dc↑↑ = dc↓↓ = 0, and we can adia- driven by a second order two-phonon pr|o↑c↓eiss←w→he|re↓↑ai
dt dt
baticallyeliminatethesetwolevels. Thelatterconditions chiral phonon is virtually created and then annihilated.
substitutedinEq.(17),giverisetothefollowingrelations Thereexisttwodifferentexchangepaths,asseeninfig.4,
depending on the left- or right-handed chiralities of the
ζ(n +1) virtual phonons involved in the process.
l
c =i (c c ),
↑↑ r 2 ↑↓− ↓↑ This effective Hamiltonian (21) can be exactly diago-
(18)
nalized yielding the eigenvalues
ζ(n +1)
r
c =i (c c ),
↓↓ r 2 ↑↓− ↓↑ Eeff :=0, Eeff :=2~ω(n n ), (22)
+nrnl −nrnl r− l
and an effective two-leveldynamics
with the following associated eigenstates
d c (t) ∆ζ 1 1 c (t)
i~dt(cid:20) c↓↑↑↓(t)(cid:21)= 2 (nr −nl)(cid:20) −1 −1 (cid:21)(cid:20) c↑↓↓↑(t) (cid:21). |E+effnrnli:=|+,nr,nli⇒nr+nl =2k+1 :k =0,1,2...
(19) Eeff := ,n ,n n +n =2k :k =0,1,2...
In this sense, we can integrate out the high-frequency | −nrnli |− r li⇒ r l (23)
modes by projecting onto the effective spin-unpolarized
invariant subspace spanned by eff := span
Hnrnl {| ↑↓ where the anti-symmetric character of the fermionic
,n ,n , ,n ,n , by means of an orthogonalprojec-
r li |↓↑ r li} states has already been considered. Therefore, the low-
tor
lying solution in the weak coupling regime can be de-
eff := ,n ,n ,n ,n + ,n ,n ,n ,n scribed by the maximally entangled Bell states in the
Pnrnl |↑↓ r lih↑↓ r l| |↓↑ r lih↓↑ r l| spin degreeof freedom, and rotationalFock states in the
(20)
orbital degree of freedom.
The z-component of the orbital angular momentum op-
esurabtsopracLezb=ec~o(ma†reasrP−neffran†llaLl)zPcnoeffrnnsltr=ain~e(dnrto−thnils)Ii2n,vawrhiaicnht pmauFirsu.trtshhIneorwmorotdhreear,ttttohheessheinostwtear-ttphesaartdtiescsulcecrhidbibestinaandcibneoguonnodlcycufearrtsmt,aiiwonnes
allows us to rewrite Eq. (19) as an effective two-level
finite values. Let us introduce the square-distance op-
Hamiltonian
erator Γ := x2 + y2, where x := (x1 x2)/√2 and
−
Heff :=ωLz(cid:20) 11 −11(cid:21)=~ω(a†rar−a†lal)(cid:20) 11 −11(cid:21). tyor:=s fo(ry1th−e yre2l)a/t√iv2e fdeernmoitoenitchedisstpaanccee.coTohrdeinexapteecotpateiroan-
− −
(21) values in the weak-coupling eigenstates (23) are
This effective interaction in the weak coupling regime
is represented in Fig. 4, where the allowed transitions ∆˜2
Γ = (1+n +n ), (24)
r l
h i± √2
which is always finite. We observe the crucial property
|↑↑i
that this system shares with a non-relativistic Cooper
pair,namely,thepairofrelativisticfermionsarebounded
in pairs even for a weak attraction ζ 1.
≪
Anotherfundamentalpropertythatoccursinstandard
Cooperpairsisthepresenceofanenergygapbetweenthe
|↑↓i |↓↑i paired energy level and the Fermi surface. This energy
gap is responsible of the stability of Cooper pairs with
respect to free fermion pairs and is proportional to the
lattice Debye frequency ∆E ~ω . In the relativistic
D
∼
regime, we observe that the energy gap with respect to
the displaced Fermi surface ( i.e. ǫ =0 ) is
′F
|↓↓i ∆E+nrnl =0, (25)
∆E =2~ω(n n ),
−nrnl r− l
FIG. 4: Fermionic spin-flip transitions in the weak coupling
regime. The low energy sector is described by an effective and therefore the only stable pair (i.e. ∆E < 0) is that
two-level system, where spin-flips occur along two different described by the spin-singlet state when n n .
l r
channelsthatincludevirtualtwo-phonontransitionsandspin- Inthissenseweobtainaspin-singletboun≥dpairwhich
polarized states becomedecoupled. clearly resembles the situation in standard Cooper pairs
5
where the fermions arealsointhe singletstate. Further- In this section we have discussed a relativistic pairing
more, we can observe from this discussion that the rela- mechanisminaweakcouplingregime. Wehavediscussed
tivistic gap is proportionalto the Dirac string frequency in detail severalanalogies with a non-relativistic Cooper
∆E ~ω, which plays the role of the usual Debye fre- pair that naturally arisein this weak coupling limit. Re-
∼
quency in superconducting materials. markably,weobtainbindingregardlessoftheinteraction
Finally, to take this comparison further, we should strength, which is a fundamental property of BCS sys-
study the properties of the stable pair eigenstates in tems. Additionally, we have shown how the relativistic
Eq.(23)andcomparethemtothenon-relativisticCooper energy gap scales with the string frequency in the same
pair features. manner as the Cooper pair gap scales with the phonon
Debye frequency. In this regard, we may conclude that
Spin degrees of freedom: In BCS theory, Cooper
thestringinteractionplaystheroleofthelatticephonons
pairsdisplayasingletstateinthespindegreeoffreedom.
thatmediatetheeffectiveattractionbetweenfermionsin
We observe in Eq. (23) that the stable bound fermionic
the BCS theory. Furthermore, we have also compared
pair state has also a spin-singlet component.
the nature of the relativistic pair eigenstates with the
Orbitaldegreesoffreedom: InBCStheory,Cooper
Cooper pair wave functions. We have seen that the rel-
pairs display a spherically symmetrical wave function
ativistic bound pair is also described by a spin-singlet
withanonion-likelayeredstructure. Wedirectlyobserve
state and a spherically symmetric onion-like state in the
fromfig.5 that relativisticbound pairprobabilitydistri-
orbital degrees of freedom. All these similarities allow
bution ρeff (r) display a similar spherically symmetric
nrnl us to state that this fermionic pairing mechanismcanbe
onion-lik−e structure.
interpretedasarelativisticCooperpair,sincewerecover
most of the usual BCS properties in the weak coupling
regime. Nonetheless, this Relativistic Cooper pair can
alsobestudiedinthestrongcouplingregime,wherenovel
properties with respect to the usual Cooper pair in BCS
theoryarise. Aswedescribebelow,whentheDiracstring
interaction becomes strong enough, phonons contribute
dynamically to the gluing mechanism.
ρe−ff11
IV. STRONG COUPLING REGIME
In this section we study the pairing properties of
the two-body relativistic system in the strong coupling
y/∆˜ regime ζ 1. In this limit we must consider the com-
x/∆˜ plete four≫-levelstructure of the system ( see fig. 2 ), and
the energy spectrum becomes clearly more involved in
Eq. (9) (see fig. 6).
In fig. 6 we have represented the different energies
for an interaction strength ζ = 5 which lays in the
strong coupling regime. We clearly see how two levels
E2,nr,nl,E3,nr,nl become stable pairs with a certain gap
∆E2,nr,nl < ∆E3,nr,nl < 0. Therefore the strong cou-
pling gives raise to a couple of stable bound fermionic
ρe−ff22 states, namely,
1
|E2,nr,nli:= Ω2[α2|−i|nr,nli+iβ2|↑↑,nr,nl+1i+
+iδ2 ,nr+1,nl ];
|↓↓ i
1
y/∆˜ x/∆˜ |E3,nr,nli:= Ω3[α3|−i|nr,nli+iβ3|↑↑,nr,nl+1i+
+iδ3 ,nr+1,nl ].
|↓↓ i
(26)
FIG.5: Spatialprobabilitydensityprofilesforweak coupling The spatial probability distribution ρ (r) of these
jnrnl
stable pairs ρe−ffnrnl(r) := Trspin(hr|E−effnrnlihE−effnrnl|ri) with stable fermionic pairs has been represented in fig. 7 in
nrbeillpi>tryesnednretn.ssiTρte−yoffp2ρ2e−(fiffr1g)1u.(rre),cworirtehspro=nd|sr1to−nrr2|/=√12,.nBlo=tto1mpfirogbuare- dltihkeenescsiattyrsuepcortofufinrelre.=pNrenosnlee=rtvhe1es.letWhssee,sncpaohnteerwcilcoeaarltrlhylyysyodbmiffsmeerrevetenrictcehsoantairtoihnsee-
6
with respect to the weak coupling regime ( compare to comes strongerand contribute to the effective attraction
the top fig. 5). inadynamicphenomenon. Thisisreminiscentofa(s,p)-
Furthermore, these two stable states form a fermionic wave symmetry of a SC order parameter. Similar types
bound pair since the inter-particle distance is finite of superconducting states appear in some quantum liq-
uids like superfluid He3: the so-called A- and B-phases
Γ 2 =∆˜2 (1+nr+nl)α22+(β22+δ22)(2+nr+nl) , exhibit different patterns of spin-orbit symmetry break-
h i
Γ 3 =∆˜2(cid:2)(1+nr+nl)α23+(β32+δ32)(2+nr+nl)(cid:3). ing [15, 16]. Layered materials like the ruthenates also
h i exhibit unusual symmetry properties like triplet super-
(cid:2) (2(cid:3)7)
conductivity [17, 18, 19].
We also observe that the strong pairing mechanism
We may conclude that the Dirac string pairing mecha-
leads to a couple of possible stable bound pairs (26),
nism leads to bound pairs in the strong coupling regime,
whereas the weak coupling only produces one stable
which display substantial differences with respect to the
bound pair. Furthermore, the energy gap displayed by
weakly coupledbound states in Eq.(23). It follows from
the bound pairsalsodepends onthe strengthofthe cou-
Eqs. (26) that the bound pairs are not in a singlet state
pling. Intheweakcouplingregime,wehavealreadyseen
but rather in a linear superposition of different spin sin-
that the energy gap scales as ∆E ~ω, whereas the
glet and triplet states entangled with different orbital
scaling in the strongly interaction l−im∼it does not present
Fockstates. Inthisregard,therelativisticpairingmecha-
such a simple scaling (see fig. 6).
nismdoes notinduce ananti-ferromagneticorderingany
longer,andcertainspin-polarizationmayarisedepending
on the value of the coupling strength ζ.
It is also instructive to compare the orbital degrees of
freedom of bound pairs in the weak and strong coupling
limits. The weakly coupled states in Eq. (23) are in or-
bital Fock states, which represent a certain number of
vibrational phonons which are frozen in this limit. On
the other hand, stronglycoupledstates in Eqs.(26) can-
not be described by a single Fock state, and therefore ρ211
the vibrational phonons acquire a dynamical behavior
n +1,n ⇆ n ,n ⇆ n ,n + 1 , which is a clear
r l r l r l
| i | i | i
signofstrongcouplinginsuperconductors. Wemaycon-
clude that the Dirac string phonons, responsible of the
gluing mechanism, become unfrozen as the coupling be-
y/∆˜
x/∆˜
E3nrnl
E+nrnl
ρ311
E1nrnl
E
mc2 E3nrnl
E+nrnl
E2nrnl y/∆˜ x/∆˜
n
l nr
FIG.7: Spatialprobabilitydensityprofilesforstrongcoupling
ζ = 5 stable pairs ρjnrnl(r) := Trspin(hr|EjnrnlihEjnrnl|ri)
FIG. 6: Energy levels of thetwo-body Dirac oscillator in the withnr =1,nl=1. Topfigurecorrespondstotheprobability
strong coupling regime ζ = 5 as a function of the different density of the stable pair ρ211(r), with r = r1 r2 /√2.
| − |
phonon number.(inset) Detail of the energies corresponding Bottomfigurerepresentstheprobabilitydensityofthestable
to levels E+nrnl,E3nrnl pair ρ311(r).
7
V. CONCLUSIONS Dirac oscillator as an instance of a relativistic Cooper
pair. Nevertheless, there may also be other types of rel-
We have studied the relativistic pairing mechanism of ativistic binding mechanisms yielding also the formation
the two-body Dirac oscillator in two dimensions, where of Cooper pairs.
a Dirac string coupling leads to fermionic bound pairs. On the other hand, a strong interaction leads to re-
We have described two different regimes where binding markable differences with respect to BCS Cooper pairs.
occurs regardless of the interaction strength. In this case, more than one bound pair can be built,
In a weak coupling regime, the fermionic pair bears which in any case is not in a singlet state but rather in
a strong resemblance to the usual Cooper pair in BCS a linear superposition of singlet and triplet states. Fur-
theory. We remarkablyfound a similarscaling of the en- thermore, the gluing phonons become unfrozen as the
ergygap,whichallowsustoidentifytheDiracstringfre- coupling strength is raised and dynamically contribute
quencyω withthelatticephononDebyefrequencyω in to the pairing mechanism.
D
superconducting materials. Additionally, we found that Acknowledgements We acknowledge financial support
the relativistic bound pair eigenstates are also in a spin from a FPU grant of the MEC (A.B.), DGS grant un-
singlet state, and presenta spherically symmetric onion- dercontractFIS2006-04885,CAM-UCMgrantunderref.
like structure in the probability distribution. All these 910758. (A.B., M.A.M.D,), and the ESF Science Pro-
remarkable analogies suggest to interpret this two-body gramme INSTANS 2005-2010(M.A.M.D.).
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