Holon Pairing Instability based on the Bethe-Salpeter Equation obtained from the t-J Hamiltonians of both U(1) and SU(2) Slave-boson Symmetries

4 Pages·0.14 MB·English

by  Sung-Sik Lee

Holon Pairing Instability based on the Bethe-Salpeter Equation obtained from the t-J
Hamiltonians of both U(1) and SU(2) Slave-boson Symmetries
Sung-Sik Lee and Sung-Ho Suck Salka
Department of Physics, Pohang University of Science and Technology,
0 Pohang, Kyoungbuk, Korea 790-784
0 a Korea Institute of Advanced Studies, Seoul 130-012, Korea
0 (February1, 2008)
2
We investigate a possibility of holon pairing for bose condensation based on the Bethe-Salpeter
n
equation obtained from the use of the t-J Hamiltonians of both the U(1) and SU(2) slave-boson
a
symmetries. It is shown that the vertex function contributed from ladder diagram series involving
J
holon-holon scattering channel in the Bethe-Salpeter equation leads to a singular behavior at a
6
critical temperature at each hole doping concentration, showing the instability of the normal state
1
against holon pairing. We find that the holon pairing instability occurs only in a limited range of

holedoping,byshowingan”arch”shapedbosecondensationlineinagreementwithobservationfor
]
n high Tc cuprates. It is revealed that thisis in agreement with a functional integral approach of the
o slave-boson theories.
c
- 1
pr theSinUc(e1)thaendadSvUen(2t)osflahvige-hboTsconsuappeprcrooancdhuecstitvoityt,hebott-hJ H =−t (c†iσcjσ +c.c.)+J (Si·Sj − 4ninj), (1)
u <Xi,j> <Xi,j>
Hamiltonian have been proposed to study superconduc-
s
at. t[1iv]-it[y3]f.orUtnhleiketwtoh-ediUm(e1n)sisolnavael-sbyosstoenmsmoeafncofipepldertohxeiodreys, wHhereereSSi·iSsj−th41eneinlejct=ro−n21s(pci†in↓c†jo↑p−erca†i↑tco†jr↓)a(tcj↑sictie↓−i,cjS↓ci↑=).
i i
m the recently proposed SU(2) theory of Wen and Lee has 12c†iασαβciβ withσαβ,thePaulispinmatrixelementand
- a merit of treating the low energy phase fluctuations of
d ni, the electron number operator at site i, ni = c†iσciσ.
order parameters [3]. Recently we proposed a theory of
n In the slave-bosonrepresentation[1]- [3] cσ, the electron
o bosecondensationby payingattentionto theholonpair- annihilation operator of spin σ can be written as a com-
c ing channel in the U(1) slave-boson theory [4], in con- posite of spinon and holon operators. That is, c =b f
1 [ throalsotntcoonodthenersaetaiornli.erInsttuhdisiepsacpoenrctehrneepdhawsiethflutchteuastiniognles winitthheαU=(11),t2h,eworhyeraenfdc(αb)=is√t1h2eh†sψpαinionnt(hheolSoUn)(2aσ)ntnhihe†oilraσy-
σ
v of the spinon pairing and hopping order parameters are
f
8 taken care of by the SU(2) theory. In addition, follow- tion operator in the U(1) theory, and ψ = 1 and
1 ing our earlier study [5] the spinon and holon degrees 1 (cid:18)f2† (cid:19)
2 f b
01 othfefrbeaedsiosmofaorne-isnitterocdhuacregde iflnutcotuthateioHnesiswenhbicehrgartiesremasona ψ2 =(cid:18)−f21† (cid:19) and h=(cid:18)b12 (cid:19) are respectively the dou-
blets of spinon and holon annihilation operators in the
0 result of site-to-site electron(and thus holon) hopping in
0 two-dimensionalquantumsystems ofhole-dopedhighT SU(2) theory.
c
/ AfterHubbardStratonovichtransformationsinassoci-
t cuprates. Most of the slave-boson theories have been
a ationwiththedirect,exchangeandpairingchannelsand
concerned with single holon bose condensation. These
m
the saddle point approximation, the effective Hamilto-
theories predicted a linear increase of bose condensation
- nian is decomposed into the spinon sector and the holon
temperature with doping concentration, contrary to the
d
sector separately. Here to explore the instability of the
n observedbosecondensationline ofan’arch’shapewhich
normal state against holon pairing, we pay attention to
o manifests the presence of optimal doping rate. Most re-
c cently Wen and Lee [3] questioned whether the single the holon sector of the Hamiltonian. The holon Hamil-
: tonian is derived to be, for the U(1) slave-boson theory
v holonortheholonpairbosecondensationisfavoredwith
i theSU(2)slave-bosontheory. Inthisworkwepayatten- [5],
X
tion to holon pair bose condensation by demonstrating
ar a possibility of holon pairing instability from the study Htb J,U(1) =−t χijb†ibj +c.c.
−
of the Bethe-Salpeter equation obtained from the use of <Xi,j>
J
tbhoesotn-JsHymammielttorineisa.nsTohfeboptrhesethnet Uap(p1)roaancdh SisU(s2h)owslnavteo- − 2 |∆fij|2b†ib†jbjbi−µ0 b†ibi, (2)
<i,j> i
X X
predict the observed characteristics of the bose conden-
sationlineofanarchshapeinexcellentagreementwitha where χij =< fj†σfiσ + J(14tδ)2b†jbi > is the hopping or-
functional integral approach to the slave-bosontheories. der parameter and ∆f =<−f f −f f >, the spinon
ij j i j i
↑ ↓ ↓ ↑
pairing order parameter, and for the SU(2) slave-boson
We write the t-J Hamiltonian, theory,
1
Htb−J,SU(2) =−2t h†iUijhj +c.c. hwohleorne Mgαaβt(sku)ba=ra G0βrdeτeeni’ks0τfu<nctTioτn[bαink(τth)be†βkS(U0()2])>s,latvhee-
<Xi,j> R
boson representation. Here k ≡ (k ,k) is the three
J 0
−2 |∆fij|2b†iαb†jβbjβbiα−µ0 h†ihi, (3) component vector of the energy-momentum of holon.
<i,j>,α,β i q = (q ,q) is the three component vector of the total
X X 0
energy-momentum of holon pair.
χ −∆f
where Ui,j = −∆ijfij∗ −χ∗iijj ! is the order parameter EqAs.f(t6e)rasnudm(m7)i,nwgeoovbetrainthteheMfoaltlsouwbinargamfarterqiuxeenqcuieastioinn
matrix [3] of hopping order, χ and spinon pairing or-
ij for the t-matrix, in the U(1) slave-bosonrepresentation,
der, ∆fij with χij =<fj†σfiσ+ J(12tδ)2(b†j1bi1−b†i2bj2)>
and ∆f =<f f −f f >. − ′
Introijducingja1 ui2niformj2hio1ppingorderparameter,χ = δk′,k′′ −mk′,k′′(q0,q) tk′′,k(q0,q)=v(k −k), (8)
χ, and a d-wave spinon pairing order parameters, ∆fji = Xk′′ (cid:16) (cid:17)
ji
±∆ (thesign+(−)isfortheijlinkparalleltoxˆ(yˆ)),we
f ′ ′
obtain from Eq.(2) the momentum space representation where tk′,k(q0,q)≡<k ,−k +q|t|k,−k+q >U(1),
of the Hamiltonian, for the U(1) theory,
1 ′ ′′ ′′ ′′
Htb J,U(1) = ǫ(k)b†kbk mk′,k′′(q0,q)≡−Nβ v(k −k )g(k )g(−k +q)
− ′′
Xk Xk0
+21N kX,k′,qv(k−k′)b†−k′+qb†k′bkb−k+q, (4) =−v(k′N−βk′′) ′′ ik0′′ −1ǫ(k′′)i(−k0′′ +q0)−1 ǫ(−k′′ +q)
whereǫ(k)=−2tχγ −µ ,theenergydispersionofholon Xk0
with γk =(coskx+kcosk0y) and v(k′ −k)=−J|∆f|2γk, = 1 v(k′ −k′′)n(ǫ(k′′))+eβǫ(−k′′+q)n(ǫ(−k′′ +q)), (9)
the momentum space representation of holon-holon in- N iq −(ǫ(−k′′ +q)+ǫ(k′′))
0
teraction. Likewise we obtain from Eq.(3),
−tχγ −µ t∆ ϕ andn(ǫ)= 1 , the bosondistribution function. Simi-
Htb J,SU(2) = h†k t∆kϕ 0 tχγ f−kµ hk larly,weobteaβiǫn−,1intheSU(2)slave-bosonrepresentation,
− f k k 0
k (cid:18) (cid:19)
X
1 ′
+2N k,kX′,q,α,βv(k−k )b†−k′+q,βb†k′,αbk,αb−k+q,β, (5) k′′,Xα′′,β′′(cid:18)δk′,k′′δα′α′′δβ′β′′ −mαk′′,βk′′α′′′β′′(q0,q)(cid:19)tkα′′′′,βk′′αβ(q0,q)
where α, β (= 1,2) represent two isospin components of ′
SU(2) holon and ϕ = (cosk −cosk ). N is the total =v(k −k)δα′αδβ′β, (10)
k x y
numberofsitesforthetwodimensionallatticeofinterest.
Considering the t-matrix of involving the particle- where tα′β′αβ(q ,q) ≡ < k′,α′;−k′ +q,β′|t|k,α;−k +
particle(holon-holon) scattering channels, we obtain the k′,k 0
′ ′ ′′ ′′
Bseentthaet-iSoanl,peter equations in the U(1) slave-boson repre- q,β >SU(2) and mαk′,β′k′′α′ β (q0,q) ≡ N1 α′1β1′ v(k′ −
<−Nk1′β,−k′ +v(kq|′t|−k,k−′′k)g+(kq′′)>gU(−(1k)=′′ +v(qk)′×−k) kq′)′U)nα†(′Eαα′′′1((kikq′′0′′)−))U(+Eβ†eα′ββ′1E′(β′k1(′′−(′−)+kkE′′+β+1q′)(n−q(kE)′.′β+1′(qH−))ekr′′e+qE))1×(kPU)α=′α′1E(kk′′−)Uµβ0′β1′(−k′′+
Xk′′ and 1E2(k) = 1−Ek − µ0 are the energy dispersions
′′ ′′
<k ,−k +q|t|k,−k+q > , (6) of the upper and lower bands of holons with E =
U(1) k
u −v
where g(k) = 0βdτeik0τ < Tτ[bk(τ)b†k(0)] >, the holon t (χγk)2+(∆fϕk)2. Uαβ(k) = vkk ukk is the
MatsubaraGreen’sfunction inthe U(1)slave-bosonrep- upnitary transformation matrix used(cid:18)for the dia(cid:19)gonaliza-
R
resentation,andintheSU(2)slave-bosonrepresentation, tion of the one-body holon Hamiltonian in Eq.(5) with
<k′,α′;−k′ +q,β′|t|k,α;−k+q,β >SU(2) uk = √12 1− tχEγkk and vk = √12 1+ tχEγkk. It is noted
=v(k′ −k)δα′,αδβ′,β thatthe tq-matrices,tk′,k(q0,q)anqdtαk′′,βk′αβ(q0,q)forthe
1 ′ ′′ ′′ ′′ U(1) and SU(2) theories respectively are independent
−Nβ k′′,Xα′′,β′′v(k −k )gα′α′′(k )gβ′β′′(−k +q) ocofntshideeMraatitosnuboafrainfsrteaqnuteannceioeusskh0′oalonnd-hko0l,onowinintegratcotitohne,
′′ ′′ ′′ ′′ ′
×<k ,α ;−k +q,β |t|k,α;−k+q,β > , (7) v(k −k).
SU(2)
2
As mentioned earlier, the t-matrix elements are ob- tor concerned with the SU(2) isospin channels α and
tained from Eqs.(6) and (7), both of which involve sum- β(α=1,2andβ =1,2)andλ,thecorrespondingeigen-
mation over the Matsubara frequencies. Using the ma- value. For the case of U(1) there are N eigenvectors
trixequations(8)and(10),thepolesofthet-matricesare corresponding to N discrete values of momenta for an
searchedforasafunctionofenergyq withq=0,i.e.,the N × N reciprocal lattice. For SU(2), 4N eigenvectors
0
zero total momentum of the holon pair. The t-matrices are available in accordance with the isospin channels of
are numerically evaluated from the use of Eqs.(8) and holon(b )-holon(b ) scattering. We search for the eigen-
α β
(10)foreachdopingrate andtemperature. The hopping vectors whose eigenvalues diverge in the limit of q = 0
0
and spinon pairing order parameters in Eqs.(2) and (3) and q = 0 at an onset(critical) temperature T , that is,
c
arethesaddlepointvaluesevaluatedfromtheusualpar- theeigenvectorscorrespondingtothepoleofthet-matrix
titionfunctionsinvolvingthefunctionalintegralsofslave- whose sign changes at T . We choose N = 10×10 for
c
boson representation [5]. For both the U(1) and SU(2) a reciprocal lattice to determine the symmetry of holon
t-matrix calculations we chooseJ/t=0.3 in Eqs.(2)and pairing at an underdoping rate, δ =0.05.
(3). The computed eigenvectors for both the U(1) and
Athightemperatures,computedpolesarefoundtobe SU(2) slave-boson approaches yielded good fits to the s-
positive and real, indicating that there exist no bound wavesymmetryofthe formcosk +cosk inmomentum
x y
states. As temperature is lowered to a critical(onset) space ; for the U(1) theory we have
value T , we find that with the U(1) theory one pole
c
changesitssignfrompositivetonegative,indicatingthat W(k)=A(cosk +cosk ), (13)
x y
their exists an instability of the normal state against
holonpairingattheonset(critical)temperatureT . Simi- where A is the normalization constant to satisfy
c
larly,withtheSU(2)slave-bosontheorytwopoleschange |W(k)|2 =1, and for the SU(2) theory,
k
theirsignsfrompositivetonegativeatT ;thesetwopoles
c
correspond to the particle-particle(holon-holon) scatter- PWαeβ(k)=Ae(coskx+cosky)(δα,1δβ,1+δα,2δβ,2), (14)
ingchannelsofb -b andb -b respectively. InFig.1. the
1 1 2 2
onset temperature T (denoted by solid square in Fig.1) and
c
for the appearance of such negative pole(s) is predicted
Wo (k)=Ao(cosk +cosk )(δ δ −δ δ ), (15)
tooccuronlyinalimitedrangeofholedopingconcentra- αβ x y α,1 β,1 α,2 β,2
tion, by revealing an ”arch” shaped feature of bose con-
densationlineinagreementwithobservationinthephase forb1 holonandb2 holonpairing,whereAe(o) is the nor-
diagram of high T cuprates. The onset temperature malization constant to satisfy |We(o)(k)|2 = 1.
c k,α,β αβ
of negative pole(s) coincide(s) surprisingly well with the Oneofthetwocomputedeigenvectorsshowsaphasedif-
P
criticaltemperature of holonpair condensationobtained ferenceofevenmultiplesofπbetweentheb andb holon
1 2
from the functional integral approach(denoted by open pairingorderparametersandiswellfittedbyEq.(14);the
square in Fig.1) of the slave-boson theories [5]. Holon- other computed eigenvector is well fitted by Eq.(15), by
holon scattering occurs above the lowest possible single showing a phase difference of odd multiples of π. The
particle energy ǫ(k=0). The negative pole corresponds divergenceof eigenvalueswasseento occurnearly atthe
tothebindingenergyoftheholonpair. Thisisanalogous sametemperature(withinanumericalaccuracyof10 4t).
−
tothebindingenergyofelectronpairswhichresultsfrom Fig.2 displays the U(1) result of the s-wavesymmetry of
the Cooper’s two particle problem of electron-electron the holon pairing order. In Figs.2(a) and (b), we show
scattering only above the Fermi energy ǫ(k ), i.e., the the SU(2) results of the s-wave symmetry of opposite
f
Fermi-surface. phase between the b and b holon pairing order param-
1 2
In order to find the symmetry of the holon pairing or- eters. Both results demonstrated excellent fits to the
derparameterwenowcomputetheeigenvectorsofthet- s-wave symmetry form of Eqs.(13) and (15). Since the
matrixwhosepoleschangetheirsignsatacritical(onset) computedresultswhichfitsEq.(14)areindistinguishable
temperature T ; the eigenvalue equations are withtheonesshowninbothFig.2andFigs.3(a)andthus
c
are not displayed. In short, both the U(1) and SU(2)
′
tk,k′(q0 =0,q=0)W(k )=λW(k), (11) theories predicted the s-wave symmetry of holon pair-
Xk′ ing order. This is equivalent to the d-wave symmetry
of hole pairing order, by noting that the hole is a com-
for the U(1) theory, where W(k) is the eigenvector and
posite of a holon(boson) of spin 0 with charge +e and a
λ, the eigenvalue and
spinon(fermion) of spin 1/2 with charge 0 and realizing
tkα,βkα′′β′(q0 =0,q=0)Wα′β′(k′)=λW(k)αβ, (12) tshpeinos-nwpavaeirihnoglo[1n]-p[a3i]ri[n5g]. in the presence of the d-wave
k′X,α′,β′ In the present study, we found that the bose conden-
for the SU(2) theory, where W (k) is the eigenvec- sation of an arch shape occurs only in a limited range
αβ
3
of hole doping concentration, by searching for the onset
temperature of the holon pairing instability. It is found
0.08
taahtpuaprtreotaihsceihnpersperodeficcibtseoedtahhgrtoehloeemnUep(na1ti)srwabnoitdshettchhoeenfSduUenn(cs2ta)itoisnolanalvtieen-mbteopgseroarn-l unit of t) 0.06 shpoinloonn ppaaoiinrriisnneggt TTT
theories. Most importantly, both of these approaches e ( 0.04 (b)
r
u
were found to satisfactorily predict the experimentally at
r
observed bose condensation (superconducting) tempera- pe 0.02
m
tureasafunctionoftheholedopingrate,byrevealingthe e
presence of the optimal doping rate in high T cuprates. T 0
c
0.1 0.2 0.3 0.4
doping rate, δ
FIG. 1. Onset temperature(denoted by (cid:4)) of the negative
energypolesrepresentingholonpairbosecondensationtemper-
atureTc asafunctionofholedopingrateforboth(a)U(1)and
(b)SU(2)slave-bosontheories.Thesymbols,(cid:13)and(cid:3)represent
respectively the spinon pairing and holon pairing temperatures
obtained from thefunctional integral approach.
W(k)
One(SHSS) of us acknowledges the generous supports 0.2
0
ofKoreaMinistryofEducation(BSRI-98and99)andthe
-0.2
Center for Molecular Science at Korea Advanced Insti-
-1
tute of Science and Technology. We thank Tae-Hyoung -0.5
Gimm, Jae-Hyun Uhm and Ki-Suk Kim for helpful dis- k / π0 1
cussions. x 0.5 0 0.5
1 -0.5 k / π
y
FIG. 2. The computed eigenvector of the U(1) t-matrix
which fitsthes-wave symmetry of holon pairing in Eq.(13).
[1] G. Kotliar and J. Liu, Phys. Rev.B 38, 5142 (1988); ref- Wo (k) (a)
erences there-in. 11
[2] a) M. U. Ubbens and P. A. Lee, Phys. Rev. B 46, 8434 0.2
(1992); b)M.U.UbbensandP.A.Lee,Phys.Rev.B49, 0
-0.2
6853 (1994); references there-in.
[3] a) X. G. Wen and P. A. Lee, Phys. Rev. Lett. 76, 503 -1
(1996); b)X.G.Wenand P.A.Lee, Phys.Rev.Lett.80, -0.5
k / π0 1
2193 (1998); references there-in. x 0.5 0 0.5
[4] T.-H. Gimm, S.-S. Lee, S.-P. Hong and Sung-Ho Suck 1 -0.5 k / π
y
Salk, Phys. Rev.B 60, 6324 (1999).
[5] S.-S. Lee and Sung-Ho Suck Salk, Int. J. Mod. Phys. in
press(cond-mat/9905268); Sung-Ho Suck Salk and S.-S.
Lee, Physica B, in press(cond-mat/9907226). Wo (k) (b)
22
0.2
0.08 0
of t) shpoinloonn ppaaiirriinngg TT -0.2
nit 0.06 onset T -1-0.5
u
ure ( 0.04 (a) kx/ π00.5 0 0.5 1
at 1 -0.5 k / π
er 0.02 y
p
m
Te 0 FIG. 3. The computed eigenvector of the SU(2) t-matrix
0.1 0.2 0.3 which fitsthe s-wavesymmetry of holon pairing in Eq.(15). (a)
doping rate, δ and (b) display thephase difference of π between theb1 and b2
holon pair order parameters.
4