Thermal conductivity in the doped two-leg ladder antiferromagnet Sr$_{14-x}$Ca$_{x}$Cu$_{24}$O$_{41}$

0.17 MB·English

by  Jihong Qin

Thermal conductivity in the doped two-leg ladder antiferromagnet Sr Ca Cu O
14−x x 24 41
Jihong Qin and Shiping Feng
Department of Physics, Beijing Normal University, Beijing 100875, China
Feng Yuan
Texas Center for Superconductivity, University of Houston, Houston, Texas 77204, U.S.A.
5
0 Wei Yeu Chen
0 Department of Physics, Tamkang University, Tamsui 25137, Taiwan
2
Withinthet-Jmodel,theheattransportofthedopedtwo-legladdermaterialSr14−xCaxCu24O41is
n studied in the doped regime where superconductivity appears under pressure in low temperatures.
a
The energy dependence of the thermal conductivity κc(ω) shows a low-energy peak, while the
J
temperature dependence of the thermal conductivity κc(T) is characterized by a broad band. In
2
particular, κc(T) increases monotonously with increasing temperature at low temperatures T <
2
0.05J, and is weakly temperature dependent in the temperature range 0.05J < T < 0.1J, then

decreases for temperatures T > 0.1J, in qualitative agreement with experiments. Our result also
]
l shows that although both dressed holons and spinons are responsible for the thermal conductivity,
e
- the contribution from thedressed spinons dominates theheat transport of Sr14−xCaxCu24O41.
r
t
s 74.25.Fy,74.62.Dh,74.72.Jt
.
t
a In recent years the in other conventional metals with charge, spin and heat
m two-leg ladder material Sr Cu O has attracted great all carried by one and the same particles.
14 24 41
- interest since its ground state may be a spin liquid state Theheattransportisoneofthebasictransportproper-
d with a finite spin gap1. This spin liquid state may play ties that provide a wealthof useful information on carri-
n a crucial role in superconductivity of doped cuprates as ersandphononsaswellastheirscatteringprocesses9–12.
o
emphasizedbyAnderson2. Whencarriersaredopedinto Intheconventionalmetals,thethermalconductivitycon-
c
[ Sr14Cu24O41,suchasthe isovalentsubstitutionofCafor tainsbothcontributionsfromcarriersandphonons9. The
Sr, a metal-insulator transition occurs1,3,4, and further, phonon contribution to the thermal conductivity is al-
2
this doped two-legladder materialSr Ca Cu O is ways present in the conventional metals, while the mag-
v 14−x x 24 41
a superconductor under pressure in low temperatures3,4. nitude of the carrier contribution depends on the type
5
Apart from the observation of superconductivity under of material because it is directly proportional to the
1
1 pressure in Sr14−xCaxCu24O41, the particular geometri- free carrier density. For the underdoped cuprates on a
6 cal arrangement of the Cu ions provides a playground square lattice, the phonon contribution to the thermal
0 for magnetic and transport studies of low-dimensional conductivity is strongly suppressed13,14. Recently, an
4 strongly correlated materials1. All cuprate supercon- unusual contribution to the thermal conductivity of the
0 ductors found up now contain square CuO planes5, doped two-leg ladder material Sr Ca Cu O has
t/ whereas Sr Ca Cu O consists of two-le2g ladders beenobserved10–12. Ithasbeenarg1u4e−dxthaxtth2i4su4n1usual
a 14−x x 24 41
of other Cu ions and edge-sharing CuO chains1,3,4, and contributionmaybeduetoanenergytransportviamag-
m 2
is the only known superconducting copper oxide with- neticexcitations10,15,andthereforecannotbe explained
-
d out a square lattice. Experimentally, it has been shown within the conventional models of phonon heat trans-
n by virtue of the nuclear magnetic resonance and nu- port based on phonon-defect scattering or conventional
o clear quadrupole resonance, particularly inelastic neu- phonon-electron scattering. In particular, in the doped
c tron scattering measurements that there is a region of regime where superconductivity appears under pressure
:
v parameter space and doping where Sr14−xCaxCu24O41 in low temperatures, this unusual contribution domi-
i in the normal state is an antiferromagnetwith commen- nates the thermal conductivity of Sr Ca Cu O .
X 14−x x 24 41
surate short-range order1,6,7. Moreover, transport mea- This is a challenge issue since it is closely related to
r
surements on Sr Ca Cu O in the same region of the doped Mott insulating state that forms the basis
a 14−x x 24 41
parameter space and doping indicate that the resistivity for superconductivity1. We16 have developed a charge-
is linear with temperatures4, one of the hallmarks of the spin separation fermion-spin theory to study the phys-
exoticnormalstatepropertiesfoundindopedcuprateson ical properties of doped Mott insulators, where the
a square lattice5. These unusual normalstate properties electron operator is decoupled as the gauge invariant
do not fit in the conventional Fermi-liquid theory8, and dressed holon and spinon. Within this theory, we17,18
may be interpreted within the framework of the charge- have discussed charge transport and spin response of
spin separation2, where the electron is separated into a Sr Ca Cu O . It has been shown that the charge
14−x x 24 41
neutral spinon and a charged holon, then the basic exci- transport is mainly governed by the scattering from the
tations of the system are not fermionic quasiparticles as dressed holons due to the dressed spinon fluctuation,
1
while the scattering from the dressed spinons due to the h h = e−iΦiσh h e−iΦi−σ = 0, are ruled out au-
iaσ ia−σ ia ia
dressedholonfluctuationdominates the spin response16. tomatically. It has been emphasized that this dressed
Inthispaper,weapplythissuccessfulapproachtodiscuss holonh isaspinlessfermionh incorporatedaspinon
iaσ ia
theheattransportofSr Ca Cu O . Withinthet-J cloud e−iΦiσ (magnetic flux), and then is a magnetic
14−x x 24 41
model, we show that although both dressed holons and dressing. In other words, the gauge invariant dressed
spinonsareresponsiblefortheheattransport,thecontri- holoncarriessomespinonmessages,i.e.,itsharesitsnon-
bution from the dressed spinons dominates the thermal trivialspinonenvironment19. Althoughincommonsense
conductivity of Sr Ca Cu O in the doped regime h is not a real spinful fermion, it behaves like a spin-
14−x x 24 41 iaσ
where superconductivity appears under pressure in low ful fermion. In this charge-spin separation fermion-spin
temperatures. representation,thelow-energybehaviorofthet-J ladder
The two-leg ladder is defined as two parallel chains (1) can be expressed as16,
of ions, with bonds among them so that the interchain
couplingiscomparableinstrengthtothecouplingsalong H =t (h†i+ηˆa↑hia↑Si+aSi−+ηˆa+h†i+ηˆa↓hia↓Si−aSi++ηˆa)
the chains, while the coupling between the two chains Xiηˆa
that participates in this structure is through rungs1. It +t (h† h S+S−+h† h S−S+)
has been argued that the essential physics of the doped i2↑ i1↑ i1 i2 i2↓ i1↓ i1 i2
X
i
two-leg ladder antiferromagnet is contained in the two-
leg ladder, and can be effectively described by the t-J +µ h†iaσhiaσ
model1, Xiaσ
+J S ·S +J S ·S , (2)
eff ia i+ηˆa eff i1 i2
H =−tk Ci†aσCi+ηˆaσ−t⊥ (Ci†1σCi2σ +Ci†2σCi1σ) Xiηˆa Xi
X X
iηˆaσ iσ
−µ C† C with Jeff = J(1 − δ)2, and δ = hh†iaσhiaσi = hh†iahiai
iaσ iaσ is the hole doping concentration. In this case, the ki-
X
iaσ
neticparthasbeenexpressedasthedressedholon-spinon
+Jk Sia·Si+ηˆa+J⊥ Si1·Si2, (1) interaction, and therefore reflect a competition between
Xiηˆa Xi the kinetic energy and magnetic energy. This compe-
tition dominates the essential physics since the dressed
supplemented by a single occupancy local constraint
holon and spinon self-energies are ascribed purely to the
σCi†aσCiaσ ≤1, where ηˆ=±c0xˆ, c0 is the lattice con- dressed holon-spinon interaction.
Pstant of the two-leg ladder lattice, which is set as unity Nowwefollowthelinearresponsetheory20,andobtain
hereafter,irunsoverallrungs,σ(=↑,↓)anda(=1,2)are the thermal conductivity along the ladder,
spinandlegindices, respectively,C† (C )is the elec-
iaσ iaσ
troncreation(annihilation)operator,Sia =Ci†a~σCia/2is κc(ω,T)=−1 ImΠQ(ω,T), (3)
T ω
the spin operator with ~σ =(σ ,σ ,σ ) as the Pauli ma-
x y z
trices, and µ is the chemical potential. In the materials with Π (ω,T) is the heat current-current correlation
Q
of interest3,6, the exchange coupling Jk along the legs is function, and is defined as,
closeto the exchangecouplingJ acrossarung,andthe
⊥
sameistrueofthehoppingtk alongthelegsandtherung ΠQ(τ −τ′)=−hTτjQ(τ)jQ(τ′)i, (4)
hopping strength t . Therefore in the following discus-
⊥ where τ and τ′ are the imaginary times and T is the
sions, we will work with the isotropic system J =J = τ
⊥ k τ order operator, while the heat current density is ob-
J, t = t = t. The strong electron correlation in the
⊥ k tained within the t-J ladder Hamiltonian (2) by using
t-J model manifests itself by the single occupancy local
Heisenberg’s equation of motion as,
constraint,andthus the crucialrequirementisto impose
thislocalconstraint. Thislocalconstraintcanbetreated j =i R [H ,H ]=j(h)+j(s), (5a)
properly within the charge-spin separation fermion-spin Q jb ia jb Q Q
XX
theory16, C = h† S−, C = h† S+, where the i,j a,b
spinful fermiiao↑n operiaat↑oriahiaσi=a↓e−iΦiσiah↓iaiadescribes the jQ(h) =−i(χkt)2 ηˆh†i+ηˆ+ηˆ′aσhiaσ
chargedegreeoffreedomtogetherwithsomeeffectsofthe iηˆXηˆ′aσ
spinonconfigurationrearrangementsdue to the presence +iχ χ t2 [(R −R −ηˆ)h† h
k ⊥ i2 i1 i+ηˆ1σ i2σ
ofthe hole itself (dressedholon),while the spinoperator X
iηˆσ
S describesthespindegreeoffreedom(dressedspinon),
ia −(R −R +ηˆ)h† h ]
then the electron local constraint for the single occu- i2 i1 i+ηˆ2σ i1σ
pancy, σCi†aσCiaσ =Si+ahia↑h†ia↑Si−a+Si−ahia↓h†ia↓Si+a = −iµχkt ηˆh†i+ηˆaσhiaσ
hiah†ia(PSi+aSi−a + Si−aSi+a) = 1 − h†iahia ≤ 1, is satis- iXηˆaσ
fied in analytical calculations, and the double spinful +iµχ t (R −R )(h† h −h† h )
fermion occupancy, h†iaσh†ia−σ = eiΦiσh†iah†iaeiΦi−σ = 0, ⊥ Xiσ i2 i1 i1σ i2σ i2σ i1σ
2
+i(χ t)2 (R −R )(h† h −h† h ), (5b) conductivityofthedopedtwo-legladderantiferromagnet
⊥ i2 i1 i2σ i2σ i1σ i1σ
Xiσ as,
1
jQ(s) =i2ǫkJe2ff (ηˆ′−ηˆ) κc(ω,T)=κc(h)(ω,T)+κ(cs)(ω,T), (6a)
iXηˆηˆ′a 1 ∞ dω′
[ ǫkSiza(Si++ηˆ′aSi−+ηˆa−Si−+ηˆ′aSi++ηˆa) κc(h)(ω,T)=−N Xkσ Z−∞ 2π
−Siz+ηˆ′a(Si+aSi−+ηˆa−Si−aSi++ηˆa) ×{Λ [A(hσ)(k,ω′+ω)A(hσ)(k,ω′)
h1 T T
+(S+S− −S−S+ )Sz ]
ia i+ηˆ′a ia i+ηˆ′a i+ηˆa +A(hσ)(k,ω′+ω)A(hσ)(k,ω′)]
1 L L
+i2Je2ffX{ηˆǫkǫ⊥[Siz1(Si++ηˆ1Si−2−Si+2Si−+ηˆ1) +Λh2[AT(hσ)(k,ω′+ω)AL(hσ)(k,ω′)
iηˆ +A(hσ)(k,ω′+ω)A(hσ)(k,ω′)]
+Sz(S+ S−−S+S− )] L T
i2 i+ηˆ2 i1 i1 i+ηˆ2 +Λ [A(hσ)(k,ω′+ω)A(hσ)(k,ω′)
+ηˆǫ [Sz(S+S− −S−S+ ) h3 T T
k i1 i2 i+ηˆ2 i2 i+ηˆ2 −A(hσ)(k,ω′+ω)A(hσ)(k,ω′)]}
+Sz(S+S− −S−S+ )] L L
i2 i1 i+ηˆ1 i1 i+ηˆ1 n (ω′+ω)−n (ω′)
+ηˆǫ [(Sz −Sz )(S+S−−S+S−)] × F F , (6b)
⊥ i+ηˆ1 i+ηˆ2 i2 i1 i1 i2 Tω
+(R −R )ǫ [Sz(S+S− −S−S+ ) 1 ∞ dω′
i2 i1 k i1 i2 i+ηˆ2 i2 i+ηˆ2 κ(s)(ω,T)=−
−Siz2(Si+1Si−+ηˆ1−Si−1Si++ηˆ1)] c 2N Xk Z−∞ 2π
+2(Ri2−Ri1)ǫkǫ⊥[Siz1Si+2Si−+ηˆ1−Siz2Si+1Si−+ηˆ2] ×{Λs1[A(Ls)(k,ω′+ω)A(Ls)(k,ω′)
+2(Ri2−Ri1)ǫ⊥[Siz+ηˆ2Si+1Si−2−Siz+ηˆ1Si+2Si−1]} +A(Ts)(k,ω′+ω)A(Ts)(k,ω′)]
+i41ǫ⊥2Je2ff (Ri2−Ri1)(Si+1Si−1−Si+2Si−2), (5c) +Λs2[A(Ls)(k,ω′+ω)A(Ts)(k,ω′)
Xi +A(s)(k,ω′+ω)A(s)(k,ω′)]
T L
where Ri1 and Ri2 are lattice sites of leg 1 and leg 2, +Λs3[A(Ls)(k,ω′+ω)A(Ls)(k,ω′)
respectively, ǫ = 1+2tφ /J , ǫ = 1+4tφ /J , the
dressed spinonk correlatiokn fueffncti⊥ons χ = h⊥S+Seff− i, −A(Ts)(k,ω′+ω)A(Ts)(k,ω′)]}
χ⊥ = hS1+iS2−ii, and the dressed holonkparticlea-ihoalei+oηˆr- × nB(ω′+ω)−nB(ω′), (6c)
der parameters φ = hh† h i, φ = hh† h i. Tω
k aiσ ai+ηˆσ ⊥ 1iσ 2iσ
Although the total heat current density jQ has been where κ(h)(ω,T) and κ(s)(ω,T) are the corresponding
separated into two parts jQ(h) and jQ(s), with jQ(h) is the contribuctions from dressced holons and spinons, respec-
dressedholonheatcurrentdensity,andj(s) isthedressed tively, N is the number of rungs, and
Q
spinon heat current density, the strong correlation be-
Λ =(Zχ t)2γ2 [(Zχ tγ +µ)2+(χ t)2], (7a)
tween dressed holons and spinons is still included self- h1 k sk k k ⊥
consistently through the dressed spinon’s order parame- Λ =2(Zχ t)2γ2 (Zχ tγ +µ)(χ t), (7b)
h2 k sk k k ⊥
ters entering in the dressed holon’s propagator, and the
Λ =−(χ t)2[(Zχ tγ +µ)2−(χ t)2], (7c)
dressed holon’s order parameters entering in the dressed h3 ⊥ k k ⊥
spinon’s propagator. In this case, the heat current- Λ =2γ2 J4 Z2{[2Zǫ (ǫ χ +C −2χ γ )
s1 sk eff k k k k k k
currentcorrelationfunction of the two-legladder system
+ǫ (ǫ χ +C )]2+(ǫ χ +ǫ χ )2}, (7d)
can be obtained in terms of the full dressed holon and ⊥ k ⊥ ⊥ k ⊥ ⊥ k
spinon Green’s functions gσ(k,ω) and D(k,ω). Because Λs2 =−4γs2kJe4ffZ2
there are two coupled t-J chains in the two-leg ladder { 2Zǫ (ǫ χ +ǫ χ )(ǫ χ +C −2χ γ )
k k ⊥ ⊥ k k k k k k
system, the energy spectrum has two branches. There-
+ǫ (ǫ χ +C )(ǫ χ +ǫ χ )}, (7e)
⊥ k ⊥ ⊥ k ⊥ ⊥ k
fore the one-particle dressed holon and spinon Green’s
1
jfu,τnc−tioτn′)s=arge m(ait−ricje,sτa−ndτ′c)a+nσbegex(pir−essje,dτ −asτg′)σ(ain−d Λs3 =−2Je4ff{[4ǫ2⊥
Lσ x Tσ
D(i−j,τ−τ′)=DL(i−j,τ −τ′)+σxDT(i−j,τ −τ′), +Zǫ⊥((ǫkχ⊥−C⊥)γk+(ǫkC⊥−χ⊥))]2
respectively,where the longitudinaland transverseparts −Z2[(ǫ χ +ǫ χ )γ −ǫ (ǫ χ +C )]2}, (7f)
are defined as g (i−j,τ −τ′)=−hT h (τ)h† (τ′)i, k ⊥ ⊥ k k k ⊥ k ⊥
Lσ τ iaσ jaσ
D (i − j,τ − τ′) = −hT S+(τ)S−(τ′)i and g (i − with γ2 = sin2k , γ = cosk , Z is the coor-
L τ ia ja Tσ sk x k x
j,τ −τ′) = −hT h (τ)h† (τ′)i, D (i−j,τ −τ′) = dination number within the leg, the dressed spinon
τ iaσ ja′σ T correlation functions C = (1/Z2) hS+ S− i,
−siohTnsτSofi+at(hτe)Schj−aa′r(gτe′)tir,anwsipthorat1′76=,wae.caFnololobwtainingtthheethdeisrcmuas-l C⊥ = (1/Z) ηˆhS2+iSk1−i+ηˆi, nF(ωP) ηˆaηˆn′d ani+Bηˆ(ωa)i+ηaˆ′re
P
3
the fermion and boson distribution functions, respec- +[Zγ +(−1)ν′+ν′′]2}
q−k
tively, and the longitudinal and transverse spectral Γ(1) (k,p,q)
functions of the dressed holon and spinon are ob- ×( νν′ν′′
tained as A(Lhσ)(k,ω) = −2ImgLσ(k,ω), AL(s)(k,ω) = ω+ξp(ν+′q) −ξq(ν)−ωk(ν+′′p)
−A(2sI)m(kD,ωL)(k,ω=) an−d2ImAD(Thσ()k(k,,ωω),) r=espe−ct2ivIemlyg,Tσ(wkh,ωer)e, − Γ(ν(ν2′ν))′ν′′(k(,νp),q) (ν′′)), (9b)
thTe full dressed holonT and spinon Green’s func- ω+ξp+q −ξq +ωk+p
tions have been obtained in Refs.17,18, and are ex- withB(ν) =B −J [χ +2χz(−1)ν][ǫ +(−1)ν],B =
pressed as, gσ−1(k,ω) = gσ(0)−1(k,ω) − Σ(h)(k,ω) and λ[(2ǫ χkz +χ )kγ −e(ffǫ χ⊥ +2χ⊥z)], λ=2⊥ZJ , and k
D−1(k,ω) = D(0)−1(k,ω) − Σ(s)(k,ω), with the lon- k k k k k k k eff
gspitiundoinnaGl raenedn’strfaunnscvteirosnesmge(0a)n(-kfi,eωld) d=res1s/e2d hol1o/n(ωan−d Fν(ν1)′ν′′ ( k,p,q)=nF(ξp(ν+′k′))[nB(ωq(ν))−nB(ωq(ν+′p))]
Lσ ν
ξ(ν)), D(0)(k,ω) = 1/2 B(ν)/(ω2−(ωP(ν))2) and +n (ω(ν′))[1+n (ω(ν))] (10a)
k L ν k k B q+p B q
g1T/(02σ)(k,ω(−) 1=)ν+11/B2(Pν)/ν((ω−21−)ν+(Pω1/(ν()ω)2−), ξwk(hν)e)r,eDνT(=0)(1k,,2ω,)an=d Fν(ν2)′ν′′ ( k,p,q)=nF(ξp(ν+′k′))[nB(ωq(ν+′p))−nB(ωq(ν))]
thePloνngitudinalkand transvekrse second-order dressed +nB(ωq(ν))[1+nB(ωq(ν+′p))] (10b)
holon and spinon self-energies are obtained by the loop F(3) ( k,p,q)=n (ξ(ν′′))[1+n (ω(ν′))
expansion to the second-order17,18 as, νν′ν′′ F p+k B q+p
+n (ω(ν))]+n (ω(ν))n (ω(ν′)) (10c)
t B q B q B q+p
Σ(Lh)(k,ω)=(N)2Xpq νXν′ν′′Ξ(νhν)′ν′′(k,p,q,ω), (8a) Fν(ν4)′ν′′ ( k,p,q)=[1+nB(ωq(ν))][1+nB(ωq(ν+′p))]
t −n (ξ(ν′′))[1+n (ω(ν′))+n (ω(ν))] (10d)
Σ(h)(k,ω)=( )2 F p+k B q+p B q
T N Xpq νXν′ν′′ Γ(ν1ν)′ν′′ ( k,p,q)=nF(ξp(ν+′q))[1−nF(ξq(ν))]
(−1)ν+ν′+ν′′+1Ξ(νhν)′ν′′(k,p,q,ω), (8b) −nB(ωk(ν+′′p))[nF(ξq(ν))−nF(ξp(ν+′q))] (10e)
t
ΣΣ(L(ss))((kk,,ωω))==((Nt ))22Xpq νXν′ν′′Ξ(νsν)′ν′′(k,p,q,ω), (8c) Γ(ν2ν)′ν′′ +( k[1,+p,qn)B=(ωnk(ν+F′′p()ξ)p(]ν[+n′q)F)[(1ξq(−ν)n)F−(ξnq(Fν)()ξ]p(ν+′q))], (10f)
T N
Xpq νXν′ν′′ andthemean-fielddressedholonandspinonexcitations,
(−1)ν+ν′+ν′′+1Ξ(s) (k,p,q,ω), (8d)
νν′ν′′ ξ(ν) =Ztχ γ +µ+χ t(−1)ν+1, (11a)
k k k ⊥
respectively, where Ξ(νhν)′ν′′(k,p,q,ω) and ω(ν)2 =αǫ λ2(1χ +ǫ χz)γ2−ǫ λ2[1α(1ǫ χ
Ξ(s) (k,p,q,ω) are given by, k k 2 k k k k k 2 2 k k
νν′ν′′ 1 1
+χz)+α(Cz + C )+ (1−α)]γ
Ξ(h) (k,p,q,ω)= Bq(ν+′p)Bq(ν) 1{[Zγ +(−1)ν+ν′′]2 1k k 2 k 4 k
νν′ν′′ 32ωq(ν+′p)ωq(ν)2 q+p+k − 2αǫ⊥λJeff(C⊥+ǫkχ⊥)γk+αλJeff(−1)ν+1
+[Zγq−k+(−1)ν′+ν′′]2} [ 12(ǫ⊥χk+ǫkχ⊥)+ǫkǫ⊥(χz⊥+χzk)]γk
×(ω+ωF(νν(ν1′))′ν−′′(ωk(,νp),−q)ξ(ν′′) −αǫkλJeff(C⊥z +χz⊥)γk+λ2[α(Ckz + 21ǫ2kCk)
q+p q p+k
1 1 1
+ Fν(ν2)′ν′′(k,p,q) + 8(1−α)(1+ǫ2k)− 2αǫk(2χk+ǫkχzk)]
ω−ω(ν′) +ω(ν)−ξ(ν′′) 1
q+p q p+k +αλJ [ǫ ǫ C +2Cz]+ J2 (ǫ2 +1)
F(3) (k,p,q) eff k ⊥ ⊥ ⊥ 4 eff ⊥
+ ω+ωq(νν+ν′p)′ν+′′ ωq(ν)−ξp(ν+′k′) − 12ǫ⊥Je2ff(−1)ν+1−αλJeff(−1)ν+1[21ǫkǫ⊥χk
1
F(4) (k,p,q) +ǫ (χz +Cz)+ ǫ C ], (11b)
+ νν′ν′′ ), (9a) ⊥ k ⊥ 2 k ⊥
ω−ω(ν′) −ω(ν)−ξ(ν′′)
q+p q p+k
where the
Ξ(s) (k,p,q,ω)= Bk(ν+′′p) {[Zγ +(−1)ν+ν′′]2 dressed spinon correlation functions χzk = hSaziSazi+ηˆi,
νν′ν′′ 16ωk(ν+′′p) q+p+k χz⊥ = hS1ziS2zii, Ckz = (1/Z2) ηˆηˆ′hSazi+ηˆSazi+ηˆ′i, and
P
4
Cz = (1/Z) hSzSz i. In order to satisfy the sum as J = J ≈ 90 meV ≈ 1000K. Therefore in the fol-
⊥ ηˆ 1i 2i+ηˆ k
rule for the Pcorrelation function hS+S−i = 1/2 in the lowing discussions, we focus on this doped regime. In
ai ai
absence of the antiferromagnetic long-range-order,a de- Fig. 1,wepresenttheresultsofthethermalconductivity
coupling parameter α has been introduced in the mean- κc(ω) in Eq. (6a) as a function of frequency at δ = 0.16
field calculation, which can be regarded as the vertex (solidline) andδ =0.20(dashedline) for t/J =2.5 with
correction21. All the above mean-field order parameters, T =0.05J. Althoughκc(ω)isnotobservablefromexper-
decoupling parameter α, and chemical potential µ, have iments, its features will have observable implications on
been determined by the self-consistent calculation. theobservableκc(T). OurresultsinFig. 1showthatthe
In the two-leg ladder material Sr Ca Cu O , frequency dependence of the thermal conductivity spec-
14−x x 24 41
the most remarkable expression of the nonconventional trumischaracterizedby arathersharplow-energypeak.
physicsisfoundinthe dopedregimewheretheholedop- Thepositionofthislow-energypeakisdopingdependent,
ingconcentrationonladdersatx=8∼12iscorrespond- and is located at a finite energy ω ∼ 0.1t = 0.25J ≈ 23
ing to δ = 0.16∼ 0.20 per Cu ladder1,3,4. In this doped meV. This low-energypeak is correspondingto the peak
regime, superconductivity appears under pressure in low observed in the conductivity spectrum22,17. Moreover,
temperatures, and the value of J has been estimated6 we also find from the above calculations that although
k
both dressed holons and spinons are responsible for the
thermal conductivity κ (ω), the contribution from the
c
dressedspinonsismuchlargerthanthatfromthedressed
2500 holons,i.e.,κ(s)(ω)≫κ(h)(ω),andthereforethethermal
c c
conductivityofthedopedtwo-legladderantiferromagnet
) 2000 in the hole doped regime δ = 0.16 ∼ 0.20 is mainly de-
s
nit termined by its dressed spinon part κ(cs)(ω).
u
1500 Now we turn to discuss the temperature dependence

.
b of the thermal conductivity κ (T), which is observable
r c
a
from experiments and can be obtained from Eq. (6a) as
( 1000
ω) κc(T) = limω→0κc(ω,T). The results of κc(T) at δ =
(c 0.16 (solid line) and δ =0.20 (dashed line) for t/J =2.5
κ 500
areplotted in Fig. 2 in comparisonwith the correspond-
ing experimental results10 taken on Sr Ca Cu O
14−x x 24 41
0 (inset), where the hole doping concentration on ladders
0.0 0.5 1.0 1.5 at x=12 is δ =0.20 per Cu ladder. The present results
ω/t indicatethatthetemperaturedependenceofthethermal
conductivity in the normal-state exhibits a broad band,
FIG.1. Thethermalconductivityofthedopedtwo-leglad-
i.e., κ (T) increases monotonously with increasing tem-
c
der material as a function of frequency at doping δ = 0.16
perature at low temperatures T < 0.05J ≈ 50K, and is
(solidline)andδ=0.20(dashedline)forparametert/J =2.5
weaklytemperaturedependent inthe temperature range
with temperature T =0.05J.
0.05J ≈50K<T <0.1J ≈100K,then decreases for tem-
peratures T > 0.1J ≈ 100K, in qualitative agreement
withtheexperimentaldata10. ForT ≤0.01J ≈10K,the
system is a superconductor under pressure. In this case,
) 105
s thethermalconductivityofthedopedtwo-legladderan-
t
ni tiferromagnet is under investigation now.
u
In the above discussions, the central concern of the
b. 104
r heat transport in the doped two-leg ladder material
a
( Sr14−xCaxCu24O41 is the charge-spin separation, then
T) the contributionfromthe dressedspinons dominates the
κ(c 103 thermal conductivity. Since κ(cs)(ω,T) in Eq. (6c) is
obtained in terms of the dressed spinon longitudinal
102 andtransverseGreen’sfunctionsDL(k,ω)andDT(k,ω),
while these dressed spinon Green’s functions are eval-
0.01 0.02 0.10 0.20
uated by considering the second-order correction due to
T/J thedressedholonpairbubble,thentheobservedunusual
FIG. 2. The thermal conductivity of the doped two-leg frequency and temperature dependence of the thermal
ladder material as a function of temperature at doping conductivity spectrum of the doped two-leg ladder anti-
δ = 0.16 (solid line) and δ = 0.20 (dashed line) for pa- ferromagnet is closely related to the commensurate spin
rameter t/J = 2.5. Inset: the experimental result on response4,18. The dynamical spin structure factor of the
Sr14−xCaxCu24O41 with x=12 taken from Ref.10. doped two-leg ladder antiferromagnethas been obtained
5
within the charge-spin separation fermion-spin theory18 dence of the thermal conductivity κ (T) is characterized
c
intermsofthefulldressedspinonlongitudinalandtrans- by a broad band. κ (T) increases monotonously with
c
verse Green’s functions as, increasing temperature at low temperatures T < 0.05J,
and is weakly temperature dependent in the tempera-
S(k,ω)=−2[1+n (ω)][2ImD (k,ω)+2ImD (k,ω)]
B L T ture range 0.05J < T < 0.1J, then decreases for tem-
=[1+n (ω)][A(s)(k,ω)+A(s)(k,ω)] peratures T > 0.1J. Our result of the temperature de-
B L T
pendence of the thermal conductivity is in qualitative
4[1+n (ω)](B(1))2ImΣ(s)(k,ω)
=− B k LT , (12a) agreement with the major experimental observations of
[W(k,ω)]2+[Bk(1)ImΣ(LsT)(k,ω)]2 Sr14−xCaxCu24O41inthenormal-state10. Ourresultalso
showsthatalthoughbothdressedholonsandspinonsare
W(k,ω)=ω2−(ω(1))2−B(1)ReΣ(s)(k,ω), (12b)
k k LT responsibleforthethermalconductivity,thecontribution
from the dressed spinons dominates the heat transport.
where ImΣ(s)(k,ω) = ImΣ(s)(k,ω) + ImΣ(s)(k,ω),
LT L T
(s) (s) (s)
ReΣ (k,ω) = ReΣ (k,ω) + ReΣ (k,ω), while
LT L T
ImΣ(s)(k,ω) (ImΣ(s)(k,ω)) ACKNOWLEDGMENTS
L T
and ReΣ(s)(k,ω) (ReΣ(s)(k,ω)) are the imaginary and
L T
real parts of the second order longitudinal (transverse) The authors would like to thank Dr. Ying Liang, Dr.
spinon self-energy in Eqs. (8c) and (8d). The renormal- Tianxing Ma, and Dr. Yun Song for the helpful discus-
ized spin excitation E2 = (ω(1))2 + B(1)ReΣ(s)(k,E ) sions. This workwas supported by the NationalNatural
k k k LT k ScienceFoundationofChinaunderGrantNos. 10125415
in S(k,ω) is doping, energy, and interchain coupling de-
and90403005,theGrantfromBeijingNormalUniversity,
pendent. In the two-leg ladder system, the quantum in-
and the National Science Council.
terference effect between the chains manifests itself by
the interchain coupling, i.e., the quantum interference
increases with increasing the strength of the interchain
coupling. In the materials of interest, J ∼ J and
⊥ k
t ∼ t , we have shown in detail in Ref.18, the dynam-
⊥ k
ical spin structure factor in Eq. (12) has a well-defined
resonancecharacter,whereS(k,ω)exhibitsthecommen- 1See, e.g., E. Dagotto and T.M. Rice, Science 271, 618
suratepeakwhentheincomingneutronenergyωisequal (1996); E. Dagotto, Rep.Prog. Phys. 62, 1525 (1999).
to the renormalized spin excitation Ek, i.e., W(kc,ω) ≡ 2P.W.Anderson,inFrontiersandBorderlinesinManyPar-
[ω2−(ω(1))2−B(1)ReΣ(s)(k ,ω)]2 =(ω2−E2 )2 ∼0 at ticle Physics, edited by R.A. Broglia and J.R. Schrieffer
antiferrokmcagnetickcwaveLvTectocrs k , then thekcheight of (North-Holland, Amsterdam, 1987), p. 1; Science 235,
AF
this commensurate peak is determined by the imaginary 1196 (1987).
partofthe spinonself-energy1/ImΣ(s)(k ,ω). Since the 3M.Uehara,T.Nagata,J.Akimitsu,H.Takahashi,N.Mori,
incoming neutron resonance energyLωT =cE is finite18, and K. Kinoshita, J. Phys.Soc. Jpn. 65, 2764 (1996).
r k 4T.Nagata,M.Uehara,J.Goto,J.Akimitsu,N.Motoyama,
then there is a spin gap in the system. This spin gap
H. Eisaki, S. Uchida, H. Takahashi, T. Nakanishi, and N.
leadstothatthelow-energypeakinκ (ω)islocatedata
c Mori, Phys.Rev.Lett. 81, 1090 (1998).
finite energy ωpeak ≈ 23meV. This anticipated spin gap 5M.A.Kastner, R.J. Birgeneau, G. Shiran, Y. Endoh, Rev.
∆ ≈23meVis nottoofarfromthespingap≈32meV
S Mod. Phys. 70, 897 (1998).
observed6 in Sr14−xCaxCu24O41. Therefore the com- 6S.Katano,T.Nagata,J.Akimitsu,M.Nishi,andK.Kaku-
mensurate spin fluctuation of the doped two-leg ladder rai,Phys.Rev.Lett.82,636(1999); M.Matsuda,K.Kat-
material Sr Ca Cu O is responsible for the heat
14−x x 24 41 sumata, H. Eisaki, N. Motoyama, and S. Uchida, Phys.
transport,in other words,the energy transportvia mag- Rev.B 54, 12199 (1996).
netic excitations dominates the thermal conductivity10. 7K. Magishi, S. Matsumoto, Y. Kitaoka, K. Ishida, K.
Based on the simple theoretical estimate11, the large Asayama, M. Uehara, T. Nagata, and J. Akimitsu, Phys.
thermal conductivity observed in the two-leg ladder ma- Rev. B 57, 11533 (1999); S. Ohsugi, K. Magishi, S. Mat-
terial Sr14Cu24O41 has been interpreted in terms of the sumoto, Y. Kitaoka, T. Nagata, and J. Akimitsu, Phys.
magneticexcitationsinthesystem11. Ourpresentexpla- Rev.Lett. 82, 4715 (1999).
nation is also consistent with theirs11. 8P.W. Anderson, The Theory of Superconductivity in the
To conclude we have discussed the heat transport of High-Tc Cuprates (Princeton, New Jersey, Princeton Uni-
the two-leg ladder material Sr Ca Cu O within versity press, 1997).
14−x x 24 41
the t-J ladder in the doped regime where superconduc- 9C.Uher,inPhysicalPropertiesofHighTemperatureSuper-
tivity appears under pressure in low temperatures. It is conductors III,edited byD.M.Ginsberg (WorldScientific,
shown that the energy dependence of the thermal con- Singapore, 1992), p.159.
ductivity spectrum κ (ω) shows a low-energy peak, and 10A.V.Sologubenko,K.Giann´o,H.R.Ott,U.Ammerahl,and
c A.Revcolevschi, Phys. Rev.Lett. 84, 2714 (2000).
thepositionofthislow-energypeakisdopingdependent,
11C. Hess, C. Baumann, U. Ammerahl, B. Bu¨chner, F.
andislocatedatafiniteenergy. Thetemperaturedepen-
6
Heidrich-Meisner, W. Brenig, and A. Revcolevschi, Phys.
Rev. B 64, 184305 (2001); F. Heidrich-Meisner, A. Ho-
necker, D.C. Cabra, and W. Brenig, Phys. Rev. B66,
140406 (2002).
12K. Kudo, S. Ishikawa, T. Noji, T. Adachi, Y. Koike, K.
Maki, S. Tsuji, and K. Kumagai, J. Phys. Soc. Jpn. 70,
437 (2001).
13Y. Nakamura, S. Uchida, T. Kimura, N. Motohira, K.
Kishio, K. Kitazawa, T. Arima, and Y. Tokura, Physica
C 185-189, 1409 (1991).
14O. Baberski, A. Lang, O. Maldonado, M. Hu¨cker, B.
Bu¨chner,andA.Freimuth,Europhys.Lett.44,335(1998).
15J.V. Alvarez and Claudius Gros, Phys. Rev. Lett. 89,
156603 (2002); X. Zotos, Phys. Rev. Lett. 92, 067202
(2004).
16ShipingFeng,JihongQin,andTianxingMa,J.Phys.: Con-
dens. Matter 16, 343 (2004); Shiping Feng, Tianxing Ma,
andJihongQin,Mod.Phys.Lett.B17,361(2003);Shiping
Feng, Z.B. Su, and L. Yu,Phys.Rev.B 49, 2368 (1994).
17Jihong Qin, Yun Song, Shiping Feng, and Wei Yue Chen,
Phys.Rev. B 65, 155117 (2002).
18Jianhui He, Shiping Feng, and Wei Yue Chen, Phys. Rev.
B 67, 094402 (2003).
19G.B. Martins, R. Eder, and E. Dagotto, Phys. Rev. B60,
R3716 (1999); G.B. Martins, J.C. Xavier, C. Gazza, M.
Vojta, and E. Dagotto, Phys.Rev. B63, 014414 (2000).
20G.D. Mahan, Many-Particle Physics (Plenum Press, New
York,1990).
21ShipingFengand YunSong,Phys.Rev.B 55, 642 (1997).
22T. Osafune, N. Motoyama, H. Eisaki, S. Uchida, and S.
Tajima, Phys. Rev. Lett. 82, 1313 (1999); T. Osafune, N.
Motoyama, H.Eisaki, andS.Uchida,Phys.Rev.Lett. 78,
1980 (1997).
7