Superconductivity in zigzag CuO chains

0.34 MB·English

by  Erez Berg

Superconductivity in zigzag CuO chains
Erez Berg1,3, Theodore H. Geballe2, and Steven A. Kivelson1
∗
1 Department of Physics, Stanford University, Stanford CA 94305-4045, USA
2 Department of Applied Physics, Stanford University, Stanford, California 94305-4045, USA
3 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, 76100, Israel
(Dated: February 1, 2008)
8
0 SuperconductivityhasrecentlybeendiscoveredinPr2Ba4Cu7O15−δ withamaximumTc ofabout
0
15K. Since the CuO planes in this material are believed to be insulating, it has been proposed
2
thatthesuperconductivityoccursin thedouble(orzigzag) CuOchainlayer. Onphenomenological

n groundsweproposeatheoretical interpretationoftheexperimentalresultsintermsofanewphase
a for the zigzag chain, labelled by C1S3. This phase has a gap in the relative charge mode and
J 2
a partial gap in the relative spin mode. It has gapless uniform charge and spin excitations and

9 can have a divergent superconducting susceptibility, even for repulsive interactions. A microscopic
2 model for the zigzag CuO chain is proposed, and on the basis of density matrix renormalization
group (DMRG) and bosonization studies, we adduce evidencethat supports our proposal.

]
n
o
Introduction - The discovery of high-T superconduc- suggests that the copper-oxide planes in these materials
c c
- tivityhasraisedthepossibilityofnewmechanismsofsu- are insulating, and that most of the conductivity occurs
r
perconductivity in strongly correlated electron systems, in the metallic copper-oxide chain layers. Since the dou-
p
u which are radically different from the well established ble (or zigzag) chains are much more structurally robust
s BCS-Eliashberg mechanism. However, the lack of the- thanthesinglechains,theyaremuchlessdisorderedand
.
t oretical techniques for obtaining well controlled solu- therefore are expected to be better conductors.
a
tions of even the simplest models of strongly correlated AssumingthattheelectricalconductivityinPr-247(as
m
fermions in dimensions greater than one has seriously inPr-248)comesmostlyfromthedoublechainsitfollows
- limited the understanding of such mechanisms. thatthesuperconductivitymustoriginateinthesechains
d
n An important exception is one dimensional systems, as well. While this assumption remains to be tested, ex-
o where powerful analytical and numerical techniques are istingNQRexperiments5whichmeasurethesite-resolved
c available. A one dimensional quantum system cannot spin-lattice relaxation rate (1/T ) have shown that only
1
[
havesuperconductinglong-rangeorderevenatzerotem- the double chain copper nuclear spins show any sharp

3 perature, as a consequence of the Mermin-Wagner theo- feature(a cusp)intheir 1/T1(T)curvesatthe supercon-
v rem. Nevertheless, such a system can have a supercon- ductingtransitiontemperature,supportingtheideathat
8 ducting susceptibility which diverges as T 0. There- the superconductivity is intimately relatedto the double
8 →
fore, an array of coupled one dimensional systems can chains. This possibility can have important implications
8
become a “bulk” superconductor with a reasonably high inothermaterialswhichalsosharethe samezigzagCuO
3
T eveninthelimitwheretheinter-chaincouplingisvery chain structure, such as YBCO-247, YBCO-248 and the
. c
8 weak. This mechanism of superconductivity has been ladder compounds.
0
studied extensively1. It serves as one of the only well- Thepurposeofthis paperisto studythe possibilityof
7
established proofs of principle for superconductivity in a superconductivity in zigzag CuO chains. This problem
0
: model with purely repulsive interactions. has been addressed in several previous papers9,10,11. In
v
Among the candidates for an experimental realiza- Ref. [9], the CuO zigzag chain was treated in the weak
i
X tion for this mechanism are the organic conductors and coupling limit and in the Hartree-Fock approximation.
r the ladder compound Sr14 xCaxCu24O41. Recently, su- Ref. [10] studied a zigzagHubbard ladder using DMRG.
a perconductivity wasdiscov−eredin Pr Ba Cu O (Pr- In Ref. [11] weak coupling fluctuation exchange (FLEX)
2 4 7 15 δ
247) in a certain region of oxygen reduction δ,−with a theorywasusedtostudysuperconductivityinthezigzag
maximumT ofabout15K2,3,4. Thismaterialisisostruc- chains.
c
tural with Y Ba Cu O (YBCO-247). However, the In this paper, we start from the experimental data,
2 4 7 15 δ
two materials display a d−ramatic difference in their elec- and analyze the experimental constraints on the zigzag
tronic behavior: the copper-oxide planes in YBCO are chain superconductivity scenario. Then, we present a
conducting, and are believed to play a crucial role in microscopic model for a single zigzag chain, which con-
the high-T superconductivity in this material. In Pr- tains in our view an important piece of physics that has
c
247(aswellasinthecloselyrelatedmaterialPr-248)the been omitted in previous studies, namely the oxygen p
y
copper-oxide planes remain antiferromagnetic with large orbitals and the ferromagnetic Hund’s rule coupling on
moments even upon doping. To the best of our knowl- the oxygen sites.30 The model is studied using numeri-
edge, the conductivity of a single crystal of Pr-247 has cal density matrix renormalization group (DMRG) cal-
not been measured yet. In Pr-248, the conductivity is culations,aswellasanalyticrenormalizationgroup(RG)
strongly anisortopicin the a b plane (with σ /σ of up andstrong coupling methods. The solutionof the model
a b
−
to1000,whereaisthechaindirection6,7,8). Thisstrongly shows features that are in agreement with experiment,
2
namely the absence (or extremely small magnitude) of An important observation is that in this scenario, the
the spin gap and a tendency towards superconductivity. spingap∆ mustbesignificantlylargerthanT . Attem-
s c
peratures higher than ∆ , the spin modes are essentially
s
gapless,andtheexponentscontrollingthesusceptibilities
I. POSSIBLE PHASES OF THE ZIGZAG CHAIN wouldcrossoverto newexponentsofagaplessphase. In
the regime T < T < ∆ “pseudogap” behavior should
c s
beobserved: agapappearsinthespectrum,butthisgap
Our main assumption is that the zigzag CuO chains
is not associated with any type of long-range order.
in Pr-247 are weakly coupled, so the basic properties of
However,NQRmeasurementsofthespin-latticerelax-
the system at T > T can be understood in terms of
c ationrate1/T aboveT shownoevidenceofgapped(i.e.
the properties of decoupled chains. Let us review briefly 1 c
thermally activated) behavior4,5. Assuming that the re-
a mechanism of superconductivity in weakly coupled 1d
laxationprocessismostlyduetolocalmagneticfieldfluc-
systems. A 1d system typically has severaltypes of fluc-
tuations(ratherthanelectricfieldgradientfluctuations),
tuating order which coexist with each other (for exam-
thisimpliesthatthereisnospingapinthezigzagchains.
ple, superconducting and spin or charge density wave
The 1/T signal seems to follow a non-trivial power law
orders). Even though none of these can truly become 1
as a function of temperature, which implies a power law
long-range ordered in the isolated 1d system at generic
decay of spin-spin correlations. For a discussion of the
(incommensurate)filling,thesusceptibilityofthesystem
interpretation of 1/T NMR measurements, see [27].
to these types for order can become large at low tem- 1
The preceding analysis leads us to the phenomenolog-
perature. Then, treating the inter-chain coupling at the
ical idea that the zigzag chain is in a phase with gapless
mean-field level, the critical temperature is determined
spin excitations. Two uncoupled chains are in a C S
bytheStonercriterion: Jχ(T )=1,whereJ istheinter- 2 2
c phase. Inter-chain interactions can gap some of these
chain coupling and χ(T) is the susceptibility to the type
modes. For example, we can consider a C S or a C S
of order considered. Assuming that the inter-chain cou- 1 2 1 1
phase. Inthesecases,thespin-spincorrelationsbehaveas
pling constants of the various types of order are all of
a power law (since the total spin mode is gapless). How-
comparable size, the type of order that is most likely to
ever, some of the relative spin and charge modes (the
be selected is the one which has the largest (i.e. most
modes that correspond to fluctuations of the charge of
divergent) susceptibility at low temperature.
the two chains relative to each other) are gapped, which
A single-component 1d electron system with repul-
can enhance superconducting (as well as other) suscep-
sive interactions can generically be described at low en-
tibilities. In the rest of this paper, we will advocate an
ergies as a Luttinger liquid with one gapless spin and
explanation of superconductivity in Pr-247 in terms of
one gapless charge mode. Such a system is usually a
such a phase, whose existence will be established in a
poor superconductor, with a superconducting suscepti-
microscopic model for the zigzag chain.
bility χSC ∼T1/Kc−1, where Kc is the charge Luttinger Themostlikelycandidatephasethatemergesfromour
parameter. Since typically K < 1, the superconducting
c analysis is what we will call a C S phase, in which the
susceptibility is non-divergentwhile the CDW andSDW 1 32
relativechargemodeand“half”oftherelativespinmode
susceptibilities both diverge.
are gapped. (We give a more precise definition for these
Multi-component 1d systems are more promising in
terms in Appendix A.) The resulting susceptibilities in
this respect. The problem of two coupled Luttinger liq-
this phase are
uidshasbeenstudiedextensively,anditisfoundtobein
a“Luther-Emery”phaseoverawiderangeofparameters. χSC T1/2K+,c−45 (1)
In this phase, only the total charge mode remains gap- ∼
less while all other spin and charge modes are gapped. χCDW,4kF ∼ T2K+,c−2 (2)
It is therefore similar to the phase of a single compo- χCDW,2kF ∼ χSDW,2kF ∼TK+,c/2−45 (3)
nent system with a spin gap caused by attractive inter-
actions, although the identity of the physical correlation Notethatthesuperconductingsusceptibilityisdivergent
functions in the two systems is somewhat different. If for K+,c >0.4, and is the dominant one when K+,c >1.
we label the 1d phases as CnSm, where n and m are the We comment that the proposed C1S23 phase is likely
numbersofgaplesschargeandspinmodesrespectively13, not to be a true T = 0 phase: it will be ultimately un-
the Luther-Emery phase is C1S0. Since the entire spin stable to formation of a C1S0 phase. Nevertheless, if the
sector is gapped, the dominant fluctuations in this sys- gap in the total spin sector is small compared to Tc, the
temaresuperconductingfluctuationsand4kF CDWfluc- physics above Tc is best described in terms of a C1S3.
2
tuations, with susceptibilities χSC T1/2K+,c−2 and Such a hierarchy of gaps is found to emerges in the mi-
∼
χCDW,4kF ∼ T2K+,c−2 respectively. (Here, K+,c is the croscopic model that will be studied here (see Section
total charge mode Luttinger parameter.) Near half fill- IVA).
ing, it has been shown15 that K 1 and therefore Finally,the Luttinger exponentK thatcontrolsthe
+,c +,c
∼
superconducting fluctuations always dominate. The in- low-energy properties of the system depends on micro-
terpretation of superconductivity in Pr-247 as deriving scopic details, and is therefore difficult to estimate theo-
from a Luther-Emery phase was proposed in Ref. [9]. retically. However, since the power with which the spin
3
into account, H is simply
0
H0 = tpdiX,α,σPˆ(cid:16)d†i,σ,αpx,i+1,σ,α−p†x,i,σ,αdi,σ,α+h.c.(cid:17)Pˆ
+ ε p†x,i,σ,αpx,i,σ,α (5)
iX,α,σ
Here d and p are hole annihilation operators
i,σ,α x,i,σ,α
in 3d Cu and 2p O orbitals respectively, Pˆ is a pro-
x
jection operator that imposes the no-double occupancy
constraint, i is the unit cell index, σ = , is the spin
↑ ↓
index and α = 1,2 is the chain index. The oxygen sites
have an on-site potential ε>0.
FIG.1: ThegeometryofthezigzagCuOchain. Squaresmark Next, we consider H . This has the form
ex
copperatomsandcirclesareoxygenatoms. Alsoshownarea
copperdx2−y2 orbitalsandanoxygenpx orbitals. Thedotted H = J S S 1n n
box marks theunit cell. ex 1 (cid:18) d,i,α· px,i,α− 4 d,i,α px,i,α
Xi,α
1
+ S S n n
susceptibility depends on temperature also depends on px,i,α· d,i+1,α− 4 px,i,α d,i+1,α(cid:19)
χsKu+sc,ce,ptiwtibeiclaietnxyt.rbaFecrtoemxthtretahcvetaelNduQefrRKommeaasmu3re/ea2ms.uernNetmotoeefn1tth/oaTft1Ttthh∼iiss − J2Xi (cid:18)Sd,i,1·Spx,i,2− 41nd,i,1npx,i,2(cid:19) (6)
SDW +,c
≈
cflourcrteusaptoionndsswtoithaχregimeTof11d/o12m.inant superconducting Here Sd = σ,σ′d†σ~σσσ′dσ′ where ~σ = (σx,σy,σz) are
SC ∼ − PaulimatricPes,nd = σd†σdσ andsimilardefinitions for
Spx, npx. J1 >0 is thPe usual antiferromagnetic superex-
change interaction. J , however, has a qualitatively dif-
2
II. THE MODEL ferent physical origin: it comes from projecting out the
state with a doubly occupied oxygen site with one p
x
state and one p state occupied by holes. Since the two
WenowpresentamodelforasinglezigzagCuOchain. y
holes belong to different orbitals, they are subject to a
The geometry of the zigzag chain is shown in Fig. 1. Of
ferromagnetic Hund’s rule interaction12. Therefore the
this structure, we will ignore the outer oxygens, leaving
effective exchange interaction is ferromagnetic (J > 0)
two copper and two oxygen atoms per unit cell. The 2
in this case. (Note the minus sign in Eq. 6.)
relevantorbitalsarethe 3d ontheCusites,andthe
x2 y2 The model we have derived for the zigzag chain has
2p and 2p orbitals on the−O sites. We assume that
x y several new features that distinguish it from the well-
the on-site Coulomb repulsion is large on both Cu and
studied Hubbard ladder model. Firstly, as we have seen,
O sites, so that doubly occupied states can be projected
theinter-chainhoppingissmallrelativetotheintra-chain
out, leaving an effective magnetic exchange interaction.
bandwidth. (It arises only from further neighbor hop-
The Hamiltonian for holes can then be written as
pings, such as the O-O hopping.) The coupling between
the two chains thus comes mostly from the electron-
H =H0+Hex (4) electron interactions16. Secondly, the inter-chain effec-
tive exchange interaction is ferromagnetic.
HereH isthehoppingHamiltonianandH isthemag-
0 ex
neticspinexchangeHamiltonian. Letusfirstconsiderthe
termsinH . A3dCuorbitalcanhybridizestronglywith III. HALF FILLING
0
thep orbitalofitsneighborOinthesamechain,orap
x y
orbital of the neighboring oxygen in the opposite chain. Itisinstructivetoconsiderthecaseofhalffillinginthe
However, it cannot hybridize with the nearest px orbital double chain, which we consider as the parent insulating
in the opposite chain or a py orbital in the same chain state. In this case (i.e. one hole per Cu site), there is
due to symmetry. Further neighbor hopping matrix ele- a charge gap of order ε. Then the problem essentially
ments(suchasadirectoxygen-oxygenhoppingtppshown reducestothatofthe zigzagHeisenbergchain,described
inFig. 1)willbe neglected,exceptinsectionIVBwhere by the effective Hamiltonian
their effect on the main results will be examined. For
simplicity, we also also project out the p orbitals, since
they essentially “belong” to the neighboyr Cu d orbital, Heff = J˜1 Sd,i,α Sd,i+1,α
·
inthe sensethatH onlyconnectsap to thatdorbital. Xi,α
0 y
The effect of the py orbitals will be reintroduced as an J˜2 (Sd,i,1 Sd,i,2+Sd,i,1 Sd,i+1,2) (7)
effective interaction in Hex. Taking these considerations − Xi · ·
4
0.8 used in Eq. (4): t = 1, ε = 0.5, and J = J = 0.5
∆ pd 1 2
0.7 ∆c The values for tpd,J1, and ε are comparable to values
∆s that are commonly used in effective t J models for the
δN=2 −
0.6 ∆ copper-oxide planes in the cuprates. In the simulations
δN=4
described in this section, the direct oxygen-oxygen hop-
0.5
ping, t , was neglected. (The effect of non-zero t is
∆ p,p p,p
p 0.4 examinedinthenextsubsection.) Uptom=2700states
a
G per blockwhere keptforthe longestsystems(L=88Cu
0.3 sites), resulting in an average truncation error of about
10 6. Theenergieswhereextrapolatedtozerotruncation
0.2 −
errors using the standard procedure22. The convergence
0.1 of the ground state energy as a function of the number
of sweeps improves dramatically when a small perpen-
0
0 0.01 0.02 0.03 0.04 0.05 dicular copper - oxygen term is added to the hopping
1/Length Hamiltonian (5), with t = 0.01. We have checked that
the physicalresults aren⊥ot affectedby addingthis term.
FIG. 2: Calculated gaps from DMRG, measured in units of In Fig. 2 we plot various gaps in the spectrum as
t , as a function of 1/L, where L is the number of Cu sites
pd a function of 1/L. ∆ = E +E 2E is the
c N+2 N 2 N
∆inδtNh=e2syanstdem∆.δN∆=c4isartehethcehagragpesgfoarp,tr∆anssifsertrhinegspoinnegaanpd, atwndo cWheardgeefignaept,heansdpin∆gsa=prEelSazt=iv2e−toEthSze=S1z−i=s−t1hestsaptein,sginacpe.
holesfromonechaintotheother(seetext). Thedashedlines
the ground state is found to have spin one for the sys-
are linear extrapolations.
tems considered here31. In addition, we have calculated
the gap to the state with one or two holes transferred
Note thatJ˜1,J˜2 aredifferentfromJ1,J2 in(6) they from one chain to the other: ∆δN=2 = EδN=2−EδN=0
− and ∆ = E E , where δN = N N
arise from projecting out the px orbitals. However, we δN=4 δN=4 − δN=0 2 − 1
stillfind thatJ˜ >0,J˜ >0. The model(7)wasstudied is the difference between the number of holes in the two
1 2
chains. The direct measurement of ∆ is possible due
in Ref. [19] by a combination of DMRG and bosoniza- δN
to the fact that N and N (the numbers of holes on
tion, for J˜ both positive and negative. It was found 1 2
2 each chain) are separately conserved. In the presence of
that for J˜ > 0, the coupling between the two chains is
2 asmallinter-chainhoppingterm,thisconservationlawis
irrelevant. This is due to the fact that the interaction
weakly violated. However, it is still possible to measure
between the two chains in Eq. (7) is geometrically frus-
∆ by applyinganinter-chainpotential difference ∆V.
trated. Later studies found that the interaction actually δN
When∆V =∆ ,oneholeistransferredfromonechain
contains a marginally relevantoperator20,21, but the RG δN
to the other. This transition is smoothed by the inter-
flowisextremelyslow,sothesystemcanstillberegarded
chainhoppingterm,withawidthproportionaltot . For
as gapless for all practical purpose20. This is an impor- t =0.01,wefoundthatthisdoesnotconsiderably⊥limit
tant difference between the zigzag ladder and the simple t⊥he accuracy in the determination of ∆ .
Hubbardladder;thelatterhasalargespingap∆ J/2 δN
s
∼ The gaps were extrapolated linearly with 1/L to the
at half filling.
thermodynamic limit (L ). The charge gap ∆
c
→ ∞
extrapolates to very close to zero. The spin gap ∆
s
also extrapolates to a very small value, which is indis-
IV. DMRG SIMULATIONS tinguishable from zero to the accuracy of our calcula-
tions. However, ∆ and ∆ clearly extrapolate
δN=2 δN=4
A. tpp =0 case to finite values. The linear extrapolation results are
∆ (1/L 0) 0.046, ∆ (1/L 0) 0.097.
δN=2 δN=4
→ ≈ → ≈
The doped zigzag chain cannot be understood as sim- Let us assume that, as the simulation suggests, ∆c =
plyasthehalffilledone. Moreover,itisnotentirelyclear ∆s = 0. In terms of the classification by the number of
what doping level leads to superconductivity in Pr-247; gaplessbosonicmodes,thisimpliesthatwemusthaveat
counting charges (and assuming that the valence of the leastonegaplesschargemodeandonegaplessspinmode.
Prionsis +3),there are9/7 2δ/7dopedholesper cop- However, since ∆δN = 0 at least some of the relative
per site on average, and sup−erconductivity occurs when spin and charge mode6s of the two chains are gapped by
δ & 0.3 (see [3]). However, these holes are distributed the inter-chain coupling. The fact that ∆δN=4 = 0 in
6
in an unknown manner between the plane, single chain, the thermodynamic limit implies that the relative charge
anddoublechainCusites. Therefore,wehaveperformed mode is gapped. This is because it involves transferring
DMRG simulations with a finite doping concentration twoholes fromone chainto the other,andtwo holes can
which we take, somewhat arbitrarily, to be 0.25 “doped be combined to a singlet and therefore carry no spin.
holes” per Cu (i.e. the density of holes is taken to be Fig. 3a shows the average hole density on the oxy-
n = 1.25 per Cu site). The following parameters were gen sites in a chain of L = 88 Cu sites. The density
5
(a)
en 0.6
g
〉n oxy0.55
〈
0.5
0 10 20 30 40 50 60 70 80
Position
4 (b)
2
0
−2
0 5 10 15 20 25 30
Position
FIG. 3: (a) The average hole density on oxygen sites as a function of position. Blue Squares (red circles) mark the upper
(lower) chain from the DMRG simulation with hni = 1.25 holes per Cu site. (b) hSzi as a function of position. Copper sites
i
are marked by squares, and oxygen sites - by circles. The size of the circles represents theaverage hole density on the oxygen
sites (bigger circles mean larger hole density).
profile shows pronounced density oscillations with a pe- expected to be robust over a finite range of t . How-
pp
riod of λ = 4a where a is the lattice constant. For the ever, this argument cannot tell us what is the range of
c
density of holes in the simulation, this corresponds to a stability. In order to study the effect of a finite t , we
pp
wavevectorof4k . Fouriertransformingthedensitypro- have performed simulations including an oxygen-oxygen
F
file revealsthatthe 2k componentis almostcompletely hopping term H . Using the same notation as in Eq.
F pp
absent,which is characteristicofsystems with strongre- (5), H is given by
pp
pulsion. Notealsothattherelativechargedensitiesinthe
twochainsappearrigidlylockedtoeachothersothatthe
cohfatrhgeeeoxsicsitlelantcieonosfaareresltartiicvtelycshtaargggeergeadp,.furtherevidence Hpp =−tppXi,σ Pˆ(cid:16)p†x,i,σ,1px,i+1,σ,2+p†x,i,σ,2px,i,σ,1+h.c.(cid:17)Pˆ
A zigzag chain model similar to Eq. (4) was studied (8)
in Ref. [17], and a spin gapwas found. However,in that In the presence of this term, the method we used pre-
study there where no oxygen sites and it was assumed viously to determine whether there is a relative charge
that J < 0 (i.e. antiferromagnetic) and J > J . The gap (based on measuring the energy gap for transferring
2 2 1
| |
ferromagneticcasewasstudied(inadifferentcontext)in one hole from one chain to the other) is no longer ap-
Ref. [18], and their results seem consistent with ours (at plicable because now the charges on the two chains are
least over a certain range of parameters). not conserved separately. Consequently, the charge dif-
In Fig. 3b we show Sz along with the oxygen site ference cannot be controlled in the calculation. Instead,
h ii
density(representedbythesizeofthecircles)inthesame we used an alternative method to determine in which
system. A weak Zeeman field H = 0.01ˆz was applied to phase the system is. The local spin and charge densities
thefirstandlastCusitesonthechaininordertoselecta on each chain where measured and Fourier transformed.
direction in spin space. The main periodicity in the spin We define
density is λ =8a, or 2k . Note that a peak in the hole
s F
numberdensityisalwaysaccompaniedbyaπ phaseshift
in the spin density wave. Some features of the pattern 1 L/2
of the chargeand spin density canbe understoodfrom a nq = L/2 e−iqxinpx,i,1
strong coupling approach, as discussed in section V A. Xi=1
p
L/2
1
Sqz = e−iqxiSdz,i,1 (9)
L/2
B. tpp 6=0 case Xi=1
p
So far, we have neglected the oxygen-oxygen hopping Here npx,i,1 = σp†x,i,σ,1px,i,σ,1, Sdz,i,1 =
tpp. This term is expected to be significantly smaller 21 σσd†i,σ,1di,σ,1 and LPis the length of the chain
than tpd, because of its longer range. Estimates from (nPumber of Cu sites). Due to the open boundary
LDA calculations23 give t 0.27t 9. conditions, n and Sz show pronounced peaks at cer-
pp ∼ pd q q
As we saw, the DMRG simulation with t = 0 re- tain wavevectors. (These are known as “Friedel-like”
pp
vealed a finite gap to the relative charge mode between oscillations24.) These peaks occur at the wavevectors
the two chains. In such a phase, single particle inter- of the gapless spin/charge modes of the system. A 1D
chain hopping is irrelevant16. Therefore, this phase is version of Luttinger’s theorem25 states that there must
6
beagaplessmodeatacertainwavevector,corresponding 0.5 (a) 4kF 2π/a−4kF tpp=0.1
in our case to 4k of a single chain, where 2k =πn (n t =0.3
F F pp
iccshouathrldgeebneourmotsbhpeeirnrogmfapohldoeelsesssmaptoedr2ekCsFuatfsoiorttehe)ea.rchIwnacavhdeavdineitc,itoeontrc,s.t)(h.e.egrI.ef, n (arb. units)q000...234 ttttpppppppp====0000....7654
there is only a single gapless charge mode, as we found
0.1
in the case t = 0, then we expect to find gapless
pp 0
modes (and therefore peaks in the Fourier transformed 0 5 10 15 qL2/02π 25 30 35 40
local spin/charge densities of a finite system with open
boundary conditions) only at a single wavevector, plus 0.8 (b) 2π/a−2kF 2kF
its harmonics. If, on the other hand, the relative charge
gap closes and there is more than one gapless charge nits) 0.6
mode, there is no reason why these modes cannot have b. u 0.4
different wavevectors (these are the analogues of the zS (arq
two Fermi wavevectors in the non-interacting ladder 0.2
problem). This way, the phase transition from a phase
0
with a single gapless charge mode to a phase with two 0 5 10 15 20 25 30 35 40
qL/2π
gapless modes can be identified.
The results of the DMRG calculations are shown in FIG. 4: (Color online.) Fourier transforms of the charge (a)
Fig. 4. No additional peaks appear at new wavevectors andspin(b)profiles(definedinEq. (9))calculatedbyDMRG
and the spin/charge profiles do not change qualitatively in a system of length L = 72 Cu sites. Various values of tpp
untiltpp 0.5 0.6tpd. Thisshowsthatthephasewitha (Eq. (8)) were used. (tpp is measured in units of tpd.) Both
∼ −
single gapless charge mode is robust at least up to t = profiles change qualitatively and new peaks appear between
pp
0.5tpd, which is higher than the value estimated from tpp =0.5tpd and 0.6tpd, which signals a phase transition to a
phase with two independentgapless charge modes.
band structure calculations for the zigzag CuO chain.
as long as they do not pass each other.
V. ANALYTICAL RESULTS IN SPECIAL
LIMITS The qualitative picture in the small tpd limit can ex-
plain some of the features in Fig. 3b. As mentioned
previously,thepeaksinthechargedensityontheOsites
A. t →0 limit
pd
are accompanied by π phase shifts in the spin density
wave on the Cu sites, which can be understood as com-
We start from a qualitative strong coupling descrip-
ing from the antiferromagnetic coupling between neigh-
tion, in which we assume that t2 /ε J ,J . As we
pd ≪ 1 2 boring Cu and O spins. The fact that the hole density
saw, at half filling the inter-chain coupling J is frus-
2 oscillationsareπphaseshiftedbetweenthetwochains,is
trated,sothelow-energyspectrumisessentiallylikethat
reminiscent of the alternating order between the chains
of two decoupled chains. A typical snapshot of the spin
described above. Moreover, the spin on the O site with
configuration in this state is depicted in Fig. 5a.
maximum local density is always parallel to the spin of
We now consider a lightly hole doped system. Since
the nearest Cu on the opposite chain, due to the ferro-
all the copper sites are occupied, the doped holes reside
magnetic coupling between them. The DMRG picture is
on oxygen sites. Fig. 5b shows two neighbor holes on
thereforequalitativelyverysimilartoFig. 5eventhough
opposite chains. Since J >0, every doped hole induces
1 t is not small.
pd
a π shift in the phase of the spin density wave around
In fact, we suspect that the present analysis is qual-
it. The doped hole locally relieves the frustration of the
itatively correct for small enough x independent of the
inter-chainferromagneticcouplingJ , causinganetcou-
2 magnitude of t . For a single hole in an antiferromag-
pd
pling between the spin fluctuations on the two chains.
netic chain, a π phase shift in the antiferromagnetic cor-
This coupling is proportional to the hole concentration
relationsappears to be generic. Beyondthat, the energy
x times J . The hole can move along the chain via a
2 whichfavorsalternatingholesonthetwochainsispurely
second order process in t without disturbing the local
pd geometric in origin, and so only requires that the holes
spin order. Note that some exchange energy is gained
be sufficiently dilute.
also near the neighbor hole in the opposite chain.
However, if two neighbor holes are in the same chain
(as shown in 5c), the situation is different. Now we can-
B. RG and Bosonization treatment
not satisfy the J terms near both holes, and the inter-
2
chaincouplingispartiallyfrustrated. Therefore,thelow-
est energy states will be ones in which the holes appear Another limit of the zigzag chain model (4) that can
in alternating order inthetwochains. Theholesarefree be treated analytically is the weak inter-chain coupling
to move so as to reduce their zero point kinetic energy, limit (J J ,t). In this limit, only the low-energy
2 1
| | ≪
7
degrees of freedom of the decoupled chains are affected
by the inter-chain coupling. These degrees of freedom
u 1
can be described as two Luttinger liquids. The inter- H = s dx K (∂ θ )2+ (∂ ϕ )2
0,s s x i,s x i,s
chain coupling can then be treated using perturbative 2 iX=1,2Z (cid:20) Ks (cid:21)
RG. The nature of the strong-coupling fixed point can (12)
beunderstoodusingabelianbosonization. Thisisastan-
dard procedure13,14. u 1
H = c dx K (∂ θ )2+ (∂ ϕ )2
Aswewillsee,inthe weakinter-chaincouplinglimita 0,c 2 Z (cid:20) c x i,c K x i,c (cid:21)
spin gap is predicted, in contradiction to DMRG results iX=1,2 c
ofsectionIVA. Thediscrepancycanbeexplainedbythe
and i = 1,2 is the chain index. SU(2) symmetry gives
fact that the inter-chain coupling is not small. Rather,
the constraint K = 1, while K < 1 depends on the
s c
we will use bosonizationto parameterizean effective low
details of the intra-chain microscopic interactions. Here
energy model which is consistent with the numerical re-
φ and θ represent dual fields, i.e. [∂ θ (x),φ (x)] =
x a b ′
sults. This model is then used to calculate low-energy
iδ δ(x x), and are related to the density and current
a,b ′
susceptibilities of the system. −
operators, e.g. ρ = 2/π∂ φ and j = u 2/π∂ θ
The Hamiltonian is written as i x i,c i c x i,c
where ρi and ji arep, respectively, the electrpon density
andcurrentdensityonchaini. Athalffilling,the charge
sector will also have a non-linear term g cos(√8πϕ ),
H =H +H (10) u c,i
LL int which corresponds to 4k umklapp scattering. The spin
F
sector should include marginally irrelevant operators,
Here H is a Luttinger liquid Hamiltonian that de-
LL
which we have not written here.
scribes the low energy degrees of freedom of each chain,
Next, we consider the interaction term between the
and H is the inter-chain coupling Hamiltonian,
int
two chains. This term is most conveniently written in
terms of the corresponding fermionic degree of freedom,
HLL =H0,s+H0,c (11) ψi,σ, with i = 1,2 for the two chains, which is in turn
written in terms of the right and left moving fermions,
where H0,s and H0,c are bosonic Hamiltonians for the ψi,σ ∼eikFxψi,R,σ+e−ikFxψi,L,σ,where2kF =πnis the
spin and charge modes of each chain: Fermiwavevector(nisthetotaldensityofholes). Taking
a naive continuum limit of the J term in (6), we get:
2
H = J a dx S (x) S (x)+S (x) S (x+a)
int 2 1 2 1 2
Z { · ·
1
[n (x)n (x)+n (x)n (x+a)] (13)
1 2 1 2
− 4 }
Here Si(x) = ψi†(x)~σψi(x), ni(x) = ψi†(x)ψi(x), and
a is the lattice constant. This form satisfies the require-
ment that the Hamiltonian is symmetric with respect to
S (x) S (x), S (x) S (x+a) and similarly for
1 2 1 2
→ →
the n ’s, which corresponds to a reflection that inter-
i
changes the two chains, followed by a translation by one
Cu-O distance.
In bosonized form, the most relevant part of (13) is
written as:
sin(δa) δa
H = 2 dxsin √4πϕ
FIG. 5: Typical pattern of the spin correlations in various int − (πa)2 Z (cid:18) −,c− 2 (cid:19)
situations. Copper (oxygen) sites are shown as square (cir-
g ˆ +g ˆ (14)
cles). (a) The undoped chain. The intra-chain interaction is 1 s1 2 s2
×h O O i
antiferromagnetic, while the ferromagnetic inter-chain inter-
action is frustrated. (b) Two doped holes, one in each chain. Here ˆ cos √4πϕ +cos √4πθ and
s1 ,s ,s
Notethatthephaseofthespinfluctuationshiftsbyπ around ˆ Ocos ≡√4π(cid:2)ϕ (cid:0)+cos−√(cid:1)4πϕ (cid:0) whe−re(cid:1)ϕ(cid:3) =
the doped holes. The spin of each doped holes is parallel Os2 ≡ −,s +,s ±,c
to the spin of the nearest copper site in the opposite chain, (ϕ1,c ϕ(cid:2)2,c)(cid:0)/√2 are t(cid:1)he even(cid:0)/odd cha(cid:1)r(cid:3)ge modes of the
±
which gains some ferromagnetic interchain exchange energy. twochains, andsimilarly for ϕ ,s andθ ,s. We havede-
(c)Twodopedholesinthesamechain. Thisissimilarto(b), finedδ =π(n 1),wherenisth±edensity±ofholesperunit
−a
except that now the inter-chain ferromagnetic exchange en- length. (δ is a wavevector whose length is proportional
ergy of thehole on the right is lost. Putting two neighboring to the amount of doping away from half filling.) At half
doped holes in the same chain is less favorable energetically. filling, additional umklapp terms appear. g and g are
1 2
8
dimensionless coupling constants, whose bare values at C. Physical susceptibilities
the initial scale are: g = g = J2a. Under RG, g and
1 − 2 2 1
g2 will flow, and need not remain equal in magnitude. GiventhatthesystemisinaC1S3 phase,thephysical
However, additional couplings are prevented as long as 2
susceptibilitiescanbe calculatedinasimilarwaytothat
the exact SU(2) spin-rotational symmetry is respected.
describedin[26],withadditionofthe chargemodes. For
Specifically,therearethreedistinctcosinesofspinfields,
an example of such a calculation, see appendix A. The
but only two independent coupling constants.
leading temperature dependence of these susceptibilities
Note thatthe inter-chaincoupling(13)vanishesinthe depends only on the total charge Luttinger parameter
limitδ 0,i.e. athalffilling. Thisisduetothe frustra- K+,c. The superconducting susceptibility is
→
tionoftheinter-chaincouplinginthatlimit. (Seesection
1 5
III. ) Then marginal operators need to be considered. χSC T2K+,c−4 (15)
∼
These produce an extremely small gap in the spectrum,
especially in the J < 0 case20, so the system can be for both singlet and triplet pairing, which gives that the
2
considered as essentially gapless at half filling19,20. superconducting susceptibility is divergent for K+,c >
0.4 (i.e. even for strongly repulsive interactions).
Awayfromhalffilling,boththeg andg termsin(14)
1 2 Other susceptibilities are 2k SDW and CDW, with
F
are relevant, and produce a gap in the spectrum. They
b(aostshumhainvge tthhaetsKame,ss=ca1li)n.gTdhiemseynsstieomn flofow1s+toKa+s,ctr<on2g χSDW,2kF ∼χCDW,2kF ∼TK+2,c−54 (16)
coupling fixed po±int at which the ϕ , ϕ and θ
,c +,s ,s and 4k CDW:
fields are pinned. Then the only ga−pless mode is t−he F
stoimtaillacrhtaorgtehemgoednee,riacnsditutahteiosnysitnemtheisHiunbabaCrd1Sl0adpdhear1s3e., χCDW,4kF ∼T2K+,c−2 (17)
The superconductingsusceptibilityis the mostdivergent
However, the DMRG result indicates that while there
one for K >1.
is a substantial gap in the relative charge mode, the gap +,c
Assuming that the relative charge mode ϕ is
in the total spin sector is either zero or very small (see gapped, it is possible to extract K from the−,dcen-
Fig. 2). Thisdiscrepancyislikelytobecausedbylessrel- +,c
sity profile in the DMRG simulation (see Fig. 3).
evant (sub-leading) operators, not included in Eq. (14),
The density profile is expected to behave as n
wnohtosineitbiaalrleycaotewffiecaieknctsouaprleinogf).orTdheer nuengitlyect(soinfctehewsee oapre- cos(4kFxi)/[Lsin(πxi/L)]K+,c, which is essentiahllyiith∼e
square root of the 4k part of the density-density corre-
erators in the initial stages of the RG transformation is F
lation function24. The amplitude of the 4k CDW near
a quantitatively unreliable approximation. Taking the F
DMRG result into account, we hypothesize that the ef- themiddle ofthe chain,Axi=L/2,thusdecaysasL−K+,c.
Fitting A of chains with different lengths to this
fective value of g at the fixed point is close to 0 (since xi=L/2
2 expression, we obtain K =0.6 0.05.
non-zero g2 is what induces a full spin-gap). If this is SinceNQRmeasureso+f,cthespin±-latticerelaxationrate
true, then over a broad intermediate range of energies,
havebeendoneonsuperconductingPr-247,itisinterest-
the system is governed by the unstable fixed point with
ing to extract the temperature dependence of this quan-
g =0 and non-zero g .
2 1 tity fromthe theory. Assuming that the main relaxation
We would like to stress that the smallness of the spin mechanismis the coupling of the nuclear spins to fluctu-
gap is not a result of fine tuning, but rather appears ations of the local magnetic field due to electronic spins,
(from the numerics) to be a ubiquitous property of the 1/T is given by27:
1
model. (The same result is obtained over a range of dif-
ferent parameters and doping levels.) In contrast, if the
1 1
inter-chaincouplingJ2 isartificiallyturnedtobeantifer- = Aα(q)2Sαα(q,ωR) (18)
T 2N | |
romagnetic, a spin gap appears in the numerical simula- 1 Xqα
tion. Therefore the small spin gap seems to be a feature
of the ferromagnetic J2 case. where Aα(q) is the form factor of the hyperfine Hamil-
tonian, ω is the nuclear spin resonance frequency, and
R
Note that the phase with g1 6= 0, g2 = 0 is unusual, Sαα(q,ωR) is the electronic spin structure factor. ωR
because neither the field ϕ nor its dual θ can be
,s ,s is typically smaller than any other energy scale in the
pinned (since Oˆs1 contains−the cosine terms−containing problem, so we will assume ωR =0.
bothfieldswithequalmagnitude). Thepropertiesofthis The spin structure factor in the C S is expected to
1 3
phase,whichfollowcloselyfromanearlieranalysisofthe 2
be of the form:
Heisenberg ladderby Shelton, NersessyanandTsvelik26,
areexploredinappendix A. Themainconclusionisthat Sαα(q,ω =0) Alnq+BqK+2,c−54 (19)
essentially“half”oftherelativespinmodeisgapped,and ∼
the other half is gapless. Therefore we denote this phase where A and B are constants. The two terms here are
as C1S3. the two maincontributionsto Sαα(q,ω =0)which come
2
9
the vicinity of the points q = 0,2k . Integrating (19) Acknowledgements. We thank S. Sasaki and Y. Ya-
F
over q, we get the dominant temperature dependence of mada for discussions and for sharing their data with us
the spin-lattice relaxation rate: before publication. S. Moukouri and S. R. White are
acknowledged for their help with setting up the DMRG
1 K+,c 1 code. Discussions with E. Altman, O. Vafek and C. Wu
AT lnT +BT 2 −4 (20)
T1 ∼ aregratefullyacknowledged. WethankA.M.Tsvelikfor
critically reading this manuscript. E.B. also thanks the
NQRmeasurements4,5showthatT1−1behavesasapower hospitality of the Weizmann institute, were part of this
law of temperature, with different exponents above and workwasdone. ThisworkwassupportedinpartbyNSF
below Tc. The fact that a power law is seen above Tc grant # DMR -551196 at Stanford.
is a clear evidence for the absence of a spin gap, and is
consistent with a C S phase. According to the mea-
1 3
2
surement, T11 ∝T0.5 above Tc. Comparing this behavior APPENDIX A: THE C1S32 PHASE
to(20), weseethatitcorrespondstoK 1.5. This is
+,c
≃
consistent with dominant superconducting correlations. We would like to describe a phase of the two-
ThevalueofK+,cextractedfromtheNQRdataissub- component electron gas with a gap in the relative spin
stantiallylargerthanthe one extractedfromourDMRG and charge sectors, but no gap in the total spin and
calculation. Clearly, this is a significant discrepancy. charge sectors. This is the phase that seems to emerge
However, it is well known that K+,c is a non-universal from the DMRG results. (See Fig. 2.) Starting from
exponent, and in this case it depends strongly on micro- the Hamiltonian H = H + H , where H is a
LL int LL
scopic details of the problem. Since the detailed aspects Luttinger liquid intra-chain Hamiltonian (11) and H
int
ofthemicroscopicmodelarecertainlynot“realistic,”we is the most relevant part of a generic SU(2) invariant
do not feel that any microscopic calculation can be ex- inter-chain interaction (Eq. 14), we see that in or-
pectedtopredicttheexperimentalvalueofK+,c reliably. der for the spin gap to vanish we must have g2 = 0
A better strategy is thus to extract K+,c directly from in (14)). The remaining term is then proportional to
experiments. As we saw, in the superconducting sample ˆ =cos √4πϕ +cos √4πθ . The total scaling
s1 ,s ,s
this yields a value of K+,c that correspondsto dominant dOimension(cid:0)of this−ter(cid:1)m with(cid:0)respec−t t(cid:1)o the Luttinger liq-
superconducting fluctuations.
uidfixedpointis1 K ,withK <1,soitwillgrow
,c ,c
The NQR measurement was done on with a sample under anRG trans−form−ation. At s−ome scale,g becomes
1
with an oxygen content δ = 0.5 which is close to the of the order of unity, and we can replace sin √4πϕ
,c
optimal value for superconductivity. We are not aware −
and cos √4πϕ by their mean value (sinc(cid:0)e the field(cid:1)
ocofnatneyntsiamndilaTrc.NWQReredasutachodnastaamopblteasinweidt,hwaedwiffouerldenptrOe- ϕH−ow,cewveirl(cid:0)l,sbinecpei−tnh,ncee(cid:1)dintteoratchteiomnicnoinmtuaimnsotfhethceospinoetsenotfiathl)e.
dict a smaller power should govern the T dependence of
conjugatefieldsϕ andθ withanequalweights,both
1/T1(T), corresponding to a lower K+,c. fields fluctuate str−o,nsglyat−t,hse fixedpoint, andneither is
pinned. Wearefacedwiththeproblemofhowtodescribe
the low-energy properties of the resulting phase.
Aremarkablyelegantsolutionforthisproblemisgiven
VI. CONCLUSIONS
in a paper by Shelton et al [26]. They considered the
problem of two weakly coupled spin 1 chains, but the
2
To summarize, we have considered the possi- results are readily generalized to our case by neglecting
bility of zigzag chain-driven superconductivity in thefluctuationsintheϕ field,replacingitbyitsmean
,c
Pr2Ba4Cu7O15 δ. Assuming that the chains are weakly value. The Hamiltonian−(10) can then be solved by re-
coupled, this im−plies that the single zigzag chain must fermionizing of the fields ϕ . We introduce two Dirac
,s
havealargesuperconductingsusceptibility. This,incon- fermions, ψ and χ, as follow±s:
junction with the fact that Cu NQR experiments do not
tshheowziagnzaygspchinaingaips ainboavepThca,seraiinsewshthicehpsoosmsieb,ilibtuytthnaott ψR,L = √21πaei√π(θ+,s±ϕ+,s)
all, of the modes of the two-component electron gas are 1
gapped. We have shown evidence for such a phase using χR,L = ei√π(θ−,s±ϕ−,s) (A1)
√2πa
a microscopic copper-oxygenmodel for the zigzag chain.
The gapping of some of the relative spin and charge where R and L denote right or left moving fields. The
modes enhances superconducting (as well as other) fluc- Hamiltonian for ψ is then just a free Dirac Hamiltonian:
tuations at low temperatures. The pairing operator is
composed of one hole from each chain with zero center 1 d 1 d
of mass momentum. Since there is no spin gap, triplet H+ =usZ dx(cid:18)ψR† i dxψR−ψL† i dxψL(cid:19) (A2)
and singlet superconducting fluctuations are equally en-
hanced. The interaction Hamiltonian H (13) becomes
int
10
quadratic in χ, giving θ is a free bosonic field, whose long-range correla-
+c
1
tions are ei√πθ+,c(x)ei√πθ+,c(0) x−2K+,c. Similarly,
H− = usZ dx(cid:18)χ†R1i ddxχR−χ†L1i ddxχL(cid:19) Dei√πϕ+,s(Dx)ei√πϕ+,s(0)E∼x−21,Esi∼nce ϕ+,s is a free field
iM withK+,s =1,asdictatedbythe SU(2)symmetry. This
+ 4 Z dx(cid:16)χ†RχL+χ†Lχ†R−H.c.(cid:17) (A3) correlation function is expected to have logarithmic cor-
rections due to marginal operators, which we have ne-
Here M depends on g1 and the average of the ϕ ,c part glected. The ϕ ,c field is massive, so at long distances
in (13). (A3) is most conveniently diagonalized b−y writ- we can replace−ei√πϕ−,c by its expectation value. The
ing χ in terms of two Majorana fermions. Adopting the treatment of ei√πθ−,s is more subtle, since as we have
notation of [26], we write mentioned,thisfieldis“halfgapped”intheC S phase.
1 3
2
Itisneverthelesspossibletocalculateitscorrelations,fol-
χ +χ
ξ = ν †ν lowing a method used in [26]. We describe this method
ν
√2 briefly here. The relative spin sector, which is described
χ χ as two independent Majorana theories, can be further
ρν = ν − †ν (A4) mapped onto two 2d Ising models. Since one of the Ma-
i√2
jorana fields is massless and the other is massive, one
with ν =R,L. Plugging this into (A3), we get: of the corresponding Ising fields is at criticality and the
other is away from criticality. It can be shown that the
H =H0[ξ]+HM[ρ] (A5) four operators sin(√πθ ,s), cos(√πθ ,s), sin(√πϕ ,s),
− cos(√πϕ ) have the −following corr−espondance to−the
,s
orderand−disorderoperatorsofthetwoIsingmodels26,28:
u 1 d 1 d
s
H [ξ]= dx ξ ξ ξ ξ (A6)
0 R R L L
2 Z (cid:18) i dx − i dx (cid:19)
cos√πϕ = µ µ , sin√πϕ =σ σ
,s 1 2 ,s 1 2
− −
cos√πθ = µ σ , sin√πθ =σ µ (A9)
,s 1 2 ,s 1 2
u 1 d 1 d − −
s
H [ρ] = dx ρ ρ ρ ρ
M R R L L
2 Z (cid:18) i dx − i dx (cid:19)
Here σ , µ , σ , µ are the order/disorder operators of
1 1 2 2
iM the two Ising models labelled 1 and 2. Note that the
+ dx(ρ ρ ρ ρ ) (A7)
R L L R
2 Z − scalingdimensionofalltheoperatorsinthelefthandside
of(A9)atcriticalityis 1,whichisconsistentwiththefact
The field ξ is thus massless, while ρ is massive, with 4
that the dimension of the Ising operatorsat criticality is
a mass M. In that sense, “half” of the field ϕ ,s is 1. Carrying out the correspondence to the Ising model
gapped. The total spin sector, on the other han−d, is 8
carefully, one finds that the Ising model labelled by 1
completely massless. We therefore denote this phase as
is in its disordered phase, so µ = 0, σ = 0. The
1 1
a C S phase. h i 6 h i
1 3 other Ising model labelled by 2 is critical, and therefore
2
emvoaSdneett,tiinngtigverign-2go6=rai0nCti1nrSa0-(c1hp3ah)ianwseiol.lpgeMraapotrooeruosvtetaorl,s(oa1d3td)hienwgtooulteladslsgsrpeeniln-- dhσis2t(axn)cσes2,(D0e)ii√∼πθx−−,s(41x,)eain√dπθs−im,s(i0la)Erly∼foxr−µ412..TThheuesxaptonloenngt
of this correlation function is just half of the exponent
erate the g term, even if it is not present in the bare
2
thatonegetsforafreeθ field. Thiscanbeunderstood
Hamiltonian. However,theDMRGcalculationpresented ,s
asaconsequenceofthefa−ctthat“half”ofthe ,smodeis
in section IVA indicates that the spin gap ∆ is much
s −
gapped. Therefore, correlation function of ∆ behaves
smaller than ∆ , the gap to transferring one hole from SC
δN
as
one chain to the other, which is related to a gap in
the relative spin/charge modes. Therefore, at interme-
diate energies (or temperatures) between ∆ and ∆ , 1
the physics is expected to be dominated by tshe unstaδbNle D∆†SC(x)∆SC(0)E∼ x2K1+,c+34 (A10)
C S fixed point.
1 3
2
Next, let us find the long-distance behavior of the
whichgivesasuperconductingsusceptibilitythatbehaves
correlation functions of physical operators in the C S
1 3 1 5
phase. For concreteness, we will focus on the pair field2 as χSC(T) T2K+,c−4. The correlation functions of
∼
other physical operators (CDW, SDW etc.) can be cal-
operator ∆ = ψ ψ . This operator creates a
SC 1R, 2L,
pair of holes in the tw↑o cha↓ins with opposite momenta culated in a similar manner.
and spins. In bosonized form, this operator becomes ItisinterestingtonotethattheHamiltonian(14)with
g = 0 is equivalent to the integrable super-symmetric
2
∆ ei√π(θ+c+θ−,s+ϕ−,c+ϕ+,s) (A8) super-Sine-Gordon model29.
SC
∼