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arXiv:cond-mat/9709216v3 [cond-mat.supr-con] 25 Jun 1999

typeset using JPSJ.sty <ver.0.7f>

Spin Gap and Superconductivity in Weakly Coupled Ladders:

Interladder One-particle vs. Two-particle Crossover

Jun-ichiro Kishine∗and Kenji Yonemitsu

Department of Theoretical Studies, Institute for Molecular Science, Okazaki 444, Japan

(Received September 17, 1997)

Effects of the interladder one-particle hopping, t⊥, on the low-energy asymptotics of a weakly

coupled Hubbard ladder system have been studied, based on the perturbative renormalization-

group approach.We found that for finite intraladder Hubbard repulsion, U, there exists a

crossover value of the interladder one-particle hopping, t⊥c. For 0 < t⊥ < t⊥c, the spin gap

metal (SGM) phase of the isolated ladder transits at a finite transition temperature, Tc, to the

d-wave superconducting (SCd) phase via a two-particle crossover. In the temperature region,

T < Tc, interladder coherent Josephson tunneling of the Cooper pairs occurs, while the inter-

ladder coherent one-particle process is strongly suppressed. For t⊥c < t⊥, around a crossover

temperature, Tcross, the system crosses over to the two-dimensional (2D) phase via a one-particle

crossover. In the temperature region, T < Tcross, the interladdercoherent band motion occurs.

KEYWORDS: doped Hubbard ladders, interladder coupling, one-particle crossover, two-particle crossover, d-wave

superconductivity, perturbative renormalization-group

Last year Uehara et al.1)discovered the superconduc-

tivity signal with Tc = 12K in the doped spin ladder,

Sr0.4Ca13.6Cu24O41.84, under a pressure of 3GPa. The

electric properties of the compound are determined by

the hole-doped ladders instead of chains.2)A remarkable

feature of the doped ladder is the existence of a spin ex-

citation gap.3)The superconducting transition under a

high pressure suggests that interladder one-particle hop-

ping induced by the applied pressure plays an important

role. Recent experiments on the resistivity along the lad-

der, ρc, of the single crystal Sr2.5Ca11.5Cu24O414)shows

that the superconductivity sets in below 10 K under

3.5GPa ∼ 8GPa with the temperature dependence of ρc

changing gradually from T-linear to T2. The anisotropy

of the resistivity also indicates the dimensional crossover

from 1D to 2D with increasing an applied pressure.

In this paper, to elucidate the nature of superconduc-

tivity in the doped ladder compound under pressure, we

consider Hubbard ladders5)coupled via a weak interlad-

der one-particle hopping. In the case of the isolated Hub-

bard ladder, the most relevant phase is characterized by

a strong coupling fixed point and is denoted by “phase I”

by Fabrizio6)and “C1S0 phase” by Balents and Fisher.7)

In this phase, only the total charge mode remains gap-

less6,7,8,9)and consequently, the d-wave superconduct-

ing correlation becomes the most dominant one as long

as the intraladder correlation is weak.6,7,8,9)From now

on we call this strong coupling phase “spin gap metal

(SGM) phase”. So far, the effects of interladder hopping

on the SGM phase have been studied through mean field

approximations10,12)and power counting arguments.11)

The central problem here is how a weak interladder

one-particle hopping, t⊥, affects the low-energy asymp-

totics of the system. Based on the perturbative renormali-

∗E-mail:kishine@ims.ac.jp

zation-group(PRG) approach, we here study one-particle

and two-particle crossovers,13)when we switch on t⊥and

the intraladder Hubbard repulsion, U, as perturbations

to the system, specified by the intraladder longitudinal

(transverse) hopping, t(t′): U,t⊥≪ t,t′. A similar ap-

proach has been considered for the problem of a cou-

pled chain system by Boies et al.14)The intraladder one-

particle process is diagonalized in terms of the bonding

(B) and antibonding (A) bands. As shown in Fig. 1(a),

we linearize the dispersions along the legs on the bond-

ing and antibonding Fermi points, ±kFm(m = A,B). In

Fig. 1, R and L denote the right-going and left-going

branches, respectively. The Fermi velocities in principle

depend on the band index as vFm= 2tsinkFm but we

assume throughout this work that vFm= vF and drop

the band index, since the difference in the Fermi veloc-

ities does not affect the asymptotic nature of the SGM

phase at least for small t′/t.6,7)In all the four branches

(LB,LA,RA,RB) of linearized bands, the energy vari-

ables, ενm (ν = R,L ; m = A,B) run over the region,

−E/2 < ενm < E/2, with E denoting the bandwidth

cutoff.

The intraladder Hubbard repulsion generates the scat-

tering processes depicted in Fig. 1(b).

are specified by dimensionless coupling constants g(1)

and g(2)

µ

denoting backward and forward scatterings, re-

spectively, with the flavor indices6)µ = 0,f,t denoting

intraband scattering, interband forward scattering and

interband tunneling processes, respectively. The usual

coupling constants with a dimension of the interaction

energy are 2πvFg(i)

We neglect the interband back-

ward processes such as (RB → RA;LA → LB) on the

grounds that these processes do not seriously modify the

asymptotic nature of the SGM phase.6,7)

When we switch on g(i)

µ

and t⊥ as perturbations to

the system with t and t′, and the temperature scale

The processes

µ

µ .

Page 2

2Jun-ichiro Kishine and Kenji Yonemitsu

Fig. 1.

with the bandwidth cutoff E and (b) intraladder two-particle

scattering vertices g(i)

µ . The solid and broken lines represent the

propagators for the right-moving and left-moving electrons. m

and ¯ m denote different bands.

(a) Four branches(LB, LA,RA,RB) of linearized bands

decreases, two kinds of dimensional crossovers, a one-

particle crossover and a two-particle crossover,13,15)oc-

cur. We illustrate one- and two-particle processes in

Fig. 2. In the case where the interladder one-particle

hopping modified by the intraladder self-energy becomes

the most relevant, a particle hops coherently from one

ladder to a neighboring one and the system crosses over

to a two-dimensional system via the one-particle crossover.

In the case where the interladder two-particle process

generated by the interladder one-particle hopping and in-

traladder two-particle scattering processes becomes the

most relevant, a pair of composite particles hops coher-

ently from one ladder to a neighboring one and the sys-

tem crosses over to a long-range-ordered phase via the

two-particle crossover. In Fig. 2, we show only the two-

particle hopping in the d-wave superconducting channel,

which corresponds to the interladder Josephson tunnel-

ing of Cooper pairs.

To study the competition between the one-particle crossover

and the two-particle crossover, we set up scaling equa-

tions for the interladder one-particle and two-particle

hopping amplitudes and study their low-energy asymp-

totic behavior. We start with the path integral repre-

sentation of the partition function of the system, Z =

?

DeS, where the action consists of four parts,

S = S(1)

?

+ S(2)

?

+ S(1)

⊥+ S(2)

⊥,(1)

where S(1)

the intraladder one-particle hopping, intraladder two-

particle scatterings, interladder one-particle and inter-

ladder two-particle hopping, respectively.

izes the measure of the path integral over the fermionic

Grassmann variables.

The idea of scaling is to eliminate the short-wavelength

degrees of freedom to relate effective actions at successive

energy scales. Based on the bandwidth cutoff regulariza-

tion scheme, we parametrize the cutoff as E(l) = E0e−l

with the scaling parameter, l, and study how the ac-

?, S(2)

?, S(1)

⊥

and S(2)

⊥

denote the action for

D symbol-

Fig. 2.

particle process (in the case of d-wave superconductivity chan-

nel). In the one-particle process, a particle hops from one ladder

to a neighboring one, while in the two-particle process, a pair of

particles hops from one ladder to a neighboring one.

Schematic illustrations of the one-particle and the two-

tion will be renormalized as l goes from zero to infinity.

The scaling equations for all the processes studied here

are depicted in Fig. 3. Diagrammatic expansions for

Fig. 3.

the intraladder scattering vertices(a), intraladder one-particle

propagator(b), interladder one-particle hopping amplitude(d),

and interladder two-particle hopping amplitude(d). A black cir-

cle, zigzag line, and shaded square represent the intraladder

two-particle scattering processes (a combination of g(i)

interladder one-particle hopping amplitude, t⊥, and the in-

terladder two-particle hopping amplitude, VSCdrespectively.

Diagrammatic representations of the scaling equations for

µ ), the

the intraladder scattering processes up to the 3rd or-

der are given in Fig.3 (a). After taking account of the

field rescaling procedure which originated from the scal-

ing of the intraladder one-particle propagators,16)given

in Fig. 3(b), we obtain the appropriate scaling equations

for g(i)

are found by setting g(1)

b

Starting with the Hubbard-type initial condition

µ . Full expressions of the scaling equations for g(i)

= g(2)

b

µ

= 0 in Eq.(A5) of ref.[6].

g(i)

µ(0) =˜U ≡ U/4πvF> 0,(2)

Page 3

Spin Gap and Superconductivity in Weakly Coupled Ladders3

the scaling equations give the fixed point

g(1)∗

0

g(2)∗

0

= −1,

= −3−2˜U

g(1)∗

f

g(2)∗

f

= 0,

=1+2˜U

g(1)∗

t

g(2)∗

t

= 1,

= 1.

4

,

4

,

(3)

Henceforth, we take˜U = 0.3 for illustration. For any

˜U > 0, the results are similar to those shown below. We

show the scaling flows for g(i)

µ in Fig. 4(a) as functions of

the scaling parameter, l. These flows scale the isolated

Hubbard ladder to the SGM phase.6,7,8)

The one-particle crossover is determined through the

scaling equation

dlnt⊥(l)

dl

= 1 − (g(1)2

0

+ g(2)2

0

+ g(1)2

f

+ g(2)2

f

+ g(1)2

t

+ g(2)2

t

− g(1)

0g(2)

0

− g(1)

fg(2)

f

− g(1)

t g(2)

t ), (4)

which is represented by Fig. 3(c). It is seen from (4)

together with the fixed point (3) that

dlnt⊥(l)/dl

l→∞

−→ −˜U2/2 − 7/8,(5)

and consequently t⊥(l) becomes always irrelevant at the

final stage of the scaling procedure. However, t⊥(l) grows

at an early stage of scaling since the intraladder cou-

plings do not grow sufficiently as yet (see Fig. 4). Once

˜t⊥(l) ≡ t⊥(l)/E0 attains an order of unity, the weakly

coupled ladder picture breaks down and the system is

scaled to a ”two-dimensional”system via the one-particle

crossover.13)In the upper planes of Figs. 4(b) and 4(c),

we show the scaling flows of˜t⊥(l) with the initial con-

ditions,˜t⊥(0) ≡˜t⊥0= 0.01 and 0.04, respectively. We

Fig. 4.

(b),(c)), and VSCd(the lower half planes of (b),(c)). For the re-

gion, l > Min(lc,lcross), the scaling flows, denoted by broken

curves, have no physical meaning, since the weak coupling pic-

ture breaks down in the region.

Scaling flows of g(i)

µ (a), ˜t⊥(l)(the upper half planes of

see that for˜t⊥0 = 0.01,˜t⊥(l) never reaches unity and

one-dimensional crossover never takes place, while for

˜t⊥0= 0.04,˜t⊥(l) exceeds unity at the scaling parameter

lcrosswhich is defined by

˜t⊥(lcross) = 1.(6)

Thus we specify the one-dimensional crossover by lcross,

although it merely has qualitative meaning.

Interladder two-particle processes are decomposed into

CDW, SDW, SS (singlet superconductivity) and TS (triplet

superconductivity) channels as in the case of the coupled

chains.13)In this case, there are additional flavor indices,

µ = 0,f,t for each channel. Then the two-particle hop-

ping amplitudes are specified as VM

SS, TS; µ = 0,f,t).The SS channel can be decom-

posed into the s-wave spuerconductivity (SCs) and d-

wave superconductivity (SCd) channels.6,7,8,9)For the

SCd channel, the action for the interladder two-particle

hopping is written in the form

⊥SCd= −πvF

2β

Q

µ

(M=CDW, SDW,

S(2)

?

VSCdO∗

SCd(Q)OSCd(Q).(7)

The SCd pair-field is given by OSCd(Q) = OBB

OAA

SS

(Q) = β−1/2?

Q)Lm′,−σ(K), with Rm,σ(Lm,σ) being the Grassmann

variable representing the right(left)-moving electron in

the band m with spin σ and K = (k?,k⊥,iεn) and Q =

(q?,q⊥,iωn) with the momentum along the leg and rung

being specified by ? and ⊥, respectively, and fermion

and boson thermal frequencies, εn = (2n + 1)π/β and

ωn= 2nπ/β, respectively.

The lowest-order scaling equation for VSCdis depicted

in Fig. 3(d), and is written as

SS(Q) −

SS(Q), where Omm′

K,σσ Rm,σ(−K +

dVSCd(l)

dl

= −?˜t⊥(l)gSCd(l)?2

+ 2gSCd(l)VSCd(l) −1

2

?VSCd(l)?2,(8)

where the transverse momentum transfer of the pair is

set at q⊥= 0. The coupling for the SCd pair field is given

by gSCd=

t

+ g(2)

t

− g(1)

straightforward manipulation, we obtain similar scaling

equations for all VM

µ(l). We have solved them with the

initial conditions

1

2(g(1)

0

− g(2)

0). By lengthy but

VM

µ(0) = 0(9)

and confirmed that VSCdalways dominates the other

channels. This situation is quite reasonable on the physi-

cal grounds that the interladder pair tunneling stabilizes

the most dominant intraladder correlation, i.e, d-wave

superconducting correlation. Below, we focus on the d-

wave superconducting channel. The third term of the

r.h.s of eq.(8) causes divergence of VSCdat a critical

scaling parameter lcdetermined by

VSCd(lc) = −∞(10)

In the lower half planes of Figs. 4(b) and 4(c), we show

the scaling flows of VSCd(l) for˜t⊥0= 0.01 and 0.04.

Figures 4(b) and 4(c) show that, in the case of˜t⊥0=

0.01, only the two-particle crossover occurs, while in the

case of˜t⊥0= 0.04, the one-particle crossover dominates

the two-particle crossover. We identify the scaling pa-

rameter with the absolute temperature as l = lnE0

Thus, based on eqs.(6) and (10), we define the one-

particle crossover temperature, Tcross, and the d-wave su-

perconducting transition temperature, Tc, as

T.

Tcross= E0e−lcross,

Tc= E0e−lc.

(11)

Page 4

4Jun-ichiro Kishine and Kenji Yonemitsu

Tcrossand Tccorrespond to Tx1and Tx2, respectively, in-

troduced by Bourbonnais and Caron13)for weakly cou-

pled chains.

By solving the scaling equations for various˜t⊥0with

the fixed value of the Hubbard repulsion,˜U = 0.3, we

obtained the phase diagram of the system in terms of

˜t⊥0 and the reduced temperature˜T = T/E0, as shown

in Fig. 5. Roughly speaking, we can regard increasing˜t⊥0

Fig. 5.

tem. SGM, SCd and 2D denote the sping gap metal phase, the

d-wave superconducting phase and the two-dimensional phase,

respectively. The interladder one-particle hopping and temper-

ature are scaled by the initial bandwidth cutoff, E0.

Phase diagram of the weakly coupled Hubbard ladder sys-

as applying the pressure under which the bulk supercon-

ductivity was actually observed. We found that there

exists a crossover value of the interladder one-particle

hopping,˜t⊥c∼ 0.025.

For 0 <˜t⊥0<˜t⊥c, the phase transition into the d-wave

superconducting (SCd) phase occurs at a finite transition

temperature, Tc, via the two-particle crossover. In the

temperature region, T < Tc, coherent Josephson tun-

neling of the Cooper pairs in the interladder transverse

direction occurs. Here we have to be careful about iden-

tification of the finite temperature phase above Tc, where

the system is in the isolated ladder regime. As the tem-

perature scale decreases, the isolated ladders are grad-

ually scaled toward their low-energy asymptotics, the

SGM phase. The gradual change of the density of dark-

ness in the SGM phase in Fig.5 schematically depicts this

situation. The SGM phase is characterized by the strong

coupling values of intraladder couplings, g(1)

and g(1)

0

= −1, (see (3)). The critical scaling parameter,

lc, is in the region, lc > 5.3, around which the intral-

adder coupling constants almost reach thier fixed point

values (see Fig.4.(a)). Thus we expect that the spin gap

is well developed near Tc. Within the framework of the

PRG approach, we cannot say for certain whether the

spin gap survives in the SCd phase or not.

For˜t⊥c<˜t⊥0, around the crossovertemperature Tcross,

the system crosses over to the 2D phase via the one-

particle crossover. The crossover value of the scaling

parameter, lcross, is in the region lcross < 4.6, around

which the intraladder coupling constants are far away

from thier fixed point values (see Fig.4(a)). Thus it is

t

= g(2)

t

= 1

disputable to assign the phase above Tcrossto the SGM

phase. In the temperature region, T < Tcross, the inter-

ladder coherent band motion occurs. Then the physical

properties of the system would strongly depend on the

shape of the 2D Fermi surface.

Finally we briefly compare the present case with the

case of coupled chains within the PRG scheme.13,14)The

scaling equation for the interchain one-particle hopping,

instead of (4), gives Tcross ∼ E0[t⊥0/E0]1/(1−θ),13,14)

where one has the exact result for the anomalous ex-

ponent, θ ≤ 1/8, for the non-half-filled Hubbard model.

17)Consequently t⊥ becomes always relevant, contrary

to the coupled ladder case. This difference reflects the

fact that the isolated Hubbard ladder belongs to the

strong coupling universality class (SGM phase), while

the isolated Hubbard chain belongs to the weak coupling

(Tomonaga-Luttinger) universality class. In that sense,

we can conclude that the spin gap opening strongly sup-

presses the one-particle crossover so that the d-wave su-

perconducting transition via the two-particle crossover is

strongly assisted in the weakly coupled ladder system.

Acknowledgements

J.K was supported by a Grant-in-Aid for Encourage-

ment for Young Scientists from the Ministry of Educa-

tion, Science, Sports and Culture, Japan.

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[5] We ignore the misfit of the rungs in the neighboring ladders

which exists in real Sr14−xCaxCu24O41 compounds, since

we take the continuum limit along the leg and consequently

the misfit never appears in the present theory.

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Sov.Phys.JETP62

Page 5

g0

(1)

Lm

Rm

Rm

Lm

Rm

Rm

Lm

Lm

Lm

Rm

Lm

Rm

Rm

Lm

Rm

Lm Lm

Rm

Rm

Lm

Rm

Lm LmRm

kFA

kFB

-kFA

-kFB

LB LA RA RB

E

(a)

(b)

gt

(2)

gt

(1)

gf

(2)

gf

(1)

g0

(2)

Page 6

One-particle Process

Two-particle Process

Page 7

+

+

d

dl

=

d

dl

=

(b)

d

dl

=

+

+

d

dl

=

(a)

(c)

(d)

Page 8

(a)

2

6

8

1.5

0.5

1

-2

-1.5

-0.5

-1

lcross

VSCd(l)

l

(c)

4

(b)

1.5

24

6

8

0.5

1

-2

-1.5

-0.5

-1

-OO

VSCd(l)

lc

l

268

-1

1

4

g(1)

0

g(2)

0

g(1)

t

g(2)

t

g(2)

f

g(1)

f

0.3

U=0.3

~

t (l)

~

t (l)

~

t =0.01

0

~

t =0.04

0

~

l

Page 9

0

0.01

0.02

00.01 0.02 0.030.04

t

2D

Tcross

t c

~

~

T~

~

Tc

~

SCd

0

SGM