Phases from Lightly-Doped Two-Legged Spin Ladders

10 Pages·0.18 MB·

by  Yujoung Bai

Various Spin Phases from Lightly-Doped Two-legged t-J Spin Ladders
Yujoung Bai
Center for Strongly-Correlated Materials Research, 27-113, Seoul National University, Seoul 151-742, Korea
Sung-Ik Lee
National Creative Research Initiative Center for Superconductivity, Department of Physics, Pohang University of Science and
Technology, Pohang, Kyungbuk 790-784, Korea
0
0
0 We study two-leg spin ladder systems analytically by using a t-J model which includes the in-
2 terchain spin exchange and the interchain hopping term. The spin part is mapped to a quantum
n sine-Gordon model via a bosonization method that fully accounts for the phase fluctuation. The
a spin gap in the Luther-Emery phase evolves to gapless Luttinger liquid phases in even-ladders at
J certaindopingconcentration. Thistransitionoccursasdopingrateincreasesand/ortheratioofthe

1 interchain spin exchange and hopping to the intrachain couplings decreases. We also estimate the
transition temperature at which the conventional electron phase can be deconfined to spinon and

] holon.
l
e
-
r
t PACS numbers: 75.10.Jm, 71.27.+a, 71.10.Hf, 74.20.Mn
s
.
t
a
m
I. INTRODUCTION
-
d
n Low-dimensional quantum magnetic systems, such as spin chains, spin ladders, and planar lattices provide very
o useful information about the physics of high T superconductors. During the past few years, spin-ladder systems
C
c have been studied extensively by various groups [1] [2] [5], especially by numerical methods or by mean-field (MF)
[
calculations [3] [6]. A“spin ladder” consists of parallel “chains” of magnetically bound Cu atoms connected by

2 interchaincouplings(“rungs”),asfound inSrn 1Cun+1O2n, La4+4nCu8+2nO14+8n (Fig. 1). Spinladdercompounds
v may become superconductors if doped with ch−arge carriers [8] [9]. Recently, superconductivity has been found in
2 Sr Ca Cu O [7] below 12 K and with external pressure of 3 GPa.
14 x x 24 41
0 Be−ginning with the half-filled ground state of a spin liquid, short-range singlets are formed predominantly across
3
therungsoftheladdersfromtheantiferromagnetic(AF)states. Theunusualspin-liquidnatureofanundopedparent
6
systemandtheevolutionofthefinitegapupondopinghavealsobeenstudiedbynumericalmethods[10]. Excitations
0
9 ofthesingletgroundstateyieldspintripletswhichpropagatealongtheladderdirection. Thestudyofeven-legladder
9 systems hasshownthatspingapsdecreaseas the number oflegs increase. ( The gapremains finite downto J =J ,
/ whichisasignatureofsuperconductivity[11]). Withinthemean-fieldtheory,innumericalworksandrenorma⊥lizatiokn
t
a group (RG) calculations, a superconducting phase with a d-wave nature emerges from the short-range RVB liquid
m
upon doping. Then, new quasiparticle (QP) states appear which carry both charge (positive) and spin.
- Intheweak-couplingapproach,onestartswithtwonon-interactingspinchainsandturnsonperturbativeinterchain
d
interactions. Forfiniteinterchaininteractions,thismethoddoesnotaccuratelydescribethecorrelationsnearthephase
n
transitions. Zhang et al. [12] have studied two-legt J ladders without the interchain exchangeby mapping the spin
o −
c part to a sine-Gordon model. Our study includes the spin exchange interaction across the rungs of the spin ladders
: andisbasedona(nearly)exactlysolvablemodel. Basedonthesolitondescription,wediscussthephasesandpossible
v
transitions between them. Exactly solving for spin ladders gives better understanding of the correlations between
i
X phaseexcitationsandtopologicaldefects, while comparingwithsomeweak-couplingstudies [4]andMFresultswhich
r partly addressed this issue [14].
a
Weuseat-J modelfortwo-legladdersinthestrongly(isotropically)coupledregimeinwhichtheinterchaininterac-
tions (t , J ) are larger(equal) than the intrachain interactions (t ,J ). Introducing weak coupling between two-leg
ladders⊥is st⊥aightforward for even-leg ladder systems. Recent studkiesk[13] have shown that ladder systems with an
odd number of legs are conducting Luttinger liquids in the odd-parity channel and spin liquids in the even parity
channel [14]. Odd-legged ladders can be mapped to a single-chain system while even-leggedladders are equivalent to
a double-chain ladder.
1
II. THE MODEL
An effective model for the t-J Hamiltonian is obtained as follows :
H = t (C+ C +h.c.)
− k X X liσ ljσ
l=1,2<ij>,σ
t (C+ C +h.c.)
− ⊥ X X liσ l′iσ
l,l′=1,2<ij>,σ
1
+J (S S n n )
liσ ljσ liσ ljσ
k X X − 4
l=1,2<ij>,σ
1
+J (S S n n )
liσ l′iσ liσ l′iσ
⊥ X X − 4
l,l′=1,2<ij>,σ
+µ C+ C (1)
liσ liσ
X X
l=1,2 σ
where<ij >isfornearest-neighbor(j =i+1)interactions,listhelegindexandσ isthespin( )index. Instrongly
↑↓
coupled t-J models with a large Coulomb repulsion (large U/t and large J /J ), we impose the single-occupancy
constraint C+C = 0 or 1 at each site. The electron chemical potent⊥ial kµ is added for the constraint in the
σ iσ iσ
form of a LPagrange multiplier. The charge(hole) concentration is related to µ(1 δ) = 1/(1+e−βµ). In order to
−
study the spinpartandthe chargepartseparately,wewrite the electronoperatorasthe productofachargecarrying
hPo=lon(1hinand a)rpesdeuudceosspthinefoopuerraotno-rsitSei:stCati↓es=0P, h>iS,i+0P, +>≡, 1h,iS>i+a,nCdi↑1=, >PhtoiSti−hePp+hy≡sihciaSli−H.ilTbehretpsproacjeecstihoonleop>e,rato>r
i, σ
and >−,th−ereby imposingsingle-occupancy|at↑each| si↓te. S|in↑ce the c|har↓gesumrule( C+C =1 δ)| holdsw|h↑en
|↓ σ iσ iσ −
the projectionoperatorsareneglected, we equatethe electronoperatorwith h S. TPhe pseudospinoperatorsacting
at site i turn into spinless fermion operators a via a Jordan-Wigner transformNation.
III. CONTINUUM LIMIT AND BOSONIZATION
In the spin exchange part, the pseudospin operators anticommute on the same site and commute on the different
sites. so, we have H(J) parts become J Ni=1(Si+Si−+1 +h.c.) and J i,l(Si+,lSi−,l+1 +h.c.). From Jordan-Wigner
k ⊥
transformation,theintrachaintermSi+Si−+P1 =a+i exp[−iπa+i ai]ai+1 =a+iP(1−2a+i ai)ai+1 forspin 12-system,whilethe
interchaintermSi+,lSi−,l+1 =a+i,la,i,l+1−2a+i,la+i,lai,lai,l+1. Weintroducespinorfieldsφα fortheslowly-varyingfieldai =
a /in whereα=evenateven(2s)siteiandoddatodd(2s+1)sitei. Theeven-siteandoddsitearecoupledbynearest-
i
e
neighborinteraction,yielding N a+a = N2 [φ+(2s) φ (2s+1) φ (2s 1) +φ+(2s+1) φ (2s) φ (2s 2) ].
i=1 i i+1 s=0 e { o − o − } o { e − e − }
In continuum, the fermion fiePld is given φα(sP) = √2a0Ψα(x) where x = 2sa0 with the lattice constant a0. Then,
φ (2s +1) φ (2s 1) 2a √2ae∂eΨ (x). The commutation relations are Ψ (x),Ψ (y) = δ δ(x y) and
o o 0 0 x 0 α β αβ
− − ≈ { } −
φ (n),φ (n) =δ δ . With the relation for exponentiated operators eAeB =e[A+B] e [A,B]/2 =eB eA e [A,B] ,
α β αβ nn − −
{we map the s′p}in exchange′ interaction terms
J SiσSjσ =J (Si+Sj−+Si−Sj++SizSjz)
k X k
<ij>,σ
=J (a+a 2a+a+a a +h.c.) H(J ). (2)
k i i+1− i i i i+1 ≡ k
Now the intrachain spin exchange paret oef Hamiletoneianebeecomes
H(J )=2a J dx
k 0 kZ
[ Ψ+(x)iσ1∂ Ψ (x)+Ψ+(x)Ψ (x)Ψ+(x)iσ1∂ Ψ (x)] (3)
η x η η η η x η
where Ψ is the Fermi field of two species(bonding, anti-bonding band) and σ1 is the Pauli matrix.
η
Then, we bosonize the Fermi field with phase fluctuation parameterswhich canvary depending onthe interactions
(ratherthantheparameterbeingaconstant√4π). EmployingtheoperatorexpansionoftheFermifield[15],Ψ (x)=
η
x
exp[iαφ (x)]exp[iβ dy∂ φ (y ,y)]wheretheBosefieldφanditscanonicalconjugatemomentumΠ=∂ φsatisfy
η 0 η 0 0
R−∞
2
[φ(x),Π(x )]=iδ(x x) and [Π(x),∂ φ(x)]= i∂ δ(x x). In order to satisfy the commutation relation of Ψ (x),
′ − ′ 0 ′ π x − ′ η
the phase fluctuation parameters α and β are related as 2αβ = π. The Fourier components of density operators
ρ (q) = Ψ+(k+q)Ψ (k) satisfy the boson commutation relat±ion [ρ (q),ρ ( q )] = Lq δ , [ρ (q),ρ ( q )] =
η k η η R R − ′ −2π qq′ L L − ′
Lq δ Pand [ρ (q),ρ ( q )]=0, where ρ (x) ρ (x)=Π andρ (x)+ρ (x)= 1∂ φ is the fermiondensity
d2πeviatqiqo′nfromRits aveLra−ge.′Introducing thReηdual−fielLdη x dyΠη (y)=R1ηθ (x) aLnηd linear−iπzinxg tηhe four branches of the
η π η
spectrum near the Fermi surface( kF), Ψη(x) = eikFRx−R∞η(x)+e−ikFxLη(x) with the right-going (R) and left-going
±
(L) components for each specie,
1
R (x)= F exp[ i(θ (x) φ (x))],
η η η η
√2πa − −
0
1
L (x)= F exp[ i(θ (x)+φ (x))] (4)
η η η η
√2πa −
0
where the Klein factors F =exp[ iθ ] are ladder operators [16] [17], raising/loweringthe η-fermion number by one.
η η
−
Theyaretakentoyieldthesamesignsforthespecies(η)thusignoredinthethermodynamiclimit,sincethetransverse
hopping and spin exchange lead to interchain pair fluctuations [4] with both specie participating. Also, the Ψ fields
arewrittenasevenΨ = 1 (R +L )andoddΨ = 1 ( R +L ). Resultantly,the intrachainexchangeJ maps
eη √2 η η oη √2 − η η k
to the form of (∂ φ )2+2cos(βφ ).
µ η 2 η
From the quartic interchain exchange J part of Hamiltonian, 2J a+a+a a =
⊥ − ⊥ i,l i,l i,l i,l+1
2iJ (2a )2[Ψ+ Ψ Ψ+ ∂ Ψ +Ψ+ Ψ Ψ+ ∂ Ψ ] (5)
− ⊥ 0 l,e l,e l,e x l+1,e l,o l,o l,o x l+1,o
There are actually ten terms giving non-zero contribution to the resultant sine-Gordon, three of which are shown
below.
H(J )= J S S = 2J a dx
⊥ − ⊥X 1iσ 2iσ − ⊥ 0Z
i,σ
[ L+∂ R R+L R+∂ L +R +L L+∂ L +..]. (6)
1 x 2− 1 1 1 x 2 2 2 2 x 1
The terms in the integrand above break the left-right (chiral) symmetry which arises from the Umklapp scatter-
ing between different chains (and backscattering between opposite spins). The interchain hopping term H(t ) =
t (C+ C +H.C.)yieldsidenticalformsinwhichweputtheintrachain chargehoppingOPh+h =⊥αδ
− ⊥ iσ 1iσ 2iσ l,i l,i+1 ≡
δ whPere 0<α<1 and interchain OP h+h =δ[1+ tk(1 α)] δ . Actual values of α, δ , δ and above rates
dkepend on the interaction strengths t , tl,i. l+1,i t⊥ − ≡ ⊥ ⊥ k
Thus, the spin part of the Hamilton⊥iankcollected from the intrachain and the interchain contributions is
1 β
H = dx [(∂2φ +∂2φ )+Wcos( )(φ +φ )] (7)
s Z 2π2 µ 1 µ 2 2 1 2
e e e e
where φ is a rescaled field from φ = (1+2Wπ)φ with the lattice constant a = 1. This is a quantum sine-Gordon
0
equation which is the equation of motion of solitons (which are magnons with spin 1 (charge 0) made of two spinons
e e
bound by a confining potential).
IV. PHASE FLUCTUATION AND THE INTERACTION STRENGTHS
The constant prefactor is the ratio of interaction strengths interchain to intrachain
δ t J
W = ⊥ ⊥− ⊥ (8)
(δ t J )
k k− k
and is related to the phase fluctuation parameter β2 =16π/(1+2W). In case of frustrated AF (which may occur in
the ground state of odd-legged ladders), the sine-Gordon equation simplifies to H = 1 [(∂ φ)2+W cos(βφ)]. For
2π2 µ
any finite value of β, the system has a zero-point fluctuation, In the dilute-soliton limit, the quantum fluctuation is
e e
dominated by the hard-core interaction between the spinless fermions [16]. Depending on the parameter β or W, we
have various soliton phases : (1) W = JJ⊥k ≡ W0 : This is undoped state whose ground state is a AF spin liquid,
3
as known from existing numerical studies [11]. (2)W W : Here we have Luther-Emery liquid with the gapped
0
≥
spinon(soliton) and gapless charge excitations. In the lower-temperature regime, excitonic soliton-antisoliton bound
statesexist. Thesecorrespondtointegerspinmagnonstateswhichareboundstatesoftwospin-1 spinons. TheCDW
2
inherited from chains exists as well as the Luttinger liquid phase holons. When temperature increases, the excitonic
bound states can be brokento spin-triplets (S =1 magnon)whichmake the systemspin-gapmetal. (3) 1 <W <W
2 0
: In this case, repulsive interaction between spinons exists, and there are no stable solitons. This is equivalent to the
strong fluctuation of a gauge field. SDW is in the background. (4) 0 < W 1 : The spinons are a Luttinger liquid
≤ 2
as shown in algebraically decaying spin-spin correlation given below. The charge part is a Luttinger liquid as in (2).
(5) W = 0 or δ J = t : This limit corresponds to single soliton(spinon gas) mode. (6)W = or δ J = t :
This correspond⊥s t⊥o the ⊥classical limit with a finite soliton density. The electron states have dimen∞sionalkcrkossovekrs
from the incommensurate 1D regime for W < 1 to commensurate 2D-like states for W <W. At low doping and at
2 0
low temperatures, the system can develop superconducting channel [10]. An excited spin phase generates magnons
fromtriplets andsolitonswhichpropagatein the backgroundofspin-density-wave(SDW).Weak but finite anisotropy
or spontaneous breaking of chiral symmetry cause spin stiffness and low-energy excitations. The interplay of the
excitationmodes,suchasorder-preservingcollectivemagnonsandorder-destroyingtopologicalsolitons,maypossibly
lead to some long-range or intermediate-range order. Considering the asymptotic behaviours shown from scaling
equations (Appendix), a schematic phase diagram is given in Fig. 2, which shows the various spinon and electron
phases with the parameter W.
V. CHARGE PART
TheseparatedchargepartoftheHamiltonianisdiagonalizedviaaBogoliubovtransformationoftheholonoperators.
H = E (f+f +constants) where the holon excitation spectrum E = 2t cosk+δ t is valid for t t (1
cos2δPπ)kfrokmkthekLuttinger sum rule. The chargephase with gaplessexkcitat−ionkforms−a T⊥o⊥monaga-Luttin⊥ge≥rgrkoun−d
state. According to numerical works [18] [11], holes have lower energies when they form pairs across the rungs of
the ladders. Thus, low-energy excitations include hole-hole pairs and hole-spin bound states (with positive charge)
propagatingalongtheladderwithakineticenergyofabout4t2/J . Theseexcitationsaswellasthefluctuationsofhole
k ⊥
pairs move against the lowest-energybackground of charge-density wave(CDW). The holon speed is v =2t sinδπ.
h
k
VI. CORRELATIONS
The asymptotic electron Green’s function can be calculated from perturbation theory excluding the vertex correc-
tions [19] [20] and is given by
ei2kFx
G (9)
e ≈ (x2 v2t2)γ(x v t)η(x+v t)η
− s − h h
where γ =(√1+W2 1)/2 with a spinon velocity v =v 1 and η =1/2. The propagatorG can be determined
s F e
− ≡
by convoluting the holon propagatorand the spinon propagatorand has no simple poles for low-energystates. Thus,
the Fermi surfaces are considered separately for the holon and for the spinon. The size of the holon Fermi surface
is found from G (k,ω) = δ (δ being the doping concentration). The ground state of the half-filled insulator
k,ω h
is a disorderedP(or dimerized) spin liquid made of spin singlets. These singlets which form across the rungs become
spin triplets as they are excited. Also, the triplets can propagate along the ladder as well as along the chain. The
propagation across the rungs and ladders are “incoherent” in the sense that there is no stable eigenstate for the
electron. The quantumnumbers, suchas spin, charge,andmomentum, are notthe same throughoutthe propagation
from one leg to another [21]. As separated phase fields, the charge and the spin degrees of freedom propagate with
different speeds. At small lengths, the phase separation may manifest itself spatially in real space, for example as
striped phases, as discussed by some authors [22]. To see whether there is any strong-coupled fixed point, consider
turning off the interchain hopping t . The renormalization group(RG) dimension of the wavefunction comes from
rescaling the variables (x,y,t) (α⊥x,y,αt). From the propagator, we have ψ α−21−η. So, the Hamiltonian H
has the RG dimension of 1 +η.→The interchain hopping is irrelevant and the Lu→ttinger liquid becomes a fixed poin⊥t,
2
if and only if η > 1. This corresponds to the case of interchain interactions being much weaker than intrachain ones
2
(t t ). However, in strongly coupled ladders, J /J remains sufficiently large to develop gaps [23] [24] and RG
flo⊥w≫is akwayfromthe unstable Luttinger liquid fixed⊥poiknt[4]. Giventhat t is greaterthanthe spin gap,there is no
strong-coupled fixed point, for even-legged ladders(η 1). The equal-time⊥spin-spin correlationis a power-law form
≤ 2
4
<S(r)S(0)> x √1+W2 cos2k x, (10)
− F
≈
which shows the spinons being a non-Fermi liquid. From second-order perturbation theory [25], the spin gap size is
J +J cosk + 1J2/J (3 cos2k ) with k =π.
⊥ k k 4 k ⊥ − k k
VII. DECONFINEMENT
In the 2D AF Heisenberg model or CP1 model, deconfinement of the electron (to a holon and spinon) occurs at
T = 0 [26]. As for spin-ladder systems, the “electron” phase is realized by having the spin and the charge phases
confined at finite T above the transition temperature T . Since the confinement-deconfinement(CD) transition
CD
occurs in the absence of ordering, it is like a Kosterlitz-Thouless transition occuring in (2+1) dimensional system.
From the partition function, T is found
CD
δ t J δ t J
k k− k <TCD < k k− k (11)
2k[δ t J ] k[δ t J ]
⊥ ⊥− ⊥ ⊥ ⊥− ⊥
where k is Boltzmann constant. Hence, for T > T , the electron propagator has the product form G (r,t)
CD s
∗
G ( r, t). Below T , the spin phase and the charge phase are separated due to the screening of the confining
h CD
− −
potential between holons and spinons. Spin-charge separation is a well-accepted fact for 1D systems which are
Luttinger liquids. Spin ladders make transition, at some doping rates δ, to a Luttinger liquid phase, where both spin
and charge excitations become gapless.
In summary, we map the spinons to solitons which are either magnons (excitons when bound) or excited phasons.
Fromthe solitongap,we have the spin-gappedmode while the charge partis gaplessLuttinger liquid. As the doping
increasestheladderscandevelopsuperconductingphaseorlosethespingapbeyondcertainholeconcentration. Also,
depending on the ratio of interchaincouplings to intraleg interactions,the spin ladders can become gaplessLuttinger
liquid. Our phase diagram is consistent with the features found in MF studies or weak-coupling approaches, such as
presence of gapless mode and superconducting phase from even-ladders. The comprehensive diagram shows possible
transitionsasanexplicitfunctionofcouplingratioanddoping. Theelectronpropagatorandthespin-spincorrelation
obey power-laws with the exponents depending on the interaction strengths and the doping rate. This shows the
correlation of topological defects and phase excitations. Constructive interferences of phase excitations (solitons and
magnons) may lead to the formation of intermediate-range or long range order, which is our forthcoming study. The
electron is seen as a confined phase of the gauge flux associated with each point. Below the estimated transition
temperature, the spin phase and the charge phase are separated. Henceforth, even-ladders show the crossover from
gapped 2D-like states to gapless 1D-nature, depending on the interleg couplings and/or the doping rate.
ACKNOWLEDGMENTS
Y. Bai is grateful to Dr. Chul Koo Kim for suggesting the problem and to M. Sigrist for invaluable discussions.
This work is supported by the Creative Research Initiative programof Ministry of Science and Technology, Korea.
APPENDIX A: RENORMALIZATION GROUP EQUATIONS
Following Fabrizio [4], Kishine [27] and Bourbonnais [28], we investigate the aymptotic behaviours in/around the
gappedphasebyRGflowofthecouplings. ThepartitionfunctionZ = DeS wheretheactionconsistsofone-particle
hopping process S and two-particle scattering S in the ladder, S =R S +S . The dispersion is linearized with
(1) (2) (1) (2)
respect to the bonding(B) and anti-bonding(A) band which lie above(inner) the bonding band(LB, LA, RA, RB),
whereL,Rdenotetheright-moving,left-movingfields. EachbandrangesbythebandwidthcutoffK . Theinterladder
0
repulsion generates scattering vertices of Fig. 3. The action for the one-particle process is given
S(1) = [G−Lη1(K )L+ησ(K)Lησ(K)+G−Rη1(K )Rη+σ(K)Rησ(K) (A1)
X X k k
K,ση=A,B
where K = (k ,k ,iǫ), K = (k ,iǫ) with the fermion frequency ǫ = (2n+1)πβ, and the Green’s functions are
n
G = [iǫ +vk (k⊥ +k k)] 1, Gk = [iǫ v (k k )] 1, with the Fermi momenta of the anti-bonding band
Lη η F Fη − Rη η F Fη −
k =k t /v kand of the bonding band−k =kk−+t /v . The action for the two-particle processes is given
FA F F FB F F
− ⊥ ⊥
5
2πv
S = F [g (σ σ σ σ )L+ (K )R+ (K )R (K )L (K )δ(K +K K K )
(2) β 0 1 2 3 4 ησ1 1 ησ2 2 ησ3 3 ησ4 4 1 2− 3− 4
X
η,K,σ
+g (σ σ σ σ )L+ (K )R+ (K )R (K )L (K )δ(K +K K K )
f 1 2 3 4 ησ1 1 η′σ2 2 η′σ3 3 ησ4 4 1 2− 3− 4
+g (σ σ σ σ )L+ (K )R+ (K )L (K )R (K )δ(K +K K K )
b 1 2 3 4 ησ1 1 η′σ2 2 ησ3 3 η′σ4 4 1 2− 3− 4
+g (σ σ σ σ )L+ (K )R+ (K )R (K )L (K )δ(K +K K K )] (A2)
t 1 2 3 4 ησ1 1 ησ2 2 η′σ3 3 η′σ4 4 1 2− 3− 4
wherethe dimensionlesscouplingsg(σ σ σ σ )=g(b)δ(σ σ )δ(σ σ ) g(f)δ(σ σ )δ(σ σ )andg isfortheintraband
1 2 3 4 1 3 2 4 1 3 2 4 0
−
process,g forinterbandforwardscattering,g forinterbandbackwardscattering,g forinterbandtunneling. Wescale
f b t
the two-spin processes to remove the high-energy components, by parametrizing the bandwidth cutoff K(l)=K e l
0 −
with length l. The scaling equations for the scattering vertices are from dlngαi(l) = zi(l) 2z(l) where i = b for
dl α −
backward scattering and i = f for forward scattering. The vertex correction diagrams to the second order of the
couplings(Zi(l)) are shownin Fig. 4. The linearizedbands consistof the slow modes(s) (from the centralportions of
α
the branches)andthe fastmodes(f) (from the outer edges). The actionis decomposedS +S +S +S .
(1),s (2),s (1),f (2),f
Expanding the fast components exp[S +S ] with respect to S and integrating out with the fast modes
(1),f (2),f (2),f
over dK(l) = 12K(l)dl, the partition function becomes Z = Dsexp[S(1),s+S(2),s+ ∞l=0 l1! <<S(l2),f >>c] where
<<>> denotes the connected diagram. The renormalized Rpropagator and hoppingPamplitudes give renormalized
c
action.
G 1(K )=[1+z(l)dl+ (dl2)]G 1(K) (A3)
−αη k O −αη
e
gi =[1+zi(l)dl+ (dl2)]gi (A4)
α α O α
e
t =[1+ (dl2)]]t (A5)
⊥ O ⊥
where α stand for (o, f, b, t) and i for backwafrd(1)and forward(2) scatterings. The Feynman diagrams contributing
to the order dl are evaluated. Those in Fig. 4 give zi(l), while the self-energy diagrams give z(l) of the scaling
α
equations (Fig. 5).
z(l)=g(1)(l)2+g(2)(l)2+g(1)(l)2+g(2)(l)2+g(1)(l)2+g(2)(l)2+g(1)(l)2
0 0 f f b b t
+g(2)(l)2 g(1) (l)g(2)(l) g(1)(l)g(2)(l) g(1)(l)g(2)(l) g(1)(l)g(2)(l) (A6)
t − 0 0 − f f − b b − t t
The scaling equations follow, provided small K <∆k ;
0 F
d
g(1)(l)= 2(g(1)(l)2+g(1)(l)g(2)(l))+2(g(1)(l)g(2)(l)2 g(1)(l)g(1)(l)g(2)(l))
dl 0 − 0 t t f t − f t t
2g(1)(l)(g(1)(l)2+g(1)(l)2+g(1)(l)2+g(2)(l)2 g(1)(l)g(2)(l)) (A7)
− 0 0 f t t − t t
d
g(2)(l)= (g(1)(l)2+g(1)(l)2+g(2)(l)2)+2(g(2)(l)g(2)(l)2 g(1)(l)g(2)(l)g(2)(l))
dl 0 − 0 t t f t − t t f
+2g(1)2(l)g(2)(l) g(1)3 g(1)2(l)g(1)(l) g(1)2(l)g(1)(l)
t f − 0 − f 0 − t f
2g(2)(l)(g(1)(l)2+g(2)(l)2 g(1)(l)g(2)(l)) (A8)
− 0 t t − t t
d
g(1)(l)= 2(g(1)(l)2+g(1)(l)2 g(1)(l)g(2)(l))+2[g(1)(l)g(2)(l)2 g(1)(l)g(1)(l)g(2)(l)]
dl f − f t − t t 0 t − 0 t t
2g(1)(l)[g(1)(l)2+g(1)(l)2+g(1)(l)2g(2)(l)2 g(1)(l)g(2)(l)] (A9)
− f 0 f t t − t t
d
g(2)(l)= (g(1)(l)2 g(2)(l)2)+2(g(2)(l)g(2)(l)2 g(1)(l)g(2)(l)g(2)(l))
dl f − f − t 0 t − t t 0
+2g(1)(l)2g(2)(l) g(1)(l)2g(1)(l) g(1)(l)2g(1)(l) g(1)(l)3
t 0 − t 0 − 0 f − f
2(g(2)(l)(g(1)(l)2+g(2)(l)2 g(1)(l)g(2)(l)) (A10)
− f t t − t t
6
d
g(1)(l)= 2(g(1)(l)g(2)(l)+g(1)(l)g(2)(l)+g(1)(l)g(1)(l) g(1)(l)g(2)(l)
dl b − b f f b b 0 − b 0
+g(1)(l)g(1)(l) g(1)(l)g(2)(l)) 2g(1)(l)[(g(2)(l) g(2)(l))2
0 b − 0 b − b b − f
g(1)(l)g(2)(l)+g(1)(l)g(2)(l)+g(1)(l)g(2)(l) g(1)(l)g(2)(l)+g(1)(l)2+g(1)(l)2]
− 0 0 0 f f 0 − f f 0 f
+2g(1)(l)(g(1)(l)2 g(1)(l)g(2)(l)+g(2)(l)2) (A11)
b t − t t t
d
g(2)(l)= 2(g(2)(l)g(2)(l) g(2)(l)g(2)(l)+g(1)(l)g(1)(l))
dl b − b f − b 0 b f
2g(1)(l)g(1)(l)g(1)(l) 2g(2)(l)[(g(1)(l) g(1)(l))2 g(1)(l)g(2)(l)
− b 0 f − b 0 − f − 0 0
+g(1)(l)g(2)(l)+g(1)(l)g(2)(l) g(1)(l)g(2)(l)+(g(2)(l) g(2)(l))2]
0 f f 0 − f f 0 − f
2g(2)(l)(g(1)(l)2 g(1)(l)g(2)(l)+g(2)(l)2) (A12)
− b t − t t t
d
g(1)(l)= 2(2g(1)(l)g(1) (l) g(1)(l)g(2)(l) g(1)(l)g(2)(l)+g(1)(l)g(2)(l)
dl t − t f − t f − f t 0 t
+g(1)(l)g(2)(l))+2(2g(1)(l)g(2)(l)g(2)(l) g(1)(l)g(1)(l)g(2)(l)
t 0 t 0 f − 0 t f
g(1)(l)g(1)(l)g(2)(l)) 2g(1)(l)(g(1)(l)2+g(2)(l)2+g(1)(l)2+g(2)(l)2
− f t 0 − t 0 0 f f
+g(1)(l)2+g(2)(l)2 g(1)(l)g(2)(l) g(1)(l)g(2)(l) g(1)(l)g(2)(l)) (A13)
t t − 0 0 − f f − t t
d
g(2)(l)= 2(g(1)(l)g(1)(l) g(2)(l)g(2)(l) g(2)(l)g(2)(l))
dl t − 0 t − 0 t − f t
+2(2g(2)(l)g(2)(l)g(2)(l) g(1)(l)g(2)(l)g(2)(l) g(1)(l)g(2)(l)g(2)(l))
0 f t − f 0 t − 0 f t
+2g(1)(l)g(1)(l)g(2)(l) g(1)(l)g(1)(l)g(1)(l)) 2g(2)(l)(g(1)(l)2+g(2)(l)2+g(1)(l)2
0 f t − t f 0 − t 0 0 f
+g(2)(l)2+g(1)(l)2+g(2)(l)2 g(1)(l)g(2)(l) g(1)(l)g(2)(l) g(1)(l)g(2)(l)) (A14)
f t t − 0 0 − f f − t t
From the initial condition g(i)(0) = t2/Jπv I 0, the scaling equations give the strong-coupling fixed point ;
α F
≡ ≥
g(1) = 1, g(1) =0, g(1) =1, g(2) =1/4(2I 3), g(2) =1/4(2I+1), g(2) =1. However, the couplings actually do not
0 − f t 0 − f f
flowto this fixedpoint, asfar asthe interchainhopping is greaterthan the gapsize, since interchainpairfluctuations
are dominant excitations [4]. The scale invariant coupling for the charge part is g =(g(1) 2g(2)) (g(1) 2g(2)).
ρ,± 0 − 0 ± f − f
Theinvariantcouplingforthespinpartg =g(1) g(1) wherethe+signisforthetotalmodeand fortherelative
σ,± 0 ± f −
mode. From aboveg’s,the spin stiffness is determined K = 1+gµ. So, K =0 means that the system developed
µ q1−gµ σ,+
the spin gapphase, while the chargepartremains gaplessfrom K =0 (Luther-Emery phase)[29]. In our equation
ρ,
(8), the sine-Gordon mass term identifies W (Jv /t2)4 g1 ±where g respresents some strong coupling value
∼ F √vF2−g12 1
resulted from the scaling equations.
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[9] S.Miyakasa et al.,preprint (1998).
7
[10] C. A. Hayward and D. Poiblanc, Phys.Rev. B 53, 11721 (1996).
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(1) t J
t
J
(2)
rung
leg(chain)
FIG.1. spin ladder example : a two-leg ladder system. The dashed circles are for spin singlets (triplets, if excited) and for
holon pairs after doped.
8
E
(f)
T
CD
(∆)
(c) L)
F S (b)
A
(
Classical
(d) limit
(a)
(e)
0 1/2 (W) W
0
FIG.2. phase diagram : the energy scale vs. the ratio W =[δ t −J ]/[δ t −J ]
⊥ ⊥ ⊥ k k k
“AF SL” stands for antiferromagnetic spin liquid which occurs at W = W = J /J with δ = 0. The spin gapsize ∆ is
0 ⊥ k
marked for the point where gapped spinon phases branch from AF SL state. (a)gapped magnons; excitonic bound states
of spinons,Luther-Emery liquid(LE) with gapless charge excitation (b) LE with gapful spinons and gapless holons, CDW (c)
unstable solitons, gauge bosons, SDW (d) Luttinger liquid (paramagnetic); gapless excitations of both-charge and spin (e)
superconducting(with d-wavesymmetry) (f) electron phase: spinon and holon confined
(1) (2)
g g g
9
FIG.3. interaction vertices. Thesolidlinesarefortheright-goingfermionsandthedashedlinesfortheleft-goingones. g(1)
and g(2) denotebackward and forward scattering couplings, respectively.
+ + +
+ + + +
+ + + +
FIG.4. vertex diagrams to thenext-to-the-leadingorder.
10