Role of Interchain Hopping in the Magnetic Susceptibility of Quasi-One-Dimensional Electron Systems

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by  Yuki Fuseya

Typeset with jpsj2.cls <ver.1.2> Full Paper
Role of Interchain Hopping in the Magnetic Susceptibility of Quasi-One-Dimensional
Electron Systems
Yuki Fuseya1,2∗, Masahisa Tsuchiizu1, Yoshikazu Suzumura1 and Claude Bourbonnais3
1 Department of Physics, Nagoya University, Nagoya 464-8602, Japan
2 Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 133-0033, Japan
7 3 Regroupement Qu´ebecois sur les Mat´eriaux de Pointe, D´epartement de physique, Universit´e de Sherbrooke, Sherbrooke,
0 Qu´ebec, Canada, J1K-2R1
0
2
The role of interchain hopping in quasi-one-dimensional (Q-1D) electron systems is investi-
n
gated by extending the Kadanoff-Wilson renormalization group of one-dimensional (1D) sys-
a
temstoQ-1Dsystems.ThisschemeisappliedtotheextendedHubbardmodeltocalculatethe
J
temperature(T)dependenceofthemagneticsusceptibility,χ(T).Thecalculationisperformed
3
bytakingintoaccountnotonlythelogarithmicCooperandPeierlschannels,butalsothenon-
2
logarithmic Landau and finite momentum Cooper channels, which give relevant contributions

] to the uniform response at finite temperatures. It is shown that the interchain hopping, t⊥,
l reduces χ(T) at low temperatures, while it enhances χ(T) at high temperatures. This notable
e
- t⊥ dependence is ascribed to the fact that t⊥ enhances the antiferromagnetic spin fluctuation
r atlowtemperatures,whileitsuppressesthe1Dfluctuationathightemperatures.Theresultis
t
s atvariancewiththerandom-phase-approximationapproach,whichpredictsanenhancementof
.
t χ(T) by t⊥ over the whole temperature range. The influence of both the long-range repulsion
a
and the nesting deviations on χ(T) is further investigated. We discuss the present results in
m
connection with the data of χ(T) in the (TMTTF)2X and (TMTSF)2X series of Q-1D or-
- ganic conductors, and propose a theoretical prediction for the effect of pressure on magnetic
d
susceptibility.
n
o
KEYWORDS: quasi-one dimension, magnetic susceptibility, renormalization group, extended-Hubbard
c
[ model, long-range interaction, nesting deviation, crossover, (TMTTF)2X, (TMTSF)2X

2
v
1. Introduction theoretically. On experimental grounds, χ(T) does not
5
9 Interacting electrons in ordinary metals and semicon- show any change around Tx,4,5 and is still temperature
7 ductors are usually described in the framework of the dependent. This indicates that the influence of the di-
6 Landau’sFermiliquid(FL)theory.1Whatliesattheroot mensional crossover may be quite specific to observable
0
oftheFermiliquidtheoryistheconceptofsingle-particle quantities.Itisofparticularinteresttoexaminehowthe
6
0 excitations that behave as effective non-interacting elec- physical quantities of the Q-1D correlated electron sys-
/ trons at low-energy. In one dimension, however, the tems are modified through Tx.
t
a FL picture is no longer valid due to the strongly en- Somebasicaspectsofthisdimensionalcrossovercanbe
m hanced correlation effects, which lead to the absence easily illustrated by first looking at the non-interacting
- of quasi-particles and the emergence of decoupled col- case. When the interchain hopping t⊥ is much smaller
d lective spin and charge excitations, which characterize than the intrachain one t, the Fermi surface consists of
n
o different quantum liquid phases such as the Tomonaga- twoslightly warpedplanesas showninFig.1,where the
c Luttingerliquid(TLL).2 While propertiesofthe FL and energy scale of the warping is of order of t⊥. Now when
v: TLL are independently well characterized, their relative the temperature scale is much higherthan t⊥, the warp-
i importance for weakly coupled chains in the quasi-one- ing is incoherent due to thermal fluctuations, the quan-
X dimensional (Q-1D) case is much less understood. In Q- tum mechanical coherence of electrons is then thermally
r 1D systems, a dimensional crossover between TLL and confined along the chains and the system is effectively
a
FL phases is expected to take place on the temperature one dimensional (1D). By contrast, for the energy scale
scale. How such a crossover is achieved in practice is much lower than t⊥, the quantum coherence of electron
an important issue in weakly coupled chains of organic extendsoverseveralchains,thewarpingoftheFermisur-
conductors(TMTCF) X (C=S, Se,X= PF ,Br,ClO , face becomes effective, and the system is considered as
2 6 4
AsF6, etc.). two-orthree-dimensional.Thescalet⊥ thuscorresponds
Sucha crossoverhas been observedin transportprop- to the crossovertemperatureTx fornon-interactingelec-
erties of these materials.3 The temperature (T) depen- trons. The presence of interactions, however, modifies
dence of resistivity (ρ) exhibits a power-law dependence this picture by reducing the crossoverscale, which leads
cρon∼sisTteαnt(αwi∼th 0a.5T)LfLorbTehaavbioovr.e2TItxis∼fo1ll0o0wKe,dwbhyicahTi2s Tx∗T∼het(mt⊥a/int)1t/hφeowrietthic0al≤diφffi<cu1lt.y6 in studying the Q-1D
dependence below T , which is expected in a FL. For systems resides in the treatment of the 1D fluctuations
x
the uniform magnetic susceptibility, χ, however, the ex- whichaffectthecouplingbetweenchainsandtherestora-
pected temperature dependence below T is not known tion of a FL behavior. At the lowest order of 1D per-
x
turbation theory, non-FL behavior basically comes from
∗E-mailaddress:[email protected]
2 J.Phys.Soc.Jpn. FullPaper AuthorName
the quantum interference between the Cooper and the 2. N-chain Renormalization Group
Peierls infrared instabilities. This is an effect that is not
2.1 Model
captured by approaches like random phase approxima-
WeconsideralineararrayofN weaklycoupledchains
tion (RPA), which are known to select only one type of
of length L at quarter filling, whose Hamiltonian H =
instability. The bosonization approach and exact solu-
H + H consists of a non interacting and interacting
0 I
tions based on the Bethe ansatz are extremely powerful
parts
in the purely 1D case, but are of limited use in higher
dimension where FL quasi-particles become relevant ex- H = tc† c t c† c +h.c. , (1)
0 − i,j,σ i+1,j,σ− ⊥ i,j,σ i,j+1,σ
citations.The renormalizationgroup(RG) technique for
iX,j,σh i
interacting electrons, however, can deal in a controlled
way with both TLL and FL behaviors in weak coupling. HI = [Uni,j,↑ni,j,↓+V1ni,jni+1,j +V2ni,jni+2,j].
In this respect, on the basis of the Kadanoff-Wilson Xi,j
(2)
RG scheme, Duprat and Bourbonnais (DB) applied a
two-variable momentum approximation for the coupling Herec (c† )isanannihilation(acreation)operator
constants, from which the competition between density i,j,σ i,j,σ
ofanelectrononthe i-thsite ofthe j-thchainwithspin
wave and superconducting states in Q-1D systems can
σ(= , ), and t (t ) is the intrachain (interchain) hop-
be analyzed.7 By calculating the instability of the nor- ↑ ↓ ⊥
ping. The intrachain interaction part is of the extended
mal state towards an ordered state below the crossover
Hubbard form with U as the onsite Coulomb repulsion,
scale T , they obtained the emergence of anisotropic su-
x and V (V ) as the nearest- (next-nearest-) neighbor re-
1 2
perconductivity from spin-density-wave correlations due
pulsion.Thedensityoperatorsaren =c† c and
to the presence of nesting deviations. These effects were i,j,σ i,j,σ i,j,σ
n =n +n .ThekinetictermH canbeexpressed
not treated in previous approaches, which are based on i,j i,j,↑ i,j,↓ 0
in the weak-coupling region as follows:
a sharp two cutoffs scaling procedure.8
At the end of the seventies, Lee et al. discussed the H0 = ξp(k)c†p,k,σcp,k,σ, (3)
roleofinterchain(Coulomb)couplingusingtheRGtech- p,k,σ
X
nique but where t was excluded.9 In their work, not
⊥
ξ (k) v (pk k )+ξ (k ), (4)
only density-wave and superconductive response func- p ≃ F − F ⊥ ⊥
tionswerediscussed,butalsotheuniformresponse,such ξ (k )= 2t cosk 2t cos2k , (5)
⊥ ⊥ ⊥ ⊥ ⊥2 ⊥
− −
asthecompressibilityandmagneticsusceptibility.When
where v is the Fermi velocity and the lattice constants
the interchain hopping is present, however, further im- F
provement of the RG method is needed to describe fluc- are set to be unity. The operator cp,k,σ (c†p,k,σ) anni-
hilates (creates) an electron with spin σ close to the
tuations that contribute to the uniform responses like
Fermi surface of the right k +k (p = +1) and left
magnetic susceptibility, compressibility and the Drude ≃ F
k k (p= 1) branches.The second-harmonicterm
weight. ≃− F −
in eq. (5), which denotes the nesting deviation, is given
Itisthepurposeofthepresentstudytofurtherdevelop
by t = t2 cosk /4tsin2k within the tight-binding
the Kadanoff-Wilson approach in the case of the Q-1D ⊥2 ⊥ F F
approximation,however,weconsidert andt asinde-
interacting electron systems to retain the full transverse ⊥ ⊥2
pendent parametersin this paper in order to distinguish
momentum-dependencefortheflowofcouplingconstants
the problem of the dimensionality and the nesting devi-
— a scheme that shall be denoted a N-chain RG in the
ations. As it will be shown later, t does not affect the
following.We shallconcentrateonthe RG calculationof ⊥2
magnetic susceptibility.
the uniform magnetic susceptibility and investigate the
The partition function Z(=Tre−βH) is written in the
dimensionalcrossoverbehaviorforthisphysicalquantity
functional integral form
thatis directly relevantto experiment.The temperature
dependence of the magnetic susceptibility χ(T) is calcu- Z = ψ∗ ψeS[ψ∗,ψ], (6)
lated to show how χ(T) varies with the interchain hop- D D
ZZ
ping t , long-range interactions and nesting deviations
⊥ where the ψ’s are the Grassmann fields. The action S is
which are well known to be relevant in organic conduc-
given by
tors.
The paper is organized as follows. In 2, the model S[ψ∗,ψ]=S0[ψ∗,ψ]+SI[ψ∗,ψ], (7)
§
Hamiltonian is introduced with the formulation of the
S [ψ∗,ψ]= G0(k˜) −1ψ∗ (k˜)ψ (k˜), (8)
N-chain RG method. The RG flow trajectories for the 0 p p,σ p,σ
couplingsconstantsacrossthe crossoverregionaregiven pX,k˜,σ(cid:2) (cid:3)
in 3.In 4,theχ(T)behaviorintheQ-1Dsystemsiscal- T
§ § S [ψ∗,ψ]= g ρ (q˜)ρ ( q˜)
culatedwithintwodifferentapproaches.Wefirstconsider I ρ p −p
−LN −
theRPAmethod,whichgivesχ(T)tolowestorderint⊥. Xp,q˜(cid:2)
The results are compared to the full N-chain RG treat- +g S (q˜) S ( q˜)
σ p −p
· −
ment, where t is treated as non perturbatively. In 5,
⊥
§ +g ρ (q˜)ρ ( q˜)+g Sz(q˜)Sz( q˜) , (9)
theresultsarediscussedinconnectionwithexperimental 4ρ p p − 4σ p p −
dataof(TMTCF)2X compoundsandweconcludein§6. wherek˜(q˜)indicatesthevectorintheFourier-Ma(cid:3)tsubara
space k˜ = (k,ω ) (q˜= (q,ω )) with ω = (2n+1)πT
n m n
J.Phys.Soc.Jpn. FullPaper AuthorName 3
(ω =2mπT).Here, the charge and spin-density opera- k
m
tors of the branch p are given by
1
ρ (q˜) 1 ψ∗ (k˜+q˜)ψ (k˜), (10) 2
p ≡ 2 p,σ p,σ
Xk˜,σ 3
1 3’ k
S (q˜) ψ∗ (k˜+q˜)σσσαβψ (k˜), (11)
p ≡ 2 p,α p,β 2’
k˜X,α,β 1’
where ψ (k˜) is the Grassmann field corresponding to
p,σ
cp,k,σ. The bare Green’s function is given by G0p(k˜) = E( )d /2: shell
[iω ξ (k)]−1.Thecouplingconstantsatquarterfilling 0
n p
−
(k =π/4,v =√2t)areexpressedintermsofU,V and
F F 1
Fig. 1. Illustration of the partial integration. The initial states
V as follows
2
(1 & 1’) are scattered to the excited states (2 & 2’), and then
g =U +4V +6V , (12) tothefinalstate(3&3’).Thecontributionoftheexcitedstate
ρ 1 2
located in the energy shell E0(ℓ)dℓ/2 are renormalized in the
gσ = U +2V2, (13) “new”renormalizedaction S[ψ∗,ψ]ℓ+dℓ.
−
g =U +4V +4V , (14)
4ρ 1 2
g = U. (15) Byassumingthepartitionfunctioninvariantunderthe
4σ
−
RG procedure, the renormalizedaction is obtained from
2.2 Renormalization Group
InthepresentN-chainRG,weachievedimprovements Z =eA(ℓ) ψ∗ ψeS[ψ∗,ψ]< ψ¯∗ ψ¯eS[ψ∗,ψ,ψ¯∗,ψ¯]dℓ
D D D D
for the Kadanoff-Wilson RG method by DB.7 The main ZZ ZZ
improvements are made on the following points. =eA(ℓ+dℓ) ψ∗ ψeS[ψ∗,ψ]ℓ+dℓ, (16)
(1) The non-logarithmic channels, which are the D D
ZZ
particle-hole (Landau) and the particle-particle where A(ℓ)correspondsto the freeenergydensity atthe
(Cooper+) channel on the same branch, are taken step ℓ. The partial integration with respect to ψ¯ and ψ¯∗
into account. The non-logarithmic channels have been can be performed by making use of the linked cluster
neglected in the previous RG, but become important theorem. Then the renormalized action is expressed as
at finite temperatures, since the logarithmic terms are
S[ψ∗,ψ] =S[ψ∗,ψ]
suppressed by the temperature and become comparable ℓ+dℓ ℓ
to the non-logarithmic terms. These non-logarithmic ∞ 1 4 n
terms contribute to the renormalization of g , and + S [ψ∗,ψ,ψ¯∗,ψ¯] ,
4ρ,4σ I,i
n!
are in turn involved in the calculation of magnetic n=1 *(cid:18)i=1 (cid:19) +
X X
susceptibility. (17)
(2) The couplings depends on full (three) momenta where S has i=1 4 ψ¯’s in the outer shell, and
I,i
(k ,k ,k ); the forth momentum k is determined ∼
1⊥ 2⊥ 3⊥ 4⊥
uniquely by the momentum conservation. Here, k1 and ψ¯∗ ψ¯(...)eS0[ψ¯∗,ψ¯]
... = D D . (18)
k2 are the incoming momenta and k3 and k4 = k1 + h i ψ¯∗ ψ¯eS0[ψ¯∗,ψ¯]
k k are the outgoing ones. When we consider the R R D D
2 3
only−fixed k = k for the Cooper channel and ne- TherenormalizationRsRofgρ,σ,4ρ,4σ attheone-looplevel
1 2
(n=2)areobtainedfromthe outer-shellaveragesofthe
glectthe non-logarithmicchannels,two variablesk and
1 quadratic term of the action, i.e., 1 (S )2 . Depending
k3 are enough to describe the momentum dependence onthe wayofchoosingtwoψ¯amon2ghfoIu,2r,Si is decom-
of the couplings. This scheme has been employed by I,2
posed into four components as
DB. However, when we consider all scattering channels,
namely the Cooper,Peierls,Landauand Cooper+chan- S =SC +SP +SL +SC+. (19)
I,2 I,2 I,2 I,2 I,2
nels,thecouplingscanbesuccessfullydescribedbythree
The corresponding diagrams are shown in Fig. 2. The
(k ,k ,k ) instead of two variables . In order to investi-
1 2 3 definitions of Sµ are given below. Each Sµ is orthog-
gatethe temperaturedependence ofthe physicalquanti- I,2 I,2
onal, so that the contractions in each channel µ can be
ties,wecalculatetheRGatthefinitetemperaturewhere
considered independently. Finally, the renormalized ac-
the Landauchannel playsan important role.So we keep
tion is obtained as
three variables in the present N-chainRG technique de-
spite the complicated formulation. S[ψ∗,ψ] S[ψ∗,ψ] = 1 (Sµ )2 , (20)
Based on the Kadanoff-Wilson RG procedure, we cal- ℓ+dℓ− ℓ 2 h I,2 i
µ
culate the effective action S[ψ∗,ψ] at the renormalized X
ℓ
whereµ=C,P,L,C+arerelatedtotheCooper,Peierls,
bandwidth E (ℓ). The partial integration is carried out
0
Landau, and Cooper+ channels, respectively.
for momentum located in the energy shell E (ℓ)dℓ on
0
The renormalized action at the one-loop level is ob-
both sides of the Fermi level (Fig. 1), where E (ℓ) =
0
E e−ℓ is the renormalized bandwidth at step ℓ of inte- tained from the shell averages of the quadratic term of
0
gration with the initial bandwidth E =2v k 2t.
0 F F
≃
4 J.Phys.Soc.Jpn. FullPaper AuthorName
SI (a) SI,2C (b) SI,2P +g4⊥(k1⊥,q⊥′ −k1⊥,k3⊥)∆¯∗4⊥,p(q˜′,k˜3)∆4⊥,p(q˜′,k˜1)
k1 k3 k1 k3 k1 k3
+c.c. , (24)
-k2 -k4 qC - k1 qC - k3 k3 - Q k1-Q where (cid:3)
different branch Cooper Peierls
(qC=k1-k2) (Q=k2+k3) ∆ss,p(q˜C,k˜) (iσy)ψ−p,α( k˜+q˜C)ψp,β(k˜), (25)
≡ −
(c) SI,2L Xα,β
k1 k1+q ∆ (q˜ ,k˜) (iσσ )ψ ( k˜+q˜ )ψ (k˜), (26)
ts,p C y −p,β C p,α
≡ −
-k2 -k2-q Xα,β
Landau
(q=k3-k1) Oc,p(Q˜,k˜)≡ ψ−∗p,σ(k˜+Q˜)ψp,σ(k˜), (27)
σ
SI (d) SI,2L (e) SI,2C+ X
k1 k3 k1 k1+q k1 k3 (Q˜,k˜) ψ∗ (k˜+Q˜)σαβψ (k˜), (28)
Os,p ≡ −p,α p,β
α,β
k2 k4 k2 k2-q q’ - k1 q’ - k3 X
Landau Cooper+ 1
same branch (q=k3-k1) (q’=k1+k2) Oρ,p(q˜,k˜)≡ 2 ψp∗,σ(k˜+q˜)ψp,σ(k˜), (29)
σ
X
Fig. 2. Diagrams of SI,2 for (a) Cooper, (b) Peierls, (c) Lan- (q˜,k˜) 1 ψ∗ (k˜+q˜)σαβψ (k˜), (30)
daurepresentationsonthedifferentbranch,and(d)Landau,(e) Oσ,p ≡ 2 p,α p,β
Cooper+representationsonthesamebranch.Thesolid(dashed) Xα,β
lineindicatesright(left)-goingelectrons.Theslashedlinesrefer
1
toaGrassmannfieldintheoutershell,ψ¯. ∆ (q˜′,k˜) τσσψ (q˜′ k˜)ψ (k˜), (31)
4||,p ≡ 2 p,σ − p,σ
σ
X
1
the action SI,2, which is given in eq. (19). The contrac- ∆4⊥,p(q˜′,k˜)≡ 2 τσσψp∗,−σ(q˜′−k˜)ψp,σ(q˜′). (32)
tion is performed by the outer shell average of all pairs σ
X
of two ψ¯’s. Each one is given by Here qC k1 k2, Q k2 + k3, q k3 k1 and
≡ − ≡ ≡ −
q′ k +k .
SIC,2 =−LTN gss(k1⊥,k1⊥−qC,k3⊥) I≡nte1rmso2feqs.(12)-(15),theabovecouplingconstants
k˜1X,3,q˜CXp (cid:2) are written as
g (k ) 1 1 3 g (k )
∆¯∗ (q˜ ,k˜ )∆ (q˜ ,k˜ ) ss 1,2,3⊥ = − ρ 1,2,3⊥ (,33)
× ss,p C 3 ss,p C 1 gts(k1,2,3⊥) 4 1 1 gσ(k1,2,3⊥)
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
+g (k ,k q ,k )∆¯∗ (q˜ ,k˜ )∆ (q˜ ,k˜ )
ts 1⊥ 1⊥− C 3⊥ ts,p C 3 ts,p C 1 gc(k1,2,3⊥) = 1 −1 −3 gρ(k1,2,3⊥)(34,)
+c.c. , (21) gs(k1,2,3⊥) 4 1 1 gσ(k1,2,3⊥)
(cid:18) (cid:19) (cid:18)− (cid:19)(cid:18) (cid:19)
g (k ) 1 1 1 g (k )
(cid:3)T 4|| 1,2,3⊥ = 4ρ 1,2,3⊥ (3.5)
SIP,2 =−LN gc(k1⊥,Q⊥−k3⊥,k3⊥) (cid:18)g4⊥(k1,2,3⊥)(cid:19) 2(cid:18)1 −1(cid:19)(cid:18)g4σ(k1,2,3⊥)(cid:19)
k˜X1,3,Q˜Xp (cid:2) 2.2.1 Cooper channel
¯∗ (Q˜,k˜ ) (Q˜,k˜ ) ThecontractionoftheCooperchannelattheone-loop
×Oc,p 3 Oc,p 1
level is carried out as
+g (k ,Q k ,k )¯∗ (Q˜,k˜ ) (Q˜,k˜ )
s 1⊥ ⊥− 3⊥ 3⊥ Os,p 3 Os,p 1 1 (SC )2
+c.c. , (22) 2h I,2 i
T dℓ
T(cid:3) = LN Nπv IC(qC,qC⊥,k⊥′ ;ℓ)
SIL,2 =−LN gρ(k1⊥,k2⊥,q⊥+k1⊥) ν=Xss,ts{k˜1X,3,q˜C}∗ F Xk⊥′
k˜X1,2,q˜Xp (cid:2) g (k ,k q ,k′ ;ℓ)g (k′ ,k′ q ,k ;ℓ)
× ν 1⊥ 1⊥− C⊥ ⊥ ν ⊥ ⊥− C⊥ 3⊥
¯∗ (q˜,k˜ ) (q˜,k˜ )
×Oρ,p 1 Oρ,−p 2 ∆∗(q˜ ,k˜ )∆ (q˜ ,k˜ ), (36)
× ν C 3⊥ ν C 1⊥
+g (k ,k ,q +k )¯∗ (q˜,k˜ ) (q˜,k˜ )
σ 1⊥ 2⊥ ⊥ 1⊥ Oσ,p 1 Oσ,−p 2 where
+g4ρ(k1⊥,k2⊥,q⊥+k1⊥)O¯ρ∗,p(q˜,k˜1)Oρ,p(q˜,k˜2) IC(qC,qC⊥,k⊥′ ;ℓ)≡ E04(ℓ) E (ℓ)+λA 1(q ,q ,k′ )
+g (k ,k ,q +k )¯z∗ (q˜,k˜ ) z (q˜,k˜ ) λX=±1 0 C C C⊥ ⊥
4σ 1⊥ 2⊥ ⊥ 1⊥ Oσ,p 1 Oσ,p 2 E (ℓ) E (ℓ)/2+λA (q ,q ,k′ )
+c.c. , (23) tanh 0 +tanh 0 C C C⊥ ⊥ ,
× 4T 2T
(cid:26) (cid:27)
(cid:3) (37)
T
SC+ = g (k ,q′ k ,k )
I,2 −LN 4|| 1⊥ ⊥− 1⊥ 3⊥ A (q ,q ,k′ ) v q +ξ (k′ ) ξ (q k′ ),
k˜1X,3,q˜′Xp (cid:2) C C C⊥ ⊥ ≡ F C ⊥ ⊥ − ⊥ C⊥− ⊥(38)
∆¯∗ (q˜′,k˜ )∆ (q˜,k˜ )
× 4||,p 3 4||,p 1 v q ξ (k q ) ξ (k ). (39)
F C ⊥ 1⊥ C⊥ ⊥ 1⊥
≡ − −
J.Phys.Soc.Jpn. FullPaper AuthorName 5
1.2 charge-couplingterms(e.g.,g g ,g g )givecontribu-
ρ 4σ 4ρ 4σ
IC, IP tionsofthesameorderofthespin-andcharge-decoupling
1 terms (e.g., g2,g g )
ρ σ 4σ
The Landau contraction [Fig. 4(c)] yields (see Ap-
0.8 pendix A for the details)
1
0.6 (SL )2
2h I,2 i
0.4 T dℓ
IL, IC+ = LN Nπv IL(q,q⊥,k⊥′ ;ℓ)
0.2 Xν,ν′{k˜1X,2,q˜}∗ F Xk⊥′
0 ×gν(k1⊥,k⊥′ −q⊥,k1⊥+q⊥;ℓ)gν′(k⊥′ ,k2⊥,k⊥′ −q⊥;ℓ)
0.01 0.1 E0( l1)/4T 10 100 ×Oν∗(q˜,k˜1⊥)Oν′(q˜,k˜2⊥), (44)
where
E (ℓ) 1
Fig. 3. PlotsofIC(qC=0),IP(Q=Q0)(solidline),andIL(q= I (q,q ,k′ ;ℓ) 0
Q0)0,I=C+(2(kqF′=,π)q′0a)nd(dqas′0h=ed(l2iknFe),πas).aWfuhnecntiothneoµfEch0a(ℓn)n/e4lTe,xwhihbeirtes L ⊥ ⊥ ≡ 4 AL(q,q⊥,k⊥′ )
completelogarithmicdivergence,Iµ reaches unity. E0(ℓ)/2+λAL(q,q⊥,k⊥)
λtanh , (45)
×" 2T #
λ=±1
X
The function I indicates the degrees of logarithmic di-
C
vergence; the case IC = 1 implies a Cooper instability AL(q,q⊥,k⊥′ )=vFq+ξ⊥(k⊥′ )−ξ⊥(k⊥′ −q⊥), (46)
showing a complete logarithmic divergence, while it is
v q =ξ (k ) ξ (k +q ). (47)
F ⊥ 1⊥ ⊥ 1⊥ ⊥
partially suppressed for I < 1 (Fig. 3). This Cooper −
C
channelisshowndiagrammaticallyinFig.4(a).We note ThefunctionILdoesnotreachtheunity,i.e.,theLandau
thatq issettobezeroinDB’sapproachassumingthat channel is always non logarithmic (Fig. 3).
C
the Cooper instability is always logarithmic. Similarly, the contraction of the Cooper+ channel at
the one-loop level [Fig. 4(d)] is
2.2.2 Peierls channel 1
(SC+)2
Attheone-looplevel[Fig.4(b)],thecontractioninthe 2h I,2 i
Peierls channel is given by
T dℓ
= I (q′,q′ ,k′ ;ℓ)
12h(SIP,2)2i LN ν,ν′X=4||,4⊥{k˜1X,3,q˜′}∗ NπvF Xk⊥′ C+ ⊥ ⊥
= LTN Ndπℓv IP(Q,Q⊥,k⊥′ ;ℓ) ×gν(k1⊥,q⊥′ −k1⊥,k⊥′ ;ℓ)gν′(k⊥′ ,q⊥′ −k⊥′ ,k3⊥;ℓ)
νX=c,s{k˜1X,3,Q˜}∗ F Xk⊥′ ×∆∗ν(q˜′,k˜3⊥)∆ν′(q˜′,k˜1⊥), (48)
g (k ,Q k′ ,k′ ;ℓ)g (k′ ,Q k ,k ;ℓ) where
× ν 1⊥ ⊥− ⊥ ⊥ ν ⊥ ⊥− 3⊥ 3⊥
×Oν∗(Q˜,k˜3⊥)Oν(Q˜,k˜1⊥), (40) IC+(q′,q⊥′ ,k⊥′ ;ℓ)≡ E04(ℓ)A (q′1,q′ ,k′ )
C+ ⊥ ⊥
where
I (Q,Q ,k′ ;ℓ) E0(ℓ) 1 λtanhE0(ℓ)/2+λAC+(q′,q⊥′ ,k⊥′ ) , (49)
P ⊥ ⊥ ≡ 4 E (ℓ)+λA (Q,Q ,k′ ) ×" 2T #
λX=±1 0 P ⊥ ⊥ λX=±1
× tanhE40T(ℓ) +tanhE0(ℓ)/2+λA2TP(Q,Q⊥,k⊥′ ) , AC+(q′,q⊥′ ,k⊥′ )=vFq′+ξ⊥(k⊥′ )+ξ⊥(k⊥′ −q⊥′ ), (50)
(cid:26) (cid:27) v q′ = ξ (k ) ξ (q′ k ). (51)
(41) F − ⊥ 1⊥ − ⊥ ⊥− 1⊥
A (Q,Q ,k′ )=v Q+ξ (k′ )+ξ (k′ Q ), (42) 2.2.4 Flow equations
P ⊥ ⊥ F ⊥ ⊥ ⊥ ⊥− ⊥ Taking into account all the above one-loop contrac-
v Q= ξ (k ) ξ (Q k ). (43)
F ⊥ 3⊥ ⊥ ⊥ 3⊥ tions (Fig. 4), we can finally write the flow equations in
− − −
The flow of I is similar to I (Fig. 3). the form
P C
2.2.3 Landau channel d 1
G (k ,k ,k )= F (k ,k ,k ,k′ ).
The one-loopcontractionof the Landau channel gives dℓ ν 1⊥ 2⊥ 3⊥ N ν 1⊥ 2⊥ 3⊥ ⊥
anon-logarithmiccontributiontotheRGequations.This Xk⊥′
(52)
channel was discarded in the previous RG. On the tem-
perature scale, however, the non-logarithmic terms be-
come effective when the Cooper and Peierls logarith-
micsingularitiesaresuppressed.Moreover,thespin-and
6 J.Phys.Soc.Jpn. FullPaper AuthorName
(a) Cooper (b) Peierls 1
+ I (q,q ,k′ )
k1 k’ k3 k1 k’ k3 8 L ⊥ ⊥
G (k ,k′ q ,q +k )G (k′ ,k ,k′ q )
- k1+ qC - k’ + qC - k3 + qC k1 - Q k’ - Q k3 - Q × σ 1⊥ ⊥− ⊥ ⊥ 1⊥ 4ρ ⊥ 2⊥ ⊥− ⊥
+(cid:2)G4ρ(k1⊥,k⊥′ −q⊥,q⊥+k1⊥)Gσ(k⊥′ ,k2⊥,k⊥′ +q⊥;ℓ)
(c) Landau G (k ,k′ q ,q +k )G (k′ ,k ,k′ q )
k1 -k’ k2 - q k1 k’ k2 - q − σ 1⊥ ⊥− ⊥ ⊥ 1⊥ 4σ ⊥ 2⊥ ⊥− ⊥
G (k ,k′ q ,q +k )G (k′ ,k ,k′ +q ) ,
− 4σ 1⊥ ⊥− ⊥ ⊥ 1⊥ σ ⊥ 2⊥ ⊥ ⊥
k1 + q - k’+ q k2 k1 + q k’+ q k2 (54)
(cid:3)
k1 -k’ -k2 - q k1 k’ -k2 - q
1
F (k ,k ,k ,k′ )= I (q,q ,k′ )
k1 + q - k’+ q -k2 k1 + q k’+ q -k2 4ρ 1⊥ 2⊥ 3⊥ ⊥ 4 L ⊥ ⊥
2G (k ,k′ q ,q +k )G (k′ ,k ,k′ q )
× − ρ 1⊥ ⊥− ⊥ ⊥ 1⊥ ρ ⊥ 2⊥ ⊥− ⊥
(d) Cooper+
k1 k’ k3 +(cid:2)G4ρ(k1⊥,k⊥′ −q⊥,q⊥+k1⊥)G4ρ(k⊥′ ,k2⊥,k⊥′ +q⊥;ℓ)
+G (k ,k′ q ,q +k )G (k′ ,k ,k′ q )
-k1 + q’ - k’ + q’ -k3 + q’ 4σ 1⊥ ⊥− ⊥ ⊥ 1⊥ 4σ ⊥ 2⊥ ⊥− ⊥
+G (k ,k′ q ,q +k )G (k′ ,k ,k′ +q )
4ρ 1⊥ ⊥− ⊥ ⊥ 1⊥ 4σ ⊥ 2⊥ ⊥ ⊥
Fig. 4. Diagrams considered in the flow equations at the one- +G (k ,k′ q ,q +k )G (k′ ,k ,k′ +q )
loop level. Here, qC =k1−k2,Q=k2+k3,q=k3−k1 and 4σ 1⊥ ⊥− ⊥ ⊥ 1⊥ 4ρ ⊥ 2⊥ ⊥ ⊥
q′=k1+k2 1I (q′,q′ ,k′ ) (cid:3)
− 4 C+ ⊥ ⊥
G (k ,q′ k′ ,k′ )G (k′ ,q′ k′ ,k )
Here,ν denotesthetypeofthecoupling,ν =ρ,σ,4ρ,4σ. × 4ρ 1⊥ ⊥− 1⊥ ⊥ 4ρ ⊥ ⊥− ⊥ 3⊥
ThefunctionFν containsthequadratictermsofthecou- +(cid:2)G4σ(k1⊥,q⊥′ −k1′⊥,k⊥′ )G4σ(k⊥′ ,q⊥′ −k⊥′ ,k3⊥) ,
plingGν(=gν/πvF)andthecontributionsoftheloopIµ. (55)
(cid:3)
The F ’s are given by
ν
1 1
Fρ(k1⊥,k2⊥,k3⊥,k⊥′ )=−4IC(qC,qC⊥,k⊥′ ) F4σ(k1⊥,k2⊥,k3⊥,k⊥′ )= 2IL(q,q⊥,k⊥′ )
× Gρ(k1⊥,k1⊥−qC⊥,k⊥′ )Gρ(k⊥′ ,k⊥′ −qC⊥,k3⊥) × −3Gσ(k1⊥,k⊥′ −q⊥,q⊥+k1⊥)Gσ(k⊥′ ,k2⊥,k⊥′ −q⊥)
+(cid:2)3Gσ(k1⊥,k1⊥−qC⊥,k⊥′ )Gσ(k⊥′ ,k⊥′ −qC⊥,k3⊥) +G(cid:2) 4ρ(k1⊥,k⊥′ −q⊥,q⊥+k1⊥)G4σ(k⊥′ ,k2⊥,k⊥′ +q⊥)
+ 1I (Q,Q ,k′ ) (cid:3) +G4σ(k1⊥,k⊥′ −q⊥,q⊥+k1⊥)G4ρ(k⊥′ ,k2⊥,k⊥′ +q⊥)
4 P ⊥ ⊥ 1I (q′,q′ ,k′ ) (cid:3)
× Gρ(k1⊥,Q⊥−k⊥′ ,k⊥′ )Gρ(k⊥′ ,Q⊥−k3⊥,k3⊥) − 4 C+ ⊥ ⊥
+(cid:2)3Gσ(k1⊥,Q⊥−k⊥′ ,k⊥′ )Gσ(k⊥′ ,Q⊥−k3⊥,k3⊥) × G4ρ(k1⊥,q⊥′ −k1′⊥,k⊥′ )G4σ(k⊥′ ,q⊥′ −k⊥′ ,k3⊥)
+ 1I (q,q ,k′ ) (cid:3) +G(cid:2) 4σ(k1⊥,q⊥′ −k1′⊥,k⊥′ )G4ρ(k⊥′ ,q⊥′ −k⊥′ ,k3⊥) ,
8 L ⊥ ⊥ (56)
(cid:3)
G (k ,k′ q ,q +k )G (k′ ,k ,k′ q ) If we neglect the k -dependence, these flow equations
× − ρ 1⊥ ⊥− ⊥ ⊥ 1⊥ 4ρ ⊥ 2⊥ ⊥− ⊥ ⊥
agree with those obtained in the 1D limit [eqs. (11)-(15)
−G(cid:2) 4ρ(k1⊥,k⊥′ −q⊥,q⊥+k1⊥)Gρ(k⊥′ ,k2⊥,k⊥′ +q⊥;ℓ) in Ref. 10]. Calculating simultaneously these differential
+G (k ,k′ q ,q +k )G (k′ ,k ,k′ q ) equations, we can obtain the renormalized couplings at
ρ 1⊥ ⊥− ⊥ ⊥ 1⊥ 4σ ⊥ 2⊥ ⊥− ⊥
eachtemperature.ForthecalculationoftheN-chainsys-
+G (k ,k′ q ,q +k )G (k′ ,k ,k′ +q ) ,
4σ 1⊥ ⊥− ⊥ ⊥ 1⊥ ρ ⊥ 2⊥ ⊥ ⊥ tem problem (2N points on the Fermi surface), we solve
(53)
(cid:3) 4N3 differential equations.
1 3. Flow of couplings & crossover behavior
F (k ,k ,k ,k′ )= I (q ,q ,k′ )
σ 1⊥ 2⊥ 3⊥ ⊥ 4 C C C⊥ ⊥
In this section, we show the results of the numerical
2G (k ,k q ,k′ )G (k′ ,k′ q ,k ) solutions of the flow equations (52) (56) for N = 31-
× σ 1⊥ 1⊥− C⊥ ⊥ σ ⊥ ⊥− C⊥ 3⊥ ∼
chain systems with U = 3t, V = V = 0, t = 0.01t
−(cid:2)Gρ(k1⊥,k1⊥−qC⊥,k⊥′ )Gσ(k⊥′ ,k⊥′ −qC⊥,k3⊥) at T =10−4t, and analyze the fl1ow of2the coup⊥lings. We
−Gσ(k1⊥,k1⊥−qC⊥,k⊥′ )Gρ(k⊥′ ,k⊥′ −qC⊥,k3⊥) soefttthheeomridneirmoufm1o0f−t8hEer,enwohrimchaliiszeldowbaenndowuigdhthf,oEr0T(ℓc=),
+ 1I (Q,Q ,k′ ) (cid:3) 10−4 10−6t. For N &09, the results become essentially
4 P ⊥ ⊥ ∼
independent of the number of chains. The present N-
× 2Gσ(k1⊥,Q⊥−k⊥′ ,k⊥′ )Gσ(k⊥′ ,Q⊥−k3⊥,k3⊥) chain RG exhibits a clear difference in the property of
spin gap between the odd-number chains (gapless) and
+(cid:2)Gρ(k1⊥,Q⊥−k⊥′ ,k⊥′ )Gσ(k⊥′ ,Q⊥−k3⊥,k3⊥) theeven-numberchains(gapped).Theaimofthepresent
+G (k ,Q k′ ,k′ )G (k′ ,Q k ,k ) paper is to investigate the many-chain systems, where
σ 1⊥ ⊥− ⊥ ⊥ ρ ⊥ ⊥− 3⊥ 3⊥
(cid:3)
J.Phys.Soc.Jpn. FullPaper AuthorName 7
(C+P) G (π/2,π/2,π/2): the scattering of the pair
ν
k π/2 and k π/2, where the Cooper
C P 1⊥ ≃ − 2⊥ ≃ −
and Peierls instabilities coexist.
(L) G (π/2,0,π/2):thescatteringofthepairk π/2
ν 1⊥
≃
and k = 0, where both the Cooper and Peierls
2⊥
−
instability are weakened.
Each pair states are illustrated in Fig. 5. Note that,
C+P in the actual calculations, π and π/2 in the arguments
L
of the above coupling constant are replaced as π
→
(30/31)π,π/2 (16/31)π due to the finite number of
→
chains.
The flows of the corresponding couplings are shown
Fig. 5. IllustrationsofthecouplingC,P,C+P,L. in Fig. 6. For E (ℓ) & 4t , each couplings are indistin-
0 ⊥
guishable and fully agree with the couplings of the 1D
case (the dash-dottedline inFig. 6).Onthe otherhand,
(a)
for E (ℓ) . 4t , a difference becomes noticeable, indi-
0 ⊥
catingthebehavioroftheinteractionisnomore1D,but
1.2
Q-1D.Thus, we can find the characteristicenergyof the
P
crossover as E = 4t . This is a clear evidence that the
1 x ⊥
structure of the interaction in Q-1D correlatedelectrons
C+P
do exhibit the crossoverbehavior.
0.8
1D Here a few remarks are in order about the crossover
0.6 behaviorandtemperatureappearingintheresponsefunc-
tions. In the presence of t , there is a characteristic
L ⊥
0.4 C temperatureforresponsefunctions,whicharecalculated
throughtheRGequations.Withdecreasingtemperature,
0.2 the effect of t⊥ on the RG equations becomes relevant,
suggestingachangeofthestatefromthe1DTLLregime
1e-05 1e-04 0.001 0.01 0.1 1
into the 2D regime. Generally, t in the presence of in-
E (l)/E ⊥
0 0 teraction is renormalized to a reduced value, and the
(b) crossover temperature is also renormalized as T∗ t1/φ
x ∼ ⊥
0.5 as obtained from the two-loop RG with the self-energy
corrections.6,11 Here we distinguish T∗ (E∗) from the
x x
0.4 P
bare one. There is no renormalization for T ( t ) at
x ⊥
∼
the one-loop level, i.e., no self-energy corrections are in-
0.3
cluded. Nevertheless, a crossover behavior does exist in
0.2 C+P the two-particlequantitiesatthis levelofapproximation
L as shown in Fig. 6. The effect of t⊥ on the two-particle
0.1 correlations such as those contributing to the magnetic
1D
susceptibility can be thus obtained even at the one-loop
C
0 level and the results are expected to carry over to the
case where T is replaced by T∗.
-0.1 x x
1e-05 1e-04 0.001 0.01 0.1 1 IntheQ-1Dregime,whereE0(ℓ).Ex,thepresentN-
E (l)/E chain RG shows that the coupling C (P) is reduced (en-
0 0
hanced)comparedwiththe1Dsituation.In1Dsystems,
(k, k) pairing contributes to both Cooper and Peierls
Fig−.G6σ. (b(C)ofolorrUon=li3n.e0)t,EVn1e=rgyV2fl=ow0,otf⊥th=e0c.o0u1tplaintgTs=Gρ10(−a)4ta.nd inst−abilities.IntheQ-1Dcase,however,theCooperpair-
ing suchasinCofFig.5 ischaracterizedbybadnesting
conditionsandcontributesweaklytothePeierlsinstabil-
ity.TheC pairingthusflowstothe attractivesectorand
thespin-gapdoesnotopen,sothatwefocusontheodd-
isattractivelyrelevant.Similarly,forsomePeierlspairing
number-chain systems in the present paper.
such as in P, the contribution to the Cooper instability
TherenormalizedcouplingsG depends onthreevari-
ν
is weaksothatthe couplingP flowsto the repulsivesec-
ables (k ,k ,k ). In order to see the essentialpoints
1⊥ 2⊥ 3⊥
tor and is repulsively relevant. On the other hand, for
of the renormalized couplings, we show the energy flow
Cooper pairs like the C+P, both the Cooper and Peierls
of the couplings for some typical pairs:
instabilitiesareinvolvedevenintheQ-1Dregime.Inthis
(C) G (0,0,0):the scatteringofthe Cooperpair k =
ν 1⊥ case,the coupling C+P becomes marginalbelow E .Fi-
x
k =0.
2⊥ nallyforthepairL,bothCooperandPeierlsinstabilities
(P) G (π,0,π): the scattering of the Peierls pair k are weakened so that coupling L is hardly renormalized.
ν 1⊥
≃
π and k2⊥ 0. Note that evenforthe coupling L orC+P,the couplings
− ≃
8 J.Phys.Soc.Jpn. FullPaper AuthorName
couplings. For the quantitative comparison of each in-
stabilities, we need to calculate the response functions,
which are presented in Appendix C
1.2
1.1 Interestingly, the structure of the present couplings
1
C+P 00..89 in Q-1D is quite similar to the f(k1,k2) for the two-
0.7 dimensional(2D)Hubbardmodelnearhalf-filling.12This
0.6
k2 0 C L P 0000....2345 steumggseshtasvtehaatlottheofesleimctirloanriitciepsrwopitehrttiehsosoef othfe2DQ-s1yDstesymss-
0.1
near half-filling. For example, the Landau parameter Fa
0
can also become positive in the Q-1D case, which sug-
gests the possible existence of the spin-dependent zero-
sound(zero-spin-sound).Thispossibilityissupportedby
0 the decrease of the magnetic susceptibility at low tem-
k
1 peratures as shown in the next section. Furthermore, it
mightbepossiblethatFsbecomesnegative,whichcorre-
1
spondstothereductionoftheDrudeweightintheQ-1D
case.
0.4
0.35 Thepossiblefixedpointsofthecouplingsintheparam-
0.3
C+P 0.25 eter space(k1⊥,k2⊥,k3⊥)aresummarizedinFig.8.The
0.2
0.15 couplingsontheCplanek1⊥ =k2⊥,areattractivelyrele-
k2 0 00..015 vantduetotheCooperinstability.OnthePplanewhere
C L P 0 k +k = π, the couplings are repulsively relevant
-0.05 2⊥ 3⊥ ±
duetothePeierlsinstability.AttheintersectionoftheC
and P planes leading to the C+P line, the couplings are
marginal below T due to the coexistence of the Cooper
x
andPeierlsinstabilities.Remaining couplings arehardly
0
renormalized in the Q-1D regime. These conditions for
k
1
the coupling constants have to be taken into account in
order to construct an effective low-energy Hamiltonian
Fig. 7. Renormalized couplings Gρ(k1⊥,k2⊥,k1⊥) (the upper in the Q-1D regime.
panel) and −Gσ(k1⊥,k2⊥,k1⊥) (the lower panel) for U/t =
3.0,V1/t=V2/t=0.0,t⊥=0.01tatT =10−4t.
plane C
(attractively relevant)
are slightly renormalized in the thermal shell E (ℓ) T
0
∼
due to the non-logarithmic terms, i.e., the Landau and
Cooper+ channels. (See E (ℓ)/E 10−4 in Fig. 6.)
0 0
∼
In Fig. 7, we plot the renormalized couplings
G (k ,k ,k ) along the plane k = k
ν 1⊥ 2⊥ 1⊥ 1⊥ 3⊥
line C+P
in the parameter-space (k ,k ,k ). The function
1⊥ 2⊥ 3⊥ (marginal)
G (k ,k ,k ) is the coupling in the limit of q 0,
ν 1⊥ 2⊥ 1⊥
→
and thus corresponds to the Landau’s interaction func-
k
tion f(k ,k ), which is the second functional derivative 3
1 2
of the energy per unit volume.1 As we can see from Fig.
7, there are two remarkablestructures.One is the valley
k
along k = k (the valley C), indicating the Cooper 1
1⊥ 2⊥
instability of which momenta are k and k . Another
1 2
is the ridge along k1⊥ = k2⊥ π (the r−idge P), indi- k2 plane P
catingthePeierlsinstabilit−yrefle±ctingthenestingvector (repulsively relevant)
Q = ( 2k , π). If the system is pure 1D (t = 0),
0 F ⊥
± ±
thevalueoftherenormalizedcouplingsare(G1D)∗ 0.7 Fig. 8. (Color online) Illustration of the total structure of the
ρ ∼
and ( G1D)∗ 0.1. The valley C is smaller and the fixedpointsforthecouplingsGρandGσ intheparameterspace
ridge −P isσlarge∼r than (G1D)∗, indicating that the valley (k1⊥,k2⊥,k3⊥).Thecouplings ontheplaneC (k1⊥ =k2⊥)are
ν attractivelyrelevantduetotheCooperinstability.Thecouplings
C is more attractive and the ridge P being more repul- on the plane P (k2⊥+k3⊥ = ±π) are repulsively relevant due
sive. Moreover, the height of the ridge is not uniform. tothePeierlsinstability.Alongthe lineC+P,thecouplings are
The cosine-like curve of the valley C suggests the d- or marginal below Tx due to the interference between Cooper and
f-wave instability, and the maximum (minimum) of the Peierlsinstability.
ridgePsuggeststhe hot(cold)spots ofthe density-wave
instability.7 Namely, we can obtain the information of
the symmetry of the Cooper pair and the structure of
the density-wavefluctuation only from the renormalized
J.Phys.Soc.Jpn. FullPaper AuthorName 9
4. Magnetic susceptibility with an infinite slope. When V increases, χ1D(T) is en-
1
hanced but the temperature profile remains essentially
In this section, we investigate the properties of the
the same; the influence of V being similar to the one
magnetic susceptibility. We first briefly review the prop- 1
ertiesofχ(T)inthe1Dlimit10 andthentacklethemain ofU.WhenV2 increases,however,χ1D(T)decreasesand
becomesflatatlowtemperature,thatisdχ1D/dT =
problem of χ(T) for Q-1D systems by first considering T→0
|
0 for U =2V (i.e., g =0).
the lowest order perturbation effect of t with respect 2 σ
⊥
to the 1D RPA solution for (t /T)2 1, and secondby In Fig. 9 (b), we compare the temperature profiles of
⊥ ≪ χ1D(T) for different approaches. The results of the RG
the full one-loop RG treatment where t is treated non
⊥
are shown by the solid lines and the 1D (renormalized)
perturbatively.
RPA form for χ1D(T),
4.1 Magnetic susceptibility in 1D 2 χ (T)
χ1D (T)= 0 , (57)
RPA πv 1+[G (T) G (T)]χ (T)/2
F σ 4⊥ 0
−
aregivenbythedashedlines.Hereχ (T)=tanh(E /4T)
(a) 0 0
is the bare magnetic susceptibility per branch p =
3.5 ±
V1 increases normalized by πvF. Gσ(T) denotes the renormalized
U=4t coupling at T, which is given by G (T) = G /[1 +
3 V1=0.5-2t σ σ
V2=0.5-2t Gσln(2T/E0)], and G4⊥ = (G4ρ −G4σ)/2. We can see
2.5 thateq.(57)includesthefluctuationbeyondtheordinary
RPA, where G (T) = G is independent of tempera-
σ σ
2 ture13 .Themaximumvaluescalculatedbythequantum
Monte Carlo(QMC)technique13 (for whichlowtemper-
1.5
atures cannot be reached due to sign problem) are in-
V2 increases dicated by the dash-dotted lines. The exact solutions at
1
T =0 are shown by the square and the circle for U =2t
0.5 and 4t, respectively.13,14
0 0.2 0.4 0.6 0.8 1 For U =2t, χ1D(T) of the RG and renormalizedRPA
T / t
results,eq.(57),quantitativelyagreeswiththatoftheex-
(b) act solutionandQMC. However,for U =4t,χ1D (T)is
RPA
1.2 sizablylargerthantheQMC,suggestingthatRPAresult
overestimatesthe effectofspincorrelations.This overes-
1
U=4t timationis notfoundin the full one-loopRGresult,and
χ1D(T)remainsclosetothatofQMC.Thisimprovement
0.8
isdue tonon-logarithmicterms,whichbecomeseffective
0.6 at high temperatures, and also comes from the mode-
U=2t
mode coupling terms.
0.4 RG Summarizing, the RG prediction for χ1D(T) agrees
RPA
QMC wellwiththeexactsolutionatT =0evenforlargeU.It
0.2
Exact , alsoagreesreasonablywellwiththeQMCresultsathigh
temperatures,thoughitonlyslightlydepartsfromQMC
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 at large U. Therefore the present RG technique yields
T / t
the mostreliableresults forthe uniformsusceptibility in
thecaseofthe1DextendedHubbardmodelforthewide
Fig. 9. (Coloronline)Magneticsusceptibilityof1D,χ1D(T),asa temperature region 0 T .t.
functionoftemperature.(a)Theblack-solidlinedenotesχ1D(T) ≤
for U = 4t, the gray-solid lines denote that for U = 4t,V1 =
0.5 ∼ 2t, the gray-dashed lines denote that for U = 4t,V2 = 4.2 RPA approach with the perturbation of t⊥
0.5∼2t.(b)Comparisonofχ1D(T)byRGtotheothermethod. First we see the variation of the couplings G on
ν
χ1D(T)byRPAisshownbydashed-lines.Themaximumvalues the basis of the perturbation of t . We adopt the 1D
of χ1D(T) by QMC locate around the dash-dotted-lines.13 The ⊥
Kadanoff-Wilson RG of Ref. 10, as a non-perturbative
exact solutionatT =0areindicatedbythesquareandcircle.
part. The functions I can be expanded in t (or
C,P,L,C+ ⊥
A ) as
µ
We plot the 1D magnetic susceptibility χ1D in Fig. 9 2
E (ℓ) A
0 C,P
(a) for (U,V1,V2)=(4t,0,0)(thick solid-line), (4t,0.5∼ IC,P ≃tanh 4T −f(E0(ℓ)/4T) 2T , (58)
2t,0) (thin solid-lines), (4t,0,0.5 2t) (thin dashed- (cid:18) (cid:19) (cid:18) (cid:19)
lines). (The results coincide with ∼those of Ref. 10, or E (ℓ) E (ℓ) A 2
I 0 cosh−2 0 g(E (ℓ)/4T) L,C+ ,
those of the following section in the limit t⊥ = 0.) For L,C+ ≃ 4T 4T − 0 2T
theHubbardmodel(U >0,V1 =V2 =0),χ1D(T)ispro- (cid:18) (cid:19) (cid:18) (59)(cid:19)
portional to 1/T at high temperatures, has a maximum
around T 0.2t, and then decreases towards the zero
temperatur∼e value χ1D(0) = (2/πv )[1 U/(2πv )]−1
F F
−
10 J.Phys.Soc.Jpn. FullPaper AuthorName
where (a) magnetic susceptiblity
k’
f(x) 1 1+ 2x 1 tanhx tanh2x 2xtanh3x , d q0 k k k
= +
≡ 4x(cid:20) (cid:18) − x(cid:19) − − (cid:21) d l
(60) k+q0 k+q0 k’+q0 k+q0
x2 cosh(2x)
g(x) − . (61) (b) density wave
≡ 3 cosh4x
k’
-Q k k k
(Here,weomitthe(q,q ,k )dependenceofI andA .) d 0
⊥ ⊥ µ µ = +
The flow equation (52) for q 0, which is relevant for d l
the magnetic susceptibility, ca→n be rewritten as follows. k-Q0 k-Q0 k’-Q0 k-Q0
G (T)=G1D(T), (62)
ρ ρ (c) superconductivity
Gσ(T)=G1σD(T)−2G2σf¯(T)(cid:18)tT⊥(cid:19)2, (63) dd l qC0 k-k=+qC0 k-k++qC0 -k’k+’q k-k+qC0
2 C0
1 t
G (T)=G1D(T)+ (G2 +G2 )g¯(T) ⊥ , (64)
4ρ 4ρ 2 4ρ 4σ T
(cid:18) (cid:19) Fig. 10. Diagrams for the renormalization of the pair vertex zµ
t 2 at the one-loop level. The triangles and the squares denote the
G (T)=G1D(T)+G G g¯(T) ⊥ , (65) pairvertexandthecouplingfor(a)the magneticsusceptibility,
4σ 4σ 4ρ 4σ T
(cid:18) (cid:19) (b) the density wave, and (c) the superconductivity. The wavy
and for G (G G )/2, linedenotes thecorrespondingsource-fieldshµ.
4⊥ 4ρ 4σ
≡ −
2
t
G (T)=G1D(T)+G2 g¯(T) ⊥ , (66)
4⊥ 4⊥ 4⊥ T Consequently,themagneticsusceptibilityisenhancedby
(cid:18) (cid:19)
t from that of 1D system on the basis of the pertur-
where ⊥
bation theory. Moreover, eq. (70) indicates that a shift
Gν(T)≡ N12 Gν(k1⊥,k2⊥,k1⊥;ℓ,T) ℓ→∞, (67) tfrhoemreχis1RDPnAoecxleiastrsfeevaetunraebsoigvneaTlxin∼g tth⊥e.Tcrhoisssosuvegrgefrsotsmth1aDt
kX1,k2 (cid:12)
(cid:12) to Q-1Din the temperature dependence of the magnetic
and f¯(T) dℓ f(E (ℓ)/4T), g¯(T) dℓ g(cid:12)(E (ℓ)/4T).
0 0 susceptibility. This is consistent with the experimental
For T → 0≡, fR¯(T) → 7ζ(3)/4π2 ≃ 0.2≡13R, g¯(T) → 0. The results for the TMTTF and TMTSF salts.4
flowequationswithfull-momentadependencearederived
Since this is the perturbation of (t /T), one may
⊥
in Appendix B. Here we use the relation
think that the perturbative form eq. (70) is not valid
A (q ,q ,k′ ) 2 = A (Q,Q ,k′ ) 2 for t⊥/T 1, namely where the crossover behavior is
{ C C C⊥ ⊥ } { P ⊥ ⊥ } expected t∼o take place. Even in such case, we can es-
kX1,k2Xk′ kX1,k2Xk′
timate χ (T) of eq. (69) from the numerical evalua-
RPA
= A (q′,q′ ,k′ ) 2 tion of G (T). As is shown later in Figs. 12 and 13, the
{ C+ ⊥ ⊥ } ν
kX1,k2Xk′ χRPA(T) so obtained qualitatively agrees with the per-
turbative form, namely, χ (T) is enhanced by t for
=8t2. (68) RPA ⊥
⊥ the whole temperature region.
Thereforethe enhancementofG ,asidefromG ,ispro-
ν ρ
portional to (t /T)2. 4.3 Renormalization group approach
⊥
Next, we estimate the magnetic susceptibility with Inapathintegralapproach,theuniformmagneticsus-
these G ’s. By extending the RPA form eq. (57), the ceptibilityortheresponsefunctionscanbecalculatedby
ν
magnetic susceptibility in Q-1D lead to be addingasetofsourcefieldsh totheaction.Thederiva-
µ
πv tionofresponsefunctionsisgiveninAppendix C.Inthe
[χ (T)]−1 = F 1+ G (T) G (T) χ (T)/2 .
RPA 2χ (T) σ − 4⊥ 0 case of the uniform magnetic susceptibility χ, we intro-
0
(cid:2) (cid:8) (cid:9) (69)(cid:3) duce the following Zeeman coupling of a source field to
the spin density operator:
By using eqs. (63) and (66), we can write χ (T) at
RPA
(t⊥/T)2 ≪1 in the form Sσh[ψ∗,ψ]= [hp(q˜)Oσ∗(q˜,k˜)+c.c.]. (73)
[χ (T)]−1 χ1D −1 πvF [∆G (T)+∆G (T)], Xk˜,q˜
RPA ≃ RPA − 4 σ 4⊥ The one-loop corrections yield
(cid:0) (cid:1) (70)
Sh[ψ∗,ψ] = h (q˜)z (k˜,k˜+q˜;ℓ)[ ∗(q˜,k˜)+c.c.]
where σ ℓ p σ,p Oσ
t 2 {kX˜,q˜}∗
∆Gσ(T)=2G2σf¯(T) T⊥ >0, (71) +χσ(ℓ)hp(q˜)hp(q˜). (74)
(cid:18) (cid:19)
t 2 where zσ,p is the pair vertex.
∆G (T)=G2 g¯(T) ⊥ >0. (72)
4⊥ 4⊥ T
(cid:18) (cid:19)