Strong-Coupling Behavior of Two $t-J$ Chains with Interchain Single-Electron Hopping

a r X i v :c o n d -m a t /9401024v 2 13 J a n 1994Strong-Coupling Behavior of Two t ?J Chains

with Interchain Single-Electron Hopping

Guang-Ming Zhang,?Shiping Feng,??and Lu Yu ?

International Center for Theoretical Physics,P.O.Box 586,34100,Trieste,Italy.

Using the fermion-spin transformation to implement spin-charge separation of constrained electrons,a model of two t ?J chains with interchain single-electron hopping is studied by abelian bosonization.After spin-charge decoupling the charge dynamics can be trivially solved,while the spin dynamics is determined by a strong-coupling ?xed point where the correlation functions can be calculated explicitly.This is a generalization of the Luther-Emery line for two-coupled t ?J chains.The interchain single-electron hopping changes the asymptotic behavior of the interchain spin-spin correlation functions and the electron Green function,but their exponents are independent of the coupling strength.PACS numbers:75.10.Lp,75.10.Jm

An important issue of current interest is whether the peculiar properties of one-dimensional(1D)Luttinger liquid(LL)[1],[2]will survive in two-and three-dimensions.The renormalization group(RG)studies seemed to indicate an instability of LL behavior with respect to the interchain single-electron hopping(SEH)t⊥[3].However,Anderson suggested that one should treat the intrachain correlations exactly including the spin-charge separation before switching on https://www.wendangku.net/doc/5115079232.html,ing the asymptotic Green functions in1D for a?nite Hubbard U,he argued that SEH is an irrelevant variable and named this property as”con?nement”of the1D Hubbard model[4].His idea has stimulated several further studies[5]–[10],most of which did not con?rm his conjecture in the strict sense.SEH is indeed renormalized to zero,but e-e or e-h pair hopping is generated,which drives the coupled chains towards a strong-coupling?xed point corresponding to superconducting or density-wave states.In fact,this type of instability was studied earlier in connection with organic superconductors [11].Nevertheless,this result is not convincing because the validity of the perturbative RG at strong-coupling?xed point with large U is questionable,as in the single-impurity Kondo problem.The Kondo physics is determined by the Wilson strong-coupling?xed point[12]. The poor-man’s scaling[13]correctly directs the RG?ow towards it,but the calculation can not be justi?ed by itself[14].

In this paper,we consider two coupled t?J chains,using a fermion-spin transformation, proposed recently by Feng et al.[15],where the charge degrees of freedom are described by spinless fermions,while the spin degrees of freedom are represented by hard-core bosons, which in turn,can be expressed as another type of spinless fermions via Jordan-Wigner transformation.The on-site local constraint for single occupancy is satis?ed even in the mean-?eld approximation(MFA)and the sum rule for physical electrons is obeyed.We combine this transformation with the abelian bosonization technique[1],[16]to consider the e?ect of SEH on the correlation functions.After spin-charge decoupling the charge dynamics can be solved trivially,while the spin dynamics can be mapped into noninteracting spinless fermions.This strong-coupling?xed point is similar to the Luther-Emery line of the single chain problem with back scattering[17].We con?rm that the spin-charge separation by itself

does not produce Anderson con?nement[8],[9].Moreover,SEH changes the asymptotic behavior of the interchain spin-spin correlation functions and the electron Green function, but their exponents are independent of the coupling strength t⊥.

We consider two coupled t?J chains

H=?t i,σ(C?1,i,σC1,i+1,σ+C?2,i,σC2,i+1,σ+h.c.)?μ i,σ(C?1,i,σC1,i,σ+C?2,i,σC2,i,σ)

+2J i( S1,i S1,i+1+ S2,i S2,i+1)?t⊥ i,σ(C?1,i,σC2,i,σ+h.c.),(1) with local constraint σC?i,σC i,σ≤1.Here C?1,i,σ(C?2,i,σ)creates an electron with spinσat site i on chain1(2),and S1,i( S2,i)is the corresponding electron spin operator;t is the intrachain hopping andμis the chemical potential.The fermion-spin transformation of constrained electrons[15]

C i,↑=P i a i S?i P?i,C i,↓=P i a i S+i P?i

can implement the spin-charge separation without additional constraints.Here a i and a?i are”holon”(or”electron”in the particle representation)operators,represented by spinless fermions.S±i and S z i are spinons or pseudo-spin operators represented by CP1hard-core bosons,di?erent from the electron spin operators in Eq.(1).P is a projection operator removing the extra degrees of freedom in the CP1representation.The anticommutation relations for constrained fermions C i,σare strictly preserved.Moreover,the local constraint is satis?ed exactly.However,the projection operator P is cumbersome to handle and in many cases,for example,MFA,we can drop it,with very good results[15].

To establish notations,consider?rst a single t?J chain

H1=?t i(a?i a i+1+h.c.)(S+i S?i+1+h.c.)?μ i a?i a i

+2J i(a?i a i) S i S i+1(a?i+1a i+1).(2) As shown in[15],using the above fermion-spin representation,the Jordan-Wigner transfor-mation S+i=f?i e iπ l

However,to obtain correct exponents for correlation functions,one has to go beyond the MFA,taking into account holon-spinon interactions.Following Weng et al.[19],this can be done by”squeezing out”holes from the spin chain,i.e.,to replace a?i a i+1(f?i f i+1+f i f?i+1) by a?i a i+1wherever there is a hole at site i and introducing the”string operators”which in our case are given by[15]

C i,↑=[a i e iπ(N? l>i a?l a l)][f i e?iπ l

C i,↓=[a i e iπ(N+ l>i a?l a l)][f?i e iπ l

In the resulting Hamiltonian,the”holon”part is free and can be easily bosonized,while the spinon part is an antiferromagnetic Heisenberg spin-1/2chain,which can also be bosonized and reduced to a standard1+1quantum sine-Gordon(SG)model[20]

H1,s= dx v s K s2K s(▽?)2?2v s K2s k2F16π? ,(3) whereαis an ultroviolate cut-o?,while the boson?eld?describes the low-energy excita-tions of spinons,Πis its conjugate momentum with a commutation relation[?(x),Π(x′)]= iδ(x?x′).The spinon velocity is v s=2J (1?δ)2? sinδπ1+4

π

)?1/2,which should be independent ofδ,and our result for K s is slightly away from the exact value derived for half-?lling[20],[21].In principle,the abelian bosoniza-tion is exact only at J z/J⊥≈0for the Heisenberg spin-1/2chain[16].However,the exact Bethe-ansatz solution does not show any singularities for?1

the SG model in order to rectify K s to be1/2after rescalingΠ→√

√

S(x i?x j,t)～cos2k F(x i?x j)

4[(x i?x j)2?(v s t)2]1

[(x i?x j)2?(v h t)2]12[(x i?x j)?(v h t)]1

(1?δ).

2

Now consider two coupled chains and use the MFA to decouple the interchain holon-spinon interaction.The Hamiltonian(1)is reduced to the following form H=H h+H s, and

H h=?t i(a?1,i a1,i+1+a?2,i a2,i+1+h.c.)?t⊥η1 i(a?1,i a2,i+h.c.)

?μ i(a?1,i a1,i+a?2,i a2,i),(6)

H s=2J eff i( S1,i S1,i+1+ S2,i S2,i+1)?t⊥η2 i(S+1,i S?2,i+h.c.),(7) where we have de?ned two MF order parametersη1andη2.

The holon Hamiltonian is trivially diagonalized by introducing A k=1

(a1,k+a2,k)and

2

B k=?1

(a1,k?a2,k)with excitation energiesεA k=?2t cosk?t⊥andεB k=?2t cosk+t⊥,

2

respectively,where we assumeη1≈1,as will be con?rmed later.The SEH splits the original holon excitation spectrum by2t⊥and in the low doping case for a?nite value t⊥>t (1?cos2δπ),only the upper band has vacancies and the lower band is fully occupied. The above condition on t⊥is usually satis?ed.Thus,it is easy to?nd the self-consistent valueη2=?(1?δ),as well as the interchain holon correlation functions using the abelian bosonization technique[1]

e?iπ l

2 l

(x i?x′j)2?(v h t)2 12,(9)

where k B F=k F?t⊥

2

Π21+

v s

2

Π22+

v s

(πα)2

cos(

√

?x

=Πand?Π=???

√√

2Π2S+

v s

2

Π2A+

v s

(πα)2

cos(

√2

[?Π2A+(▽??A)2]+

(1?δ)t⊥2π

2

,the coupling strength of the SG model isβ2=4π.This is nothing but a free massive Thirring model[22]with a mass gap in the excitation spectrum △s≈2(1?δ)t⊥.It is known that t⊥is a relevant variable in the range0<β2<8π.Some years ago,Haldane[23]considered the renormalization of the Bethe ansatz equations for the massive Thirring model,equivalent to the1+1SG model[24].He extracted a quantum ?uctuation parameter that controlled the correlation functions of this model,and found that atβ2=4πthe renormalization of the model stops and it corresponds to a free spinless fermion?eld.This means thatβ2=4πis just the strong-coupling?xed point,analogous to the Toulouse limit of the single-impurity Kondo problem[25].It is the?xed point that

controls the properties of this model in the whole region0<β2<8π.It is remarkable that after correcting the K s value using the Bethe ansatz solution for the single chain,we end up exactly at this?xed point for two coupled t-J chains.If we did not rectify the parameter K s in the absence of the SEH,the coupling strength of the above SG model would be at (β′)2<4π,and it should be renormalized to strong-coupling?xed pointβ2=4π,while the parameter K s is renormalized to1

[(x i?x j′)2?(v h t)2]14

,(14)

T C1,i,σ(t)C?2,j,σ(0)

～e ik B F(x i?x j ′)

64 (x i?x′j)?(v s t) 12.(15) The parameterη1≈1,as mentioned earlier.As compared with(4)and(5)for a single t?J chain,the SEH has generated new exponents,independent of t⊥.The singularity is weaker due to the presence of a gap in one of the excitation branches.The”spinon”

exponent is?1

instead of

4

?1

REFERENCES

?Permanent address:Pohl Institute of Solid State Physics,Tongji University,Shanghai 200092,P.R.China.

??Permanent address:Department of Physics,Beijing Normal University,Beijing100875, P.R.China.

?Permanent address:Institute of Theoretical Physics,Academia Sinica,P.O.Box2735, P.R.China.

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