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Confinement-deconfinement transition in two-coupled chains with umklapp scattering

A role of umklapp scattering has been examined for two-coupled chains with both forward and backward scatterings by applying renormalization group method to bosonized Hamiltonian. It has been found that a state with relevant interchain hopping changes into a state with irrelevant (confined) one when the magnitude of umklapp scattering becomes larger than that of interchain hopping. Critical value of umklapp scattering for such a confinement-deconfinement transition is calculated as the function of interchain hopping and intrachain interactions. A crossover from one-dimensional regime into that of coupled chains is also shown with decreasing temperature.


I. INTRODUCTION
Quasi-one-dimensional organic conductors, (TMTSF)2X and (TMTTF)2X salts, exhibit instabilities toward spin Peierls state, spin density wave (SDW) state and superconducting (SC) state, where the phase diagram has been displayed on the plane of effective pressures and temperature. 1,2 The interplay of low-dimensionality and repulsive interaction is important for the SDW state which indicates onedimensional fluctuations. 3,4 There are also some evidences for dimensional crossover. 5 Crystal structure shows quarter-filling for conduction electrons but the existence of dimerization leads to half-filled band. 6 A crossover from half-filled band to quarter-filled one has been found by decreasing dimerization under effective pressure, i.e., the variation of anions, X. Electronic properties, which suggest a role of the dimerization, have been reported recently at temperatures just above the SDW state. 7,8 Optical experiments on a series of above materials, which have different values of interchain electron transfer energy, show a correlation gap due to umklapp scattering and a crossover from metallic state to insulating state with increasing the anisotropy. An insulator to metal transition followed by the deconfinement of interchain hopping has been observed when the interchain transfer energy exceeds a critical value with a magnitude of the order of the gap.
Theoretical studies of these conductors have been explored by use of quasi-one-dimensional model consisting of an array of chains coupled by interchain hopping. For repulsive intrachain interaction and incommensurate band, the transverse hopping is always relevant for the weak interaction 9 but there is a reduction of transverse hopping by one-dimensional fluctuation. 10 Two-coupled chains is a basic model for quasione-dimensional system since both one-dimensional fluctuation and transverse hopping can be studied on the same footing. In a Tomonaga-Luttinger model with only forward scattering, the dominant state remains the same as that of one-dimensional system but the degeneracy of in-phase and out-of-phase pairings is removed. [11][12][13][14] When backward scattering is added, the phase diagram becomes quite different from that of a single chain. In a Hubbard model with repulsive interaction and without umklapp scattering, the ground state of two-coupled chains is the d-wave like SC state 12,[15][16][17] although that of a single chain is the SDW state. 18 The effect of interchain hopping is much strong compared with the intrachain interaction since the transverse hopping is relevant except for extremely large intrachain interaction. 12 However intrachain interaction becomes important as well as the interchain hopping for the case of the spin anisotropic backward scattering where a spin gap induced in a single chain leads to a competition between SDW state and SC state. 19 Recently confinement which denotes incoherence of single particle hopping between Luttinger liquids has been maintained, 20,21 where there is no coherence of hopping and then no split Fermi surface below a critical value of single particle hopping. The confinement has been argued for the metallic state of organic conductor (TMTSF)2X under a magnetic field, which is close to coherence-incoherence transition. 22 The role of umklapp scattering, which leads to the relevance and the irrelevance of the correlation gap, has been examined for organic conductors. 23 In terms of a Mott gap, the irrelevance of single particle hopping has been discussed in quasione-dimensional system. 24,25 A confinement has been demonstrated in two-coupled chains with half-filled band 26 in order to understand a crossover from metallic state to insulating state, which has been found at temperatures just above SDW state of organic conductors. 8 The weakly coupled half-filled chains with infinite numbers have been also studied by a perturbative renormalization group approach. 23,27 In the present paper, such a deconfinement-confinement transition due to umklapp scattering is studied in details for two-coupled chains with half-filled band by developing the previous work. 26 In section II, formulation is given in terms of bosonized phase Hamiltonian. Renormalization group equations are derived for coupling constants and response functions. In section III, the critical value for confinement is calculated. A crossover at finite temperatures is also shown. In section IV, we discuss the validity of our present calculation and examine an effect of forward scattering within the same branch.
where t is the interchain hoping energy. The quantity a † k,p,σ,i denotes a creation operator for the electron with momentum k, spin σ(=↑, ↓ or +, −) and chain index i(= 1, 2). The symbol p = +(−) represents the right-going (left-going) state. In Eq. (2.1), ǫ k,p (= vF(pk − kF)) is the linearized kinetic energy with Fermi velocity vF and Fermi momentum kF. Quantities g2, g1 and g3 are coupling constants of intrachain interactions for forward scattering, backward scattering and umklapp scattering, respectively.
The diagonalization of the first and the second terms in Eq. (2.1) is performed by making use of an unitary transformation, c k,p,σ,µ = (−µa k,p,σ,1 + a k,p,σ,2 )/ √ 2 with µ = ±. After the bosonization of electrons around the new Fermi point, kFµ ≡ kF − µt/vF, we define the phase variables, θρ+ and θσ+ (θC+ and θS+), which express fluctuations of the total (transverse) charge density and spin density. 14 They are given by where the suffix − denotes the canonically conjugate variable. In terms of these phase variables, the field operator is expressed as where α is of the order of the lattice constant. The phase factor, πΞp,σ,µ, in Eq. (2.6), which is introduced for the anticommutation relation, is taken as Ξ2n+j =N1 + · · · +N2n + (−1) j+1 2 (N2n+1 +N2n+2), We make use of renormalization group method for response functions, [28][29][30] which are assumed to be invariant for scaling, α → α ′ = αe dl . We express the nonlinear terms in Eq. (2.9) as g νp,   − Gν+,C− GC−,Sp, (2.14) wheret(l) = t(l)/ǫF, ǫF ≡ vFα −1 ,ṽν = vν /vF, ν = ρ, σ and p, p ′ = ±. In these equations, the l-dependence is not written explicitly and Jn (n = 0, 1) is the n-th order Bessel function. 31 Initial conditions are given by The second order renormalization group equations with respect to all the coupling constants are derived by expanding as K ±1 ν (l) = 1 ∓ Gν(l) + · · ·. In case of g3 = 0, 19 these equations become equal to those of Fabrizio, 12 which satisfy the SU(2) symmetry with respect to spin rotation. Although such a symmetry is satisfied only approximately for Eqs. (2.10)-(2.12), the difference is very small within the present choice of parameters as is shown later. The renormalization equations of Eqs. (2.10)-(2.12) determine the fluctuations of the total charge, total spin, transverse charge and transverse spin density respectively. Equation (2.15) corresponds to forward scattering and backward scattering with parallel spins. In r.h.s. of these equations, there are bilinear terms with respect to G νp,ν ′ p ′ (l), which appear in the presence of umklapp scattering and/or backward scattering while they are absent for only forward scattering. Equations (2.13) and (2.14) with ν = ρ (ν = σ) correspond to umklapp scattering (backward scattering with opposite spins). It is found that there is a symmetry between equations of the total charge and those of the total spin, i.e., the renormalization equations remain the same for the replacement given by Kρ ↔ Kσ, vρ ↔ vσ and (Gρ+,C+, Gρ+,C−, Gρ+,S+, Gρ+,S−) ↔ (Gσ+,C+, Gσ+,C−, Gσ+,S+, Gσ+,S−). Equation (2.16) is the scaling equation for the interchain hopping. It is noted that these equations with t = 0 is reduced to those of a single chain. 18 We examine order parameters for the possible states in case of repulsive interaction. In terms of phase variables, order parameters are expressed as (Appendix A)  where r = [x 2 + (vFτ ) 2 ] 1/2 and the quantities Kν(l) (ν = ρ, σ, C and S) and G νp,ν ′ p ′ (l) (p, p ′ = ±) are calculated from Eqs. (2.10)-(2.15). In these equations, the renormalization of the velocity 28 has been discarded in a way similar to the spinless case. 30

III. CONFINEMENT VS. DECONFINEMENT
We examine confinement-deconfinement transition by calculating the renormalization group equations for interactions of both Hubbard model and general model with g1 = g2. The scaling quantity l (= ln r/α) is related to energy ω and/or temperature T by the relation that l = ln(ǫF/ω) = ln(ǫF/T ). Numerical calculation is performed by use of normalized quantitiesgj ≡ gj /(2πvF) for j = 1 ∼ 3. In Fig. 1(a), quantitiest(l) and 1/KC(l) as a function of l are shown by solid curves and dashed curves, respectively with the fixedg3 = 0.05,g3c (=0.119) and 0.3 where t/ǫF = 0.1 andg1 =g2 = 0.4.
Both mutual interactions and umklapp scattering suppress the increase of t(l) as is seen from Eq. (2.16). In case ofg3 =0.05,t(l) (solid curve (1)) increases rapidly. Such a behavior oft(l) denotes the deconfinement of the transverse hopping. The corresponding 1/KC(l) shown by dashed curve (I) decreases monotonically to zero indicating a formation of the transverse charge gap. In the present case, some of G νp,ν ′ p ′ (l) diverge at finite l and then solutions stop due to the second order renormalization group equations. It is expected that the calculation with third order equations gives finite value of G νp,ν ′ p ′ (l) for all value of l. 12 A noticeable difference appears for large value of the umklapp scattering as is shown for g3 = 0.3 (curves (3) and (III)). With increasing l,t(l) (curve (3)) takes a maximum and reduces to zero and 1/KC(l) (curve (III)) remains finite even at the limiting value of l. Such a behavior oft(l) indicates the absence of interchain hopping which leads to confinement of electrons within a single chain. There is no transverse charge gap due to finite KC(l), where the oscillatory behavior comes from the Bessel functions in (1) (a) The l-dependences oft(l) (solid curves) and 1/KC(l) (dashed curves) forg3= 0.05 ((1) and (I)), g3 =g3c(= 0.119) ( (2) and (II)) andg3 = 0.3 ( (3) and (III)), respectively where t/ǫF = 0.1 andg1 =g2 = 0.4. In the inset, curves (1), (2) and (3) denote t(l)/te l forg3 = 0.05 (1),g3c (2) and 0.3 (3). (b) Theg3-dependences of t min and 1/K asym C . The quantity t min denotes a minimum of t(l)/te l and the quantity 1/K asym C is the limiting value of 1/KC(l). The arrow denotes the critical value,g3 =g3c, corresponding to a boundary between deconfinement and confinement.
Eqs. (2.10)-(2.15) obtained with use of the sharp cutoff in the formulation of renormalization group technique. 28 The fact that Gρ+,C+(l)/Gρ+,C−(l) ≃ Gσ+,C+(l)/Gσ+,C−(l) ≃ GC+,S+(l)/GC−,S+(l) ≃ GC+,S−(l)/GC−,S−(l) ≃ 1/KC(l) for the limiting value is consistent with the irrelevance of the interchain hopping. At a critical value given byg3 =g3c, a transition from deconfinement to confinement takes place where botht(l) and 1/KC(l) reduce to zero at the limiting value of l (curves (2) and (II)). In the inset, the normalized interchain hopping, t(l)/t e l , is shown forg3 = 0.1 (1), 0.119 (2) and 0.3 (3) where t e l denotes the value for non-interacting one. The limiting behavior of curve (1), which remains constant for large l, indicates deconfinement. In Fig. 1(b), theg3dependences of 1/K asym C and t min are shown where 1/K asym C is the limiting value of KC. The quantity t min denotes a minimum of t(l)/te l , which is found with increasing l from zero (e.g. curve (1) in the inset of Fig. 1(a)), and is essentially the same as the limiting value. Deconfinement is obtained for finite t min while confinement is found for finite 1/K asym C .
Both t min and 1/K asym C are reduced to zero atg3 =g3c, which denotes a critical value for deconfinement-confinement transition. We note that the Bessel function J1(4t(l)) in Eq. (2.16) is crucial to obtain such a transition. Actually, in r.h.s. of Eq. (2.16), the second term becomes negligible for deconfinement but the second term becomes larger than the first term for confinement.
In Fig. 2, the corresponding l-dependences for Kρ(l), Kσ(l) and KS(l) are shown by solid curves, dotted curves and dashed curves respectively where numerical results are shown for |G νp,ν ′ p ′ (l)| < 10. Curves (1), (4) and (7) are forg3 = 0.05, curves (2), (5) and (8) are forg3 =g3c and curves (3), (6) and (9) are forg3 = 0.3. The quantity Kρ(l) as a function of l decreases to zero. A charge gap is formed for Kρ(l) ≃ Kρ/2, which gives a result consistent with that of the Hubbard model. 32 The transverse spin fluctuation is also suppressed by umklapp scattering because KS(l) with the fixed l is reduced byg3. However theg3-dependence of Kσ(l) is very small, i.e., the l-dependence of Kσ(l) is similar to one-dimensional case. Therefore there is no behavior of spin gap for the total spin fluctuation except for very low energy. We note that, for single chain, Kσ(l) decreases monotonically to Kσ(l → ∞) → 1 and that KS(l) = 1 for all l. From these l-dependences, it is found that a separation of freedoms of charge and spin still exists at energy corresponding to formation of the charge gap.
), respectively where parameters are the same as those of Fig. 1(a). In Fig. 3, the t-dependence ofg3c, is shown forg1 = g2 ≡g = 0, 0.2 and 0.4 where confinement (deconfinement) is obtained forg3 >g3c (g3 <g3c). Theg3c-dependence of t forg = 0 is expressed as where K1 ≃ 1.2. The intrachain interaction enhances the confined region. The presence ofg leads to a different behavior for t with smallg3. From the inset which is calculated for g = 0.2 (1) and 0.3 (2), it turns out that theg3c-dependence of t forg = 0 is obtained as where K2 ≃ 0.2, K3 ≃ 0.2 andg3c ≪ 1. The range ofg3c, in which Eq. (3.2) is valid, decreases with decreasingg, e.g. the upper bound ofg3c is given by 0.15, 0.04 and 0 forg = 0.4, 0.2 and 0 respectively. Coefficients K1, K2 and K3 have been derived numerically since analytical treatment is very complicated. We note that the magnitude ofg3c is determined mainly by the balance between the charge gap created by the umklapp scattering and the energy of interchain hopping as is shown later. Coefficients K1, K2 and K3 have been derived numerically since analytical treatment is very complicated. We note that the magnitude ofg3c is determined mainly by the balance between the charge gap created by the umklapp scattering and the energy of interchain hopping as is shown later. As t goes to zero,g3c reduces to zero and then the interchain hopping becomes always relevant in the absence of umklapp scattering within the present choice of intrachain interaction. Now we examine the boundary between confinement and deconfinement as the function ofg1 andg2 with the fixed t. From the calculation with some choices of t/ǫF = 0.1, we find it a good approximation that the boundary with fixed g3 depends only on 2g2 −g1. Therefore,g3c is determined essentially as the function of 2g2 −g1, i.e., Kρ. Such a result originates in the fact that Gρ+,C+(l)-term gives a dominant contribution and the effect of gσ is negligibly small for other coupling constants in r.h.s. of Eq. (2.16). In Fig. 4, the quan-tityg3c as the function of 2g2 −g1 is shown by the solid curves with choices of t/ǫF = 0.1 and 0.01 whereg1 = 0.2. The region ofg3 >g3c (g3 <g3c) corresponds to the confinement (deconfinement). The dotted curves is explained in section IV. With increasing 2g2 −g1 (i.e., decreasing Kρ), the region for the confinement is enhanced.
By use of response functions for order parameters (Eqs. (2.17)-(2.19)), we calculate states at finite temperatures where a crossover is shown on the plane ofg3 and normalized temperature, T /ǫF, (or energy) in Fig. 5. The dotted curve denotes the temperature corresponding to the gap for the total charge, ∆, where ∆ ≡ ǫF exp[−lg] and Kρ(lg) ≡ Kρ/2. Around the energy corresponding to the charge gap, the total spin shows a behavior similar to one-dimensional case with the gapless excitation. The solid curve which is obtained from ǫF exp[−lt] witht(lt) = 1 denotes the crossover temperature, below which the state reveals the property of two-coupled chains. Such a temperature becomes lower than the bare interchain hopping energy, t/ǫF = 0.1, due to the renormalization by the intrachain interaction. removes the degeneracy of out-of-phase state and in-phase state, 30 but the energy is still high compared with the gaps for the transverse fluctuations, which develop just above ∆ (dotted curve). In this region, one finds a crossover into the out-of-phase SDW state. The present calculation indicates a short range correlation for the SCd state in a certain domain with smallg3(< 0.001) and finite temperatures just above the dotted curve. This could retain a trace that the ground state forg3 = 0 is given by the SC state. 12,16,17 In region (III), the gap of the total charge fluctuation develops. At very low temperatures, all the fluctuations become gapful due to relevant interchain hopping. In this case, the correlation of the SCd state as well as the SDW state decays exponentially. We note that such a state in the limit of low energy corresponds to the "C0S0" phase which has been obtained at half-filled band by Balents and Fisher. 17 In region (IV), the gap of the total charge is so large that the interchain hopping becomes irrelevant leading to the isolated chains 24 and then absence of other gaps. The state in this region has a resemblance to that of the half-filled one-dimensional chain. 18 In the inset, we show a phase diagram of confinement and deconfinement on the plane of t/ǫF and ∆/ǫF in the limit of absolute zero temperature, where dash-dotted curve (solid curve) corresponds tog = 0 (0.4) in Fig. 3. The region for confinement increases by the increase of intrachain interaction,g. The ratio in the interval region of 0.01 < t/ǫF < 0.3 is given by ∆/t ≃ 1.1 (0.7 < ∆/t < 0.9) forg = 0 (0.4). 31 These results indicate the fact that the deconfinement-confinement transition is determined essentially by the competition between the charge gap and the interchain hopping energy.

IV. DISCUSSION
In the present paper, the effect of umklapp scattering on two-coupled chains has been examined by calculating the boundary for confinement and deconfinement as the function of umklapp scattering, interchain hopping and intrachain interactions. It has been shown that electrons are confined in a single chain when interchain hopping becomes smaller than a critical value of the order of the charge gap or the umklapp scattering exceeds a threshold.
We discuss the validity of our calculation of renormalization group equations given by Eqs. (2.10)-(2.16). Here we calculate these equations by making use of the expansion, K ±1 ν (l) = 1 ∓ Gν (l) + · · · (ν = ρ, σ, C and S), where the initial conditions are Gν (0) = gν/2πvF. Since such a method leads to a solution with the SU(2) symmetry, we have done following two kinds of evaluations. One of them is shown in Fig. 4 by the dotted curve which denotesg3c as a function of intrachain interaction with fixed t/ǫF = 0.1 and 0.01. A good coincidence between solid curve and dotted curve is obtained for 2g2 −g1 > 0. The other is shown in Fig. 6 by the dotted curve which denotesg3c as a function of t/ǫF for g1 =g2 =g = 0.3. There is a small difference between the result without the expansion of Kν (solid curve) and that with the expansion of Kν. Thus, such a calculation may be justified within the present choice of parameters.
Instead of treating two-coupled chains with the split Fermi surface, Khveshchenko and Rice applied the bosonization method to two degenerate bands and enlarged the parameter space of the renormalization group to examine the pairhopping for the case of the relevant interchain hopping. 15 We discuss the effect of the pair-hopping process on the present result with the half-filled band. Expressing the pair-hopping in terms of Eqs. (2.2)-(2.5), we obtain nonlinear terms consisting of θρ+, θσ+, θC+ and θS+, which are also found in Eq. (2.9) except for a new term given by g8 cos( √ 2θρ+) cos( √ 2θσ+). 15 When the interchain hopping is larger than the charge gap, the pair-hopping becomes relevant. In such a region,t(l) and Kρ(l) of the present paper exhibit trajectories similar to those of the reference 15 and the differences in magnitudes are small, although the g8-term gives rise to a visible enhancement of the spin gap obtained by Kσ(l). It is noted that the interchain hopping is not renormalized directly by the g8-term since the r.h.s. of Eq. (2.16) is determined only by terms including θC+. When the interchain hopping is smaller than the charge gap, the pair-hopping be-comes irrelevant due to confinement, e.g., g8 ≃ 0 at the energy of the charge gap. Therefore it is considered that the effect of the pair-hopping is small for the boundary between confinement and deconfinement. Now we examine the effect of the forward scattering with the same branch, where the coupling constant is given by g4.
The Hamiltonian corresponding to g4 is expressed as which has two kinds effects. One of them appears in Kρ, Kσ, vρ and vσ which are written as Kρ 1/2 with gν = gν/2πvF. Another is the nonlinear terms given by where gC+,C− = −gS+,S− = g4. Equation (4.2) leads to additional terms to renormalization group equations fort(l) and KC(l), which are of the order of o(t 3 ) and o(t 4 ) respectively. There is also renormalization for GC+,C− and GS+.S−. However we found that, in the present calculation, the latter effect given by Eq. (4.2) is negligibly small. On the other hand, in the former case, there is a noticeable effect of g4, which comes from the variation of Kρ. Note that Kρ(< 1) increases with increasingg4, since the effect of g4 is equivalent to replacing gρ bygρ/(1 +g4) in Kρ. The increase of Kρ(< 1) reduces the renormalization of umklapp scattering, Gρ+,C+. Then one needs largerg3 to obtain the confinement. An example withg = 0.3 is shown in Fig. 6 whereg3c including theg4term (dashed curve) is compared with that withg4 = 0 (solid curve). The quantityg3c is increased by g4, i.e., the suppression of the region of the confinement on the plane of t/ǫF and g3. In the inset, a phase diagram of confinement and deconfinement is shown on the plane of t/ǫF and ∆/ǫF for both g4 = 0 (solid curve) andg4 = 0 (dashed curve). The fact that the critical value of ∆ is reduced by g4 is understood as follows. The magnitude of Kρ(0) forg4 = 0 is larger than that forg4 = 0, although the difference between these two cases is very small as for r.h.s. of Eq. (2.10). Since larger Kρ(0) leads to slower decrease of Kρ(l), one obtains smaller gap, ∆, in the presence of g4. Thus, it is found that g4 reduces the effects of both intrachain interaction and umklapp scattering. Based on the present calculation, we comment on electronic states for TMTSF and TMTTF salts, both of which show the correlation gap above the SDW state. Such a gap is possible for a choice of T /ǫF ≃ 10 −2 in Fig. 5 where ǫF ≃ 10 3 K and T ≃ 10K. 1 The fact that the plasma edge perpendicular to chains is present for TMTSF salt (absent for TMTTF salt) 8 suggests the relevance (irrelevance) of the interchain hopping. These behaviors of interchain hopping are found when the umklapp scattering satisfiesg3 < ∼g 3c for TMTSF salt andg3 > ∼g 3c for the TMTTF salt. Thus the existence of dimerization, which increasesg3, is crucial to understand property of these salts above the SDW state.
For the metallic state, the above organic conductors may be regarded as a doped system rather half-filling since oneparticle hopping between chains leads to small deviation from commensurate one. 24 In this case, the metallic state is expected with increasing the doping rate as is shown in a onedimensional case. 33 Actually a crossover from confinement to deconfinement has been obtained in the presence of doping even for two-coupled chains. 34 It will be of interest to study such an effect of doping on many coupled chains.

ACKNOWLEDGMENTS
The authors thank G. Grüner and H. Yoshioka for useful discussions. This work was partially supported by a Grantin-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture, (No.09640429) Japan.