Ann. Physik 3 (1994) 163-180 Annalen der Physik 0 Johann Ambrosius Barth 1994 Electrons with correlated hopping interaction in one dimension1 G. Japaridze’ and E. Muller-Hartmann Institute of Theoretical Physics, University of Cologne, D-50937 Kdln, Germany Received 9 February 1994, accepted 25 February 1994 Abstract. The tD system of correlated electrons characterized in addition to the usual on-site (V)and nearest-neighbour ( V ) repulsion by a correlated-hopping term ( t * ) is considered. The ground state phase diagram is studied within the framework of the weak-coupling continuum-limit approach. At filling v the effective interaction originating from the correlated-hopping term which appears in the continuum-limit theory is given by t* cos (nv). Being repulsive for v< 1/2 and attractive for v > 1/2, this interaction leads to a characteristic band-filling dependence of the phase diagram of the system. For v f 1/2, depending on the relation between the bare values of the coupling constants (U,V , t * cos (nv))and, hence, on the band-filling, the system shows three different phases in the infrared limit: a Luttinger metal, a nonmagnetic metal and a singlet superconducting phase. For v = 1/2, the correlated-hopping interaction is dynamically trivial, leading only to a renormalization of the oneelectron hopping amplitude and the phase diagram of the model coincides with that of the extended Hubbard model. Keywords: Bond-charge repulsion; Correlated hopping; Superconductivity. 1 Introduction The discovery of high temperature superconductivity [l] has renewed the interest in the physics of correlated electrons in low dimensions. Although the main interest is centered around the physics of two-dimensional highly correlated electron systems, the onedimensional analogues of models related to high temperature superconductivity are very popular, mainly due to the conjecture [2] that properties of the 1D and 2D variants of certain models have common aspects. Moreover, exact or analytical solutions of the one-dimensional models usually serve as a good guide for understanding more complex cases. One of the challenging properties of the high-T, superconductors is the strong bandfilling dependence of the low-temperature properties of these materials. Within a quite narrow range of values of the band-filling, controlled by doping, these materials manifest properties of magnetic insulators, metals and superconductors [3 J.The complicated character of the low-energy phase diagram of the high-T, superconductors -~ Research performed within the program of the Sonderforschungsbereich 341 supported by the Deutsche Forschungsgemeinschaft. Permanent address: Institute of Physics, Georgian Academy, TbiIisi, Georgia. 164 AM. Physik 3 (1994) which exhibit (via doping) a sequence of phase transitions between states of various symmetries led to an increased interest in models containing unusual correlation mechanisms, which would produce strong band-filling dependence as well as superconducting instabilities. Among other scenarios, the so called bond-charge repulsion is a subject of current considerations [4, 5 , 61. This describes the interaction between charges located on bonds and on sites. The simplest 1D tight-binding Hamiltonian including interaction with charges located on bonds contains, in addition to the usual terms describing interaction between electrons on the same site (U)and on neighboring sites (V),a term describing the modification of the electron hopping by the presence of other particles on the sites [7]: Here C ~ , ~ ( C ~is, the ~ ) creation (annihilation) operator for an electron with spin CJ at site n and pn,a = C ~ , ~ C , , , . There are Np particles, L sites and the band filling v = Np/2Lis controlled by the chemical potential p. In the standard case of Coulomb repulsion all model parameters in (1) are positive, moreover in this article we will study the weak-coupling case only, where to%U,V,t* >O will be assumed. The correlated-hopping interaction, as a source for superconducting instabilities of electron systems, was first discussed by Hirsch, two studied the three-dimensional version of the model (1) [4]. Hirsch pointed out that the correlated-hopping interaction produces effective interactions of different sign at the bottom and the top of the band which are most attractive near the top. Assuming proper band-fillings which ensure an attractive character of the correlated-hopping interaction Hirsch found, within the standard BCS type mean-field approach, that superconductivity exists in the system for 221 t* I > U+z V, where z is the number of nearest neighbours. However, the BCS-type analysis is based on the assumption of Fermi liquid behaviour of the electron system and completely ignores the interference between instabilities of different type, two essential points possibly distinguishing the high-T, superconductors from the usual ones [2]. Therefore, an exact or analytical solution of the 1D version of the correlatedhopping model, incorporating the non-Fermi-liquid character of the behaviour of the electron system as well as the strong interference between correlations of different symmetries, is of great interest. All recently proposed integrable models of strongly correlated electrons showing superconducting transitions in the case of repulsive interaction [5, 61 contain as the essential part the correlated-hopping interaction. The supersymmetric extended Hubbard model proposed by EBler, Korepin and Schoutens [ 5 ] (EKS-model) contains all interactions of the correlated-hopping model (1) as well as the pair-hopping term [8] and nearest-neighbour spin-spin interaction. Unfortunately, the symmetry conditions ensuring integrability of the model, strongly restrict the values of the various coupling constants and leave, besides the on-site (Hubbard) coupling constant U only a single free model parameter in the theory. This restriction makes it impossible to separate the various sources of correlation from each other and, hence, pinpoint the one which is G. Japaridze, E. Miiller-Hartmann, Electrons with correlated hopping interaction 165 responsible for superconductivity. Moreover, since the number of electron pairs located on a site is conserved in the EKS-model, the Hubbard interaction looses its role as the source of various dynamical instabilities and simply serves as a chemical potential for local pairs. Another integrable version of the correlated-hopping model, showing superconductivity, was recently proposed by Bariev, Kliimper, Schadschneider and Zittartz (BKSZmodel) (61. The Hamiltonian of the BKSZ-model has the form: The exact solution of the model (2) [61 confirms Hirsch’s prediction about superconductivity originating from the correlated-hopping interaction and provides a critical value of the particle concentration v, for the transition from a state characterized by dominating density-density correlations into a state with most divergent correlations of the superconducting type. For t : > O the superconducting transition takes place for v > v,, where v, varies monotonically from 0.5 to 0.5858 with increasing coupling constant t:. Unfortunately, the Hamiltonian (2) contains only half of the correlated-hopping interaction and thereby has a reduced U(1)-spin symmetry. This results, for any nonzero t r , into the existence of a gap in the spin’excitation spectrum [6] and hence to different low-energy properties of the model. Moreover the BKSZ-model does not contain the on-site (Hubbard) interaction and therefore does not shed light on the question of the comDetition between on-site and correlated-hopping interactions. One of the very profound tools for the investigation of correlations in one-dimensional systems is the field theory approach based on the construction of the corresponding continuum-limit Hamiltonian with subsequent use of the bosonization procedure and renormalization group analysis. This approach was widely used for the investigation of one-dimensional systems of correlated electrons and various spin-chain models (for reviews see [9- 1I]). Although the field theory treatment is based on the weakcoupling limit, where the bare values of coupling constants are assumed to be much smaller than the bandwidth (t *, V ,Uez to), the essential features characterizing the behaviour of the system, received within this description, can be expected to remain valid even in the strong coupling limit. In this paper we study the correlated-hopping model (1) using field theory techniques. In what follows we will briefly describe some qualitative features of the model we will obtain. In the low energy limit, when the physics is controlled by states near the Fermi points, our renormalization-group and bosonization methods indicate that the strength and the sign of the effective interaction provided by the correlated-hopping term strongly depend on the band-filling. The effective interaction originating from the bond-charge repulsion which appears in the continuum-limit theory, is given by t* cos (nv). This is repulsive for v < 1/2 and attractive for v > 1/2 and determines the principal dependence of the ground state phase diagram of the system on the band-filling. At half-filling, v = 1/2, there is no cross coupling between the bond and site charge densities [7] and the correlated-hopping interaction leads only to a renormalization of 166 Ann. Physik 3 (1994) the one-electron hopping amplitude. In this case the phase diagram of the model (1) coincides with that of the extended Hubbard model [9,121 corresponding to the case t * = 0. For v # 1/2,depending on the relation between the bare values of the coupling constants (U, V,t* cos (xv)) and, hence, on the band-filling, the system shows three different phases in the infrared limit: a Luttinger metal, a nonmagnetic metal and a singlet superconducting phase. For 4 t * > U + 6 V, depending on the band-filling, all these phases are realized: for band-fillings v < vC1= arccos ( - (U-2 V)/4 t * ) / x the ground state corresponds to a Luttinger metal phase (gapless charge and spin excitation spectra, dominating fluctuations of the charge-density wave (CDW) and spin-density wave (SDW) type), for v,, € v < vC2= arccos (- (U+6 V)/4t*)/7r to a nonmagnetic metal phase (gap in the spin excitation spectrum and dominating CDW fluctuations) and for v > vC2to a singlet superconducting phase (i.e. gap in the spin excitation spectrum and dominating singlet superconducting fluctuations). For U + 6 1/>4t* the superconducting transition is absent, but the transition from the Luttinger metal into the nonmagnetic metal state remains for any I U-2 Vl < 4 t * . For I U- 2 V l > 4 t * the phase diagram of the model (1) coincides with that of the extended Hubbard model. The range of applicability of the continuum-theory description is restricted by the condition U,V ,t * 4t sin (nv ) such that the linearization of the spectrum around the Fermi points is justified. Nevertheless, qualitative features of the phase diagram obtained from the weak coupling theory are expected to remain valid also for U,V,t * t. In a forthcoming paper we will consider, within the same weak-coupling approach, the integrable version of the correlated-hopping model, the BKSZ-model governed by the Hamiltonian (2). Comparison with the results of the exact (Bethe-ansatz) solution [6] allows us to clarify the accuracy of our description and to identify the features of the weak coupling theory surviving in the strong coupling limit. Our paper is organized as follows: in the following section we will review the model and its symmetries. In Section 3 we construct the continuum-limit version of the model (1) and perform a renormalization-group analysis of the corresponding field theory. In Section 4 we study the large-scale behaviour of the various correlation functions. In Section 5 the ground state phase diagram of the model is presented. Finally, in Section 6 summary of results is given. - 2 Review of the model and its symmetries In the absence of the correlated-hopping interaction (t* = 0) the Hamiltonian (1) is the Hamiltonian of the extended Hubbard model. The features of the extended Hubbard model are well studied [9,121. Away from the half-filled case the charge excitation spectrum of this model is always gapless, while the spin excitation spectrum is gapless for U >2 V and has a gap for U <2 V. The line U = 2 V marks the crossover from the Luttinger metal phase for U > 2 Vinto the nonmagnetic metal phase for U < 2 V. In the half-filled case there is a gap in the charge excitation spectrum for any U,V > O and the U = 2 V line serves as the boundary between the CDW insulating phase (Uc2V) and the SDW insulating phase (U>2 V). As we see, in the case of repulsive interaction, there is no trace of superconductivity in the phase diagram of the extended Hubbard model. Moreover, except for the special case of the haif-filled band, features of the model remain essentially the same for any value of the band-filling. G. Japaridze, E. Miiller-Hartmann, Electrons with correlated hopping interaction 167 Let us analyse, from the symmetry point of view, the difference between the extended Hubbard and the correlated-hopping model. The extended Hubbard model has two important symmetry properties, the S U(2)-spin invariance and invariance with respect to particle-hole transformation transforming &'(to,U,V) for the band-filling v (in what follows refered to as X(to,U,V ,v ) ) into &'(to,U,V, 1 - v ) and thus indicating the essential equivalence of descriptions in terms of particles or holes. The three generators of the spin-SU(2) algebra commute with the Hamiltonian (1) which shows its .SU(2)-spin invariance. The particle-hole transformation (3) converts &'(to,U,V,t*,v)-+&'(t0-2vt*,U,V,--tt*,l-v) , (5) which indicates two different roles played by the correlated-hopping interaction. The first is trivial - renormalization of the onelelectron hopping amplitude, the second is dynamical - source of correlations. To separate these two effects it is convenient to subtract out the Hartree part of the interaction, rewriting the Hamiltonian (I) in the following way: where t(v) = t , - 2 v t * (9) is the renormalized hopping, p- is the renormalized chemical potential, and normal ordering symbols :. . .: denote that the vacuum average has been subtracted out. For the given value of the band-filling v, we obtain 168 Ann. Phvsik 3 (19941 The band-filling dependence of the one-particle hopping amplitude in (7) will introduce an additional band-filling dependence of nonuniversal quantities, for instance, of the conductivity of the system in the metallic state, but does not change the ground state phase diagram, determined by the relationship between the small dimensionless coupling constants U/t, V / t ,t*/te1. In terms of the Hamiltonian (6),the result of the particle-hole transformation (3) reads f l t ( v ) ,u, V , t * , v ) - + f l t ( l v- ) , u,V , - t*, 1 - v ) , (12) from which one concludes that the particle-hole symmetry is violated by the correlatedhopping interaction, but the ground state phase diagrams of the correlated-hopping model for t * >0 at band-filling v and for t * <0 at band-filling 1 - v coincide. As we will show below, the effective interaction derived from the bond-charge repulsion in the continuum-limit theory is given by t * cos (nv) which fulfills the symmetry condition. 3 Continuum-limit theory and renormalization-groupanalysis Although the derivation of the continuum-limit versions of ID lattice Hamiltonians is quite standard and has been reviewed in many places [9, 10, 111 the special case of the correlated-hopping interaction has not been described before. We will stress here the main points only. For details of the method we refer the reader to the excellent paper of Shankar [ i l l . The field theory treatment of the model (6) is based on the weak-coupling limit t*, V, U g t . The free particle Hamiltonian A$ is diagonalized by Fourier transform, to give the bare particle spectrum ~ ( k=) - t [cos ( k a )- cos (k F a )],where a is the lattice spacing and kF = n v / a . Assuming that the low energy physics is controlled by states near the Fermi points f kF we linearize the spectrum around these points and obtain two species (for each spin projection a), y1(2),,(n), which describe excitations = k uFp. The Fermi velocity is uF = ta sin (nv), and with dispersion relations El,cz) the momentum p is measured from the two Fermi points. More explicitly, one decomposes the momentum expansion in (13) into two parts, centered around +kF, where the Fermion fields yl,,(n) and y2,,(n)describe right-moving and left-moving particles, respectively, and are assumed to be smooth on the scale of the lattice spacing. This allows us to define the continuum fields G. Japaridze, E. Milller-Hartmann, Electrons with correlated hopping interaction 169 The effective Hamiltonian describing the low-energy excitations with linear spectrum, X ) , the form written in terms of continuum fields W ~ ( ~ ) , ~ ( has which is recognized as the Hamiltonian of a free massless Dirac field. Now let us turn to qnt. Using Eqs. (14) and (15) we show that t : c n , u ~ n , u := :wT,u(n) + ei2nnv cy,,,(n): v2Jn): - w:,u(n>W2,u(n)+e -+ab,,u(x) + + :w;&) - i2nnv +P2,U(X) i2nnv wzt,,(n)y/,,,(n) + ei2nnv v:,,(X) wI,u(x) wl,u(x)l W2,AX) (17) ? and :ctn,ucn+,,u: = einv:v:,u(n)vl,u(n+1 ) : + e - ' * ~ : ~ / t 2 , ~ ( n ) ~ ~1): ,~(n+ + ei(2n 3 a + 1)nv einv ( [ w:,,(n)w2,,(n+ l)+e P1,u(X)+e-iXvP2,u(X)+e +,-i(2n+t)nv -i(2n+l)nv + 1) i(2ni1 ) ~ v w:.u(x)w2,u(x) +O(a2)1 v;,u(x)vl,u(x)l wt2,u(4 Wl,u(n (18) 7 where P1(2),u t (19) = :v1(2),uW1(2),u(X): Using (17) and (18) we now will rewrite A&t in terms of continuum fields w ~ ( ~ ) , ~ ( x ) . The whole information about the lattice structure of the initial model (6) is contained in the oscillating phase factors eiinV". On the grounds that the fields I , u ~ , ~ ~are ,(~) dx. All terms in smooth on the lattice, one can convert sums into integrals: Xntthat contain the rapidly oscillating factors e *""' and e*14nnV will drop out. However, in the case of a half-filled band, the Umklapp terms, which go as * i 4 n n v = 1, are not oscillating and have to be kept. Performing this procedure, and keeping only first order terms with respect to a, one arrives at: qnt =a 1dx C (A, <p:,u(x> +~ 3 , u ( x ) + ) ~0 .j. @ i , u ( ~ ) ~ --u(x) ,, t- P ~ , ~ , ( x )-,(XI> A, U + ~ l , , P l , U ( ~ ) P 2 , U ( X ) +A1 L P l , U ( X ) P 2 . t t -&I t + A B S , , ( v 1 , a v 2 , u vv,,uw2,u + h.c.)+ 'ZBSL (VLU t v2,- u v1,- u v 2 , u + h.c.1 170 - . YI Y2 --p---->- vz *I .Y2 Forward scattering in the vicinity of one Fermi point --<; 7 Y --+-*2 7.T--*)+- --*)->- *2 91 YT ?* -2Pz *I *I Backward scattering Yl -- -- YI Forward scattering in the vicinity of different Fermi points = -*? ~ 'f-* ~ - - *- --*- ---p*I - - *- - - *I ~ *Z YZ *Z Ann. Physik 3 (1994) 12 Y2 Umklapp processes 'I Fig. 1 Classification of the scattering processes found when interaction terms are expressed in a basis of left (" 1") and right ("2") moving Fermions. As we see qnt contains eight scattering terms (see Fig. 1) including forward scattering in the vicinity of one Fermi point (Aoll,.), forward scattering in the vicinity of different Fermi points (Alll,L), backward scattering (ABsll,L) and also, in the case of a half-filled band, Umklapp processes (Aul~,.). The bare values of the coupling constants are given by: 1 2 Lo. = - A * . ABS. 1 =-(V+2V+4t*cos(nv)) , 2 1 1 =--(U--2Y+4t*cos(7tv)) , A " . = Z ( u - 2 1 / ) . 2 As we see the lattice features of the initial model are reflected in the continuum-limit Hamiltonian in two ways: a) via the selection of scattering processes relevant for the physics of the system at large distances at the given value of the band-filling; and b) via the bare values of the coupling constants of the corresponding scattering processes, i.e. via normalization conditions for the effective interactions. Contrary to the extended Hubbard model case (t* = 0), when the band-filling dependence appears in the continuum theory only through the selection of the scattering processes (Umklapp processes, responsible for the dynamical generation of a gap in the charge excitation spectrum for any V, Y> 0, appear in the continuum theory only for the commensurate case of a half-filled band) we see from (18) and (21), that the site off-diagonal nature of the correlated-hopping interaction results in a band-filling dependence of the bare coupling constants of the effective continuum-limit Hamiltonian. Only in the special case of a half-filled band the role of the correlated-hopping interaction is reduced to a trivial renormalization of the electron hopping amplitude given by (9) and the dynamical features of the model coincide with those of the extended Hubbard model. In what follows, if not indicated specifically, we will assume bandfillings not close to 1/2 and neglect Umklapp processes. Moreover, the terms corre- G. Japaridze, E. Miiller-Hartmann, Electrons with correlated hopping interaction 171 sponding to the forward scattering in the vicinity of a single Fermi point, (Aorl,+) leading only to a renormalization of the Fermi velocity, will be omitted. Taking thls into account, we deal with the following Hamiltonian, easily recognized as the LutherEmery backscattering model [14]: Here 1 Pl(2) =f i c W:(2),aW1(2),u 7 g vF is the Fermi velocity and the small dimensionless coupling constants are given by 1 g,= -- ( 4 1 1+ L l L - - 2 L B S ( ( ) = -- (O+6 V + 4 t * cos (nv)) , ? VF 71 VF go= 1 1 -- 1 (All[ - A * , - 2 & q ) = -(U-2 nVF g, =-21, 1 72 OF Y + 4 t * cos (7cv)) , AVF 1 =-(LI-2V+4t*cos(7cv)) . 7.c VF The model (22) is characterized by a complete decoupling of charge (p = p1+p2) and spin (a = o1+02)degrees of freedom. The most convenient way to analyze it is to use the bosonization procedure by which this decoupling becomes more transparent. [I31 In the boson representation of the Fermi fields ~1(2),o(x) (here a is a cut-off parameter of the order of the lattice constant) the Hamiltonian (22) splits into two decoupled parts, describing, respectively, charge- and spin-density degrees of freedom [14]: Let us first consider the charge sector. As seen in (31), away from half-filling the charge degrees of freedom of the system are described in terms of the Tomonaga model [15]. The coupling constants in the Tomonaga model are not renormalized and 8 is easily diagonalized by use of the Bogoliubov transformations 1161 leading to the free bose field Hamiltonian Here up is the sound velocity and p,,(x) and P,(x) are a scalar field and its conjugate momentum, respectively, related to the charge and current density in following way: with G= 8 n (l++gp)1/2(l-;gp) - 1/2 - (I++.> * Thus, it follows from (33) that for arbitrary values of the coupling constants the model is characterized by a gapless charge excitation spectrum. The effective interaction remains weak and, since it is included in (33) via renormalization of the sound velocity and via rescaled values of the p,(x) and P,(x) fields, it manifests itself in a slight renormalization of the charge excitation spectrum and in the renormalization of the critical indices describing the power-law decay of the various charge-charge and current-current correlation functions. Let us now consider the spin channel. In analogy to the case of the charge channel, we rewrite 2, in the standard form of a quantum Sine-Gordon Hamiltonian where v, is the spin wave velocity, p,(x) and P,(x) are a scalar field and its conjugate momentum, respectively, related to the spin and spin-current density in the following way: G. Japaridze, E. Miiller-Hartmann, Electrons with correlated hopping interaction 173 with In contrast to the case of the charge sector, the coupling constants in the Sine-Gordon theory are renormalized. Fortunately, in the weak-coupling limit (lg,l, Jg, 1 Q 1) the properties of the Sine-Gordon model are well known [17, 181. The infrared behaviour of the Sine-Gordon Hamiltonian 2, is described by the corresponding pair of renormalization group equations for the effective coupling constants I'i : where L = In ( a ) and rj(0)= gi [17]. The Eq. (41) describe a Kosterlitz-Thouless transition [19] with a flow diagram as given in Fig. 2. The flow lines lie on the hyperbolae r2,-r: = g2,- g t = constant , (42) and, depending on the relation between the bare coupling constants gu and g,, show two different regimes of behaviour. For g, 2 I g 1 we have a weak coupling regime; the effective mass f I 40, and the fixed-point Hamiltonian, describing the low-energy behaviour of the system, coincides with the free bose field Hamiltonian. The spin excitation spectrum in this case is gapless and the values of the corresponding critical indices describing the power-law decay of , Fig. 2 The renormalization-group flow diagram; the arrows denote the direction of flow with increasing length scale. I 174 Ann. Physik 3 (1994) the spin density-density and spin current-current correlations are determined by the fixed-point value of the effective coupling constant r,. For g,< lg, 1 the system scales into a strong coupling regime; depending on the sign of the bare mass mu,the effective mass f +m , which signals the cross-over into a strong coupling regime and indicates the dynamical generation of a commensurability gap in the spin excitation spectrum. The field pu gets ordered with the vacuum expectation values , -+ In the case of the correlated-hopping model (6) which we are considering here due to the S U(2)-spin symmetry the flow lines are restricted to moving along the bisectrix gu = g,. A) For g,>O i.e. U-2 V + 4 t * cos (nv)>O the system renormalizes into the weakcoupling regime, with fixed-point values of the effective coupling constants ru=r,=o. (45) B) For gu<O i.e. 11-2 V + 4 t * cos (nv)<O it moves towards the strong-coupling regime in the spin-channel, accompanied by the ordering of the field pUwith vacuum expectation value (p,)=O * (46) The ordering of the p0 field strongly influences the behaviour of the correlation functions and determines the symmetry properties of the ordered ground states of the Fermionic system. 4 Correlation functions To clarify the symmetry properties of the various ground states of the system let us introduce the following order parameters describing the short wave-length fluctuations of the charge density and of the spin density as well as two superconducting order parameters corresponding to singlet (Ass) and triplet (ATS)superconductivity: G. Japaridze, E. Mliller-Hartmann, Electrons with correlated hopping interaction A&) = c v:,u(x)wtz,-&I - exp (iBp@)*COS (p,(4,/2) , U 175 (49) where Pp =g w p 9 and QP is the field dual to pp: = Pp,aXpp= Pp. Let us first consider the sector of the coupling constants U-2V+4t* c0s(lrv)~0 (A) corresponding to the case of the weak-coupling regime in the spin channel. In this sector, charge and spin channel are both gapless and the behaviour of the corresponding degrees of freedom is governed by effective free bose field Hamiltonians. Using the expressions for free bose field correlators (exp w,,,, [vo, ( x ) - v,,, WI/W - Ix -x‘ I - v ~n, ,/ ~ (52) and and the fixed-point value of the effective coupling constant p$,, = 87c, one can easily obtain the following behaviour of the correlations at large distances: 176 Ann. Physik 3 (1994) where gp = (U+6V + 4 t * cos (~cv))/TcvF>O . (58) Let us now consider the sector of the coupling constants u-2 V + 4 t * cos(nv)<O (B) corresponding to the case of the strong-coupling regime in the spin-channel. In this case the ordering of the field pa with vacuum expectation value (p,) = 0 completely suppresses the SDW and TS instabilities and changes the power-law decay of the CDW and SS correlations into and Here only the gapless charge degrees of freedom were relevant. 5 Phase diagram Using the results of the previous section about the excitation spectrum of the system and about the large.distance behaviour of the correlation functions, we will now discuss the ground state phase diagram of the model (1). Let us first consider the case U = V = 0, where the system is characterized only by the correlated-hopping interaction with coupling constant t* >0. The effective interaction appearing in the continuum-limit theory is given by t* cos (nv) and, depending on the band-filling, leads to two different regimes of behaviour of the system: for v< 1/2, the effective interaction is repulsive, both charge and spin excitations are gapless and the system manifests properties of a Luttinger liquid. The ground state of the system can be characterized as a paramagnetic metal with strongest instabilities of the densitydensity type, characterized by the critical indices 2t* acD,=asD,-2--cos(nv) . ZVF The superconducting correlations, characterized by critical indices - + -cos (nv) ass= aTS 2 &b XVF , G. Japaridze, E. Muller-Hartmann, Electrons with correlated hopping interaction 177 decay faster. We call this phase the Luttinger metal phase. With increasing band-filling v the effective interaction decreases. For v = 1/2 the correlated-hopping interaction becomes dynamically trivial, leading only to a renormalization of the electron hopping amplitude (9). At this point all critical indices are equal to 2 and coincide with those of the free electron system. For v > 1/2 the effective interaction is attractive and a transition into the state characterized by a gap in the spin excitation spectrum occurs. The freezing out of the spin excitations, accompanied by the ordering of the field pGwith vacuum expectation value (p,) = 0, completely suppresses the SDW and TS instabilities. The power-law decay of the CDW and SS correlations is changed. The corresponding critical indices are given by 2t* 1+-cos(xv) 1 ass=-= aCDW , ZUF which implies that the SS correlations are most divergent for v > 1/2. Thus, for bandfillings v > v, = 1/2, the correlated-hopping model manifests properties of a nonmagnetic metal with strongly developed instabilities of the singlet superconducting type (see Fig. 3a). 1 I Luttinger I I I I I metal CDW Luttinger I I I SDW SDW ss I I I I I I I I I I 'V I Nonmagnetic metal I CDW superconductor I I metal Singlet CD W I I Singlet I superconductor I I I I I I I ss 178 Ann. Physik 3 (1994) Let us now turn on the on-site repulsion U >0. In this case the role of the correlatedhopping interaction remains nontrivial only for U <4 t* and in the regime v > v c =1 - ~ c c o ~ ( - ~ ), 71 where the system is in a superconducting state (see Fig. 3 b). Outside of this regime, the phase diagram ,coincides with that of the Hubbard model, with the coupling constant replaced 'by an effective coupling constant Ueff= U + 4 t * cos (nv ) >0. The nearest-neighbour interaction V leads to a competition with the on-site term and makes the phase diagram more rich. Let us first consider the case 4t*> U + 6 V , (65) where, depending on the band-filling, the system shows three different regimes of behaviour corresponding to a Luttinger metal, a nonmagnetic metal and a singlet superconducting state. For 1 u-2v , v<vcl=-arccos(--;;;-) n we have a Luttinger metal phase. The critical indices of the correlations are a C D W = aSD W - - ass = aTs 2 + - I/+ 6 V + 4t* cos (nv) , 2nvF U+ 6 V + 4t* cos (nv) 2nvp At the band-filling v = vcl a transition into a nonmagnetic metal phase takes place. This state is characterized by a gap in the spin excitation spectrum and most divergent CD W fluctuations. The corresponding critical indices are 1 aco,=--=lass U + 6 V + 4 t * cos (nv) 2nvF As follows from (69), for U + 6 V + 4 t * cos (nv)<O , (70) the SS fluctuations become most divergent and thus, at the band-filling vc2 = Iarccos (-7 ) U+6V , R a crossover from the state with dominating CDW fluctuations (vc,<v<vc2) into a state with dominating SS fluctuations (v > vc2) occurs (see Fig. 3 c). Outside the regime G. Japaridze, E. Miiller-Hartmann, Electrons with correlated hopping interaction 179 of coupling constants given by (70) the superconducting transition does not occur, but the transition from the Luttinger metal into the nonmagnetic metal state exists as long as IU-2VI < 4 t * . (72) For I U - 2 VI >4 t * the phase diagram of the model (6) coincides with that of the extended Hubbard model: for arbitrary v # 1/2 and u-2 V>4t* the system shows the properties of a Luttinger metal state and for u-2v<-4t* of a nonmagnetic metal state. 6 Summary To summarize, we have considered the one-dimensional correIated-hopping model (1) which describes a ID system of correlated electrons incorporating, in addition to the usual repulsive on-site (U)and nearest-neighbour (V) interactions, an interaction between charges located on sites and on bonds (the correlated-hopping interaction). The ground state phase diagram was studied within the framework of the weak-coupling continuum-limit approach. The effective interaction originating from the correlatedhopping term which appears in the continuum-limit theory is given by t* cos ( K V ) . Repulsive for v < 1/2 and attractive for v > 1/2, this interaction leads to a characteristic band-filling dependence of the ground state phase diagram of the system. It was shown that for v # 1/2, depending on the relation between the bare values of the coupling constants (U,V ,t* cos (nv))and, hence, on the band-filling, the system shows three different regimes of behaviour in the infrared limit. These regimes correspond to a Luttinger metal, a nonmagnetic metal and a singlet superconducting state. For v = 1/2, the correlated-hopping interaction becomes dynamically trivial, leading only to a renormalization of the one-electron hopping amplitude and the phase diagram of the model coincides with that of the extended Hubbard model. The excitation spectrum of the system as well as the behaviour of the various correlation functions was studied and the detailed ground state phase diagram was constructed. One of the authors (G.J.) greatfully acknowledges S. b t l u n d and H. Johannesson for their kind hospitality and support during his stay at Chalmers University of Technology, where this work was started. He also would like to thank R. Bariev, H. Johannesson and A. Schadschneider for many helpful discussions. 180 Ann. Physik 3 (1994) References [l] J.G. Bednorz, K.A. Miiller, Z. Phys. B64 (1986) 189 [2] P.W. Anderson, Science 235 (1987) 1196; Phys. Rev. Letters 64 (1990) 1839 13) B. Batlogg, in: High Temperature Superconductivity, K. S. Bedell (ed.), Addison-Wesley, Redwood City 1990 [4] J.E. Hirsch, Physica C158 (1989) 326; Phys. Letters A138 (1989) 83 [5] F.B. Essler, V.E. Korepin, K. Schoutens, Phys. Rev. Letters 68 (1992) 2960 [6] R.Z. Bariev, A. Kltimper, A. Schadschneider, J. Zittartz, J. Phys. A: Math. Gen. 26 (1993) 1249 [7] S. Kivelson, W.-P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. Letters 58 (1987) 1899 [8] K.A. Penson, M. Kolb, Phys. Rev. B33 (1986) 1633 [9] V. J. Emergy, in: Highly Conducting One-Dimensional Solids, J. T. De Vreese, R. 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