Orbital antiferromagnetic ordering in a two-chain model of interacting fermions

Volume 153, number 1

PHYSICS LETrERS A

18 February 1991

Orbital antiferromagnetic ordering in a two-chain model of interacting fermions A.A. Nersesyan Institute of Physics, Academy ofSciences of the Georgian SSR, Tamarashvili 6, 380077 Tbilisi, USSR Received 27 November 1990; accepted for publication 13 December 1990 Communicatedby A.A. Maradudin

A ground state with orbital antiferromagnetic long-range order is shown to exist in the system of weakly interacting spinless fermions on two coupled chains.

1. Recent studies [1,2] of weakly interacting electhe basis of a continuum weak-coupling Hamiltotron systems on a square lattice with a half-filled ennian that involves only those electron states close to ergy band have shown that the perfect nesting propthe Van Hove saddle points of the spectrum [6,1]. erty of the square Fermi surface can lead not only to However, mean-field theory totally ignores the inthe charge-density-wave (CDW) or spin-densityterference between competing fluctuations of differwave (SDW) instabilities of the system, but to orent symmetries. On the other hand, a consistent perbital antiferromagnet (OAF) and spin nematic (SN) turbative approach, outside the scope of the saddle[31types of ordering as well. These two phases can point (or the leading log2 T [61) approximation, is occur as a result of anisotropic, spin-singlet or -tripstill lacking due to nonrenormalizability of the thelet electron—hole pairing on the neighboring sites and ory in 2 + 1 dimensions. are characterized by nonzero local charge or spin In view of these difficulties, it seems useful to start currents, which circulate around plaquettes with a out with the simplest model that preserves the posdistribution corresponding to the unit cell doubling. sibility for circulating currents and, at the same time, Current states of this kind have been first discussed makes the problem solvable. Such a model decribes by Halperin and Rice [4] in a two-band model ofan weakly interacting spinless fermions on a strip of excitonic insulator. A mean-field description of the plaquettes formed by two coupled chains (fig. 1). 2D OAF and SN phases [5] has revealed a number Although the spinless case is, of course, an oversimof interesting low-temperature properties of these plification, it deals with fewer degrees of freedom and states. allows one to decrease the number of competing inMeanwhile, the conditions under which the OAF stabilities: for repulsive interactions, the only posor SN instabilities could predominate over the CDW and SDW ones have not yet been clarified. Apparently, the OAF and SN orderings could only be re1 1+1 alized as a combined effect of various interactions, necessarily including non-point-like ones, since in these states the point symmetry of the square lattice ___________________________ is broken. So far, dynamical instabilities in the 2D Hubbard model, supplemented with finite-range and I exchange interactions, were studied either in the Fig. 1. A two-chain lattice with numeration of sites indicated. mean-field approximation [2], or perturbatively, on The arrows show the OAF type distribution oflocal currents. .

Elsevier Science Publishers B.V. (North-Holland)

.

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Volume 153, number 1

PHYSICS LETTERS A

sible orderings are different kinds ofthe CDW (both site-diagonal and off-diagonal) and the OAF. The purpose ofthis Letter is to show explicitly that, in a two-chain model which will be specified below, with half a spinless fermion per site, the OAF is indeed one of the stable ground states with true longrange order, and to clarify the role of different interactions that make such an ordering possible.

OCDW(fl) =

18 February 1991

(—1

)“

~ ~ a

In the region U> 2 Vthe SDW phase is realized, with a gapless spin excitation spectrum and an identical power-law decay of the three correlation functions (a=x, y, z), where OsDw~(n)=(—l)~>ac~cna,

2. The Hamiltonian I consider is given by

+

~

(4)

ia

OsDwy(n)=—i(_l)>aCn.JCn,_a.

(~Unjani,_a+V1njanj+1,a+V2nicnj+1,_a) .

Here fl,~=C~C10,a= ±1 is the chain index, = t~and t~are the intrachain and interchain hopping amplitudes, respectively. The last three terms in (1) include repulsive interactions between particles on the neighboring sites belonging either to the same chain (J”1), or to different ones (U), and the interaction along diagonals of the plaquettes (J72). It will be shown that the “frustrating” interaction ~72, together with the interchain hopping (ti), is crucial in suppressing the CDW instabilities in favor of the OAF. The origin of the OAF ordering in the two-chain system of spinless fermions can qualitatively be understood by using the equivalence between the original model (1) and a purely one-dimensional system of fermions with spin When the chain index a is treated as a spin variable, Hamiltonian (1) is recognized as a 1 D, half-filled Hubbard model extended to include U (1)-symmetric interaction on the nearest-neighbor sites, giving rise to the exchange anisotropy term 2 ( V1 V2 )S~S~~1,where S~=~(n1÷— n1_). The interchain hopping term reduces to the interaction with a “magnetic field”, H= 2t~I,perpendicular to the spin anisotropy (z) axis. Spin rotational symmetry is thus totally broken, and Sz= >, S~is not conserved. The special case t = 0, V1 = V2 V corresponds to the SU (2)-symmetric extended Hubbard model which, for U, V> 0, is known [71 to have two phases. At U< 2 V the long-range ordered CDW occurs, with nonzero vacuum expectation value of the operator ~‘

~.

—

(5)

C

(1)

50

(3)

a

~ (c~c~+1,~+h.c.)—t1 ~ C~Cj,_a i,a

(2)

In terms of the original two-chain model, the CDW and SDW correspond to charge-density waves on the two chains, with relative phases 0 and ~t, respectively. They will be denoted by CDW~.The offdiagonal operators, OSDwC and °SDWy, describe, respectively, a modulation of the vertical bonds of the two-chain lattice (bond wave, BW ), and a modulation of currents on vertical links (OAF ~). Clearly, the SDW phase exists in the anisotropic case, V1 ~ V2, too (in the weak-coupling limit, this will be shown to occur at U> mm (2 V1, 2 V2)). However, the degeneracy between the longitudinal (SDWZ) and transverse (SDWX, SDW~)spin-density waves is removed by the exchange anisotropy. When increasing V2 (at t =0) from the region V2 < V1 to the region V2> V1, the spin anisotropy parameter changes its sign, and the system undergoes a spin-flop transition from an ordered, Ising-like, SDWZ(CDW) phase to a disordered, XY-like phase with degenerate SDWX(BW±)and SDW~’(OAF1) correlations. Applying in the region V2> V1 a homogeneous magnetic field H= 2t~ along the x~axis in spin space results in suppressing the SDWX( BW ~) correlations and the onset of long-range order in the SDW~(OAFI). The corresponding alternative alignment of the currents on the horizontal links simply follows from the conservation of the total current at each site of the two-chain lattice (see fig. 1), thus indicating the OAF ordering of the system. In the rest of this Letter I will advance scaling arguments to support this conclusion by making use of known results [8] related to a specific 1 D Fermi gas model with non-spin-conserving scattering processes.

Volume 153, number 1

PHYSICS LETTERS A

18 February 1991

3. Consider model (1) in the case of weak interaction and interchain coupling: U, ~“1,2, t~ t~.Then the bare particle spectrum may be linearized near two Fermi points, ±kF (kF= pt), and the continuum limit of Hamiltonian (1) can be taken by using the correspondence C~0—’i~v~~(X) + (—i)~~20(x);the

tinuum Hamiltonian (6), (7) to a quantum theory of two independent sine-Gordon fields, 0,, and Ø~:

fermion fields ~1’i0(X) and W2~(X) describe rightmoving and left-moving particles, respectively. One obtains

(a=p, a), where ~a are the momenta conjugateto a0, ~ is a cutoff parameter of the order of the lattice constant,

‘~

~.

a WtaWi,_a

+

W2g~1’2,_g)]

+ *nt,

(6)

where v~is the Fermi velocity. The interaction *nt has the structure of the Luther—Emery backscattering model [9], supplemented with umklapp processes: )~nt~VF

$

~(_~spip2_gu

= ~VF[Pa(X)

+ (ma/7ta)

+ ((9xOa)2]

COS[fiaOa(X)]

(9)

2a)g~, m fi~/87t=l+~g,,fi~/87t=l+~g11, m~=(vF/ 0=(vF/2a)g1

~Jdx[_ivF(w~ôxwIa_w~8x~2a)

— ~H(

~(x)

The relation between /3,, (/3,,), in,. (rn,) and g~(g), g~(g1) is universal for lg,l <<1. Bosonizingthe fermion field W1,2;a [9] leads in the continuum limit to factorization of the order parameter operators (2)—(5),

°CDW(CDW+)(X) OSDW.(CDW-)(X)

a1 a2

~S,.(x)S,,(x),

OsDwBw±(x)—’CP(x)C,,(x)

+ ~

a

OSDWy(OAF~)(X)—~CP(X)~,,(X) ,

(7)

where Cacos(~fia0a) ,

where p3 and a3 (j= 1, 2) stand for the “charge”-density and “spin”-density operators. The small dimensionless coupling constants are given by g~= — (U+ 4 V 1 + 2 V2 ) /JtVF, 2V g11 =(U—4V1 + 2)/7tvF, g1=g~=(U—2V2)hrvF. (8) In obtaining (7), terms of the form P~P,—~ leading to renormalization of the Fermi velocities in second order in g, are omitted. Several operators with nonlocal structure, w~(x)~t’(x)w~(x+a)ci,(x+a), that originate from the last two terms in (1), have free anomalous dimensions 4 + 0(g) and, for I gil ~ 1, are strongly irrelevant. (All nearly marginal operators with dimensions 2+0(g) are, of course, retamed in (7).) Neglecting these terms results in a complete decoupling between the “charge” (p) and “spin” (a) degrees of freedom. Let us first consider the case H~21~=0. The standard bosonization procedure [10] converts the con-

(10)

Ca =cos~

=

~)

Sa=sin(~fia0a)

S~.=sin(~/~3a);

8xOaand = ‘5a ~ We shall needdual two more 87t/fia, is aalso field to opOa: = ‘~a’

erators describing off-diagonal SDWX and SDW~orderings (denoted by SDWX and SDW~’): O~~(n,n+l)=(—l)”~(c~,,c~~1,_,,+h.c.), O~~’(n,n+ 1) =i( —1 )fl ~ In terms ofthe two-chain model, these operators correspond to a modulation of effective bonds and currents along diagonals of the plaquettes (BWd, OAFd), respectively. Their continuum local form is O~~(Bw,)(X) —S,.(x)C,,(x) (11) The renormalization group equation for the effective couplingsz, of the Hamiltonians ~,, are those of the Kosterlitz—Thouless (KT) transition: O~EWY(OAFd)(X)~Sp(X)S,,(X) .

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dz,/dl=

—

PHYSICS LETTERS A

z~, dz~/d1=

—

zszu,

dz11/dl=—z~ dz1/dl=—z11z1 , ,

(12) (13)

18 February 1991

where G11 =g1, G1 = ~(g11 +g1) and Gf= ~(g1~—g1). As a result, ,~transforms as follows, )~= ~V~r[Pg+

where l=ln(ct), z,(0) =g,. For repulsive interactions, the condition g~< I g~I is always satisfied, in which case eqs. (12) show the development of the strong-coupling regime in the p-channel. As a result, a commensurability gap (or soliton mass) is dynamically generated in the “charge”-density excitation spectrum, and the field 0,. gets ordered, with a vacuum value bare expectation mass m~:(Ø,~> = 0depending at U< 2 Von the sign of the 2 <0,,) = it/fl,, at U> 2 V2. In the the a-channel, eqs. (13) indicate the weak-coupling gapless regime at g11> I g1 I~ which corresponds to the region V2> V1, U> 2 V1. In other cases, a “spin” gap is generated, resulting in longrange order in the O~-field: <Ø~~> = 0 at U< 2 V2 <~~> =it//3,, at U> 2V2. Using (10) and (11) and the vacuum values of the fields 0,, and 0~(cf. ref. [8]), one obtains the phase diagram at H= 2l~= 0. In the region V1> V2 there are two ordered commensurate phases in the ground state: the CDW(CDW~) at U’<2V2, and SDWZ(CDW_) at U>-2V2. In the opposite case, V1 < ~72, the CDW(CDW~)phase occurs at U<2V1, while in the region U> 2 V2 the system is characterized 5(OAF by coexistence of the SDWX ( BW 1) and SDW 1) correlations at 2 VI < U< 2 V2, or the SDWX( BWd) and SDW~ ( OAFd) correlations at U> 2 V2, showing a power-law decay at large distances. 4. Having found the region 1”2> V1, U> 2 V1, characterized by degeneracy between SDWX ( BW) and SDW~(OAF) correlations, I consider now the role of a finite “magnetic field”, H= 2t1. It is convenient to perform a ~it rotation in “spin” space around the yaxis to have the new z-axis along the magnetic field. The rotation has no effect on the “charge” degrees of freedom; however, it gives rise to non-spin-conserving (Ge) scattering processes in the a-channel:

(l9XOa)2]

—

(fi~HI47t)ô~O,,

+(m/ita)cos(fiaØa)+(rn/ita)cos(ffa~,,), (14) where /3~/8it=1+~G11, rn—(vF/2a)Gl and Fñ=(vF/ 2a ) Gf. Under the ~it rotation the OAF order parameters in (10) and (11) preserve their form, while the BW operators transform to tBwd) _*S~S~. (15) OSDW’(BWi) CpSa~ ~~E~ The scaling properties of model (14) were previously analyzed [8] in connection with the study of spin—orbit effects in 1 D conductors. The magnetic field introduces into the theory a new length scale, a11— VF/ I HI, and a two-cutoff renormalization procedure distinguishes between the cases 1< h and 1> h, where h=ln(aH). Within logarithmic accuracy, the effective couplings Z, do not depend on h in the region lh, the magnetic field suppresses G1 processes, but has no effect on Gf processes (this can be checked by considering the energy and momentum conservation laws for these processes at H~0). Therefore, at 1> h one is left with effective couplings Z~1and Zf satisfying the KT equations [8] —

—

dZ11 /d1 Z~, dZf/d1 Z11 Zf. (17) Eqs. (17) are only applicable, if the new “bare” couplings, Z11 (h) = z1 (h) and Zf(h) = ~[z11 (h) z1 (h)], are small. As shown above, this indeed is the case in —

the V2> V1, U> 2 V1. As follows from eqs. (11),region z11(h)> Iz±(h)I at any h implying that —~

—G11 a1 a2 +G1

c//ICI//2CWlC1,U2C

—Gfw~w2,~W1,_C,

Z11(h)> IZf(h) I. This condition means that, at 1> h, eqs. (17) describe the development of a strongcoupling regime, with Z11—+ IZf I in the infrared —

—~

limit. At l>h the rn-term in (14) becomes irrele52

Volume 153, number 1

vant, and dual field

PHYSICS LETTERS A

18 February 1991

scales to a sine-Gordon model for the with a term linear in P~: 2] (flH/4it)J5 J~ta~VF[Pg+ (ôx~a) .~

(+1

~

—

+(rn/ita) cos(~~). As a result, 9x0a and 0,, acquire vacuum expectation values: <13X0a> =H/3a/4IEVF, = It/Pa, whereas correlations of the field 0~.decay exponentially. One then concludes that, at t~0, the BW correlations are suppressed in the whole region V 2> V1, U> 2 V~, while the order parameters °OAF1 and °OAF~ have nonzero average values at 2 V~< U< 2 V2 and U> 2172, respectively. In the range 2 V1 < U< 2 V2, one thus finds a twochain analog of 2D OAF ordering on a square lattice [5] (fig. 1). The onset of long-range order in the interchain (vertical) currents (OAF1) is indeed accompanied by alternative alignment of the inchain (horizontal) currents described by the order parameter OOAF1(fl,

n+l)-..i(—l)”

~

a(c~cn+i,a—h.c.)

.

In the continuum limit, after ~it rotation in “spin” space, this operator takes the form

i

i+I

Fig. 2. Second type of OAF ordering. Local currents effectively flowing along diagonals of the plaquettes are shown.

fective currents flowing along diagonals of the plaquettes, as shown in fig. 2. 5. The above analysis was concerned with the weakcoupling limit of the two-chain model (1). This model still remains tractable if U>> t~,t~, V~,V2, in which case Hamiltonian (1) (in the subspace with the constraint >a fl,a 1, V1) maps onto the XXZ spin-i chain in a magnetic field perpendicular to the spin anisotropy axis. Here again one can find conditions for the OAF ordering of the two-chain system (details will be given elsewhere). An extension of the two-chain model to a quasi-one-dimensional case, as well as taking into account the spin of the particles is now under study. I would like to express my sincere gratitude to N.

OOAE1(X)’~ffpPp(X)5’p(X)Ca(X)

+ fi~(ôxOa) C,.(x)~a(x) .

(18)

In the region J”2> V~,2V~< U-<2U2, the secondterm in (18) has a nonzero expectation value. The ratio between the average local currents on the horizontal and vertical links of the two-chain lattice is proportional to ~ as it should be due to conservation of the total current at each lattice site. To calculate this ratio exactly and, more generally, to describe the ground state properties of the OAF phase in the twochain system, one would need an exact solution of model (14) at H= 2t~~ 0 in the strong-coupling regime (the Bethe-ansatz solution of this model is only known [11] for the case H= 0). At U> 21/2, another OAF phase with <°OAFd> ~ 0 is realized. Note that, in contrast with the previous OAF state, the current conservation law does not require long-range ordering of the currents flowing through horizontal and vertical links ofthe two-chain lattice (in fact, <°OAFi > = =0 in the OAFd state), although they contribute to formation of ef-

Andrei, G.I. Japaridze, H.J. Schulz, G.E. Vachnadze and especially to A. Luther for helpful discussions on two-chain models of spinless fermions and their relation to spin chain models.

References [1] A.A. Nersesyan and A. Luther, Preprint F1’T-9, Institute of Physics, Thilisi (1988). [2]H.J. Schulz, Phys. Rev. B 39 (1989) 2940. [3] A.F. Andreev and IA. Grishchuk, Zh. Eksp. Teor. Fiz. 87 (1984) 467. [4] B.I. Rice,and in: Solid state physics, Vol.21, eds. Halperin F. Seitz, and D. T.M. Turnbull H. Ehrenreich (Academic Press, New York, 1968) p. 116. [5] A.A. Nersesyan and G.E. Vachnadze, J. Low Temp. Phys. 77 (1989) 293; A.A. Nersesyan, G.I. Japaridze and 1G. Kimeridze, to be [6] published. I.E. Dzyaloshinskii, Pis’ma Zh. Eksp. Teor. Fiz. 46 (1987) 110; H.J. Schulz, Europhys. Lett. 4 (1987) 609.

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Hirsch, Phys. Rev. B 33 (1986) 8155; J.W. Cannon and E. Fradkin, Urbana preprint (1989). [8] T. Giamarchi and H.J. Schulz, J. Phys. (Paris) 49 (1988) 819; K.A. Muttalib and V.J. Emery, Phys. Rev. Lett. 57 (1986) 1370. [9]A. Lutherand V.J. Emery, Phys. Rev. LeIt. 33(1974) 589.

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[101 V.J. Emery, in: Highly conducting one-dimensional solids, eds. J.T. Devreese, R.P. Evrard and V.E. van Doren (Plenum, New York, 1979); J.Solyom,Adv.Phys.28 (1979) 201. [11] T.T. Truong and K.D. Schotte, Phys. Rev. Lett. 47 (1981) 285.