An exactly soluble model describing a spin S=1 antiferromagnetic chain doped
with mobile S=1/2 carriers is constructed. In its continuum limit the undoped
state is described by three gapless Majorana fermions composing the SU(2)
triplet. Doping adds to this a scalar charge field and a singlet Majorana
fermion with different velocity. We argue that this mode survives when the
Haldane gap is added.Comment: RevTeX, 6 pages, 3 figures; final version, to appear in PR

A. B. Zamolodchikov

A. Foerster

A. M. Tsvelik

B. Sutherland

D. G. Shelton

E. Dagotto

F. H. L. Eßler

H. J. de Vega

H. M. Babujian

Holger Frahm

J. F. DiTusa

J. Riera

K. Penc

L. Takhtajan

M. Marcu

M. P. Pfannmüller

M. Scheunert

M. Takahashi

Markus P. Pfannmüller

N. Andrei

P. Coleman

P. P. Kulish

P. Schlottmann

S. R. Aladim

V. E. Korepin

V. J. Emery

English

arXiv

An exactly soluble model describing a spin S=1 antiferromagnetic chain doped with mobile S=1/2 carriers is constructed. In its continuum limit the undoped state is described by three gapless Majorana fermions composing the SU(2) triplet. Doping adds a scalar charge field and a singlet Majorana fermion with different velocity. We argue that this mode survives when the Haldane gap is added

Frahm, Holger

Pfannmüller, M.P.

Tsvelik, A.M.

Institutional Repository of Leibniz Universität Hannover

VOLUME 81, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 7 SEPTEMBER1998mplessjo-dded.2116Doping of a Spin-1 Chain: An Integrable ModelHolger Frahm,1 Markus P. Pfannmüller,1 and A. M. Tsvelik21Institut für Theoretische Physik, Universität Hannover, D-30167 Hannover, Germany2Department of Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, United Kingdo(Received 12 March 1998)An exactly soluble model describing a spinS ­ 1 antiferromagnetic chain doped with mobileS ­ 1y2 carriers is constructed. In its continuum limit the undoped state is described by three gaMajorana fermions composing the SUs2d triplet. Doping adds a scalar charge field and a singlet Marana fermion with different velocity. We argue that this mode survives when the Haldane gap is a[S0031-9007(98)07027-6]PACS numbers: 71.10.Pm, 75.10.Jm, 75.50.Eetheael.anrestnceleene)c-le-mi-d,ec-anatoennIt is well known that mobile holes introduced intoantiferromagnetic Mott insulators by means of dopincause frustration which seriously affects the magnetproperties of the system. In this paper we studydoped spin-1 chain which in the presence of mobiholes interpolates betweenS ­ 1 and S ­ 1y2 states.According to Refs. [1–3] this situation is realized inthe carrier doped Haldane system Y22xCaxBaNiO5 [4]:For x ­ 0 the interactions between the spin-1 Ni21 ionsare well described by a Heisenberg model, upon dopinmixing of the S ­ 1y2 holes on the Oxygen sites leadsto a low energy doublet state in an effective one-banHamiltonian which can move in theS ­ 1 background.The model we study is an integrable model whosHamiltonian closely resembles the Hamiltonian introduced in [2]:H ­LXn­1hH exchn,n11 1 Hhoppn,n11j ,H exchij ­12√1SiSjSi ? Sj 2 11 dSiSj ,1f1 2 sSi ? Sjd2g!,(1)Hhoppij ­ 2s1 2 dSi,Sj dPijsSi ? Sjd .Here S2i ­ SisSi 1 1d with Si ­ 1 or 1y2 and Pijpermutes the spins on sitesi andj. Comparing (1) to theeffective one-band Hamiltonian for the doped Ni oxidgiven in [2],Hij ­ dSiSj ,1JSi ? Sj11 2 PijµSi ? Sj 112∂, (2)we find several differences. First, the spin exchangbetweenS ­ 1 sites contains biquadratic terms giving thespin-1 Takhtajan-Babujian chain for hole concentratiox ­ 0. Therefore in the undoped limit the spectrum igapless and one may think that the most spectacular fture of the dopedS ­ 1 chain—filling the gap with holes,is lost in our model. We shall see later, however, thatis possible to reintroduce a gap in the continuous limwhere we have a field theoretical description of the modeSecond, due to a hidden supersymmetry of the model (s0031-9007y98y81(10)y2116(4)$15.00gicalegde-eensea-ititl.eebelow) the ratio between the exchange integral andhopping is fixed in (1). We do not expect this to beserious restriction on physical properties of the modThird, hopping processes such asj1i # jl ! j # i1jlare allowed. Finally, the integrable model containsantiferromagnetic exchange interaction between neaneighbor hole states which is responsible for the abseof a ferromagnetic phase of (2) for sufficiently large hoconcentration. Such an interaction has, however, beconsidered in Ref. [3] to improve agreement with thexperiments atx ­ 1.In the following we will use the exact solution of (1to deduce an effective field theory for the low-energy setor of the system which will enable us to study possibrelevant perturbations. The low-energy limit of the undoped system (x ­ 0) is well known to be a realizationof an SUs2d level k ­ 2 Wess-Zumino-Novikov-Witten(WZNW) model with central chargec ­ 3y2. This modelis equivalent to a model of three massless Majorana ferons composing an SUs2d triplet [5]. A completely filled(x ­ 1) band corresponds to theS ­ 1y2 chain which inthe low-energy limit is equivalent to the SUs2d1 WZNWmodel. For finite doping we find, as may be expecteone free bosonic mode in the charge sector. The spin stor, however, turns out to be rather unusual containingdirect sum ofc ­ 3y2 and c ­ 1y2 models with differ-ent velocities. Thus doping generates the fourth Majorafermion—a feature observed in various models relatedtwo-channel Kondo physics [6,7].The model (1) is constructed from solutions to thYang Baxter (YBE) equation invariant under the actioof the graded Lie algebragls2 j 1d in the “atypical”representationfSg1 [8]. These representations contaitwo multiplets of spinS andsS 2 1y2d with respect to theSUs2d-subalgebra ofgls2 j 1d. A well known example fora lattice model obtained from thef1y2g1-representation inthis approach is the supersymmetrict-J model (see, e.g.,[9]). Denoting the generators of the SUs2d subalgebraby Sa, the Us1d-charge operator byB and the remainingfermionic generators ofgls2 j 1d by V 6, W6 (which createand annihilate the holes, see [8]) the localL operator ofthe quantum inverse scattering method reads© 1998 The American Physical SocietyVOLUME 81, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 7 SEPTEMBER1998L smd ~0B@ m 1 2iB ip2W2 ip2W1ip2V 1 m 1 isB 1 Szd 2iS12ip2V 2 2iS2 m 1 isB 2 Szd1CA . (3)s,r-mrdThe spectrum of the transfer matrix for a vertex modwith L of these weights per row is obtained by meanof the algebraic Bethe ansatz [10]. Starting from thfully polarized state of spin-S multiplets on each sitewe consider states withNh holes (generating sites withspin S 2 1y2) and magnetizationMz ­ LS 2 12 Nh 2N#. This leads to an auxiliary eigenvalue problem foan inhomogenous graded six-vertex model on a lattiof n ­ Nh 1 N# sites which is solved by means of asecond Bethe Ansatz through addition of states wfermionic grading (holes) to its eigenstate withn ­ N#(for details see, e.g., [9]). As a result the spectrum of tsupersymmetric vertex model is parametrized by the roof the following set of Bethe-ansatz equations (BAE)√lj 1 iSlj 2 iS!L­Nh1N#Ykfijlj 2 lk 1 ilj 2 lk 2 iNhYa­1lj 2 na 2i2lj 2 na 1i2,j ­ 1, . . . , Nh 1 N#1 ­Nh1N#Yk­1na 2 lk 1i2na 2 lk 2i2,(4)a ­ 1, . . . , Nh .For S ­ 1y2, where the model becomes the supersymmetric t-J model, and Eqs. (4) are Sutherland’s form othe BAE [11]. The grading of the underlying algebrallows to construct two more sets of (equivalent) BA[9,12]. Equations (4) are most convenient for our analsis, however.To construct alocal Hamiltonian forS ­ 1 we projectthe graded tensor productLasm 1i2 db≠Lbsm 2 i2 d intwo copiesVa, Vb of the matrix space of (3) onto thef1g1 representation inVa ≠ Vb. The result Lf1g1 smdof this fusion [13,14] defines a new vertex model witeigenstates again parametrized by the roots of (4). Tlogarithmic derivative of its transfer matrix atm ­ 0 isthe Hamiltonian (1) with eigenvaluesEshljj, hnajd 2 HMz 2 m̃Nh ­Nh1N#Xk­1√H 22l2k 1 1!2NhXa­1√m 112H!2 LH (5)[we have added an external magneticH field and a (hole)chemical potential̃m ­ m 1 1 to the Hamiltonian].In the thermodynamic limit the solutions of the BAE(4) are real hole rapiditiesn, whose density we denoteas rsnd and complexl strings with densitiessnsld(n ­ 1, 2, . . .). The equations for the densities aredn,2ssxd ­ snsxd 1 Cnm p s̃msxd 2 dn,1s p rsxd ,a2ssxd ­ r̃sxd 1 f1 1 a2g21 p rsxd 1 s p s̃1sxd ,(6)elserceithheots-faEy-hhewhere all densities marked by tilde correspond to holef p gsxd denotes a convolution,ssxd ­ 1ys2 coshpxd,ansxd ­ s2nypdys4x2 1 n2d andCnmsxd ­ dnmdsxd 2°dn11,m 1 dn21,m¢ssxd .Writing the energy (5) in terms of densities ofl holes andr we obtainEyL ­ 22 2Zdxf2psa2 p sd 1 mgrsxd1Zdx 2pssxds̃2sxd 2 limn!`HnZdx s̃nsxd .(7)By minimization of the free energy we obtain the ther-modynamic Bethe ansatz (TBA) equations for the enegies en ­ T lnss̃nysnd of l strings andk ­ T lnsr̃yrdfor the hole rapiditiesensxd ­ Ts p lnf1 1 een21sxdyT g f1 1 een11sxdyT g2 2pdn,2ssxd 2 dn,1Ts p lnf1 1 e2ksxdyT g , (8)subject to the condition limn!`senynd ­ H and2f2pa2 p ssxd 1 mg 2 Ts p lnf1 1 ee1sxdyT g­ ksxd 1 TR p lnf1 1 e2ksxdyT g , (9)whereR ­ a2 p s1 1 a2d21. In terms of these functionsthe free energy readsFyL ­ 22 2 TZdx ssxd lnf1 1 ee2sxdyT g2 TZdxsa2 p sdsxd lnf1 1 e2ksxdyT g . (10)For temperaturesT ø H the energiesen.2 can beliminated from (8), giving the following equation fore2:e2sxd 2 TR p lnf1 1 ee2sxdyT g ­ 22pssxd 1 Hy21 Ts p lnf1 1 ee1sxdyT g .(11)Together with then ­ 1 equation from (8) and (9) thisequation determines the low-temperature phase diagraof the system (Fig. 1). For sufficiently large negativemEq. (9) impliesk . 0 corresponding to vanishing holedensity. The remaining TBA coincide with those for theintegrableS ­ 1 magnet [15,16]. Choosingm . Hy2we find k , 0 and e1 , 0 which implies a hole densityx ­ 1. The resulting TBA foren$2 are those of theS ­ 1y2 Heisenberg chain. Similar considerations fointermediatem yield the complete phase diagram and leaus to identifyk and e1 as the energy of the charge andspin modes associated with the mobile carriers whilee2 isthe magnetic mode of the background spins.2117VOLUME 81, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 7 SEPTEMBER1998hmenilehegnthhoftoe-r-f-x-si-lansFIG. 1. Low temperature phase diagram of the doped chaina magnetic fieldH ¿ T : the saturated ferromagnetic phase ilabeled FM. At intermediate fields the low-energy propertieof the system are determined by gapless bosonic modes wenergiesk sA, B1d, e1 sA, B2, FMd, and e2 sA, B1,2, Cd. TheH ­ 0 phase is discussed in the main text.We now concentrate on the case of finite doping invanishing magnetic field which corresponds to chemicpotentialsp 2 4 , m , 0. In this regime we havek ,0 for jxj , Q. For temperatures smaller than the Fermenergy of holesT ø 2m one can replacek in Eq. (8)by its zero temperature valuek0sxd and the free terms bytheir asymptotics. As a result we getensxd ­ Ts p lnf1 1 een21sxdyT g f1 1 een11sxdyT g2 2pdn,2e2pjxj 2 2pAdn,1e2pjxj, (12)where A ­ 2s2pd21RQ2Q dy epyk0s yd. To study thespecific thermodynamic properties of the present modwe have to separate the contributions to the free enerstemming from the charge sector from those duethe en. Considering low temperatures again the leadincontributions tok come from the vicinity of the Fermiwave vectors6Q. In this region one can safely negleccontributions to Eq. (9) frome1 and rewrite it as2f2pa2 p ssxd 1 mg 2 TR p lnf1 1 e2jksxdjyT g­ ksxd 2Z Q2Qdy Rsx 2 ydks yd , (13)where Q is determined by the conditionks6Qd ­ 0.Using the procedure introduced by Takahashi [17], we carewrite the free energy (10) asFyL ­ E0yL 1 f1 1 f2whereE0 is the ground-state energy andf1 ­ 2TZdx r0sxd lnf1 1 e2jk0sxdjyT g ø 2pT2y6y ,(14)f2 ­ 2TZdx ssxd lnf1 1 ee2sxdyT g2 TZdxss p r0d sxd lnf1 1 ee1sxdyT g . (15)2118inssithaalielgytogtnHerer0sxd is the density of hole rapidities atT ­ 0r0sxd 2Z Q2Qdy Rsx 2 ydr0s yd ­ a2 p ssxdandy is the velocity of the charge modeky ­12pr0sQd≠k0≠xÉx­Q. (16)In the low-T limit T ø 2m where this form isvalid one may replacessxd and sr0sxd in (15) by theirasymptotics. Now we have TBA equations in sucform that the spin sector is manifestly decoupled frothe charge one. The free energy of the latter is givby Eq. (14) representing a scalar bosonic mode, whEqs. (12) and (15) describe the thermodynamics of tspin sector. Such TBA equations (i.e., with two drivinterms) arise also in other systems combining differespins [18] and in the two-channel Kondo model wita channel anisotropy [19]. As shown in Ref. [7] suca two-channel Kondo model can be written in termsfour Majorana fermions. These considerations lead ushypothesize that the effective low-energy theory for thHamiltonian (1) at small doping isHeff ­ZdxfH 1 H gH ­ 2i23Xa­1y2xa≠xxa 2i2y1x0≠xx0 (17)1 gx0x1x2x3and a similar expressionH shxjd for the left movers.The fieldsx0, xa are Majorana fermions and the marginal coupling g between two sectors in (17) is theonly one possible by symmetry (we have omitted maginally irrelevant interactions between sectors with diferent chirality). A magnetic field couples to the termiRsbx0xa 1 eabcxbxcd quadratic in the fermions withsome constantb.The parameters of the effective theory have to be etracted from the TBA at small doping. This correspondto A ø 1 and implies that the two terms in (15) are domnated by contributions from the regionpx , lns2pyT d(whereje2j , T ) and bypx , lns2pAyT d, respectively.This separation of scales allows one to obtain the smalAbehavior of (15) at low temperaturesf2 ­ 2pT26y1µ1223A4pln A∂2pT26y2µ3213A4pln A∂1 . . . .For A ­ 0 this is the free energy of a single Majoranfermion [x0 in (17)] with velocity y1 and that of theSUs2d2 WZNW model with velocityy2 which can be ex-pressed in terms of massless triplet of Majorana fermio[5]. The ratio of the velocities is given byy1yy2 ­ 212pRQ2Q dy epyk0s ydRQ2Q dy epyr0s yd. (18)VOLUME 81, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 7 SEPTEMBER1998rei-htede...,,.The coupling constantg needs to be chosen such thait produces theA dependence of 2. Similarly, theparameterb can be determined from theT ­ 0 magneticsusceptibility. At smallA ø 1 we getx ­1pµ1y221y1A2pln A 1 . . .∂giving b2 ­ Ay2p. In Fig. 2 some of these quantitiesare given as a function of the hole concentrationx.The simplest SUs2d invariant perturbation of (17) is amass termiRdxsmP3a­1 xaxa 1 Mx0x0d for the par-ticles in the spin sector (recall that the model of thremassive Majorana fermions is a good continuous appromation for theS ­ 1 Heisenberg antiferromagnet [20]).ChoosingM ­ 0 we get a situation with a Haldane gapin the triplet sector and gapless singlet excitations. Thmay be a good description for the magnetic states insithe Haldane gap observed in the doped Ni oxide [4].To summarize, we have introduced an integrable moddescribing a magnetic system which interpolates betweFIG. 2. Parameters of the effective model (17): (a) Velocitiey of the charge mode (16) andy1 of the fermionx0 (18) and(b) the parameterA in the TBA (12) versus hole concentrationx. The inset showsA for small x. Note thaty2 ; p for allfillings.texi-isdeelensthe integrableS ­ 1 Takhtajan-Babujian and theS ­1y2 Heisenberg chain. For finite doping we find up tothree massless modes determining the low temperatuthermodynamics of the system. In the SUs2d invariantcase (H ­ 0) these are a bosonic charge mode and a drect sum of a SUs2d2 WZNW model and one Majoranafermion with different velocities in the spin sector. Fur-ther analysis of the TBA is necessary to gain more insigin the nature of the phase diagram of this and the relatS . 1 systems.This work has been supported by the DeutschForschungsgemeinschaft under Grant No. Fr 737y2–2.[1] K. Penc and H. Shiba, Phys. Rev. B52, R175 (1995).[2] E. Dagotto, J. Riera, A. Sandvik, and A. Moreo, PhysRev. Lett.76, 1731 (1996).[3] J. Riera, K. Hallberg, and E. Dagotto, Phys. Rev. Lett.79,713 (1997).[4] J. F. DiTusaet al., Phys. Rev. Lett.73, 1857 (1994).[5] A. B. Zamolodchikov and V. A. Fateev, Sov. J. Nucl. Phys43, 657 (1986).[6] V. J. Emery and S. Kivelson, Phys. Rev. B46, 10 812(1992); P. Coleman, L. B. Ioffe, and A. M. Tsvelik,Phys. Rev. B52, 6611 (1995); P. Azaria, C. Hooley,P. Lecheminant, C. Lhuilier, and A. M. Tsvelik (to bepublished).[7] P. Coleman and A. J. Schofield, Phys. Rev. Lett.75, 2184(1995).[8] M. Scheunert, W. Nahm, and V. Rittenberg, J. Math. Phys18, 155 (1977); M. Marcu, J. Math. Phys.21, 1277 (1980).[9] F. H. L. Eßler and V. E. Korepin, Phys. Rev. B46, 9147(1992); A. Foerster and M. Karowski, Nucl. Phys.B396,611 (1993).[10] V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin,Quantum Inverse Scattering Method and CorrelationFunctions (Cambridge University Press, Cambridge1993).[11] B. Sutherland, Phys. Rev. B12, 3795 (1975).[12] P. P. Kulish, J. Sov. Math.35, 2648 (1986) [Z. Nauchn.Semin. LOMI145, 140 (1985)].[13] P. P. Kulish and E. K. Sklyanin, inIntegrable QuantumField Theories, Lecture Notes in Physics Vol. 151,edited by J. Hietarinta and C. Montonen (Springer-VerlagBerlin, 1982), pp. 61–119.[14] M. P. Pfannmüller and H. Frahm, Nucl. Phys.B479, 575(1996).[15] L. Takhtajan, Phys. Lett. A87, 479 (1982).[16] H. M. Babujian, Phys. Lett. A90, 479 (1982); Nucl. Phys.B215, 317 (1983).[17] M. Takahashi, Prog. Theor. Phys.46, 401 (1971).[18] H. J. de Vega and F. Woynarovich, J. Phys. A25,4499 (1992); S. R. Aladim and M. J. Martins, J. Phys. A26, L529 (1993); H. J. de Vega, L. Mezincescu, andR. I. Nepomechie, Phys. Rev. B49, 13 223 (1994);P. Schlottmann, Phys. Rev. B49, 9202 (1994).[19] N. Andrei and A. Jerez, Phys. Rev. Lett.74, 4507 (1995);A. Jerez, N. Andrei, and G. Zarand, cond-mat/9803137.[20] A. M. Tsvelik, Phys. Rev. B42, 10 499 (1990); D. G.Shelton, A. A. Nersesyan, and A. M. Tsvelik, PhysRev. B53, 8521 (1996).2119

Doping of a Spin-1 Chain: An Integrable Model

Crossref

Physical Review Letters

https://repo.uni-hannover.de/bitstreams/09f6a86e-c0da-4d42-a5cd-198bc1f3ecb4/download

Doping of a spin-1 chain: integrable model

Doping of a spin-1 chain: integrable model

Abstract

Similar works

Full text

Available Versions

Institutional Repository of Leibniz Universität Hannover

Crossref