8/3/2019 A. Bjelis and D. Zanchi- Effects of Zeeman splitting on spin density waves

1/4

JOURNAL DE PHYSIQUE IVColloque C2,supplkment au Journal de Physique I, Volume 3,juillet 1993

Effects of Zeeman splitting on spin density wavesA. BJELISand D. ZANCHIDepartment ofPhysics, Faculty ofScience, UniversityofZagreb,PO. Box 162,41001 Zagreb, Croatia

Starting from the Hubbard model with the repulsive Coulomb interaction andimperfectly nested Fermi surfaces we calculate the total, i.e. spin and charge densitywave, 2kF-susceptibility above the critical temperature, taking into account both,orbital and spin coupling of electrons to the external magnetic field H. The influenceof the magnetic field on the collective modes, and in particular the emerging splittingof degeneracies present in the isotropic case (H = O), are also discussed.

1. IntroductionThe external magnetic field H influences the spin density wave (SDW) order through the

orbital and Pauli coupling. The orbital coupling may lead to the field induced stabilization ofSDW [l - 31, or to the increase of critical temperature in systems with SDW already existing atH = 0 [4]. Furthermore, this mechanism also causes the decrease of the transverse correlationlengths (even if the nesting is perfect)[5] and the angular resonances of Lebed's [6 ] type wheneverthe magnetic lengths in the two transverse directions are commensurate [7,8]. These effects weremostly observed in SDW Bechgaard salts, although the orbital mechanism does not distinguishSDW from charge density wave (CDW) systems. More precisely, it does not lift the orientationaldegeneracy of spin in the perfect SDW order without internal spin anisotropies. This is howevernot the case with the Pauli coupling which introduces the Zeeman splitting in the electronic spinorientations. At first, it favors the spin orientation of SDW in the plane perpendicular to H [9],so that the SDW order takes place in this plane whenever the external field is stronger than theinternal spin flop field. This has immediate consequences on six SDW collective modes. Withinthe mean-field approximation and in the isotropic (H = 0) case they are degenerate above thecritical temperature T,. At T < Tc three of them acquire gaps, while another three are acousticGoldstone modes representing two spin wave and one phason branches. The finite H at T > Tcshifts two of six modes upwards. Below Tc the spin wave mode representing the fluctuations ofthe spin component perpendicular to the easy plane acquires a gap proportional to the magneticfield. In this work we analyse these effects quantitatively by calculating the RPA susceptibility atT > T, and the corresponding lowest order harmonic term in the Landau free energy.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:1993266

http://www.edpsciences.org/http://dx.doi.org/10.1051/jp4:1993266http://dx.doi.org/10.1051/jp4:1993266http://www.edpsciences.org/

8/3/2019 A. Bjelis and D. Zanchi- Effects of Zeeman splitting on spin density waves

2/4

JOURNAL DE PHYSIQUEIV

2. R P A SusceptibilityIn the discussion of all collective modes it is necessary to take into account that the SDW is a

complex three-dimensional vector. Its three components are

where a;are Pauli matrices and IE' is the four-component fermion field

with the first index denoting two sheets of the quasi-one-dimensional Fermi surface. Furthermore,the Zeernan spitting causes the mixing of the SD-W component parallel to H (chosen to be M3)and the CDW amplitude

0 IC M , = B ' [ ~ o ] l . (3)The full 2kF - susceptibility above T, s thus represented by the 4 x 4 matrix with the components

The straightforward calculation of ~ ; jithin RPA for the usual Hubbard model with the imperfectnesting leads to the expression

Here X, = I ; 6 = Ixt(s) - xl(q)l/2xg and XT.I = xO(ql* 271, where fi denotes thewave number in the chain direction and 7 = pBH/vF (pB - Bohr magneton, VF - Fermi velocity).x0 s the scalar Hartree-Fock (bubble) susceptibility which contains only orbital contributions fromthe magnetic field [7];

with

8/3/2019 A. Bjelis and D. Zanchi- Effects of Zeeman splitting on spin density waves

3/4

X- is the branch which for U > 0 reduces to x~~n the limit ]HI -+0 Its poles are therefore thoseSDW collective modes which are renormalized due to the Zeeman splitting. Note also that in thelong-wavelength limit M3 and C are decoupled, since 6(q = 0) = 0 ( but xg(q = 0) # xo(q = 0)). Comparing l/xL and 1iX!! , t comes out that , irrespectively of the value of H, the absoluteminimum of at q = 0 is at all temperatures higher than the absolute minimum of 1iX!(which can be at a finite value of q for pgH 2 0.34. 2xTc). The mean-field critical temperatureis therefore not affected by the Zeernan splitting.

3. Collective ModesLet us consider the simplest case of the perfect one-dimensional nesting (all ta's equal to

zero). The long-wavelength propagator of the two degenerate modes from X is then given by theexpansion

I Ix-(ql, w) 2~ (iw t all + t!q:)-wherefl s proportional to the gap at ql = 0, a ?r/8T and (! is the longitudinal correlationlength. The dependence of these coefficients on h p ~ H / 2 x T t T = T,and for NFU = 0.16is shown in Fig.1. a' and 6; are the corresponding quantities from the - dispersion (a' = 0at T = T,). he value all depends parabolically on the magnetic field up to h = h, = 0.34. Atthis value of h the longitudinal correlation length # approaches zero, signifying that by a furtherincrease of the magnetic field the maximum of X shifts to the finite point q = tc with tc d m ,with the asymptotic behavior vFtc/4aT = h. In Fig.1 we also show all and the correlation lengthmeasured with respect to the new maximum at h > h,.

Some hints on the collective modes below T, may be drawn from the Landau expansion for thefree energy. Expecting that main effects of Zeeman splitting enter through the second-order term(i. e. the susceptibility (5)), we complete this expansion by the fourth-order term from Ref.10.This term introduces an important constraint on the SDW order in the isotropic (H = 0) case,namely all three components M; have a common phase in the equilibrium. As a consequence, thesix collective modes split into three Goldstone branches mentioned in the Introduction and threemodes with the gaps. One of the latter modes is the usual amplitudon, while the remaining twoare the fluctuations of the relative phases with respect to their zero values imposed by the aboveconstraint. The Zeeman splitting introduces a finite gap into the Goldstone mode representingfluctuations of the spin orientation out of the easy plane. The gap of this mode is determined byd n . nother mode influenced by the Zeeman splitting is associated to the fluctuations ofthe phase of M3 with respect to the common phase of the order parameter. Here the dependenceon H enters mainly through the correlation length ti1.

The finite transverse hoppings do not alter significantly the results of Fig.1, and vice versa,the effects of Zeeman splitting on the transverse correlation lengths ti,, of X - modes are weakerthan those originating from the orbital contribution (6) [5,11]. Finally, the imperfect nestingbrings a new type of angular resonances [12] in the gap and the correlation lengths of all modesinfluenced by the Zeeman splitting. They appear at the angles cos0 = 4pBH/nwb (n - integer), asthe combined effect of Pauli and orbital coupling. The quantitative analysis of these resonanceswill be given elsewhere [l l ] .

8/3/2019 A. Bjelis and D. Zanchi- Effects of Zeeman splitting on spin density waves

4/4

JOURNALDEPHYSIQUEIV

Fig.1. (a) all a t T = Tc as the function of h. Th e lower curve at h > hcrepresen ts all(n).(b) T h e co rresp ond ing r at io (i1(h)/t:. he insert represents th e dependence of (v~ /4.lrT,)non h.

References[l] GOR 'KOV L. P. an d LEBE D' A. G ., J. Ph ysique Lett.45 (1984) L433..[2] HE RIT IER M., M ONTAM BAUX G. and LED ERE R P. , J. Physique Le tt .45 (1984) L943.[3] M A KI K., P hys . Rev.B 33 (1986 ) 4826.[4] BJEL IS A. an d MA KI K., Phys . Rev.B 42 (1990) 10275.[5] BJE LIS A. and M AK I K., Phys. Rev.B 4 4 (1991) 6799.[6] LEBED' A. G., Pi sm a Zh.Eksp.Teor.Fiz.43 (1986) 137 [ JE T P L e t t. 43 (1986) 1741.[7] BJE LIS A. an d MA KI K., Phys. Rev.B 45 (1992) 12887; Sy nth . M etals 55-57 (1993) 2749.[8] SUN Y. a nd MA KI K., preprint.[9] POILB LAN C D. an d LED ERE R P. ,Phys. Rev.B 37 (1988) 9650.[lo] HASEGAWA Y. an d FUKUY AMA H., J. Phys . Soc. Jap an 5 5 (1986) 3978.[I l l BJELIS A. and ZAN CHI D., to be published.[12] LEBED' A. G., J. Physique 2 (1992) no 11.