STATIC AND DYNAMIC ASPECTS OF SPIN-LATrlCE PEIERLS INSTABILITIES IN QUASI.ONE-DIMENSIONAL SYSTEMS

A. HOLZ Universitlit des Saarlandes, Theoretische Physik, D-6600 Saarbriicken, Fed. Rep. Germany

K. A. PENSON Universit~ Paris-Sua~ B~t. 510, F-91405 Orsay, France

K. H. BENNEMANN Freie Universitlit Berlin, lnstitut fiJr Theoretische Physik, D-IO00 Berlin, Germany

A qualitative theory of lattice Peierls instabilities for electronic- and spin-systems is developed. The physical situations where s u c h instabilities lead to continuous and discontinuous phase transitions (PT) are discussed. The possibility of a sliding spin density wave in compressible Heisenberg chains and in the presence of a magnetic field is mentioned.

I The theoretical study of the s =~ antiferromag- netic Heisenberg chain coupled to the lattice shows that a uniform magnetic chain may be unstable against dimerisation [1]. PyRe [1] obtained this result by going over to a Fermion representation of the Heisenberg Hamiltonian and within the Hartree-Fock approximation. In a three-dimen- sional (3D) lattice a PT into a dimerized phase is then postulated due to the 3D nature of the prob- lem induced by the phonons. Experimentally the spin-Peierls transition has been demonstrated by Bray et al. [2] in the isostructural compounds TTF • MS4C4(CF3) 4 with M = Cu and Au. Although the PT in these systems is continuous there is also an example of a discontinuous spin-Peierls transition which has been observed by Pouget et al. [3] on the zig-zag chains of VO 2. Further examples are dis- cussed by Holz et al. [4]. Because the alternating bond lengths in the dimerized state imply a mag- netic excitation gap, one may think that the reverse is also true. This is, however, not necessarily the case. In uniformly stacked T M P D - T C N Q crystals a magnetic excitation gap is observed without structural dimerisation. In this system it is pos- tulated in ref. [4] that a spin polarisation instability takes place. Here the electric polarisation plays the phonons part in the instability.

Although to date no rigorous treatment of the spin-Peierls instability exists the problem has been attacked in ref. [4] by rather powerful methods allowing rather general conclusions about the physics of that transition. Consider for instance the

1 anisotropic S - - 7 Heisenberg magnet on a

harmonic chain N

H -'~ E (Jo ..t- (--1)iAJl)[Ot3izoiZ+l i - l

+ 2(oi+Oi+l + oZai+,)] + gw0 A2. (1)

Here o~ z, ol +- are Pauli operators, Jo > 0 is the exchange constant, J~ its derivative and h the am- plitude of the lattice distortion along the chains. The last term of eq. (1) represents the harmonic lattice energy. Furthermore, p < - 1, p = 0 and p > - 1 represent the anisotropic ferromagnets, the XY-model and the anisotropic antiferromag- nets, respectively. The ground-state energy of H is plotted schematically in fig. 1. For p < - 1 i t can be shown rigorously that no magnetic energy can be gained for distortions AJ < Jo and this produces the three well-shaped e,o(p, Jo, AJI) ground-state energy curves. For i < - 2 , - ~ P ~ < 6 the two well- shaped curves are obtained and for - 1 < p ~< - ~, and 0 ~> 6 the four well-shaped curves. ¢0~ divides the phase diagram (PD) into dimerized and undimerized regions at T = 0. Fig. 1 is derived in ref. [4] using rigorous arguments.

Consider now the situation for T > 0 and o~ o < o~. In the 1D case symmetry will be immediately restored by domain formation. As an effective Hamiltonian to describe the system one may use

Ho,f = E s,J"(p, J0, J,, T)Sj ( i , j )

+ £ J0, J,, ,%, T)g. (2) J

Journal of Magnetism and Magnetic Materials 15-18 (1980) 1015-1016 ©North Holland 1015

1016 A. Holz et al./ Spin Peierls instabilities

% (g ,Jo, AJ~)

~.~. /il? ~- - I / / / / I _.f// \ .--_/, .," /

j

/ " - . . . . -" -V2s ~ ~ 6 9>6

< _< q ~ -1/2

Fig. 1. Right-hand side of symmetrical ground-state energy curve e~,(p, Jo, AJl) vs. dimer order parameter AJ v # < - 1 labels three well-shaped e,~ curves for anisotropic ferromagnets; - ½ ~ p ~ 6 labels two well-shaped % curve for anisotropic antiferromagnets, w~ divides the PD into dimerized and undim-

erized regions [4].

Here Sy may assume n different values, if n wells are present leading to n different domains. In vec- tor notat ion ~ is a n-state-vector, jeff a n × n coupling constant matrix and /t a n-vector repre- senting the chemical potentials. The sum in eq. (2) is performed over nearest neighbors and the lattice unit of the chain on which eq. (2) is defined is of the o rder of the d o m a i n b o u n d a r y width

6(p, Jo, Jl, ~o, T). In general eq. (2) describes two types of transi-

tion phenomena. The first is a Ising-type transition a round T *I below which domains get rather large and a round the second transit ion temperature T *D domain format ion sets in. Clearly T *I < T *o has to hold. The second transit ion is a consequence of the fact that for T > 0, e,~(p, Jo, AJl) in fig. 1 has to be subst i tu ted by the free energy f,~(P, Jo, AJx, T). In ref. [4] it has been shown that due to the gap in the excitation spectrum of the Heisenberg magnet which is p roduced by the dis- tortion A the free energies of the wells in fig. 1 get lifted relative to the state A -- 0 and the free energy barrier in front of the origin (in case it exists) gets reduced. Consequently, at a certain temperature T *D domain format ion must set in and if the free energy barrier is still rather large the transition should be rather well defined.

In the 3D case, eq. (2) is supplemented by interchain couplings which are bilinear in Sj. If a

3D phonon exists whose symmetry allows each chain to dimerize then fig. 1 still applies. The interchain coupling J~ should be propor t ional to the width of the phonon spectrum with wave length

= 2 along the chains. F o r [IJi[[ << [IJeff[[ a Ising- type PT occurs at T c << T *l. The temperature inter- val between Tc and T *I defines then the regime where strong 1D fluctuations occur. For IlJ;[I ~> I[Jeff[[, 1D fluctuations are strongly suppressed and in the presence of a free energy barrier a discon- t inuous PT may result. It follows f rom fig. l that for the isotropic s = ½ Heisenberg model (p = l) and XY-model (p = 0) cont inuous PT arise. The

1 discontinuous PT of the isotropic s = i Heisenberg chain in VO 2 is a consequence of the free energy barrier produced by a " anha rmon ic" lattice energy as explained in ref. [4].

A similar analysis can be applied to the H u b b a r d model linearly coupled to the lattice. Here only cont inuous Peierls transitions are possi- ble. Only if two-band H u b b a r d models are used a discontinuous metal insulator PT may occur as, e.g., in VO 2 [4].

Finally, we would like to ment ion the possibility of sliding spin density waves in the f luctuation regime of the spin-Peier ls transit ion in the pres- ence of a magnetic field h [5]. A strong magnetic field will change the periodicity of the magnet ic ground state such that it is incommensurable with the lattice. Strong local incommensurabi l i ty may be released, however, by the lattice by enforcing phase jumps into the ground-state wave funct ion of the magnetic system which will act as intrinsic pinning centers for the spin density wave. A theory treating that effect has so far not been worked out.

R e f e r e n c e s

[1] E. Pytte, Phys. Rev. B10 (1974) 4637. [2] J. W. Bray, H. R. Hart, Jr, L. V. Interrante, I. S. Jacobs, J. S.

Kasper, G. D. Watkins, S. H. Wee and J. C. Bonner, Phys. Rev. Lett. 35 (1975) 744.

[3] J. P. Pouget, H. Launois, J. P. D. Haenes, P. Merenda and T. M. Rice, Phys. Rev. Lett. 35 (1975) 873.

[4] A. Holz, K. A. Penson and K. H. Bennemann, Phys. Rev. Bl6 (1977) 3999.

[5] A. Holz, in: Quasi One-Dimensional Conductors II, ed. S. Bari~i~ (Springer-Verlag, Berlin, 1979) p. 109.