Journal of Magnetism and Magnetic Materials 54-57 (1986) 1261-1262

SPIN-PEIERLS STATE VS. MAGNETIC STATE IN THE QUASI-ONE-DIMENSIONAL HEISENBERG ANTIFERROMAGNET

1261

Satoru INAGAKI

Facult_v of Engineering Meljr University, Tama ku, Kawasaki 214, Japan

and Hidetoshi FU KUYAMA

Insiitule for Solid State Physics, Umuersity of Tokyo, Roppong, Minato ku, Tok_yo 106, Japcm

The competition between the spin-Peierls state and magnetic states in the quasi-one-dimensional Heisenberg antiferromagnet is discussed on the basis of the phase Hamiltonian. which takes proper account of the quantum fluctuations essential to the spin-Peierls transition. Various phase transitions induced by the magnetic field are also pointed out.

The spin-Peierls (SP) transition reflects the subtle feature of the ground state of the one-dimensional (ID) Heisenberg antiferromagnet, i.e., the coexistence of vari- ous states including the nonmagnetic singlet state and magnetic NCel (N) state due to the large quantum fluctuations. Owing to the existence of singlet compo- nent the nonmagnetic SP state is always stabilized in strictly 1D systems how small the coupling with the 3D lattice may be [l]. On the other hand, the weak inter- chain exchange interaction present in actual quasi-1D systems favors the magnetic N state, and therefore the competition between the SP state and the N state is important to discuss the properties of actual quasi-1D spin systems. This competition problem is studied ex- plicitly on the basis of the phase Hamiltonian intro- duced by the bosonization of the spinless fermions.

By taking into account the weak interchain exchange interaction as the mean field after Scalapino, Imry and Pincus [2], the quasi-lD Heisenberg antiferromagnet reduces to the 1D one in an effective field whose magnitude is determined self-consistently:

-z~J’~~{(s,--)s,z-~(s~)2}. /

(1)

where J > 0 is the antiferromagnetic intrachain ex- change interaction, J’ the interchain interaction be- tween nearest neighbor chains (the number is given by z). We have assumed ]J’] <J, and u,, a, h and K denote the lattice distortion, the lattice spacing, the spin-lattice coupling and the elastic constant, respec- tively. By following Nakano and Fukuyama [3], the Hamiltonian (1) is transformed into the following phase Hamiltonian [4]

H = dx A( v~(x))‘+ Cp’(x) 1 [

-BsinB(x)-Dcos20(x)

+$u’(x)-Fcos e(x)+;(cos B(x)) , 1 (2) with 0(x) and p(x) being the canonical conjugates and

A, B, C, D and F are given by A = Ja/S, B = JXu/a*,

C = =*Ja/2, D = 2,n2Ad/a2 and F=

z 1 J’ 1 (cos O( x,)/a, respectively. The values of A and C have been adjusted following

Cross and Fisher [S] to yield the known exact results of the spin wave velocity and the critical exponent of the spin correlation function. The D term comes from the umklapp scattering of the spinless fermions (d is its characteristic parameter. 0 Q d < l/2). The B term comes from the coupling of the spins to the dimerized lattice distortion u,, which is given by uI = (- 1)‘~ at the Ith lattice site.

The phase variable 0(x) is related to the spin oper- ators in such a way that the z-component of the spin operator is given by

SZ(x)=(l/a)(-l)‘cos~(x)+(l/271)V~(x), (3)

at x = lu. We note that the state with 0 = 0 corresponds to the N state and that with 0 = n/2 to the SP state. The B term in (2) favors the SP state while the D and F

terms favor the N state. In the case of the 1D undistorted Heisenberg model

where B = F = 0, the quantum fluctuations renormalize the D term resulting in vanishing contributions, i.e., effectively D = 0 [3]. Thus 6’ is arbitrary in the ground state in this case and the coexistence of various states due to the large quantum fluctuations, a unique feature of the 1D Heisenberg antiferromagnet, is properly taken into account by the phase Hamiltonian.

We treat the general Hamiltonian (2) in the self-con- sistent harmonic approximation [3]. Then the ground state of the Hamiltonian is found to be the SP state for q/j’ > ((1 + d)/(l - d))“’ and the N state otherwise (fig. l), where r~ and j’ are the dimensionless spin-lattice coupling and the interchain exchange interaction de-

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1262 S. Inagaki, H. Fukuyuma / Spin - Peierls slate vs. mugnerrc stute

I/ 0 1

Fig. 1. The phase diagram of the stable state in the presence of weak interchain exchange interaction. The ordinate q and abscissa j’ are the dimensionless spin-lattice coupling and interchain exchange interaction. The phase boundary between the spin-Peierls (SP) state and the NCel (N) state is given by the full line. For the dashed line and A, B and C. see fig. 2.

fined by 1) = JA2/4a2Ka and j’ = z / J’~/IT~J.

The transition temperatures of the undistorted paramagnetic (P) state to the two states are obtained by calculating the appropriate correlation functions in the P state. The result is given by T,,/J = C,TJ and TN/J =

C, j’, where C, and C, and numerical constants of order unity. The essentially same result was obtained by Cross and Fisher for Tsp [5] and by Fowler for TN [6]. It is consistent with the phase diagram at T = 0 K that the P state is transformed into the SP state or the N state according as 7 is larger or smaller than j’.

In the presence of the magnetic field we have the Zeeman energy, which is given by -gp,H/2/@(x)dx in the phase representation [3]. This allows for the phase variable 0(x) varying spatially due to the field, which results in the commensurate-incommensurate transition for the SP state: the SP state is transformed into the state with spin-solitons above some critical value of the magnetic field. This state may be called the ‘magnetic discommensurate’ (MD) state. The energy of the MD state is calculated by using the phase Hamiltonian. On the other hand, the N state changes into the spin-flop state (SF) under the field in the absence of the spin-ani- sotropy energies. The energy of the SF state is estimated by using the relation Es, = E, - ixH2 where the spin susceptibility x is approximately given by the suscept- ibility of the 1D Heisenberg antiferromagnet exactly known at T = 0 K.

Energy comparison of these states at T = 0 K de- termines the phase diagram of the stable states, which is

shown in fig. 2 [7]. On increasing the magnitude of the magnetic field (h is the dimensionless field defined by h = gpBH/T2J) the following three types of phase tran- sitions occur depending on the ratio j’/q of the inter-

SF -

h 0 I I I

0 0.2 0.4 O.’ jR? CBA

Fig. 2. Phase diagram of stable states in the presence of the magnetic field. The ordinate h/T denotes the field vs. spin- lattice coupling, and the abscissa J’/V the interchain interac- tion vs spin-lattice coupling (d = 0.5).

chain interaction to the spin-lattice coupling: (A) N + SF (B) SP ---) SF (c) SP + MD + SF. The three types of transitions are indicated by A. B and C and the region corresponding to each case is indicated schematically also in fig. 1.

The phase transitions of SP + SF at h = h, and MD + SF at h = ht2 are of first order, and the one of SP+MD at h=h,, is of second order in clean sys- tems. However. it may well be also of first order in real systems considering the presence of lattice imperfec- tions or impurities. Fig. 2 also shows that it is rather difficult to obtain the SF state from the SP state by applying the magnetic field: This occurs only in the narrow region of ~‘/TJ around case B where the SP state and the N state are very close in energy. This result is

consistent with the experimental observations of the actual SP systems [l] at low temperatures where only the transition of SP + MD seems to be observed [8].

Experimentally the ratio j’/q can be varied by applying the pressure or by modifying the local environment of

the spin chains. It is interesting to realize situations where the energy of the SP state and the N state are close so that the above prediction can be tested.

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For the review of the SP transition, see, for instance, J.W. Bray, L.V. Interrante. IS. Jacobs and J.C. Banner, in: Extended Linear Chain Compounds, vol. III. ed. J.S. Miller (Plenum Press, New York, 19X3). D.J. Scalapino, Y. Imry and P. Pincus: Phys. Rev. Bll (1975) 2042. T. Nakano and H. Fukuyama, J. Phys. Sot. Japan 49 (1980) 1679. 50 (1981) 2489. S. Inagaki and H. Fukuyama, J. Phys. Sot. Japan 52 (1983) 3620. M.C. Cross and D.S. Fisher, Rhys. Rev. B19 (1979) 402. M. Fowler, Phys. Rev. B17 (1978) 2989. S. lnagaki and H. Fukuyama, J. Phys. Sot. Japan 53 (1984) 4386. T.W. Hijmans, H.B. Brom and L.J. de Jongh. Phys. Rev. Lett. 54 (1985) 1714.