low-temperature properties of one-dimensional heisenberg systems with random anistropy

R. S. GEKHT : Low-Temperature Properties of 1D Heisenberg Systems

phys. stat. sol. (b) 122, 509 (1984)

Subject classification: 13 and 18.2

L. V . Kirenskii Institute of Physics, Academy of Sciences of the USSR, Siberian Branch, Krasnoyarsk‘)

Low - Temperature Proper ties of One -Dimensional Heisenberg Systems with Random Anisotropy BY R. S. GEKHT

509

Thermodynamic properties of classical one-dimensional Heisenberg systems with random V (of the “easy axis” type) and regular D (of the “easy plane” type) anisotropies are considered. The expression for the free energy is obtained by the transfer-integral techniques and by the solution of the Fokker-Planck equations. Low-temperature dependences of susceptibility and correlation lengths are found for a random disordered system. The expression for the square of fluctuation of the S, spin component is obtained in dependence on the parameters V , D and temperature T.

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1. Introduction At present there is a number of magnetics in which exchange interaction in a spin chain is considerably larger than the interaction between the spins of different chains. One-dimensional magnetics in the regular anisotropy field are theoretically studied in many papers. Using the transfer-integral techniques, McGurn and Scalapino [l] and separately Loveluck et al. [ 2 ] have obtained the thermodynamic properties of such systems by solving numerically the Heisenberg model with classical spins. The analytical expressions for low temperature and correlation lengths are obtained in the nearest-neighbour approximation by Feigelman [3], Nakamura and Sasada [4], and by the author [5] with allowance for long-range interaction.

I n recent years an increasing interest has appeared in the study of disordered one- dimensional systems with manifold degenerate ground state in the presence of a large number of frustrations [6, 71. Feigelman and Vinokur et al. [8] examined the properties of one-dimensional systems at T = 0 for the xy model in the random anisotropy field (which is equivalent to the problem on the charge density wave in the random potential of impurities [9 to 111).

I n the present paper the thermodynamic properties of one-dimensional Heisenberg systems with both random anisotropy V (> 0) of the “easy axis” type and constant anisotropy D (< 0) of the “easy plane” type will be considered. The behaviour of such systems substantially depends on the relation between the exchange interaction energy J and the random anisotropy energy V . In what follows the condition J > V under which random space axis directions in each site are independent is assumed to be fulfilled (this corresponds to “weak pinning” of the charge-density wave to impurities [lo, 111).

__ - l) 860 036 Krasnoyarsk, USSR,.

510 R. S. GEKHT

2. Fokker-Planck Equation

We consider a chain of spins with random anisotropy of the "easy axis" type (V > 0) and constant anisotropy of the "easy plane" type (D < 0 ) in the external field h, whose Hamiltonian is given in the form

H = - C [ ISt a + V(St - Z i ) 2 + D ( S i . n)z + h - SJ , (2.1) i

where S , = (sin €Ji cos v,, sin Oi sin ~ 1 $ , cos 6,) is the classical spin of unit magnitude of the i-th particle; Z, = (sin 43% COSLX~, sin $3* sinat, cos $3,) and n are unit vectors along the random axis and the z-axis, respectively.

The partition function 2 for the system of interacting particles is expressed as

N

i=l 2 = J dSi exp (- H/T) ,

where dSt = sin Oi d6, dvi. Using the transfer-integral techniques [12] and following [8] to calculate the values of interest, we introduce the partition function of the chain x N ( O N , P ) N ) which depends on the angles O N , qN in its last points (and integrated with respect to all 6,, vi with i < N ) . Then ~ ~ + 1 ( 6 N + 1 , Q ) N + ~ ) is connected with x N ( O n ; , vN) in the following way:

Using the representation X N ( ~ ~ , qN) in the form

x N ( e N , P ) N ) = exp [- & N ( e N , Q)N)/TI

and substituting it jnto (2.2), we get in the temperature interval T < I the following recurrent relation for the free energy & , ( O N , vN) depending on the angle coordinates oh', Q)N

1 ^ T * 21 2 1

- h * S - V ( S - ZN+1)2 - I + T In ( I /ZZT) ,

& N + 1 ( 0 , v) - E N ( 8 , v) = - ( L E N ) 2 - - L2 EN + \Dl (s ' m)2 -

(2.3) A

where L = - i [S x a/aS] is the angular momentum operator. In the case of weak anisotropy V < I one may regard the random angles @,, at in each site as independent and uniformly distributed on the sphere [lo, 81. Equation (2.3) in which random walks are given by the term with the factor V will define the Markovjan process for cN.

We introduce now the angles y N , G N for which the energy E N ( 6 , q) is minimal, i.e.

(2.4)

and the quantities

Low-Temperature Properties of One-Dimensional Heisenberg Systems 511

Then in terms of Y N , a N , B N , @ N we easily obtain the recurrent relations (Langevin equations) presented in the Appendix. For a set of random values y , a, g, e the cor- responding distribution function must obey a Fokker-Planck equation. We are interested in the functions Wl(a, e) and Wl, (y , g) averaged over the rest variables /Y$m) = an+"eN/ay" aam with n > 2 or m > 2. Then one easily obtains with the help of Langevin equations (AZ), (A3) describing the Markovian process, stationary equations for the distribution functions [13, 8, 141

where the function f ( e ) is expressed as

(2.8) T + + 1.6 ~ 2 ; es = ( I v 2 ) 1 / 3

= 1 + ( e M 3 es and

<F-'> = j g-lW,\(B, F) dg dy; 7 = 7 - n/2 . First we consider (2.6), (2.7) a t hz = h, = 0. I n this case (2.6) represents the equa-

tion of stationary diffusion in a potential field. In zero magnetic field the distribution function W y ) is independent of a, and from (2.6) we have (x =

wY)(x) = A , exp [-u(z)I ,

where Al is a normalization constant determined by the condition J WL(e) de = 1. We consider now (2.7) which in terms of dimensionless variables [ = Ill3 IDI/84/3,

y = g/es, and z = T/es is

(2.10)

For [> 1 the range of W f ' different from zero is near f = 0. The approximate solution of (2.10) is given in this range in the form

512 R. S. GEKHT -

The distribution function Wf’ has a sharp niaximum a t the point y = 1/25, = 0. Random fluctuations of the anisotropy field and the temperature lead to smearing of the distribution function.

Knowing Wf’(/?, 7) one can determine now the average (b-’) in (2.9) for Wy’(e) in the limit ,o> es. Averaging with the aid of (2.11) and neglecting the temperature corrections z / ic which are smaller by an order of magnitude compared to the temperature term in the exponent of (2.11), we obtain

(2.12)

Substituting the average (8-l) into (2.9) and using (2.8) in the limiting cases x> 1, x < 1 we have

(2.13) w(0) ; x > l , I = A 1

x < 1 , where the numerical calculations of A 1 yield

A , = 0.82 (1 - 0.6[-’12 + 0 . 6 ~ ) ec1 . (2.14)

3. Free Energy and the Parameter (S:)

The free energy F per particle is determined as follows:

F = (&.v+I (~ ,v+I , ~ , v + l ) > - ( E K ( Y K , O N ) ) 9 (3.1)

where averaging is done both over the distribution functions Wy)(p) , W;p’(b, y ) and the random angles 8, a. Using (2.3) a t h = 0 and averaging over the random angles of anisotropy axes, we have in the limit < $ l

Averaging (3.2) with the help of the functions Wy)(p) , Wf’(8, 9 ) from (2.11) and (2.13) finally we get

(3.3) As V + 0 (3.3) transforms into the expression for F obtained in [4] (free energy

of spin waves). The energy of the ground state a t T = 0 is determined as

The two last terms here give the contribution to E due to nonlocal formation of the ground state. The term El - V(V/1)1/3 in order of magnitude corresponds to the one obtained in [lo, 81 for the case of the charge-density wave in the field of random impurities when I is replaced by eF, where E~ is the Fernii energy, or for a chain of xy spins with random anisotropy. The term E, - V 2 / i m is due to spin three- dimensionality in the Heisenberg model.


From (3.3) we get the average of the square of the longitudinal component of spin (Sz) = -8F/aD in the form

4. Susceptibilities and Correlation Functions

In order to calculate the longitudinal components of linear susceptibility we consider the stationary Fokker-Planck equations (2.6), (2.7) when h + 0. Assuming the spin energy in the external field hx, h , to be small (this will be seen later in comparison with V ( V/I)1/3 and V(IDl/I)l/*, respectively) we confine ourselves to a linear approximation in the perturbation theory while finding the functions W,(e, u) and Wll(B, f).

and transverse

Substituting the solution of (2.6) in the form

one may easily obtain v l = 15x/8 in the limit x> 1 and v l = I/Lfor x < 1. The free energy F , due to the presence of the field h, is given as

Fx = - h x (8,) x - h x (COS O) . (4.2) Averaging (4.2) over the function W l from (4.1) we obt'ain

where the symbol (...),, denotes averaging over the function W y ) . For the transverse susceptibility x1 = -p2 a2Px/8h: ( p is the magnetic moment of a particle) we easily obtain

With decreasing ID1 and T the transverse susceptibility increases. As T ---f 0 the existence of anisotropy fluctuations leads to a finite value [6].

we represent the solution of (2.7) in the form

To get the longitudinal susceptibility

WII = W? (1 - 7 YVll (Y) - h z l - e s 1 (4.5)

Substituting (4.5) into (2.7) and making allowance for the fact that the function of (2.11) has a sharp maximum a t the point y = VE, we get

~ l l ( y ) % -3.75 (1 + 0.35t[-1/2) y .

F , = M 7 ) = - 7 (?2ql(Y))o *

(4.6) The free energy F , = -h, (8,) due to the presence of the field h, is

h," I (4.7)

(after inserting (4.6) into (4.7) and sub- es

The expression for the susceptibility sequently averaging over the function W f ' ) then becomes

R. S. GERHT 514

Here it is used that (?2y), = (15[)-1. In the approximation [> 1 under study the longitudinal susceptibility does not depend on random anisotropy fluctuations and coincides with the expression for derived in 141.

Let us consider now the transverse correlation function g l in the absence of an external stationary magnetic field, i.e. g1 = g,, = guy. For 7 <l we have

9 1 (cos (TN ’ cos 00) =

= J ON, Q N , N ; GO, Po) G N cos 0 0 dciv do, deN deo 9 (4-9) where for a random Markovian process the unit normalized two-particle function 9, is expressed via a conditional probability P I and a one-particle distribution function W I in the form [13,8]

(4.10) The function is determined from the nonstationary Fokker-Planck equation. This

9JL = p I ( G N , @ N , 3 I 60, @o) Wl(ao , @o) .

equation, with allowance for the initial conditions, has a Green function form

__- ap, Y ; P I = g(z - xo) ~ ( a - Go) g(r) ; r = N (Ey, (4.1 1)

where the operator Y i is defined by (3.1) at h, = 0 ; in terms of the dimensionless variable z we get 2’; after substitution (8-1) = ~ ; ~ ( 2 5 ) - l / ~ as

ar

Transformation of 2’; to the Schrodinger operator d p L = exp ( U / 2 ) 2‘; exp (- U / 2 ) with U from (2.9) yields

Yl = 2y + Y y + = Y y ,

(4.13)

Then the expression for the Green function becomes

PI = exp __ + q) E yv(z, a) yy”(z0, 0,) exp (- ~ , r ) , (4.14)

where the eigenfunctions yy and eigenvalues A, are found from the Schrodinger equation

YLYY = - A v Y v f (4.15)

r:) Y

Inserting (4.10) into (4.9) and taking (4.14) into account we obtain

gl = 2 Cc-)C$+) exp (-Avr) , V

where Ck-) = J cos (T, exp [ --U(2,)/2] yY”(zo, a,) dz, do, , C$+) = J cos o exp [ U ( z ) / 2 ] yv(x, a) dz da .

(4.16)


The eigenfunctions yv(x, a) = p,l(z) eizO, 1 = 0, f l , &2, ... satisfy the solution of (5.7). Nonvanishing values CLT) exist only for I = f l . For large distances r the main contribution to the correlation function (4.16) gives the term with a minimum eigenvalue 2,” = 10, +I. Then po, satisfies

(4.17)

For [-lj2 = 0 and t = 0 the eigenvalue of (4.17) is determined as (0)

& , + 1 = 12.61 . At nonvanishing values

(4.18) The transverse correlation length El =&l+l(~s/V)z, where the lattice spacing a is equal to unity, is defined according to (4.18) in the form

and z we have from perturbation theory that Ao 3 - +1 = 12.61 - 0.03<-1/2 + 0.22.

61 = 0.079 ($r3 + 0.002 i& - 0.016 (4.19)

The first term in the right-hand side of (4.19) is essentially the same as the result in the zy-model [8] apart from a factor. From the expression for [I it follows that if the correlation length decreases with temperature increase, then a decrease of ID1 in its turn (under the effect of random fluctuations the spins S turns off the plane in the Heisenberg model) leads to an increase of 61.

Similar calculations for the longitudinal correlation length show its independence of the random anisotropy V in the limit [> 1 and is determined as

This is in agreement with the results of Nakamura and Sasada 141.

5. Conclusions

Low-temperature properties of a one-dimensional Heisenberg system of classical spins are studied with random (“easy axis” type) and regular (“easy plane” type) anisotropy. The properties of the disordered system are considered provided the condition I > V is fulfilled, under which random space axes of easy magnetization are independent in each site (zero correlation radius). The problem of finding thermodynamic quantities is solved in the temperature region T < ( IV2)1/3 by the transfer- integral techniques and the Fokker-Planck equations for disordered systems. The results obtained are valid for values of dimensionless parameter < = Ill3 IDI/V4/3> 1. The expression for the free energy is obtained. The nonlocal ground state formation in the Heisenberg model is due both to random and regular anisotropy. The susceptibilities and correlation lengths are calculated a t nonvanishing temperatures. The behaviour of the longitudinal susceptibility X I I and the longitudinal correlation length 511 is shown in the limit <> 1 to be due mainly to the regular anisotropy D and the transverse values XI, 61 are caused by the random anisotropy with increasing temperature. The transverse susceptibility ~1 and the correlation length 61 unlike the longitudinal components XI/, 611 decrease. The mean value of the square of spin component S,, as it should be, increases with increasing random anisotropy fluctuations and temperature. The contribution to (St> due to random field fluctuations is of the order VV2/1/I/IDl3 and that due to temperature fluctuations of the order T / i f l .

516 R. S. GEKHT

Acknowledgement

The author is grateful to V. A. Ignatchenko for useful discussions on the subject treated here.

Appendix

To obtain the recurrent relation for the quantities y N , aN, B N , eN of interest we expand the function ~ ~ ( 0 , y ) into a series near its minimum with angular coordinates 8 = yn., 9' = GN,

SY1) 0 - Y N ) (9, - ON) + ~ n ( 8 , cp) = E N ( Y N , G N ) + 7 IBN' (0 - Y N ) ~ + 28 ( 1 (2 0 )

Y

+ BY?"(cp - o37)21 + ... , pp0)+v, /?N ( O , % - = e N .

Differentiating (2.3) with respect to obtain with allowance for (A.l ) that

ID1 . BN

YN+I - y~ = ~ sin 2yN+l

0 and cp and assuming 8 = YN+1, cp = oAv+l we

v . + - [sm 2yN+l sin2 8 N + l cos2 ( ( T N + ~ - an;+l) - sin 2 y ~ + l cosz 8N+, + B N


In the derivation of (A.2) and (A.3) we have taken into account the fact that the fluctuations of spin along the z-axis are small, i.e. the spin lies preferably in the easy plane (xy) with the values Y N f l = n12. The cross terms j3$*m) with n, m not equal to zero are neglected in (2.4), (2.5). These terms along with the terms with j3F+ljo) and /?$,2m+1) to the first power have a lower order of magnitude in powers of the small parameter ( V / I ) 1 / 3 .

References [l] A. R. MCGURN and D. I. SCALAPINO, Phys. Rev. B 11, 2552 (1975). [2] J. M. LOVELUCK, S. W. LOVESEY, and S. AUBRY, J. Phys. C 8, 3841 (1975). [3] 11. V. FEIQELMAN, Fiz. tverd. Tela 18, 2454 (1976). [4] Ti. NAKAMURA and T. SASADA, J. Phys. C 11, 331 (1978). (51 R. S. GEKHT, Teor. mat. Fiz. 63,401 (1982). [6] Ti. BINDER, Proc. EPS Conf. Condensed Matter, Antwerp., Plenum Press, New York (p. 695).

P. J. FORD, Contemp. Phys. 23, 141 (1982). [7] A. APOSWILOV, H. HRISTOV, M. MIRHOV, and V. SKUMRIEV, phys. stat. sol. (a) 75, 401 (1983). [8] RI. V. FEIGELMAN, Zh. eksper. teor. Fiz. 79, 1095 (1980).

[9] P. A. LEE, T. M. RICE, and P. W. ANDERSON, Solid State Commun. 17, 1089 (1975). V. M. VINOKUR, M. B. MINEEV, and M. V. FEIQELMAN, Zh. eksper. teor. Fiz. 81, 2142 (1981).

[lo] H. FUHUYAMA and P. A. LEE, Phys. Rev. B 17, 535 (1977). [113 H. MIZUBAYASHI, T. ARAI, and S. OKUDA, phys. stat. sol. (a) 76, 165 (1983). [12] D. Y. SCALAPINO, M. SEARS, and It. A. FERRELL, Phys. Rev. B 6, 3409 (1972). [13] R. L. STRATONOVICH, Topics in the Theory of Random Noise, Vol. 1, Gordon & Breach,

[14] R. S. GEKRT, V. A. IGNATCHENKO, Yu. L. RAIKHER, and M. I. SCHLIOMIS, Zh. eksper. teor.

(Received October 3, 1983)

New York/London 1963.

Fiz. 50, 1300 (1976).

34 pliysira (b) 122/?

low-temperature properties of one-dimensional heisenberg systems with random anistropy

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