Time-dependent solitary solutions of the quantum-mechanical anisotropic Heisenberg chain

A. FREUDENHAMMER : Time-Dependent Solitary Solutions of the Heisenberg Chain 153

phys. stat. sol. (b) 134, 153 (1986)

Subject classification: 75.10

Theoretische Festkorperphysik, Fachbereich 10, Universitat-Gesamthochschule- Duisburgl)

Time-Dependent Solitary Solutions of the Quantum-Mechanical Anisotropic Heisenberg Chain

BY A. FREUDENHAMMER

A solitary time-dependent solution for a linear chain of quantum-mechanical Heisenberg spins is given. The interaction is of the anisotropic exchange type. An approximation to a one-spin-flip of the Hamiltonian on the basis of local spin quantization directions leads to the result that kink solutions only exist for a static kink.

Berechnet werden solitare zeitabhangige Losungen fur eine quantenmechanische lineare Kette von Heisenberg-Spins. Es wird eine anisotrope Austauschwechselwirkung angenommen. Eine Ein- spin-flip- Approximation des Hamilton-Operators auf der Basis einer lokalen Spinquantisierungs- richtung fuhrt zu dem Resultat, daJ3 nur statische Kink-Losungen existieren.

1. Introduction Solutions of an infinite linear chain of quantum-mechanical spins have been presented for different spin interactions. Most solutions use a fixed quantization direction so that the deviations of the spins from this direction have to be small in order to use the approximate Holstein-Primakoff transformation (see, for example, [ 1 to 31).

The above restriction of small spin deviations is not necessary if a local quantization direction is introduced as for instance done by Fischer and Heber [4] and Bishop et al. [5]. Starting with a solution of a static kink structure the development in the course of time with finite velocity of the kink structure is calculated. The time-dependent state vector of the spin system is restricted to the time dependence of the local quantization direction and a common time-dependent phase factor. As will be discussed in Section 3, the movement of the solitary wave is caused by local one-spin-flip excita- tions. The dependence of the quantization direction of the static kink on the contin- uous chain parameter x is deformed because of the finite velocity.

The quantization direction is taken to be always orthogonal to the chain axes x, and the boundary conditions for the angle q of the n-kink are given by q ( x + +a) = = 0 and q ( x 4 -a ) = n. As will be shown in the next sections only one of these boundary conditions can be fulfilled at finite velocities. A general proof of the existence of quantum-mechanical solitary waves has been given in [6] for s = +. Because of the lack of boundary conditions the type of the solitary solution is not fixed.

A critical velocity appears in the equation of motion so that all numerical calcula- tions have to be done for both regions (w < w,, and w > wc).

The paper is organized as follows: The approximation of the Hamiltonian and the equation of motion is given in Section 2 . The singular points and the structure of these points are discussed in Section 3. With the results of Section 3 the global first integral of the equations of motion is calculated in Section 4. The solutions with given boundary conditions are discussed in Section 5 and a summary is given in Section 6.

Postfach 101 629, D-4100 Duisburg, FRG.

154 A. FREUDENXAMMER

2. The Equation of Motion The Hamiltonian of an exchange anisotropic spin chain is

H = - J C (fj’kfi;+l + fj%Si,”) - J” C SiSi+l . z z

i i

In order to investigate a kink-like soliton the quantization axis rotates [4,5] according to

(2.2) Inserting (2.2) in (2.1) yields a long expression, which will not be given here. Instead, the action of the resultant Hamiltonian on the state

8, = 8,. , 8, = cos yS,? - sin yS,. , 8, = sin y8,. + cos yS,? .

10) = n 0 Iz;(y,), S ) 9

sj,,[z;(y,), 8 ) = 0 ,

(2-3) j

where 8l+qy,), 8 ) = hSlZ,(y,), 8 )

is considered. Because of (2.3) terms proportional to the transformed Hamiltonian acting on the state (2.3) can be written as

have no contribution so that

HIO) = (Ha + + HA 10) (2.4)

(2.5)

with

H , = - c ~ ~ [ S Z J sin y , sin p?i+l + ~ S Z cosy, cos p j + l ~ , j

1 i JS + sin y , (cos yj-l + cos yj+l) 85, ,

sin y , sin y j + 1 ~ ~ ~ 3 ‘ . H , = - C - (1 - cos y , cos yj+1) X i r S j i ; l - -

(2.7) In order to obtain a closed form of equations of motion we neglect the operator H, in (2.4). The operator Hl (2.6) flips the spin in the local frame whereas Ha maps the state (2.3) onto itself. The spin operators Si, are omitted because the state (2.3) is an eigen- state of S$.

1 J 3 . j 4 4

Therefore, the time-dependent ansatz

is made for the solution of the Schrodinger equation d dt -It> = (Ha + Hl) It) , (2-9)

Eo = H a . (2.10) where

The time derivative in (2.9) yields two terms. The first term leads to (2.10) and the second to the equation

(2.11) j

Time-Dependent Solitary Solutions of the Anisotropic Heisenberg Chain 155

Equation (2.11) can be reduced further to

(2.12)

with - B k = hJS sin q k (cos v k - 1 + cos q ~ k + ~ ) - hJX cos q k (sin q k - 1 + sin q k + l ) .

(2.13) According to [4,5] the angle q k describes a rotation around the x-axis, therefore we set

(2.14) with

(2.15)

The ansatz (2.15) leads to a differential equation for qk(t) which is a differential equation with complex coefficients.

IZ;(qk), 8 ) == Rg(qk) I z k , 8 )

(k) R^(qk) ei9ksz /fi .

Hence, arbitrary real phase factors q(m) are introduced by

(2.16)

(2.17)

With the changed rotation operator R-, (2.16) the derivative with respect to qk in (2.12) leads to

(2.19)

Hence, the Schrodinger equation (2.12) becomes

If we fix the phase y ( S ) = 0, then y(X - 1) = *n/2 is obtained from the condition that the differential equation should be real,

(2.21)

Introducing the continuum limit (ak - x) and searching for a solitary solution q(x, t) = p(z - vt) = p(l) we obtain the differential equation

(2.22) where

V J J

and A = - - 1 . OL=*------ a h JS

(2.23)

156 A. FREUDENHAMMER

Two possibilities of transforming (2.22) are used (5 + x), (i) z(x) = cosy(s) , (2.24) (ii) y(x) = siny(z) . (2.25)

y(x) is inversible in the range -n/2 < y < n / 2 or a12 < y < 3 ~ 1 2 , whereas a unique inversion is possible for z(x) in the range 0 < y <

Introducing the transformation (2.24) and the inversion of z(x) to x = x ( z ) we define

(2.26)

Equation (2.22) becomes

or n < q < 2n.

p = z’(x(2)) 3 p ( z ) .

zpz - a(l - 9) p + 2 4 4 1 - p’(z ) = A ( 9 - 1) [(A + l ) / O - z23p ’ (2.27)

where the prime means differentiation with respect to the argument of the function. On the other hand, the transformation (2.25) with the inversion x = ~ ( y ) yields

(2.28)

(2.29)

3. Singularities of the Equation of Motion The behaviour of our solutions can be better overlooked by examining the neighbourhood of the singularities.

The differential equations (2.27) and (2.28) show three singularities a t y, z = &I, 0 and q , p = 0.

In order to obtain a unique solut,ion, both differential equations have to be considered simultaneously. Only from the values of z and y the angle y can be determined uniquely.

Because of theLipschitzconditionfor ordinary differential equations the denominator and the numerator have to be zero a t the singular point in order to produce non- unique solutions.

Therefore, we have to investigate the equations of motion for small deviations from the singular points, i.e. the differential equation is approximated by the leading terms of the denominator and the numerator.

First (2.27) for p = p ( z ) is investigated. The approximation of (2.27) for small deviations from the singular point p = z = 0 is given by

Searching for solutions of (3.1) of the form p = az, which we call “linear solutions”, we obtain

(3-2) (a f - SA(1 +d)) z

2(A + 1) P* =

The solution (3.2) exists only if ~.

a > a,; a, = JFATl- + A )

for A > 0.

(3-3)


z -

Fig. 1. The global solution for the first integral p = p ( z ) in the range - I < z < 1 and a > a,

The structure of the singular point can be seen in Fig. 1 for a > a,. The whole picture is calculated with the exact differential equation (2.27). The structure changes considerably if the parameter a is taken to be smaller than the critical value a, (Fig. 2). The parameter a is proportional to the velocity so that a critical velocity

occurs. Inspection of (2.27) yields that the figure for - 1011 can be obtained from that with la1 by reflection a t the abscissa.

Froin symmetry consideration we conclude that the singular points f 1 are equivalent.

Fig. 2. The global solution for the first integral p = p ( z ) in the range -1 < z < 1 and a < a,

-05 0 05 10 2-


For small deviation around z = 1 the differential equation (2.27) is approximated by the leading terms of the numerator and denominator as

p 2 + 2a(z - 1) p + 8A(z -- 1)2 pr = ~ _ _ _ 2 k - 1 ) P

The “linear solutions’’ of (3.4) are ~~

p t = (a & /a2 + 8 A ) ( z - 1).

This equation shows no critical value 01 as long as A > 0 (see Fig. 1 and 2).

singular point q, y = 0 the approximate equation of (2.28) is obtained as In the next sections the differential equation (2.28) will be investigated. A t the

with the “linear solutions”

= + ( a * 1 / o i 2 + ~ A ) Y . (3.7) For A > 0 no critical a exists. This structure can be detected in Fig. 5 and 8 a t q = = y = 0 (which are discussed further in Section 4). The situation becomes again different if the singular point a t q = 0, y = 1 is considered. The approximation of (2.28) yields

(3.8) ( A + 1) q2 - 8A(y - q’ = + 24Y - 1) q

2(A + 1) (Y - 1) 4 and the “linear solutions” are

Again for a > a, two “linear solutions” appear (Fig. 3a) and for a < ac no such solution exists (Fig. 3 b).

Y- Fig. 3. The structure of the singular point a t q = O and y = 1 for a) a > a, and b) a < ac


We have considered throughout this section the case A > 0. For A < 0 the role of p and p is changed.

4. Global Solution of p and q 2,

Returning to the exact differential equations (2.27) and (2.28) we try to get the solution of (2.27) for LX > 01,. The result is shown in Fig. 1. The structure of the singular points has been discussed in the last section. For& < IX, the spiral structure is obtained (Fig. 2). Considering the global solution only of (2.27) leads to a unique solution in the range -1 < z < 1. The continuation of the solution through the singular points fl is undefined. We have to consider the global solutions of the second differential equation (2.28) in order to get the right continuation.

As mentioned before the solutions for p ( z ) are unique in the range -1 < z < 1 and q ( y ) in the range -1 < y < 1. Therefore, we have to fix the rotational sense of the angle ~ ( 2 ) . This is done by defining

y = - sign ( z ) /I - 22, z = sign (y) 1/1 - y2.

(4-1)

(4.2)

-;3 -05 0 05 m Y -

Fig. 4 Fig. 5 Fig. 4. The first integral p = p ( z ) for a = 0.5 and A = 0.5. The flow direction for a fixed initial value is given by increasing numbers. Different initial values are indicated by different types of lines. __ curve 1, --- curve 2, -1.- curve 3

Fig. 5. The same solutions as given in Fig. 4, but in tha representation q = q ( y ) and in addition curve4 (.-.-a )

~~

2, The integration of this and the next section has been performed numerically.


I I-

1 0 -0.5 0 a5 z -

Fig. 6

t ’4

I

-70 -05 0 0.5 20 Y-

Fig. 7 Fig. 6. The first integral p = p ( z ) for a = 2.5 and d = 0.5. The flow direction for a fixed initial value is given by increasing numbers. Different initial values are indicated by different types of lines. -..-.. curve 1, .-.-. curve 2, - -- curve 3, ~ curve4. Curve 1 andcurve 3 are separatrixes. Insert: Enlargement of the region p = -1, 1 and z = 0, 1

Fig. 7. The same solution as given in Fig. 6, but in the representation q = q(y)

From z = cos p(x), y = sin p(x) we obtain

All examples given in the next sections will have a A = 0.5 and therefore i t is 01, = = 2.449.

In Fig. 4 and 5 01 is set ‘to 01 = 0.5 < 01,, whereas in Fig. 6 to 8 01 = 2.5 > 01, is chosen. The integration is performed in the clockwise sense changing from Fig. 4 to 5 and vice versa.

As an example for illustration, curves 1 of Fig. 4 and 5 are taken: From the initial value 2 (of curve 1 in Fig. 4) both singular points a t z = *1 ( p = 0) are reached through integration. An arbitrary point 3 in the clockwise sense is chosen (curve 1 in Fig. 4). With the values of point 3 we calculate through (4.1) and (4.3) the equivalent point 3 in the q ( y ) representation (Fig. 5). In this representation we integrate to both singular points y = 5 1 and determine arbitrarily the point 4. With (4.2) and (4.3) the equivalent point 4 for the p ( z ) representation is calculated (Fig. 4), and so on. Thus, the whole solution for a given initial value can be evaluated.

Whereas in Fig. 4 all curves seem to have a flow parallel to each other (terminating in a spiral fixed point), Fig. 5 shows that this is not the case. The curve 1 plays the role of a separatrix. Curves that are on the right-hand side (in the direction from

Time-Dependent SoIitary Solutions of the Anisotropic Heisenberg Chain 161

I Fig. 8. The same as Fig. 7 but on a smaller scale and curve 5 (-----a). Curves 3 and 5 are separatrixes

large to smallnumbers) of curve 1 terminate in a spiral a t y = +1 (Fig. 5) and that on the left in a spiral a t y = -1 (q = 0). The curve 4 of Fig. 5 represents the four initial branches of the separatrix.

Therefore, starting with a “linear solution” at y = 0, as is thecasefor curve 1, we never terminate a t the second “linear solution” a t this point. Instead the solution separates more and more from the singular point y = q = 0. Hence, the boundary conditions = 0 for x + +co and = z for x - -00 cannot be fulfilled as will be shown in the next section. In the next ex-

-4 . - I -10 4 5 0 05 which is larger than the critical velocity

(01 >ac) will be considered. We have set

The flow of the solutions is again clockwise in Fig. 6 to 8. We have taken four dif-

In Fig. 6 to 8 the flow away from the “linear solution’) is shown and in Fig. 8 the

The separatrixes in Fig. 6 are curves 1 and 3, whereas in Fig. 8 the separatrixes are

By inspection of Fig. 6 to 8 we conclude again that for 01 > ac the curves with a linear solution” as a starting point never terminate in a “linear solution’). Hence,

again p(x + -00) = 0 but not p(x - +a) = n can be fulfilled as is shown in the next sect,ion.

-2 -

-3 -

amples (Fig. 6 to 8) the case of a velocity

y - 01 = 2.5 > OC, = 2.449.

ferent curves that are calculated in the same manner as in the case 01 < 01,.

neighbourhood of the “linear solution’’ of Fig. 7 is shown on a small scale.

curves 3 and 5.

“

5. Solutions and Boundary Conditions

In order to obtain the final solution (i.e. the second integral z = z(x)) we have to integrate p ( z ) = dz/dx,

z

20

The integration over p ( z ) in the vicinity of a singular point, i.e. a “linear solution”, results in

z

20

11 physica (b) 134/1


x -

Fig. 9. The solution z = cos q(z) = z(z) for the boundary condition z (z ---f -co) = 0 and a) a = = 2.5, A = 0.5; b) a = 0.5, d = 0.5

Therefore, z = zo exp [a(% - z,,)]

z = 1 + zo exp [a(. - zo)] .

(5.3)

(5.4)

or, if this type of initial condition is taken a t the singular point z = +1,

Hence, the “linear solution” yields the right boundary condition z ( z + - 00) = 1 for a > 0 or z(z + m) = 1 for a < 0. But since only one “linear solution” can be reached, the second condition z(x + 00) = 1 cannot be fulfilled in a global solution z = z(x) . Besides the “linear solution” a t z = *l also a power law of the form

p(2 ) - (z - l)l/r; n > 1 (5.5)

occurs in Fig. 4 to 8, which gives no proper boundary conditions as can be verified. Performing the second integral with the values of curve 1 of Fig. 6 we obtain the

solution shown in Fig. 9 a for 01 = 2.5 and A = 0.5 ; and the values of curve 2 of Fig. 4 vield that of Fig. 9 b for 01 < 01,.

The only solution which connects two different “linear solutions’’ a t the singular points z = 1 is obtained for zero velocity shown in Fig. 10a. The second integration gives the right result depicted in Fig. lob.

-1.0 -05 0 05 1.0 0 2 4 6 8 10 z- x -

Fig. 10. The solution for the static kink, a = 0 and d = 0.5. a) First integral p = p ( z ) ; b) second integral z = z(z) with the boundary conditions z(z - -co) = -1 and z(z - +co) = 1


Hence, already small velocities (a =+ 0) lead to a deformed first integral which

The helical structure obtained in Fig. 2 for a =+ 0 (a < aC) degenerates to an ellipse therefore prevents the second boundary condition to be fulfilled.

in the case of a = 0 (this structure has not been shown in Fig. 10a a t z = 0).

6. Summary and Discussion Considering a linear chain with an exchange anisotropy of Heisenberg spins the time- dependent Schrodinger equation in the one-spin-flip approximation has been solved.

The time-dependent spin state has the form of a product ansatz (2.8) in the basis of a locally rotating quantization direction. Therefore, no spin waves as solutions of the time-dependent Schrodinger equation occur.

The main purpose of this work is to obtain soliton-like solutions with given boundary conditions at infinity. For that reason the singular points of the differential equation were analysed. We obtained in the first integral two “linear solutions”, which provided us after the second integration of the equation of motion with one boundary condition a t +cc or - co. But analysing the flow of solutions no proper second boundary condition could be reached because the flow of solutions always ran away from the singular points where a “linear solution” could be reached.

Therefore, no solution exists with the boundary condition q ( x ---+ +a) = 0 and p(z + -cc ) = n for finite velocity of the n-kink. The same is true for nn-kinks.

The only n-kink solution which exists is the static n-kink solution (i.e. v = 0).

Acknowledgement

I am much indebted to Mr. H. Beyer who performed t.he drawing of the figures on our plotter.

References [l] D. I. PUSHKAROV and KH. I. PUSHKAROV, phys. stat. sol. (b) 81, 703 (1977). [2] D. I. PUSHKAROV and KH. I. PUSHKAROV, Phys. Letters A 61, 339 (1977). [3] R. CALVANESE and G. VITIELLO, Phys. Letters A 100, 161 (1984).

E. MAGYARI and H. THOMAS, 5. Phys. C 15, L333 (1982). K. A. LONG and A. R. BISHOP, J. Phys. A 12, 1325 (1979).

[4] K. 0. FISCHER and G. HEBER, phys. stat. sol. (b) 129, 649 (1985). [5] A. R. BISHOP, E. DOMANY, and J. A. KRUMHANSL, Ferroelectrics 16, 183 (1977). [6] L. A. TAKHTADZHAN and L. D. FADDEEV, Russian Math. Surveys 34, 11 (1979).

(Received October 14, 1985)

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Time-dependent solitary solutions of the quantum-mechanical anisotropic Heisenberg chain