Comment on relativistic ideal soliton-gas approximations

SoLid State Communications, Vol. 38, pp. 521-525. Pergamon Press Ltd. 1981. Printed m Great Britain.

0038-1098/81 / 180521-05 S02.00/0

COMMENT ON RELATIVISTIC IDEAL SOLITON-GAS APPROXIMATIONS S.E. Trullinger* Department of Physics, University of Southern California, Los Angeles, CA 90007, U.S.A. and A.R. Bishop* Statistical Physics and Materials Theory Group, Los Alamos Scientific Laboratory, Los Alamos, NM 87545, U.S.A.

(Received 15 August 1980;revised 18 November 1980 b.v A.A. Maradudin) We comment on the validity of using relativistic, ideal, Boltzmann-kinkgas phenomenology involving "'bare" kinks for interpretation of experiments and molecular dynamics simulations. An improved velocity distribution is obtained for sine-Gordon and ~ f o u r kink gases which incorporates the kink renormalization found to be essential in obtaining the correct total kink density. We argue, however, that even with this improvement, current ideal gas phenomenologies can only be trusted at tow temperatures compared to the kink rest energy, where the majority of "kinks have low velocity in any case. RECENTLY there has been increasing emphasis [ 1-41 placed on the use of relativistic soliton- (or kink-) gas phenomenologies for interpreting results of experiments on soliton-bearing quasi-one-dimensional systems, such as the magnetic chain materials [5, 61 CsNiF3 and TMMC, as well as molecular dynamics studies [4] of model systems such as the sine-Gordon (SG) and 0-four chains [7]. Although such phenomenologies provide a very simple and physically appealing framework for calculating quantities such as dynamic response functions [e.g. S(q, co)] there are several pitfalls associated with the naive use of what we shall refer to as "bare" phenomenologles which are formally identical to the one-dimensional relativistic Boltzmann gas theory for a collection of independent particles (solitons or kinks) having tile simple energy spectrum, E,~ ) = 7EJg ),

7 = (1--v2/c~) -1,2,

(1)

where Co is the characteristic limiting velocity and EtK°~ is the rest energy [7] of the relevant kink excitation. First of all, there are corrections to equation (I) arising from the particular nature of the physical system in question. The fundamental discreteness of lattice systems leads to a modification of the soliton energy and its propagation behavior [8]. For the magnetic systems * Research supported by the National Science Foundation under Grant No. DMR-7908920. -t Work performed under the auspices of the U.S. Department of Energy.

~(CsNiF3 and "I3,1MC) governed approximately [9-11] by the sine-Gordon (SG) equation, the applied magnetic field can be adjusted to reduce the discreteness corrections [I, t0] t'or slowly-moving solitons to a negligible level, but they are dramatically enhanced for fast-moving solitons [10] due to "Lorentz contraction'" of the soliton width. Another example of an "intrinsic" type of correction in the magnetic chains is that due to large tipping of the spins out of the easy plane [1, 10]. Here again the corrections to equation (1) may be small for tow-velocity solitons, but become enhanced [10] at high velocity. In addition to corrections intrinsic to the physical system in question, there are important corrections to "bare" phenomenologies used to date which must be considered even when the ideal nonlinear equation of motion (e.g. SG or 0-four) is assumed to be exact. These corrections are the subject of the remainder of this comment. The modifications with which we are concerned can be generally thought of as due to competition between the excitations (nonlinear and linear)for the available degrees o]'freedom. Even for completely integrable systems such as SG, where direct m o d e - m o d e interactions result only in phase shifts, there nevertheless exists an indirect, non-trivial interaction arising from this competition (see Section IV of [7]). The practical consequence of this effect is that, even at low temperatures where the kink density is low [7], the kinks influence the density of other excitations and must therefore be assigned a renormalized energy [7] E~ 521

Vdl. 38,No. 6

RELATIVISTIC IDEAL SOLITON-GAS APPROXIMATIONS

522

3.0

k._2

ksT (o) "2T6~= 0.15 Ew

3.0 i (b) E~OJ: 0.45 l=(v),P*(v) P(o)

P(v),P'~(v)

P(o)

2.0

2.0

...-~-..--~,

k' . . ~ : . ~

""".,

1.(3

o.oo ,

1

I

I

l

0.2_

0.4

0.6

0.8

v/c o

1.0

~/co

ksT (c) --~'IKO~= 0.75 3.0

r,

P(v),P*(v)

/.....I

/:: ~m

P (o)

2 .,"

2.C

I.C

O.

.0

1

I

I

I

0.2

0.4

0.6

0.8

V/c o

1.0

Fig. 1. Normalized velocity distributions for an ideal, relativistic kink gas [P(v)] and for a non-interacting kink gas but including interactions with linear extended modes [P*(v)] - as explained in the text. Solid line:P(v)/P(O); dashed line: P*(v)/P(O) for SG; dotted line: P*(v)/P(O) for ~-four. k B T [ E ~ ~ =or -l - (a) 0.15, (b) 0.45, (c) 0.75. to account for these influences. Although the calculation of E~ for arbitrary temperature is quite difficult, it has been possible [ 12] to obtain E~c as a function of velocity in the low temperature limit where only the influence of kinks on the density of states for linear (phonon) excitations need to be considered. By assigning [7] to the kink the change it induces in the (classical) phonon free energy, one finds [12] that equation (1) should be replaced in the SG case by E~-(7) = 7 E ~ ) - - 3 - ' In [3h~o(1 + 3')]

(SG),

(2a)

where coo is the characteristic frequency in the notation

of [7], and 3 - ( k B T ) -l. A similar expression holds [13] in the case of ~ f o u r kinks:

(q-four).

(2b)

Both of these formulas [(2a) and (2b)l are valid only in the temperature range [7] tacoo '~ kBr. The use of the renormalized kink energy E~ in ideal-gas phenomenologies has been shown [7, 12] to

Vol. 38, No. 6

RELATIVISTIC IDEAL SOLITON-GAS APPROXIMATIONS I

l

523

l

1.0

0.8 ~ O.6 :E 0.4 ().2 0.0

I

I

I

1.0

2.0

3.0

I

i

4.0

E~?k aT

Fig. 2. Velocities (v m, v~) at which the distributions P(v) and P*(v) have maxima, as functions of t:'~)/ka T (= a). Nonzero values of vm(v*) imply a double-peaked velocity distribution.

8.0

8.C

og(nKd)-

log(nKd)-' 6.0

6.C

4..0

4.0

2.0

2.0

o.%'o

I

I

0.1

0.2

I

I

I

0.3

0.4

0.5

0.6

0-%:0

0.I

k TIc{O~ 8 /L- K

0.2

A 0.3 0.4 k8T/E °~

0.5

0.6

Fig. 3. Kink density predictions vs a -I = kBT/E~ ) for (a) SG and (b) @four. A: general theory including kink interactions with linear extended modes; B: asymptotic low-Tlimit of A; C: "exact" result as deduced from numerical transfer operator results; D: ideal kink gas prediction, neglecting all mode interactions; See text for details. give correct expressions for the low-temperature density of kinks, by comparison with exact results of the transfer-operator method for computing the classical free energy and equal-time correlation functions. To obtain the correct low-temperature kink density one may neglect the velocity dependence of the correction terms in equations (2a) and (2b), since very few fastmoving kinks are expected at low temperature. This is the approach taken in [7], where it is found that the low-temperature kink (plus antikink) densities for the SG and C-four cases are: =

and

(31'/22aU2e-~

(C-four),

(3b)

where d -Co/~o and t~ -/~E~ ). We emphasize that these

formulas were obtained in a non-relativistic approximation where all kinks were assumed to be slowly moving. Recently, however, use [ 1-4] has been made of a relativistic Boltzmann gas phenomenology at higher temperatures where significant numbers of fast-moving kinks are expected, and emphasis has been placed [2-4] on using the velocity distribution P(v) for an ideal gas of relativistic particles with energies given by equation (1): .y3 e - 7 ~

P(v) - 2coKt(a),

(4)

where Kt is the modified Bessel function. We must take exception to the use of equation (4) for the kink velocity distribution for several reasons. First, since the calculation ofP(v)using equation (1) neglects the renormalization of the kink energy [e.g. equation (2)], it is clearly inconsistent with the corresponding calculation of(nK), for which the inclusion of

524

RELATIVISTIC IDEAL SOLITON-GAS APPROXIMATIONS

the kink influence on the density of states of the extended modes is essential, Second, P(v) does not include the effects of kink-kink interactions which may be expected to become important for the larger kink densities at temperatures high enough to give significant numbers of fast-moving kinks in the first place. Finally, equation (4) completely ignores the intrinsic corrections discussed above (such as discreteness effects) which become important for the fast-moving kinks which equation (4) is purported to govern. Although we cannot offer remedies for these last two ailments until much more understanding is gained regarding these effects, we can offer at least a partial improvement on equation (4) which accounts for the "kink energy renormalization arising from its influence on linear extended modes. We emphasize, however, that the improved velocity distribution is still only valid for low temperatures because it is assumed that oscillations in the presence of kinks are harmonic. The inclusion of the velocity dependence of the energy renormalizations in equation (2) leads to modified formulas [12, 13] for the average kink densities:

2/

(nK) = ~ and

o~[Ko(a)+ Kl(a)] + Kl(a)

1

<,z~> --- ~-~ a[Ko(~) + K~(a)] + ~KL(,')

(SG),

(5a)

(C-four). (Sb)

"l-he corresponding normalized velocity distributions are P*(v) =

~'73( 1 + 7) e-'ra

2Co{a[Ko(a) + Kl(a)l + Kl(a)}

(SG),

(6a)

and a7~(1 + 7X~ + 7) e - ' ~ e*(v) = 3Co{a[Ko(a) + K~(a)] + ]K,(a)}

(C-four). (6b)

In Fig. 1 we give plots of both the "bare" [P(v)] and "renormalized" [P*(v)] velocity distributions for various temperatures. Note that the same vertical scale normalization [P(0)] is used to facilitate comparison of the shapes of these distributions. The maxima in these distributions occur at non-zero velocities only for temperatures above certain minimum values. The locations of the maxima are given by

Iv~l/co =

{0{, 1-~az}

vz,

,',>~3

a~3,

for the bare distribution, and by

(7a)

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IvLI/co = O, {32 --4~ -- 2a 2 + 2(4--~){a: + 4a + 16] ~'"-}~': 4 - - ~ +[e2 + 4 ~ + 16] ~'/ (7b) ct~>7/2 (SG),

a~7/2 for the SG renormalized distribution. For the O-four renormalized distribution, we have

Iv,~llco--

0, {1_7m2},;2 '

c, t> ~ ~<~

(7c) (0-four),

where 3'm is the positive real root of the oubic equation. -

vk+

In Fig. 2 we plot [vm[/co vs a for the pure relativistic Boltzmann gas [equation (7a)] and Iv*l/co vs ct for the renormalized SG [equation (7b)] and O-four [equation (7c)] gases. By including the kink-energy renormalization, we see that the double-peaked structure (including negative v values) develops at a lower temperature than for the bare distribution. However, as we mentioned above, the quantitative results for P*(v) should not be taken too seriously, except at low temperature (roughly kB T <~ 0.2E~)), since we have neglected the anharmonic nature of fluctuations about the kinks as well as kinkkink interaction effects, both of which are expected to become important as the temperature is increased. We can gain some feeling for the importance of these effects by comparing the above formulas for kink densities with appropriate inverse correlation lengths [7] available from numerical implementation [14] of the transfer-operator method [7] at higher temperatures. In Fig. 3 we plot the inverse kink density as a function of temperature according to equations (3) and (5), and as obtained from numerical transfer-operator results. Also included for comparison is the prediction of the "bare" ideal gas theory in the form proposed by Krumhansl and Schrieffer [ 15] (no kink-energy renormalization). As can be seen from these curves, the formulas (3) and (5) give much better agreement with the "exact" result than the bare theory [ 15], although expressions (5) obtained using the full velocity dependence of the renormalization give worse agreement than expressions (3) which were obtained in the lowvelocity approximation [7]. This clearly indicates the need for a more precise theory at high temperatures

goal. 38, No. 6

RELATIVISTIC IDEAL SOLITON-GAS APOROXIMATIONS

which includes all mode interactions. Unfortunately, the "'true" velocity distribution cannot be extracted from "exact" transfer-operator results for static correlation lengths. Thus, for the present, interpretations of experiments (and molecular dynamics simulations) can only be made at low temperatures (k 8 T-< 0.2E~°)) where the theory is reliable. In this note we have introduced one more correction to ideal gas kink theories. As we emphasized earlier, there are several such corrections which are essential to a quantitatively successful theory. It may there(ore be helpful to briefly summarize the status of kink theories. First we must distinguish between application to SG or C-four molecular dynamics simulations versus proposed applications of these models to specific materials, e.g. CsNiF3 or TMMC. The latter are mitigated by realistic material perturbations (e.g. quantum spins or nonlinear out-of-plane spin motions). Before we can assess these we must understand the intrinsic corrections to ideal gas kink theory (on a discrete lattice). For this purpose there are now careful molecular dynamics simulations of the SG [4, 16] and ~four [4, 17] chains, studying central peak structures. For SG, kink theory applied to appropriate correlations (e.g. ~-¢~ or sin c-sin ~) is qualitatively very successful in that a split central peak is observed whose location scales linearly with q (for q < d-~). However, the Tdependence, intensity and location of the split peak are very badly predicted by ideal kink theory. In addition weight is observed for co > c o q . These failings of ideal kink theory, can be substantially remedied [4, 10, 16] by a combination of rescalings of the effective kink energy and of the Lorentz-contracted kink width;both corrections stron~y enhance splitting at lower Tand co. A priori calculations of these corrections (i.e. m o d e mode interactions and discrete lattice effects) are not available beyond low-order perturbation theories (e.g. here and [ 10]) which are quantitatively inadequate; using the kink energy and width as fitting parameters should be successful, however [16]. q~-¢ correlations in the C-four model are rather different. The appropri-

525

ate ideal kink theory is of a slightly modified Ising variety where the mean kink-antikink collision time plays a crucial role [I 8]. In practice [4, 17], long-lived breather modes have a severe effect on ideal kink dynamics (unlike in the near-integrable SG case), dominate kink "lifetimes", and remove any significant central peak splitting predicted in ideal gas theory. These effects on kink dynamics have not been accounted for to date and are so qualitatively important that gentler mode-mode interactions or discrete lattice modifications cannot yet be identified separately. REFERENCES 1. 2.

3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18.

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