Volume 106A, number 1,2 PHYSICS LETI'ERS 19 November 1984

COMMENT ON FRACTIONALLY CHARGED SOLITONS AT FINITE TEMPERATURE

AND CHEMICAL POTENTIAL "~"

Wemer KEIL Department o f Physics, University of British Columbia, Vancouver, BC, Ca,~do V6T 2A6

Received 18 June 1984

The influence of t-mite temperature and chemical potential on fermion fractionization is discussed. An explicit expres- sion for the charge of two locally different soliton prof'fles in a one-dimensional soliton-fermion system realized by linearly conjugated polymers is found and evaluated numerically. A rapid change in the soliton charge is observed in an experimental- ly accessible domain of the chemical potentials.

During the last year fractionally charged quantum states were discovered as an interesting feature of topological- ly non-trivial physical systems. Particularly interesting is the case of a one-dimensional fermion-soliton system, since it can be realized by linearly conjugated polymers and thus allows a direct experimental study of fermion frac- tionization. A simple, well understood example of a polymer with fractional fermion number (=charge) is polya- cetylene: its domain wall solitons exhibit fractional charge ½ per spin degree of freedom.

The previous studies (see refs. [1 -7] ) were mostly concerned with fermion fractionization at zero temperature. Real physical systems like polymers in a lab, however, always have a finite density and temperature background. Since the actual measurements also depend on these parameters [5,8] it is important to include their effects into a description of the polymer and to compare these with the previous results.

A general discussion of this problem is given by Niemi and Semenoff [9] for finite temperature and by Niemi [10] for finite temperature and chemical potential. Niemi and Semenoff [9] showed that at finite temperature (as at zero temperature) the fermion number associated with a soliton is determined only by its asymptotic, i.e. topological characteristics. However, Niemi [10] has shown that, with finite chemical potential, a nontopological contribution appears in the expression for the fermion number.

In the present paper we examine the importance of these nontopological contributions by evaluating the fer- mien number for two different soliton profiles. We shall find that the qualitative features of fermion fractioniza- tion are rather insensitive to local variations of the soliton profile. Furthermore, we observe a rapid change in the soliton charge (per spin degree of freedom) from negative to positive values as the chemical potential increases. This effect occurs in an experimentally accessible range of temperature and chemical potential.

The dynamics of the electron in the polymer is effectively described by a 1-dim Dirac hamiltonian H(O) which "represents the continuum limit of a lattice hamiltonian for a diatomic polymer containing the interaction with background soliton potential O. Its form is given by:

H(~b) = 0 2 i-10x + o1~ + o3e , (1)

with e E I:l +, o / , / = 1, 2, 3, Pauli matrices and ~: I:l ~ R the soliton-background field with asymptotic value +q~0 = limx-. ± . 0 (x ) , but without any other specification (besides the necessary integrability and differentiability proper- ties required for the following calculations).

This work is supported in part by the National Sciences and Engineering Research Council of Canada and a University of British Columbia Summer Fellowship.

0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. 89 (North-Holland Physics Publishing Division)

Volume 106A, number 1,2 PHYSICS LETTERS 19 November 1984

The following analysis is then carried out under the assumption that only electrons have a thermal energy distri- bution but the soliton field remains temperature independent. This is physically justified since the typical tempera- ture values for polyacetylene are very small compared to the magnitude of the asymptotic soliton amplitude ¢0 with an energy in the range of 104 K (see ref. [5]). By the same argument the chemical potential and e, the charge- conjugation-symmetry-breaking parameter in (1) can be assumed to be temperature independent.

Following the method of Jackiw and Semenoff [6], we derive now a general expression for the charge (= fermion number) of a soliton. The electron states in a system with hamiltonian (1) are given by the 2-dim Dirac spinors

b e ( x , t) : e -lEt erE(X), (2a)

with fie solution of the Dirac equation

H(¢)~O E = Ed/E . (2b)

The charge density of the soliton is defined as the charge density in the presence of a soliton ¢ minus the vacuum charge density for ¢(x) = ~0 = const for all x ~ R:

E

wheie Ca E, ~a O are solutions of(2b) for H(~), n ( ¢ = ¢0) respectively, f ( E ) = {exp[•(E - bt)] + 1} -1 is the Fermi distribution function for the occupation of the energy levels E at temperature T = 1 [[Jk B and chemical potential /a and the symbolic sum X; E extends over the discrete and continuous parts of the spectrum. The total charge per spin degree of freedom is given by the space integral

o o

r): f dx a.,r(x). (4) t2(u, - - o n

The general solution ¢E for (2) is given by Jackiw and Semenoff [6] : (a) The vacuum case ~ : Oo is trivial: the spectrum of H(Oo) is continuous (E = ±(k 2 + e 2 + ~)1/2 Ik E R) and

the components of ~O are proportional to plane waves uO(x) = dkx; (b) for nontrivial ~, (2) has (i) one bound state solution

× ( x ) ~ ( e x p [ - f x ~(x') dx'] ) , (5)

0

with energy eigenvalue E : e; (ii) a continuous spectrum (E = ±(k 2 + e 2 + ~2)1/2 Ik ~ R) with corresponding spinors

( x/(E +--O/2EUk(X) ) ~kE(x) : (sign E ) V 2 E ( E + e) [(b x + ¢) u k l (x) :

(6)

with u k a normalized solution of the SchriSdinger equation

(--a 2 - (ax~b ) + ~ 2 ) u k =(E 2 - e 2)u k ,

with potential 0 2 - a x ¢ . Using these results we can perform the sum in (3) and obtain

(7)

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Volume 106A, number 1,2 PHYSICS LETTERS 19 November 1984

o~, r (x) = f ( E = e)×f (x) ×(x)

+ ~ i ' ~ n f(Ek)( luk(x)'2 -luO(x)12* sign E _oo

and for the soliton charge

1 Q(u, T) = exp[O(e- ta)] + 1

a x [a x luk(x ) 12 + 2 lUk(X)12~(x)]

4ek(ek + 0 !

+ f d~ Z~ :(~k)[I.k(X)l 2 -lup,(x)l 2] sign E

- - o o - - o o

sign e 4Ek(E k + e) x = - - - o o

The second integral depends only on the spatial asymptotics of the integral; it can be regarded as a global or topo- logical quantity. Since u k is a "scattering solution" of(6) we can use its asymptotic behaviour

Uk(X ) -+ e ikx + Re ikx , x -+ _o0,

-~ T (ak x x + oo

and the unitarity condition [R [2 + [ T[2 = 1 for the reflection and transmission coefficients R, T. We obtain

a x lUk(X ) 12 + 2 lUk(X) 12~(x)Ix--_ - 00 = 4¢0,

and thus for the soliton charge per spin degree of freedom

1 f Q(#, T) =exp[O(e ta)] + 1 + dx ~ f (Ek)[ lUk(X) l 2 --lu0(x)121 -- sign E

- - ~ - - o o

(8) ~ E f dk <Po + 2-~ f(Ek) Ek(E k ÷ e)"

si -oo

For f ( E ) = 0 ( - E ) , i.e. zero temperature and chemical potential, the first integral can be reduced to - 1 by com- pleteness arguments (see ref. [6]) and the fermion number becomes a purely topological quantity. Furthermore, for finite temperature but zero chemical potential the Fermi distribution reduces to

f ( E k ) _ 1 1 =1 sign E exp(~Tk) + 1 + exp(--~Tk) + 1 '

and the same result follows. However, for a general Fermi distribution f ( E ) with nonzero chemical potential this argument no longer holds;

the integral will depend on local properties of the integral, thus adding a nontopological term. We examine now the actual importance of this local term by evaluating the integrals for two locally differing soliton profiles and comparing the results.

A standard soliton profile is

O(x) = q~0 tanh ~0 x ,

where u k is explicitly known to be (see ref. [6])

tanh ¢~0 x - ik/~ 0 Uk(X) = - 1 + ik/~ o exp(ikx),

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Volume 106A. number 1,2 PHYSICS LETTERS 19 November 1984

and (8) reduces to

2,0 f a,:(ek) 1 +¢o ~ dkf(Ek)E k +e) +expt~(e #)1 + 1 " (9) Q(/~ 7') - 27r sign E -** k2 +'~0 21r sign e -**

A second soliton profile is the usual step function ¢(x) = sign(x) *0 which can be regarded as the infinite-slope lim- it of the previous hyperbolic soliton. Eq. (6) has the form of an ordinary Schrbdinger equation with Dirac 8-poten- tial ax,(X ) = 2¢0~(x ) with exact plane wave solutions

Uk(X )=d kx +R(k) e -ikx, x ~ O ,

= T(k) e ikx , x > 0 , (10a)

where the reflection coefficient is determined to

~ k ) = (_ ,2 + V,,0)/(,02 + k2), (10b)

and transmission coefficient T = 1 + R. With these results the spatial part of the first integral

e o

f dx(luk(x)l 2 - 1)

reduces to

0 f dx [2 Re (R (k) e-2ikx)] ,

which has to be treated in the distribution theoretical sense, i.e. we regularize the integral by e -ax, ~ E 8 and take the limit ~ -~ O:

0 2*0 2 a k*0 2k a--0 *2rr k*0 1 f 2 - - - - - - - - - + ~ ( k ) - - p . v . d dxtRe(R(k)e-2ikx)l ~ 0 + k 2 ot2+4k 2 ~ 0 + k 2 ~2+4k2 ~ 0 + k 2 *02+k 2 ~T'

- - o o

which give us the final result for (8) o o

I ' e(v,n=_ 33 1 _*o f dkf(Ek)k2+,2 2 sign g exp[/~(Ek -/z)] + 1 k=0 2n sign E _0o

¢0 : ( ~ 1 (11) +'2"~ signE~ dkf(Ek)E k +e) + exp[~(e- /~)] + 1"

Comparing this result to (9) we observe that, besides multiplying the first integral by a factor of ~, the local varia- tion of the soliton field results in adding two exp-functions. To determine, however, the actual change of Q a numerical evaluation of the integrals is needed.

Contour maps of Q [given by (9) and (11)] versus the fermion-mass-normalized chemical potential ~ = #[(e 2 + ¢~)1/2 and temperature r = T/kn(e 2 + ¢~a) t12 for various values of the charge.conjugation-symmetry-breaking

v 2 + 2 1 / 2 - ' i - parameter ~'= e/(e ¢6) are dtsplayed in figs. l a - l f . We find that, although the exact form depends on the chosen form for the soliton, Q(~, r) has a qualitatively similar behaviour in both cases.

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Volume 106A, number 1,2 PHYSICS LETTERS 19 November 1984

T too

.75

.5o

.25

!

a

f i ' 3 3 ~

-! 0 I p.

T .75

"50 t .25

' '

-I O I #

'I"

too

.75

.5o

.25

' c '

- . ~ ¢ •

-I o I #

T

t£x)

.7~ =

.5¢

.2.~

d

.03

-.14 . ~ ~ . ~

-I 0

T

" 7 5 ~ 1 - . ~ ' I '

-I 0 I

.:o -.,, f /.,,-.

-I 0 I

Fig. 1. Contour maps of charge (per spin degree of freedom) Q versus normalized chemical potential ~'and temperature ~ for: ~" = 0 for (a) the tanh-soliton; (b) the 0-sofiton; ~"= 0.1 for (c) the tanh-soliton; (d) the O-soUton; ~'= 0.5 for (e) the tanh-soliton; (f) the 0-soliton.

Furthermore, we observe a rapid change of Q from negative to positive values as ~increases from ~ e to ~ ~.. This transition occurs in the experimentally relevant low temperature domain ~ 4{ 1; an experimental confirmation of our results should thus be possible.

In the course of this work we became aware of a preprint by Soni and Baskaran [11] some of whose analysis and conclusions are similar to ours.

I would like to thank Dr.G. Semenoff for many helpful discussions and suggestions. I also wish to thank Dr. Eric Hiob who provided the computer graphics for this work.

References

[1] R. Jackiw and C. Rebbi, Phys. Rev. D13 (1976) 3398.

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Volume 106A, number 1,2 PHYSICS LETTERS 19 November 1984

[2] J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47 (1981) 968. [3] R. Jackiw and J.R. Schrieffer, Nucl. Phys. B190 (1981) 253. [4] W. Su, J. Schrieffcr and A. Heeger, Phys. Rev. Lett. 42 (1979) 1698; Phys. Rev. B22 (1980) 2099. [5] M. Rice and E. Melt, Phys. Rev. Lett. 49 (1982) 1455. [6] R. Jackiw and G. Semenoff, Phys. Roy. Lett. 50 (1983) 439. [7 ] R. Jackiw, in; Proc. Third Nuffield Workshop (Cambridge Univ. Press, London). [8] B.R. Wcinberger, Phys. Rev. Lett. 50 (1983) 1693. [9] A. Niemi and G. Semcnoff, MIT preprint CTP 1086 (May 1983).

[10] A. Niemi, IAS pccprint (March 1984). [11 ] V. Soni and G. Baskazan, CERN preprint TH 3869 (April 1984).

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