8
Physica 127B(1,984) 3-10 North-Holland, Amsterdam STRAP~GE QUANTUM IO~ERS ON SOLi~x'ONS IN BROKEN SYMIE~TRY SYSTEMS LR, SCHRFEFFER Departm~!nt of Physics and .rt~stiture .for T~!enrctical Ph~sic,~, Uniw~.'sityof California, Santa Barbara, CA 9"3106. USA 1 wouk[ YX~e to ,~!scusstwo questions which deal with the nature of elementary excitations in wel]-dcfined cor.densed mutter sy~a~.n~s. Tt<, first question is, 'Cau a system made v~ o:I pal~icles each of whicl~ have integer cl~,arg¢ (±e)have exci~atlonsox fraetional charge O ~ -.evewhhout violating quantum mcetxanlcsT ~rhe second question is, 'Can a system c,f particles interzetln;zwith forces which are itwariant under space and titan inversion have excitatlons which nan move in only a sh~gle ;lireel/onT The answer to both of these questions is 'affin~aative. It is interesting to see how these {3eenlia~" resalts cOme about in specific ~ e s , 1. Quasi one-dimer~onal ¢ondudors To illustrate how fractional charge can come about, consider the quasi one-dimensional trans- polyacetylene [1], shown in fig. 1. Polyacetytene is simply a linear chain of CH groups, which has a zig-zag structure. Each CI-t group has two so-called cr bonds extending to the neighboring grou!as, where these bonds are given as linear combinations of p orbitals in the x-y plane and the s orbital of the carbon. These bonds are symmetric under rotation about the axis joining the CH groups and form a stable backbone structure for the molecule. There is a remaining bond namely, t~he p~ or ¢r bond which holds one electron per group on average. Since a filled ~r band has two electrons per group, it is clear that in the absence of distortion of the molecule this system would be a one-dimensional conductor. However, ~Lhirty years ago R.E. Peierls [2] p~oved that such a one-dimensional metal is anstable with reg~d to a static distortion of the lattice and in this case the period of the distor- tion is such tha~ the system repeats its structure every two carbon atoms. Thus, the unit cell has been doubled il~ size and a Br/Ilouin zone has been halved in size due to this Peierls distortion or as the chemists call il, dimerization. In fig. la, one sees that the upper CH groups move to the ~¢fl: and the lower ones move to the right, leading to double bonds which slope up- ward to the right. In fig. lb the reverse is true, with the bonds sloping downward to the ~vight. These two different states are illustrated in fig. 2 where the potential energy is plotted vs, t]ae staggered displacement, 4,,, = ( - 1)"u~. (1) When d~, is positive, one has the A phase while if ~b, is negative, one has the B phase as shown in fig. 1. The undistorted system is that given by <k = 0. In either the A or B phase, one has an energy gap of width 2 A in the electronic excitation spectrum as shown in fig. 3. This gap., termed the Peierls gap, is the origin of the distor- tien, and is in essence a Ja_hn-Teller effect in the molecule. "ll~e distortion lowers the energy of the states which are occupied and raises states which are unoccupied leading to an overaL! energy re- duction so long as tiffs effect is l~ot cx~unter balanced by the increa~ of elastic energy of the or bonds which accompanies the distortion. One can construct a very simple one- dimensional electron-phonon Hamiltoniarl [3] =- Z t.(c:~i,:,L, " ' -b cn, ~c.÷l. ~.) P,~+ ) z ~ -- zc(u.- u~,,,) , t,, -- tn'-,x(~.~, -u,), (2) wheze M is the CH mass. 0378-4363/84/$03.00 (~) Elsevier Science Publishers 13.V. (Norti~-Hollaad Physics Publishing Division)

Strange quantum numbers on solitons in broken symmetry systems

Embed Size (px)

Citation preview

Page 1: Strange quantum numbers on solitons in broken symmetry systems

Physica 127B (1,984) 3-10 North-Holland, Amsterdam

STRAP~GE Q U A N T U M I O ~ E R S ON SOLi~x'ONS IN B R O K E N SYMIE~TRY SYSTEMS

LR, SCHRFEFFER Departm~!nt of Physics and .rt~stiture .for T~!enrctical Ph~sic,~, Uniw~.'sity of California, Santa Barbara, CA 9"3106. USA

1 wouk[ YX~e to ,~!scuss two questions which deal with the nature of elementary excitations in wel]-dcfined cor.densed mutter sy~a~.n~s. Tt<, first question is, 'Cau a system made v~ o:I pal~icles each of whicl~ have integer cl~,arg¢ (±e)have exci~atlons ox fraetional charge O ~ -.eve whhout violating quantum mcetxanlcsT ~rhe second question is, 'Can a system c,f particles interzetln;z with forces which are itwariant under space and titan inversion have excitatlons which nan move in only a sh~gle ;lireel/onT The answer to both of these questions is 'affin~aative. It is interesting to see how these {3eenlia~" resalts cOme about in specific ~es ,

1. Quasi one -d imer~ona l ¢ondudors

To illustrate how fractional charge can come about, consider the quasi one-dimensional trans- polyacetylene [1], shown in fig. 1. Polyacetytene is simply a linear chain of C H groups, which has a zig-zag structure. Each CI-t group has two so-called cr bonds extending to the neighboring grou!as, where these bonds are given as linear combinat ions of p orbitals in the x - y plane and the s orbital of the carbon. These bonds are symmetr ic under rotation about the axis joining the C H groups and form a stable backbone structure for the molecule. There is a remaining bond namely, t~he p~ or ¢r bond which holds one electron per group on average. Since a filled ~r band has two electrons per group, it is clear that in the absence of distortion of the molecule this system would be a one-dimensional conductor. However, ~Lhirty years ago R.E. Peierls [2] p~oved that such a one-dimensional metal is anstable with r e g ~ d to a static distortion of the lattice and in this case the period of the distor- tion is such tha~ the system repeats its structure every two carbon atoms. Thus, the unit cell has been doubled il~ size and a Br/Ilouin zone has been halved in size due to this Peierls distortion or as the chemists call il, dimerization.

In fig. la, one sees that the upper CH groups move to the ~¢fl: and the lower ones move to the right, leading to double bonds which slope up- ward to the right. In fig. lb the reverse is true,

with the bonds sloping downward to the ~vight. These two different states are illustrated in fig. 2 where the potential energy is plotted vs, t]ae staggered displacement,

4,,, = ( - 1)"u~. (1)

W hen d~, is positive, one has the A phase while if ~b, is negative, one has the B phase as shown in fig. 1. The undistorted system is that given by <k = 0. In either the A or B phase, one has an energy gap of width 2 A in the electronic excitation spect rum as shown in fig. 3. This gap., termed the Peierls gap, is the origin of the distor- tien, and is in essence a Ja_hn-Teller effect in the molecule. "ll~e distortion lowers the energy of the states which are occupied and raises states which are unoccupied leading to an overaL! energy re- duction so long as tiffs effect is l~ot cx~unter balanced by the inc rea~ of elastic energy of the or bonds which accompanies the distortion.

One can construct a very simple one- dimensional e lectron-phonon Hamiltoniarl [3]

= - Z t . (c:~i , : ,L, " ' -b cn, ~c.÷l. ~.)

P,~+ ) z ~ - - z c ( u . - u~,,,) ,

t,, -- tn ' - , x (~ .~ , - u , ) ,

(2)

wheze M is the CH mass.

0378-4363/84/$03.00 (~) Elsevier Science Publishers 13. V. (Norti~-Hollaad Physics Publishing Division)

Page 2: Strange quantum numbers on solitons in broken symmetry systems

4 J.R, Schrieffer / Strange aucntuat rtwnbers on svlitons

H H H H H J I u~ 4 I ) C C ~ ' ¢ C ¢

' / \ / \ & \d."/x? k h ~ ° " ,V ''÷' ~ ,,,

( a )

H H H H H , , u . , , ( b )

C C C'--" C C / I I ~ t l -P i ~ n * l I t J H H H H H H

Fig. I, a) A ptm.~e or trarm (Cl'I)~ ~ b) B phase ot trans [CH),,

Fig. 2. "lbml energy ptmled as a funcdon of the dlnterization amrlitude,

? , / / , ~ . /

~77~, 2~/7Z Fig. 3. Energy gap in ll',e one.electron spectrum.

Within the mean field approximation, namely, replacing the u,':; by their :;elf-consistent average vJues , one obtains th<~ results diseu~ed above. A~! this point the prop,-rtlcs of the system would appear to be those of a conventional semi- coaductor with energ5 gap 2,4. This is not the case as seen in ~he iollowing section.

2. Solitoas ~n trans-(CH)~

As illustrated in fi~. 4, consider a state of the chain in which the system is in the B phase on

B A H H H H H

I I i i J

C ''-> C--~" C + - C * " C

I i I I t I H H H H H H

Fig. 4. A narrow st>~iton S separating flxe B and A phasos.

the left and A phase on the rfghL We see that fine displacements in the two phases are just the mirror images of each other, and, therefore, the displacement of the a tom in the center is zero.

in fig. 5 is plotted the staggered displacement field $,, showing that it goes from - u o as n tends to minus infinity and to +uu as n tends to plus infinity. The displacement uo of atoms i n ei ther phase is approximately 0,03 A. Calcula- tions by Su, Heeger, and the author [3] as well as by Rice [4) show thin the form of the domain wall or soliton is give~ by

4'. = uo tanh (r"O. (3)

The width of the soliton, 2l, is of order 14 lattice spacings, i ,e., 16a. The soliton mass is given by equat ing the kinetic energy of the system with a solitou drifting at speed V, to the quantity M~V~d2. One finds that M, is approximately equal to 6 t imes the electron mass. Thus, the soliton is indeed very light despite the fact that its inertia is determined in essence by the nuclear mass. The reason for this is that the lattice displacement is a very small fraction of the lat- tice spacing a, and it is the square of this ~atio which comes in to determine the inertial parame- ter, We note that the anti-soliton has the struc- '.~ure shown in fig. 6, namely, d~,, is u0 as n goes to

f °° _ _ _ . .>

- U 0

Fig. 5. The st0ggered or0t:r paramt:ter d,,, for a soliton S.

Page 3: Strange quantum numbers on solitons in broken symmetry systems

J.R, SehriefIrr / Strange quantum numbers an solit~ns

~bl~ I U o

--U 0

Fig. 6. ~6,, for an aati-.soliton S,

- ~ and - u p as n goes to o~. We denote soliton and anti-st,iiton by S and S, respectively.

3. E~ect~on~c speet~mm ot the; sof i ton

In fig. 7a, the electronic spectrum of the undis- torted (CH)~ chain is shown for t h e ' s band, showing that one has in fact a metallic conductor due to the finite density of state.s at the Fermi surface. However, in fig. 7b, we sec that the dimerization opens up a gap 2A at the Fermi suxtace and the system would appear ~o become a semi-conductor. This is true of either the A or B phase, i t is important for us to keep track of the total state count and this is il~ustr~ed for the uadimerized and dimerized spectra showing that the valence band and c~nduction bands each have exactly half the number o~ states if we start

(n} Ib)

(c) (a)

Fig. 7. a) Electronic sl~cttum % for a tight binding b~nd without dimerizatlon; b) The spectrum E~, wifl~ dimerlz~ttion, leading to an energy ~p 2 A at /¢ = ~12a; e)/,~ ring with an even number N of CH groups; d) The ¢c~duelion and valence band each having N/2 states.

& Fig 8. Evcn N ring, with t~,. plotted radialry, showing a widcly spaced SS pair; b) State count for ::~e SS syStem, exhibithtg two gap states.

out with a ring having an even number , say N, CE groups as illustrated in fig. 7c. W e have ptoltted the order parameter 4~ radially, where ~b,, outs ide the ring i~dicates the A phase while ~k~ inside the ring ind, icates the B phase. In fig. 8a we distort ~ such 'that the A phase is eon- verl:ed rote B phase and then back to the A phase again, ~ r respondhag to a soliton and an anti-soliton as on~ moves around the ring. For topological reasons it is impossible to have only a solitc, a or anti-so,iron i~ one has an even N ring. O~ the other hand for an odd N ring one can only h~ve an odd number of solitons plus anti- solitons, again for topological reasons. In fig. 7d we see that the valence and conduction bands each have NI2 states while tor ~ , in fig. 8 this number has geen reduced to N ] 2 - 1 for each conduction and the valence band. Two new states a~,pear in the gap, which for widely spaced S and

we associate with a state localized at the sol- iton and a state localized at the antbsoliton. This assignment is correct o~tly in the limit of the spacing ber, veen S and S approaching infinity so that the two levels are degenerate. More fgener- ally, as S and S approach each other, the~;e levels interact and produce the s3anmetric and anti- symmetr ic combinations corresponding to bond- ing and anti-bonding with energies spli~ by the hybridization matrix e lemenls connecting them. Typically th,.'.~ is a small effect except when S and

are withi,:t 21 of each other.

Consider the process in which we begin w~th the system in the perfect A phase as shown in fig. 9a. Suppose the staggered ozd~r paraa'~eter ~,, is slowly distorted to have the fo~ra shown in fig.

Page 4: Strange quantum numbers on solitons in broken symmetry systems

J.R. Sehrieffet i Strange quantum n,mbers on soliJ'ons

u i ., n> N>

Fig. 9 a) d', for the uniform A phase; b) ~, for a widely spaced soliton and anti-soliton pair.

9b where there is a region which is perfect B phase ~nd this switches to the A phase, as one goes to ±:o. As the nuclei deform to move from the perfect A phase to that shown in fig. 9b t.~e electronic states also deform. However, if this is done very slowly the electrons will never ,:ross the energy gap, However, due to the fae,: the states are moving there is, of course, an electron current flow which one ca,a view as a flow of current in the filled valence band. At fix'st sight, this is contrary to conventional band theory; however , we realize that band theory typically speaks about configurations in which the ntlelei remain at rest. In our case, it is the motion of the nuclei which gives rise to the interesting electronic current flow in the system. In fig. I0a we see the charge density in the filled valence band of the A phase is uniform while in fig. lob we see that this density has dips in the vlcin~ty of the soliton and the anti-soliton, even though the valence band is filled and the conduc- tion band is empty in both eases, An important quay'ties is the magnitude of the depletion of the density in the vielnky in each of the excitations. In the limit that S and ~i are widely spaced compared to their intrinsic width 2t, we can prove that the depletion is precisely 1/2 an elec- tron per spin erientation. "Ihus, She depletion of tile valence band is a total of else state per spin

pIx} 1 pixY!"l

) x I x~" (a} (hi

Fig. I(). a) Th,: average ¢r eleetYon density ior the unil~.~l'm A pha~c; b) The "rr electron density m the presence of S and ~;.

with I /2 of this depletion being localized around each of the two excitations, Were there only up spin ¢" electrons in polyacetylene this would lead to the conclusion that the soliton would have charge e l 2 since one has removed a charge of - e l 2 f rom the vicinity of each excitation due to vacuum flow.

One simple way to see this result is to use t~- ~ fact that the integral of the local density of states on each site is conserved, Since a state is p ro - duced at mid-gap in the vicinity of the soliton, it follows that the integrated depletion of the cozl- duetion and valence bawd is precisely one state. However, by charge conjugation symmetry, a property which one can easily prove for the above Hamiltonian, it follows that the density of states :in the conduction and valence bands i,~ symmetr ic about the center" of the gap. There- fore, the depletion of the valence band is pre.- cisely half o~ a single state, i.e., 112 an electron is removed from the rifled valence band in the vicinity of the soliton. The same reasoning shows that the depletion of charge nabs the anti-soliton is also half an electron. Therefore, in the absence of filling the gap center states near S and S we would have two objects each of charge 1/2 were there only one spin orientation of ~'he ~r elec- trons of polyacetylene. Of course, in real (CH)= both spins are present and, therefore, the total depletion of electrons in the valence band near the soliton is one electron. The same situation holds near S. These results are summarized in fig. 11 where in fig. l l a we see a depletion of a total of one electron f rom the valence band while the gap center state is unoccupied. This leads to a

{o) (b) (c)

....... _4_.. - ¢ - ~$_.

O.,e O=O O=-e S=O S=1/2 S =0

Fig. I I. a) Occupation of states for a po:~itivcly charged solhon S'" b) Oce~patJon slat¢~ foJ" S°; c) OccupatioeJ of stales for S".

Page 5: Strange quantum numbers on solitons in broken symmetry systems

Table 1

Ch',~ge Sp~n

Electron -e 1/2 S'" or ~- -e 0 Hole +e t/2 $+ or S-~ +e 0 Excit0n 0 0 $" or ~o 0 t/2

net davrge on the soliton of +e. The spin of the soliton is clearly zero since the charge hvz been removed symmetrically from the up from the down spin bands, tn fig. l l h the gaI~ center state is occupied by one electron (either spin up or down) so the charge has been reduced to O = 0, while the spin in this case is 1/2. Finally, ~f two electrons are placed in the gap center state as in fig. l l c the charge is now - e and once again the spin is zero since the gap center state electrons form a singlet. The charge and spin assignments are shown in table I for both the soliton and anti-soliton and compared with the correspond- ing quantit ies for an electron, a bole and an e,~citon. The crucial point concerns the traditional relationship between charge and spin usually as- cribed to Kramer ' s theorem, i.e., an even number of fermions has integer spin while an odd number of fermions has half odd integer spin. This theorem is apparently violated by the solitons. Of course, Kramer ' s theorem really applies only to the entire system rather than a portion of the system such as that surrounding the soliton. The topological constraint requiring that a soliton and an anti-soliton be simultane- ously created allows the peculiar spin and charge relationships in table I for S and S to be ob- served despite the fact that the overall ,~pecimen does satisfy Kramer ' s theorem.

The conclusion is that were there only one spin orientation of the ~- elections in polyacetylene the excitations would have frac- tional charge, namely, ±e/2. Solitons with frac- tional fermion numbers such as this were disco-. vered by Jackiw and Rebbi [5] in the context of a model relativistic field theory model in one space and one time dimension. In a different eont.ex't

J.Ft. Sehritffer / Strange quttntUm numbers on solitons 7

[5e] Hubbard found excitations of charge I12 in a v e r T :~.trong Coulomb interaction limit of a one-dinz~;nsionat model . Their results and those of SSH rely on the same fundamental mechan- ism. Namely, the deformation of the vacuum in creating the soliton and anti-soliton and the flow of quan tum numbers such as fermion number , charge and spin in going from the odginal vacuum to the distorted vacuum which is present after the soliton and anti-soliton are created.

The existence of such objects with strange charge and spin relations were observed by Heeger et al, in experiments on undopcd and doped t r a n s - p o l y a c e t y l e n e [6, 7, 8]. In the an - doped specimens nuclear magnetic r~sonance and electron spin re'sonanee measttrem~nts have established that there is approximately one highly mobile spin one-half object per 3,000 CH groups. The g value of these objects is approxi- mately that for a ~r electron on .a CH group in a typical organic molecule. Since the electrieam conductivity is extremely small in undoped trans CH.~ it follows that these carders are electrically neutral, consistent with Q = 0 and S = 112.

In exper iments on doped tran:~ (CI-1) x for dop- ants in the range of 1 -5%, one finds a very high conductivity and yet the spin susceptibility is immeasurably small. This suggests that in the doped material the solitong are charged but have spin zero consistent with the above picture.

Ano the r picture of the elecw~n~e structure of the soliton is given by the chemist 's view which assumes the width 21 is of order unity rather than of order ~14 which is the actual situation. In the simplified lixnit which is shown in fig. 12, the neutral soliton in panel b is shown as an unpaired electron with pair bonds on either side of the central site being decoupled from the free spin at the origin. Iv, a and c are shown the cases where the unpaired orbital is empty or doubly oc- cupied. Thus , the unpaired orbital i~ the ana3og

-/- • oe

Hg. 12. a) Chemical designation for a ~ery narrow ~oliton 3c'; b) A very narrow .~olLton S~; c) A '~ery nan'ow soliton S%

Page 6: Strange quantum numbers on solitons in broken symmetry systems

J.R. Schrleffer f SZrang¢ q~tanl!um numbera' on solitons

of the soliton state occurring at mid-gap discus- sed above. For the width of the soliton actually one or two lattice sites the activation energy for motion of these objects would be very high and one would not see mobile spin transport at low temperature or l~igh eleer..dcal conductivity at room temperature in doped materials, The large width of the soliton reduces the pe:riodic poten- tial effects on the soliton energy and makes them relatively fTee so that high mobilJties can occur in practice.

We note that ENDOR experiments by Dalton and co-workers [9] have shown the oscillatory character of the gap center state. Since the gap center wave function g,~ vanishes on odd sites for a soliton and on even sites for an anti-soliton, one wou~d expect the spin distribution to be spin up on the even sites but to be zero on the odd sites. In fact, there is a substantial induced spin polarization on the odd sites due to the electron electron Coulomb interactions and this induced polarization is, analogous to the RKKY pol~riza-

J tion familiar m ordinary metals.

I

~ o Fig. 14. The sam.p!Jng function f(x) for measuring the sollton charge, compared with O(x).

and the author [10] in the following way. In fig. 14 one sees a sampling function f(x) whose value is unity in the vicinity of the .,;oliton while far from the solilon it drops to zero on a length scale L One finds that the soliton charge operator ,defined as the integral of the szmpling function f multiplied by the charge density p is such that the root mean square fluctuations of this quantity vanish as e -t" L as L tends to infinity. This proves that the fractional charge of the above excita- tions is in fact sharp, i.e., a so)iron is asymptomi- rally an eigenstate of the charge operator with fractional quantum number.

6. U n m m ~ n g the tractional charge

5. S h a r p n e s s o f iraefion~l charge

Having discussed the origin and nature of frac- tional charge we turn to the question of whether fractional charge is in fact a sh~-rp quantum observable or whether it is simply a fluctuating quantity whose average or expectation value is fractional This is illustrated in fig, 13 where in panel a one sees a 6 function probability at O/e = 1/3 while in b one has a strongly fluctuating fractional charge with probability 1/3 of observing O/e = t and probability 213 of observing zero charge. This issue was first resolved by Kivelson

t - I I = ;, 113 Q / : 0 1/3 2 /3 I O/e

Filz. ] 3. a) 9rob;;bili W P(Oi of a charge 0 being measured if dlc fractional charge is a sh;irp qiianlum obi~el~,'ahlc; b} P(Q) ~Or ~luctltaling [r~ctJonal char~,e, where only the mean charge i~; fr~ctional.

Ferromagnetic polyacetylcne would have frac- tional charge. The actual solitoas have integer charge due to the spin masking. Sa and the author [11] considered a three-fold degenerate gromld state, that is, a charge density wave of commensurability 3, as illustrated in fig. 15. We term the three degenerate ground states as the A, B, and C phases in this case. The charge density wave is of the form

o(;0 = p2+ Acos(~ax+O(x) ), (4)

i I i , , A I ~ , - - I - ~ l - - [ - - - - I - - i - -

Q I I I I ~ {

, - - / - - l = l - - ~ - - i = i - - a 1 i \ / I I r I t r I I V I I I ! t !

c l------l----l--l~l--i---f Fig, 15. The bond representation of thing phases+ A , B and C. for a charp, e de~lsity wave of coramcn.surability three.

Page 7: Strange quantum numbers on solitons in broken symmetry systems

J.R. Schrieffer I Strcnb~o qua,it~d~ ~ldgtbers or, .¢olitons 9

where a is the spacing between two adjacetat groups or monomers and 0 is the CDW pha~¢ angle. 0 ~- 0, 2~r/3 and 4~'/3 correspond to the A, B, and C phases, respectively, while 0-~6~r13 corresponds to the A phase again.

As in the discussion above where we made soliton and an anli-soliton, suppose we very slowly deform O(x) such that as we move from minus infinity to infinity along the chain axis the 0 goes from zero to 2~'13, to 4¢r13, and then to 6¢r/3, rema~.uing at this vahm to infinity. Thu~, we have created a sequence of phases corres. ponding to A, ]3, C, A as one moves from - ~ tv ~. 'rite boundaries between AB, BC, and CA correspond to three solitons. We minimize the energy by adjusting the 0 in their vicinity aml find to the state of lowest energy of the system. It is clear that the process of deforming 0 leads to a motion of the pair bonds or charge density of the ~r electrons as shown by the dashed line i~ fig. 15 steadily shifting pairs to the righ~ as one moves A to B to C and back to A agin. The net result is a transfer of a pair of electrons from a given site to three sites further along the chain. Thus, if we construct a Gaus,~ian surface at some large distance to the right of the CA botmdar~ we see that two electrons have passed through this surface to infinity leaving the region contain, ing the three solitons with a net charge of +2e, Furthermore, one can show from the invarianc~ of the I-Iamiltoniaa under translations which are multiples of the bond length a that each of the three solitons will have the same charge. Thns~ three solitons share two electronic charge units or the charge of a single stilton is given by 213e. In addition, the flow has occurred without spin polar~ations so the spin of the soliton is zero. Th~s simple argument is supported by detailed calculations. Other charge states are given hy adding an electron to the soliton and the possible charges are shown in table IL Anti-solitons ar.z given by constructing the reverse phase shift corresponding to addir~g two electrons. Their charges are also given ia table II,

These considerations were generalized to three dimensions and reIativil;tic field theory by Gold- stone anti ~ilczek [12]. Again, the results de- pend only on the asymptotic nature of the state~;

Table. ii

Charge Spin

S 2et3 0 -el3 t/2 -4/3e 0

g -2e/3 0 el3 1/2 4e13 0

h~ ih~oity and the vacuum current flow corms- BorJqJhg to a shift of one vacu-am to anot~,er in B¢bdl~Ong the excitations. These results do not q 6 ~ O upon the shape of the soliton.

I[ ~¢~1 .-.at have time to go into detail concern- ihg I~,¢ solitons are generated. Suffice it to ,say tha~ if one introdt~tces a donor or an acceptor a~ta~ the low energy slate is not the conventienal ~nak~ndnctor electron or hole, but rather sol- ilt0h~ ~tgd anti-solitons which are ehar~.ed eiflaer pqMt~¢~ly or negatively. Another way of generat- Jhff t ~ e topological excitations is by absorbing lisht ~tbove the energy gap 2z~ and creating an a~l}0tl hole pair which decays into a soliton ~t~tp~pliton pair ir~ a time scale of c.rder 10-~ s. 51'1~ ~ ~xcitations m'e observed to contribute to ~bqfqg~Tnductivity [6].

q, ~ way exeilmiom

~/V~ torn now to the second ,question raised in ~h~ i~ftoduction, namely, 'Can a system of parti- cles i~t~racting with forces which are invariant ~hdCt' ~face and time inversion have excitations which ~ n move in only one: direction?' This l ? ~ b l ~ has been recently studied by T.L Ho, J. (hJqP~ ~. Wilezek and the author [13]. They ~c~o~jq# ~ the quasi particle spectrum associatexl WSth igl0nar solitons in supelt]uid "~He-A. By ~ol'¢i~ the Bogoliubov de Gennes equations for ~:la~ ~%~ in which the orbited angular momentum ~" ¢h~q.j~¢s direction as one moves through the w~tlI, IF~Y find the spectrum exhibits bound sl:ates Iv~al~tJ at the wall, much as the gap c~nter ~t~t~ ~/pb is localized on the (C/-I)~ ~olitot~., In

Page 8: Strange quantum numbers on solitons in broken symmetry systems

10 J.R. Schrieffer / Strange quamum numbers on solitons

~He.-A the bound quas i par t ic les a re e igenfunc- t ions of m o m e n t u m para l le l to the wal l b u t re- m a r k a b l y the b o u n d s t a t e spec t rum d o e s no t show inve r s ion (or t ime r~wersal) s y m m e t r y . Th is leads to the p red ic t ion of a quas i par t ic le cu r r en t f lowing in a un ique d i rec t ion para l le l to the wall wi th the fo rmat ion of vor t ices s ince the cur ren t flows i~J the oppos i t e d i rec t ion on ne ighbo r ing ami,soli . to~s. The r eade r is r e fe r red to the l i t e ra - ture for de ta i l s of this work .

Th is w o r k was suppor t ed in pa r t by the Na t iona l Science Founda t ion through G r a n t No, D M R 8 2 - 16285.

R e f e r e n c e s

I[1] A gcncrat overview of the area of quasi one- dimensional conductors is provided in: (a) Cmasi One- Dimensional Conductors. Procoedings, Dubrovnik, 1978; tecture notes in Physic,~ (Springer, Berlin, 1979): (b~ P~oc, Int. Conf. Low Dimensional Synthetic Metals, Helsit~gar (1980); Chcmica Scripts. rot. 17 (1981); (c) Proc, Conf. IJ.>W Dimensioual Conductors. Boulder (198i)~ Molecttlar Crystal.~ ~nd Liquid C rfstals 77 (1981) 1-356" (d) Pro¢, Int. Conf. Physi~ and Chemis- try of Polymeric Conductors. Les Arcs, J. Physique (1083): (e) J.T. Dcvrcose and V,E, van Doren, FIighly Conducting Om:-Dimensional Solids (Plenul~, New York, 1979): if) The Physics and Chemistry of Low

Dimensional Solids, L. AIcacer, ed, (ReldeL Dordrecht, t~80),

[2} R.E. P¢icrts, Quantum Theory of Solids (Oxford Univ. Press, London, 19,55) p, 108,

[.3] W,P. So, A.J. Heeger and J.R. Schri=ffer, Phi;. Rev, ld,~tt. 42 (I979) 1698: Phys. Rev. B 22 (1980) 2099.

~4] M.J. Rice. Phys. Lett. 71 (1979) 152. [5] (a) R. Jackiw and C. Rabbi, Phys. Ray. D 13 (1.976)

3398; (b) R. Jackiw and J.R. Sclnqeffer, Nucl. Phys. B 190, (1981) 253; (e) J,R. $chrieffer, Mol. Cryst. & L/quid Cryst. 77 (t9:~.2) 209.

[6] (a) A.L I.ieeger and A.G. MacDiarmid, Mot Cryst. Liq. Cryst. 77 (t981) 1; (b) A.I. Heeger, 'Editorial', Synthe- tic Metals 9, i-vlil. January/March (1984).

[7] N. Suzuki, M. Ozaki, S. Etcmad, A.J. Hecgcr and A,G. MacDiannid, Phys. Ray, Lott. 45 (1980) 1209,

[8] (a) Proc. Int. Conf, Low Dimensional ConductorS, Boul- der (1981): Molecular Crystals and Liquid Ca3~stals 77 (1981) 1-356; (b) Prec, Int. Coal. Physi~ and Chemis- try of Polymeric ConductorS. L~s Ares, J. Physique (I 983).

[9] L,R. Dalton. H, ']'homann, Y. Tomkiewiez and N. Shi~ ran, Phys, Ray. Lett. 50 (1983) 533.

[10]" S. Kivelson mid J.R. Schrieffer, Phys. Rev. B 2¢ (1982) 6447,

[I t] Ca) W.P, Stt and J.R. Schricffer. Phys, Rcv. Lctt. 46 (1981) 738; (b} J,R. Schrieffer. Mol Cryst. & Liquid Cryst. 77 t1982) 209.

[12~' J. Goldstone and F, Wflczek, Phys. Rev. Lctt. [13] T,L, He, J,R. Fnlco, ].R, Schricffer and F, Wilczek.

$olitons in Superf]uid 3He~A: Boond States on Domain Walls. Phys. guy. L.¢tt. 32 (1984} 1524.