- Email: [email protected]
Non-identical anyons and new statistics for spinons and holons
Physics Letters A 184 ( 1993 ) 29-36 North-Holland
PHYSICS LETTERS A
Non-identical anyons and new statistics for spinons and holons T. M o r Raymond and Beverly 3ackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel Aviv University, Ramat Aviv 69978, Israel Received 24 September 1993; accepted for publication 18 October 1993 Communicatedby V.M. Agranovich
We discuss the various existingproposals for the statistics of holons and spinons (the proposed fundamental excitationsof the tJ model for HTSC), and we present new anyonic alternatives. We generalize Wilczek's realization of anyons to include nonidentical anyons with mutual statistics and concentrate on particular cases which preserve time-reversal symmetry. Under the restriction of time-reversal symmetry, we find two additional possibilities for the statistics of holons and spinons - either both holons and spinons are bosom or both are fermions. In addition they obey an antisymmetric mutual statistics. We discuss the pairing mechanism of holons in this model, and the microscopicorigins for this statistical behavior.
1. Introduction
The phenomenon of high transition temperatures in the doped copper-oxides superconductors has inspired searches for an unconventional mechanism of superconductivity. Some of the fundamental properties of these materials can be explained [ 1,2] by the tJ model and the resonating valence bond state. The excitations in this model are speculated to be [ 3] a spinless charge + e quasiparticle - the holon, and a spin-half chargeless quasiparticle - the spinon. There is a long controversy regarding the statistics of these excitations. In refs. [ 3-7 ] the holons were identified as bosons and the spinons as fermions (in accordance to their spin). However, a system of charge e bosons can condensate into a superfluid state with flux quantum h c / e (a fluxon), while the flux in high-To superconducting materials was shortly thereafter found to be quantized in units of h c / 2 e as in standard superconductors. In refs. [ 8-12 ] the holons were identified as fermions and the spinons as bosons. This alternative is consistent with the units of flux quantization, and seems to be energetically favored, but still a suitable pairing mechanism does not arise automatically in this model and more assumptions need to be made.
The assignment of bosonic characteristics to a spinhalf spinon seems to contradict the spin-statistics theorem (and the same problem holds for a fermionic spin-zero holon), however the spin-statistics theorem is questionable when we consider two-dimensional excitations, and the choice of statistics is a matter of convenience [ 11,12 ] as long as the composite o f a spinon and a holon (which is a real threedimensional hole) continues to be a fermion. A different approach was taken by the authors of refs. [ 8,9 ] who tried to justify the assignment of this nonnatural statistics by speculating that both holons and spinons transmute their statistics by attachment to fictitious half-fluxons which interact with a fictitious charge (a topological charge of an RVB state) common to both holons and spinons. Statistics transmutation is speculated to occur in both the "shortrange RVB" model [ 9 ] and the "dimers RVB" model [8]. The appearance of the effective topological gauge field may be related to the f l u x phase [ 13-15 ] in two-dimensional spin systems. The viability of these scenarios (RVB and flux phase) remains, however, still far from being established, and the mechanism responsible for the flux attachment is somewhat obscured. The possibility that only one type of excitations transmutes its statistics (even though it is discussed in refs. [8,9] ) seems to be irrelevant
0375-9601/93/$ 06.00 © 1993 ElsevierSciencePublishers B.V. All fights reserved.
29
Volume 184, number 1
PHYSICS LETTERSA
since the (fermionic) anticommutation relations of their composite are violated (see ref. [ 11 ] for a discussion on slave fermion / Schwinger boson techniques). The proposition that the holons and the spinons are anyons (particles that obey fractional statistics [ 16-18] ) - or more precisely semions (with statistical phase 0 = ½n) - first raised by Laughlin [ 19,20], captured the imagination of physicists [ 21,22 ] due to its exotic nature and new pairing mechanism based on topological considerations (which stems from the effective model with no need for additional interactions). Furthermore, Wen, Wilczek and Zee [23] discuss some possible microscopic origins for this statistical behaviour using concepts which resemble the flux phase but for a model which contains next to nearest neighbour (nnn) antiferromagnetic interactions. However, the possibility that excitations responsible for high temperature superconductivity (HTSC) are semions can probably be dismissed due to the failure [24,25 ] of the main conjecture of the semionic model - the breakdown of time-reversal (Treversal) symmetry in high-To superconductors. Aharonov [26] (and Wilczek [27]) suggested a T-reversal symmetric anyonic model based on a generalized anyonic behavior. Following his idea, the primary purposes of this paper are to generalize the concept of statistics and present new alternatives for the statistics of holons and spinons. The new alternatives maintain the benefits of the semionic model while avoiding its main problem (i.e. the prediction of T-reversal symmetry breaking). With this motivation, we will not be very specific about the precise microscopic origin of our model. For the sake of generality, we will only assume that - HTSC is described by the tJ model (or an extended version of it). - The ground state of the tJ model is an RVB state. - The parameters of the model are such as stabilize the flux phase. The first two assumptions are required to create localized excitations - the spinon and the holon, and the third assumption is required for the discussion of the microscopic origins of the model. Nevertheless, the effective Hamiltonian for the new alternatives is independent of the details of the model, and it is possible to justify the occurrence of the new statistics at least in some of the various detailed models. 30
27 December 1993
In section 2 of this paper we present the concept of non-identical anyons [ 26-28 ], by which we mean a generalized class of particles that includes both identical particles with a regular (and in general anyonic) statistics, and non-identical particles which obey mutual statistical behavior (under a double-exchange), a property of any non-identical particles with hard-core repulsion in two space dimensions. We show that the original notion of anyon (a composite of fictitious charge and flux-tube [ 17,18 ] ) is not sufficient to deal with non-identical particles, hence we must generalize Wilczek's realization in an appropriate way. The corresponding generalization of the braid group to colored braids [29] is briefly discussed. In section 3 we discuss T-reversal symmetry in anyonic models and we focus on some special cases in which non-identical anyons obey this symmetry [26,27 ]. The statistics of a composite of two types of anyons is calculated to find which of the various possibilities is suitable for our purpose, leaving only two non-trivial options. In section 4 we discuss the consequences of the two new alternatives for the statistics of the spinons and the holons. We present their effective Hamiltonian and discuss pairing in this model. I f we ignore spinstatistics restrictions for each type of particles then the new options are as legitimate as any other possibilities shown before, hence yielding a new starting point for further investigation. However, following refs. [ 8,9 ], we derive this statistical behaviour from microscopic considerations without violating the spin-statistics theorem, using the statistical transmutation only for one of the two types of particles (after showing that the electron is still a fermion in this case). In the last section we present our conclusions.
2.
Non-identical
anyons
The existence of identical particles obeying fractional statistics (under adiabatic exchange) is proven in two space dimensions for two particles with hardcore repulsion, using the fact that the path of full rotation in the configuration space •2× (~2_ origin)/ Z2 is not contractible. However, this fact leads to an additional (and certainly not less radical) conclu-
Volume 184,number I
PHYSICS LETTERSA
sion: A loop in the configuration space of two nonidentical particles, ~2× (~2_origin), is also uncontractible, hence rotating one particle around another particle (of a different type) multiplies the wavefunction by a general phase X,
27 December 1993
so that it interacts with all particles via AB [ 30 ] and AC [31 ] effects ~x. The Hamiltonian is
lffi X
~u(R, r, ~)=eiX¥(R, r, ~ + 2 n ) .
(I)
This new statistical phase is named "mutual statistics" [28,27 ] and the most important examples are z = 2 n x - the trivial mutual statistics, and X= ( 2 n + 1 ) x - the antisymmetric mutual statistics. Only the trivial mutual statistics exists in three dimensions. Consider a general system of N particles (in two dimensions) of M different types, N~ +... + ArM=N. Hence, in addition to the information about the identical particles, we have to consider the special properties of a wavefunction of any two non-identical particles. As we see later, by using a generalized form of the braid group, even exchanging two identical particles will depend on the number of particles of any other type which were wound by the path of exchange. A system composed of two or more different types of identical particles in two dimensions may exhibit "'anyonic" behavior even when each type is a boson or a fermion. Is Wilczek's realization of fictitious charge and flux-tube attached together applicable for our system of M types of non-identical anyons? Obviously it is not. Each individual particle type has its own fractional statistical phase Oi ( M free parameters) and each pair obeys mutual statistics with ½ M ( M - 1 ) free parameters ZiJ, altogether requiring ½( M 2 + M )
(2)
free parameters. Wilczek's classification contains only 2M free parameters, and hence must be generalized. Let us associate a fictitious charge e z with each particle according to its type I. In the anyon gauge this system is governed by a Hamiltonian of N free particles with a multivalued wavefunction that obeys the required boundary conditions (according to the parameters 0z and )~zJ). In the radial gauge the above system is transformed to a system of bosons with flux-tubes attached to them. However, each particle carries M flux-tubes corresponding to the M types of charges
if I
1
M
- =
E
C
2n
/ e i ~ t +e~q~J N~
/
c J = l,Jd:I \
' E
j= 1
E v ,j
j = ],j,,i
\q2~ /I
J
(3)
where • / is the flux-tube of type I carried by a particle of type J, and where V~0o=g× ( r ; - r j ) / I r~-rjl 2 The first term contains a summation over all particles of the same type and the second term contains a summation over all particles of all other types. Note that each charge interacts only with one type of fluxtube. The phase acquired by the wavefunction under exchange of two particles of the same type I is ~ e I~ z I Zn e -~oo ,
(4)
where ~o = hc/e is a fluxon. The phase acquired by the wavefunction under full rotation of non-identical particles of types I and J is ez ~ r eJ ~ 2n e ~ - o + e ~ - o } '
(5)
and this phase reduces to 2n[2(eZ/e)~I/q~o] when the two particles are identical (from which we reproduce the correct phase for exchange). Actually, this new notation has some redundancy (in comparison to the required number of parameters ( 2 ) ) , so let us choose e ~. . . . . e ~ = e (we could also fix the flux-tubes to be ~i = ¢~di but we prefer to keep this freedom since sometimes only particles of one type are attached to flux-tubes). While the statistics of identical particles in three space dimensions is represented by the permutation group, the statistics of hard core identical particles in two space dimensions is represented by the braid group [32]. The braid group of N strands, BN, is an infinite group which is generated by N - 1 elementary moves trl, ..., a~_ ~ satisfying #] For identical particles the AC effect leads to the same phase as the AB effect, hence multiplies the phase by a factor o f two, but this is not true for non-identical particles.
31
Volume 184, number 1
PHYSICS LETTERS A
{Ti{~i+lai=ai+l~i~i+l
for i = 1, ..., N - 2,
(6)
tricrj=trja~
for l i - j l >12
(7)
(tri is the generator of an exchange of the strand at site i with the strand at site i + 1 in an anti-clockwise direction, and its inverse is or;-~ ). The strands of non-identical anyons must have an additional index, say color, to indicate the type of the anyon. This group is still defined by eqs. (6) and (7) without refering to the indices of color. However, there are now "color limitations" on the allowed braids: The initial and the final configurations must have the same colors at all sites (i.e. the closure of the braids must not mix different colors [29] ), since a final configuration with some exchange of non-identical particles is not a closed loop in the configuration space. Hence, the colored braid is restricted by an additional condition: Let n (cri) czJ) be the number of operations of the type tr~ exchanging two strands of colors I and J in an anti-clockwise direction (before the replacement the particle of type I is on site i and the particle of type J is on site i + 1 ), and let a(tr;-~ )(zJ) be the number of operations of the type a;- ~ exchanging the same strands in a clockwise direction. The condition on
h} m = n( ai) ~iJ) + a( ai-~ )cis)
(8)
is that for each site i and color I M
E
J= I,J#Z
M
fi}m=
E
J= l,J#I
•}gz),
(9)
that is to say, strands of color I change places from the ith site to the ( i + 1 )st site and vice versa an equal number of times. This condition is necessary and sufficient to achieve the desired colors at the end. Let nCU)= X,-n (a,-) on) (sum over all sites) and the same with ~c/t), n (m, etc. The phase of these braidings is then
i~=l(OZ(n(ZZ)-n(zz))
+j~>1½Zij(n(ZS)+n(JZ)-n(zY)-n(SZ)) .
(10)
For example, exchange of two identical particles in a way that one particle of a second type is wound by 32
27 December 1993
the path of exchange is described by the braid tr}~A) tr~A) at As) (where we added the indices of color to the generators to make counting easier) and the resulting phase is OA+ZAS. The generalization of the braid group can have a natural use in knot theory if we consider a link which is made from few knots (closed loops) as a system of colored strands where each loop has a different color. This way the braid contains exactly the relevant information about the link.
3. Time-reversal symmetric anyonie model for HTSC After Laughlin [19,20] suggested that anyons might play an important role in explaining HTSC, it was generally agreed that the test of the model will be whether or not high-To superconductors obey Treversal symmetry. Later on it was shown that other models might also predict the breakdown of this symmetry, and on the other hand, anyons models that do not break this symmetry were formulated. However, T-reversal symmetry still remains a major issue. It can be seen immediately that identical anyons break T-reversal symmetry: suppose that two identical anyons are adiabatically exchanged in the clockwise direction leading to a statistical phase factor 0. The time reversed operation - exchanging them in the opposite direction - will lead to a phase - 0 which is different from 0 except for special cases of the fermions (0=re) or the bosons ( 0 = 2 n ) . Breakdown of T-reversal symmetry can be also proven using the antisymmetric nature of the Chern-Simons term if we examine the Lagrangian of an anyonic system in QFT. After a long debate and conflicting experimental results it is now accepted that high-To superconducting materials do not break T-reversal symmetry [24,25]. The dielectric tensor in a T-reversal symmetry breaking model has some antisymmetric terms, leading to "circular birefringence" which was found (experimentally) to be zero in high- T¢ superconductors. Semenoffand Weiss [33] and Wilczek [34] presented an anyonic system of two types of anyons with opposite statistical phases 01 and 02,
Volume 184,number 1
PHYSICS LETTERSA
o, = - 0 5
(II)
(and no mutual statistics).Their model is symmetricunder an "enhanced T-reversaloperation" that in addition to changing the directionof time also transforms all particlesof the firsttype into particlesof the second type and vice versa. This option can be relevant to two types of semions which have some similarity,such as up and down spinons. Nevertheless,it seems to be irrelevant for an anyonic model based on two distinctexcitationssuch as the spinons and the holons. We now present a T-reversal symmetric anyonic model that does not demand an "enhanced" T-reversal operation [26,27 ], and may be relevant to a system of holons and spinons. It uses the generalization of statistics for nonidentical anyons. Let us concentrate on the simplest case which seems to be most applicable in solid state systems - a system of two types as of anyons (generalization of the T-reversal symmetric model to M types is straightforward). Using eq. (3), the most general Hamiltonian for two species of free anyons A and B is A
NA
Hf
l--~---z., P i -
2 m A iffil
2.,
C j f f i l d ~ i 211;
e ~B O A + O ~, + Ck ~ 1
+--
2~ 2VB f
~2 V~ik
2e
Y~ ~pk-- c
2rob k ~ l
~ij
) NB ~
2,
OBB
-=-V~0k~
l=l,lvtk ~'~ 2
e N^ ~ + ~ A all" C i~l Ere
V~k'~j
(12)
where we used the redundancy of free parameters to fLx the charges eA----eB----e.Then the exchange phase of A is 0A=2XOA/O0, the exchange phase of B is 0e=2nOea/Oo and the mutual statistical phase is X=2x( o A + O~,)/Oo (recall that OA is a flux-tube of type A carded by particles of type B). In order that T-reversal symmetry would be preserved both types must be fermions or both must be #z A system o f two types o f a n y o n s can be described by the standard notation since Wilczek's realization [ 17,18 ] still provides enough parameters. Nevertheless, we stick to our new notation which provides a more realistic picture in which the three statistical phases are not necessarily related to each other.
27 December 1993
bosons. Nevertheless, this still leaves a non-trivial possibility for the T-reversal symmetric model: If the resulting phase under full rotations of non-identical particles is either n or -re (antisymmetric mutual statistics) T-reversal symmetry is not broken. This restricts the flux-tubes to satisfy: • A + ~ ~ = ½~b0. In general, our two-species anyonic system obeys T-reversal symmetry only when 0A= n 1g, 6~ = n2n and x=n3n with nl, n2 and n3 integers. We see that there are eight different possibilities for a T-reversal symmetric anyonic model (from which four are trivial), since each of the parameters nl, n2 and n3 can be either even or odd (phases are defined modulo 2n). Let us now apply the various alternatives to hoIons and spinons. An electron is composed of a spinon and a holon. Since the electron is a "real" (i.e. three-dimensional) object it is still a fermion. A composite made of the two non-identical anyons will have the statistical phase (for exchange) 0eom= 0A "t"0B -]-X •
(13)
(This is SObecause a full rotation can be sub-divided into four separate contributions: 20A, 20B, X, X.) With 0¢,,m= n, the number of different possibilities for a non-trivial T-reversal symmetric anyonic model now reduces to two: (1) Both the spinon and holon are bosons with antisymmetric mutual statistics. (2) Both the spinon and holon are fermions with antisymmetric mutual statistics. The number of possibilities of trivial mutual statistics also reduces to two, recovering the two standard approaches discussed in the introduction.
4. Consequences and origins of the T-reversal symmetric anyonic model The Hamiltonian of a T-reversal symmetric model of holons and spinons which are both bosons with antisymmetric mutual statistics is a simplified version of (12): 1
Nh ( P i - - e N,
+ ~Zmsk~l - - - - g~r° ( p k - - -e ~ o C i~l
2
V~k, )2 ,
(14)
33
Volume 184, number 1
PHYSICS LETTERS A
where h stands for holons and s for spinons, and where we used hs + ~ hs = ½~0 as before. In a case that the two types are fermions we can write them as bosons with an additional vector potential (i.e. additional flux-tubes) • ~ = • ] = ½~o. Standard arguments for anyonic superconductivity [ 20-22 ] and their generalization to non-identical anyons [ 27 ] are not applicable for this system of two completely distinct types of particles unless the densities of the two fluids oscillate together, which certainly is not the case here. In addition to this, those arguments are based on approximating the anyons to be fermions with small flux-tubes attached to them (then the gauge interaction can be replaced by an average magnetic field), but - as was already said by those authors and others - the validity of this approximation to semions is questionable. The same is true for the particles which are of concern here. Nevertheless, we believe that the topological interaction in this model may give rise to a pairing mechanism. Our system is analogous to a system of charges (with no Coulomb interaction) and halffluxons. Similar problems were solved as a function of flux in the AB effect, and we can make use of this knowledge. The following examples describe situations which are different from our system but still may give some indication for pairing (and possibility of superconductivity). In these examples we consider • ~, the flux-tube carried by the spinon, to be equal to half a fluxon and g~[ to be zero. Scattering problem. While there is a non-trivial scattering of a single holon on a single spinon (due to the AB effect) a pair of holons is moving as a free particle with no interaction at all, since all flux-tubes which are carried by the spinons are quantized in charge units of 2e. Approximating this problem to a one-dimensional scattering problem (taken from ref. [ 35 ] ), shows that the holon is "rejected" (i.e. scattered backwards) by a single spinon (see fig. 1 ), while a pair moves freely. Bound state problem. Compare the energy of two holons in a background of a free spinon as a function of the distance R between the holons. We discuss only the two limits of R ~ 0 and R-~oo. In these two limits we consider "closing" a spinon in a round box (twodimensional well with radial symmetry) and calculating the energy levels. When the holons' relative distance is zero their 34
scalterincj of one holon
27 December 1993
scattering of two holons
Fig. 1. Scattering of one holon versus scattering of two holons on a single spinon: while one holon is scattered backwards, two hoIons move freely ( ( 0 ) holon, ( × ) spinon).
charge is 2e hence the gauge potential ½qbo can be eliminated by a gauge transformation so the spinon behaves as a free particle. The term 12/r 2 (where r is the distance from the center of the box) contributes zero energy at the ground state (lz=0). When the distance is very large we consider interaction with one holon at a time, and neglect the interaction with the second holon which is far away (since the interaction is proportional to [rholon__rspinon[-l ). Hence, we close the spinon and (only) one holon in a box. We now approximate this problem to a different one which is trivial: Suppose that one holon is fixed at the center of the well, the wave function of the spinon in the presence of one holon at the center of the box is equivalent to that of an electron with half a fluxon at the center of the box in which the ground state energy levels are higher due to the centrifugal barrier [36] ( h = 1 ): lz (l= - ~ 1 ~ o ) 1 r-~=~ r2
(15)
This could account for an effective attractive "force" between the holons in the spinon background which is generated here by the presence of dynamical particles of the other type. Is there any advantage of one of the alternative models, with both holons and spinons being bosons versus the ease when both are fermions, over the other? These topological considerations equally account for both cases, but if both particles are bosons, the generation of pairs is easier due to an effective attractive interaction between two bosons. The ground state will not be a Bose condensate of single holons, but of pairs of holons, due to the mutual statistical interaction that prevents the free movement of a single particle, while it does not disturb the motion of a pair, as shown by the first example. Su-
Volume 184,number 1
PHYSICSLETTERSA
perfluidity may be generated here due to the (speculated) attractive interaction between two particles of the same kind as shown by the second example. Could these models prefer pairing of holons? This is a crucial point since if only spinons create pairs, the flux (which corresponds to the "real" electric charge) will be still quantized in units of hc/e. Hence the model must prefer pairing of holons or of both holons and spinons. To the extent that pairing is mediated by the mutual statistics, the model is symmetric (as for the spinons and the holons) so pairing of both holons and spinons is expected (and then a "spontaneous symmetry breaking" [37] might occur, so that only one type will form pairs). However, taking into account a specific microscopic model, it may be possible to obtain preference for pairing of holons. The derivation of the microscopic origins of this statistical behaviour is based on the statistical transmutation in the flux phase [ 8,9 ]. In both references the authors discuss the statistical transmutation of holons and spinons, and raise the possibility that only spinons transmute their statistics since holons unbind the half-fluxons at large kinetic energy t. The issue of statistics in this case is not discussed. Recall that the spin-statistics theorem implies that the bare holon is a boson, and a spinon with half a fluxon attached, is also a boson. We can therefore deduce that, in this case, the holons and spinons will be both bosons with additional mutual statistics n, hence yielding the desired realization of a T-reversal symmetric anyonic model: The resulting mutual statistics is due to the fact that the two species of particles, the hoIons and the spinons, interact with the same type of flux-tubes (i.e. they have the same type of statistical charge). The bare holon feels the flux-tube carried by the spinon, and therefore, under full rotation of the holon around the spinon (or vice versa), the wavefunction of the system will acquire a statistical phase n. The opposite alternative - that only holons bind the half-fluxons - will lead to the second case of interest since now both species will be fermions with mutual statistics n. We did not find a solid proposition for the derivation of this behaviour from microscopic considerations. However, in the "long range RVB" the spinons are gapless excitations hence created and annihilated as vacuum fluctuations from
27 December 1993
the "RVB vacuum", in which case it is impossible for them to bind the half-fluxons. This might be a good starting point for the search for this phenomenon. Note that an electron carries a fictitious charge of 2e, hence the creation of half-fluxons is a pure gauge transformation, and the electrons do not transmute their statistics when only spinons (or only holons) are attached to half-fluxons.
5. Conclusions In this paper we generalized the concept of anyonic behaviour to include mutual statistics between non-identical anyons. We discussed all the possibilities of a T-reversal symmetric anyonic model. We derived four possibilities for the statistics of holons and spinons, from which two are standard (i.e. show trivial anyonic behaviour) and have been thoroughly investigated in the literature. Two are new possibilities which are presented here for the first time. We described these new possibilities by a model Hamiltonian which is independent of the specific microscopic origins of the statistical behaviour. Finally we discussed pairing in this model and showed some possible microscopic origins for this statistical behaviour. Eventually, model dependent microscopic considerations may determine which among the two possibilities will be realized. We tried to present a model independent general case, and we leave this question open. We believe that the new options that are given here open interesting possibilities for further research, and should revive the possible role of anyons in HTSC, since T-reversal symmetry is not broken.
Acknowledgement The author is greatly indebted to Yakir Aharonov for providing the initial motivation for this work and for many valuable discussions. It is also a pleasure to thank Aharon Casher, Nissan Itzhaki, Sandu Popescu and Alexander Shnirrnan for very helpful discussions. The work was supported in part by grant 425/91-1 of the Basic Research Foundation (ad35
Volume 184, number 1
PHYSICS LETTERS A
ministered by the Israel Academy of Sciences and Humanities).
References [ 1 ] P.W. Anderson, Science 235 ( 1987 ) 1196. [2]G. Baskaran, Z. Zou and P.W. Anderson, Solid State Commun. 63 (1987) 973. [3] S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Phys. Rev. B 35 (1987) 8865. [4] P.W. Anderson, G. Baskaran, Z. Zou and T. Hsu, Phys. Rev. Lett. 58 (1987) 2790. [5] Z. Zou and P.W. Anderson, Phys. Rev. B 37 (1988) 627. [6] P.B. Wiegmann, Phys. Rev. Lett. 60 (1988) 821. [ 7 ] Y. Suzumuram, Y. Hasegawa and H. Fukuyama, J. Phys. Soc. Japan 57 (1988) 2768. [8] S. Kivelson, Phys. Rev. B 39 (1989) 259. [ 9 ] N. Read and B. Chakraborty, Phys. Rev. B 40 ( 1989 ) 7133. [10] F.D.M. Haldane and H. Levine, Phys. Rev. B 40 (1989) 7340. [ 11 ] C.L. Kane, P.A. Lee and N. Read, Phys. Rev. B 39 (1989) 6880. [12] D. Yoshioka, J. Phys. Soc. Japan 58 (1989) 32. [ 13] I. Affleck and B.J. Marston, Phys. Rev. B 37 (1988) 3744. [ 14 ] I. Affleck and B.J. Marston, Phys. Rev. B 39 (1989) 11538. [ 15 ] G. Kotlier, Phys. Rev. B 37 (1988) 3664. [ 16 ] J.M. Leinaas and J. Myrheim, Nuovo Cimento 37B (1977) 1.
36
27 December 1993
[ 17] F. Wilczek, Phys. Rev. Lett. 48 (1982) 1144. [18] F. Wilczek, Phys. Rev. Lett. 49 (1982) 957. [ 19] R.B. Laughlin, Science 242 ( 1988 ) 525. [20] R.B. Laughlin, Phys. Rev. Lett. 60 (1988) 2677. [21 ] Y. Chert, F. Wilczek, E. Witten and B.I. Halperin, Int. J. Mod. Phys. B 3 (1989) 1001. [22] A.L. Fetter, C.B. Hanna and R.B. Laughlin, Phys. Rev. B 39 (1989) 9679. [23] X.G. Wen, F. Wilczek and A. Zee, Phys. Rev. B 39 (1989) 11413. [24] S. Spielman et al., Phys. Rev. Lett. 65 (1990) 123. [25 ] T.W. Lawrence, A. Szoke and R.B. Laughlin, Phys. Rev. Lett. 69 (1992) 1439. [ 26 ] Y. Aharonov, private communication ( 1991 ). [27] F. Wilczek, Phys. Rev. Lett. 69 (1992) 132. [ 28 ] C.R. Hu, Int. J. Mod. Phys. B 5 ( 1991 ) 1771. [ 29] L. Brckke, Phys. Lett. B 271 (1991 ) 73. [ 30 ] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [31] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53 (1984) 319. [32 ] Y.S. Wu, Phys. Rev. Lett. 52 (1984) 2103. [ 33 ] G.W. Semenoff and N. Weiss, Phys. Lett. B 250 (1990) 117. [ 34 ] F. Wilczek, Fractional statistics and anyon superconductivity (World Scientific, Singapore, 1990) p. 16. [ 35 ] A.G. Aronov and Yu.V. Sharvin, Rev. Mod. Phys. 59 ( 1987 ) 755. [36] M. Peshkin, Phys. Rep. 80 (1981) 375. [ 37 ] S. Popeseu, private communication (1992).
PHYSICS LETTERS A
Non-identical anyons and new statistics for spinons and holons T. M o r Raymond and Beverly 3ackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel Aviv University, Ramat Aviv 69978, Israel Received 24 September 1993; accepted for publication 18 October 1993 Communicatedby V.M. Agranovich
We discuss the various existingproposals for the statistics of holons and spinons (the proposed fundamental excitationsof the tJ model for HTSC), and we present new anyonic alternatives. We generalize Wilczek's realization of anyons to include nonidentical anyons with mutual statistics and concentrate on particular cases which preserve time-reversal symmetry. Under the restriction of time-reversal symmetry, we find two additional possibilities for the statistics of holons and spinons - either both holons and spinons are bosom or both are fermions. In addition they obey an antisymmetric mutual statistics. We discuss the pairing mechanism of holons in this model, and the microscopicorigins for this statistical behavior.
1. Introduction
The phenomenon of high transition temperatures in the doped copper-oxides superconductors has inspired searches for an unconventional mechanism of superconductivity. Some of the fundamental properties of these materials can be explained [ 1,2] by the tJ model and the resonating valence bond state. The excitations in this model are speculated to be [ 3] a spinless charge + e quasiparticle - the holon, and a spin-half chargeless quasiparticle - the spinon. There is a long controversy regarding the statistics of these excitations. In refs. [ 3-7 ] the holons were identified as bosons and the spinons as fermions (in accordance to their spin). However, a system of charge e bosons can condensate into a superfluid state with flux quantum h c / e (a fluxon), while the flux in high-To superconducting materials was shortly thereafter found to be quantized in units of h c / 2 e as in standard superconductors. In refs. [ 8-12 ] the holons were identified as fermions and the spinons as bosons. This alternative is consistent with the units of flux quantization, and seems to be energetically favored, but still a suitable pairing mechanism does not arise automatically in this model and more assumptions need to be made.
The assignment of bosonic characteristics to a spinhalf spinon seems to contradict the spin-statistics theorem (and the same problem holds for a fermionic spin-zero holon), however the spin-statistics theorem is questionable when we consider two-dimensional excitations, and the choice of statistics is a matter of convenience [ 11,12 ] as long as the composite o f a spinon and a holon (which is a real threedimensional hole) continues to be a fermion. A different approach was taken by the authors of refs. [ 8,9 ] who tried to justify the assignment of this nonnatural statistics by speculating that both holons and spinons transmute their statistics by attachment to fictitious half-fluxons which interact with a fictitious charge (a topological charge of an RVB state) common to both holons and spinons. Statistics transmutation is speculated to occur in both the "shortrange RVB" model [ 9 ] and the "dimers RVB" model [8]. The appearance of the effective topological gauge field may be related to the f l u x phase [ 13-15 ] in two-dimensional spin systems. The viability of these scenarios (RVB and flux phase) remains, however, still far from being established, and the mechanism responsible for the flux attachment is somewhat obscured. The possibility that only one type of excitations transmutes its statistics (even though it is discussed in refs. [8,9] ) seems to be irrelevant
0375-9601/93/$ 06.00 © 1993 ElsevierSciencePublishers B.V. All fights reserved.
29
Volume 184, number 1
PHYSICS LETTERSA
since the (fermionic) anticommutation relations of their composite are violated (see ref. [ 11 ] for a discussion on slave fermion / Schwinger boson techniques). The proposition that the holons and the spinons are anyons (particles that obey fractional statistics [ 16-18] ) - or more precisely semions (with statistical phase 0 = ½n) - first raised by Laughlin [ 19,20], captured the imagination of physicists [ 21,22 ] due to its exotic nature and new pairing mechanism based on topological considerations (which stems from the effective model with no need for additional interactions). Furthermore, Wen, Wilczek and Zee [23] discuss some possible microscopic origins for this statistical behaviour using concepts which resemble the flux phase but for a model which contains next to nearest neighbour (nnn) antiferromagnetic interactions. However, the possibility that excitations responsible for high temperature superconductivity (HTSC) are semions can probably be dismissed due to the failure [24,25 ] of the main conjecture of the semionic model - the breakdown of time-reversal (Treversal) symmetry in high-To superconductors. Aharonov [26] (and Wilczek [27]) suggested a T-reversal symmetric anyonic model based on a generalized anyonic behavior. Following his idea, the primary purposes of this paper are to generalize the concept of statistics and present new alternatives for the statistics of holons and spinons. The new alternatives maintain the benefits of the semionic model while avoiding its main problem (i.e. the prediction of T-reversal symmetry breaking). With this motivation, we will not be very specific about the precise microscopic origin of our model. For the sake of generality, we will only assume that - HTSC is described by the tJ model (or an extended version of it). - The ground state of the tJ model is an RVB state. - The parameters of the model are such as stabilize the flux phase. The first two assumptions are required to create localized excitations - the spinon and the holon, and the third assumption is required for the discussion of the microscopic origins of the model. Nevertheless, the effective Hamiltonian for the new alternatives is independent of the details of the model, and it is possible to justify the occurrence of the new statistics at least in some of the various detailed models. 30
27 December 1993
In section 2 of this paper we present the concept of non-identical anyons [ 26-28 ], by which we mean a generalized class of particles that includes both identical particles with a regular (and in general anyonic) statistics, and non-identical particles which obey mutual statistical behavior (under a double-exchange), a property of any non-identical particles with hard-core repulsion in two space dimensions. We show that the original notion of anyon (a composite of fictitious charge and flux-tube [ 17,18 ] ) is not sufficient to deal with non-identical particles, hence we must generalize Wilczek's realization in an appropriate way. The corresponding generalization of the braid group to colored braids [29] is briefly discussed. In section 3 we discuss T-reversal symmetry in anyonic models and we focus on some special cases in which non-identical anyons obey this symmetry [26,27 ]. The statistics of a composite of two types of anyons is calculated to find which of the various possibilities is suitable for our purpose, leaving only two non-trivial options. In section 4 we discuss the consequences of the two new alternatives for the statistics of the spinons and the holons. We present their effective Hamiltonian and discuss pairing in this model. I f we ignore spinstatistics restrictions for each type of particles then the new options are as legitimate as any other possibilities shown before, hence yielding a new starting point for further investigation. However, following refs. [ 8,9 ], we derive this statistical behaviour from microscopic considerations without violating the spin-statistics theorem, using the statistical transmutation only for one of the two types of particles (after showing that the electron is still a fermion in this case). In the last section we present our conclusions.
2.
Non-identical
anyons
The existence of identical particles obeying fractional statistics (under adiabatic exchange) is proven in two space dimensions for two particles with hardcore repulsion, using the fact that the path of full rotation in the configuration space •2× (~2_ origin)/ Z2 is not contractible. However, this fact leads to an additional (and certainly not less radical) conclu-
Volume 184,number I
PHYSICS LETTERSA
sion: A loop in the configuration space of two nonidentical particles, ~2× (~2_origin), is also uncontractible, hence rotating one particle around another particle (of a different type) multiplies the wavefunction by a general phase X,
27 December 1993
so that it interacts with all particles via AB [ 30 ] and AC [31 ] effects ~x. The Hamiltonian is
lffi X
~u(R, r, ~)=eiX¥(R, r, ~ + 2 n ) .
(I)
This new statistical phase is named "mutual statistics" [28,27 ] and the most important examples are z = 2 n x - the trivial mutual statistics, and X= ( 2 n + 1 ) x - the antisymmetric mutual statistics. Only the trivial mutual statistics exists in three dimensions. Consider a general system of N particles (in two dimensions) of M different types, N~ +... + ArM=N. Hence, in addition to the information about the identical particles, we have to consider the special properties of a wavefunction of any two non-identical particles. As we see later, by using a generalized form of the braid group, even exchanging two identical particles will depend on the number of particles of any other type which were wound by the path of exchange. A system composed of two or more different types of identical particles in two dimensions may exhibit "'anyonic" behavior even when each type is a boson or a fermion. Is Wilczek's realization of fictitious charge and flux-tube attached together applicable for our system of M types of non-identical anyons? Obviously it is not. Each individual particle type has its own fractional statistical phase Oi ( M free parameters) and each pair obeys mutual statistics with ½ M ( M - 1 ) free parameters ZiJ, altogether requiring ½( M 2 + M )
(2)
free parameters. Wilczek's classification contains only 2M free parameters, and hence must be generalized. Let us associate a fictitious charge e z with each particle according to its type I. In the anyon gauge this system is governed by a Hamiltonian of N free particles with a multivalued wavefunction that obeys the required boundary conditions (according to the parameters 0z and )~zJ). In the radial gauge the above system is transformed to a system of bosons with flux-tubes attached to them. However, each particle carries M flux-tubes corresponding to the M types of charges
if I
1
M
- =
E
C
2n
/ e i ~ t +e~q~J N~
/
c J = l,Jd:I \
' E
j= 1
E v ,j
j = ],j,,i
\q2~ /I
J
(3)
where • / is the flux-tube of type I carried by a particle of type J, and where V~0o=g× ( r ; - r j ) / I r~-rjl 2 The first term contains a summation over all particles of the same type and the second term contains a summation over all particles of all other types. Note that each charge interacts only with one type of fluxtube. The phase acquired by the wavefunction under exchange of two particles of the same type I is ~ e I~ z I Zn e -~oo ,
(4)
where ~o = hc/e is a fluxon. The phase acquired by the wavefunction under full rotation of non-identical particles of types I and J is ez ~ r eJ ~ 2n e ~ - o + e ~ - o } '
(5)
and this phase reduces to 2n[2(eZ/e)~I/q~o] when the two particles are identical (from which we reproduce the correct phase for exchange). Actually, this new notation has some redundancy (in comparison to the required number of parameters ( 2 ) ) , so let us choose e ~. . . . . e ~ = e (we could also fix the flux-tubes to be ~i = ¢~di but we prefer to keep this freedom since sometimes only particles of one type are attached to flux-tubes). While the statistics of identical particles in three space dimensions is represented by the permutation group, the statistics of hard core identical particles in two space dimensions is represented by the braid group [32]. The braid group of N strands, BN, is an infinite group which is generated by N - 1 elementary moves trl, ..., a~_ ~ satisfying #] For identical particles the AC effect leads to the same phase as the AB effect, hence multiplies the phase by a factor o f two, but this is not true for non-identical particles.
31
Volume 184, number 1
PHYSICS LETTERS A
{Ti{~i+lai=ai+l~i~i+l
for i = 1, ..., N - 2,
(6)
tricrj=trja~
for l i - j l >12
(7)
(tri is the generator of an exchange of the strand at site i with the strand at site i + 1 in an anti-clockwise direction, and its inverse is or;-~ ). The strands of non-identical anyons must have an additional index, say color, to indicate the type of the anyon. This group is still defined by eqs. (6) and (7) without refering to the indices of color. However, there are now "color limitations" on the allowed braids: The initial and the final configurations must have the same colors at all sites (i.e. the closure of the braids must not mix different colors [29] ), since a final configuration with some exchange of non-identical particles is not a closed loop in the configuration space. Hence, the colored braid is restricted by an additional condition: Let n (cri) czJ) be the number of operations of the type tr~ exchanging two strands of colors I and J in an anti-clockwise direction (before the replacement the particle of type I is on site i and the particle of type J is on site i + 1 ), and let a(tr;-~ )(zJ) be the number of operations of the type a;- ~ exchanging the same strands in a clockwise direction. The condition on
h} m = n( ai) ~iJ) + a( ai-~ )cis)
(8)
is that for each site i and color I M
E
J= I,J#Z
M
fi}m=
E
J= l,J#I
•}gz),
(9)
that is to say, strands of color I change places from the ith site to the ( i + 1 )st site and vice versa an equal number of times. This condition is necessary and sufficient to achieve the desired colors at the end. Let nCU)= X,-n (a,-) on) (sum over all sites) and the same with ~c/t), n (m, etc. The phase of these braidings is then
i~=l(OZ(n(ZZ)-n(zz))
+j~>1½Zij(n(ZS)+n(JZ)-n(zY)-n(SZ)) .
(10)
For example, exchange of two identical particles in a way that one particle of a second type is wound by 32
27 December 1993
the path of exchange is described by the braid tr}~A) tr~A) at As) (where we added the indices of color to the generators to make counting easier) and the resulting phase is OA+ZAS. The generalization of the braid group can have a natural use in knot theory if we consider a link which is made from few knots (closed loops) as a system of colored strands where each loop has a different color. This way the braid contains exactly the relevant information about the link.
3. Time-reversal symmetric anyonie model for HTSC After Laughlin [19,20] suggested that anyons might play an important role in explaining HTSC, it was generally agreed that the test of the model will be whether or not high-To superconductors obey Treversal symmetry. Later on it was shown that other models might also predict the breakdown of this symmetry, and on the other hand, anyons models that do not break this symmetry were formulated. However, T-reversal symmetry still remains a major issue. It can be seen immediately that identical anyons break T-reversal symmetry: suppose that two identical anyons are adiabatically exchanged in the clockwise direction leading to a statistical phase factor 0. The time reversed operation - exchanging them in the opposite direction - will lead to a phase - 0 which is different from 0 except for special cases of the fermions (0=re) or the bosons ( 0 = 2 n ) . Breakdown of T-reversal symmetry can be also proven using the antisymmetric nature of the Chern-Simons term if we examine the Lagrangian of an anyonic system in QFT. After a long debate and conflicting experimental results it is now accepted that high-To superconducting materials do not break T-reversal symmetry [24,25]. The dielectric tensor in a T-reversal symmetry breaking model has some antisymmetric terms, leading to "circular birefringence" which was found (experimentally) to be zero in high- T¢ superconductors. Semenoffand Weiss [33] and Wilczek [34] presented an anyonic system of two types of anyons with opposite statistical phases 01 and 02,
Volume 184,number 1
PHYSICS LETTERSA
o, = - 0 5
(II)
(and no mutual statistics).Their model is symmetricunder an "enhanced T-reversaloperation" that in addition to changing the directionof time also transforms all particlesof the firsttype into particlesof the second type and vice versa. This option can be relevant to two types of semions which have some similarity,such as up and down spinons. Nevertheless,it seems to be irrelevant for an anyonic model based on two distinctexcitationssuch as the spinons and the holons. We now present a T-reversal symmetric anyonic model that does not demand an "enhanced" T-reversal operation [26,27 ], and may be relevant to a system of holons and spinons. It uses the generalization of statistics for nonidentical anyons. Let us concentrate on the simplest case which seems to be most applicable in solid state systems - a system of two types as of anyons (generalization of the T-reversal symmetric model to M types is straightforward). Using eq. (3), the most general Hamiltonian for two species of free anyons A and B is A
NA
Hf
l--~---z., P i -
2 m A iffil
2.,
C j f f i l d ~ i 211;
e ~B O A + O ~, + Ck ~ 1
+--
2~ 2VB f
~2 V~ik
2e
Y~ ~pk-- c
2rob k ~ l
~ij
) NB ~
2,
OBB
-=-V~0k~
l=l,lvtk ~'~ 2
e N^ ~ + ~ A all" C i~l Ere
V~k'~j
(12)
where we used the redundancy of free parameters to fLx the charges eA----eB----e.Then the exchange phase of A is 0A=2XOA/O0, the exchange phase of B is 0e=2nOea/Oo and the mutual statistical phase is X=2x( o A + O~,)/Oo (recall that OA is a flux-tube of type A carded by particles of type B). In order that T-reversal symmetry would be preserved both types must be fermions or both must be #z A system o f two types o f a n y o n s can be described by the standard notation since Wilczek's realization [ 17,18 ] still provides enough parameters. Nevertheless, we stick to our new notation which provides a more realistic picture in which the three statistical phases are not necessarily related to each other.
27 December 1993
bosons. Nevertheless, this still leaves a non-trivial possibility for the T-reversal symmetric model: If the resulting phase under full rotations of non-identical particles is either n or -re (antisymmetric mutual statistics) T-reversal symmetry is not broken. This restricts the flux-tubes to satisfy: • A + ~ ~ = ½~b0. In general, our two-species anyonic system obeys T-reversal symmetry only when 0A= n 1g, 6~ = n2n and x=n3n with nl, n2 and n3 integers. We see that there are eight different possibilities for a T-reversal symmetric anyonic model (from which four are trivial), since each of the parameters nl, n2 and n3 can be either even or odd (phases are defined modulo 2n). Let us now apply the various alternatives to hoIons and spinons. An electron is composed of a spinon and a holon. Since the electron is a "real" (i.e. three-dimensional) object it is still a fermion. A composite made of the two non-identical anyons will have the statistical phase (for exchange) 0eom= 0A "t"0B -]-X •
(13)
(This is SObecause a full rotation can be sub-divided into four separate contributions: 20A, 20B, X, X.) With 0¢,,m= n, the number of different possibilities for a non-trivial T-reversal symmetric anyonic model now reduces to two: (1) Both the spinon and holon are bosons with antisymmetric mutual statistics. (2) Both the spinon and holon are fermions with antisymmetric mutual statistics. The number of possibilities of trivial mutual statistics also reduces to two, recovering the two standard approaches discussed in the introduction.
4. Consequences and origins of the T-reversal symmetric anyonic model The Hamiltonian of a T-reversal symmetric model of holons and spinons which are both bosons with antisymmetric mutual statistics is a simplified version of (12): 1
Nh ( P i - - e N,
+ ~Zmsk~l - - - - g~r° ( p k - - -e ~ o C i~l
2
V~k, )2 ,
(14)
33
Volume 184, number 1
PHYSICS LETTERS A
where h stands for holons and s for spinons, and where we used hs + ~ hs = ½~0 as before. In a case that the two types are fermions we can write them as bosons with an additional vector potential (i.e. additional flux-tubes) • ~ = • ] = ½~o. Standard arguments for anyonic superconductivity [ 20-22 ] and their generalization to non-identical anyons [ 27 ] are not applicable for this system of two completely distinct types of particles unless the densities of the two fluids oscillate together, which certainly is not the case here. In addition to this, those arguments are based on approximating the anyons to be fermions with small flux-tubes attached to them (then the gauge interaction can be replaced by an average magnetic field), but - as was already said by those authors and others - the validity of this approximation to semions is questionable. The same is true for the particles which are of concern here. Nevertheless, we believe that the topological interaction in this model may give rise to a pairing mechanism. Our system is analogous to a system of charges (with no Coulomb interaction) and halffluxons. Similar problems were solved as a function of flux in the AB effect, and we can make use of this knowledge. The following examples describe situations which are different from our system but still may give some indication for pairing (and possibility of superconductivity). In these examples we consider • ~, the flux-tube carried by the spinon, to be equal to half a fluxon and g~[ to be zero. Scattering problem. While there is a non-trivial scattering of a single holon on a single spinon (due to the AB effect) a pair of holons is moving as a free particle with no interaction at all, since all flux-tubes which are carried by the spinons are quantized in charge units of 2e. Approximating this problem to a one-dimensional scattering problem (taken from ref. [ 35 ] ), shows that the holon is "rejected" (i.e. scattered backwards) by a single spinon (see fig. 1 ), while a pair moves freely. Bound state problem. Compare the energy of two holons in a background of a free spinon as a function of the distance R between the holons. We discuss only the two limits of R ~ 0 and R-~oo. In these two limits we consider "closing" a spinon in a round box (twodimensional well with radial symmetry) and calculating the energy levels. When the holons' relative distance is zero their 34
scalterincj of one holon
27 December 1993
scattering of two holons
Fig. 1. Scattering of one holon versus scattering of two holons on a single spinon: while one holon is scattered backwards, two hoIons move freely ( ( 0 ) holon, ( × ) spinon).
charge is 2e hence the gauge potential ½qbo can be eliminated by a gauge transformation so the spinon behaves as a free particle. The term 12/r 2 (where r is the distance from the center of the box) contributes zero energy at the ground state (lz=0). When the distance is very large we consider interaction with one holon at a time, and neglect the interaction with the second holon which is far away (since the interaction is proportional to [rholon__rspinon[-l ). Hence, we close the spinon and (only) one holon in a box. We now approximate this problem to a different one which is trivial: Suppose that one holon is fixed at the center of the well, the wave function of the spinon in the presence of one holon at the center of the box is equivalent to that of an electron with half a fluxon at the center of the box in which the ground state energy levels are higher due to the centrifugal barrier [36] ( h = 1 ): lz (l= - ~ 1 ~ o ) 1 r-~=~ r2
(15)
This could account for an effective attractive "force" between the holons in the spinon background which is generated here by the presence of dynamical particles of the other type. Is there any advantage of one of the alternative models, with both holons and spinons being bosons versus the ease when both are fermions, over the other? These topological considerations equally account for both cases, but if both particles are bosons, the generation of pairs is easier due to an effective attractive interaction between two bosons. The ground state will not be a Bose condensate of single holons, but of pairs of holons, due to the mutual statistical interaction that prevents the free movement of a single particle, while it does not disturb the motion of a pair, as shown by the first example. Su-
Volume 184,number 1
PHYSICSLETTERSA
perfluidity may be generated here due to the (speculated) attractive interaction between two particles of the same kind as shown by the second example. Could these models prefer pairing of holons? This is a crucial point since if only spinons create pairs, the flux (which corresponds to the "real" electric charge) will be still quantized in units of hc/e. Hence the model must prefer pairing of holons or of both holons and spinons. To the extent that pairing is mediated by the mutual statistics, the model is symmetric (as for the spinons and the holons) so pairing of both holons and spinons is expected (and then a "spontaneous symmetry breaking" [37] might occur, so that only one type will form pairs). However, taking into account a specific microscopic model, it may be possible to obtain preference for pairing of holons. The derivation of the microscopic origins of this statistical behaviour is based on the statistical transmutation in the flux phase [ 8,9 ]. In both references the authors discuss the statistical transmutation of holons and spinons, and raise the possibility that only spinons transmute their statistics since holons unbind the half-fluxons at large kinetic energy t. The issue of statistics in this case is not discussed. Recall that the spin-statistics theorem implies that the bare holon is a boson, and a spinon with half a fluxon attached, is also a boson. We can therefore deduce that, in this case, the holons and spinons will be both bosons with additional mutual statistics n, hence yielding the desired realization of a T-reversal symmetric anyonic model: The resulting mutual statistics is due to the fact that the two species of particles, the hoIons and the spinons, interact with the same type of flux-tubes (i.e. they have the same type of statistical charge). The bare holon feels the flux-tube carried by the spinon, and therefore, under full rotation of the holon around the spinon (or vice versa), the wavefunction of the system will acquire a statistical phase n. The opposite alternative - that only holons bind the half-fluxons - will lead to the second case of interest since now both species will be fermions with mutual statistics n. We did not find a solid proposition for the derivation of this behaviour from microscopic considerations. However, in the "long range RVB" the spinons are gapless excitations hence created and annihilated as vacuum fluctuations from
27 December 1993
the "RVB vacuum", in which case it is impossible for them to bind the half-fluxons. This might be a good starting point for the search for this phenomenon. Note that an electron carries a fictitious charge of 2e, hence the creation of half-fluxons is a pure gauge transformation, and the electrons do not transmute their statistics when only spinons (or only holons) are attached to half-fluxons.
5. Conclusions In this paper we generalized the concept of anyonic behaviour to include mutual statistics between non-identical anyons. We discussed all the possibilities of a T-reversal symmetric anyonic model. We derived four possibilities for the statistics of holons and spinons, from which two are standard (i.e. show trivial anyonic behaviour) and have been thoroughly investigated in the literature. Two are new possibilities which are presented here for the first time. We described these new possibilities by a model Hamiltonian which is independent of the specific microscopic origins of the statistical behaviour. Finally we discussed pairing in this model and showed some possible microscopic origins for this statistical behaviour. Eventually, model dependent microscopic considerations may determine which among the two possibilities will be realized. We tried to present a model independent general case, and we leave this question open. We believe that the new options that are given here open interesting possibilities for further research, and should revive the possible role of anyons in HTSC, since T-reversal symmetry is not broken.
Acknowledgement The author is greatly indebted to Yakir Aharonov for providing the initial motivation for this work and for many valuable discussions. It is also a pleasure to thank Aharon Casher, Nissan Itzhaki, Sandu Popescu and Alexander Shnirrnan for very helpful discussions. The work was supported in part by grant 425/91-1 of the Basic Research Foundation (ad35
Volume 184, number 1
PHYSICS LETTERS A
ministered by the Israel Academy of Sciences and Humanities).
References [ 1 ] P.W. Anderson, Science 235 ( 1987 ) 1196. [2]G. Baskaran, Z. Zou and P.W. Anderson, Solid State Commun. 63 (1987) 973. [3] S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Phys. Rev. B 35 (1987) 8865. [4] P.W. Anderson, G. Baskaran, Z. Zou and T. Hsu, Phys. Rev. Lett. 58 (1987) 2790. [5] Z. Zou and P.W. Anderson, Phys. Rev. B 37 (1988) 627. [6] P.B. Wiegmann, Phys. Rev. Lett. 60 (1988) 821. [ 7 ] Y. Suzumuram, Y. Hasegawa and H. Fukuyama, J. Phys. Soc. Japan 57 (1988) 2768. [8] S. Kivelson, Phys. Rev. B 39 (1989) 259. [ 9 ] N. Read and B. Chakraborty, Phys. Rev. B 40 ( 1989 ) 7133. [10] F.D.M. Haldane and H. Levine, Phys. Rev. B 40 (1989) 7340. [ 11 ] C.L. Kane, P.A. Lee and N. Read, Phys. Rev. B 39 (1989) 6880. [12] D. Yoshioka, J. Phys. Soc. Japan 58 (1989) 32. [ 13] I. Affleck and B.J. Marston, Phys. Rev. B 37 (1988) 3744. [ 14 ] I. Affleck and B.J. Marston, Phys. Rev. B 39 (1989) 11538. [ 15 ] G. Kotlier, Phys. Rev. B 37 (1988) 3664. [ 16 ] J.M. Leinaas and J. Myrheim, Nuovo Cimento 37B (1977) 1.
36
27 December 1993
[ 17] F. Wilczek, Phys. Rev. Lett. 48 (1982) 1144. [18] F. Wilczek, Phys. Rev. Lett. 49 (1982) 957. [ 19] R.B. Laughlin, Science 242 ( 1988 ) 525. [20] R.B. Laughlin, Phys. Rev. Lett. 60 (1988) 2677. [21 ] Y. Chert, F. Wilczek, E. Witten and B.I. Halperin, Int. J. Mod. Phys. B 3 (1989) 1001. [22] A.L. Fetter, C.B. Hanna and R.B. Laughlin, Phys. Rev. B 39 (1989) 9679. [23] X.G. Wen, F. Wilczek and A. Zee, Phys. Rev. B 39 (1989) 11413. [24] S. Spielman et al., Phys. Rev. Lett. 65 (1990) 123. [25 ] T.W. Lawrence, A. Szoke and R.B. Laughlin, Phys. Rev. Lett. 69 (1992) 1439. [ 26 ] Y. Aharonov, private communication ( 1991 ). [27] F. Wilczek, Phys. Rev. Lett. 69 (1992) 132. [ 28 ] C.R. Hu, Int. J. Mod. Phys. B 5 ( 1991 ) 1771. [ 29] L. Brckke, Phys. Lett. B 271 (1991 ) 73. [ 30 ] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [31] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53 (1984) 319. [32 ] Y.S. Wu, Phys. Rev. Lett. 52 (1984) 2103. [ 33 ] G.W. Semenoff and N. Weiss, Phys. Lett. B 250 (1990) 117. [ 34 ] F. Wilczek, Fractional statistics and anyon superconductivity (World Scientific, Singapore, 1990) p. 16. [ 35 ] A.G. Aronov and Yu.V. Sharvin, Rev. Mod. Phys. 59 ( 1987 ) 755. [36] M. Peshkin, Phys. Rep. 80 (1981) 375. [ 37 ] S. Popeseu, private communication (1992).