On the possibility of a statistics-changing phase transition

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1 On the possibility of a statistics-changing phase transition X.G. Wen, A. Zee To cite this version: X.G. Wen, A. Zee. On the possibility of a statistics-changing phase transition. Journal de Physique, 1989, 50 (13), pp < /jphys: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1989 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Nous J. Phys. France 50 (1989) er JUILLET 1989, 1623 Classification Physics Abstracts d On the possibility of a statistics-changing phase transition X. G. Wen and A. Zee Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, U.S.A. (Reçu le 17 janvier 1989, accepté le 30 mars 1989) 2014 Résumé. discutons quelques propriétés surprenantes qu aurait une transition de phase changeant la statistique. Abstract We discuss some of the puzzling features associated with the possibility of a statisticschanging phase transition. The physics underlying the possibility of having fractional statistics in (2 + 1 ) dimensional spacetime [1-3] is exceedingly simple. We begin with a brief review. Given a Lagrangian Co with a conserved current Ju, we can always introduce additional gauge interactions [4] according to (Note that Au is not to be identified with the electromagnetic gauge potential. It is simply a gauge potential responsible for generating fractional statistics.) To define a, we assume that Ju, is suitably normalized. The gauge potential is then determined by Let there be an object which carries qo units of the charge qo d2x JO. Take the object as sitting at rest. Far away from this object F12 0 according to equation (2) and so the gauge potential A. is a pure gauge. However, it must be topologically nontrivial since [4] where C is a contour encircling the object. Thus, the gauge potential Article published online by EDP Sciences and available at

3 1624 where Q is the azimuthal angle around the object. If we now move another object of charge q 0 0 slowly around this object, a phase proportional to dx Ai is induced in the wave c function. This phase is the origin of fractional statistics. To calculate this phase, we can simply integrate out Au (in the Landau gauge, say) and obtain an effective non-local action S fd3x Co + H, where We now evaluate this expression for with the particle 2 sitting at rest x02 (T ) T, x2 0 and particle 1 going around particle 2. If we allow particle 1 to go half way around particle 2 and then apply a suitable translation, we would have in effect exchanged the two particles. Quantum statistics is identified as the net phase in eis remaining even when the two particles are arbitrarily far from each other. Since we are interested in the effect due to the joint presence of both particles, we can simply calculate the term proportional to qo q in H. We find that it is equal to qo q 18 a when particle 1 goes half way around particle 2. Thus, if qo q 0 1, we have the phase factor ei/8 a. This is conventionally defined as eio. Thus we have 03C0 (21k + 1) with k an integer, these objects, if they In particular, for a 8 0 7T 2 k + 1 ) the Lagrangian 0, would behave as fermions, and vicie versa. Physically, the charged particle is «dressed» by its coupling to the gauge potential and carries with it a long ranged and topologically non-trivial Ai. Strictly speaking, H should be regulated, as had been discussed recently [5, 6]. One approach would be to «spread out» the delta function in equation (6). In particular, we can take [2] J IL (x ) to be the topological current of an SO(3) nonlinear o-model and then H would be the Hopf invariant discussed in references [2, 7]. The value of H can then be determined by relating the Hopf invariant to the linking number [2] between the trajectories of particle 1 and particle 2. In the above we have calculated directly the phase associated with the exchange of two particles. It is also instructive to determine this phase by calculating separately the interaction of one particle with the gauge potential of the other, along the line suggested by equation (3) and equation (4). We define a total current J tot as the coefficient of auf, action under the change Au -> Au + 8,,f. Thus we have were bosonic in in the variation of the The difference between Jutot and Ju, may be thought of as the effect of vacuum polarization due to the presence of the charged object. Another way of seeing the necessity of the second term in equation (8) is simply that according to the last term in equation (1) the gauge potential

4 1625 generated by one particle interacts also with the gauge potential generated by the other particle. The total charge of the object is then Effectively the gauge field also carries charge and the total charge is equal to 1/2 of the bare charge qo. We must keep track of this factor of 2 lest we run into potential confusion. We now calculate the phase factor induced in the wave function when an object of total charge q moves slowly around this object of total charge q. The contribution due to the charge q moving around in the gauge potential generated by the charge q (with Ao 0 and Ai given as in equation (4)) is given by We have to multiply the phase by two to take into account the effect of gauge potential generated by the moving charge q on the charge q. Thus, the actual phase factor is eiqóqo/4 a. Let qo q 0 2 q 1 as before. Under interchange of the two objects (half-way around the contour), we have the phase factor e i/8 a. This agrees with our preceding calculation. If we want to regard particles carrying fractional statistics as particles with charge q and flux 4J, so that the phase factor when a particle with charge and flux equal to (q, 4J) goes around another with (q, 0 ) is equal to q 0 + q 4J, then charge must be defined as in equation (9). The argument is completely general and goes through regardless of whether J IL is the current of a point particle, the U(1 ) current of a field theory, or the topological current of a theory with solitons. Thus, the Lagrangian should describe a theory of particle with fractional statistics. Similarly, for aq 1/8 TT with D u ~ a, u~ - ia u. ~ should describe a theory of fermions. Note that we can absorb a ~1/2 in Au, and we see that the coupling e a ~- 1/2 is in the strong coupling regime for a. 1/8 w. Note that for equation (12) the relevant conserved current is the gauge invariant current (~ a, 0 - a u ~ t ~ ). The and not the non-conserved gauge non-invariant current ju i particles with fractional statistics are those carrying the gauge invariant current. The transmutation of fermion into boson is possible because a massive fermion in (2 + 1 ) dimensions has only one spin degree of freedom. In general, the fermion mass term [4] violates P and T. In this paper we would like to suggest the possibility of a novel statistics-changing phase transition. When 0 acquires a non-zero vacuum expectation value v in 4,, writing ~ (x) (v + h (x ) ) e in (x), we have

5 1626 where W IL A IL + 8,q and W IL v a IL W v - a v W IL. gauge boson has a propagator given in momentum space by The gauge invariance is broken. The It has become massive, with mass squared v 4 /4 a 2. The theory is now described by a massive scalar boson coupled to a massive vector boson. Since the vector boson has a short ranged interaction, it can not alter the statistics of the scalar boson. By definition, quantum statistics is associated with the exchange of two particles in the limit when the two particles are infinitely far apart. We can imagine the phase transition occurring at some temperature Tc. As the temperature is lowered, the system would show a strange behavior, changing abruptly from a Fermi gas (for a 1/8 TT) to a gas of a scalar and a vector boson. Note that the number of degrees of freedom is conserved : we can verify that the gauge boson, though massive, has only one spin degree of freedom. It appears quite puzzling, but evidently some linear combinations of the fermion and antifermion fields have become the scalar and vector boson fields. From the point of view of renormalizable field theory, we should have introduced a potential V (0 ) m K ( ~ )4 + À 1 cf> 16. The phase transition could occur for m 2 negative or for m2 positive but K negative and large. In the latter case, even at zero temperature there could be a metastable phase (0) > 0) with massive fermions and a stable phase ( (0 ) # 0 ) with massive bosons. To get a better understanding about the relation between and C~ we may add a term Ju Au to,03c8 where Ju is the current of a test charge. After integrating out the fermion field [8, 9] we obtain for m large Interchanging two identical unit test charges, we get a phase factor where e depends on the statistics of the «bare» test charge (03BE + 1 for bosons, - 1 for fermions and e" in general). If we had N fermion fields in 03C8 (with the same sign mass), the expression in the rounded parenthesis in equation (17) is modified to a - N m. Thus, even when «0, the test charge g would still have fractional 16 03C0 m statistics if N> 3. Similarly, if we add Ju A, to Q with fil the current of a test charge, the test charge would acquire a statistics governed by the phase factor e e i/8 O!q,. Thus, we expect that at low energies (compared to m) or at large distances Q and,03c8 are equivalent when The introduction of a test particle underlines the distinction between the phase transition considered and more «standard» phase transitions. In a standard phase transition, the relevant quasiparticle may have different characters in different phases. Here, however, the

6 1627 physical character of test particles also changes through the phase transition. This point is also made clear by considering the model Lagrangian Note equation (19) has a conserved charge, called ~ charge, associated with a global symmetry ~-> ei8 ~. Above the phase transition temperature (T> Tc), cf> > 0 and a quasi-particle with unit ~ charge has fractional statistics determined by ak. Below the phase transition, température (~) v and the quasi-particle with unit ~ charge behaves as an ordinary boson. In our discussion, we have put in the Chern-Simon term [4, 8] ILVÀ A,uFvA by hand. It could also be induced by coupling to a fermion field as mentioned above. Thus, if instead of the Chern-Simon term we add to equation (11) the interaction then after integrating out q the effective term appears. The coefficient is, however, temperature dependent. At finite temperature [9], the coefficient in (21) is to be multiplied by tanh ( 1 2 M T 1 ). As the tem erature T tends to infinity, the Chern-Simon term disappears. This suggests an even stranger picture. After the phase transition from bosons to fermions at some finite temperature, at even higher temperatures (on a scale determined by M) the fermions could gradually change back into bosons. In our discussion, we have assumed that the current is normalized so that the «elementary excitation» of the theory has unit charge. With gauge symmetry breaking, other excitations are possible in 0. In particular, for the Nielsen-Olesen vortex [10] with 0 (x) > V ei cp as x - oo, finite energy considerations force the gauge potential Ai to be equal to ai6 as x - oo rather than as given in equation (4). Their charges are forced to be equal to according to equation (3). Thus, when we interchange two vortices, we obtain a phase of ei 8 ff2 a. For a 8 ir (2 1 k this phase is equal to eiffl(2 k +1). Except for k 0 they do not 8 03C0 (2 k+1 ) act like fermions. Note that the 0 parameter of the vortex is Finally, following Coleman and Weinberg, we calculate the one-loop correction V1(~) to the potential due to the fluctuations in the gauge fields by using Cartesian coordinates 0 Q1 + i cp 2 and setting Q2) 0 and Q 1) v. By scaling cp cp /A 1/4 we 1 i 1 see that the one loop correction is an expansion in À 2 if K À - 2 and a À 2 are fixed. This corresponds to the large a limit. We fix the gauge by adding the term y (aa )2 to equation (12)

7 1628 (i a -1 êp,và qà + y-1 qi-l qv/q2)/q2. so that the gauge boson propagator is Both the Au2O/1 and the Au, ~1 a u, ~2 vertices enter. We find, after rotating to Euclidean space Assuming m 2 K 0, we find that for small ~ at zero temperature T 0, where A is the cut-off. After subtracting the counter terms, we obtain,. 1 where a and b are undetermined constants. Therefore if we choose Aa + ba -2 symmetry breaking at zero temperature. At high temperature 0 there is a We see the symmetry is restored. It is far from clear whether the considerations presented here will remain a field theoretic phenomenon or will be of relevance to real condensed matter systems [11]. For one thing, the circumstances that may allow a condensed matter system to be described by equations (11) or (12) may no longer hold at high temperatures. Acknowledgments. We thank J. R. Schrieffer, M. Srenicki and F. Wilczek for helpful conversations. This research was supported in part by the National Science Foundation under Grant No. PHY , supplemented by funds from the National Aeronautics and Space Administration, at the University of California at Santa Barbara. Note added in proof The possibility of statistics-changing phase transition may be realized in the order-disorder transition associated with high temperature superconductivity. (See X. G. Wen and A. Zee, Phys. Rev. Lett. 62 (1989) 1937, and Santa Barbara ITP preprint NSF-ITP ) References [1] WILCZEK F., Phys. Rev. Lett. 48 (1982) 1144 ; 49 (1982) 957. [2] WILCZEK F. and ZEE A., Phys. Rev. Lett. 51 (1984) 2250 ; Santa Barbara ITP Preprint (1984). [3] AROVAS D. P., SCHRIEFFER R., WILCZEK F. and ZEE A., Nucl. Phys. B 251 [FS13] (1985) 117. [4] DESER S., JACKIW R. and TEMPLETON S., Phys. Rev. Lett. 48 (1982) 975 ; Ann. Phys. N.Y. 140 (1982) 372 ; SCHONFELD J., Nucl. Phys. B 185 (1981) 157. [5] TZE H. C. and NAM S., Phys. Lett. 210B (1988) 76.

8 1629 [6] Wu Y. S. and ZEE A., Phys. Lett. 207B (1988) 39. [7] Wu Y. S. and ZEE A., Phys. Lett. 147B (1984) 325. [8] REDLICH A. N., Phys. Rev. D 29 (1984) [9] BABU K. S., DAS A. and PANIGRAHI P., Phys. Rev. D 36 (1987) [10] PAUL S. K. and KHARE A., Phys. Lett. B 174 (1986) 420 ; B 193 (1987) 253 ; DE VEGA H. J. and SCHAPOSNIK F. A., Phys. Rev. Lett. 56 (1986) 2564 ; One of us (X.G.W.) understands that similar observations have been made by S. KIVELSON and S. C. ZHANG. [11] SEMENOFF G. W. and SODANO P., Phys. Rev. Lett. 57 (1986) 1195 ; and references therein.