Vortex unbinding transitions and electrons in disordered systems

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VORTEX U N B I N D I N G T R A N S I T I O N S AND ELECTRONS IN D I S O R D E R E D SYSTEMS

D.J. THOULESS C a v e n d i s h Laboratory, M a d i n g l e y Road, Cambridge C B 3 0 H E , England and Dept. of Physics FM-15, University of Washington, Seattle, WA98195, U%A It is a great honour for me to be selected to receive the Fritz L o n d o n Award. London died w h i l e I was still an u n d e r g r a d u a t e , but his b o o k on s u p e r f l u i d s has b e e n a continual source of insight for me ever since I first turned my a t t e n t i o n to low t e m p e r a t u r e physics twenty five years ago. The analogies and d i f f e r e n c e s between superconductivity and superfluidity w e r e u n d e r s t o o d and explained by London, and he showed h o w far it was p o s s i b l e to go on the basis of the existence of a condensate wave function. This a p p r o a c h to p r o b l e m s in physics, w i t h conclusions drawn by manipulating concepts w h o s e fundamental basis is only poorly if at all understood, was one that I found difficult to accept w h e n I first met it. Perhaps my unease w i t h this approach explains in part why the w o r k for w h i c h I am b e i n g honoured today was left in an embryonic state by me, and the full p o s s i b i l i t i e s of it w e r e only seen by others. What I shall be talking about today is not the w o r k as I did it myself, but as it was developed into a form w h e r e it was possible to compare w i t h experiment and to open up new areas of low t e m p e r a t u r e experimental research. 1 am going to give a brief summary of two topics. The first is the theory of transitions such as the v o r t e x u n b i n d i n g transition in helium films. This I worked on in c o l l a b o r a t i o n w i t h M i c h a e l K o s t e r l i t z ( l , 2 ) , and it was a n a t u r a l g e n e r a l i z a t i o n of w o r k on a similar t r a n s i t i o n in a one d i m e n s i o n a l Ising m o d e l with long range interactions(3). M a n y of our ideas w e r e a n t i c i p a t e d by Berezinskii(4,5). A recent review of this has been given by Nelson(6), who, in c o l l a b o r a t i o n w i t h H a l p e r i n and others, has done so m u c h to develop the theory. The other topic I want to discuss is the theory of the l o c a l i z a t i o n of electrons in d i s o r d e r e d systems. H e n c e my w o r k d e v e l o p e d from m u c h earlier w o r k by Philip Anderson(7) and Sir N e v i l l Mott(8,9), and 1 had the benefit of n u m e r o u s d i s c u s s i o n s w i t h both of them, and I w o r k e d at one time in close c o l l a b o r a t i o n w i t h Don Licciardello. It seemed that this subject would became manageable when the analogies with critical phenomena were exploited. I really only h e l p e d to lay the foundations of this, and successful scaling

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theories were developed and by Wegner(ll).

by

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In the v o r t e x u n b i n d i n g theory of the superfluid - normal fluid t r a n s i t i o n (1,2,4,5) it is assumed that there is some condensate in w h i c h vortices, w h i c h are singularities of the condensate around w h i c h the phase of the wave function changes by 24, can exist. At low temperatures isolated v o r t i c e s are unstable, but v o r t e x pairs can exist in thermodynamic e q u i l i b r i u m at any temperature. Because both the energy and entropy of an isolated vortex diverge l o g a r i t h m i c a l l y with the size of the s y s t e m there is a critical temperature.

c27-ff B above which vortex pairs dissociate s p o n t a n e o u s l y and isolated vortices are stable. The s u p e r f l u i d . d e n s i t y jumps abruptly to zero at this temperature unless something else intervenes at a lower temperature. The value of p in this equation is the measured ~> , even s on an inhomogeneous substrate, because s it is the long range b e h a v i o u r that determines the strength of the logarithmic dependence on size. Real experiments involve finite sizes and finite m e a s u r i n g times, and this results in an upward shift of the observed transition temperature as c a l c u l a t e d bv A m b e g a o k a r et al(12). ] show the results of Bishop and Reppy(13) for the superfluid density, but many other similar results have been obtained. There were already third sound m e a s u r e m e n t by R u d n i c k et al(14) w h i c h we were unaware nf when we w o r k e d out the theory, and there have been several m e a s u r e m e n t s on $He diluted by 3He(15). Many other possible a p p l i c a t i o n s of these ideas have been suggested, but there Js no case w h e r e the experimental evidence is as strong as it is for superfluid helium films. We initially thought that the transition could not occur in s u p e r c o n d u c t i n g films(2), because the finite penetration depth would lead to a finite, rather than l o g a r i t h m i c a l l y divergent, energy of flux lines. Beasley et al(]6) pointed out that in a very thin film the p e n e t r a t i o n depth is so large that it is of no practical importance the cut-off it introduces in the logarithm may be less

D.J. Thouless / Vortex unbinding transitions

important than the cut-off introduced by finite sample size or finite measuring time. Recent work seems to show a transition in superconducting thin films driven by flux line pair unbindlng(17,18). Another intriguing example of a possible transition is the melting of two-dimensional solids. Here it is the dislocation that plays the role played by the vortex in a superfluid. Dislocations have an energy that depends logarithmically on the size of the system, although the coefficient in front of the logarithm is not simply proportional to the rigidity modulus, but depends also on the Poisson ratio. The fact that the dislocation has a directional property (the direction of the Burgers vector) and is not a pure scalar is an additional complication. Dislocation pairs modify the rigidity modulus just as vortex pairs modify the superfluid density in a helium film. The detailed theory of this transition was worked out by Halperin and Nelson(19) and by Young(20). If the dislocation unbinding transition occurs it leads to a transition to a state in which free dislocations migrate in response to shear stresses, so that the phase is viscous rather than rigid, but do not destroy orientational order. This is known as the hexatic phase, and is a sort of liquid crystal. In this phase isolated disclinatlons have a logarithmically divergent energy - a dislocation can be regarded as a close pair of opposite disclinations. At some higher temperature a discllnation unbinding transition to the unoriented fluid phase should occur. Unfortunately it seems that this beautiful picture does not correspond to reality in simple solids. Adsorbed gases which have a low temperature solid phase incommensurate with the substrate mostly seem to have a first order melting transition. Numerical simulations of two dimensional solids have shown a solid phase which appears to be stable up to the theoretical dislocation unbinding transition, but there is then an abrupt transition to a two phase system of lower free energy(21). Not only is it clear that the dislocation unbinding temperature is above the true equilibrium melting temperature, but the high temperature phase seems to be a true fluid rather than a hexatlc fluid. It seems that the dislocation unbinding has something to do with the transition, but something else intervenes to give a first order transition. One suggestion that has been made by Chin(22) and a number of other people is that the melting transition may be due to spontaneous formation of small angle grain boundaries, which can be regarded as a row of dislocations of the same sign.

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One candidate still remains for the dislocation unbinding transition. It is known that the two dimensional Coulomb solid melts at a temperature close to the dislocation unbinding temperature(23,24), but it is still not established whether it has the expected continuous transition or is a first order transition. Guo et ai(25) have studied the response of electrons trapped on helium to an ac electric field. They find an adsorption signal at melting very similar to the signal seen from superfluid helium films, and a frequency dependence of the apparent melting temperature which is in good agreement with the form expected from a dislocation unbinding transition. Recent work has put unbinding transitions on basis(26).

the theory of vortex a sound mathematical

The theory of localization of electrons by a random potential was first developed more than twenty-five years ago. Anderson(7) made a detailed study of a tight binding model of an electron in which the site energies were spread over a range W. He showed that if the ratio of W to the matrix element V for hops of the electron from one site to a neighbouring one was sufficiently large the electron eigenstates would be localized. Each wave function would have a maximum at a different point in the system, and fall off exponentially from that maximum. There should be a sharp transition, as the disorder W is reduced, between localized states and extended states, and as the transition is approached the localization length should diverge. Shortly afterwards Mott and Twose(8), and Landauer(27) argued that in a disordered one-dimenslonal system all eigenstates should be localized however weak the disorder. For fifteen years very little progress was made in understanding these theories at a fundamental level or in combining them, but some important results were obtained ten or twelve years ago by using methods developed for the theory of critical phenomena. One idea was that of block scaling. Blocks of random material of size L can be characterized by two parameters, V(L), the strength of the coupling between wave functions on one block and on another, and the mismatch in energy W(L) between an energy level on one block and the nearest level on a neighbourlng block; W(L) is essentially the spacing between l e v i s in a block and is proportional to L in d dimensions(28). Just as localization in the original Anderson model depended on W/V, so localization in the rescaled model depends on W(L)/V(L). This idea led Abrahams et al(10) to the scaling relation which can be written as

D.J. Thouless / Vortex ttnb#lding tratlsitioHs

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V(2L)=V(L)fd(V(L)/W(L)), W(2L)=2-dw(L)

(2)

Another important ingredient of this scaling theory is that V(L)/W(L) is directly related to the conductance of a block of size L, so that the scaling relation can be rewritten as a scaling relation for the conductance. In fact we got the expression for the conductance wrong(29,30) it is not proportional to V/W but to V2/W 2. From this scaling relation dlng/dlne=Bd(g),

(3)

dependence because the inelastic scattering length is smaller than the localization length, but at lower temperatures the electrons are trapped within a localization length and only diffuse further because they are freed by inelastic scattering. Experiments by Bergman(36) and by White et ai(37) on twodimensional films have shown that where ordinary scattering dominates the resistance increases as the temperature or magnetic field is lowered (thus increasing the inelastic scattering length or magnetic length), but where spin-orbit scattering dominates the resistance decreases with decreasing field or temperature.

where g(L) is the conductance of a block divided by e2/h and ~d is a function of g that depends on the number of space dimensions d, and from a knowledge of the limiting form of BN(g) for strongly localized systems (g÷o) and w~akly disordered systems (g÷~) several important conclusions were drawn(10). In one and two dimensions all states are localized for real (time reversal invariant) scatterers. In three dimensions there is a mobility edge, and the resistivity goes to infinity as the mobility edge is approached from the metallic side - in fact the resistivity times e2/h plays the role of the correlation length on the metallic side. In a two dimensional system with spin-orbit coupling the conductance increases logarithmically with length scale if the disorder is weak(31).

In recent years this topic has also been the subject of mathematically rigorous investigation, and a number of results originally obtained by heuristic arguments have now been confirmed(38,39).

At first sight localization theory appears to present a number of paradoxical results. For example the resistance of a wire should increase exponentially with length rather than linearly, once its resistance is above 26 kOhms, and the resistance of a square should increase logarithmically with its size. It must be realized that this is a zero temperature theory, and that one of the most important effects of nonzero temperature is to cause the electrons to change their energy so that the phase coherence effects that produce localization are destroyed. These inelastic scattering events, such as electron-phonon or electron-electron scattering, give a limiting time, or a corresponding limiting length scale, beyond which localization theory is not applicable(32,33). The effects of magnetic field can be understood in the same way beyond the radius of the lowest quantized cyclotron orbit the magnetic field dephases the wave function and destroys locallzation(31,34). There have now been many observations of localization made in this manner, particularly in one and two dimensional systems. For example, recent work by Dean and Pepper(35) shows one-dimensional localization in a MOS device with a constricted channel. At higher temperatures there is a weak temperature

(6)

References (I) (2) (3) (4)

(5)

(7) (8) (9)

(10)

(II) (12) (13)

(14) (15) (16) (17) (18)

J.M. Kosterl.itz and D.J. Thouless, J.Phys. C 5 (1972) L124 J M Kosterlitz and D.J. Thouless, .T.Phys.C 6 (1973) 1181 D.J. Thouless, Phys. Rev. 187 (1969) 732 V.L. Berezinskii, Zh. Eksp. Teor. Fiz. 59 (1970) 907 [translation in Soviet Phys. JETP 32 (1971) 493] V.L. Berezinskii, Zh. Eksp. Teor. Fiz. 61 (1971) 1144 [translation in Soviet Phys. JETP 34 (1972) 610] D.R. Nelson in Phase Transitions and Critical Phenomena, Vol.7, ed. C. Domb and J.L. Lebowitz (Academic Press, London, 1983) pp 1-99 P.W. Anderson, Phys.Rev. 109 (1958) 1492 N.F. Mott and W.D. Twose, Adv. Phys. I0 (1960) 107 See N.F. Mott and E. Davis, "Electronic Processes in Non-Crystalline Materials" (Oxford University Press, 1979) E.A. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishman, Phys.Rev.Lett. 42 (1979) 673 F. Wegner, Z.f.Phys. B35 (1979) 207 V. Ambegaokar, B.I. Halperin, D.R. Nelson and E.D. Siggia, Phys. Rev. B21 (1980) 1806 D.J. Bishop and J. Reppy, Phys. Rev. Lett. 40 (1978) 1727 I. Rudnick, R.S. Kagiwada, J.C. Fraser and E. Guyon, Phys.Rev.Lett. 20 (1968) 430 E. Webster, M.Chester, G.D.L. Webster and T. Oestereich, Phys.Rev. B22 (1980) 5186 M.R. Beasley, J.E. Mooij and T.P. Orlando, Phys.Rev.Lett. 42 (1979) 1165 K. Epstein, A.M. Goldman and A.M. Kadin, Phys. Rev.Lett. 47 (1981) 534 A.F. Hebard and A.T. Fiory, Phys.Rev. Lett. 50 (1983) 1603

D.J. Thouless / Vortex unbinding transitions

(19) B.I. Halperln and D.R. Nelson, Phys.Rev.Lett. 41 (1978) 121 and 519 (20) A.P. Young, Phys.Rev. BI9 (1979) 1855 (21) F.F. Abraham, Phys.Rev.Lett. 44(1980) 463 (22) S.T. Chui, Phys.Rev.Lett. 48 (1982) 933 (23) D.J. Thouless, J.Phys.C. Ii (1978) L189 (24) C.C. Grimes and G Adams, Phys.Rev.Lett. 42 (1979) 795 (25) C.J. Guo, D.B. Mast, Y-Z. Ruan, M.A. Stan, and A.J. Dahm, Phys.Rev.Lett. 51 (1983) 1461 (26) J. FrShllch and T. Spencer in "Mathematical Problems in Theoretical Physics. Proceedings of the Sixth International Conference on Mathematical Physics, Berlin 1981" ed. R. Schrader, R. Seller and D.A. Uhlenbrock (Springer-Verlag, Berlin, 1982) (27) Landauer's work was published much later in R. Landauer, Phil. Mag. 21 (1970) 863 (28) J.T. Edwards and D.J. Thouless, J.Phys. C5 (1982) 807 (29) D.C. Licclardello and D.J. Thouless, Phys.Rev.Lett. 35 (1975) 1475 (30) P.W. Anderson and P.A. Lee, Progr.Theor.Phys.Suppl. (Japan) 69 (1980) 212 (31) S. Hikami, A.I. Larkin and Y. Nagaoka' Progr.Theor.Phys. (Japan) 63 (1980) 5142 (32) D.J. Thouless, Phys.Rev.Lett. 39(1977) 1167 (33) D.J. Thouless, Solid St. Commun. 34 (1980) 683 (34) L.P. Gorkov, A.I. Larkln and D. Khmel'nltskil, Pis'ma Zh. Eksp.Teor. Fiz. 30 (1979) 248 [translation in JETP Lett. 30 (1979) 228] (35) C.C. Dean and M. Pepper, preprint (36) G. Bergman, Phys.Rev. B28 (1983) 513 (37) A.E. White, R.C. Dynes and J.P. Garno, Phys.Rev. B29 (1984) 3694 (38) H. Kunz and B. Souillard, Commun.Math.Phys. 78 (1980) 201 (39) J. FrBhllch and T. Spencer, Phys.Repts. 103 (1984) 9

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