Revival of the Kondo effect

Transcription

1 Nnotechnology hs rekindled interest in the Kondo effect, one of the most widely studied phenomen in condensed-mtter physics Revivl of the Kondo effect WHY would nyone still wnt to study physicl phenomenon tht ws discovered in the 1930s, explined in the 1960s nd hs een the suject of numerous reviews since the 1970s? Although the Kondo effect is well known nd widely studied phenomenon in condensedmtter physics, it continues to cpture the imgintion of experimentlists nd theorists like. The effect rises from the interctions etween single mgnetic tom, such s colt, nd the mny electrons in n otherwise non-mgnetic metl. Such n impurity typiclly hs n intrinsic ngulr momentum or spin tht intercts with the electrons. As result, the mthemticl description of the system is difficult mny-ody prolem. However, the Kondo prolem is well defined, mking it n ttrctive testing ground for the new numericl nd nlyticl tools tht hve een developed to ttck other chllenging mny-ody prolems. Interest in the Kondo effect hs therefore persisted ecuse it provides clues to understnding the electronic properties of wide vriety of mterils where the interctions etween electrons re prticulrly strong, for instnce in hevy-fermion mterils nd hightemperture superconductors. Physicists fscintion with the phenomenon hs continued since it ws first explined y Jpnese theorist Jun Kondo in In fct, interest in the Kondo effect hs recently peked thnks to new experimentl techniques from the rpidly developing field of nnotechnology, which hve given us unprecedented control over Kondo systems. Electron trnsport t low tempertures The electricl resistnce of pure metl usully drops s its temperture is lowered ecuse electrons cn trvel through metllic crystl more esily when the virtions of the toms re smll. However, the resistnce sturtes s the temperture is lowered elow out 10 K due to sttic defects in the mteril (figure 1). Some metls for exmple led, nioium nd luminium cn suddenly lose ll their resistnce to electricl current nd ecome superconducting. This phse trnsition from conducting to superconducting stte occurs t so-clled criticl temperture, elow which the electrons Leo Kouwenhoven nd Leonid Glzmn The theory tht descries the scttering of electrons from loclized mgnetic impurity ws initited y the work of Jun Kondo in 1964 ehve s single entity. Indeed, superconductivity is prime exmple of mny-electron phenomenon. Other metls, like copper nd gold, remin conducting nd hve constnt finite resistnce, even t the lowest ccessile tempertures. The vlue of the low-temperture resistnce depends on the numer of defects in the mteril. Adding defects increses the vlue of this sturtion resistnce ut the chrcter of the temperture dependence remins the sme. However, this ehviour chnges drmticlly when mgnetic toms, such s colt, re dded. Rther thn sturting, the electricl resistnce increses s the temperture is lowered further. Although this ehviour does not involve phse trnsition, the so-clled Kondo temperture roughly speking the temperture t which the resistnce strts to increse gin completely determines the low-temperture electronic properties of the mteril. Since the 1930s there hve een mny oservtions of n nomlous increse in the resistnce of metls t low temperture. Yet it took until 1964 for stisfctory explntion to pper. Electricl resistnce is relted to the mount of ck scttering from defects, which hinders the motion of the electrons through the crystl. Theorists cn redily clculte the proility with which n electron will e scttered when the defect is smll. However, for lrger defects, the clcultion cn only e performed using perturtion theory n itertive process in which the nswer is usully written s series of smller nd smller terms. In 1964 Kondo mde strtling discovery when considering the scttering from mgnetic ion tht intercts with the spins of the conducting electrons. He found tht the second term in the clcultion could e much lrger thn the first. The upshot of this result is tht the resistnce of metl increses logrithmiclly when the temperture is lowered. Kondo s theory correctly descries the oserved upturn of the resistnce t low tempertures. However, it lso mkes the unphysicl prediction tht the resistnce will e infinite t even lower tempertures. It turns out tht Kondo s result is correct only ove certin temperture, which ecme known s the Kondo temperture, T K. The theoreticl frmework for understnding the physics P HYSICS W ORLD J ANUARY ASAHI SHIMBUN

2 resistnce conductnce 1 The Kondo effect in metls nd in quntum dots 2e 2 /h ~ 10 K ~ 0.5 K temperture temperture () As the temperture of metl is lowered, its resistnce decreses until it sturtes t some residul vlue (lue). Some metls ecome superconducting t criticl temperture (green). However, in metls tht contin smll frction of mgnetic impurities, such s colt-in-copper systems, the resistnce increses t low tempertures due to the Kondo effect (red). () A system tht hs loclized spin emedded etween metl leds cn e creted rtificilly in semiconductor quntum-dot device contining controllle numer of electrons. If the numer of electrons confined in the dot is odd, then the conductnce mesured etween the two leds increses due to the Kondo effect t low temperture (red). In contrst, the Kondo effect does not occur when the dot contins n even numer of electrons nd the totl spin dds up to zero. In this cse, the conductnce continuously decreses with temperture (lue). elow T K emerged in the lte 1960s from Phil Anderson s ide of scling in the Kondo prolem. Scling ssumes tht the low-temperture properties of rel system re dequtely represented y corse-grined model. As the temperture is lowered, the model ecomes corser nd the numer of degrees of freedom it contins is reduced. This pproch cn e used to predict the properties of rel system close to solute zero. Lter, in 1974, Kenneth Wilson, who ws then t Cornell University in the US, devised method known s numericl renormliztion tht overcme the shortcomings of conventionl perturtion theory, nd confirmed the scling hypothesis. His work proved tht t tempertures well elow T K, the mgnetic moment of the impurity ion is screened entirely y the spins of the electrons in the metl. Roughly speking, this spin-screening is nlogous to the screening of n electric chrge inside metl, lthough the microscopic processes re very different. The role of spin The Kondo effect only rises when the defects re mgnetic in other words, when the totl spin of ll the electrons in the impurity tom is non-zero. These electrons coexist with the moile electrons in the host metl, which ehve like se tht fills the entire smple. In such Fermi se, ll the sttes with energies elow the so-clled Fermi level re occupied, while the higher-energy sttes re empty. The simplest model of mgnetic impurity, which ws introduced y Anderson in 1961, hs only one electron level with energy ε o. In this cse, the electron cn quntummechniclly tunnel from the impurity nd escpe provided its energy lies ove the Fermi level, otherwise it remins trpped. In this picture, the defect hs spin of 1 / 2 nd its z-component is fixed s either spin up or spin down. However, so-clled exchnge processes cn tke plce tht effectively flip the spin of the impurity from spin up to spin down, or vice vers, while simultneously creting spin excittion in the Fermi se. Figure 2 illustrtes wht hppens when n electron is tken from the loclized impurity stte nd put into n unoccupied energy stte t the surfce of the 34 Fermi se. The energy needed for such process is lrge, etween out 1 nd 10 electronvolts for mgnetic impurities. Clssiclly, it is foridden to tke n electron from the defect without putting energy into the system. In quntum mechnics, however, the Heisenerg uncertinty principle llows such configurtion to exist for very short time round h/ ε o, where h is the Plnck constnt. Within this timescle, nother electron must tunnel from the Fermi se ck towrds the impurity. However, the spin of this electron my point in the opposite direction. In other words, the initil nd finl sttes of the impurity cn hve different spins. This spin exchnge qulittively chnges the energy spectrum of the system (figure 2c). When mny such processes re tken together, one finds tht new stte known s the Kondo resonnce is generted with exctly the sme energy s the Fermi level. The low-temperture increse in resistnce ws the first hint of the existence of the new stte. Such resonnce is very effective t scttering electrons with energies close to the Fermi level. Since the sme electrons re responsile for the low-temperture conductivity of metl, the strong scttering contriutes gretly to the resistnce. The Kondo resonnce is unusul. Energy eigensttes usully correspond to wves for which n integer numer of hlf wvelengths fits precisely inside quntum ox, or round the oritl of n tom. In contrst, the Kondo stte is generted y exchnge processes etween loclized electron nd free-electron sttes. Since mny electrons need to e involved, the Kondo effect is mny-ody phenomenon. It is importnt to note tht tht the Kondo stte is lwys on resonnce since it is fixed to the Fermi energy. Even though the system my strt with n energy, ε o, tht is very fr wy from the Fermi energy, the Kondo effect lters the energy of the system so tht it is lwys on resonnce. The only requirement for the effect to occur is tht the metl is cooled to sufficiently low tempertures elow the Kondo temperture T K. Bck in 1978 Duncn Hldne, now t Princeton University in the US, showed tht T K ws relted to the prmeters of the Anderson model y T K = 1 / 2(ΓU ) 1/2 exp[πε o (ε o +U )/ΓU ], where Γ is the width of the impurity s energy level, which P HYSICS W ORLD J ANUARY 2001

3 2 Spin flips initil stte virtul stte finl stte density of sttes U T k U energy ε 0 Γ () The Anderson model of mgnetic impurity ssumes tht it hs just one electron level with energy ε 0 elow the Fermi energy of the metl (red). This level is occupied y one spin-up electron (lue). Adding nother electron is prohiited y the Coulom energy, U, while it would cost t lest ε o to remove the electron. Being quntum prticle, the spin-up electron my tunnel out of the impurity site to riefly occupy clssiclly foridden virtul stte outside the impurity, nd then e replced y n electron from the metl. This cn effectively flip the spin of the impurity. () Mny such events comine to produce the Kondo effect, which leds to the ppernce of n extr resonnce t the Fermi energy. Since trnsport properties, such s conductnce, re determined y electrons with energies close to the Fermi level, the extr resonnce cn drmticlly chnge the conductnce. is rodened y electrons tunnelling from it, nd U is the Coulom repulsion energy etween two electrons t the site of the impurity. Due to the exponentil dependence on the prmeters, the Kondo temperture cn vry, in prctice, from K. Remrkly, the rtio of the resistnce, R, divided y the vlue t solute zero, R 0, depends only on the temperture divided y the Kondo temperture, i.e. R/R 0 = f (T/T K ). Moreover, ll mterils tht contin spin- 1 / 2 impurities cn e descried y the sme temperture-dependent function, f (T/T K ). So the prmeters tht chrcterize the system U, ε o nd Γ cn e replced y single prmeter, T K. Scnning tunnelling microscopy Nnotechnology ims to mnipulte nd control mtter t the tomic scle. One of the centrl tools in the field is the scnning tunnelling microscope (STM), which cn imge surfce with tomic resolution, move individul toms cross surfce nd mesure the energy spectrum t prticulr loctions. Recently, the STM hs een used to imge nd mnipulte mgnetic impurities on the surfce of metls, opening new venue of reserch into the Kondo effect. Previously, physicists could only infer the role of the Kondo effect from mesurements of resistnce nd mgnetic susceptiility. With the dvent of the STM, however, physicists cn now simply photogrph the surfce nd therey resolve the position of the toms prior to studying the phenomenon. The first results cme simultneously in 1998 from Mike Crommie nd collegues, then t Boston University in the US, nd from Wolf-Dieter Schneider s group t the University of Lusnne in Switzerlnd. Both groups used n STM to imge prticulr mgnetic tom nd then to mesure the Kondo resonnce from the current-versus-voltge chrcteristics. More recently, group led y Don Eigler t IBM s Almden Reserch Center in Cliforni hs uilt n ellipse of toms round colt impurity, which ws plced t one of the two focl points of the ellipse (see figure 3). Next, they used n STM to mesure the energy spectrum of the colt impurity nd found lrge pek tht corresponded to the Kondo resonnce. The symmetry of n ellipse is such tht electron wves pssing through one focus inevitly converge t the second one, thus creting mirror imge of the Kondo resonnce. The energy spectrum mesured t the second focus lso hs Kondo-like pek, in spite of the fct tht there is no mgnetic impurity t tht point. The IBM tem hs referred to this seemingly unrel sitution s quntum mirge (see Mnohrn et l. in further reding). Menwhile, Crommie, now t the University of Cliforni t Berkeley, nd co-workers hve moved two mgnetic impurities towrds ech other nd studied the interction etween them s function of their seprtion (figure 3). Scnning tunnel microscopy hs moved the Kondo revivl in the direction of tom imging nd mnipultion, s well s sptilly dependent spectroscopy. Wht n STM cnnot do t lest not yet is lter the properties of the mgnetic impurity nd its coupling to the metl. In other words, these experiments cnnot continully trnsform one type of mgnetic impurity into nother with different chrcteristics. However, this is precisely the direction in which physicists studying quntum-dot devices re moving. Quntum dots s rtificil mgnetic elements Vrious groups round the world hve exploited chip technology to fricte smll semiconductor devices for investigting fundmentl prolems in physics. One such device is the quntum dot little semiconductor ox tht cn hold smll numer of electrons (see Kouwenhoven nd Mrcus in further reding). Quntum dots re often clled rtificil toms since their electronic properties resemle those of rel toms. A voltge pplied to one of the gte electrodes of the device controls the numer of electrons, N, tht re confined in the dot (figure 4). If n odd numer of electrons is trpped within the dot, the totl spin of the dot, S, is necessrily non-zero nd hs minimum vlue of S = 1 / 2. This loclized spin, emedded etween lrge electron ses in the two leds, mimics the colt-in-copper system. And mny of the known Kondo P HYSICS W ORLD J ANUARY

4 3 Single mgnetic impurities under the microscope () By mnipulting colt toms on copper surfce, Don Eigler nd collegues t IBM hve plced single colt tom t the focl point of n ellipse uilt from other colt toms (ottom). The density of sttes (top) mesured t this focus revels the Kondo resonnce (left pek). However, ellipticl confinement lso gives rise to second smller Kondo resonnce t the other focl point (right) even though there is no colt tom there. () Menwhile, Mike Crommie nd co-workers hve mesured two Kondo resonnces produced y two seprte colt toms on gold surfce (top). When two colt toms re moved close together using n STM, the mutul interction etween them cuses the Kondo effect to vnish (dt not shown). phenomen cn e expected to occur in these trnsistor-type devices, s ws pointed out ck in 1988 (see Glzmn nd Rikh, nd Ng nd Lee in further reding). One of the min distinctions etween quntum dot nd rel metl is relted to their different geometries. In metl, the electron sttes re plne wves, nd scttering from impurities in the metl mixes electron wves with different moment. This momentum trnsfer increses the resistnce. In quntum dot, however, ll the electrons hve to trvel through the device, s there is no electricl pth round it. In this cse, the Kondo resonnce mkes it esier for sttes elonging to the two opposite electrodes to mix. This mixing increses the conductnce (i.e. decreses the resistnce). In other words, the Kondo effect produces the opposite ehviour in quntum dot to tht of ulk metl. The dvntge of quntum dots is the ese with which the prmeters of these rtificil toms cn e controlled. Externl knos llow the discrete energy-level structure of the device to e vried, s well s the numer of electrons trpped within the dot. In terms of the Anderson impurity model, the energy, ε o, of the single electron level, its width, Γ, nd the Coulom repulsion energy, U, cn ll e vried y simply djusting the voltges on the gtes. Like the resistnce of ulk smple in the Kondo regime, the conductnce of quntum dot depends only on T/T K. With quntum dots, this universlity cn e redily checked, ecuse the prmeters tht define T K cn e esily chnged with the turn of kno. These remrks cn e illustrted y some recent results otined y one of us (LK) nd collortors t Delft University in the Netherlnds, NTT in Jpn nd Tokyo University. Similr experiments hve previously een crried out y Dvid Goldher-Gordon nd co-workers t the D EIGLER, IBM M CROMMIE, UNIVERSITY OF CALIFORNIA AT BERKELEY Msschusetts Institute of Technology in collortion with reserchers t the Weizmnn Institute of Science in Isrel (see Goldher-Gordon et l. in further reding). At Delft, the conductnce of the device ws mesured s function of the gte voltge, which chnges the numer of electrons confined within the quntum dot (figure 5). For n even numer of electrons, the conductnce decreses s the temperture is lowered from 1 K to 25 mk. This ehviour indictes tht the Kondo effect disppers when the numer of electrons is even. In contrst, when there is n odd numer of electrons, the Kondo effect produces the opposite ehviour, i.e. the conductnce increses t low tempertures. Moreover, t the lowest tempertures, the conductnce pproches the quntum limit of conductnce 2e 2 /h, where e is the chrge of n electron. To nlyse these dt, we concentrted on the region with N + 1 electrons nd plotted the conductnce s function of temperture for three different vlues of gte voltge, i.e. three different vlues of ε o (figure 5). The temperture dependence of the conductnce is clerly different for the vrious vlues of ε o lthough the ehviour of the conductnce is similr in ech cse. The precise temperture dependence ws fitted to function with T K s free prmeter, which llowed the conductnce to e replotted s function of T/T K for different vlues of ε o (figure 5c). In this so-clled normliztion plot, the different curves ll lie on top of ech other, i.e. the dt exhiit universl scling elow T K. The low-temperture increse in conductnce nd the sturtion t 2e 2 /h re in some sense strnge, even though the ehviour is in complete greement with theory. The system initilly contined two potentil rriers nd lrge energy scle, U, which tries to lock electrons from tunnelling into or out of the dot. Also, the energy ε o is fr from the Fermi level, i.e. the system is off resonnce. As result, the set-up is highly unfvourle for electron trnsport. However, the higher-order spin-flip processes tht led to the Kondo effect completely turn the sitution round nd increse the conductnce until it reches its ultimte limit. Indeed, the fct tht the conductnce reches 2e 2 /h implies tht the electrons re trnsmitted perfectly through the dot somehow the Kondo effect is le to mke the dot completely trnsprent. Artificil toms: going eyond rel toms Quntum dots hve provided new opportunities to control the Kondo effect experimentlly. Yet in mny wys the results descried so fr for systems hving n odd numer of electrons nd spin of 1 / 2 re similr to the old coltin-copper systems. However, quntum dots cn lso push reserch into the Kondo effect in new directions, where rtificil structures cn e exploited in regimes tht re inccessile with mgnetic impurities. 36 P HYSICS W ORLD J ANUARY 2001

5 The Kondo effect cn lso occur for impurities nd quntum dots tht hve spin of 1, or higher. While the spin of n tom is determined y the electronic structure, the spin of quntum dots cn e ltered more esily. The spectrum of energy levels in quntum dot cn e chnged y, for exmple, pplying mgnetic field of round 1 tesl to force trnsition etween singlet (S = 0) nd triplet (S = 1) stte. Although the sme trnsition cn occur in rel toms, it requires mgnetic field of out 10 6 tesl, which cnnot e generted in the l. Exctly t the singlet triplet trnsition, we find tht severl sttes with different spin nd oritl chrcteristics hve the sme energy, i.e. the sttes re sid to e degenerte. Exchnge processes, like those in figure 2, mix the degenerte sttes, including those with different oritl ngulr moment. The new mny-ody effect tht rises from the degenercy nd exchnge interctions is similr to the conventionl Kondo effect in mny respects. An importnt difference, however, is tht the mgnetic field fcilittes the effect, rther thn destroys it. Recently, the conductnce nomlies ssocited with the singlet triplet trnsition were oserved in two very different types of devices (see Pustilnik et l. in further reding). First they were found in rectngulr quntum dot mde from semiconductor (figure 4c). More recently, Poul Erik Lindelof nd co-workers t the University of Copenhgen hve oserved these conductnce nomlies in moleculr electronic device consisting of cron nnotue thin rolled-up sheet of grphite just few nnometres in dimeter (see Nygård et l. in further reding). And Chrles Lieer s group t Hrvrd University hs lso recently reported the Kondo effect in cron nnotues. The mesurements were mde with n STM ner colt clusters tht were deposited on the nnotues. These ltest experiments illustrte the generlity of the Kondo effect nd its importnce to nnoelectronic devices. Whenever smll system with well defined numer of electrons is connected to electrodes, Kondo physics ffects the low-temperture electronic properties of the device. Nnotechnology llows physicists to engineer n rtificil tom nd lso design its environment. For exmple, it is possile to insert quntum dot into the rm of n electron interferometer (see figure 4). Such ring-shped devices were pioneered y Moty Heilum nd co-workers t the Weizmnn Institute. These two-slit devices enle one to split n electron wve nd mesure the resulting interference pttern t the point where the two rms reconnect. For device with quntum dot in one rm, the wvefunction of n incoming electron is split into two prts, one prt trvels through the rm without the dot, while other hs to trverse the quntum dot where it experiences the spin interctions of the Kondo effect. Does this mjor difference in the two pths destroy the interference pttern? The nswer should e no. Unlike detector, the Kondo effect does not ct s n oserver who cn pss on informtion out the pth tken y the electron; the quntum dot is n integrl prt of the lrger quntum-mechnicl system. As long s no one interferes, the interference pttern is preserved. Indeed, recent experiments t Delft nd the Weizmnn Institute indicte tht the interference pttern is not destroyed when the Kondo effect is ctive on only one of the two slits. 4 Quntum-dot devices c () A quntum dot cn e defined y pplying voltges to the surrounding gte electrodes (yellow). The tunnelling etween the dot nd the externl electrodes (top left) is controlled y chnging the voltges on the lower-left nd lower-right gtes. This coupling defines the lifetime rodening, Γ, of the quntum stte in the dot. The numer of electrons nd the energy levels re tuned y the voltge on the lower-centrl gte. The puddle of electrons (confined red region) is out 0.5 microns in dimeter. () Quntum dots cn e plced in oth rms of two-slit interference device. Such device hs een used to investigte whether this scttering destroys the interference pttern. (c) Three quntum dots tht hve een used to compre the Kondo effect for singlet, doulet nd triplet spin-sttes. Kondo s future Scnning tunnel microscopy nd quntum-dot devices hve provided new tools for studying the Kondo effect from different perspectives nd with unprecedented control. Some of the recent studies hve counterprts in conventionl metl mgnetic-impurity systems, nd some re unique to rtificil nnostructures. Investigtions into the Kondo effect re fr from complete. One ongoing dete concerns the so-clled Kondo cloud. The mny electrons tht re involved in the spin-flip processes in figure 2 comine to uild the Kondo resonnce. The Kondo cloud consists of electrons tht hve previously intercted with the sme mgnetic impurity. Since ech of these electrons contins informtion out the sme impurity, they effectively hve informtion out ech other. In other words, the electrons re mutully correlted. The holy gril for reserch on the Kondo effect is to know whether it is possile to mesure nd control the Kondo cloud. But perhps n eqully importnt quest is to understnd the time evolution of such mny-ody quntum stte. For exmple, how does the stte uild up? Is it possile to suddenly switch on the exchnge interction in quntum-dot experiment? Would such experiments llow us to mesure how the ccompnying Kondo cloud forms? The Kondo cloud lso provides possile mechnism to investigte the interctions etween mgnetic impurities. For exmple, how do the two mny-ody sttes tht re formed round two seprted loclized mgnetic moments merge? A well controlled study of intercting loclized spins could provide us with new view on extended Kondo systems, such s spin glsses. The sic technology for fricting intercting Kondo systems now exists, nd my soon give irth to yet nother Kondo revivl. P HYSICS W ORLD J ANUARY

6 5 Universl scling 2 N = even N+4 conductnce (e 2 /h) 1 N+1 N+2 N+3 0 gte voltge 2 c 2 conductnce (e 2 /h) conductnce (e 2 /h) T (K) T/T K () The conductnce (y-xis) s function of the gte voltge, which chnges the numer of electrons, N, confined in quntum dot. When n even numer of electrons is trpped, the conductnce decreses s the temperture is lowered from 1 K (ornge) to 25 mk (light lue). This ehviour illustrtes tht there is no Kondo effect when N is even. The opposite temperture dependence is oserved for n odd numer of electrons, i.e. when there is Kondo effect. () The conductnce for N+1 electrons t three different fixed gte voltges indicted y the coloured rrows in (). The Kondo temperture, T K, for the different gte voltges cn e clculted y fitting the theory to the dt. (c) When the sme dt re replotted s function of temperture divided y the respective Kondo temperture, the different curves lie on top of ech other, illustrting tht electronic trnsport in the Kondo regime is descried y universl function tht depends only on T/T K. Further reding L I Glzmn nd M E Rikh 1988 Resonnt Kondo trnsprency of rrier with qusilocl impurity sttes JETP Lett D Goldher-Gordon et l Kondo effect in single-electron trnsistor Nture L P Kouwenhoven nd C M Mrcus 1998 Quntum dots Physics World June pp35 39 H C Mnohrn, C P Lutz nd D M Eigler 2000 Quntum mirges formed y coherent projection of electronic structure Nture T K Ng nd P A Lee 1988 On-site Coulom repulsion nd resonnt tunneling Phys. Rev. Lett J Nygård, D H Coden nd P E Lindelof 2000 Kondo physics in cron nnotues Nture M Pustilnik et l Mgnetic field-induced Kondo effects in Coulom lockde systems rxiv.org/s/cond-mt/ W G vn der Wiel et l The Kondo effect in the unitry limit Science Leo Kouwenhoven is in the Deprtment of Applied Physics nd the ERATO project on Mesoscopic Correltions, Delft University of Technology, PO Box 5046, 2600-GA Delft, The Netherlnds, e-mil leo@qt.tn.tudelft.nl. Leonid Glzmn is in the Theoreticl Physics Institute, University of Minnesot, 116 Church Street SE, Minnepolis MN 55455, USA, e-mil glzmn@tpi.umn.edu 38 P HYSICS W ORLD J ANUARY 2001