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PUPT 1516, IASSNS 94/108 cond-mat/9501050 December 1994 Space-Time Aspects of Quasiparticle Propagation ? 5 Richard Levien 9 n a J 9 y 1 Chetan Nayak 0 Department of Physics 5 0 Joseph Henry Laboratories 1 0 Princeton University 5 9 Princeton, N.J. 08544 / t a m - d z n Frank Wilczek o c School of Natural Sciences Institute for Advanced Study Olden Lane Princeton, N.J. 08540 ? [email protected] y Research supported in part by a Fannie and John Hertz Foundation fellow- ship. [email protected] z ResearchsupportedinpartbyDOEgrantDE-FG02-90ER40542. [email protected] ABSTRACT Highlycorrelatedstates of electrons are thought to produce quasiparticles with very unusual properties. Here we consider how these properties are manifested in space-time propagation. Speci(cid:12)cally, we discuss how spin-charge separation is realized in a simple double-layer geometry, how mass renormalization a(cid:11)ects time-of-(cid:13)ight in compressible Hall states, and how quantum drifts can reveal the e(cid:11)ective charge, mass, and quantum statistics in incompressible Hall states. We also discuss the possibility of observing the e(cid:11)ect of fractional statistics directly in scattering. Finally we propose that, as a result of incompressibility and the fundamental charge-(cid:13)ux relation, charged probes induce macroscopic, measurable rotation of Hall (cid:13)uids. 2 1. Introduction Over the past few years, qualitatively new behaviors of interacting electrons have been observed in deep quantum regimes involving some combination of low temperature, low dimensionality, and large magnetic (cid:12)elds. The theoretical description of these states predicts that their low-energy excitations can be described as quasiparticles, which as their name implies have particle-like properties { but with several unusual twists including in various cases fractional charge, fractional statistics, singular mass renormalizationand spin-charge separation. Although the theory as a whole is compelling, and consequences of it have been tested both by experiments and by extensive numerical work, for the most part these dramatic predictions for exotic properties of quasiparticles have been exhibited only rather indirectly. Asexperimentaltechniqueandsamplepurityimprove,onemightantici- pate that conceptually ideal experimentsthat directly probe ballistic quasiparticle propagation will become feasible. In this note, we will explore a few especially interesting issues that arise in interpreting such propagation. We also suggest a possible macroscopic manifestation of the charge-(cid:13)ux relation for quasiparticles, involving the rotation of the entire Hall (cid:13)uid when an external charge is brought near. 2. Spin-Charge Separation Theseparation ofspinandchargequantumnumbersisaphenomenonof funda- mentalinterest. It is (cid:12)rmlypredicted to occur in realistic 1+1-dimensional models involving solitons (e. g. polyacetylene) or non-trivial long wavelength dynamics (e. g. 1d metals). It is also known to occur in a wide variety of (cid:12)eld-theory models, including some in higher dimensions, and has been vigorously discussed as a possibility for the normal state of copper oxide superconductors [1]. Here we propose a simple, direct, and seemingly accessible space-time manifestation of quantum number separation in layered Hall systems. 3 Consider a system of two nearby layers of electron (cid:13)uid, subject to a strong perpendicularmagnetic(cid:12)eld. Oneknowsthat insuitablecircumstancestheground state of the electrons is an incompressible correlated state. The low-energy excitations of a bounded sample are then described as surface waves of the (two-layered) electron droplet. We will consider the case of a simple edge, such that there is just one mode in each separate layer. The dynamics of these waves, at long wave- lengths, is governed by the1+1-dimensional (cid:12)eld theory with a Lagrangian density of the form [2] ij i j ij j j L = K @t(cid:30) @x(cid:30) + V @x(cid:30) @x(cid:30) (2:1) Inthisexpressioni;j arelabelsforthelayer(orquasi-spin)andK;V aresymmetric numericalmatrices. (Inprincipletheycoulddependonxort,butthiscomplication is inessential for us.) This is the universal Hamiltonian symmetric between the i layers,containing the leadingtermsat long wavelength. The (cid:30) are invariant under R i addition of a constant, which implies number conservation for the charges @x(cid:30) . The form of K is determined by the bulk theory. The V matrix is not universal. One source for it, presumably mainly supplying the diagonal terms, is E(cid:2)B drift (the o(cid:11)-diagonal terms in K re(cid:13)ect the fact that this interaction serves to induce the velocities characteristic of drift in the e(cid:11)ective magnetic (cid:12)elds, which include contributions from correlation phases). V is also a(cid:11)ected, presumably mainly in its o(cid:11)-diagonal pieces, by ordinary Coulomb repulsion. In the simple case that the layers are symmetric,so K11 = K22 and V11 = V22, 1 2 the propagating modes with de(cid:12)nite velocities are made up from (cid:30) (cid:6) (cid:30) . The upper sign represents a holon { charge without spin { while the lower represents a spinon { spin without net charge. They propagate at velocities V11 +V12 vholon = K11 +K12 (2:2) V11 (cid:0)V12 vspinon = : K11 (cid:0)K12 Now consider the e(cid:11)ect of injecting an electron at the edge of one layer, as 4 exhibited in Figure 1. It will decompose into a spinon and holon, which propagate at di(cid:11)erent speeds. Downstream of the injection point, then, one will (cid:12)rst see a pulse of voltage di(cid:11)erence between the two layers, with no net charge, and then a pulse of charge, with no voltage di(cid:11)erence. (This is for V12 small. For V12 large, the order will be reversed.) This is spin-charge separation in a very tangible form. A nice example for this framework is provided by the Halperin-Laughlin (m,m,n) states [3]. They have K11=K12 = m=n. Thus in the limit where V is approximately diagonal (that is, for sharp (cid:12)eld gradients near the edge, or weak coupling between the layers) the ratio of holon to spinon velocities is expressed directly in terms of the fundamental integers characterizing the state, i.e. vholon m+n ! : (2:3) vspinon m(cid:0)n Under certain circumstances one can induce phase transitions between states at di(cid:11)erent values of m;n but with the same (cid:12)lling fraction 2=(m+n); our consider- ations implythat there are qualitative changes in the velocity spectra through the transition. 3. Mass Renormalization and Time-of-Flight Recently Halperin, Lee, and Read [4] proposed a striking description of the compressible quantum Hall state near (cid:12)lling fraction (cid:23) = 1=2, which has gained some experimental support. Their theory is based on the idea, which sounds startling on (cid:12)rst hearing, of approximating the electrons by fermions in zero magnetic (cid:12)eld. The rationale for this idea is that correlations between the electrons are such as to cancel the e(cid:11)ect of the imposed magnetic (cid:12)eld, so that the dressed quasiparticles have the quantum numbers of electrons but travel in straight lines. The special feature of (cid:23) = 1=2 is that attaching notional two-unit (cid:13)ux tubes to electrons formally does not alter their properties, yet it involves a vector potential which cancels o(cid:11) the growing part of the real imposed vector potential. Thus one 5 might hope to treat the di(cid:11)erence as a perturbation, especially since it is at least quasi-local and the Fermi liquid one is perturbing away from, though gapless, is quite robust. The perturbative treatmentof the statistical gauge (cid:12)eld is intrinsicallylimited, however, because the coupling is neither small nor truly short-range. Nor, since it is essentially a magnetic coupling, is it e(cid:11)ectively screened. Two of us have presented a renormalization group analysis of this problem, with the following main result (for alternative discussions, see [6]. In [5], we described our results in equivalent but more technical language). The dangerous interaction is sensitive to long-wavelength density (cid:13)uctuations. In considering these (cid:13)uctuations, it is important to include the e(cid:11)ect of the Coulomb repulsion (which suppresses them); x for the sake of generality, one considers a 1=jkj interaction, where x = 1 is the Coulomb case. Among the couplings in a non-relativistic Lagrangian is the mass 1 2 term 2m(cid:3)(r ) , and it turns out that the crucial renormalization for the gauge (cid:3) theory is the renormalizationof m as one one scales toward energiesand momenta on the Fermi surface. Indeed since the fundamental coupling is magnetic, an (cid:3) increase in m tends to suppress it. For general x we (cid:12)nd a (cid:12)xed point for the (cid:3) e(cid:11)ective coupling at (cid:11) = (1 (cid:0)x)=4; for the Coulomb case there is a logarithmic approach to zero coupling. For the e(cid:11)ective mass in this case we have 1 (cid:3) m (!) ! const: ln (3:1) j!j as one approaches the nominal Fermi surface. Notice that this mass diverges at the Fermisurface, so that strictly speaking the Fermiliquid theory does not apply, though many of its qualitative features survive in a modi(cid:12)ed analysis. Clearly the predicted behavior (3.1) is of fundamental interest, as is the question whether it can be seen directly in space-time. Actuallytodo thiswould seemto requireonlythat thebeautifulmeasurements of Goldman, Su, and Jain [7] be extended. These authors tested somefundamental 6 properties of thequasiparticlesby workingin magnetic(cid:12)elds B1=2+(cid:14)B slightlydif- ferentfromthose whichgiveexactly(cid:23) = 1=2. The quasiparticlesare thensupposed to see the e(cid:11)ective (cid:12)eld (cid:14)B, which causes them to move in large cyclotron orbits. In fact since the cyclotron radius is r = pc=(e(cid:14)B) and the momentumis cut o(cid:11) at the Fermi surface there is predicted to be a minimum radius, a prediction which was veri(cid:12)ed experimentallyusing a two-slit arrangement. Now what is required to test (3.1) is clearly that the time of (cid:13)ight should be measured as a function of the radius; from it one readily obtains the velocity and, knowing the momentum, the e(cid:11)ective mass. 4. Quantum Drifts: E(cid:11)ective Field, Charge, Mass, and Statistics We would likenow to discuss how the properties of quantized Hall e(cid:11)ect quasi- particlesarere(cid:13)ectedintheirballisticpropagation inslowlyvarying external(cid:12)elds. In the incompressible Hall states, as opposed to the (cid:23) = 1=2 state just discussed, the e(cid:11)ective (cid:12)eld seen by the quasiparticles is not zero (though in general it is modi(cid:12)ed from the imposed (cid:12)eld by de(cid:12)nite, substantial multiplicative factors, for example a factor 1/3 in the classic (cid:23) = 1=3 Laughlin state.) The calculation of the motion of charged particles in a strong, nearly homogeneous magnetic (cid:12)eld and additional small perturbations is a classic chapter of plasma physics [8]. The leading idea is that to a (cid:12)rst approximation in strong magnetic (cid:12)eld the motion is qB simply circular cyclotron motion with the characteristic frequency !c = mc , and radius pc rcl = : (4:1) qB When there are additional weak forces that vary slowly in time and space (on the scales de(cid:12)ned by !c; rcl) then in addition to this fast motion one (cid:12)nds slow drifts. The drifts may be calculated by averaging the dynamical equations over the fast motion, and (cid:12)nding the residual motion. Now when quantum mechanics is taken into account one (cid:12)nds that there is another natural length scale, the magnetic 7 length s h(cid:22)c lB (cid:17) : (4:2) qB When this magneticlength becomes comparable to the length scale over which the weak (cid:12)elds vary, clearly the classical averaging procedure is inadequate and one must use a di(cid:11)erent method to (cid:12)nd the drift motions. These quantum drifts will allow one to measure the charge and mass of the quasiparticles separately. We will present results for the quantum drifts of a particle with charge q and mass m subject to magnetic (cid:12)eld B. We note that in general the magnetic (cid:12)eld B seen by a quasiparticle is not the same as the external magnetic (cid:12)eld, and of course the charge q is some rational multiple of e: the microscopic theory suggests de(cid:12)nite values for q and B at each (cid:12)lling fraction. Formalisms have been elaborated to deal with motion in this deep quantum regime,but theyare quitecomplexand do not easilyleadto explicitresultsbeyond the leading order. The fundamental di(cid:14)culty, which makes it plausible that this complexityis intrinsic, is the vast degeneracy present in the unperturbed problem. Onecanreadilyobtainresultsofinterestforourpurposes byspecializingtothecase ofcircularsymmetryintwospatialdimensions. Thisspecializationvastlysimpli(cid:12)es the problem, because there is at most one state of a given angular momentum in each Landau level. Thus in calculating corrections due to circularly symmetric perturbations one is doing non-degenerate perturbation theory. A detailed account of the calculations will be presented elsewhere; here we record a few key results. First consider the e(cid:11)ectof a constant radial electric(cid:12)eld, yieldingthe potential V = qEr. One then (cid:12)nds for the radius, current, and velocity in the lth partial 8 wave: (cid:20) (cid:21) 2 3l mE B hri = r0 1+ 2 + 2 +(cid:1)(cid:1)(cid:1) 4r0 qB r0 (cid:20) (cid:21) 2 q E 3E l B J(cid:18) = (cid:0) + 2 +(cid:1)(cid:1)(cid:1) (4:3) 2(cid:25)r0 B 4B r0 2 E mE v(cid:18) = (cid:0) (cid:0) 3 +(cid:1)(cid:1)(cid:1) ; B qB r0 p where r0 (cid:17) 2llB is the radius at which the unperturbed lowest Landau level wavefunction is a maximum. J(cid:18) is the integrated current through a line (cid:18) = 1 constant, and v(cid:18) is simply qJ(cid:18) (cid:2) 2(cid:25)hri. There is actually a double expansion in equation (4.3): (cid:12)rstly the usual perturbation expansion in powers of V; and lB secondly an expansion in , which tells us how well localized the unperturbed r0 wavefunction (which has radial width (cid:24) lB) is. Note that the leading term in v(cid:18) is the usual E (cid:2)B drift, whose form is basically (cid:12)xed by Galilean invariance. The second term is in fact a classical correction (there are no h(cid:22)'s!), the so-called q \polarizationdrift",whichallowsonetomeasuretheratio . Quantumcorrections m appear in the directly measurable quantities hri and J(cid:18), though to this order they cancel in the product v(cid:18). 0 7E lB lB We also (cid:12)nd a drift induced by electric (cid:12)eld inhomogeneities, v(cid:18) = (cid:0)4 B r0, (cid:12) 0 dE(cid:12) where E = . This drift, which is intrinsically quantum mechanical (i.e. dr r=r0 it vanishes in the h(cid:22) ! 0 limit), allows one to measure directly the quasiparticle charge! This is impossible classically since the charge and mass enter the equa- q tions of motion only in the combination . By measuring the drift due to an m electric (cid:12)eld gradient and combining it with a measurement of the constant electric (cid:12)eld polarization drift above, one can determine both the charge and mass of quasiparticles. Although we shall not attempt to design experiments in any realistic detail, it seems appropriate brie(cid:13)y to describe extremely simple and powerful methods to produce, guide, and detect the quasiparticles. An idealized production process, shown inFigure 3a, follows fromthe fact that the quasiparticlesembodythe lowest 9 energy deviations from charge homogeneity. Thus if a sharply localized external charge is brought near the incompressible Hall (cid:13)uid, it will be energetically favor- able to produce quasiparticles of the appropriate charge, to take advantage of the imposed (cid:12)eld (see below). If the external charge is then movedaway, the produced quasiparticles will be free to move. An idealizedguidance process, shown in Figure 3b, follows from the fact that the quasiparticles carry charge and are subject to a large(e(cid:11)ective)magnetic(cid:12)eld. To a (cid:12)rst approximation,theirmotionissimplythe classicE(cid:2)B drift. Thus the quasiparticleswilltend to movealong electricequipo- tentials. One can create approximate equipotentials, and thus preferred paths for quasiparticle motion, by bringing a conductor of the desired shape nearby, in the presence of sources that would otherwise create a smoothly varying E (cid:12)eld. By providing entry and exit arms of known orientation emerging from a scattering region, as depicted in Figure 3b, one could measure scattering probabilities as a functionofangle,andthus(forexample)checkforthee(cid:11)ectsofthepredictedexotic quantum statistics. Perhaps the most notable such e(cid:11)ect is the \anyon swerve": the cross-section is markedly di(cid:11)erent at scattering angles (cid:30) and (cid:0)(cid:30). This e(cid:11)ect, and others of interest, are beautifully illustrated in the explicit Mott scattering cross-section (including exchange) calculated in [9]. With production occurring at a known location at a known time,and subsequent space-timepropagation at least roughly under control, the problem of detecting the arrival of electric charge, or voltage, at a known place and time perhaps becomes manageable. A further note is in order regarding the production process for quasiparticles, which as we have said above could involve bringing an external charge close to the Hall surface. As we create one or more quasiparticles at the origin and (cid:12)x them there (by not removing the external charge), other quasiparticles (or ordinary electrons) nearby will experience two kinds of azimuthal drift: (cid:12)rstly due to any uncanceled Coulomb charge, and secondly due to the (cid:12)ctitious (cid:13)ux tubes attached to the newly created quasiparticles. By comparing the orbital period of say quasiparticles and ordinary electrons, it is in principle possible to disentangle these two e(cid:11)ects, so that the charge and statistics of the quasiparticles may be 10