University of AmsterdamScience Park 904, 1098 XH Amsterdam

A Historical Perspective on the Quantum Hall Effect

Wout Neutkens

Amsterdam, 21 january 2013

Institute for Theoretical Physics (ITFA)

Master Theoretical Physics, Track CE

Supervisor: AMM Pruisken

Examinator: B. Nienhuis

2nd Examinator: A. de Visser

Student: W. Neutkens

Student number: 0306347

Abstract

Around 1980, von Klitzing et al.1 discovered the quantized behav-ior of a cold, 2D electron gas in a randomly disordered potential thatis placed in a high magnetic �eld perpendicular to the surface. At thattime, the �eld theoretical description of this electron setting was incom-plete and the quantum Hall e�ect could only be explained by semi-classicalarguments. These ad-hock arguments became very popular in the liter-ature because they provided a good temporal framework for research onthe quantum Hall e�ect. When, four years later, the fundamental prob-lems of the �eld theoretical description were solved by Levine, Libby andPruisken2�4 the physics community coudn't accomodate a new, �eld the-oretical framework. `A historical perspective on the quantum Hall e�ect'gives a brief overview of the semi-classical arguments and then emphasizeson the historical formation of a (modern) �eld theoretical perspective andits relation to experiment.

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Contents

1 The Early Quantum Hall E�ect 4

1.1 The Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Extended versus Localized States . . . . . . . . . . . . . . . . . 51.3 Quantized Conductance . . . . . . . . . . . . . . . . . . . . . . . 61.4 The Percolation Picture . . . . . . . . . . . . . . . . . . . . . . . 81.5 The True Quantum Hall Regime . . . . . . . . . . . . . . . . . . 9

2 The Modern Quantum Hall E�ect 10

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Anderson Localization . . . . . . . . . . . . . . . . . . . . . . . . 122.3 No QHE from Anderson Localization . . . . . . . . . . . . . . . . 132.4 No alternative for Anderson Localization . . . . . . . . . . . . . . 142.5 The Non-Linear Sigma Model and Disorder . . . . . . . . . . . . 152.6 The Instanton Vacuum . . . . . . . . . . . . . . . . . . . . . . . . 162.7 Importance of the Edge . . . . . . . . . . . . . . . . . . . . . . . 172.8 General Features are Super-Universal . . . . . . . . . . . . . . . . 18

3 Experiments on Critical Behavior 20

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Quantum Hall Laboratory . . . . . . . . . . . . . . . . . . . . . 213.3 The Critical Exponent . . . . . . . . . . . . . . . . . . . . . . . 223.4 Fermi-liquid principles . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Other issues 26

4.1 Coulomb Interactions . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Super-Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Summary 28

References 29

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1 The Early Quantum Hall E�ect

1.1 The Discovery

The quantum Hall e�ect (QHE) occurs when a 2D electron gas is cooled downto low temperatures and placed in a high magnetic �eld. When we �x a currentin the longitudinal direction of a rectangular sample, the electrons start to moveand a Lorentz force de�ects them in the perpendicular direction. This causesan electric force in the perpendicular direction that eventually compensates theLorentz force. We measure the voltages in both directions and obtain the resis-tance tensor. The perpendicular voltage di�erence divided by the longitudinalcurrent is then called the `Hall resistance'. If the magnetic �eld and the tem-perature are tuned to the quantum regime, the Hall resistance shows a series ofplateaus as a function of the magnetic �eld (�gure 1.1). On these plateaus thelongitudinal resistance vanishes, but there are peaks at the transitions betweenplateaus. As �rst observed by von Klitzing et al.1 in 1980, the values for theHall resistance at which the plateaus occur are extremely accurate. These valueswere somehow not in�uenced by minor variations in di�erent samples, but theywere precisely determined by just integer multiples of the fundamental units ofelectrical charge and Plank's constant. Therefore, the QHE could be used toprovide an independent measurement of the fundamental �ne structure couplingconstant appearing in electromagnetic interactions and even as a standard ofresistance. But truly interesting for theoretical physics was the universality, thesample-independence of this result.

Figure 1.1: The quantum Hall e�ect in a GaAs-GaAlAs heterojunction, recordedat 30mK. The Hall resistance ρxy forms a `staircase' of plateaus. The longitu-dinal resistance ρXX forms peaks at the transitions between plateaus. (ImageCourtesy of D.R. Leadley, Warwick University 1997)

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How can measurements on di�erent samples return exactly the same, univer-sal values? These samples all have di�erent boundaries, electron densities andother features that should in�uence the measured resistance. In condensed mat-ter, we often start with an in�nite, homogeneous system without boundaries.Any universal result that is obtained from this theory would not be immediatelythe result that one measures. Experimentalists use tricks to extract the universalresult from their measurements. In the quantum Hall case, it is the universalresult that is directly obtained from the di�erent samples. Are we somehowvery lucky here? For example, the longitudinal resistance is proportional tothe length of the sample. Di�erent samples will therefore yield di�erent resultsthat are all proportional to the presumed universal result. The length of everysample has to be measured very carefully to extract this result, which wouldbe completely impossible in the Quantum Hall case. But, the Hall resistivity isthe perpendicular voltage di�erence divided by the longitudinal current. If werescale the length, the longitudinal current and perpendicular voltage remainthe same. If on the other hand we rescale the width, the longitudinal currentand the perpendicular voltage di�erence both change, but they change equally.The longitudinal resistance will change upon rescaling. It is only because oftwo-dimensions and it is only the Hall resistance that doesn't depend on sampledimensions. The precision of measuring a universal result for the longitudinalresistance will always be limited by the small variations in sample dimensionsthat are present if we compare di�erent samples. It is therefore at least pos-sible that our two-dimensional setting provides us a result that is independentof small variations in the dimensions of the sample. The fact that the result isalso independent of the entire geometry of the 2D samples as a whole, and moreimportantly, that it is even independent of the many kinds of disorder in thesamples is not at all explained!

1.2 Extended versus Localized States

As it is well known, a magnetic �eld makes a moving electron to circle aroundsome (possibly un�xed) guiding center. In the two-dimensional electron gas, thiscauses the energy levels of a free electron to become quantized. This is visiblein the quantum regime, where the thermal energy is smaller than the levelseparation. In the staircase picture (�gure 1.1), the horizontal plateaus do not

occur because the energies are quantized. In any disordered sample, the energylevels (called Landau Levels) are broadened and the disorder broadening of theenergy levels is so large that their tails are overlapping. It is not a band gap,but the immobility of the carrier system that prohibits a change in resistance.The horizontal plateaus indicate that the carrier system has to be extremelyimmobile. But surprisingly, the sudden changes between those plateaus and thepeaks in the longitudinal resistance indicate that the carrier system can also bevery mobile. Here, the Fermi energy approaches the center of a Landau band.In other words, near the band center the localization lengths of the electronsare macroscopically large, while in the tails of the Landau bands, they aremicroscopically small (see �gure 1.2). Obviously, we are dealing with some sort

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of phase transition at (or near) every band center. The entire staircase picture,consisting of very �at plateaus and phase transitions in between, is visible asa function of the magnetic �eld. Increasing the magnetic �eld introduces morepossibilities to form electrons per Landau level. Since the total number ofelectrons remains constant, increasing the magnetic �eld sweeps us through thestaircase picture.

Figure 1.2: A Landau band. The states in the centre (indicated by a blackline) are completely extended, while the rest of the states (shaded area) arecompletely localized.

The Hall resistance remains �nite after the Fermi energy is again in a regionof localized states, because even when all states are immobile in the bulk of thesystem, current is still able to �ow through electron states that extend along theedge of the sample. In our two-dimensional system the conductance is relatedto the resistance by matrix inversion. Since at the plateaus the longitudinalresistance is zero, the Hall conductance is just the inverse of the Hall resistance.Every time we shift the Fermi energy over the center of a Landau band, theHall conductance increases exactly with e2/h. We will �rst discuss the exactdi�erence between the conductance plateaus (subsection 1.3) and then we willdiscuss why there can be very localized states as well as very extended states inthe system (subsection 1.4).

1.3 Quantized Conductance

An argument that became famous in the literature on the QHE, Laughlin'sgauge argument, assumes the existence of extended states (which is a very non-trivial assumption). Laughlin5 considers the 2D electron gas in the geometryof a loop, where the magnetic �eld points perpendicular to the surface. Hethen inserts a magnetic �ux through the loop adiabatically. This changes thephase of the quantum mechanical wave functions. The extended states, thatgo all the way around the loop, can be gauge transformed back to the original

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situation if their phase has been changed by any multiples of 2π. This phasechange alters the potential energy in the direction along the loop and if a gaugetransformation was possible, the total electronic energy must have remained thesame. But, it can be compensated by a change in energy over a perpendicularpotential. Thus, the wave phase is allowed to change by a multiple of 2π andthen the Hall conductance changes with an integer multiple of e2/h.

It is important that the Fermi energy is in a region of well-localized states,otherwise the wave phases are allowed to change by non-integer multiples of2π. What is then the mechanism by which electric charge is transported acrossthe ribbon? It is possible that there is no such mechanism and the `integer'equals zero. But just after Laughlin's publication, Halperin6 clari�es the chargetransfer across the ribbon. At an edge, the con�ning potential shifts up toin�nity and the Landau levels will cross the Fermi energy (see �gure 1.3). An

Figure 1.3: The Landau levels that are under the Fermi energy, will cross thisenergy at the edges

electron in an extended state can get excited into a localized state just abovethe Fermi energy on one edge of the sample. This leaves a hole behind froma state that extends to the other edge of the sample. A localized electron justbelow the Fermi energy on the other edge can thus �ll this hole and e�ectively,charge has been transferred across the ribbon.

The geometry of a loop can be related to the rectangular Hall bar by con-sidering the formation of extended edge states. These edge states should besomehow not in�uenced by all the microscopic bumps and so on in the sample.Butticker7 explained this due to the absence of back scattering at the edge.Instead of going around in a cyclotron motion, a particle reverses its motionevery time it scatters from the edge resulting in series of semicircles followingthe edge equipotential line. This motion was called a `skipping orbit' and itwas shown that disorder on the edges, even large disorder from the contactsused to measure the voltages, altered only the shape of those orbits, but not theextended edge states themselves.

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1.4 The Percolation Picture

We have seen that there had to be extended states near the center of the Lan-dau bands as well as localized states elsewhere. To understand this a percola-tion concept was introduced8;9. Percolation can have di�erent manifestations,ranging from current percolation to the percolation of rigidity. In physics it isalways a critical phenomenon. In the context of the quantum Hall e�ect, one�rst assumes that the spatial correlation of the disorder potential is large com-pared to the magnetic length. Thus, we have a very smooth potential landscapeand since an electron cannot change its energy, its guiding center moves alongequipotential lines. The disorder potential can be compared to a landscape withmountains and valleys. At the tail of a Landau band, we are either near the topof a mountain, or near the bottom of a valley. In this situation, the equipoten-tial lines are closed shapes with a mean radius. But, if we approach the centerof a Landau band, the radius of the shapes becomes bigger and when valleysturn into mountains, the radius of a valley circle exceeds the system size andelectrons can only move along mountain circles or along the edges. Thus, in thetails of a Landau band (the valleys or mountains), the electrons are localizedand in the center of the band and at the edges, the states are extended (see�gure 1.4). Since the equipotential circles are very big near the center of a Lan-dau band, there are many points where they are close to each other. Quantumtunneling then causes di�usion of electrons over multiple valleys or mountains.This implies that the critical aspects of the valley/mountain phase transitionare not determined by classical percolation, but by quantum percolation.

Figure 1.4: A random contour map. The black dots indicate areas where anelectron cyclotron motion takes place. Such an area cannot be too large forthe percolation picture to apply. The black line indicates a percolating shapewith a radius that is larger than the sample size. For larger samples, the energywindow where percolating states occur becomes smaller.

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1.5 The True Quantum Hall Regime

Conclusively, we obtain a picture where for every disorder broadened Landauband under the Fermi energy, there are extended states near the band centerthat give rise to a quantization condition. At the edges, the Landau level shiftsup to the Fermi energy allowing an edge current to �ow and resulting in aquantized Hall conductance.

To explain the Quantum Hall E�ect at a fundamental level, we will needour modern techniques of Quantum Field Theory. The argumentations fromabove only clarify the �rst measurements on the quantum Hall e�ect. Theirmajor shortcoming is that they are formulated entirely from a semi-classicalperspective. A statistical �eld theory describing the quantum Hall e�ect re-quires the formulation of certain abstract operators acting on �elds that livein a particular space. These operators are then weighed by a set of couplingconstants that is evaluated from a perturbative (renormalization) scheme. Thisgives rise to an entirely di�erent language and perspective. Take for examplelocalization, quantization and edge currents. In the percolation picture, local-ization of the electron states was due to closed equipotential shapes while inthe �eld theoretical language, localization is described by the general theory ofAnderson localization - that also applies to non-smooth randomness. Also, wehave seen that the quantization of the Hall current is explained with Laughlin'sgauge argument, while in �eld theory, the quantization arises due to topologicalinvariants in the mapping structure of the �elds. Finally, the edge currents aredescribed by one-dimensional `massless chiral edge excitations', these are verydi�erent from the edge currents described by Butticker.

Anderson localization in the presence of a magnetic �eld, topological invari-ants and massless chiral edge excitations have all been subject of Pruisken'swork. It is this work that is the starting point of a mathematical descriptionof the QHE. For example, by understanding more about his Quantum FieldTheory, we may infer what happens to the percolation picture when the disor-der correlation length is of the order of the magnetic length, instead of muchlarger. Although such an understanding may lead to a clear mental picture ofthe percolation mechanism (which may be important for education, nummerics,etc..) the path of theoretical physics narrows down to the most fundamental as-pect of the Quantum Hall measurement. Since the renormalization behavior ofthe conductance parameter describes this measurement from a Quantum FieldTheory perspective, we will have to take a look at the the transitions that occurin between the staircase plateaus (In 1988, it would be experimentally veri�ed10

that these were true quantum phase transitions).Unfortunately, `looking at the phase transitions' was often done from the

perspective of the percolation picture, which also happens to be a critical phe-nomenon. In smoothly disordered samples, there is indeed a low temperatureregime where power-law scaling reminiscent of this picture can be observed,But in the limit of zero temperature, or in�nite system size, the critical aspectsare di�erent. Although this research can be meaningful for the same practicalapplications, it is obviously not the same study of critical aspects that we are

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interested in. Still, these di�erent kinds of research are often intertwined in theliterature. We have to remember that the percolation picture is irrelevant inthe sense of a �eld theoretical description of the Quantum Hall E�ect.

This statement about percolation is actually very similar to an earlier state-ment by Mott and Thouless even before the QHE was discovered. In thatcontext, the same two-dimensional, disordered electron system as the QHE wasstudied, but then without a magnetic �eld (we will look at this situation in moredetail later on). To investigate the critical aspects of the QHE from a theoreticalperspective we will have to learn more about Pruisken's work on the subject.Since the appropriate �eld theory was already found by him three decades agoand he and his collaborators have been working on it ever since, we can onlylightly delve into his work in the present thesis.

Also, there are real measurements on the phase transitions possible in theQuantum Hall laboratory. These subtle measurements are currently the deep-est connection of the theoretical framework with its laboratory. But, the mea-surements on the QHE phase transitions are technically very challenging. Insmoothly disordered samples we have already stated that the percolation criti-cal phenomenon interferes with the true quantum phase transition. Therefore,these smoothly disordered samples have to be of subkelvin temperatures in or-der to measure the critical exponents of interest. But, from the perspective of�eld theory, there is more to quantum phase transitions than just their criticalexponents. For example, since both the Hall conductance and the longitudinalconductance undergo the same phase transition, their behavior has been studiedin a renormalization group �ow diagram (see �gure 2.3 on page 19). Even lesscold, or less ideal samples are then suitable to study critical behavior, albeitonly in a more general sense.

Today it is found11 that for long-ranged disorder correlations there are tworegimes with power law scaling behavior as the temperature is lowered. In-deed, the regime of lowest temperature has a critical exponent that is univer-sal and independent of the disorder correlation length (see �gure 1.5). In thezero-temperature limit the smoothly disordered system always �ows to the truequantum regime. In this limit, percolation breaks down because length scalesmuch larger than the separation between tunneling points will be revealed. Thetunneling ensemble can then again be imagined as a disordered potential andwe end up with an Anderson transition, just like in the non-smoothly disor-dered system. We can therefore say that, in the disordered electron gas, thepercolation picture is `unstable' with respect to scaling.

2 The Modern Quantum Hall E�ect

2.1 Introduction

The modern language of the Quantum Hall E�ect has its roots in a �eld theo-retical description of localization in electron systems. Just before the quantumHall e�ect was discovered, it was strongly believed that all electronic states in

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Figure 1.5: Two regimes of criticality for a smoothly disordered system (recentexperimental results11). One with critical exponent κ = 0.58 and one withκ = 0.42. Above: For the lowest temperatures, the exponent saturated becauseof �nite sample size. Below: The squared dots indicate a system with a shorterdisorder correlation length. Its critical exponent remains κ = 0.42 for highertemperatures.

two dimensions had to be localized by the mechanism of Anderson localization.This highly nontrivial and fundamental result was based on a novel formulationof Anderson localization, many years after Anderson's original discovery12. Thesuccess of Anderson localization was based on the possibility to apply renormal-ization group ideas on the disordered electron system13. Because the problemof localization in electron systems was described in the modern language of �eldtheory, the results of Anderson localization were believed to be `proven facts'instead of theoretical pictures (like the percolation picture) or other models thatcould be very close to the reality of the experiment.

The most practical and interesting result, was that in two dimensions, allelectronic states had to be localized. This means that even a very good twodimensional conductor, with macroscopically large localization lengths in thequantum regime, will always be a perfect insulator on an in�nite sample. Thelittle disorder that is present in the `conductor' will localize the electronic statesfor some length that is large enough. Such a result was considered a victory

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for the renormalization group theory, but the Quantum Hall E�ect was still tobe discovered. It doesn't take much imagination to understand that the QHEcannot be in accordance with `all states are localized in 2D'. It did take a lot ofimagination to disprove the `proven facts' of Anderson localization.

If Anderson localization is wrong, then either quantum �eld theory or renor-malization group theory must be wrong and that is quite an alarming situa-tion in theoretical physics. But this situation was almost three decades ago,when topological issues were only beginning to become important in statisticalphysics. Pruisken and his colleagues were at the frontier of understanding thatthose issues are as important as we know today. As we will see, their unexpectedsolution to the `extended states' problem is in fact topological in nature and itis able to reconcile Anderson localization with the results of the Quantum HallE�ect.

2.2 Anderson Localization

The theory of Anderson localization is a statistical quantum model of electronsin the presence of disorder. Since every sample in reality is somewhat disor-dered, we can imagine that Anderson localization is in fact a vast subject instatistical physics. Localization of electrons in general can be due to the re-pulsive Coulomb interactions (`Mott localization' ) as well as due to disorder(`Anderson localization'). There is weak and strong Anderson localization. Inthe path integral formulation we can imagine an electron to follow many pathsof which some return to the original location. For every such self-returning path,there is a path in the opposite direction which is otherwise exactly the sameand the electron traversing this path has the same phase shift as the electrontraversing the original path. Thus, after scattering, the electron constructivelyself-interferes at its origin, causing the weak localization. For more heavy dis-order, wave interferences everywhere around the electron stop the wave frompropagating through the sample, and there is strong localization.

With the renormalization group, the Anderson localization problem could beformulated as a scaling theory. This was motivated by a relation on a quantummechanical level found by Edwards and Thouless14 between the conductanceand the response to perturbations of boundary conditions. As for scaling, thismeans that the change in the conductance with the system size only dependson the conductance itself. This gives rise to a one-parameter scaling theory anda beta function can be calculated perturbatively. In the Anderson theory, itwas found that for an initially large conductance (little disorder), the scalingcorrections are very small but negative in two dimensions. Since the correctionsare small, scaling the system only slightly alters the conductance. In this regimethe electrons are weakly localized. But, the beta function is always negative intwo dimensions and scaling to larger sizes will decrease the conductivity. Forlittle conductance (initially strong disorder) the beta function is very negativeand scaling the system induces large, negative corrections to the conductance.This is the regime of strong localization and the two-dimensional system willalways �ow to a perfect insulator (see �gure 2.1). Therefore the conclusion could

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be reached that for any given amount of disorder, no states with in�nitely largelocalization lengths exist in two dimensions.

Figure 2.1: Beta functions of some d dimensional disordered systems. Arrowsindicate the renormalization �ow upon scaling to larger system sizes. In twodimensions, corrections to the conductance can be small, but they always remainnegative. (Image courtesy E. Abrahams15)

2.3 No QHE from Anderson Localization

By renormalizing the general action (formula 2.1 on page 16) for the disorderedelectron gas, the beta function as in �gure 2.1 can be obtained and hence thefamous result of Anderson localization theory that `all states are localized in

two dimensions'. Around 1980, when the quantum Hall e�ect was discovered,the e�ect of the magnetic �eld was included in Anderson theory by consideringthe symmetry requirement for the �eld matrices not to be orthogonal, but onlyof unitary nature. Then, weak localization becomes even weaker because self-interference requires time-reversal symmetry, which is broken in the unitaryensemble. In this case, the localization lengths of the electrons can becomemacroscopically large and the �nite Hall conductance was naively explained.But in fact, the Hall conductance is so robustly quantized that it doesn't changewithin measurable precision for larger system sizes. Then the mean localizationlength, which is of Gaussian curvature, has to be so extremely large that this isirreconcilable with the results of unitary Anderson models. In other words, thequantum Hall e�ect says that `there are extended states in two dimensions'.

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The fundamental issue here is that the statements `all states are localized'

and `there are extended states' (one from �eld theory and one from experiment),are contradicting. From our modern perspective on this situation we know thattopological considerations will solve this kind of issue. Around 1980, manykinds of `pictures', `arguments' or `mechanisms' were invented to understandor describe the Quantum Hall E�ect, but these approaches obviously couldn't(and didn't) solve our fundamental issue.

2.4 No alternative for Anderson Localization

In our introduction to the Quantum Hall E�ect, a number of such explanationslike Lauglin's gauge argument or Landau level quantization were reviewed andthe problem with the percolation picture was already indicated. In order toclearly separate those early inventions (that are still appearing in the modernliterature on the QHE) from the modern perspective we will again indicate thedi�erence between the percolation picture and Anderson localization before wecontinue.

In the context of the QHE, percolation ideas were introduced in attempts8;16;17

to understand the delocalization mechanism of the electrons in their disorderedlandscape (see subsection 1.4). However, even before the discovery of the quan-tum Hall e�ect, practically the same percolation ideas were introduced in thedisordered electron gas without a magnetic �eld and these ideas were alreadyfound to be wrong! From the percolation perspective, the conclusion that ex-tended states could exist in two dimensions was reached by Cohen and Jortner18

and that there was a mobility edge near half-�lling of the disordered potential.Then, after Cohen and Jortner, Thouless and Mott independently published the-oretical papers19;20 that demonstrated the invalidity of percolation and othermere intuitive ideas about the mobility edge.

Without a magnetic �eld, percolation predicts that the electron states arelocalized in valleys (lakes) for low �lling fractions and that for high �lling frac-tions all states are extended (a big, `extended' sea with only a few islands, see�gure 2.2). This perspective radically changes when we let �eld theory de-scribe the system and thereby include all quantum e�ects like tunneling andself-interference. As we have seen in subsection 2.2, renormalization predicts noextended states in two dimensions and no phase transition. Even for high �llingfractions (the big sea with little islands) the states are essentially localized, buttheir localization lengths are so large that we have to imagine the system onmuch larger length scales to get the picture. The results of �eld theory are there-fore the very non-intuitive ideas of Anderson localization, where all states areessentially localized in two dimensions and the former `mobility edge' becomesa �nite-size (or �nite-temperature) e�ect.

So before the quantum Hall e�ect was invented, the conclusion was alreadyreached that the localization problem could not at all be described by anypercolation mechanisms. Thouless and other theoreticians that reached thoseconclusions about the invalidity of percolation21, still continued to investigatethe same percolation ideas after the quantum Hall e�ects discovery. Because

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the Quantum Hall E�ect could not at all be explained from the perspective ofAnderson �eld theory, they tried to describe the electron gas again with thepercolation picture. The disagreement between percolation and �eld theorywas apparently less impressive than the problem of `No QHE from Andersonlocalization'.

Figure 2.2: with only few islands as a metaphor for a disordered system withhigh �lling fraction. The fact that the electrons in the sea are able to perco-late through the system is pretty obvious from this picture. Still, Andersonlocalization predicts that there are no extended states in any 2D disorderedsystem. This shows that the percolation picture is not predictive at all in thetrue (in�nite) quantum regime.

2.5 The Non-Linear Sigma Model and Disorder

To describe the disordered electron gas as an e�ective �eld theory, a non-linearsigma model can be used22. The non-linear sigma model is actually a verygeneral model, where the target manifold has any nontrivial curvature (often asphere). It can be used when a global symmetry group is spontaneously brokenby the ground states, which possess only the symmetry of a subgroup. In themodern context of the quantum Hall e�ect, the special unitary SU(N +M) Liegroup is being investigated and the vacuum possesses only a S(U(N)⊗ U(M))symmetry. The Grassmanian �eld matrices Q have N +M components andsatisfy the non-linear constraint of unitarity. The action that describes the

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dynamics of the isolated system without a magnetic �eld is given by

S = −σ8

ˆdrtr(∇Q)2 (2.1)

Where σ is the unrenormalized conductance. Besides the disordered electrongas, where the phenomenon of Anderson localization occurs, the non-linearsigma model contains other systems like anti-ferromagnetic spin chains in theSO(3) model. This system is obtained by setting N = M = 1 (SU(2) is thedouble covering group of SO(3)).

The two dimensional electron gas without disorder has a vector �eld withonly two components. The N+M component �eld matrices are studied becausethe disorder must be taken into account. This can be done by averaging thelogarithm of the partition function (the entropy) over the disorder potential.This is possible because any observables are expected to be self-averaging withrespect to the disorder. The entropy, or the logarithm of the partition function,can be transformed into a polynomial function by performing the `replica trick'.The problem of averaging the logarithm of the partition function can be reducedto the problem of averaging a sum of powers of the partition function. By thede�nition of the partition function, taking powers of the partition function withitself is like taking `replicas' of the same system. It is often su�cient to justconsider this set of replicas as a grand system, of which the dimension equalsthe amounth of replicas.

The two-dimensional electron system is therefore studied as an N +M di-mensional system. The N +M components from the �elds in the SU(N +M)Lie group as mentioned above are the replicas of the two-dimensional �elds. Inthe end, analytic continuation to zero replicas (N,M → 0) has to be performedto obtain the real, two-dimensional electron gas. Sometimes this analytic con-tinuation mechanism introduces extra parameters in the model and this is called`replica symmetry breaking'. If there is replica symmetry breaking, the phasesin the replicated system, do not guarantee the same phases real system. Forthe Quantum Hall E�ect, this doesn't matter, because its most prominent fea-tures are already apparent in the replicated model (see subsection 2.8). Byconsidering the replica trick, it is possible to include a general disorder distri-bution function. Not an unknown, speci�c realization of the disorder potentialhas to be provided, but a general distribution function su�ces, with certaincharacteristics such as the disorder correlation length.

2.6 The Instanton Vacuum

Levine, Libby and Pruisken2�4 managed to solve the fundamental shortcomingof localization theory by introducing the theta angle in the Anderson model.The theta angle came from an unexpected `angle' (Quantum Chromodynam-ics, QCD) because such a bizarre topological issue wasn't considered by thenin any of the statistical �eld theories of �nite samples. Apart from topology,the theories of QCD and Anderson theory share quite fundamental similarities.

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For example, we have seen that in Anderson theory there is a crossover be-tween weak localization at small scales to strong localization at large scales. InQuantum Chromodynamics, there is a crossover between weak con�nement atsmall scales to strong con�nement at large scales. This phenomenon is called`asymptotic freedom' and strong con�nement is a necessary ingredient for anyrealistic quantum �eld theory about quarks.

The similarity that those theories didn't share at that time was the thetaangle. This angle is equal to the weight of an operator that measures a purelytopological property of the �eld. It can therefore be used as a label for thedi�erent topological sectors in the �nite system. But, in QCD, its physicalinterpretation remained completely mysterious. Under the normal conditionswhere QCD applies, the theta angle is very small and has no e�ect on the quarkcon�nement, but it was argued by 't Hooft that above a critical value θ = πthe con�nement breaks down and a new phase appears. The meaning of thisphase was unknown but it was clear the critical value could never be observedin the universe. It was therefore considered a very uninteresting mathematicalartifact of some particular QCD model. But, when we apply this uninteresting`artifact' to the �eld theory of Anderson localization23, it does exactly what thistheory couldn't describe to be happening for the quantum Hall e�ect! Here, inthe very `realistic' situation of the QHE, the theta angle obtains a clear physicalmeaning: it is the unrenormalized Hall conductance σxy itself. The action ofthe modern Anderson model including its topological properties is then writtenas24

S = −σxx8

ˆdrtr(∇Q)2 +

σxy8

ˆdrtr[Q(∇Q ∧∇Q)] (2.2)

The �rst part including the longitudinal conductance σxx is the same as inequation 2.1(no magnetic �eld). The integration corresponding to the thetaangle is an integration over the �eld variables of a modern version of 't Hooft'sinstanton vacuum, called the theta vacuum. Since the theta vacuum is topolog-ical in origin, any continuous deformations around the instanton �eld will leaveit invariant. The unconventional renormalization procedure of the theta anglethat encompasses the non-perturbative sectors of the original Anderson theoryis originally invented by 't Hooft, due to the concept of instantons.

The interesting properties of the laboratory system at criticality are then de-scribed by a topological picture of the instanton vacuum. According to Pruisken,the topological theta vacuum picture can be used to describe the phase transi-tion that appears in the 2D quantum electron system.

2.7 Importance of the Edge

Without the need to understand the QHE with Anderson's localization theory,the important features of the instanton vacuum didn't get uncovered by thephysics community. Early attempts25�27 to understand the scaling behaviorof the instanton vacuum failed because the topology of the system was notappropriately included. This led to the idea that 't Hooft's instanton approachwasn't interesting in the �eld theory of QCD.

17

The reasoning was globaly that by considering the large N expansion of theCPN−1 model, which is a also a SU(N) non-linear sigma model, the theory be-comes exactly solvable and a certain electromagnetic �eld appears (in analogyto the Hall current). But the electromagnetic �eld that was obtained is a free�eld with a charge that can take any value and an energy that is non-periodicin theta. In the Hall context this would mean that the Hall conductivity isunquantized and with no phase transitions. In the QCD context this wouldmean that no phase transitions can occur and that the theta angle is just anextra parameter which doesn't change the fundamental properties of the quarksystem. In this sense, the entire QHE didn't appear in the sigma model. Toexplain this, it was argued that the QHE would have to be a very specializedfeature of the theta angle concept, which only appears after analytic continua-tion, by replica symmetry breaking. The unknown replica symmetry breakingmechanism was supposed to introduce extra parameters in the theory allowingthe systems criticallity. Because the whole situation could now be explained,the theta angle wasn't supposed to provide any fundamental information aboutthe instanton vacuum or QCD. But (and that is a very big `but'), by introduc-ing the boundary of the system, Pruisken was able to show explicitly24 thatfor any replica limit (also the large N expansion of the CPN−1 model) the 2Dsigma model displays the same scaling behavior as observed in the QHE. Thisshould have fundamentally altered the physics communities vision on the basicproperties of the (in)famous non-linear sigma model.

2.8 General Features are Super-Universal

For the general action given by equation 2.2, the topological term can be rewrit-ten as a one-dimensional integral over the boundary of the system. It turns out28

that the instanton vacuum generically displays massless chiral edge excitations.The �eld con�gurations at the edge classify topologically di�erent sectors. Thebulk variables of the system can be integrated out for each topological sector.Hereby, an e�ective action for the edge can be formulated, allowing the one-dimensional edge to be renormalized separately. This edge is the edge of thebulk of the system. In a 2D simply connected system such an edge is topologi-cally equivalent to a circle. The edge itself as an object is obviously a non-simplyconnected system. It isn't an extra `shell' around the bulk, or any other sepa-rate channel, but it is the natural edge of the e�ective system (quantum �eld).Renormalizaton by instantons of the critical action for the bulk-edge providesus with scaling functions that can be combined in a general �ow diagram. This�ow diagram consists of �xed points that are determined by the topology of theedge. For the quantum Hall e�ect, this general �ow diagram actually explainsthe staircase picture (�gure 1.1)! On larger scales, conductances renormalizeto the attractive �xed points (that are in the strong coupling regime) and theHall plateaus are formed (see �gure 2.3). The robustness of these plateaus isthus an emergent property of the large-scale limit. The plateaus don't `blur' be-cause of the disorder as would be the case with energy levels. It is the oppositethat is true. If the disorder correlation length becomes smaller, we are driven

18

closer to the attractive �xed points, enhancing the precision of the plateaus1.Conclusively, because of the nontrivial topology of the systems edge, the scaleinvariance property of the system becomes detectable in the macroscopic Hallconductance. The scaling behavior thereby captures all the basic features of thequantum Hall e�ect.

Figure 2.3: Flow diagram in the plane of conductances, with σH in units ofe²/h. As the system �ows to the strong coupling regime, the Hall conductanceaquires a quantized value.

It is established29 that the structure of �xed points doesn't apply speci�-cally to the QHE, but that it is a general future of the instanton vacuum. Asmentioned, the non-linear sigma model with arbitrary number of �eld compo-nents is a very general theory about spontaneous symmetry breaking. Since itis found that the number of �eld components don't alter the scaling functions,it is even more exciting that the �xed point structure can be directly measuredin a quantum Hall experiment. Particular realizations of the instanton vacuumwith a certain number of �eld components correspond to systems in variousuniversality classes, with di�erent critical exponents. The general features ofthe theta vacuum apply to all these classes and are thus called `super-universal'.The most notable super-universal features are: massless chiral edge excitations,the robust quantization of the Hall current (or more generally, the topologi-cal charge) and the scaling behavior governed by a periodic structure of �xedpoints.

In short, Pruisken found that, by super-universality, the non-linear sigma

1There are many kinds of disorder. Here we refer to disorder with a small correlation

length. This is like comparing a polished surface with a rough one.

19

model with theta angle describes and explains the QHE. Because it now doesn'tmatter whichM,N are chosen in the SU(M+N)model, we can takeM = N−1and N → ∞ (the large N expansion) to obtain the same general features asin the replica limit M,N → 0. This is a great advantage, because the CPN−1

model is exactly solvable and thereby we have a very unexpected model thatis able to describe the physics of the theta vacuum from the strong couplingside. At criticality, the electron gas delocalizes and the edge excitations becomerelevant. Therefore, the symmetry of the theta vacuum is broken and multiplephases appear. In the strong coupling regime (the true quantum regime) thephases and their transitions become visible!

In �gure 2.3, we see how the conductances renormalize from weak to strongcoupling. The periodicity in theta (in units of θ = π, θ = 2π, etc..) is an exactresult of the CPN−1 model. The staircase picture (�gure 1.1) can be understoodas follows. When we start with any non-critical value for the unrenormalizedHall conductance, the renormalized or measured Hall conductance takes exactvalues, independent of sample-geometry. The electrons here are strongly local-ized except for massless chiral edge excitations that form the Hall conductance.At criticality, the unrenormalized conductances experience the in�uence of asaddle-point (θ = 1

2πn) and we have a true quantum phase transition. Such atransition is characterized by a unique critical exponent that can be measured inthe laboratory. The value of this critical exponent is not a super-universal fea-ture of the instanton vacuum. In Pruisken's modern work, the critical aspectsof the delocalization mechanism are being investigated. Because of the largesensitivity of the critical system, even the in�nitely long ranged Coulomb in-teractions become relevant. The Fermi-liquid approximation is then inaccuratemaking the theory extremely complicated.

3 Experiments on Critical Behavior

3.1 Introduction

In collaboration with Pruisken, H.P. Wei30;31 investigated the scaling behaviorof the QHE. By lowering the temperature, the system scales to larger e�ectivesystem sizes and a �ow diagram in accordance with �gure 2.3 was observed. Af-ter more precise measurements10 Wei observed power law scaling and universalexponent values for the �rst three plateau transitions. This was in agreementwith the scaling behavior that was derived by Pruisken32 on the basis of therenormalization group theory in the presence of a theta angle. Eventually, themeasurements demonstrated that it is possible to observe true quantum criti-cal behavior in the QHE and hence to research the instanton vacuum and itssuper-universal aspects experimentally.

20

3.2 Quantum Hall Laboratory

It is not an easy task to perform measurements on criticality. The narrowregime approaching self-similarity comes with an exponential sensitivity for theobservables. Any critical exponent has to be �tted on a log-log plot requir-ing a window that spans decades of the involved physical quantities (insteadof multiples) if it is to be unambiguously determined. The quantum Hall ef-fect is actually only a very rare occasion where the theory of continuous phasetransitions can even be tested. For example, measuring the critical exponent ofa normal liquid-gas transition involves a trip to outer space, just because theearth gravitational �eld introduces a small density gradient enough to make thecritical point vary too signi�cantly. A similar situation happens in the QHE,when there are macroscopic inhomogeneities that cause variations in the elec-tron density over the sample (in the lowest phase transition of the QHE thereis a clever way to avoid this problem33). If the width of the uncertainty in thecritical point's value isn't much smaller than the width of the phase transitionitself, criticality cannot be observed.

The accessibility of the scaling regime then depends sensitively on the sam-ple choice. In the past decades, a lot of technological progress has been madein the fabrication of all kinds of samples. There are very clean materials andsamples with a high electron mobility, but also samples with controlled impu-rity additions and samples with a more homogeneous electron density. As wehave seen, the correlation length of the disorder potential always plays a crucialrole in observing quantum scaling phenomena34. If the disorder is too smooth,one observes traces of the percolation mechanism (see �gure 1.5) in�uencing thecritical exponent. This is the case for a GaAs-AlGaAs heterostructure where theionized impurities are in the opposite layer, away from the two-dimensional elec-tron gas and therefore inducing a smooth disorder potential landscape. Indeedthe measurements on scaling that were being performed on the GaAs-AlGaAsheterostructure, yielded non-universal critical exponents35 or even the absenceof scaling36. On the opposite side, when the disorder correlation length is toosmall, the states are not su�ciently localized to observe them delocalize. Inboth cases, very large e�ective system sizes, at inaccessibly low temperatures,are needed to obtain true quantum criticallity.

A modern material, graphene, has such a high electron mobility that thequantum Hall e�ect can even be measured at room temperature. It would bea great opportunity to measure the critical behavior from room temperaturedown to millikelvins, which is a truly huge temperature window. But again thisis going to be di�cult, if not impossible. At high temperatures, the Landaulevels in graphene are valley- as well as spin-degenerate, which means that wewould be measuring transitions between two or four levels simultaneously. Thedegeneracy alters the scaling behavior in an uncontrollable fashion. Only at lowtemperatures thermal excitations between the levels are rare enough to studyphase transitions independently.

In the Quantum Hall laboratory, strong coupling problems can be researchedfrom a unique perspective. It might seem strange that such a large, `dirty' sys-

21

tem provides such `pure' and fundamental information. But also in the classicalregime we see a surprisingly universal result arising from the Hall experiment.Over a relatively large range of magnetic �eld strengths, the longitudinal re-sistance is independent of the magnetic �eld up to a precision of a few partsper-mil 33. This means that in the σxx, σxy plane, the experimental data exactlyfollows a semi-circle. For the lowest electron density, we see in the inset of �gure3.1 that the system is departing the semi-circle for lower temperatures and is�owing to the strong localization regime, where eventually the quantum Halle�ect appears (which is not at all visible on this scale).

Figure 3.1: In the classical regime, the conductivities follow a semi-circle withgreat precision. Inset: the colder system departs from the semi-circle. (Imagecourtesy, Pruisken et al.33)

3.3 The Critical Exponent

In 1988, After testing many samples, H.P. Wei's choice to measure scaling inthe quantum regime, fell on an InGaAs-InP sample where electron scatteringoccurs due to the di�erent atoms in the alloy. It was therefore expected that thedisorder correlation length was of the order of the lattice spacing. To extractthe critical exponent, temperature scaling of the e�ective sample size was used.The following steps that were derived by Pruisken32, illustrate how the criticalexponent can be identi�ed. Here, we can see how renormalization group theorypoints out the critical exponents in the phase transitions of the QHE.

The �eld theory of Anderson localization implies that the scaling dependenceof the conductance is a universal function F of only the conductance itself.The Anderson theory with theta angle implies this for the combination of thelongitudinal conductance σ0 and the Hall conductance σH . Near a �xed point,

22

the conductances scale algebraically with length. In general, the environment ofthe �xed point is nonlinear and we must work within curvilinear coordinates37.Suppose we have a starting point for scaling L0 and we denote the correspondingconductances as:

σ0H ≡ σH(L0, B) (3.1)

σ00 ≡ σ0(L0, B)

(Where σ0H can be identi�ed as the �lling fraction of the Landau band) Then

near the �xed point ( 12 , σ∗0) we can express the curvilinear coordinates (µH , µ0)

as Taylor series around the �xed point:

µH = (σ0H −

1

2)

[1 + α(σ0

0 − σ∗0) + β(σ0

H −1

2)2 + ...

](3.2)

µ0 = (σ00 − σ∗

0)

[1 + γ(σ0

0 − σ∗0) + δ(σ0

H −1

2)2 + ...

](Where the expansion is di�erentiable). As in the linear case, these nonlinearcoordinates also scale algebraically with length in the large scale limit37. Thus,if we change the length scale L0 → L we will have:

µH →(L

L0

)yHµH (3.3)

µ0 →(L

L0

)−y0µ0

Which allows us to conveniently de�ne the scaling �elds, that are the renormal-ized coordinates (�elds):

X ≡(L

L0

)yHµH (3.4)

Y ≡(L

L0

)−y0µ0

We can then express the macroscopic or measured conductances in terms ofuniversal scaling functions FH,0:

σH(B,L) = FH(X,Y ) (3.5)

σ0(B,L) = F0(X,Y )

The experimentally measured resistances RH,0 also depend on these scalingvariables since they are directly related to the conductances by matrix inversion.We can thus di�erentiate them with respect to the magnetic �eld:

∂RH,0∂B

=∂RH,0∂X

∂X

∂B+∂RH,0∂Y

∂Y

∂B(3.6)

23

Because Y scales as an irrelevant parameter, we take Y → 0 as L → ∞ whileX is �xed. Here, B enters the equations via the starting points for scaling, thebare conductances σ0

0 , σ0H . Also derivatives renormalize and we can again apply

the renormalization trick (including an arbitrary function f):

∂X

∂B(B,L) =

(L

L0

)yHf(

(L

L0

)−yHX) =

(L

L0

)yHf(0) (3.7)

Where in the last equality we used L→∞. If we now analyze at the maximumslope of RH (which is not the critical point):[

∂RH∂B

]max=

(L

L0

)yH [∂RH∂X

(X)f(0)

]max(3.8)

The expression between square brackets on the RHS does not depend on B orL anymore, because X is at a �xed value that ensures the maximum (and Band L enter the equations only via X).

For R0, the maximum slope is related to the half width of the visible peak(see �gure 1.1), which is approximately Gaussian32. This gives a width, 4B,and we can write:

4B ≈ 4µH∂B

∂µH=

(L

L0

)−yH [∂X

∂B

]−1

(3.9)

And, as in equation 3.7, the expression between square brackets doesn't dependon B or L.

The e�ective sample size scales with temperature provided Leff � L andwe can substitute temperatures for lengths to obtain:[

∂RH∂B

]max∝

(T

T0

)−κ

(3.10)

4B ∝(T

T0

)−κ

We therefore have two independent measurements that give the value of thecritical exponent κ. For a list of di�erent temperatures we can sweep B over thephase transition, �nd the maximum and measure the critical exponent withoutknowing the location of the critical point. This provides access to the scalingregime and allows the observation and proof of criticality.

H.P. Wei was able to measure a critical exponent κ = 0.4210that was thesame in the transitions at the centers of the �rst three Landau levels. Thegeneral way in which scaling theory introduces itself in a model implies in thiscase that the critical exponent is the same for the longitudinal as well as theHall resistance. Luckily the mixing of data that appears because the resistancesare taken from di�erent parts of the sample doesn't spoil the opportunity tomeasure a critical exponent.

24

Over a decade later, de Visser et al.38 preformed measurements on the sameInGaAs-InP heterostructure but found κ = 0.57 for the critical exponent. Acalculation on mesoscopic level33 could explain the di�erence with the value κ =0.42 due to electron density inhomogeneities. Those density inhomogeneitiesintroduce a kind of disorder with a macroscopic localization lenght and thereforethey have to be treated in an entirely di�erent fashion. The critical exponentvalue that H.P. Wei was measuring had a lower value because he was measuringan e�ective exponent.

But still, the true critical exponent wasn't found. In very recent years,the Princeton group11 measured the exponents on state-of-the-art AlxGa1-xAs-AlGaAs samples with a controllable alloy scattering and very little sample in-homogeneities. Adjusting the alloy, they found critical exponents between thevalues 0.42 and 0.58. Crucially, the data of the di�erent alloys all converged toκ = 0.42 (the �rst `H.P. Wei value') for the zero temperature limit. With thisexperiment we can clearly observe that if the temperature is lowered to makethe e�ective sample size large enough, the true quantum critical behavior of thesystem becomes visible and the critical exponent value becomes unambiguous(see �gure 1.5).

Conclusively, De Visser et al. measured a larger value κ = 0.57 in theInGaAs-InP sample. This had to be accounted for by an unforeseen, long rangeddisorder correlation (again a disorder of a di�erent kind). The fact that H.P.Wei found a value already in 1988 that was equal to the value of the latestmeasurements on scaling in the QHE today might seem mysterious, but inphysics we must consider it a complete coincidence...

3.4 Fermi-liquid principles

It is standard in scaling experiments on the QHE, to interpret the correlationlength exponent using Fermi liquid ideas (or free electron theory). As we haveseen in subsection 3.3, the canonical correlation length exponent can be ex-tracted by assuming �nitely ranged Coulomb interactions. We assume thatthe only e�ect of the Coulomb interactions is to dephase the electrons afterthey collide. The radius of the electrons quantum coherent �eld is then lim-ited by its inelastic scattering behavior. The inelastic scattering length has tobe much larger than the localization length for Anderson theory to apply. Ifthis is the case, the length scale with which the measured conductance scales iswell-determined by an e�ective system size. This e�ective system size inducedby Coulomb scattering scales with temperature. The critical scaling exponentthat can be associated is called the inelastic scattering exponent. Unfortunatelythis exponent cannot be calculated, but has to be taken as a phenomenologicalquantity.

The assumption of an e�ective system size is generally believed to be valid inthe regime of strong localization. We can now identify the critical exponent κ asthe ratio of the inelastic scattering length exponent p (how length relates to tem-perature) and the localization (correlation) length exponent ν = 1/yH and writeκ = p/2ν. In a modern context, the dephasing mechanism is called dynamical

25

screening and a dynamical critical exponent z is used to describe how lengthrelates to time instead of temperature. Because the time between electron-electron collisions is inversely proportional to the temperature, the measuredcritical exponent κ is identi�ed as κ = 1/zν.

The canonical correlation length exponent was calculated numerically39;40

as well as analytically41 and values were found between ν = 2.3 ∼ 2.4. This wasconsidered to be in agreement (or at least not in disagreement) with experimentson �nite size scaling42 and current scaling43, where ν can be measured directly.Because we have κ = 1/zν, agreement of the calculated value ν = 2.3 ∼ 2.4 withthe well-known experimental value κ = 0.42 required z ≈ 1. But if we assumethat Coulomb interactions only dephase electrons, z = 2 is the appropriate `freeelectron' value to work with. Then, a number of calculations on the dynamicaldimension of interacting electrons44;45 suggested that it is possible to put z ≈ 1,while maintaining the Fermi-liquid properties of free electrons. Therefore ν =2.3 ∼ 2.4 was accepted for the quantum Hall regime and Fermi-liquid principles(as well as the percolation picture) remained to be pursued in the context ofcritical behavior in the QHE.

More recently, in 2009, Li et al.46 published new results (see �gure 1.5) onscaling in the quantum Hall regime using state-of-the-art GaAs-AlGaAs sam-ples with a controllable alloy scattering and very little sample inhomogeneities.Measuring κ = 0.42 and assuming z ≈ 1, they obtained ν = 2.4 con�rming theoriginal, accepted value for correlation length exponent in the Quantum Hallregime. However, in the same year, the correlation length exponent was calcu-lated again in a Chalker-Coddington network model47 and it was demonstratedthat in fact ν = 2.6. We must have z > 148, so when using κ = 1/zν, thevalue ν = 2.6 is in disagreement with the value κ = 0.42. This indicates thatthe theory of free electrons is not su�cient when the system becomes quantumcritical.

4 Other issues

4.1 Coulomb Interactions

The quantum Hall e�ect cannot be in the Fermi-liquid universality class andin�nitely ranged Coulomb interactions have to be taken into account. At criti-cality, the localization length becomes in�nite and thereby at least comparableto the inelastic scattering length even for very low temperatures. This is notin accordance with the assumption of a localization length that is much smallerthan the inelastic scattering length. The reason why the concept of dynamicalscreening is going to help us out here remains unclear. If we want to work in acomplete, �eld theoretical language we must forget about inelastic scattering orCoulomb screening in the critical regime and work with a theory including in-�nitely ranged Coulomb interactions. The numerical network simulations thatare based on percolation ideas are calculating a canonical correlation lengthexponent that can only be related to the quantum Hall laboratory due to the

26

concept of dynamical screening.The very di�cult task of including interactions was undertaken by Pruisken

and his collaborators. Their starting point was an obscure method used byFinkel'stein49 involving �eld matrices of in�nite size. Basic solutions like topo-logical excitations, �nding global symmetries and identifying physical observ-ables that were obtained in the context of Fermi-liquid theory become mean-ingless when in�nite interactions are included. Pruiskens ground work28;50�52 isthe starting point for studying Coulomb interactions in the quantum Hall e�ect,and hence for studying its critical regime. In Pruiskens work, the new (quantumHall) universality class is called the F − invariant universality class.

4.2 Super-Symmetry

Besides constructing a fundamental theory for the quantum Hall e�ect, it can beimportant to invent mathematical toy models that are useful during the di�er-ent phases in the research on the quantum Hall e�ect. For example, to calculatethe exact value of the critical exponent Zirnbauer53 tried to construct a discretevertex model that falls in the same universality class as the quantum Hall tran-sition. The super-symmetric versions of the non-linear sigma model with thetaangle and the Chalker-Coddington network model are shown by Zirnbauer topossess the same global symmetries at criticality. Global symmetries de�ne auniversality class, so both models must be in the same class. Since the super-symmetric Chalker-Coddington network model does not �t within the usualframework of integrable models, he needs to construct another vertex modelthat is in the same class and integrable. With this vertex model the criticalexponent could be calculated. But, since Zirnbauer used the non-linear sigmamodel without Coulomb interactions, this critical exponent is unfortunately stillthe Fermi-liquid universality class. Constructing the Zirnbauer vertex model istherefore unimportant in the context of the quantum Hall e�ect.

Interestingly, because of the super-symmetric relation between the non-linearsigma model with theta angle and the Chalker-Coddington vertex model, thesigma model is a continuum version of the discrete, Chalker-Coddington vertexmodel. Since this vertex model is a random hopping model including a phasedependence, it represents the percolation picture including quantum tunneling.This explains why semi-classical lattice percolation was so close to explainingthe quantum Hall e�ects critical behavior even though it didn't make any sensein the general explanation of the staircase picture. At the Fermi-liquid crit-ical point, the non-linear sigma model without Coulomb interactions and thepercolation mechanism share the same universality class. By considering onlythe critical exponent itself, the semi-classical percolation mechanism appearedto be a valid substitute for the Anderson localization mechanism, but sincethe universality class of the quantum Hall transition is the formerly unknownF − invariant universality class, there is no such valid substitute for studyingAnderson theory in the QHE.

27

5 Summary

The quantum Hall e�ect has an important and fundamental role in moderncondensed matter physics. Its history of over tree decades involved thoroughtheoretical and experimental research. The QHE involves some of the mostinteresting features in todays laboratory, like a quantum �eld of 2D electrons,quantum magnetization, phase transitions in a random potential and edge cur-rents around an insulating bulk system. Semi-classical arguments like perco-lation of electronic states, skipping orbits resulting in conducting edge statesand a Fermi-liquid approximation for the critical exponent can never providea complete, theoretically justi�ed framework. This is a general problem in thearea of topological insulators.

For the QHE, the modern picture is based on the notion of instantons inthe theory of Anderson localization. The non-linear sigma model with M +Ncomponents describes the 2D disordered electron gas. With the renomaliza-tion group theory, Anderson localization can be proven, including the fact that,without a magnetic �eld, `All 2D states are localized'. The de-localization mech-anism is constructed by including the theta angle, a topological property of 2Delectrons in a magnetic �eld. Renormalization by instantons of the theta angleprovides an e�ective theory for the edge of the system. This edge-theory hassuper-universal features (massless chiral edge excitations, robust quantizationof the Hall conductance, etc..) that explain the quantum Hall e�ect.

The calculation of the exact value of the critical exponent remains very di�-cult. In the critical regime, the Fermi-liquid approximation becomes invalid andhence electrons do not behave like free particles that scatter o� each other aftertraveling their inelastic scattering length. In the future, a lot of interesting the-oretical work can be done on including in�nitely ranged Coulomb interactionsin the non-linear sigma model. Experimentally, there are many opportunitiesto continue on the QHE, like the QHE in graphene and studying other topo-logical insulators. The critical behavior of such mesoscopic quantum devices isin my opinion the most interesting phenomenon in all of our condensed matterlaboratories.

28

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