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Physical principles underlying the quantum Hall effect

In this contribution, we present an introduction to the physical principles underlying the quantum Hall effect. The field theoretic approach to the integral and fractional effect is sketched, with some emphasis on the mechanism of electromagnetic gauge anomaly cancellation by chiral degrees of freedom living on the edge of the sample. Applications of this formalism to the design and theoretical interpretation of interference experiments are outlined.

The reason the quantum Hall effect (QHE) is relevant to the subject of this colloquium, metrology, lies in the circumstance that it yields a highly precise experimental value for the von Klitzing constant This constant plays a fundamental role in the QHE: The Hall conductance of a twodimensional incompressible electron gas (2DEG) exhibiting the QHE turns out to be an integral or rational multiple of R −1 K . Its significance for metrology is clearly an important aspect of the QHE. Apart from that, the QHE is a fascinating phenomenon, because its theoretical description is related to quite fundamental and abstract concepts in mathematics and theoretical physics, such as fractional-or braid statistics, tensor categories, knot theory, 2D conformal field theory (CFT), and 3D topological field theory (TFT); see Fig. 1. In these notes, we present a short introduction to some of the concepts underlying the theory of the QHE. We also provide a list of important references, with emphasis on our own contributions. Experimentally, one can measure the longitudinal resistance, R L , as well as the transverse Hall resistance, R H : Let n denote the density of electrons in the 2DEG, and let be the quantum of magnetic flux. The dimensionless quantity is called the filling factor. The filling factor corresponds to the number of filled Landau levels for a gas of free spinless fermions of charge −e. In Eq. (3), B 0⊥ is the component of the external magnetic field perpendicular to the plane of the 2DEG.

Classical theory
We start by studying the classical mechanics of a 2DEG exhibiting the Hall effect. In a steady state, where the electrons in the 2DEG have a constant velocity, the total force on an electron must vanish. Hence and the Hall conductivity, σ H , is apparently given by We observe that classical theory predicts a linear relation between the Hall conductivity and the filling factor ν, with a factor of proportionality given by R −1 K = e 2 /h. Interestingly, experiments with Hall samples at low temperature and in strong magnetic fields yield a behaviour of σ H that deviates from the classical linear relation in (7). Experimental data, sketched in Fig. 3, show plateaux where σ H is very nearly constant. Whenever (ν, σ H ) lies on a plateau, the longitudinal resistivity vanishes and σ H only takes certain values (see Fig. 4). There is ample experimental evidence for the following claims.
(IV) If R K σ H / ∈ (fractional QHE), some of the quasi-particles observed in the sample appear to carry fractional electric charges [25][26][27]. The precision of the integral plateau heights is of the order of 10 −9 . Thus, systems exhibiting the QHE allow for an extremely accurate determination of R K = h/e 2 . Together with Josephson junction experiments measuring the fundamental quantity K J = e/(hc) and quantum pumps, which determine the elementary charge e, the metrological triangle closes [15].

Tasks for theorists
Given these experimental findings, the following theoretical questions arise: Answers to these questions would have to be based on a detailed understanding of the quantum many-body problem in the presence of disorder and interactions. In situations relevant for the fractional QHE, quantitative insights are primarily based on large-scale computer simulations [22][23][24]; but see [18][19][20][21]. However, for a 2DEG consisting of non-interacting electrons in a random external potential, one only 5 observes the integral QHE, and the theory of this phenomenon is well understood [49,50]. Questions of this sort can be studied and answered with the help of an elegant effective field-theory approach. In the following, we outline this approach.

Applications
The QHE has many important (or potentially important) applications, such as: • Metrology, determination of fundamental constants of nature, definition of a resistance standard, [16].

Electrodynamics of an incompressible Hall fluid
Consider a 2DEG confined to a planar region Ω and subject to a strong, uniform external magnetic field B 0 transversal to Ω. In such a system, the vanishing of the longitudinal resistance R L is a signal for the existence of a mobility gap in the bulk. One then speaks of an incompressible Hall fluid. Let us consider the response of the system to a small, slowly time-dependent perturbation of the electromagnetic (EM) fields, with The orbital dynamics of electrons in the region Ω (assumed to be contained in the x-y plane) only depends on B total 3 = ( B 0 + B(x)) · e z and E = E(x) = (E 1 (x), E 2 (x)). We set B = B · e z and introduce a vector potential, for the electromagnetic field tensor in 2+1 dimensions, The expectation value of operators in a (quasi-stationary) state of the 2DEG in an external vector potential A is denoted by (·) A . For example, the electric charge-and current density is given by with µ = 0, 1, 2, where J µ (x) is the quantum-mechanical current density.
From phenomenological and fundamental laws of physics the following equations can be derived: The electric current is perpendicular to the electric field, i.e., with k, l = 1, 2, where ǫ kl is the sign of the permutation (kl) of (12), and (ii) Charge conservation Charge-and current density in Λ satisfy the continuity equation (iii) Faraday's induction law The laws (i) through (iii) imply that We integrate Eq. (16) in time, with integration constants chosen such that where −en is the charge density of a homogenous 2DEG in a constant magnetic field B 0 . We then arrive at (iv) "Chern-Simons Gauss law" [48] Next, we propose to show that the laws (i) through (iv) imply the existence of anomalous chiral currents circulating at edges of the incompressible Hall fluid. Faraday's induction law (iii) says that which (by Poincaré's lemma) implies that the EM field tensor can be derived from a vector potential, In compact notation, laws (i) and (iv) can be written as Whenever σ H is constant, the current (21) satisfies the continuity equation (ii), i.e., However, wherever the value of σ H jumps, e.g., at the boundary of the sample, the current Then we have that which violates the law (ii)! The apparent contradiction between (24) and the continuity equation (ii) disappears when one notices that the current (21) is not the total current. Apparently, there must be an additional current supported on Σ that cancels the anomaly (24): with Equation (24) for the bulk current (21) then implies that, on the "edge" Σ, where E | Σ denotes the electric field "parallel" to Σ (i.e., the component of E| Σ parallel to the contour lines of σ H ) and ∆σ H is the discontinuity of σ H across Σ. This nonconservation of the edge current is called chiral anomaly in 1 + 1 dimensions. The chiral anomaly (in 3+1 dimensions) is a well-known phenomenon in gauge theories of elementary particles. It plays an important role in various physical processes; see Refs. [9,31]. In Fig. 5, an illustration of the edge current in a quantum Hall sample is given. The velocity v of an electron at the edge can be calculated by equating the Lorentz force and the confining force, In classical physics, a phenomenon analogous to the chiral edge currents in an incompressible Hall fluid are the hurricanes in the atmosphere of the earth! In this case, the magnetic field B is replaced by the angular velocity of the earth, ω earth , and the role of the Lorentz force is played by the Coriolis force. The confining force, − ∇V edge , in a Hall fluid is replaced by the gradient of the air pressure, − ∇P . 8

Chiral anomaly in 1 + 1 dimensions
An anomalous current satisfying (26) is carried by charged, chiral, gapless "modes", i.e., by particles traveling with a certain velocity along the edge. Let us suppose that the current j µ edge is carried by N species of chiral modes. We denote their coupling constants to the EM field by eQ 1 , . . . , eQ N . The anomaly of j µ edge is then described by (see our discussion in Sect. 5) Combining this equation with (26) (with ∆σ H = σ H ) it follows that the (dimensionless) Hall conductivity is given by One can convince oneself that, in the integral QHE, each filled Landau level gives rise to exactly one species of electrons circulating at the edge and thereby contributing to the edge current j µ edge . Therefore, R K σ H = N is the number of filled Landau levels. For an incompressible Hall fluid exhibiting the fractional QHE, with R K σ H / ∈ Z, it follows that at least one of the "charges" eQ i must be a fraction of the elementary charge e. Arguments similar to the ones reported here can be found, e.g., in [3,9,32].

Effective action of an incompressible Hall fluid and topological field theory
In this section, we determine the effective action of an incompressible Hall fluid (IHF). Here, and in the following section, we express the Hall conductivity in units of e 2 /h, i.e., where σ is dimensionless. The space-time of the sample is the cylinder Λ = R × Ω.
For simplicity, we assume that the support of ∇σ is Σ = ∂Ω; (of course, this is an idealization of what one encounters in real samples). We denote the surface of the 9 cylinder by ∂Λ = R × ∂Ω. The quantum-mechanical current operator is J µ (x), and (·) A is the expectation value in a stationary state of the IHF in an external EM field with vector potential A.
The effective action of an IHF, Γ Λ [A], is the generating functional of the current Green functions. Hence it satisfies The current in the bulk of the Hall sample, Eq. (21), is given by After integration, (31) and (32) then yield the following expression for the effective action.
where a := A| ∂Λ , and Γ[a] is the generating functional of the edge-current Green functions (up to terms local in a).
The Chern-Simons action Λ A ∧ dA in (33) is not invariant under a gauge transformation, A → A + dα with α| ∂Λ = 0. In fact, we find that where we have used Stokes' theorem. In our case of a time-independent sample Ω, we have that da = ∂ µ a ν dx µ ∧ dx ν = E dt ∧ dξ, where ξ is a coordinate parametrizing ∂Ω, and is the electric field parallel to the edge. The total action Γ Λ , however, must be gauge invariant (conservation of electric charge). Therefore, the violation of gauge invariance described in (34) must be cancelled by the edge action Γ[a], which is then found to be given by (up to manifestly gauge-invariant terms), assuming that all chiral edge modes have the same propagation speed and direction; (the general case will be discussed in Sect. 5). By (31) and (33), where µ, ν ∈ {0, 1}, and, using (36), in accordance with Eq. (26).
We would like to emphasize that, up to gauge invariant terms, the effective action for the edge current, (36), is uniquely determined by the requirement of electric charge conservation. The only possible generalization is to consider several independent edge channels of charged quasi-particles. In contrast to the bulk contribution to the total action (33), the edge action is not topological, i.e., it depends on the space-time metric of the edge. Therefore, in general, each edge channel may couple to a different space-time metric (i.e., exhibit a different propagation speed). We will discuss this point in more detail in the next section.
The total electric current, J µ , is conserved, By Poincaré's lemma, it can therefore be derived from a vector potential, which we denote by B, The potential B in (40) gives rise to the gauge symmetry B µ → B µ + ∂ µ β: J µ does not change under a gauge transformation of B. The action describes the theory of the gauge potential B coupled to an EM vector potential A.
In (41) Thus, the exponential of Q O , is a Wilson loop operator for the field B associated with the contour ∂O. Wilson loops and -networks furnish the "observables" in a 3D topological field theory (TFT). Static sources of B inserted in the bulk at a point z ∈ Ω are described by vectors in a Hilbert space, |(q, λ), z ∈ [(q, λ), z] with whenever O contains the insertion point z. Here, q is the flux of the field B, and λ is some additional "internal" quantum number needed to label the sectors of the TFT describing the bulk of an IHF. The state vectors |(q, λ), z are elements of a sector (subspace) of the total state space denoted by [(q, λ), z] .
Sectors are thus labeled by (q, λ) and an insertion point z ∈ Ω. The fact that the bulk theory has trivial dynamics (static sources, purely local current correlators) is a consequence of the mobility gap in the bulk, after passing to the scaling limit. 11

Fusion of sources
Next, we discuss properties of states in a TFT with several distinct sources. The sources in a TFT constitute a fusion algebra. This means the following: Consider the tensor product space corresponding to two sources located at z 1 and z 2 , denoted by [(q 1 , λ 1 ), z 1 ] ⊗ [(q 2 , λ 2 ), z 2 ]. As the locations z 1 and z 2 of the two sources approach the same point z, the tensor product can be written as a direct sum, where γ i = (q i , λ i ) and q = q 1 + q 2 . The non-negative integers N γ γ1γ2 are called fusion rules; they are the multiplicities of the spaces [(q, λ), z] in the tensor product space. The morphisms, F γ,a γ1γ2 , from [γ 1 , z 1 ] ⊗ [γ 2 , z 2 ] to the space [γ, z] are called fusion matrices, with a = 1, . . . , N γ γ1γ2 ; (see Fig. 6). In a "physical" theory, i.e., for a quasi-rational TFT, all multiplicities in the "Clebsch-Gordan series" (47) must be finite, more precisely, γ N γ γ1γ2 < ∞, for all pairs {γ 1 , γ 2 }. If N γ γ1γ2 > 0, then γ is called a fusion channel for γ 1 and γ 2 . A TFT is abelian if Figure 6: Fusion of two sources in a topological field theory; ("b.c." stands for a boundary condition). γ N γ γ1γ2 = 1, for all pairs {γ 1 , γ 2 }. In Sect. 5, we will focus on edge theories dual to abelian TFTs in the bulk.

Constraints on physical TFTs
The spin s q,λ of a state is determined by considering a rotation in the plane through an angle of 2π. Let U rot (2π) represent the rotation by 2π around the origin. Then U rot (2π)|(q, λ), z = e 2πis q,λ |(q, λ), z , (49) and the spin s q,λ is given by (see, e.g., [1,11]) where, for a quasi-rational TFT, as defined above, Vafa's theorem [33] implies that So, in general, s q,λ / ∈ 1 2 , and the theory may have quasi-particles in its spectrum, that have fractional spin and are neither bosons nor fermions (so-called anyons).
In a theory describing a physical IHF, there must exist bulk states with the quantum numbers and properties of one-electron states. Suppose that the state |(q * , λ * ), z is obtained by adding a single electron at the point z ∈ Ω to the groundstate of the IHF. Then we have the following constraints: The charge of this state, see (45), must be √ σ q * = −1 .
Furthermore, the spin, see (50), must be half-integer, i.e., with l ∈ . From Vafa's theorem we know that ∆ λ * is rational. It thus follows that the Hall conductivity, σ, is rational, There is a third constraint on physical theories: the so-called relative locality of all quasi-particle states with respect to electron insertions. We will not discuss it here; but see Refs. [6,11] for more information. Using these three constraints on a theory describing a physical IHF, one can show that the smallest electric charge of a quasi-particle that can appear in an IHF is given by where f ∈ AE is an integer (namely the order of the simple current corresponding to the insertion of an electron); see, e.g., [11].
To conclude this section, we remark that one may view the spaces labeled by (q, λ) as sectors of a chiral algebra describing some chiral conformal field theory (CFT) [34]. Abstractly, they can be understood as the "irreducible objects" of a braided tensor category [35,36].

Monodromy and braiding
Let us consider the transformation of a state describing two sources when the sources are adiabatically carried around one another, as depicted in Fig. 7. This corresponds to a rotation of the two sources through an angle 2π. After subtracting the contribution of the spins of the sources, the monodromy matrix, M , is defined by ,λ2),z2] = e 2πi(s q 1 ,λ 1 +s q 2 ,λ 2 ) M (q1,λ1)(q2,λ2) (56) 13 We may fuse the tensor product states on both sides. Using (48) and (49), we get λ1)(q2,λ2) . (57) This shows that the monodromy matrix M (q1,λ1)(q2,λ2) is block-diagonal in the decomposition of the tensor product space into the subspaces [(q, λ), z], and its eigenvalues on these subspaces are given by The factor e 2πiq1q2 corresponds to the well-known Aharonov-Bohm phase for carrying a charged particle around an insertion of magnetic flux. In general, it may happen that M = ½, for some pairs of quasi-particles. The particles then exhibit braid statistics.
Braid statistics is an interesting phenomenon only encountered in two-dimensional systems. In dimensions larger than two, quantum statistics is always described by representations of the permutation group; see, e.g., [37] and refs. given there. The theoretical possibility of braid statistics in 2D systems appears to be realized in IHFs at certain fractional plateaux.

The edge of an incompressible Hall fluid
Next, we propose to find an action, S, for matter fields located on the edge, ∂Λ, that describe chiral modes coupled to the electromagnetic field. (In addition, there may be neutral modes, which we omit here.) The main constraint on the edge action is its gauge variation; i.e., under a → a + dα| ∂Λ , we find [cf. Eq. (34)] that Furthermore, the edge current, J µ edge , must satisfy the anomaly equation [see (38)] Since the edge degrees of freedom of an IHF form a system in 1+1 dimensions, we can make use of bosonization techniques. A current carried by gapless quasi-particles can be decomposed into left-and right-moving currents with opposite propagation directions, J L and J R . The vector current, J = J L + J R , is always conserved. This means that the anomalous edge current, J edge , must be chiral, i.e., there is an imbalance between leftand right-moving modes.
Conservation of the vector current, ∂ µ J µ = 0, allows us to introduce a (possibly multi-valued) scalar potential φ, i.e., In the absence of an external electric field, let us write the chiral edge current in terms of the scalar potential as When the external field vanishes, the edge current is conserved, which is the equation of motion for a massless free Bose field, φ, with action We choose a metric g µν on ∂Λ with g = det(g µν ) = −1. More precisely, where u is the propagation speed of φ. Next, we introduce the edge action, S[φ, a], for a non-zero vector potential a µ : S[φ, a] is required to yield the effective edge action (36), after functional integration over the matter field φ, i.e., This uniquely fixes S[φ, a] (up to gauge-invariant terms). It is found to be given by with ∂ µ φ = g µα ∂ α φ and a µ = g µα a α . The edge current in the presence of external fields [see (37)] is It exhibits the correct anomaly (61), on a solution to the equation of motion for φ.
To generalize our construction, N > 1 conserved vector currents can be introduced, The action for the fields φ = (φ 1 , . . . , φ N ) generalizing (68) is then given by where χ i ∈ {+, −} is the chirality of the edge current carried by the field φ i , and Q i ∈ R are some constants. The metrics, g i , may be different for each field, (g µν where u i is the propagation speed of J i . The action (71) has the correct gauge variation, and the edge current, exhibits the expected anomaly, provided that Using (71), the equation of motion for the field φ i is given by By inspection, under a gauge transformation a µ → a µ + ∂ µ α, solutions to (74) transform as This shows that the edge currents in (72) [and, in particular, (69)] are gauge-invariant objects.
Canonical quantization of the action (71) yields the equal-time commutators Hence, the currents J χi generate N chiral U(1) Kac-Moody algebras [38]. Using (76) and (73), the edge current (72) satisfies the commutation relation The cancellation of the gauge anomaly of the electromagnetic effective (Chern-Simons) action in the bulk by appropriate massless chiral field theories on the edge of the Hall sample is an example of the holographic principle (applied, here, to gapless quantum field theories in two dimensions and three-dimensional TFTs). A more conventional version of this principle tells us that there is a correspondence between certain 3D TFTs and 2D chiral conformal field theories (CFTs). This formulation is somewhat misleading, though, since (as our example shows) the massless edge modes may have different propagation speeds, i.e., the conformal symmetry may be broken. This is usually the case in realistic IHFs.
For abelian IHFs with N conserved currents, (71), the family of physical theories has been classified mathematically, [5][6][7][8]. For each fluid, it is possible to enumerate all quasi-particle excitations. For a given Hall conductivity, σ, and a certain number, N , of currents, the quasi-particles are labeled by vertices of a lattice, Γ * , of dimension N , dual to an odd integral lattice, Γ, of (multi-)electron excitations. This again implies that the dimensionless Hall conductivity σ is a rational number. Steps towards a generalization of this approach to non-abelian IHFs have been undertaken in [11,12,39,40].

Chiral vertex operators
So far, we have introduced an algebra of chiral currents, {J µ χj }, generating U(1) Kac-Moody algebras; Eq. (76). The next step is to construct quasi-particle creation-and annihilation operators as chiral vertex operators. For simplicity, let us discuss the case of an abelian IHF with only a single edge degree of freedom. The operator creates a charged quasi-particle at a point ξ = (ξ µ ) ∈ ∂Λ of the edge. N denotes normal ordering, and J µ edge is given in (69). The starting point of the line integral in (78) is some reference point, usually taken to be an ohmic contact. Note that a continuous deformation of the path in the line integral in (78) leaves ψ q (ξ) invariant. In other words, the vertex operators only depend on the homotopy class of the path. This is because the curl of the integrated vector field vanishes, The electric charge is measured by the operator The charge of the quasi-particle created by ψ q is obtained from the commutator Hence, the electric charge deposited by (78) is equal to √ σ q. This suggests that the vertex operators ψ q (ξ) are in one-to-one correspondence with bulk states |(q, λ), z introduced in the previous section.
Commuting two vertex operators yields the "statistical phase", The sign of the phase depends on the relative positions of ξ 1 , ξ 2 , and the starting point of the line integral (and on the homotopy classes of the paths). The conformal spin of the vertex operator (78) is given by These results are in accordance with the properties of the states |(q, λ), z of the bulk TFT discussed in Sect. 4, for ∆ λ = 0. In fact, the statistical phase appearing in (82) corresponds to "half-monodromies" in the bulk, see Eq. (58). The conformal spin (83) coincides with the spin of the bulk state, Eq. (50). We note that, under a transformation a → a + dα, the vertex operator transforms like ψ q (ξ) → ψ q (ξ) e −2πi √ σ q α(ξ) , as expected of an operator creating a particle with electric charge √ σ q. The quantum field theory for the edge of an IHF can be used to predict observable effects that can be tested in beam-splitting interference experiments using electronic versions of Mach-Zehnder or Fabry-Pérot interferometers (Fig. 8) [10,41]. For interference effects to appear, excitations need to be allowed to tunnel between different edges of the sample. This may be modeled by adding tunneling terms of the form V q (x; y) ∝ dt ψ † q (x, t) e 2πi y x a1(ξ,t) dξ ψ q (y, t)

Inter-edge tunneling and interference experiments
to the edge action. Important phenomena are: • The tunneling current from one component of the Hall edge to another one through a quantum point contact is related to the electric charge of the particles transmitted through the contact and to the scaling dimension of the tunneling operator (exp. [42]; theory [12]). 18 • Aharonov-Bohm oscillations in the tunneling current are studied in [13,41,43]. Remarkably, they have the electronic period, Φ 0 , if external flux tubes are added (topological screening), but may have quasi-particle period, Φ 0 /q min , if edges are deformed by a modulation gate; (q min is the smallest fractional charge observed in the fluid).