Abstract
We discuss the ordinary quantum Hall effect and a higherdimensional cousin. We consider the dimensional reduction of these effects to and spacetime dimensions, respectively. After dimensional reduction, an axion field appears, which plays the rôle of a spacetime dependent difference of chemical potentials of chiral modes. As applications, we sketch a theory of quantum pumps and a mechanism for the generation of primeval magnetic fields in the early universe.
Axions, Quantum Mechanical Pumping,
[5mm] and Primeval Magnetic Fields
[22mm] Jürg Fröhlich , Bill Pedrini
[7mm] Institut für Theoretische Physik
ETH Hönggerberg
CH – 8093 Zürich
[5mm] I.H.E.S.
35, Rte de Chartres,
F91440 BuressurYvette
[7mm]
[3mm] x
1 Introduction
In these notes, we clarify the rôle played by certain pseudoscalar fields related to “axions” in some transport or pumping processes in semiconductor devices and in the early universe. These processes are similar to ones observed in quantum Hall systems. We therefore start by recalling some key features of the theory of the quantum Hall effect. We then consider transport processes in very long, narrow rectangular Hall samples with constrictions, as shown in Figure 1.
For samples of this kind, filled with an incompressible Hall fluid, the component, , of the electromagnetic vector potential, , parallel to the short axis, , of the sample can be interpreted as a pseudoscalar field analogous to the axion known from elemantary particle physics [1]. In the region where the sample has a constriction, tunnelling processes between the chiral edge modes on the upper and lower edge of the sample may occur. It is interesting to consider the effect of turning on a timedependent voltage drop in the direction. Not surprisingly, we find that when such a voltage drop, , with
is turned on, an electric charge proportional to is transported through the constriction from the left, , to the right, . This system thus realizes a simple “quantum pump”. Due to the tunnelling processes between edge states of opposite chirality, the state of the pump exhibits a periodicity in proportional to the inverse electric charge of the charge carriers in the sample. Thus, such a pump can be used, in principle, to explore properties of the quasi particles in incompressible quantum Hall fluids, such as their electric charges, [2].
Our model can also be used to describe quantum wires carrying a Luttinger liquid. The rôle of the constriction is then played by impurities mixing left and right movers.
We then proceed to studying a fivedimensional analogue of the quantum Hall effect. If fourdimensional physics is described by dimensional reduction from a fivedimensional slab to two parallel boundary “3branes”, the axion can be interpreted as the component of the fivedimensional electromagnetic vector potential transversal to the branes. Tunnelling of chiral fermions from one to the other brane, due e.g. to a mass term, generates a periodic axion potential. It is then argued that the dynamics of the axion may trigger the growth of largescale primeval magnetic fields in the early universe. In other words, axion dynamics  which is coupled to the dynamics of the curvature tensor of spacetime  can be viewed as a realization of a quantumfield theoretical “pump” driving the growth of largescale primeval magnetic fields, [3][4][5]; see also [6]. Whether this mechanism plays a rôle in explaining the observed largescale magnetic fields in the universe is, however, still uncertain; see [7].
2 Brief Recap of the Quantum Hall Effect
We consider a uniform dimensional electron gas of density forming at the interface between a semiconductor and an insulator when a gate voltage is applied in the direction perpendicular to the interface. We imagine that a homogeneous magnetic field, , perpendicular to the interface is turned on. Let denote the “filling factor”. From the experiments of von Klitzing et al. [8] and Tsui et al. [9] we have learnt that, for certain values of , the dimensional electron gas forms an incompressible fluid, in the sense that the longitudinal resistence, , of the system vanishes. We consider the response of such a system to turning on a small external electromagnetic field , where denotes the inplane component of the electric field, and is the component of the total magnetic field perpendicular to the plane of the fluid. By we denote the current density in the plane of the dimensional electron gas, and by the deviation of the electric charge density from the uniform background charge density, ; (here , where is a point in the sample and is time).
By combining Hall’s law (for ), i.e.,
(1) 
where is the bulk Hall conductance, with the continuity equation for and and Faraday’s induction law, one easily finds that
(2) 
see [10]. Denoting by the electromagnetic field tensor over the dimensional spacetime, , of the sample and by the form dual to the chargecurrent density , eqs. (1) and (2) can be summarized in
(3) 
the field equation of “ChernSimons electrodynamics” [11]. Defining the dimensionless Hall conductivity, , by
(4) 
and using units such that , the field equations of ChernSimons electrodynamics are
(5) 
Taking the exterior derivative of eq. (5), we find that
(6) 
because . The gradient is transversal to the boundary, , of the sample’s spacetime. Eq. (6) tries to tell us that electric charge is not conserved in an incompressible Hall fluid, because , the dual of , does not vanish. The origin of this false impression is that, so far, we have neglected the diamagnetic edge current, , in our equations. This current is localized on and is dual to a vector field with support on and parallel to . The edge current saves electric charge conservation:
(7) 
(8) 
where is the component of the electric field parallel to the boundary of the sample, and the “edge” conductivity, , is equal to , the “bulk” conductivity, as follows from (6). Eq. (8) describes the dimensional chiral anomaly [12]. Apparently, the edge current, , is an anomalous (chiral) electric current localized on the boundary of the sample; (the chirality of depends on the direction of and the sign of the electric charge of the fundamental charge carriers).
Equations (8) and (5) can be derived from an action principle. If denotes the effective action, i.e., the generating functional of the current Green functions, of an incompressible Hall fluid confined to a threedimensional spacetime region , in the presence of an external electromagnetic field with vector potential , then
(9) 
where is the restriction of to the boundary, , of , and “” means that only the leading contributions (in the sense of dimensional analysis) to the effective action are displayed on the R.S. The first (bulk) term on the R.S. of (9) is the ChernSimons action, the second (edge) term turns out to be the anomalous chiral action [12] in two spacetime dimensions. Its gauge variation fixes the value of by
(10) 
Electromagnetic gauge invariance is a fundamental property of nonrelativistic manybody theory. Thus, must be gauge invariant, i.e.,
(11) 
for an arbitrary function on . Individually, the ChernSimons
action,
, and the boundary action
are not invariant under a gauge
transformation, , not vanishing on the boundary
; but the R.S. of (9) is gauge
invariant precisely if .
Since is the generating functional of the current Green functions, we have that
(12) 
These expressions, togheter with eq. (9) for , reproduce the basic equations (5) and (8).
The boundary action is known to be the generating functional of the chiral KacMoody current operators of current algebra with gauge group . It is then a natural idea [13][14] that the boundary degrees of freedom of an incompressible Hall fluid are described by a chiral conformal field theory. Under the natural assumptions that

sectors of physical states of this theory are labelled by their electric charge and, possibly, finitely many further quantum numbers (e.g. spin) with finitely many possible values; and

excitations of this theory with even/odd electric charge (in units where ) obey Bose/Fermi statistics,
one shows that is necessarily a rational number, and one obtains a table of values of that compares well with those of the dimensionless Hall conductivity of experimentally established incompressible Hall fluids, [13][14]. Moreover, one can systematically work out the spectrum of fractionally charged quasiparticles propagating along the edge of the sample. The smallest fractional electric charge turns out to be given by , where is a positive integer  and, for many fluids,  and is the integer denominator of , (writing , with and relatively prime integers); see [13][14].
3 Hall Samples with Constriction
and Quantum Wires
In this section we consider a very long, narrow rectangular Hall sample, as shown in Figure 1. The axis parallel to the long side of the sample is taken to be the axis, the one parallel to the short side is the axis, and we set . We define a field by
(13) 
where is a point on the lower edge of the sample, is a point on the upper edge, and is the straight line from to . We assume that the component,
(14) 
of the inplane electric field, , is independent of . It is convenient to choose a gauge such that is independent of . Then the effective action in equation (9) becomes
(15) 
where is the interval on the axis which the Hall sample is confined to. The terms corresponding to the upper and the lower edge in the boundary action on the R.S. of (9) cancel each other, because are independent of , up to a manifestly gaugeinvariant term proportional to . The action describes the coupling of an “axion field” to the electric field of dimensional QED. For the current, , through the sample and the charge density, , in an external axion field configuration , we find the expressions
(16) 
(17) 
provided , so that there are no contributions from the boundary action. (Here , . We observe that (16) and (17) imply the continuity equation .)
The action in eq. (15) yields an accurate description of charge transport in a long, narrow sample filled with an incompressible Hall fluid with Hall conductivity if the electric field in the direction vanishes (so that the term proportional to does not contribute), as long as tunnelling processes between the upper and the lower edge can be neglected. However, for a sample with a constriction, as shown in Figure 1, such tunnelling processes do occur. In a description of the Hall fluid in terms of an action that displays the edge degrees of freedom explicitly, tunnelling processes between the two edges are described by terms of the form
(18)  
where labels the different species of charged quasiparticles described by left chiral fields, , on the upper edge and by right chiral fields , on the lower edge, and is the electric charge of a quasiparticle of species . Setting
(19) 
and recalling eq. (13), the term (18) can be written as
(20) 
The function is a measure for the strength of the amplitude of tunnelling between the two edges; is “large” for close to the constriction, and tends to rapidly, as the distance of to the constriction increases.
Besides (20), the action for the edge degrees of freedom contains terms not mixing the left and rightmoving degrees of freedom. These terms do not depend on . Integrating (or “tracing”) out all edge degrees of freedom, we obtain an effective “boundary action”, , which now depends on ! It is periodic in : if is the smallest real number such that
(21) 
for all species , then
(22) 
This follows immediately from the form of (20) of the tunnelling terms in the boundary action.
The remarks on the relation between fractional charges and the value of the Hall conductivity at the very end of Section 2 lead to the equation
(23) 
where is the Hall numerator and is an integer, (see [13]; actually , for the Laughlin and the simple Jain fluids with , ).
The total effective action is given by
(24) 
The periodicity property (22) of implies that, if the component of the electric field vanishes , the macroscopic state of this system depends periodically on the external “axion field” , with period , and that eqs. (16) and (17) for the electric current and the charge density continue to hold in average when the system is driven through several cycles. Indeed, because of the invariance of under a gauge transformation , one has
(25) 
and one can write
(26) 
where the function , which depends on the axion field configuration and the spacetime point , is given by
(27) 
The function is periodic in , with period ,
(28) 
and does not depend on time explicitely,
(29) 
Consider a pump which works with a period , i.e., a pump driven by an axion field which fulfills
(30) 
for some integer . One then finds that the charge transport due to the second term on the R.H.S. of (24) vanishes, since
(31) 
We now recall the physical meaning of the axion field . By eq. (13),
(32) 
where is the voltage drop at between the lower and the upper edge of the sample; (we are using that , because is independent of ). Let be an independent configuration of the “axion field”, with
(33) 
Then eq. (16) tells us that the total amount, , of electric charge transported from the left to the right of the sample is given by
(34) 
(and , by eq. (17)). Thus, a sample with a timedependent voltage drop between the upper and lower edge can be viewed as a “quantum pump” transporting electric charge from the left to the right. The macroscopic state of this pump is periodic in with period . Thus, when the pump is operated over cycles, a total amount, , of electric charge
(35) 
is transported from the left to the right; (here we have used eq. (23)). Since is the smallest fractional electric charge of a quasiparticle tunnelling through the constriction, a measurement of this charge can be obtained from independent measurement of the charge transported from the left to the right in cycles and of . Whether a given voltage pulse corresponds to an integer number of cycles of the pump can be inferred from the fluctuations of the charge, , transported from the left to the right during that pulse around its mean value : if, on the left and right ends the sample is connected to freeelectron leads then (independently of ) must be an integer (multiple of ). If is not an integer then will exhibit fluctuations around its mean value . But if corresponds to exactly cycles, , then is an integer, and hence the fluctuations of in this process will essentially vanish.
Typical features of the effective action , with , can be determined by measuring the tunnelling current through the constriction: when
(36) 
where is the width of the sample at . A tunnelling current can be generated e.g. by a modulation of the magnetic field perpendicular to the plane of the sample. The expression for the tunnelling current in terms of given above shows that, from measurements of and of the voltage drop as functions of time, one can infer the period of and, hence, the smallest fractional electric charge of the quasiparticles. Furthermore, one can argue that the fluctuations of are proportional to the fractional charge of the quasiparticles tunnelling through the constriction  an effect used in the experiments described in [2] to measure the fractional charges of quasiparticles.
If the magnetic field is set to our considerations can also be used to describe quantum wires. Then has the interpretation of a (spacetime dependent) difference of chemical potentials between left and rightmoving modes in the wire. Eq. (16) then says that if there isn’t any chiralityreversing scattering in the wire (i.e. ), and for ,
(37) 
where now has the interpretation of a longitudinal conductance. If all quasiparticles in the wire have integer electric charge then is an integer multiple of ; (see [15][16]).
If there are tunnelling processes mixing left and rightmovers, due e.g. to impurities in the wire, then the term on the R.S. of (24) does not vanish, even if . The general expression for the current in the wire is given by the equation
(38) 
with and . If scattering at the impurities converts left into rightmovers, and conversely, the second term on the R.S. of eq. (38) does not vanish even if , and hence conductance is not quantized, anymore, in accordance with experiment. However, charge transport over long periods of time still exhibits “quantization”, provided , due to the periodicity of in .
A more detailed account of our results and an analysis of the Luttinger liquids in quantum wires in the presence of impurities will be given elsewhere.
4 A 5dimensional analogue of the Quantum Hall Effect, and Primeval Magnetic Fields in the Early Universe
Imagine, for a moment, that our world corresponds to a stack of branes in a dimensional spacetime. We suppose that all electrically charged modes propagating through the dimensional bulk have a large mass (comparable, e.g., to the Planck mass) and have parityviolating dynamics. We may then ask whether there is an analogue of the quantum Hall effect in the dimensional bulk. To be specific, we assume that there are two parallel, flat branes separated by a dimensional slab of width representing the bulk of the system. Let denote the dimensional electromagnetic vector potential and the restriction to the boundary, , of the slab of the components of parallel to . Assuming that only the graviton and the photon are massless modes, and dropping the gravitational contribution, the effective action of such a system is given by
(39)  
with , where is the coordinate perpendicular to the boundary branes, which are located at , respectively, and is a dimensionless constant. The first term on the R.S. of (39) is a Maxwell term, which is the dominant term, the second term is the dimensional ChernSimons action, the last term is the dimensional anomalous chiral (boundary) action, which ensures that is gaugeinvariant. From the theory of the chiral anomaly we infer that
(40) 
where the ’s are the charges of the chiral fermions propagating along . The action is the dimensional version of the boundary action in eq. (9). It is the generating functional of the Green functions of chiral currents satisfying
(41) 
where is the electromagnetic field on the boundary branes. Modes of opposite chirality are localized on the two opposite branes, (at and , respectively).
Imagine that the fields , are independent of . We define the axion field, , by
(42) 
After dimensional reduction to the boundary branes, , the effective action in (39) becomes
(45)  
If tunnelling between the two boundary branes is suppressed completely the boundary action is independent of and can be combined with the Maxwell term to renormalize its coefficient. But if tunnelling processes mixing fermions of opposite chirality are present then depends on and is even if . Tunnelling processes generate a small mass, proportional to , of boundary fermions; (here is a typical bulk mass scale, and is the Planck length). By repeating the arguments explained in Section 3, one finds that is periodic in with period proportional to , where is the smallest electric charge of modes propagating along the branes.
We recall that is the generating functional of the Green functions of the pseudoscalar density and the electric current density; in particular, . Plugging the expressions for and for obtained from (45) into Maxwell’s equations and the equations of motion for the axion field, we find the following equations of motion:
(46)  
where and are dimensionsless constants, and the term , where is the Riemann tensor, comes from a term in the effective action describing the coupling of the axion to spacetime curvature (which has not been displayed in eq. (45)). If there exist magnetic monopoles the first equation in (46) must be replaced by , where is the magnetic current 3form. In conventional vector analysis notation, eqs. (46) take the form
(47) 
where, in the fourth equation of (47), the term has been added to describe a dissipative current parallel to , with the longitudinal conductivity; (Ohm’s law).
It is clear from eq. (42) that the time derivative, , of the axion field has the interpretation of a (spacetime dependent) difference of chemical potentials of righthanded and lefthanded charged modes propagating on the “upper” and the “lower” brane, respectively.
Absorbing the leading dependent contribution to into a renormalization of the constant , the leading term in is independent of and has the form
(48) 
where is a (temperaturedependent) periodic function with period . Plugging eq. (48) into (47), we find that a special solution of (47) is given by and solving the the equation
(49) 
As mentioned above, actually depends on the the temperature of the universe: at temperatures well above the electroweak phase transition; while, at temperatures below the electroweak scale, is a nonconstant, periodic function of with minima at , . Thus, at the time of the electroweak phase transition, the configuration corresponds, approximatively, to a solution of
(50) 
and there is no reason why should be close to a minimum of the function , or why should be small. The source term on the R.S. of (50) does not vanish, provided there are gravitational waves propagating through the universe. For a Friedman universe, it is proportional to the amplitude of gravitational waves: thus, such waves can, in principle, feed the growth of the axion field.
At times , the equation of motion of the axion is given by
(51) 
with constant. Assuming that gravitational waves eventually disperse away, the term proportional to will approach , for a Friedman universe. Let us suppose that, after inflation, varies slowly over space. Then eq. (51) reduces to an ordinary differential equation
(52) 
to be solved for essentially random initial conditions, , . Eq. (52) is the equation of motion for a pendulum in a potential force field, . Solutions of (52) are given by
(53) 
where is a periodic function of .
Next, we linearize eqs (47) around the special solution , as in (53). This is not a difficult task. One finds that for sufficiently small wave vectors, , (, for ), there are exponentially growing transverse modes, , of the magnetic field with nonvanishing magnetic helicity. One expects that axion field configurations which are slowly varying in space lead to qualitatively similar instabilities. When combined with the galactic dynamo mechanism they might provide an explanation of the largescale magnetic fields observed in the universe; (but see [7] for discussion of some of the difficulties with this and other scenarios). We hope to present a more detailed account of our results, in particular of the possible rôle of gravitational waves, elsewhere.
References

[1]
R. D. Peccei and H. R. Quinn,
Phys. Rev. Lett. 38, 1440 (1977).
J. E. Kim, “Cosmic axion”, 2 Intl. Workshop on Gravitation and Astrophysics, Univ. of Tokyo, 1997, astroph/9802061.  [2] L. Saminadayar, D. C. Glattli, Y. Jin and B. Etienne, Phys. Rev. Lett. 79, 2526 (1997).
 [3] I. I. Tkachev, Sov. Astronom. Lett. 12, 305 (1986).
 [4] M. Turner and B. Widrow, Phys. Rev. D 37, 2743 (1988).
 [5] J. Fröhlich and B. Pedrini, in “Mathematical Physics 2000”, A. Fokas, A. Grigorian, T. Kibble and B. Zegarlinsky (eds.), Imperial College Press, London, 2000.
 [6] M. Joyce and M. Shaposhnikov, Phys. Rev. Lett. 79, 1193 (1997).

[7]
M. Giovannini, Phys. Rev. D 61, 063502 (2000).
D. Grasso and H. R. Rubinstein, “Magnetic Fields in the Early Universe”, preprint astroph/0009061.  [8] K. Von Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
 [9] D. C. Tsui, H. L. Störmer and A. C. Gossard, Phys. Rev. B 48, 1559 (1982).
 [10] J. Fröhlich and T. Kerler, Nucl. Phys. B 354, 369 (1991).
 [11] S. Deser, R. Jackiw and S. Templeton, Ann. of Phys. 140, 372 (1982).
 [12] R. Jackiw, in “Current Algebras and Its Applications”, S. B. Treiman, R. Jackiw, D. J. Gross (eds.), Princeton University Press, Princeton NJ, 1972.

[13]
J. Fröhlich and E. Thiran, J. Stat. Phys. 76, 209 (1994).
J. Fröhlich, U. M. Studer and E. Thiran, J. Stat. Phys. 86, 821 (1997).  [14] J. Fröhlich, B. Pedrini, C. Schweigert and J. Walcher, J. Stat. Phys 103, 527 (2001).
 [15] B. J. van Wees et al., Phys. Rev. Lett. 60, 848 (1988).
 [16] A. J. Alekseev, V. V. Cheianov and J. Fröhlich, Phys. Rev. B 54, 320 (1996).
 [17] J. E. Avron, A. Elgart, G. M. Graf and L.Sadun, Phys. Rev. B 62, R10618 (2000), and Phys. Rev. Lett. 87, 236601 (2001).