Two Approaches to Fractional Statistics in the Quantum Hall Effect: Idealizations and the Curious Case of the Anyon

Abstract

This paper looks at the nature of idealizations and representational structures appealed to in the context of the fractional quantum Hall effect, specifically, with respect to the emergence of anyons and fractional statistics. Drawing on an analogy with the Aharonov–Bohm effect, it is suggested that the standard approach to the effects—(what we may call) the topological approach to fractional statistics—relies essentially on problematic idealizations that need to be revised in order for the theory to be explanatory. An alternative geometric approach is outlined and endorsed. Roles for idealizations in science, as well as consequences for the debate revolving around so-called essential idealizations, are discussed.

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Notes

  1. 1.

    In addition, anyons have “fractional charge,” “fractional spin,” and can be thought of as point charged vortices (i.e., point particles with both electric charge and magnetic flux), but we set aside such details here. See footnote 3.

  2. 2.

    The name is due to Frank Wilczek in his [108]. Note that anyons and fractional statistics have nothing to do with so-called paraparticles and parastatistics (which arise from higher dimensional representations of the permutation group; see Sect. 4).

  3. 3.

    For standard textbook accounts and introductions to anyons and fractional statistics see [58, 69, 83, 86, 90, 100, 109].

  4. 4.

    For more on the integer and fractional quantum Hall effects see [29, 35, 41, 89, 100, 113]. For a history and introductory overview see [60]. Also see Sects. 3, 4 and 6.

  5. 5.

    The principle is based on Earman ([39], p. 191]) and Ruetsche ([92], p. 336]). It is worthwhile to note that many may consider the principle to be too strong or just plainly false. In this paper I set this issue aside since my goal is to motivate the idea that there may be something peculiar about anyons, which merits philosophical attention.

  6. 6.

    To be clear, it is not my intention to reject any or all idealizations that may be dubbed “topological,” and I concede that topology certainly plays an important role in science. Instead, my claim is that the type of topological idealizations appealed to in the standard account of fractional statistics are inconsistent with the sound principle and hence need to be revised in order for the theory for be explanatory.

  7. 7.

    Butterfield [24], Section 3] discusses similar distinctions. In particular, he makes a distinction between a system \(\sigma (N)\) that depends on some parameter \(N\) (let \(\{\sigma (N)\)} denote a sequence of such systems), a quantity defined on the system \(f(\sigma ( N))\) (let \(\{f(\sigma ( N))\}\) denote a sequence of quantities on successive systems), and a (real number) value \(v(f( {\sigma ( N)}))\) of quantities on successive systems (where a sequence of states on \(\sigma (N)\) is implicitly understood; let \(\{v(f( {\sigma ( N)}))\}\) denote a sequence of values on successive systems). A limit system \(\sigma (\infty )\) arises when \(\mathop {\lim }\limits _{N\rightarrow \infty } \{\sigma (N)\}\) is well-defined-otherwise there is no limit system. A property of a limit system refers to the value \(v(f( {\sigma ( \infty )})\) of the (natural) limit quantity\(f( {\sigma ( \infty )})\) (in the natural limit state) on \(\sigma (\infty )\). A limit property \(v(f( {\sigma ( N)})\) is a limit of a sequence of values of quantities on successive systems (or, values on the systems on the way to the limit) and is well-defined when \(\mathop {\lim }\limits _{N\rightarrow \infty } \{v(f( {\sigma ( N)}))\}\) exists. The question that I will be discussing is whether a property of a limit system equals the system’s limit property. More precisely, the question asks whether \(v(f(\sigma (\infty ))=\mathop {\lim }\limits _{N\rightarrow \infty } \{v(f( {\sigma ( N)}))\}\) (assuming both are well-defined).

  8. 8.

    For more on faithful representation see Shech [96].

  9. 9.

    These are all rough and intuitive characterizations of the notions of connectedness, homotopy, fundamental group, etc. They will do for my purposes, and I will only introduce further details when necessary. For precise characterization see standard textbooks on topology and algebraic topology, e.g., [47, 8082].

  10. 10.

    If, in turn, the properties lost are used by our best scientific theories to predict genuine physical effects, then the sound principle is wrong, and this is what I mean by a “failure of the sound principle.”

  11. 11.

    One may wonder why my example of a sequence of cuboid-models ought to be labeled an “idealization,” when I have not identified what in the physical world is being idealized. My point is that given some physical cubical system with an impenetrable object in its interior, all the cuboids in the sequence except for the first are idealized models of this original physical system. See [24, 84] for further details and discussion.

  12. 12.

    It is worthwhile to clarify in order to avoid a possible point of confusion. There are two senses of “essentialness” at play in this paper. In the sense discussed here, an “idealization” in Norton’s [84], Section 3] terminology, viz., an appeal to a limit system, is essential if there is a mismatch between a property of the limit system and the corresponding limit property. In other words, if we are interested in a property of a limit system that does not have a corresponding limit property then it is essential that we appeal to the idealized limit system. Such an idealization is pathological if it turns out that it is not possible to render it consistent with the sound principle. More generally, In Sect. 7 I will argue that idealizations, broadly construed, are essential to science for explanatory, methodological, and pedagogical purposes.

  13. 13.

    What I mean by transitioning from a coarse-grained to a fine-grained distinction, in Butterfield’s [23] terminology, is to transition from looking at a particular value of a quantity on successive systems to a different one that sheds light on the first. See Butterfield [23], pp. 1078–1079] for an illuminating and simple example.

  14. 14.

    By “approximately” we mean that for any non-zero epsilon of maximal error about the volume one may care about, we can find a de-idealization with volume less than this epsilon.

  15. 15.

    See footnotes 6 for qualification.

  16. 16.

    See footnotes 3 and 4.

  17. 17.

    See [5] for an overview of the electric properties of such two-dimensional systems.

  18. 18.

    The cyclotron frequency is the frequency of a charged particle (in our case, the charge of the electron \(e)\) with mass \(m_e \), moving in the presence of a perpendicular and uniform magnetic field with value \(B\). Note not to confuse the use of \(m\) or \(m_e \) to signify the mass of a particle such as the electron, with the odd integer \(m\) commonly used in the discussion of the FQHE in the context of the fractional filling factor \(\nu =\frac{1}{m}\).

  19. 19.

    The expression used here for Laughlin’s wavefunction is the one that appears in [7], p. 282], but also see [6, 66].

  20. 20.

    Originally, [65] postulated that excited FQHE states were bosons and [45] thought they were fermions. See [37, 87] for background.

  21. 21.

    A mere glance at the literature will confirm that the standard account off the nature of idealizations that arise in the FQHE is topological in this sense. For example, Khare ([58], p. 5]; original emphasis) states “...in two dimensions, the space is multiply connected which results in the possibility of ... intermediate statistics” (where intermediate statistics refer to fractional statistics).

  22. 22.

    The type of “justification” that I am rejecting is a main component of the topological approach to fractional statistics and anyons, and is regularly found in canonical textbooks as can be seen by a glance at a lengthy quote from Khare ([58], p. 2]; emphasis mine):

    The point is that because of the third law of thermodynamics, which states that all the degrees of freedom freeze out in the limit of zero temperature, it is possible to strictly confine the electrons to surfaces, or even to lines or points. Thus it may happen that in a strongly confining potential, or at sufficiently low temperature [(both such conditions are satisfied in FQHE experiments)], the excitation energy in one or more direction may be much higher than the average thermal energy of the particles so that those dimensions are effectively frozen out. An illustration might be worth while here. Consider a two dimensional electron gas... The electrons are confined to the surface of a semiconductor by a strong electric field, and they move more or less freely along the surface. On the other hand, the energy \(E\) reguired to excit motion in the direction perpendicular to the surface is of the order of several milli-electron-Volt (meV). Now at a temperature of say \(T=1K\), the thermal energy is \(kT\), where \(k\) is the Boltzmann constant. Thus if the transverse excitation energy is say 10 meV, then the fraction of electrons in the lowest excited transverse energy level is

    $$\begin{aligned} e^{-\frac{E}{kT}}=e^{-100}\approx 10^{-44} \end{aligned}$$
    (3.1)

    which is zero for all practical purposes. Thus the electron gas is truly a two-dimensional gas.

    Clearly there is some tension—as well as confusion—regarding whether or not the dynamically two-dimensional system is a two-dimensional system in the strict or approximate sense. Talk of “strictly” and “truly ... two-dimensional” seem to confirm the former, while “effectively” and “for all practical purposes” confirm the later. Certainly, the above calculation will not do as a justification for the idea that FQHE systems are, strictly speaking, dynamically two-dimensional. The fact remains that there is a non-zero probability for an excitation in the third-dimension, and this is all that is needed to cancel out the emergence of anyons vis-à-vis the topological approach to fractional statistics. Having said that, I am willing to concede that more may be able to be said to defend and flesh out a coherent stance on the idea of a dynamically two-dimensional system, but I hold that the burden of proof is not my own and belongs to a defender of the topological approach. In either case, I leave this issue for future study.

  23. 23.

    See, for example, [36, 55] for an identification of the fallacy.

  24. 24.

    Still, it is worthwhile to note that if the operator framework turns out to be the correct foundation for permutation invariance in QM, we will lack a corresponding conceptual foundation for the notion of fractional statistics.

  25. 25.

    This is a somewhat ad hoc justification. Nevertheless, it is given by many of the pioneers of the configuration space framework. For example, Laidlaw and DeWitt [62], p. 1377]:

    Whether or not two point particles can simultaneously occupy the same point in space is not a question that we wish to settle here. We are only saying that by excluding points of coincidence from the configuration space, the resulting topology leads to meaningful physical results.

  26. 26.

    \(C\) is not necessarily a closed curve. If the bundle is not curved than \(C\) is closed but if the bundle is curved than \(C\) is not closed.

  27. 27.

    See [18] for a recent survey.

  28. 28.

    Such an illustration arises in standard introductory accounts of fractional statistics such as [58]. Note that Z is the cyclic group of order one, i.e., the infinite group of integers under addition. Z\(_2\) is the cyclic group of order two, i.e., it is the multiplicative group of, say, 1 and \(-\)1. RP\(_1\) and RP\(_2\) are the real projective one- and two-dimensional spaces, respectively.

  29. 29.

    I elaborate on this point in Shech [97]. See [88] for a review of the effect and its experimental confirmation [103, 104], along with a selective review of the debate that arose with respect to the reality of the effect in the physics literature. Also see [101], Ch. 6, p. 100] for a recent overview of experimental confirmation. Ehrenberg and Siday [38] are usually credited with first noting the effect and Chalmers [30] with being the first experimental confirmation. See [28] for recent experimental confirmation emphasizing the absence of a force in the manifestation of the effect (i.e., there is no unknown force that is responsible for shifting the electron wave packet).

    The AB effect is of special interest to philosophers because it seems to portray a type of quantum non-locality (since the magnetic field affects the beam while being confined to a region where the beam is not) and raises a host of interpretive issues regarding ontology and indeterminism in classical electromagnetism and quantum mechanics. See, for example, [15, 19, 4850, 67, 70, 71, 73, 74, 85] for some of the philosophical literature.

  30. 30.

    See Healey [50], Ch. 1–2; Appendix B] for a philosophical friendly introduction to the fiber bundles.

  31. 31.

    Sentiments of this sort arise in both the physic and philosophical literature, e.g.: [15], pp. 543]; [19], pp. 544]; [47], pp. 42]; [82], pp. 193]; [67]; [68]; [73], V]; [74], [78], pp. 356–359]; [79], pp. 301]; [77, 81], [82], pp. 193]; [96]].

  32. 32.

    Specifically, the assumption of impenetrability, for a continuously differentiable wave function, means that the electron probability \(j\) current must vanish at the solenoid boundary (where\(j\equiv -i({\Psi }^*\nabla {\Psi }-{\Psi }\nabla {\Psi }^*)\).This can be implemented using different boundary conditions including the Dirichlet boundary conditions (\({\Psi }=0)\) that Aharonov and Bohm [2] used, Neumann boundary conditions (\(\nabla {\Psi }=0)\), or Robin boundary conditions (\(\nabla {\Psi }=r{\Psi },r\in \mathbb {R})\) de Oliveira and Pereira [33]. What de Oliveira and Pereira [32] show is that Dirichlet boundary conditions are singled out if we represent the assumption of infinite impenetrability with step-function potentials \(V_{n,L} \) and take the infinite limit \(n\rightarrow \infty \). What is important from our perspective is that although it does not matter which boundary condition one uses in order to attain the interference shifts associated with the AB effect, still, different boundary conditions correspond to different physics (i.e., different energy eigenvalues and unitary dynamic) along with different empirically confirmable predictions for scattering experiments. So there is a real need to justify the appeal to the AB effect idealizations and tell a story about why the AB effect, conventionally defined on a non-simply configuration space, has anything to do with the AB effect as it is manifested in the laboratory.

  33. 33.

    Specifically, [1113] provide a quantitative error bound for the difference in norm between the exact solution and the Aharonov–Bohm Ansatz, thereby showing that the Aharonov–Bohm Ansatz is a good approximation of the exact solution. In addition, [9, 91] consider the solenoid of finite length limit, while [61] also considered the finite potential barrier limit. An additional idealization that has been studied via a limiting process by [106] concerns the fact that the electromagnetic field associated with the solenoid will interact with surrounding particles when being turned on and off.

  34. 34.

    In order to preempt objections to the effect that the geometric phase itself is topological in nature, I refer to reader to [57] who gives a thoroughly geometrical interpretation of effect.

  35. 35.

    I use the \(H_{FQHE}^I \)notation to emphasize that this is the idealized (I) Hamiltonian corresponding to FQHE systems.

  36. 36.

    An exception to this lack of attention includes Earman [40], to which this paper is heavily in debt.

  37. 37.

    Where some go so far as to reject the notion altogether, e.g., [99].

  38. 38.

    See [27].

  39. 39.

    See [31, 5153].

  40. 40.

    See [20, 88] and references therein for discussion.

  41. 41.

    See Shech [96].

  42. 42.

    E.g., [1416].

  43. 43.

    Assuming that said former features are necessary for a particular instance of a pattern to obtain.

  44. 44.

    See Shech [97].

  45. 45.

    Especially Norton [84].

  46. 46.

    Especially Batterman [1416].

  47. 47.

    If we symbolize this by \(\pi _{1} ( {Path})\) we get that \(\pi _{1} ( {PathA})=0\) for the trivial homotopy class, but the rest of the paths will be elements of non-trivial homotopy classes: \(\pi _{1} ( {PathB})=1,\pi _{1} ( {PathC})=2,\ldots \) and so on, so that we generate all of the integers \(Z\). Negative integers corresponding to traversal of the circular configuration space in the opposite direction.

  48. 48.

    The adiabatic theorem is originally due to Born and Fock [23]. See, for instance, [17], Sect. 9.4] for a proof.

  49. 49.

    See, for instance,[44], p. 373] for the adiabatic approximation. Basically, we drop terms that depend on the time or parameter derivative of \(H( {\varvec{R}( t)})\) for, by assumption, the change is minute.

  50. 50.

    I use the \(H_{FQHE}^I \) notation to emphasize that this is the idealized (I) Hamiltonian corresponding to FQHE systems.

  51. 51.

    See [95], pp. 307–309] and [109], pp. 300–301] for an explicit calculation

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Acknowledgments

Parts of this paper were presented at the “Bucharest Colloquium in Analytic Philosophy: New Directions in the Philosophy of Physics” conference at University of Bucharest. I am grateful to the audience for stimulating discussion, to Iulian Toader for editing this volume of Foundations of Physics, and to two anonymous referees for helpful comments. I am especially grateful to John Earman and John D. Norton for comments on earlier versions of this paper, numerous insightful discussions, and their constant support. This paper is heavily in debt to work done by John Earman, who initially drew my attention to the main issues discussed in this paper, and has been especially kind in helping me work through the finer details. Special thanks to Naharin Shech for her help with figures.

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Appendices

Appendix 1: Visualizing Fractional Statistics on the Configuration Space Approach

The fundamental group of the configuration space of the simplest scenario of two particles \(N=2\) in the \(d=2\) and \(d=3\) cases is as follows:

$$\begin{aligned} \pi _{1} \left( {\frac{\mathbb {R}^2\setminus \Delta }{S_2 }}\right)= & {} \pi _{1} ( {RP_1 })=Z\,\,\text {for}\,\,d=2\\ \pi _{1} \left( {\frac{\mathbb {R}^3\setminus \Delta }{S_2 }}\right)= & {} \pi _{1} ( {RP_2 })=Z_2 \,\,\text {for}\,\,d=3 \end{aligned}$$

Where \(Z\) is the cyclic group of order one, i.e., the infinite group of integers under addition. \(Z_2 \) is the cyclic group of order two, i.e., it is the multiplicative group of, say, 1 and \(-\)1. \(RP_1\) and \(RP_2 \) are the real projective one- and two-dimensional spaces, respectively.

Pictorially, for the \(d=3\) case the configuration space reduces to the real projective space in two dimensions \(RP_2\). This can be visualized as the surface of a three-dimensional sphere with diametrically opposite points identified (see Fig. 13). Consider three scenarios, corresponding to three paths \(A\), \(B\), and \(C\) in configuration space including no exchange (Fig. 13a), exchange (Fig. 13b), and a double exchange (Fig. 13c), respectively.

Fig. 13
figure13

The real projective space in two dimensions \(RP_2 \), represented by a sphere with diametrically opposite points identified. Cases (a), (b), and (c) correspond to no exchange, exchange, and double exchange, respectively

Concentrating on the no exchange case (Fig. 13a). We trace a path \(A\) in configuration space in which the two particles move and return to their original positions. Path \(A\) is a loop in configuration space, with the same fixed start and end points, which can be shrunk to a point. This correspond to a trivial homotopy class in which the phase factor is trivial.

Moving onto the exchange case (Fig. 13b), we start at one end of the configuration space and trace a path \(B\) to its diametrically opposite point. This represents an exchange or permutation between the two particles. Notice that since diametrically opposite points are identified (because the particles are identical), this path is actually a closed loop in configuration space. However, since the start and end points of Fig. 15b are fixed, the loop cannot be shrunk to point. This corresponds to a non-trivial homotopy class with a non-trivial phase factor.

The double exchange (Fig. 13c) case includes tracing a path \(C\) in configuration space similar to that of \(B\), but then tracing around the sphere back to the original starting point. Path \(C\) is a closed loop in configuration space that can be shrunk to a point, and so it is in the same homotopy class of path \(A\) with a corresponding trivial phase factor. Equivalently, we may visualize the paths \(A\), \(B\), \(C\) on a hemisphere with opposite points on the equator identified as in Fig.14, where paths \(A\) and \(C\) can be continuously deformed to a point but path B cannot because of the diametrically opposed fixed start and end point on the equator.

Fig. 14
figure14

The real projective space in two dimensions \(RP_2 \), represented by the northern hemisphere with opposite point on the equator identified

On the other hand, in the context of the \(d=2\) case, we are dealing with the real projective space in one dimension \(RP_1 \). We can visualize this configuration space as a circle with diametrically opposite points identified (see Fig. 15). Again, consider three paths \(A\), \(B\), and \(C\) in configuration space that correspond to no exchange (Fig. 15a), exchange (Fig. 15b), and a double exchange (Fig. 15c), respectively. Path \(A\) traces a closed loop in configuration space (where the particles move but then return to their original positions with no exchange) which can be continuously shrunk to a point and has a corresponding trivial phase factor (as in the \(d=3\) case of Fig. 14a). Next, we trace a path \(B\) across half the circumference of the circle. Since diametrically opposed points are identified, this represents a particle exchange (Fig. 15b). Path B traces a closed loop in configuration space that cannot be continuously shrunk to a point and has a corresponding non-trivial phase factor (as in the \(d=3\) case of Fig. 14b).

Fig. 15
figure15

The real projective space in one dimension \(RP_1\), represented by a circle with diametrically opposite points identified. Cases (a), (b), and (c) correspond to no exchange, exchange, and double exchange, respectively

The main difference between the \(d=3\) and \(d=2\) cases arises when we consider path \(C\) (Fig. 15c), in which the particles are permuted twice, represented by traversing the entire circular configuration space. Path \(C\) is a closed loop in configuration space but, unlike the \(d=3\) case, it cannot be shrunk to a point because the circle itself (so to speak) acts as an obstructive barrier. Moreover, path \(C\) cannot even be continuously deformed to overlap with path B. This means that, not only is the phase factor corresponding to the two paths non-trivial, but each path has a different phase factor for each path belongs to a different homotopy class. In fact, for every traversal (in configuration space) of half a circle, we get a closed loop that is in its own homotopy class.Footnote 47 In other words, by transitioning from three dimensions to two dimensions, we have transitioned from a doubly connected space to an infinitely connected space, and it is this change in topology that allows for intermediate statistics.

Appendix 2: A Geometric Approach to Fractional Statistics

We follow Berry’s [21] original paper on the non-dynamical phase factor accompanying cyclic evolutions of quantum systems. To begin, consider some system, such as a spinless electrically charged particle in a box, with a corresponding Hamiltonian \(H( {\varvec{R}( t)})\) that depends on a set of parameters \(\varvec{R}=(X,Y,\ldots )\) and can be altered over time by varying said parameters. We can view the alteration of \(H( {\varvec{R}( t)})\) as a path in parameter space. If the system starts at some time \(t=0\), and is gradually changed over time \(t\) so that the parameter values are returned to their original values \(\varvec{R}(0)=\varvec{R}(t)\) then this maps out a closed curve \(C\) in parameter space. According to the adiabatic theorem, if the system was originally (at time \(t=0)\) in the \(n\)th eigenstate \({\psi }_n (\varvec{R}(0))\) of \(H( {\varvec{R}( 0)})\), if \(H( {\varvec{R}( t)})\) is non-degenerate, and if the excursion in parameter space is sufficiently slow, then the system will transition (under Schrödinger evolution) into the \(n\)th eigenstate of \({\psi }_n ( {\varvec{R}(t)})\) of \(H( {\varvec{R}( t)})\) (with some added phase factor).Footnote 48

The general state of the system \({\Psi }( t)\) evolves according to the time-dependent Schrödinger equation, and atany instant \(t\) the eigenstates of the time-independent Schrödinger equation form a natural basis satisfying:

$$\begin{aligned} H( {\varvec{R}( t)}){\psi }_n ( {\varvec{R}(t)})=E_n (\varvec{R}( t)){\psi }_n ( {\varvec{R}(t)}) \end{aligned}$$

According to the adiabatic approximation then, the general state of the system \({\Psi }( t)\) at some time \(t\) can be expressed as follows:Footnote 49

$$\begin{aligned} {\Psi }( t)={\psi }_n ( {\varvec{R}(t)})e^{i\theta } \end{aligned}$$

Where the exchange phase \(\theta \) has two components \(\theta =\theta _D +\theta _G \) such that:

$$\begin{aligned} \theta _D =-\frac{1}{\hbar }\mathop \smallint \limits _0^t E_n ( t)dt\,\,\,\text {and}\,\,\theta _G =i\mathop \smallint \limits _0^t {\psi }_n ( t){\vert }\frac{\partial }{\partial t}{\psi }_n ( t)dt \end{aligned}$$
(7.1)

\(\theta _D \) corresponds to the usual dynamical phase (accompanying the Schrödinger evolution of any stationary state) and \(\theta _G \)is called the geometric phase or Berry’s phase (where I have used Dirac’s bra-ket notation and hid the parameter dependence for convenience). It can be expressed more generally as a quantity dependent on both the closed curve \(C\) in parameter space and the parameters \({\varvec{R}}=(X,Y,\ldots )\):

$$\begin{aligned} \theta _G (C)=i\mathop \oint \limits _C {\psi }_n ({\varvec{R}}){\vert }\nabla _{\varvec{R}} {\psi }_n ( R)\cdot d{\varvec{R}} \end{aligned}$$
(7.2)

Where \(\nabla _R \) is the gradient with respect to the parameters \({\varvec{R}}=(X,Y,\ldots )\) (and assuming that \({\varvec{R}}( 0)=R({\varvec{t}})\) so that \(C\) forms a closed curve). There are similar results for degenerate systems [110] and for a cyclic evolution that is not necessarily adiabatic [1].

Next, my goal in the rest of this appendix is solely to repeat some of the steps taken by Arovas, Schrieffer, and Wilczek [7] to derive fractional statistics in order to emphasize the disconnect between this geometric approach and the topological (and pathological) approach discussed in Sect. 4. Many steps will be skipped, and I refer the reader interested in a more details to explicit calculations made by Arovas in [95], pp. 284–322] and Laughlin in [109], pp. 262–303].

Following [6, 7, 6466] closely, let us consider a FQHE system with filling factor\(\nu =\frac{1}{m}\) where \(m\) is an odd integer, and the applied strong magnetic field \(B\) is in the \(z\)-axis direction corresponding to magnetic flux \({\Phi }.\) In such a situation, the Hamiltonian governing the system isFootnote 50:

$$\begin{aligned} H_{FQHE}^I =\mathop \sum \limits _j \frac{( {{\varvec{p}}_j -q{\varvec{A}}_{\varvec{j}} })^2}{2m_e }+{\varvec{V}}(z_j )+\mathop \sum \limits _{j>k} \frac{e^2}{\left| {z_j -z_k } \right| } \end{aligned}$$

Recall, \(z_j \equiv x_j +iy_i \) are in units of magnetic length \(l_B \equiv \sqrt{\hbar c/eB} \), which have been set to equal one, \(e\) is the charge of the electron, and \(j\) and \(k\) run over \(N\) particles. The \(\frac{( {p_j -qA_j })^2}{2m}\) term signifies the kinetic energy of charged particles in a magnetic field, \(\varvec{V}(z_j )\) is average background potential, and \(\frac{e^2}{\left| {z_j -z_k } \right| }\) is the Coulomb interaction between particles. Laughlin’s [64, 65] celebrated wavefunction for the ground state of \(H_{FQHE}^I \) is:

$$\begin{aligned} {\Psi }_m =\mathop \prod \limits _{j<k}^N (z_j -z_k )^me^{\left( -\frac{1}{4}\mathop \sum \limits _l^N \left| {z_l } \right| ^2\right) } \end{aligned}$$

The state function of two excited states (quasiholes) \(a\) and \(b\) located at positions \(z_a \) and \(z_b \), respectively, is represented by

$$\begin{aligned} {\Psi }_m^{z_a z_b } =N_{ab} \mathop \prod \limits _i (z_i -z_a )(z_i -z_b ){\Psi }_m \end{aligned}$$
(7.3)

where \(N_{ab} \) is a normalizing factor.

We can determine the quantum statistics associated with exchanging quasiholes \(a\) and \(b\) by calculating the geometric phase associated with carrying quasihole \(a\) adiabatically around a closed loop \(C\), thereby adding time dependence to \(z_a =z_a ( t),\) and identifying the geometric phase with the exchange phase. The geometric phase \(\theta _G \) can be calculated by plugging Eq. 7.3 into Eq. 7.1 or 7.2 as follows:

$$\begin{aligned} \frac{d\theta _G }{dt}=i{\Psi }_m^{z_a z_b } (z_a ( t),z_b ){\vert }\frac{d}{dt}{\vert \Psi }_m^{z_a z_b } (z_a ( t),z_b ) \end{aligned}$$
(7.4)

Denoting the mean number of electrons inside loop \(C\) with\(\langle {n_{e}}\rangle _C \), it turns out that solving Eq. 7.4 leads to the following expression for the geometric phase \(\theta _G =-2\pi \langle {n_{e}}\rangle _C \).Footnote 51 If quasihole \(b\) is outside the loop then \(\langle {n_{e}}\rangle _C \) is equal to \(\frac{\nu {\Phi }}{{\Phi }_0 }\), where \(\nu \) is the filling factor, \({\Phi }\) is the magnetic flux corresponding to the strong magnetic field applied in FQHE systems, and the constant \({\Phi }_0 \equiv \frac{hc}{e}\) is the “flux quanta,” so that \(\theta _G =-2\pi \frac{\nu {\Phi }}{{\Phi }_0 }\). However, if quasihole \(b\) is inside the loop then there is a deficit in mean number of electrons by an amount –\(\nu \) so that \(\theta _G =-2\pi \frac{\nu {\Phi }}{{\Phi }_0 }+2\pi \nu \). The difference in geometric phase between the two scenarios is \({\Delta }\theta _G =2\pi \nu \).

In other words, when quasihole \(a\) encircles quasihole \(b\), the new doubly permuted wavefunction \(\psi _m^{{'}z_a z_b } \)gains an extra geometric phase \({\Delta }\theta _G =2\pi \nu \):

$$\begin{aligned} {\Psi }_m^{'z_a z_b } =e^{i2\pi \nu }{\Psi }_m^{z_a z_b } \end{aligned}$$

But recall from Sect. 1 that double permutation leads to general phase factor with an exchange phase \(\theta :\)

$$\begin{aligned} {\Psi }_m^{'z_a z_b } =e^{i2\theta }{\Psi }_m^{z_a z_b } =e^{i2\pi \alpha }{\Psi }_m^{z_a z_b } \end{aligned}$$

Where we have introduced the “statistical parameter” defined as \(\alpha \equiv \frac{\theta }{\pi }\). We see that \(\alpha =\nu \) and recalling that \(\nu =\frac{1}{m}\) where \(m\) is an odd integer, it follows that \(\theta =\frac{\pi }{m}\) . For the m=1 case, \(\theta =\pi \) corresponding to Fermi–Dirac statistics. But for other values of \(m\), \(\theta \) corresponds to anyonic statistics.

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Shech, E. Two Approaches to Fractional Statistics in the Quantum Hall Effect: Idealizations and the Curious Case of the Anyon. Found Phys 45, 1063–1100 (2015). https://doi.org/10.1007/s10701-015-9899-0

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Keywords

  • Idealization
  • Approximation
  • Emergence
  • Reduction
  • Aharonov–Bohm effect
  • Quantum Hall effect
  • Anyons
  • Fractional Statistics
  • Representation
  • Explanation