## Abstract

Various claims regarding intertheoretic reduction, weak and strong notions of emergence, and explanatory fictions have been made in the context of first-order thermodynamic phase transitions. By appealing to John Norton’s recent distinction between approximation and idealization, I argue that the case study of anyons and fractional statistics, which has received little attention in the philosophy of science literature, is more hospitable to such claims. In doing so, I also identify three novel roles that explanatory fictions fulfill in science. Furthermore, I scrutinize the claim that anyons, as they are ostensibly manifested in the fractional quantum Hall effect, are emergent entities and urge caution. Consequently, it is suggested that a particular notion of strong emergence signals the need for the development of novel physical–mathematical research programs.

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## Notes

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Butterfield (2011, Section 3) discusses similar distinctions. In particular, he makes a distinction between a

*system*\( \sigma \left( N \right) \) that depends on some parameter \( N \) (let \( \{ \sigma \left( N \right) \)} denote a sequence of such systems), a*quantity*defined on the system \( f\left( {\sigma \left( N \right)} \right) \) (let \( \left\{ {f\left( {\sigma \left( N \right)} \right)} \right\} \) denote a sequence of quantities on successive systems), and a (real number)*value*\( v\left( {f\left( {\sigma \left( N \right)} \right)} \right) \) of quantities on successive systems (where a sequence of states on \( \sigma \left( N \right) \) is implicitly understood; let \( \left\{ {v\left( {f\left( {\sigma \left( N \right)} \right)} \right)} \right\} \) denote a sequence of values on successive systems). A*limit system*\( \sigma \left( \infty \right) \) arises when \( \lim_{N \to \infty } \left\{ {\sigma \left( N \right)} \right\} \) is well-defined—otherwise there is no limit system. A*property of a limit system*refers to the value \( v(f\left( {\sigma \left( \infty \right)} \right) \) of the (natural) limit quantity \( f\left( {\sigma \left( \infty \right)} \right) \) (in the natural limit state) on \( \sigma \left( \infty \right) \). A*limit property*\( v(f\left( {\sigma \left( N \right)} \right) \) 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 \( \lim_{N \to \infty } \left\{ {v\left( {f\left( {\sigma \left( N \right)} \right)} \right)} \right\} \) 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 \( v(f\left( {\sigma \left( \infty \right)} \right) = \lim_{N \to \infty } \left\{ {v\left( {f\left( {\sigma \left( N \right)} \right)} \right)} \right\} \) (assuming both are well-defined). - 6.
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As an anonymous reviewer notes, at this point in the history of the discipline most theoreticians that would accept particles in their ontology at all would also accept anyons based on the fractional quantum Hall effect. Experimentalists may be more divided since the concept of effective theories has not reached the same centrality in this community.

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See Earman (2010) for a discussion.

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The name is due to Noble laureate Wilczek (1982). 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 in the context of the operator approach. For more on anyons see Wilczek (1990), Khare (2005), Stern (2008) and references therein.

- 11.
It may be incorrect to say that fractional statistics cannot arise on the operator approach. As is noted by an anonymous reviewer, the operator approach views permutation invariance as a constraint imposed directly on an algebra of local observables, and fractional statistics can be expressed in the algebraic formalism. A recent discussion can be found in Naaijkens (2015). Similarly, although Landsman (2016) does say that the equivalence between the two approaches fails in two dimensions since there are technical problems with identifying all the irreducible representations of the braid group, it may be possible to evade this problem by knowing ahead of time (as in fractional quantum Hall systems) all those irreducible representations that are relevant to one’s particular system. In any case, I leave these issues aside for future study, and for the purposes of this paper I will work with the standard assumption that anyons only arise in the context of the configuration space framework.

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Recall Norton’s distinction between approximation and idealization from Sect. 2.2. and see Norton (2012, 215–219) for details: The TDL as the continuum limit (Compagner 1989), the Boltzmann–Grad limit (Lanford 1975, 70–89, 1981), or the “weak” TDL (Le Bellac et al. 2004, 112) is an approximation, while the TDL as the “strong” TDL corresponding to a well-defined infinite system (e.g., Ruelle 1999/2007, 2004; Lanford 1975, Section 4) is an idealization.

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The non-analyticities of the thermodynamic potentials correspond to a mismatch between properties of the limit system and the corresponding limit properties (cf. Norton (2012)).

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For more on fictions in science see contributions in Suarez (2009) and my own Shech (2016). Batterman (2002), Batterman and Rice (2014), Bokulich (2008) all develop accounts of explanation for which fictions (or idealizations and abstractions) are indispensible, and see Earman (2017), Shech (2015, 2017) and Shech and Gelfert (2016) form more on the exploratory role of fictions.

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This is similar to Norton’s (2012, 213) point that there is no such thing as an infinitely large sphere. To be clear: A disk is an object with a circular circumference that is composed of a set of points equally far away from the center. But there are no such points for an infinite radius, and so there is no corresponding limit system.

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Gelfert (2016, 83–94) calls this role a “proof-of-principle demonstration” in the context of the exploratory models.

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In order to avoid a confusion, let me stress that the distinction between anyons

_{T}and anyons_{FQHE}is not one of epistemology versus ontology. Rather, it is of abstract or fictional entities versus concrete entities: anyons_{T}concern an abstract or fictional, theoretical structure, while anyons_{FQHE}concern the concrete, phenomenological entities arising in FQHE on the Laughlin ground state and Topological order accounts. - 20.
Thanks to Jonathan Bain for suggesting how this argument should go. All mistakes are my own of course.

- 21.
Excising talk of “conventionality,” Lancaster and Pexton (2015, Section 4) word similar worries in discussing possible entity emergentism in the FQHE. In addition, note that the analogy with parastatistics is weaker in the context of relativistic quantum mechanics, and so it should be taken with a grain of salt, so to speak.

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While \( {\mathcal{L}} \) describes a non-relativistic quantum field theory with an implicit spacetime metric such that it is invariant under Galilean symmetries, \( {\mathcal{L}}_{eff} \) describes a topological quantum field theory with no spacetime metric, so that it is not invariant under said symmetries. Thus, \( {\mathcal{L}}_{eff} \) is “both

*dynamically independent*of, and*dynamically robust*with respect to” \( {\mathcal{L}} \) (Bain 2016, 21; original emphasis). - 25.
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## References

Ando, T., Fowler, A. B., & Stern, F. (1982). Electronic properties of two-dimensional systems.

*Reviews of Modern Physics,**54,*437–672.Arovas, D., Schrieffer, J. R., & Wilczek, F. (1984). Fractional statistics and the quantum Hall effect.

*Physical Review Letters,**53,*722–723.Artin, E. (1947). Theory of braids.

*Annals of Mathematics,**48*(1), 101–126.Bain, J. (2013). Emergence in effective field theories.

*European Journal for Philosophy of Science,**3,*257–273.Bain, J. (2016). Emergence and mechanism in the fractional quantum Hall effect.

*Studies in History and Philosophy of Modern Physics*,*56*, 27–38.Baker, D. J., Halvorson, H., & Swanson, N. (2015). The conventionality of parastatistics.

*British Journal for the Philosophy of Science,**66*(4), 929–976.Bangu, S. (2009). Understanding thermodynamic singularities: phase transitions, date and phenomena.

*Philosophy of Science,**76,*488–505.Bangu, S. (2011). On the role of bridge laws in intertheoretic relations.

*Philosophy of Science,**78*(5), 1108–1119.Bangu, S. (2015a). Neither weak, no strong? Emergence and functional reduction. In B. Falkenburg & M. Morrison (Eds.),

*Why more is different: Philosophical issues in condensed matter physics and complex systems*(pp. 153–164). Heidelberg: Springer.Bangu, S. (2015b). Why does water boil? Fictions in scientific explanation. In U. Mäki., I. Votsis., S. Ruphy. & G. Schurz (Eds.),

*Recent developments in the philosophy of science: EPSA13 Helsinki. European studies in philosophy of science*(Vol. 1, pp. 319–330). Cham: Springer.Batterman, R. (2002).

*The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence*. London: Oxford University Press.Batterman, R. (2005). Critical phenomena and breaking drops: Infinite idealizations in physics.

*Studies in History and Philosophy of Modern Physics,**36,*225–244.Batterman, R., & Rice, C. (2014). Minimal model explanations.

*Philosophy of Science,**81*(3), 349–376.Blythe, R. A., & Evans, M. R. (2003). The Lee–Yang theory of equilibrium and nonequilibrium phase transitions.

*The Brazilian Journal of Physics,**33*(3), 464–475.Bokulich, A. (2008).

*Re-examining the quantum-classical relation: Beyond reductionism and pluralism*. Cambridge: Cambridge University Press.Borrmann, P., Mülken, O., & Harting, J. (2000). Classification of phase transitions in small systems.

*Physical Review Letters,**84,*3511–3514.Butterfield, J. (2011). Less is different: Emergence and reduction reconciled.

*Foundations of Physics,**41*(6), 1065–1135.Camino, F. E., Zhou, W., & Goldman, V. J. (2005). Realization of a Laughlin quasiparticle interferometer: Observation of fractional statistics.

*Physical Review B,**72,*075342.Chakraborty, T., & Pietilinen, P. (1995).

*The quantum Hall effects*. Berlin: Springer.Chalmers, D. J. (2006). Strong and weak emergences. In P. Clayton & P. Davies (Eds.),

*The re-emergence of emergence: The emergentist hypothesis from science to religion*(pp. 244–257). Oxford: Oxford University Press.Chomaz, P., Gulminelli, F., & Duflot, V. (2001). Topology of event distributions as a generalized definition of phase transitions in finite systems.

*Physical Review E,**64,*046114.Compagner, A. (1989). Thermodynamics as the continuum limit of statistical mechanics.

*American Journal of Physics,**57,*106–117.Earman, J. (2010). Understanding permutation invariance in quantum mechanics. Unpublished manuscript.

Earman, J. (2017). The role of idealizations in the Aharonov–Bohm effect.

*Synthese*. https://doi.org/10.1007/s11229-017-1522-9.Emch, G. (2006). Quantum statistical physics. In Butterfield, J., & Earman, J. (Eds.),

*Philosophy of physics, part B, a volume of the handbook of the philosophy of science*(pp. 1075–1182). North Holland.Ezawa, Z. F. (2013).

*Quantum Hall effects: Recent theoretical and experimental developments*. Singapore: World Scientific.Fadell, E., & Neuwirth, L. (1962). Configuration Spaces.

*Mathematica Scandinavica,**10,*111–118.Falkenburg, B. (2015). How do quasi-particles exist? In B. Falkenburg & M. Morrison (Eds.),

*Why more is different: Philosophical issues in condensed matter physics and complex systems*(pp. 227–249). Heidelberg: Springer.Fox, R., & Neuwirth, L. (1962). The braid groups.

*Mathematica Scandinavica,**10,*119–126.Fradkin, E. (2013).

*Field theories of condensed matter physics*(2nd ed.). Cambridge: Cambridge University Press.Franzosi, R., & Pettini, M. (2004). Theorem on the origin of phase transitions.

*Physical Review Letters,**92,*060601.Franzosi, R., Pettini, M., & Spinelli, L. (2000). Topology and phase transitions: Paradigmatic evidence.

*Physical Review Letters,**84,*2774–2777.Gelfert, A. (2016).

*How to do science with models: A philosophical primer*. Cham: Springer.Gross, D. H. E., & Votyakov, E. V. (2000). Phase transitions in “small” systems.

*The European Physical Journal B—Condensed Matter and Complex Systems,**15,*115–126.Guay, A., & Sartenaer, O. (2016a). A new look at emergence. Or when after is different.

*European Journal for Philosophy of Science,**6,*297–322.Guay, A., & Sartenaer, O. (2016b). Emergent quasiparticles. Or how to get a rich physics from a sober metaphysics. In: O. Bueno, R. Chen, & M. B. Fagan (Eds.),

*Individuation across experimental and theoretical sciences*. Oxford: Oxford University Press. http://hdl.handle.net/2078.1/179059.Halliday, D., Resnick, R., & Walker, J. (2011).

*Fundamental of physics*(9th ed.). Hoboken, NJ: Wiley.Hempel, C. (1965).

*Aspects of scientific explanation and other essays in the philosophy of science*. New York: Free Press.Hendry, R. F. (2010). Ontological reduction and molecular structure.

*Studies in History and Philosophy of Modern Physics,**41,*183–191.Huang, Wung-Hong. (1995). Boson-fermion transmutation and the statistics of anyons.

*Physical Review E,**51*(4), 3729–3730.Jain, J. (1989). Composite-fermion approach for the fractional quantum Hall effect.

*Physical Review Letters,**63,*199–202.Jian, C.-M., & Qi, X.-L. (2014). Layer construction of 3D topological states and string braiding statistics.

*Physical Review X,**4,*041043.Jiang, S., Mesaros, A., & Ran, Y. (2014). Generalized modular transformations in (3 + 1)D topologically ordered phases and triple linking invariant of loop braiding.

*Physical Review X,**4,*031048.Kadanoff, L. P. (2000).

*Statistical physics: Statics, dynamics and renormalization*. Singapore: World Scientific.Khare, A. (2005).

*Fractional statistics and quantum theory*. New Jersey: World Scientific.Kim, J. (1998).

*Mind in a physical world*. Cambridge: MIT Press.Kim, J. (1999). Making sense of emergence.

*Philosophical Studies,**95,*3–36.Kim, J. (2006). Emergence: Core ideas and issues.

*Synthese,**151*(3), 347–354.Kitcher, P. (1981). Explanatory unification.

*Philosophy of Science,**48,*507–531.Klitzing, K. V., Dorda, G., & Pepper, M. (1980). New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance.

*Physical Review Letters,**45,*494.Laidlaw, M. G., & DeWitt, C. M. (1971). Feyman functional integrals for system of indistinguishable particles.

*Physical Review D,**3,*1375–1378.Lancaster, T., & Blundell, S. (2014).

*Quantum field theory for the gifted amateur*. Oxford: Oxford University Press.Lancaster, T., & Pexton, M. (2015). Reduction and emergence in the fractional quantum Hall state.

*Studies in History and Philosophy of Modern Physics,**52,*343–357.Landsman, N. P. (2016). Quantization and superselection III: Mutliply connected spaces and indistinguishable particles.

*Reviews in Mathematical Physics,**28,*1650019.Lanford, O. (1975). Time evolution of large classical systems. In J. Moser (Ed.),

*Dynamical systems, theory and applications: Lecture notes in theoretical physics*(Vol. 38, pp. 1–111). Heidelberg: Springer.Lanford, O. (1981). The hard sphere gas in the Boltzmann–Grad limit.

*Physica A,**106,*70–76.Laughlin, R. (1983). Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations.

*Physical Review Letters,**50,*1395–1398.Laughlin, R. B. (1999). Nobel lecture: fractional quantization.

*Reviews of Modern Physics,**71,*863–874.Laughlin, R. B. (2005).

*A different universe: Reinventing physics from the bottom down*. New York: Basic Books.Le Bellac, M., Mortessagne, F., & Batrouni, G. G. (2004).

*Equilibrium and non-equilibrium statistical thermodynamics*. Cambridge: Cambridge University Press.Lederer, P. (2015). The quantum Hall effects: Philosophical approach.

*Studies in History and Philosophy of Modern Physics,**50,*25–42.Leinaas, J. M., & Myrheim, J. (1977). On the theory of identical particles.

*Nuovo Cimento,**37B,*1–23.Masenes, L., & Oppenheim, J. (2017). A general derivation and quantification of the third law of thermodynamics.

*Nature Communications,**8,*14538.McMullin, E. (1985). Galilean Idealization.

*Studies in the History and Philosophy of Science,**16,*247–273.Menon, T., & Callender, C. (2013). Turn and face the strange… Ch-ch-changes: Philosophical questions raised by phase transitions. In R. Batterman (Ed.),

*The oxford handbook to philosophy of physics*(pp. 189–223). Oxford: Oxford University Press.Messiah, A. M. (1962).

*Quantum mechanics*. New York, NY: Wiley.Messiah, A. M., & Greenberg, O. W. (1964). Symmetrization postulate and its experimental foundation.

*Physical Review B,**136,*248–267.Mitchell, S. D. (2012). Emergence: Logical, functional and dynamical.

*Synthese,**185,*171–186.Morandi, G. (1992).

*The role of topology in classical and quantum mechanics*. Berlin: Springer.Morrison, M. (2012). Emergent physics and micro-ontology.

*Philosophy of Science,**79,*141–166.Naaijkens, P. (2015). Kitaev’s quantum double model from a local quantum physics point of view. In R. Brunetti et al. (Eds.),

*Advances in algebraic quantum field theory*(p. 365). New York: Springer.Nagel, E. (1961).

*The structure of science*. New York: Harcourt, Brace and World.Norton, J. D. (2012). Approximations and Idealizations: Why the difference matters.

*Philosophy of Science,**79,*207–232.O’Connor, T., & Wong, H. Y. (2005). The metaphysics of emergence.

*Noûs,**39*(658), 678.Rueger, A. (2000). Physical emergence, diachronic and synchronic.

*Synthese,**124,*297–322.Ruelle, D. (1999/2007).

*Statistical mechanics: Rigorous results*. Repr. Singapore: World Scientific.Ruelle, D. (2004).

*Thermodynamic formalism*(2nd ed.). Cambridge: Cambridge University Press.Salmon, W. (1984).

*Scientific explanation and the causal structure of the world*. Princeton: Princeton University Press.Santos, G. C. (2015). Ontological emergence: How is that possible? Towards a new relational ontology.

*Foundations of Science,**20*(4), 429–446.Shech, E. (2013). What is the ‘paradox of phase transitions?’.

*Philosophy of Science,**80,*1170–1181.Shech, E. (2015). Two approaches to fractional statistics in the quantum Hall effect: Idealizations and the curious case of the anyon.

*Foundations of Physics,**45*(9), 1063–1110.Shech, E. (2016). Fiction, depiction, and the complementarity thesis in art and science.

*The Monist,**99*(3), 311–332.Shech, E. (2017). Idealizations, essential self-adjointness, and minimal model explanation in the Aharonov–Bohm effect.

*Synthese*. https://doi.org/10.1007/s11229-017-1428-6.Shech, E. (2018). Infinitesimal idealization, easy road nominalism, and fractional quantum statistics.

*Synthese*. https://doi.org/10.1007/s11229-018-1680-4.Shech, E., & Gelfert, A. (2016).

*The exploratory role of idealizations and limiting cases in models*. http://philsci-archive.pitt.edu/13338/.Stamerjohanns, H., Oliver Mülken, O., & Borrmann, P. (2002). Deceptive signals of phase transitions in small magnetic clusters.

*Physical Review Letters,**88*(5), 053401–053414.Stanley, H. E. (1971).

*Introduction to phase transitions and critical phenomena*. New York and Oxford: Oxford University Press.Stern, A. (2008). Anyons and the quantum Hall effect-a pedagogical review.

*Annalen der Physik,**323,*204–249.Suarez, M. (Ed.). (2009).

*Fictions in science: Essays on idealization and modeling*. London: Routledge.Tsui, D. C., Stormer, H. L., & Gossard, A. C. (1982). Two-dimensional magnetotransport in the extreme quantum limit.

*Physical Review Letters,**48*(22), 1559.Wales, D. J., & Berry, R. S. (1994). Coexistence in finite systems.

*Physical Review Letters,**73,*2875–2878.Wang, C., & Levin, M. (2014). Braiding statistics of loop excitations in three dimensions.

*Physical Review Letters,**113,*080403.Wang, J. C., & Wen, X.-G. (2015). NonAbelian string and particle braiding in topological order: Modular SL(3,\( {\mathbb{Z}} \)) representation and (3 + 1)-dimensional twisted gauge theory.

*Physical Review B, 91*, 035134.Weisberg, M. (2013).

*Simulation and similarity: Using models to understand the world*. New York: Oxford University Press.Wen, X.-G. (1990). Topological orders in rigid states.

*International Journal of Modern Physics B,**4,*239–271.Wen, X.-G. (2004).

*Quantum field theory of many-body systems*. Oxford: Oxford University Press.Wilczek, F. (1982). Quantum mechanics of fractional-spin particles.

*Physical Review Letters,**49,*957–959.Wilczek, F. (Ed.). (1990).

*Fractional statistics and anyon superconductivity*. Singapore: World Scientific.Winsberg, E. (2009). A function for fictions: Expanding the scope of science. In M. Suárez (Ed.),

*Fictions in science*(pp. 179–192). London: Routledge.Yang, C. N., & Lee, T. D. (1952). Statistical theory of equations of state and phase transitions. I. Theory of condensation.

*Physical Review,**97,*404.Zee, A. (1995). Quantum Hall fluids. In H. Geyer (Ed.),

*Field theory, topology, and condensed matter physics*(pp. 99–153). Berlin: Springer.Zhang, S.-C., Hansson, T., & Kivelson, S. (1989). Effective-field-theory model for the fractional quantum Hall effect.

*Physical Review Letters,**62,*82–85.

## Acknowledgements

I gratefully acknowledge useful discussion with Jonathan Bain, Tom Lancaster, Robin Hendry, and Olivier Sartenear. Previous versions of this paper were presented at the “Emergence and the Limit: A Workshop in Philosophy of Physics” at London School of Economics on 11/25/2016, “Workshop on Symmetry and Symmetry Breaking in Fundamental Physics” at Paris Centre for Quantum Computing on 12/02/2016, and “Scale and the Sciences: A One-day Workshop” at The Institute for advanced Study at Durham University on 12/07/2016 7. I thank the participants for helpful comments. Thanks also to Narin Shech and Isabel Ranner for assistance with figures and editing. This work was produced as part of a Senior Research Fellowship at Durham University generously funded by the Institute for Advanced Study.

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Shech, E. Philosophical Issues Concerning Phase Transitions and Anyons: Emergence, Reduction, and Explanatory Fictions.
*Erkenn* **84, **585–615 (2019). https://doi.org/10.1007/s10670-018-9973-z

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