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Philosophical Issues Concerning Phase Transitions and Anyons: Emergence, Reduction, and Explanatory Fictions

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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

  1. For more on the philosophical debate revolving around phase transitions see Batterman (2005), Butterfield (2011), Norton (2012), Menon and Callender (2013), Shech (2013) and references within.

  2. See Kim (2006) for various problems associated with the status of bridge laws and Butterfield (2011) for a recent defense of Nagelian-style reduction.

  3. Kim (1999, 10–11) identifies three steps but I follow Bangu (2015a, 156) in collapsing Kim’s steps two and three into one.

  4. For various well-received characterizations and taxonomies of idealizations and approximations see McMullin (1985) and Weisberg (2013).

  5. 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. For the theory of phase transitions see Stanley (1971) and Kadanoff (2000).

  7. For pedagogical but more precise articulation of the need of the TDL for non-analyticities see Blythe and Evans (2003) on Yang and Lee’s (1952) theory of PT in the context of classical SM, and see Emch (2006, Section 5.4) for the case of quantum SM.

  8. 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.

  9. See Earman (2010) for a discussion.

  10. 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.

  12. 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.

  13. 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)).

  14. For an example of some standard accounts, see Hempel’s covering law account (Hempel 1965), Salmon’s causal-mechanistic account (Salmon 1984), and Kitcher’s unificationist (Kitcher 1981).

  15. 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.

  16. 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.

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

  18. Thanks to Olivier Sartenear for helpful discussion pertinent to this subsection.

  19. For more on the quantum Hall effect see Chakraborty and Pietilinen (1995), Fradkin (2013, Ch. 13), Ezawa (2013) and Stern (2008), and for recent philosophical assessments see Bain (2013, 2016), Guay and Sartenaer’s (2016a, b), Lancaster and Pexton (2015), Lederer (2015), and Shech (2015, 2018).

  20. In order to avoid a confusion, let me stress that the distinction between anyonsT and anyonsFQHE is not one of epistemology versus ontology. Rather, it is of abstract or fictional entities versus concrete entities: anyonsT concern an abstract or fictional, theoretical structure, while anyonsFQHE concern the concrete, phenomenological entities arising in FQHE on the Laughlin ground state and Topological order accounts.

  21. Thanks to Jonathan Bain for suggesting how this argument should go. All mistakes are my own of course.

  22. 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.

  23. Here I follow Bain (2016, 9) in order to connect with Guay and Sartenaer (2016a, b)’s discussion, but also see Fradkin (2013, 502) and Zee (1995).

  24. See Lancaster and Blundell (2014, 419), Wen (2004, 298), and Zee (1995, 110) for details.

  25. 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).

  26. See Zee (1995, 110) and Wen (2004, 298) for the details.

  27. See Menon and Callender (2013) and Bangu (2015a, b) for philosophical a philosophical assessment.

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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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