Different particles can be represented as different irreducible representations of Poincare group. Can we classify quasiparticles using irreducible representations of some group? Also, quasiparticles are low energy excitations of the system, and it is closely related to the ground states of the system, so I am wondering if there is any relations we can use to help us classify quasiparticles from ground state symmetry. Also, is the irreducible representation of some group argument can apply to those excitations of topological origin?
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2$\begingroup$ What is your precise definition of "quasiparticles" here? $\endgroup$– ACuriousMind ♦Jun 6, 2017 at 11:27
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$\begingroup$ @ACuriousMind Sticking to standard vocabulary, he means the like of holes in semi-conductor, phonons in solids, etc. Actually maybe not the latter as sometimes quasi-particle is restricted to fermions. $\endgroup$– user154997Jun 6, 2017 at 12:07
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1$\begingroup$ @LucJ.Bourhis See, that always happens - someone asks about the classifications of quasiparticles (or solitions, or "topological excitations"), I ask for a precise definition so we know what we actually want to classify, and the response is a list of examples. You'll have to admit that a list of examples is not a definition. $\endgroup$– ACuriousMind ♦Jun 6, 2017 at 12:09
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1$\begingroup$ Fair enough, the concept is ill-defined. But it is not the OP being sloppy here, so asking for a definition is too stringent here. At best, we could ask for a set of examples. $\endgroup$– user154997Jun 6, 2017 at 12:18
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$\begingroup$ @ACuriousMind I agree with you on the lack of rigorous definition, but I'd like to try :-) Why not defining quasi-particle (QP) as the elementary excitations on top of a spontaneous symmetry broken ground state, and considered as a free gas. I'm not sure one needs to require mean-field approximation to make it more sound. In any case, one can now give a few examples: Bogoliubov-QP for U(1) gauge redundancy breaking in superconductor, Galilean relativity breaking for neutral QP in superfluid, phonon-QP for translation invariance breaking, magnons for rotation breaking of ferromagnetic systems.. $\endgroup$– FraSchelleJun 7, 2017 at 6:47
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The group representation argument can not be applied to those excitations of topological origin. Group theory is not the most general classification criterion of quasiparticles. Group theory is useful when symmetry plays an important role in the discussion. For example, Goldstone modes in the symmetry breaking phases can be classified by group representation theory, boundary modes in the symmetry protected topological phases can be classified by group cohomology theory. But if the system has no symmetry (or the symmetry is not relevant to the discussion), then group theory may not be very useful. For example, different anyon excitations in topologically ordered phases are not classified by their group representations: they may all belong to the trivial representation for example, but still, they are distinct topological excitations and can not transit from one to another. In this case, we use category theory to classify different topological excitations. In fact, category theory is a more general classification criterion of quasiparticles. Because the representations of a group also form a category, so the group representation classification is part of the categorical classification. Moreover, the categorical classification also captures the topological excitations nicely, which demonstrate its advantage over the group theory classification.
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6$\begingroup$ This answer sounds very interesting, but it stops short of actually telling the reader what the classification is. What categories classify quasiparticles and why? $\endgroup$– ACuriousMind ♦Sep 11, 2017 at 7:49