Abstract
We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space X. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of X which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.
Highlights
Cn(X ) := (X ×n − Δn)/Sn, where Δn := {(x1, . . . , xn) ∈ X ×n : ∃i= j xi = x j } and Sn is the permutation group that acts on X ×n by permuting coordinates [1]
Quantum statistics can be viewed as a flat connection on the configuration space Cn(X ) that modifies definition of the momentum operator according to minimal coupling principle
The most challenging part in formulating such a framework is to avoid the language of differential geometry, as graph configuration spaces are not manifolds, whereas the great majority of results in the field concerns quantum statistics on manifolds
Summary
It is easy to see that exchanges of particles on X correspond to closed loops in Cn(X ) [1,2,3] Under this identification all possible quantum statistics (QS) are classified by unitary representations of the fundamental group π1(Cn(X )). Because the relevant literature is rather scarce, it was a nontrivial task to make such a meta analysis and we consider it an essential step in describing our results This is because we see the need of introducing in a systematic and concise way the framework for studying quantum statistics which is designed for graphs. We emphasise the important role of nontrivial flat vector bundles that can lead to spontaneously occurring non-abelian quantum statistics This is motivated by the fact that in R3 fermions and bosons correspond to two non-isomorphic vector bundles that admit flat connections. For general paracompact X , this point can be phrased as classification of the U (k) - representations of the corresponding braid group, i.e. the fundamental group of Cn(X )
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