quantum statistics on graphs

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1 n-particle 1, J.P. Keating 2, J.M. Robbins 2 and A. Sawicki 1 Baylor University, 2 University of Bristol, M.I.T. TexAMP 11/14

2 Quantum statistics Single particle space configuration space X. Two particle statistics - alternative approaches: Quantize X 2 and restrict Hilbert space to the symmetric or anti-symmetric subspace. Bose-Einstein/Fermi-Dirac statistics. ψ(x 1, x 2 ) = ±ψ(x 2, x 1 ) (1)

3 Quantum statistics Single particle space configuration space X. Two particle statistics - alternative approaches: Quantize X 2 and restrict Hilbert space to the symmetric or anti-symmetric subspace. Bose-Einstein/Fermi-Dirac statistics. ψ(x 1, x 2 ) = ±ψ(x 2, x 1 ) (1) (Leinaas and Myrheim 77) Treat particles as indistinguishable, ψ(x 1, x 2 ) ψ(x 2, x 1 ). Quantize two particle configuration space.

4 Definition Configuration space of n indistinguishable particles in X, C n (X ) = (X n n )/S n where n = {x 1,..., x n x i = x j for some i j}.

5 Definition Configuration space of n indistinguishable particles in X, C n (X ) = (X n n )/S n where n = {x 1,..., x n x i = x j for some i j}. 1st homology groups of C n (R d ): H 1 (C n (R d )) = Z 2 for d. 2 abelian irreps. corresponding to Bose-Einstein & Fermi-Dirac statistics.

6 Definition Configuration space of n indistinguishable particles in X, C n (X ) = (X n n )/S n where n = {x 1,..., x n x i = x j for some i j}. 1st homology groups of C n (R d ): H 1 (C n (R d )) = Z 2 for d. 2 abelian irreps. corresponding to Bose-Einstein & Fermi-Dirac statistics. H 1 (C n (R 2 )) = Z Any single phase e iθ can be associated to every primitive exchange path anyon statistics.

7 Definition Configuration space of n indistinguishable particles in X, C n (X ) = (X n n )/S n where n = {x 1,..., x n x i = x j for some i j}. 1st homology groups of C n (R d ): H 1 (C n (R d )) = Z 2 for d. 2 abelian irreps. corresponding to Bose-Einstein & Fermi-Dirac statistics. H 1 (C n (R 2 )) = Z Any single phase e iθ can be associated to every primitive exchange path anyon statistics. H 1 (C n (R)) = 1 particles cannot be exchanged.

8 What happens on a graph where the underlying space has arbitrarily complex topology?

9 Graph connectivity Given a connected graph Γ a k-cut is a set of k vertices whose removal makes Γ disconnected. Γ is k-connected if the minimal cut is size k. Theorem (Menger) For a k-connected graph there exist at least k independent paths between every pair of vertices. Example: v u Two cut

10 Graph connectivity Given a connected graph Γ a k-cut is a set of k vertices whose removal makes Γ disconnected. Γ is k-connected if the minimal cut is size k. Theorem (Menger) For a k-connected graph there exist at least k independent paths between every pair of vertices. Example: v u Two cut

11 Graph connectivity Given a connected graph Γ a k-cut is a set of k vertices whose removal makes Γ disconnected. Γ is k-connected if the minimal cut is size k. Theorem (Menger) For a k-connected graph there exist at least k independent paths between every pair of vertices. Example: v u Two independent paths joining u and v.

12 Features of graph statistics On statistics only depend on whether the graph is planar (Anyons) or non-planar (Bosons/Fermions).

13 Features of graph statistics On statistics only depend on whether the graph is planar (Anyons) or non-planar (Bosons/Fermions). A two dimensional lattice with a small section that is non-planar is locally planar but has Bose/Fermi statistics.

14 On 2-connected graphs statistics are independent of the number of particles.

15 On 2-connected graphs statistics are independent of the number of particles. F B F B F For example, one could construct a chain of -connected non-planar components where particles behave with alternating Bose/Fermi statistics.

16 On 1-connected graphs the statistics depends on the no. of particles n.

17 On 1-connected graphs the statistics depends on the no. of particles n. Example, star with E edges. no. of anyon phases ( ) ( ) n + E 2 n + E 2 (E 2) + 1. E 1 E 2

18 1st homology group of graph By the structure theorem for finitely generated modules (for a suitably subdivided graph Γ) where n i n i+1. H 1 (C n (Γ)) = Z k Z n1... Z nl, (2) So H 1 (C n (Γ)) is determined by k free (anyon) phases {φ 1,..., φ k } and l discrete phases {ψ 1,..., ψ l } such that for each i {1,... l} n i ψ i = 0 mod 2π, n i N and n i n i+1. ()

19 Basic cases Quantum statistics For 2 particles. Γ 1 2 C 2 (Γ) (1) (12) (2) around loop c; one free phase φ c (1) (2) (12) (4) (14) (24) at Y-junction; one free phase φ Y.

20 Basic cases Quantum statistics For 2 particles. Γ C 2 (Γ) (1) 1 2 (12) (2) around loop c; one free phase φ c (1) (2) (12) (4) (14) (24) at Y-junction; one free phase φ Y.

21 Basic cases Quantum statistics For 2 particles. Γ 1 2 C 2 (Γ) (1) (12) (2) around loop c; one free phase φ c (1) (2) (12) (4) (14) (24) at Y-junction; one free phase φ Y.

22 Basic cases Quantum statistics For 2 particles. Γ C 2 (Γ) (1) 1 2 (12) (2) around loop c; one free phase φ c (1) (2) (12) (4) (14) (24) at Y-junction; one free phase φ Y.

23 Basic cases Quantum statistics For 2 particles. Γ C 2 (Γ) (1) 1 2 (12) (2) around loop c; one free phase φ c (1) (2) (12) (4) (14) (24) at Y-junction; one free phase φ Y.

24 Basic cases Quantum statistics For 2 particles. Γ 1 2 C 2 (Γ) (1) (12) (2) around loop c; one free phase φ c (1) (2) (12) (4) (14) (24) at Y-junction; one free phase φ Y.

25 Basic cases Quantum statistics For 2 particles. Γ 1 2 C 2 (Γ) (1) (12) (2) around loop c; one free phase φ c (1) (2) (12) (4) (14) (24) at Y-junction; one free phase φ Y.

26 Basic cases Quantum statistics For 2 particles. Γ 1 2 C 2 (Γ) (1) (12) (2) around loop c; one free phase φ c (1) (2) (12) (4) (14) (24) at Y-junction; one free phase φ Y.

27 Basic cases Quantum statistics For 2 particles. Γ 1 2 C 2 (Γ) (1) (12) (2) around loop c; one free phase φ c (1) (2) (12) (4) (14) (24) at Y-junction; one free phase φ Y.

28 Basic cases Quantum statistics For 2 particles. Γ 1 2 C 2 (Γ) (1) (12) (2) around loop c; one free phase φ c (1) (2) (12) (4) (14) (24) at Y-junction; one free phase φ Y.

29 Basic cases Quantum statistics For 2 particles. Γ 1 2 C 2 (Γ) (1) (12) (2) around loop c; one free phase φ c (1) (2) (12) (4) (14) (24) at Y-junction; one free phase φ Y.

30 Basic cases Quantum statistics For 2 particles. Γ 1 2 C 2 (Γ) (1) (12) (2) around loop c; one free phase φ c (1) (2) (12) (4) (14) (24) at Y-junction; one free phase φ Y.

31 Lasso graph Quantum statistics 4 (1) (2) 2 (12) (4) 1 (14) (24) Identify three 2-particle cycles: (i) Rotate both particles around loop c; phase φ c,2. (ii) Exchange particles on Y-subgraph; phase φ Y. (iii) Rotate one particle around loop c other particle at vertex 1; (1, 2) (1, ) (1, 4) (1, 2), phase φ 1 c,1. Relation from contactable 2-cell φ c,2 = φ 1 c,1 + φ Y.

32 Lasso graph Quantum statistics 4 (1) (2) 2 (12) (4) 1 (14) (24) Identify three 2-particle cycles: (i) Rotate both particles around loop c; phase φ c,2. (ii) Exchange particles on Y-subgraph; phase φ Y. (iii) Rotate one particle around loop c other particle at vertex 1; (1, 2) (1, ) (1, 4) (1, 2), phase φ 1 c,1. Relation from contactable 2-cell φ c,2 = φ 1 c,1 + φ Y.

33 Let c be a loop. What is the relation between φ u c,1 and φv c,1? (a) u and v joined by path disjoint with c. φ u c,1 = φv c,1 as exchange cycles homotopy equivalent. (b) u and v only joined by paths through c. Two lasso graphs so φ c,2 = φ u c,1 + φ Y 1 & φ c,2 = φ v c,1 + φ Y 2. Hence φ u c,1 φv c,1 = φ Y 2 φ Y1. (a) (b) Y 1 c Y 2 Y 1 c Y 2 u v u v

34 Let c be a loop. What is the relation between φ u c,1 and φv c,1? (a) u and v joined by path disjoint with c. φ u c,1 = φv c,1 as exchange cycles homotopy equivalent. (b) u and v only joined by paths through c. Two lasso graphs so φ c,2 = φ u c,1 + φ Y 1 & φ c,2 = φ v c,1 + φ Y 2. Hence φ u c,1 φv c,1 = φ Y 2 φ Y1. (a) (b) Y 1 c Y 2 Y 1 c Y 2 u v u v Relations between phases involving c encoded in phases φ Y. H 1 (C 2 (Γ)) = Z β 1(Γ) A, where A determined by Y-cycles. In (a) we have a B subgraph & using (b) also φ Y1 = φ Y2.

35 The prototypical -connected graph is a wheel W k. W 5 Theorem (Wheel theorem) Let Γ be a simple -connected graph different from a wheel. Then for some edge e Γ either Γ \ e or Γ/e is simple and -connected. Γ \ e is Γ with the edge e removed. Γ/e is Γ with e contracted to identify its vertices.

36 Lemma For -connected simple graphs all phases φ Y are equal up to a sign. Sketch proof. The lemma holds on K 4 (minimal wheel). By wheel thm we only need to show that adding an edge or expanding a vertex any new phases φ Y are the same as the original phase. Adding an edge: Γ e Γ e Using -connectedness identify independent paths in Γ to make B. Then φ Y = φ Y.

37 Lemma For -connected simple graphs all phases φ Y are equal up to a sign. Sketch proof. The lemma holds on K 4 (minimal wheel). By wheel thm we only need to show that adding an edge or expanding a vertex any new phases φ Y are the same as the original phase. Adding an edge: Γ e Γ e Using -connectedness identify independent paths in Γ to make B. Then φ Y = φ Y.

38 Lemma For -connected simple graphs all phases φ Y are equal up to a sign. Sketch proof. The lemma holds on K 4 (minimal wheel). By wheel thm we only need to show that adding an edge or expanding a vertex any new phases φ Y are the same as the original phase. Vertex expansion: Split vertex of degree > into two vertices u and v joined by a new edge e. Γ v e u Using -connectedness identify independent paths in Γ to make B. Then φ Y = φ Y.

39 Lemma For -connected simple graphs all phases φ Y are equal up to a sign. Sketch proof. The lemma holds on K 4 (minimal wheel). By wheel thm we only need to show that adding an edge or expanding a vertex any new phases φ Y are the same as the original phase. Vertex expansion: Split vertex of degree > into two vertices u and v joined by a new edge e. Γ v e u Using -connectedness identify independent paths in Γ to make B. Then φ Y = φ Y.

40 Theorem For a -connected simple graph, H 1 (C 2 (Γ)) = Z β 1(Γ) A, where A = Z 2 for non-planar graphs and A = Z for planar graphs.

41 Theorem For a -connected simple graph, H 1 (C 2 (Γ)) = Z β 1(Γ) A, where A = Z 2 for non-planar graphs and A = Z for planar graphs. Proof. For K 5 and K, every phase φ Y = 0 or π. By Kuratowski s theorem a non-planar graph contains a subgraph which is isomorphic to K 5 or K,.

42 Theorem For a -connected simple graph, H 1 (C 2 (Γ)) = Z β 1(Γ) A, where A = Z 2 for non-planar graphs and A = Z for planar graphs. Proof. For K 5 and K, every phase φ Y = 0 or π. By Kuratowski s theorem a non-planar graph contains a subgraph which is isomorphic to K 5 or K,. For planar graphs the anyon phase can be introduced by drawing the graph in the plane and integrating the anyon α vector potential 2π ẑ r 1 r 2 r 1 r 2 along the edges of the 2 two-particle graph, where r 1 and r 2 are the positions of the particles.

43 Summary Quantum statistics Full classification of abelian by decomposing graph in 1-, 2- and -connected components. Physical insight into dependance of statistics on graph connectivity. Interesting new features of graph statistics. Statistics incorporated in gauge potential. JH, JP Keating, JM Robbins and A Sawicki, n-particle, Commun. Math. Phys. (2014) arxiv: JH, JP Keating and JM Robbins, Quantum statistics on graphs, Proc. R. Soc. A (2010) doi: /rspa arxiv:

Topology and quantum mechanics

Topology and quantum mechanics Topology, homology and quantum mechanics 1, J.P. Keating 2, J.M. Robbins 2 and A. Sawicki 2 1 Baylor University, 2 University of Bristol Baylor 9/27/12 Outline Topology in QM 1 Topology in QM 2 3 Wills