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 building Bristol University
Outline Topology in QM 1 Topology in QM 2 3 But they remember America
Quantum mechanics Two slit experiment wave-particle duality Schrödinger equation i ( ) 2 t ψ(r, t) = 2m 2 + V (r) ψ(r, t) ψ 2 probability density for particle.
Quantum mechanics Two slit experiment wave-particle duality Schrödinger equation i ( ) 2 t ψ(r, t) = 2m 2 + V (r) ψ(r, t) ψ 2 probability density for particle.
Quantum mechanics Two slit experiment wave-particle duality Schrödinger equation i ( ) 2 t ψ(r, t) = 2m 2 + V (r) ψ(r, t) ψ 2 probability density for particle.
Quantum mechanics Two slit experiment wave-particle duality Schrödinger equation i ( ) 2 t ψ(r, t) = 2m 2 + V (r) ψ(r, t) ψ 2 probability density for particle.
Quantum mechanics Two slit experiment wave-particle duality Schrödinger equation i ( ) 2 t ψ(x, t) = 2m 2 + V (x) ψ(x, t) ψ 2 probability density for particle.
Feynman path integral Action S(p) = T mẋ 2 0 2 V (x)dt the integral of the classical Lagrangian along path p. Contribution of p to probability amplitude for transition between two states is e i S(p). Total probability amplitude obtained by summing over all possible paths connecting A & B. B A
Feynman path integral Action S(p) = T mẋ 2 0 2 V (x)dt the integral of the classical Lagrangian along path p. Contribution of p to probability amplitude for transition between two states is e i S(p). Total probability amplitude obtained by summing over all possible paths connecting A & B. B A Formal solution to Schrödinger eqn at time T is exp( i HT )ψ 0 where H = 2 2m 2 + V (x). Transition amplitude from ψ 0 to final state ψ f is ψf exp( i HT )ψ 0. Breaking T into infinitesimal time steps leads to the path integral.
Aharonov-Bohm effect Turn on magnetic field B in region inaccessible to particle.
Aharonov-Bohm effect Turn on magnetic field B in region inaccessible to particle.
Aharonov-Bohm effect Path integral formulation. B = A. Contribution from path enclosing B acquires a phase e iθ where θ = A ds, as A cannot be zero everywhere on path enclosing B.
Aharonov-Bohm effect Path integral formulation. B = A. Contribution from paths enclosing B acquires a phase e iθ where θ = A ds, as A cannot be zero everywhere on path enclosing B.
Gauge transformations Wavefunction ψ(x) for x R n. In general we are free to change ψ by a phase factor. ψ(x) e iφ(x) ψ(x) p i + φ Let φ(x) = x 0 A ds where A = 0
Two particles in space X. Alternative approaches: Quantize X 2 and restrict Hilbert space to symmetric and anti-symmetric subspaces. Bose-Einstein/Fermi-Dirac statistics. ψ(x 1, x 2 ) = ±ψ(x 2, x 1 ) (1)
Two particles in space X. Alternative approaches: Quantize X 2 and restrict Hilbert space to symmetric and anti-symmetric subspaces. Bose-Einstein/Fermi-Dirac statistics. ψ(x 1, x 2 ) = ±ψ(x 2, x 1 ) (1) Treat particles as indistinguishable, ψ(x 1, x 2 ) ψ(x 2, x 1 ). Quantize configuration space, C 2 (X ) = ( X 2 ) /S 2. (2)
Bose-Einstein and Fermi-Dirac statistics In R 3 using the relative coordinate, at a constant separation the configuration space C 2 (R 3 ) is the projective plane. Exchanging particles corresponds to traveling around a closed loop p in C 2. On the projective plane p is not contractible but p 2 is contractible. To associate a phase e iθ to p requires (e iθ ) 2 = 1 or e iθ = ±1. Quantizing C 2 with a phase 1 associated to exchange paths corresponds to Fermi-Dirac statistics while a phase +1 corresponds to Bose-Einstein statistics.
Fermi-Dirac statistics are only consistent with multi-particle states constructed from different single-particle states. The electron has Fermi-Dirac statistics
Fermi-Dirac statistics are only consistent with multi-particle states constructed from different single-particle states. The electron has Fermi-Dirac statistics chemistry!
Anyon statistics Topology in QM Pair of indistinguishable particles in R 2. Any phase e iθ can be associated with a primitive exchange path.
Anyon statistics Topology in QM Pair of indistinguishable particles in R 2. Two particles not coincident. Any phase e iθ can be associated with a primitive exchange path.
Anyon statistics Topology in QM Pair of indistinguishable particles in R 2. Two particles not coincident. Relative position coordinate x R 2 \ 0. Any phase e iθ can be associated with a primitive exchange path.
Anyon statistics Topology in QM Pair of indistinguishable particles in R 2. Two particles not coincident. Relative position coordinate x R 2 \ 0. Topology of configuration space the same as puncturing space with a magnetic field in A-B experiment. Any phase e iθ can be associated with a primitive exchange path.
Anyon statistics Topology in QM Pair of indistinguishable particles in R 2. Two particles not coincident. Relative position coordinate x R 2 \ 0. Topology of configuration space the same as puncturing space with a magnetic field in A-B experiment. Exchange paths; closed loops about 0. Any phase e iθ can be associated with a primitive exchange path.
Anyon statistics Topology in QM Pair of indistinguishable particles in R 2. Two particles not coincident. Relative position coordinate x R 2 \ 0. Topology of configuration space the same as puncturing space with a magnetic field in A-B experiment. Exchange paths; closed loops about 0. A phase change around loops can be encoded in a vector potential. Any phase e iθ can be associated with a primitive exchange path.
Fractional quantum Hall effect Hall effect (1879) Quantum Hall effect (1975) Low temp & large B. Conductance quantized E n = eb (n + 1/2). m Fractional quantum Hall effect(tsui and Str omer 1982) Gallium arsenide. Integers can be replaced with rational numbers p/q. Explained using Laughlin wavefn, composite particles with anyon statistics.
Fractional quantum Hall effect Hall effect (1879) Quantum Hall effect (1975) Low temp & large B. Conductance quantized E n = eb (n + 1/2). m Fractional quantum Hall effect(tsui and Str omer 1982) Gallium arsenide. Integers can be replaced with rational numbers p/q. Explained using Laughlin wavefn, composite particles with anyon statistics.
Fractional quantum Hall effect Hall effect (1879) Quantum Hall effect (1975) Low temp & large B. Conductance quantized E n = eb (n + 1/2). m Fractional quantum Hall effect (Tsui and Strömer 1982) Gallium arsenide. Integers can be replaced with rational numbers p/q. Explained using Laughlin wavefn, composite particles with anyon statistics.
Configuration space of n particles X one particle configuration space. 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}.
Configuration space of n particles X one particle configuration space. 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}. π 1 (C n (R d )) = S n for d 3 abelianization H 1 (C n (R d )) = Z/2
Configuration space of n particles X one particle configuration space. 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}. π 1 (C n (R d )) = S n for d 3 abelianization H 1 (C n (R d )) = Z/2 π 1 (C n (R 2 )) = B n braid group of n strands, H 1 (C n (R 2 )) = Z.
Configuration space of n particles X one particle configuration space. 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}. π 1 (C n (R d )) = S n for d 3 abelianization H 1 (C n (R d )) = Z/2 π 1 (C n (R 2 )) = B n braid group of n strands, H 1 (C n (R 2 )) = Z. π 1 (C n (R)) = 1
What happens on a graph where the underlying space has arbitrarily complex topology?
Quantum graph Topology in QM x 1 x 2 Graph G Vertices {1,..., v} Edges E = {(i, j)} Adjacency matrix A, A jk = 1 if j k, A jk = 0 otherwise. v j valency of vertex j. For many particles on a metric graph boundary conditions are hard to incorporate as particles become coincident.
Single particle QM on combinatorial graph ψ = v j=1 ψ j j in Hilbert space C v. Hamiltonian H, v v Hermitian matrix. e.g. discrete Laplacian H = A D, where D = diag{v 1,..., v v }. Transitions possible between adjacent vertices.
Single particle QM on combinatorial graph ψ = v j=1 ψ j j in Hilbert space C v. Hamiltonian H, v v Hermitian matrix. e.g. discrete Laplacian H = A D, where D = diag{v 1,..., v v }. Transitions possible between adjacent vertices. Gauge potential Ω is a v v real antisymmetric matrix, Ω ij = 0 if (ij) / E. Incorporate gauge potential in Hamiltonian, H H Ω where H Ω ij = e iω ij H ij.
Gauge transformation Let U = diag{e iθ 1,..., e iθv }, ψ U ψ H UHU A gauge potential Ω is trivial if H Ω = UHU for some U, i.e. H Ω is generated by a gauge transformation. For a trivial gauge potential { θ k θ j + 2πM jk, j k, Ω jk = (3) 0, otherwise, where M is an antisymmetric integer matrix.
Let T be a spanning tree of G, a connected subgraph whose cycles are all self-retracing. Index every edge not in T with φ {1,..., f }. c φ ( ) denotes fundamental cycle using edge e φ based at. π 1 (G) generated by {c φ ( )}. H 1 (G) = π 1 (G)/(c i c j c j c i ) abelianized homotopy group.
Let T be a spanning tree of G, a connected subgraph whose cycles are all self-retracing. e 3 e 1 e 4 e 2 Index every edge not in T with φ {1,..., f }. c φ ( ) denotes fundamental cycle using edge e φ based at. π 1 (G) generated by {c φ ( )}. H 1 (G) = π 1 (G)/(c i c j c j c i ) abelianized homotopy group.
Let T be a spanning tree of G, a connected subgraph whose cycles are all self-retracing. e 3 e 1 e 4 e 2 Index every edge not in T with φ {1,..., f }. c φ ( ) denotes fundamental cycle using edge e φ based at. π 1 (G) generated by {c φ ( )}. H 1 (G) = π 1 (G)/(c i c j c j c i ) abelianized homotopy group.
Two-particle graph C 2 (G) = (V 2 )/S 2 can be regarded as graph G 2. An edge in G 2 corresponds to keeping one particle fixed while the other is moved along an edge of G. A 2 is adjacency matrix of G 2. Example: K 3 G = K 3 G 2
Two-particle graph C 2 (G) = (V 2 )/S 2 can be regarded as graph G 2. An edge in G 2 corresponds to keeping one particle fixed while the other is moved along an edge of G. A 2 is adjacency matrix of G 2. Example: K 3 G = K 3 G 2
Two-particle graph C 2 (G) = (V 2 )/S 2 can be regarded as graph G 2. An edge in G 2 corresponds to keeping one particle fixed while the other is moved along an edge of G. A 2 is adjacency matrix of G 2. Example: K 3 G = K 3 G 2
Two-particle graph C 2 (G) = (V 2 )/S 2 can be regarded as graph G 2. An edge in G 2 corresponds to keeping one particle fixed while the other is moved along an edge of G. A 2 is adjacency matrix of G 2. Example: K 3 G = K 3 G 2
Two-particle graph C 2 (G) = (V 2 )/S 2 can be regarded as graph G 2. An edge in G 2 corresponds to keeping one particle fixed while the other is moved along an edge of G. A 2 is adjacency matrix of G 2. Example: K 3 G = K 3 G 2
Contractible cycles Definition Contactable cycles are cycles on G 2 that are not self-retracing but are metrically contractible for two particles on Γ. Pairs of disjoint edges in G generate contractible cycles.
Contractible cycles Definition Contactable cycles are cycles on G 2 that are not self-retracing but are metrically contractible for two particles on Γ. Pairs of disjoint edges in G generate contractible cycles.
Contractible cycles Definition Contactable cycles are cycles on G 2 that are not self-retracing but are metrically contractible for two particles on Γ. Pairs of disjoint edges in G generate contractible cycles.
Contractible cycles Definition Contactable cycles are cycles on G 2 that are not self-retracing but are metrically contractible for two particles on Γ. Pairs of disjoint edges in G generate contractible cycles.
Contractible cycles Definition Contactable cycles are cycles on G 2 that are not self-retracing but are metrically contractible for two particles on Γ. Pairs of disjoint edges in G generate contractible cycles.
Contractible cycles Definition Contactable cycles are cycles on G 2 that are not self-retracing but are metrically contractible for two particles on Γ. Pairs of disjoint edges in G generate contractible cycles.
Contractible cycles Definition Contactable cycles are cycles on G 2 that are not self-retracing but are metrically contractible for two particles on Γ. Pairs of disjoint edges in G generate contractible cycles. Let K(G 2 ) C(G 2 ) be set of contractible cycles mod self-retracing. C(G 2 )/K(G 2 ) = π 1 (C 2 (Γ)) H 1 (C 2 (Γ)) = π 1 (C 2 (Γ))/(c i c j c j c i )
Example: Lasso graph G G 2 H 1 (G 2 ) Z 2
Example: Lasso graph G G 2 H 1 (G 2 ) Z 2
Topological gauge potentials Definition Ω 2 is a topological gauge potential for G 2 if Ω 2 (c) = 0 mod 2π for every contractible loop c on G 2.
Topological gauge potentials Definition Ω 2 is a topological gauge potential for G 2 if Ω 2 (c) = 0 mod 2π for every contractible loop c on G 2. Classification: Choose spanning tree T 2 of G 2.
Topological gauge potentials Definition Ω 2 is a topological gauge potential for G 2 if Ω 2 (c) = 0 mod 2π for every contractible loop c on G 2. Classification: Choose spanning tree T 2 of G 2. Fix gauge: Ω e = 0 for e T 2.
Topological gauge potentials Definition Ω 2 is a topological gauge potential for G 2 if Ω 2 (c) = 0 mod 2π for every contractible loop c on G 2. Classification: Choose spanning tree T 2 of G 2. Fix gauge: Ω e = 0 for e T 2. ω := (ω 1,..., ω f2 ) where ω φ2 is flux of Ω 2 through c φ2.
Topological gauge potentials Definition Ω 2 is a topological gauge potential for G 2 if Ω 2 (c) = 0 mod 2π for every contractible loop c on G 2. Classification: Choose spanning tree T 2 of G 2. Fix gauge: Ω e = 0 for e T 2. ω := (ω 1,..., ω f2 ) where ω φ2 is flux of Ω 2 through c φ2. Constraints from contractible loops Rω = 0 mod 2π, where R is integer matrix.
Aharonov-Bohm phases and two-body phases Aharonov-Bohm phase A free statistics phase that corresponds to taking one particle around a cycle in G with the other fixed. An Aharonov-Bohm phase can be produced by threading a cycle of G with a magnetic flux.
Aharonov-Bohm phases and two-body phases Aharonov-Bohm phase A free statistics phase that corresponds to taking one particle around a cycle in G with the other fixed. An Aharonov-Bohm phase can be produced by threading a cycle of G with a magnetic flux. Free phases not parameterized by Aharov-Bohm phases and discrete phases are two body phases which characterize Abelian graph statistics.
G G 2 1 A-B phase. 1 free 2-body phase. ( E 1)( E 2) 2 free 2-body phases. 3 A-B & 1 free 2-body phase.
Non-planar graphs Theorem (Kuratowski) Every nonplanar graph a graph which cannot be drawn in the plane without edges crossing contains K 5 or K 3,3 as a subgraph or contains a subgraph that is homeomorphic to K 5 or K 3,3.
Non-planar graphs Theorem (Kuratowski) Every nonplanar graph a graph which cannot be drawn in the plane without edges crossing contains K 5 or K 3,3 as a subgraph or contains a subgraph that is homeomorphic to K 5 or K 3,3. 6 A-B phases & 1 discrete phase of 0 or π. 4 A-B phases & 1 discrete phase of 0 or π. 12 A-B, 6 2-body & 2 phases of 0 or π.
Figure: Configuration space graph G 2 of K 3,3, edges shown as solid lines are in the spanning subtree with root (1, 2). Open edges are joined left to right and top to bottom.
Counting phases Ko & Park (2011) H 1 (C n (G)) = Z N 1(G)+N 2 (G)+N 3 (G)+β 1 (G) Z N 3 (G) 2 (4) N 1 (G) sum over one cuts j of N(n, G, j). ( ) ( ) n + µj 2 n + µj 2 N(n, G, j) = (µ(j) 2) (v j µ j 1) n 1 n µ j # components of G \ j. N 2 (G) sum over two connected components of G. N 3 (G) # 3-connected planar components of G. N 3 (G) # 3-connected non-planar components of G. β 1 (G) # of loops of G.
3-connected components Lemma A 3-connected graph has a single 2-body exchange phase. Sketch proof: We already know lemma holds for K 4 the simplest 3-connected graph.
3-connected components Lemma A 3-connected graph has a single 2-body exchange phase. Sketch proof: We already know lemma holds for K 4 the simplest 3-connected graph.
3-connected components Lemma A 3-connected graph has a single 2-body exchange phase. Sketch proof: We already know lemma holds for K 4 the simplest 3-connected graph.
3-connected components Lemma A 3-connected graph has a single 2-body exchange phase. Sketch proof: We already know lemma holds for K 4 the simplest 3-connected graph.
Conclusions Topology in QM Formulated multi-particle QM on combinatorial graphs. Vector field used to incorporate statistics in Hamiltonian. Demonstrated new forms of quantum statistics for graphs. Graph statistics allow multiple anyon phases and discrete phases.
Conclusions Topology in QM Formulated multi-particle QM on combinatorial graphs. Vector field used to incorporate statistics in Hamiltonian. Demonstrated new forms of quantum statistics for graphs. Graph statistics allow multiple anyon phases and discrete phases. Outlook Metric graphs Non-abelian statistics Many body properties: transport, analogue of fractional quantum Hall effect, Hartree-Fock. Physical mechanism for exotic graph statistics.
References Topology in QM JH, JM Robbins and A Sawicki, Many particle anyon statistics on graphs, (in preparation) JH and B Winn, Intermediate statistics for a system with symplectic symmetry: the Dirac rose graph, (to appear in) J. Phys. A: Math. Theor. arxiv:1205.6073 JH and K Kirsten, Vacuum energy of Schrödinger operators on metric graphs, in: M Asorey, M Bordag and E Elizalde (ed. s) Quantum field theory under the influence of external conditions (QFEXT11), Int. J. Mod. Phys. Conf. Ser. 14 (2012) 357 366. R Band, JH and CH Joyner, Finite pseudo orbit expansions for spectral quantities of quantum graphs, J. Phys. A: Math. Theor. 45 (2012) 325204. JH, K Kirsten and C Texier, Spectral determinants and zeta functions of Schrödinger operators on metric graphs, J. Phys. A: Math. Theor. 45 (2012) 125206. JH, JP Keating and JM Robbins, on graphs, Proc. R. Soc. A (2010) doi:10.1098/rspa.2010.0254
Thank You What you were missing in Bristol