Spectral properties of quantum graphs via pseudo orbits

Spectral properties of quantum graphs via pseudo orbits 1, Ram Band 2, Tori Hudgins 1, Christopher Joyner 3, Mark Sepanski 1 1 Baylor, 2 Technion, 3 Queen Mary University of London Cardiff 12/6/17

Outline Quantum graphs 1 Quantum graphs 2 3

Quantum graphs Metric graph: associate each edge e to interval [0, l e ]. Total length L = E e=1 l e. Laplace equation on [0, l e ], d2 dxe 2 f e (x e ) = k 2 f e (x e ). (1) Hilbert space H := E e=1 L2( [0, l e ] ). Vertex conditions chosen s.t. op. self-adjoint.

Applications Free electron model in organic molecules (1930 s) Anderson localization Nanotechnology Photonic crystals Quantum chaos Wave guides

Scattering approach f e (x e ) = a in e e ikxe + a out e e ikxe (2) At vertex v, d v incoming and outgoing plane waves related by vertex scattering matrix, a v out = σ (v) a v in. Example: Neumann conditions f is continuous at v and e v f e (v) = 0. [σ (v) ] ij = 2 d v δ ij

Scattering approach Alternatively, to quantize graph assign a unitary vertex scattering matrix σ (v) to each vertex v. Example: A democratic choice is the discrete Fourier transform matrix, 1 1 1... 1 σ (v) e,e = 1 1 ω ω 2 dv... ω 1 1 ω 2 ω 4 2(dv... ω 1) dv....... 1 ω dv 1 ω 2(dv 1) (dv 1)(dv... ω 1) ω = e 2πi dv a primitive d v root of unity.

Spectrum Quantum graphs Combine vertex scattering matrices into an E E matrix Σ, { σ (v) Σ e,e = e,e v = t(e ) = o(e), (3) 0 otherwise Let L = diag{l 1,..., l E }, the quantum evolution op., U (k) = e ikl Σ. (4) Spectrum { k 2 det (I U (k)) = 0 }

Characteristic polynomial Characteristic polynomial of U(k) F ξ (k) = det (ξi U (k)) = 2B n=0 a n ξ 2B n Note: spectrum corresponds to roots of F 1 (k) = 0.

Riemann-Siegel lookalike formula Unitarity of U connects pairs of coefficients of characteristic polynomial. ( ) F ξ (k) = det ( ξu (k)) det ξ 1 I U (k) (5) = det ( ξu (k)) = det (U (k)) E anξ n E (6) n=0 E anξ n (7) n=0 Compare with F ξ (k) = E n=0 a nξ 2B n and use a E = det (U (k)). a n = a E a E n. (8)

Periodic orbits on a quantum graph To periodic orbit γ = (e 1,..., e m ) on a quantum graph associate, topological length E γ = m. metric length l γ = e j γ l e j. stability amplitude A γ = Σ e2 e 1 Σ e3 e 2... Σ enen 1 Σ e1 e m.

Pseudo orbits on a quantum graph To pseudo orbit γ = {γ 1,..., γ M } associate, m γ = M no. of periodic orbits in γ. topological length E γ = M j=1 E γ j. metric length l γ = M j=1 l γ j. stability amplitude A γ = M j=1 A γ j.

Theorem 1 (Band,H.,Joyner) Coefficients of the characteristic polynomial F ξ (k) are given by, a n = γ E γ=n ( 1) m γ A γ exp (ikl γ ), where the finite sum is over all primitive pseudo orbits topological length n. Idea Expand det (ξi U (k)) as a sum over permutations. A permutation ρ S E can contribute iff ρ(e) is connected to e for all e in ρ. Representing ρ as a product of disjoint cycles each cycle is a primitive periodic orbit.

Variance of coefficients of the characteristic polynomial a n k = a n 2 k = γ E γ=n = ( 1) m γ A γ γ, γ E γ=e γ =n γ, γ E γ=e γ =n Diagonal contribution lim K { 1 K 1 n = 0 e ikl γ dk = K 0 0 otherwise ( 1) m γ+m γ A γ Ā γ lim K 1 K e ik(l γ l γ ) dk K 0 ( 1) m γ+m γ A γ Ā γ δ l γ,l γ (9) a n 2 diag = γ E γ=n A γ 2

Background Variance of coeffs of the characteristic polynomial of graphs Kottos and Smilansky (1999). Spectral statistics of binary graphs Tanner (2000)&(2001). Variance of coeffs of the characteristic polynomial of binary graphs studied via permanent of transition matrix Tanner (2002) Random matrix variance a n 2 COE = 1 + a n 2 CUE = 1 n(e n) E + 1

q-nary graphs V = q m vertices, labeled by words length m on A. E = q m+1 directed edges e, each labeled by word w length m + 1. Origin vertex o(e), first m letters of w. Terminal vertex t(e), last m letters of w. Each vertex has q incoming and q outgoing edges. Spectral gap: adjacency matrix has simple eval. 1 and eval. 0 with multiplicity V 1, i.e. maximal spectral gap and maximally mixing.

Example: binary graph with 2 3 vertices 001 0011 011 0001 0010 1010 1011 0111 000 010 0100 0101 101 0000 1001 0110 1111 1000 111 1101 1110 100 1100 110

Example: binary graph with 2 3 vertices 001 0011 011 0001 0010 1010 1011 0111 000 010 0100 0101 101 0000 1001 0110 1111 1000 111 1101 1110 100 1100 110

Example: binary graph with 2 3 vertices 001 0011 011 0001 0010 1010 1011 0111 000 010 0100 0101 101 0000 1001 0110 1111 1000 111 1101 1110 100 1100 110

Example: binary graph with 2 3 vertices 001 0011 011 0001 0010 1010 1011 0111 000 010 0100 0101 101 0000 1001 0110 1111 1000 111 1101 1110 100 1100 110

Example: ternary graph with 3 2 vertices 20 200 01 201 120 012 12 220 001 011 112 122 000 11 222 202 010 101 121 212 020 00 10 21 22 110 211 100 221 111 210 002 102 021 022 02

Lyndon words Lexicographic order w = a 1 a 2... a l (10) w = b 1 b 2... b k (11) with a i, b j A a totally ordered alphabet of q letters. w > lex w iff there exists i st a 1 = b 1,..., a i 1 = b i 1 and a i > b i or l > k and a 1 = b 1,..., a k = b k. A word is a Lyndon word if it is strictly smaller in lexicographic order than all its cyclic shifts. Example: binary Lyndon words length 3, 0 < lex 001 < lex 01 < lex 011 < lex 1.

The standard decomposition Theorem 2 (Chen, Fox, Lyndon) Every word w can be uniquely written as a concatenation of Lyndon words in non-increasing lexicographic order, the standard decomposition of w. Example: standard decompositions of binary words length 3, (0)(0)(0) (01)(0) (1)(0)(0) (1)(1)(0) (001) (011) (1)(01) (1)(1)(1) A standard decomposition w = v 1 v 2... v k with v j a Lyndon word and v j lex v j+1 is strictly decreasing if v j > lex v j+1.

The standard decomposition Theorem 2 (Chen, Fox, Lyndon) Every word w can be uniquely written as a concatenation of Lyndon words in non-increasing lexicographic order, the standard decomposition of w. Example: standard decompositions of binary words length 3, (0)(0)(0) (01)(0) (1)(0)(0) (1)(1)(0) (001) (011) (1)(01) (1)(1)(1) A standard decomposition w = v 1 v 2... v k with v j a Lyndon word and v j lex v j+1 is strictly decreasing if v j > lex v j+1.

Periodic orbits A path length l is labeled by a word w = a 1,..., a l+m. A closed path length l is labeled by w = a 1,..., a l. A periodic orbit γ is the equivalence class of closed paths under cyclic shifts. A primitive periodic orbit is a periodic orbit that is not a repartition of a shorter orbit. Primitive periodic orbits length l are in 1-to-1 correspondence with Lyndon words length l. Example: 0011 is a primitive periodic orbit length 4. 001 011 0011 1001 0110 1100 100 110

Pseudo orbits A pseudo orbit γ = {γ 1,..., γ M } is a set of periodic orbits. A primitive pseudo orbit γ is a set of primitive periodic orbits where no periodic orbit appears more than once. Note: there is a bijection between primitive pseudo orbits and strictly decreasing standard decompositions. Example: 011010 has strictly decreasing standard decomposition (011)(01)(0). 001 011 000 010 101 111 100 110

Diagonal contribution Transition probability σ (v) e,e 2 = 1 q. No. of primitive pseudo orbits length n equal to no. of strictly decreasing standard decompositions of words length n, denoted Str q (n). Diagonal contribution a n 2 diag = γ E γ=n = γ E γ=n = Str q(n) q n A γ 2 1 q n

Binary words length 4 and 5. 0000 0100 1000 1100 0001 0101 1001 1101 0010 0110 1010 1110 0011 0111 1011 1111 00000 01000 10000 11000 00001 01001 10001 11001 00010 01010 10010 11010 00011 01011 10011 11011 00100 01100 10100 11100 00101 01101 10101 11101 00110 01110 10110 11110 00111 01111 10111 11111

Binary words length 4 and 5. (0)(0)(0)(0) (01)(0)(0) (1)(0)(0)(0) (1)(1)(0)(0) (0001) (01)(01) (1)(001) (1)(1)(01) (001)(0) (011)(0) (1)(01)(0) (1)(1)(1)(0) (0011) (0111) (1)(011) (1)(1)(1)(1) (0)(0)(0)(0)(0) (01)(0)(0)(0) (1)(0)(0)(0)(0) (1)(1)(0)(0)(0) (00001) (01)(001) (1)(0001) (1)(1)(001) (0001)(0) (01)(01)(0) (1)(001)(0) (1)(1)(01)(0) (00011) (01011) (1)(0011) (1)(1)(011) (001)(0)(0) (011)(0)(0) (1)(01)(0)(0) (1)(1)(1)(0)(0) (00101) (011)(01) (1)(01)(01) (1)(1)(1)(01) (0011)(0) (0111)(0) (1)(011)(0) (1)(1)(1)(1)(0) (00111) (01111) (1)(0111) (1)(1)(1)(1)(1)

Theorem 3 (Band, H., Sepanski) For words of length n 2 the no. of strictly decreasing standard decompositions is, Str q (n) = (q 1)q n 1. Note: Hence the proportion of words length n with strictly decreasing standard decompositions is q 1 q. i.e. half of binary words have strictly decreasing standard decompositions.

Comparison with RMT Diagonal contribution a n 2 diag = Str q(n) (q 1) q n = q RMT without time-reversal symmetry a n 2 CUE = 1 Diagonal contribution deviates from RMT as seen in numerical results for binary graphs (Tanner). For a sequence of graphs with increasing connectivity q the diagonal contribution would approach the random matrix result.

Counting Lyndon words Let L q (n) be the no. of Lyndon words length n. Lemma 4 A classic result about Lyndon words is, ll q (l) = q m l m Proof. Every word length l is a Lyndon word, a cyclic shift of a Lyndon word, or a repartition of such a word.

Proof of Theorem 3 Let Str q (0) = 1, Str q (1) = q and define two functions, p(x) = Str q (n) x n (12) n=0 f (x) = qx 2 1 qx 1 = 1 + qx + (q 1)q n 1 x n (13) We show p = f. As p(0) = f (0) = 1 note p = f if n=2 d dx log p = d dx log f = q 1 qx 2qx 1 qx 2 = 1 (qx) m 2 (qx 2 ) m x x m=1 m=1

Proof of Theorem 3 Let Str q (0) = 1, Str q (1) = q and define two functions, p(x) = Str q (n) x n = (1 + x l ) Lq(l) (14) n=0 l=1 f (x) = qx 2 1 qx 1 = 1 + qx + (q 1)q n 1 x n (15) We show p = f. As p(0) = f (0) = 1 note p = f if n=2 d dx log p = d dx log f = q 1 qx 2qx 1 qx 2 = 1 (qx) m 2 (qx 2 ) m x x m=1 m=1

log p = ( 1) j+1 L q (l) j l=1 d dx log p = 1 x = 1 x = 1 x = 1 x j=1 l=1 j=1 x lj ( 1) j ll q (l)x lj ( 1) m l ll q (l)x m m=1 l m ll q (l)x m 1 x m=1 l m ll q (n)x m 2 x m=1 l m m=1 l 2m ( 2m ) 1 + ( 1) l llq (n)x 2m ll q (n)x 2m m=1 l m Then applying Lemma 4 proves the result.

Summary Quantum graphs Graph spectrum encoded in finite number of short primitive pseudo orbits. Investigating characteristic polynomial led to new questions for Lyndon words. Obtained proportion of standard decompositions that are strictly decreasing. Suggests RMT behavior of coefficients may be obtained under stronger conditions. R. Band, J. M. Harrison and C. H. Joyner, Finite pseudo orbit expansions for spectral quantities of quantum graphs, J. Phys. A: Math. Theor. 45 (2012) 325204 arxiv:1205.4214 R. Band, J. M. Harrison and M. Sepanski, Lyndon word decompositions and pseudo orbits on q-nary graphs, arxiv:1610:03808