A new type of PT-symmetric random matrix ensembles

Transcription

1 A new type of PT-symmetric random matrix ensembles Eva-Maria Graefe Department of Mathematics, Imperial College London, UK joint work with Steve Mudute-Ndumbe and Matthew Taylor Department of Mathematics, Imperial College London, UK PHHQP 15 University of Palermo, May 2015

2 Bohigas, Giannoni, Schmit, PRL 52 (1984) 1, Haake, quantum signatures of chaos, Springer, 2010 Random Matrix Theory (RMT) «Matrices whose elements are random numbers «Dyson & Wigner: Spectral properties of sufficiently complicated systems described by random matrices Bohigas-Giannoni-Schmit conjecture: Spectral fluctuations of quantum system with chaotic classical counterpart similar to those of certain Hermitian random matrices. «Dyson s threefold way: Gaussian symmetric, unitary and symplectic ensembles PT-symmetric RMT?

3 Outline «Standard Random Matrix Theory: Gaussian ensembles, spectral features, Ginibre ensemble «PT-symmetric as split-hermitian systems: Split-complex numbers, split-quaternions, split- Hermitian matrices i 2 = 1 j 2 =k 2 =ijk=+1 «Split-Hermitian Gaussian ensembles Probability distribution on space of matrices, 2x2: Analytical results, relation to Ginibre ensemble

4 Outline «Standard Random Matrix Theory: Gaussian ensembles, spectral features, Ginibre ensemble «PT-symmetric as split-hermitian systems: Split-complex numbers, split-quaternions, split- Hermitian matrices i 2 = 1 j 2 =k 2 =ijk=+1 «Split-Hermitian Gaussian ensembles Probability distribution on space of matrices, 2x2: Analytical results, relation to Ginibre ensemble

5 Dyson s threefold way «Conventional closed quantum systems described by Hermitian Hamiltonians «Three important universality classes depending on timereversal properties: «Real symmetric, invariant under orthogonal transformations: timereversal symmetric with T 2 =1 «Complex Hermitian, invariant under unitary transformations: no time-reversal symmetry «Quaternionic Hermitian, invariant under symplectic transformations: time-reversal symmetric with T 2 = 1 «Gaussian probability distributions on space of these matrices describe universal spectral features

6 Dyson s threefold way «Gaussian orthogonal/unitary/symplectic ensembles: H = A + A 2 A mn : independently distributed normal random variables over the real/complex numbers / quaternions «Probability distribution on space of matrices: 8 >< e 1 2 Tr(H H), GOE P(H) / e >: Tr(H H), GUE e 2Tr(H H), GSE «Invariant under orthogonal/unitary/symplectic transformations «Spectral properties analytically known for arbitrary matrix size

7 2x2 Gaussian ensembles «One-level distributions: GOE GUE GSE «Level spacing distributions: 8 2 s e 4 s2, GOE >< P (s) = 32 s 2 e 4 2 s2, GUE >: s 4 e s2, GSE R 1 ( )= 8 3 p 2 ( )e 2 2

8 The Ginibre ensembles «Gaussian random matrices without Hermiticity constraint «Real Ginibre ensemble: Matrices with independently distributed real normal random elements «Invariant under orthogonal transformations «Real or complex conjugate eigenvalues «Analytically challenging, but many properties known

9 C. M. Bender and P. Mannheim, Phys. Lett. A 374 (2010) 1616 PT-symmetric random matrix theory? «What about PT-symmetric random matrices? «Several attempts, mostly 2x2 (Jain, Ahmed, Wang et al.), beta type ensembles (Pato et. al.) «Universality and invariance classes? «Natural parameterisation of PT-symmetric matrices? Bender and Mannheim 2010: PT-symmetric matrices as complex matrices with real characteristic polynomial «PT-symmetric N N matrices can be parameterised by 2N 2 N real parameters

10 Outline «Standard Random Matrix Theory: Gaussian ensembles, spectral features, Ginibre ensemble «PT-symmetric as split-hermitian systems: Split-complex numbers, split-quaternions, split- Hermitian matrices i 2 = 1 j 2 =k 2 =ijk=+1 «Split-Hermitian Gaussian enesmbles Probability distribution on space of matrices, 2x2: Analytical results, relation to Ginibre ensemble D. C. Brody and EMG, JPA (2011)

11 Split-complex numbers «Hyperbolic version of complex numbers imaginary unit squares to plus one z = x +jy x, y 2 R j 2 =+1 «Conjugate: z = x jy «Representation as real 2x2 matrix: z $ «Indefinite norm : z 2 x = z z =det y x y y x y x = x 2 y 2

12 (Split)-quaternions z = z 0 +iz 1 +jz 2 +kz 3 z j 2 R Sir William Rowan Hamilton

13 Split-quaternions z = z 0 +iz 1 +jz 2 +kz 3 z j 2 R i 2 = 1 j 2 =k 2 =ijk=+1 Sir James Cockle «Conjugate: «2x2 matrix representation: z = z 0 iz 1 jz 2 kz 3 z $ z0 +iz 1 z 2 +iz 3 z 2 iz 3 z 0 iz 1 «Indefinite norm : z z = z z 2 1 z 2 2 z 2 3

14 Split-Hermitian matrices «Inner product on split-quaternionic vector space: (~u, ~v) = NX ū n v n n=1 «Adjoint of split-quaternionic matrix: (~u, A~v) =(A ~u, ~v) = transpose and split-quaternionic conjugate Split-Hermitian matrices: H = H Invariant under unitary transformations!

15 Split-Hermitian matrices «Space of split-hermitian real dimensions N N matrices has 2N 2 N «Use 2x2 matrix representation to define eigenvalues & eigenvectors ) ) Real characteristic polynomial Eigenvalues doubly degenerate in «Split-complex Hermitian 2N 2N Split-Hermitian matrices can be viewed as a representation of PT-symmetric matrices! $ problem real PT-symmetric matrices «Zero inner product between eigenvectors belonging to distinct eigenvalues

16 Outline «Standard Random Matrix Theory: Motivation, Gaussian ensembles, Ginibre ensemble «PT-symmetric as split-hermitian systems: Split-complex numbers, split-quaternions, split- Hermitian matrices i 2 = 1 j 2 =k 2 =ijk=+1 «Split-Hermitian Gaussian ensembles Probability distribution on space of matrices, 2x2: Analytical results, relation to Ginibre ensemble

17 EMG, S Mudute-Ndumbe, M Taylor, arxiv: Split-Hermitian Gaussian ensembles «Construct split versions of Gaussian unitary and symplectic ensembles: H = A + A 2 A mn : independently distributed normal random variables over the split-complex numbers / splitquaternions «Probability distributions on space of split-hermitian matrices: Split-complex: Split-quaternionic: P(H) = P(H) = 1 N N(N 1) e Tr(HHT ) N 2 2 p 2N(N 1) e Tr(HH I +H I H) transpose & complex conjugation «Invariant under orthogonal/unitary transformations!

18 EMG, S Mudute-Ndumbe, M Taylor, arxiv: «2x2 split-complex Hermitian ensemble 2 2 split-complex Hermitian matrix: 1 j H = 1,2,, 2 R +j 2 «Probability distribution: P(H) = 2 2 e Tr(HHT) = 2 2 e ( ) «Related to Ginibre ensemble in real H $ B A = OT representation: C A O

19 EMG, S Mudute-Ndumbe, M Taylor, arxiv: x2 split-quaternionic Hermitian ensemble «2 2 H = split-quaternionic Hermitian matrix: 1 iµ j k +iµ +j +k 2, µ,, 2 R «Probability distribution: P(H) = 32 3 e 2( ( 2 +µ )) Joint probability of eigenvalues, one-level densities, level spacings for real eigenvalues etc?

20 EMG, S Mudute-Ndumbe, M Taylor, arxiv: x2 split-hermitian ensembles «Analytic expressions for spectral properties «One-level distribution: Split-complex Hermitian ensemble R 1 ( )= 2 =( ) p e 2(<( )2 =( ) 2) erfc(2 =( ) )! 2 e + (=( )) erf( )+ e p «Probability that eigenvalues are real: P ( 2 R) = 1 p 2

21 EMG, S Mudute-Ndumbe, M Taylor, arxiv: x2 split-hermitian ensembles «Analytic expressions for spectral properties «One-level distribution: Split-complex Hermitian ensemble R R 1 ( )= e 2 2 erf( )+ e p «Probability that eigenvalues are real: P ( 2 R) = 1 p 2

22 EMG, S Mudute-Ndumbe, M Taylor, arxiv: x2 split-hermitian ensembles «Analytic expressions for spectral properties «One-level distribution: Split-complex Hermitian ensemble R R 1 ( )= e 2 2 erf( )+ e p GOE:

23 EMG, S Mudute-Ndumbe, M Taylor, arxiv: x2 split-hermitian ensembles «Analytic expressions for spectral properties «One-level distribution: Split-complex Hermitian ensemble R I 1( )= 2 =( ) p e 2(<( )2 =( ) 2) erfc(2 =( ) )

24 2x2 split-hermitian ensembles «Analytic expressions for spectral properties «One-level distribution: Split-quaternionic Hermitian ensemble r 2 R 1 ( )=2 4 (<( )) =( ) e 2 +(=( )) 2 + (=( )) e p 2 + e 2 p h ! 1 i «Probability that eigenvalues are real: P ( 2 R) =1 2 p 2 EMG, S Mudute-Ndumbe, M Taylor, arxiv:

25 EMG, S Mudute-Ndumbe, M Taylor, arxiv: x2 split-hermitian ensembles «Analytic expressions for spectral properties «One-level distribution: Split-quaternionic Hermitian ensemble R1 R ( )= e p 2 + e 2 2 p h i 8 2 «Probability that eigenvalues are real: P ( 2 R) =1 1 2 p 2

26 EMG, S Mudute-Ndumbe, M Taylor, arxiv: x2 split-hermitian ensembles «Analytic expressions for spectral properties «One-level distribution: Split-quaternionic Hermitian ensemble R1 R ( )= e p 2 + e 2 2 p h i 8 2 GUE:

27 EMG, S Mudute-Ndumbe, M Taylor, arxiv: x2 split-hermitian ensembles «Analytic expressions for spectral properties «One-level distribution: Split-quaternionic Hermitian ensemble r 2 R1( I )=2 4 (<( )) =( ) e 2 +(=( )) 2

28 EMG, S Mudute-Ndumbe, M Taylor, arxiv: x2 split-hermitian ensembles Level spacing distributions for real eigenvalues split-complex split-quaternionic P (s) = 2 se 4 s2 split-quaternionic: P (s) = 2p a p 2 as 2 e as2 p + p as e as 2 erfc( p! 2as) 2 p 2

29 EMG, S Mudute-Ndumbe, M Taylor, arxiv: Summary «Hermitian Gaussian random matrices describe universal features of quantum systems with chaotic classical counterpart H = H «Split-quaternionic Hermitian matrices as parameterisation of PT-symmetric matrices «Split-Hermitian Gaussian ensembles as new universality classes for PTsymmetric systems?

30 EMG, S Mudute-Ndumbe, M Taylor, arxiv: Summary «Hermitian Gaussian random matrices describe universal features of quantum systems with chaotic classical counterpart H = H Thank you for your attention! «Split-quaternionic Hermitian matrices as parameterisation of PT-symmetric matrices «Split-Hermitian Gaussian ensembles as new universality classes for PTsymmetric systems?