PT-symmetric random matrix ensembles Eva... · 2018. 11. 15. · Random Matrix Theory (RMT) Bohigas, Giannoni, Schmit, PRL 52 (1984) 1, Haake, quantum signatures of chaos, Springer,

PT-symmetric random matrix ensembles Eva-Maria Graefe

Department of Mathematics, Imperial College London, UK

Dissipative Quantum Chaos: from Semi-Groups to QED Experiments PCS IBS, Daejeon, South Korea, October 2017

Random Matrix Theory (RMT)

Bohigas, Giannoni, Schmit, PRL 52 (1984) 1, Haake, quantum signatures of chaos, Springer, 2010

« 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

Random Matrix Theory (RMT)

Bohigas, Giannoni, Schmit, PRL 52 (1984) 1, Haake, quantum signatures of chaos, Springer, 2010

« 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?

Outline

« Split-Hermitian Gaussian ensembles Probability distribution on space of matrices, 2x2 analytical results

« Standard Random Matrix Theory: Gaussian ensembles, spectral features

« PT-symmetric as split-Hermitian systems: Split-complex numbers, split-quaternions, split-Hermitian matrices

i2 = �1

j2 = k2 = ijk = +1

« Quantum systems with loss and gain: PT-symmetric QM - introduction and example

Outline




i2 = �1

j2 = k2 = ijk = +1


PT symmetry

Non-Hermitian Hamiltonians with anti-unitary symmetry can have purely real eigenvalues!

«  PT: originally parity and time reversal, now general anti-unitary symmetry

Bender at al 90’s

PT symmetry

Non-Hermitian Hamiltonians with anti-unitary symmetry can have purely real eigenvalues!

«  PT: originally parity and time reversal, now general anti-unitary symmetry

«  Two-level model:

«  PT: interchange levels and loss and gain

�± = ±p

v2 � �2

H =

✓i� vv �i�

◆

«  Balanced loss and gain

PT symmetry in coupled optical wave guides


Coupled optical wave guides:

Loss

�± = ±p

v2 � �2

H =

✓i� vv �i�

◆

Gain




Loss

H =

✓0 vv �2i�

◆

�± = �i� ±p

v2 � �2



Loss


H =

✓0 vv �2i�

◆

�± = �i� ±p

v2 � �2



Loss

A. Guo et al., Phys. Rev. Lett. 103, 093902 (2009)


H =

✓0 vv �2i�

◆

�± = �i� ±p

v2 � �2

PT symmetry

1998

2016 250

50

200

150

100

Published items with topics “PT symmetry” and “PT-symmetric”

year

Isi web of knowledge

PT symmetry

PT symmetry in optics, among Nature’s Invisible acoustic sensor

based on PT symmetry Single-mode laser by PT symmetry breaking

1998

2016 250

50

200

150

100

Cham, Nature Phys. 11, 799 (2015) Feng et.al, Science (2014) Fleury et.al. Nature Comm. (2015)

Outline




i2 = �1

j2 = k2 = ijk = +1


Dyson’s threefold way

« Standard QM: Hermitian Hamiltonians

« Universality classes - time-reversal properties:

« GOE: Real symmetric, invariant under orthogonal transformations: time-reversal symmetry, T 2 = 1

« GUE: Complex Hermitian, invariant under unitary transformations: no time-reversal symmetry

T 2 = �1

« GSE: Quaternionic Hermitian, invariant under symplectic transformations: time-reversal symmetric,

Dyson’s Gaussian ensembles

« Gaussian orthogonal/unitary/symplectic ensembles:

H =A† +A

2

: independently distributed normal random variables over the real/complex numbers / quaternions

Amn



H =A† +A

2


Amn

« Probability distribution on space of matrices:

« Invariant under orthogonal/unitary/symplectic transformations

P(H) /

8><

>:

e�12Tr(H

2), GOE

e�Tr(H2), GUE

e�2Tr(H2), GSE



H =A† +A

2


Amn

« Probability distribution on space of matrices:

« Invariant under orthogonal/unitary/symplectic transformations

« Spectral properties analytically known for arbitrary matrix size

P(H) /

8><

>:

e�12Tr(H

2), GOE

e�Tr(H2), GUE

e�2Tr(H2), GSE

2x2 Gaussian ensembles « One-level distributions:

R1(�)=8

3p2⇡

(�4+3

2�2+

3

16)e�2�2

GOE GUE GSE

2x2 Gaussian ensembles « One-level distributions:

R1(�)=8

3p2⇡

(�4+3

2�2+

3

16)e�2�2

GOE GUE GSE

« Level spacing distributions:

P (s) =

8>>>><

>>>>:

⇡2 s e

�⇡4 s2 , GOE

32⇡2 s2 e�

4⇡ s2 , GUE

218

36⇡3 s4 e�649⇡ s2 , GSE

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

Outline




i2 = �1

j2 = k2 = ijk = +1


PT-symmetric random matrix theory?

« Several attempts, mostly 2x2 (Jain, Ahmed, Wang et al.), beta type ensembles (Pato et. al.)

« What about PT-symmetric random matrices?


« Universality and invariance classes?

« Natural parameterisation of PT-symmetric matrices?







Bender and Mannheim 2010: PT-symmetric matrices as complex matrices with real characteristic polynomial


C. M. Bender and P. Mannheim, Phys. Lett. A 374 (2010) 1616





Bender and Mannheim 2010: PT-symmetric matrices as complex matrices with real characteristic polynomial


C. M. Bender and P. Mannheim, Phys. Lett. A 374 (2010) 1616

« PT-symmetric matrices can be parameterised by real parameters 2N2 �N

N ⇥N

Split-complex numbers

« Hyperbolic version of complex numbers – imaginary unit squares to plus one

z = x+ jyx, y 2 R j2 = +1

z $✓x y

y x

◆« Representation as real 2x2 matrix:

« Conjugate: z = x� jy

« Indefinite “norm”: |z|2 = zz = det

✓x y

y x

◆= x

2 � y

2

(Split)-quaternions

Sir William Rowan Hamilton

1805 - 1865

z = z0 + iz1 + jz2 + kz3zj 2 R

Split-quaternions

Sir James Cockle 1819 - 1895

z = z0 + iz1 + jz2 + kz3zj 2 R

i2 = �1

j2 = k2 = ijk = +1

z = z0 � iz1 � jz2 � kz3« Conjugate:

Split-quaternions


z = z0 + iz1 + jz2 + kz3zj 2 R

i2 = �1

j2 = k2 = ijk = +1

« 2x2 matrix representation: z $✓z0 + iz1 z2 + iz3z2 � iz3 z0 � iz1

◆z = z0 � iz1 � jz2 � kz3« Conjugate:

Split-quaternions


z = z0 + iz1 + jz2 + kz3zj 2 R

i2 = �1

j2 = k2 = ijk = +1

« 2x2 matrix representation:

« Indefinite “norm”:

z $✓z0 + iz1 z2 + iz3z2 � iz3 z0 � iz1

◆z = z0 � iz1 � jz2 � kz3« Conjugate:

zz = z20 + z21 � z22 � z23

Split-Hermitian matrices

« “Inner product” on split-quaternionic vector space:

« Adjoint of split-quaternionic matrix: (~u,A~v) = (A†~u,~v)

= transpose and split-quaternionic conjugate

“Split-Hermitian” matrices: H† = HInvariant under unitary transformations!

(~u,~v) =NX

n=1

unvn

Split-Hermitian matrices

« Use 2x2 matrix representation to define eigenvalues & eigenvectors

Real characteristic polynomial

Eigenvalues doubly degenerate in problem 2N ⇥ 2N

« Split-complex Hermitian real PT-symmetric matrices

« Space of split-Hermitian matrices has real dimensions

2N2 �NN ⇥N

)

)

Split-Hermitian matrices can be viewed as a representation of PT-symmetric matrices!

$

Outline




i2 = �1

j2 = k2 = ijk = +1


Split-Hermitian Gaussian ensembles

« Construct split versions of Gaussian unitary and symplectic ensembles:

H =A† +A

2

: independently distributed normal random variables over the split-complex numbers / split-quaternions

Amn

EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751

Split-Hermitian Gaussian ensembles

« Construct split versions of Gaussian unitary and symplectic ensembles:

H =A† +A

2

: independently distributed normal random variables over the split-complex numbers / split-quaternions

Amn

« Probability distributions on space of split-Hermitian matrices:

P(H) =⇣ 1

⇡

⌘N2⇣ 2

⇡

⌘ 12N(N�1)

e�Tr(HHT )

P(H) =� 2⇡

�N2� 2p

⇡

�2N(N�1)e�Tr(HHI+HIH)

transpose & complex conjugation

Split-complex:

Split-quaternionic:


2x2 split-complex Hermitian ensemble

H =

✓⇤1 � � j�

� + j� ⇤2

◆⇤1,2, �, � 2 R

P(H) =2

⇡2e�Tr(HHT) =

2

⇡2e�(⇤2

1+⇤22+2�2+2�2)

«  split-complex Hermitian matrix: 2⇥ 2

« Probability distribution:


2x2 split-complex Hermitian ensemble

H =

✓⇤1 � � j�

� + j� ⇤2

◆⇤1,2, �, � 2 R

P(H) =2

⇡2e�Tr(HHT) =

2

⇡2e�(⇤2

1+⇤22+2�2+2�2)

«  split-complex Hermitian matrix: 2⇥ 2


« Related to Ginibre ensemble in real representation: 4⇥ 4


H $

0

BB@

⇤1 0 � �0 ⇤1 � �� ⇤2 0�� 0 ⇤2

1

CCA = OT

0

BB@

⇤2 � � � 0 0� + � ⇤1 0 00 0 ⇤2 � + �0 0 � � � ⇤1

1

CCAO

2x2 split-quaternionic Hermitian ensemble

«  split-quaternionic Hermitian matrix:


2⇥ 2

H =

✓⇤1 ✓ � iµ� j⌫ � k�

✓ + iµ+ j⌫ + k� ⇤2

◆✓, µ, ⌫, � 2 R

P(H) =32

⇡3e�2(⇤2

1+⇤22+2(✓2+µ2+⌫2+�2))

Joint probability of eigenvalues, one-level densities, level spacings for real eigenvalues etc?


2x2 split-Hermitian ensembles

« Analytic expressions for spectral properties

« One-level distribution:

R1(�) =2|=(�)|p

⇡e�2(<(�)2�=(�)2)erfc(2|=(�)|)

+ �(=(�)) �e��2

2erf(�) +

e�2�2

2p⇡

!

Split-complex Hermitian ensemble





« Probability that eigenvalues are real:

R1(�) =2|=(�)|p

⇡e�2(<(�)2�=(�)2)erfc(2|=(�)|)

+ �(=(�)) �e��2

2erf(�) +

e�2�2

2p⇡

!

P (� 2 R) = 1p2






« Probability that eigenvalues are real: P (� 2 R) = 1p2


RR1 (�) =

�e��2

2erf(�) +

e�2�2

2p⇡






RR1 (�) =

�e��2

2erf(�) +

e�2�2

2p⇡

GOE:






RI1(�) =

2|=(�)|p⇡

e�2(<(�)2�=(�)2)erfc(2|=(�)|)





Split-quaternionic Hermitian ensemble

R1(�) =2

r2

⇡|=(�)|e�4

�(<(�))2+(=(�))2

�

+ �(=(�))

e�4�2

8�2p2⇡

+e�2�2

p2⇡

h2�2 + 1� 1

8�2

i!







P (� 2 R) = 1� 1

2p2

R1(�) =2

r2

⇡|=(�)|e�4

�(<(�))2+(=(�))2

�

+ �(=(�))

e�4�2

8�2p2⇡

+e�2�2

p2⇡

h2�2 + 1� 1

8�2

i!







P (� 2 R) = 1� 1

2p2

RR1 (�) =

e�4�2

8�2p2⇡

+e�2�2

p2⇡

h2�2 + 1� 1

8�2

i






RR1 (�) =

e�4�2

8�2p2⇡

+e�2�2

p2⇡

h2�2 + 1� 1

8�2

i

GUE:






RI1(�) = 2

r2

⇡|=(�)|e�4

�(<(�))2+(=(�))2

�



Level spacing distributions for real eigenvalues

split-complex split-quaternionic

P (s) =⇡

2se�

⇡4 s2

split-quaternionic: P (s) =2pa

1� 12p2

as2e�as2

p⇡

+

pas eas

2

erfc(p2a s)

2p2

!


GOE

Outlook

«  Special features of split-Gaussian ensembles (compared to general complex random matrices)?

«  Mapping between PT-symmetric and split-Hermitian systems?

«  Use split-Hermiticity to make progress on real Ginibre ensemble

«  Compare to spectral features of PT-symmetric chaotic quantum systems

«  Universal behaviour?

Outlook

«  Special features of split-Gaussian ensembles (compared to general complex random matrices)?

«  Mapping between PT-symmetric and split-Hermitian systems?

«  Use split-Hermiticity to make progress on real Ginibre ensemble

«  Compare to spectral features of PT-symmetric chaotic quantum systems

«  Universal behaviour?

Postdoc and PhD positions available in 2018!

Summary

« Split-Hermitian Gaussian ensembles New universality classes for PT-symmetric systems?

« Quantum systems with loss and gain: PT-symmetric QM useful!

« Hermitian Gaussian random matrices: universal features of quantum chaotic systems

« Split-quaternionic Hermitian matrices: Parameterisation of PT-symmetric matrices

H† = H


Summary

« Split-Hermitian Gaussian ensembles New universality classes for PT-symmetric systems?

« Quantum systems with loss and gain: PT-symmetric QM useful!

« Hermitian Gaussian random matrices: universal features of quantum chaotic systems

« Split-quaternionic Hermitian matrices: Parameterisation of PT-symmetric matrices

H† = HThank you for your attention!


Documents

PT-symmetric random matrix ensembles Eva... · 2018. 11. 15. · Random Matrix Theory (RMT) Bohigas, Giannoni, Schmit, PRL 52 (1984) 1, Haake, quantum signatures of chaos, Springer,