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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
« 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
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
« Two-level model:
Coupled optical wave guides:
Loss
�± = ±p
v2 � �2
H =
✓i� vv �i�
◆
Gain
PT symmetry in coupled optical wave guides
« Two-level model:
Coupled optical wave guides:
Loss
H =
✓0 vv �2i�
◆
�± = �i� ±p
v2 � �2
« Two-level model:
Coupled optical wave guides:
Loss
PT symmetry in coupled optical wave guides
H =
✓0 vv �2i�
◆
�± = �i� ±p
v2 � �2
« Two-level model:
Coupled optical wave guides:
Loss
A. Guo et al., Phys. Rev. Lett. 103, 093902 (2009)
PT symmetry in coupled optical wave guides
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
« 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
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
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
« 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
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
« 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
« 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
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?
PT-symmetric random matrix theory?
« Universality and invariance classes?
« Natural parameterisation of PT-symmetric matrices?
« Several attempts, mostly 2x2 (Jain, Ahmed, Wang et al.), beta type ensembles (Pato et. al.)
« What about PT-symmetric random matrices?
PT-symmetric random matrix theory?
« Universality and invariance classes?
« Natural parameterisation of PT-symmetric matrices?
« Several attempts, mostly 2x2 (Jain, Ahmed, Wang et al.), beta type ensembles (Pato et. al.)
Bender and Mannheim 2010: PT-symmetric matrices as complex matrices with real characteristic polynomial
« What about PT-symmetric random matrices?
C. M. Bender and P. Mannheim, Phys. Lett. A 374 (2010) 1616
PT-symmetric random matrix theory?
« Universality and invariance classes?
« Natural parameterisation of PT-symmetric matrices?
« Several attempts, mostly 2x2 (Jain, Ahmed, Wang et al.), beta type ensembles (Pato et. al.)
Bender and Mannheim 2010: PT-symmetric matrices as complex matrices with real characteristic polynomial
« What about PT-symmetric random matrices?
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
Sir James Cockle 1819 - 1895
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
Sir James Cockle 1819 - 1895
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
« 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
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:
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
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:
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
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:
« Related to Ginibre ensemble in real representation: 4⇥ 4
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
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:
« Probability distribution:
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?
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
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
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
« 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
Split-complex Hermitian ensemble
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
« Probability that eigenvalues are real: P (� 2 R) = 1p2
Split-complex Hermitian ensemble
RR1 (�) =
�e��2
2erf(�) +
e�2�2
2p⇡
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-complex Hermitian ensemble
RR1 (�) =
�e��2
2erf(�) +
e�2�2
2p⇡
GOE:
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-complex Hermitian ensemble
RI1(�) =
2|=(�)|p⇡
e�2(<(�)2�=(�)2)erfc(2|=(�)|)
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
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!
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
« Probability that eigenvalues are real:
Split-quaternionic Hermitian ensemble
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!
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
« Probability that eigenvalues are real:
Split-quaternionic Hermitian ensemble
P (� 2 R) = 1� 1
2p2
RR1 (�) =
e�4�2
8�2p2⇡
+e�2�2
p2⇡
h2�2 + 1� 1
8�2
i
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-quaternionic Hermitian ensemble
RR1 (�) =
e�4�2
8�2p2⇡
+e�2�2
p2⇡
h2�2 + 1� 1
8�2
i
GUE:
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-quaternionic Hermitian ensemble
RI1(�) = 2
r2
⇡|=(�)|e�4
�(<(�))2+(=(�))2
�
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
2x2 split-Hermitian ensembles
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
!
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
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
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
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!
EMG, S Mudute-Ndumbe, M Taylor, J. Phys. A 48 (2015) 1751
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