Exploring new random matrix ensembles
for PT-symmetric quantum systems
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
Quantum and Classical Physics with non-Hermitian Operators (PHHQP13)
Israel Institute for Advanced Studies, July 2015
Random Matrix Theory (RMT)
« Matrices whose elements are random numbers
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, 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.
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
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?
Bohigas, Giannoni, Schmit, PRL 52 (1984) 1, Haake, quantum signatures of chaos, Springer, 2010
Outline
« Standard Random Matrix Theory:
Gaussian ensembles, spectral features,
Ginibre ensemble
« PT-symmetric as split-Hermitian systems:
Split-complex numbers, split-quaternions, splitHermitian matrices
i2 =
1
j2 = k2 = ijk = +1
« Split-Hermitian Gaussian ensembles
Probability distribution on space of matrices,
2x2: Analytical results, relation to Ginibre
ensemble
Outline
« Standard Random Matrix Theory:
Gaussian ensembles, spectral features,
Ginibre ensemble
« PT-symmetric as split-Hermitian systems:
Split-complex numbers, split-quaternions, splitHermitian matrices
i2 =
1
j2 = k2 = ijk = +1
« Split-Hermitian Gaussian ensembles
Probability distribution on space of matrices,
2x2: Analytical results, relation to Ginibre
ensemble
Dyson’s threefold way
« Conventional closed quantum systems described by
Hermitian Hamiltonians
Dyson’s threefold way
« Conventional closed quantum systems described by
Hermitian Hamiltonians
« Three important universality classes depending on time-
reversal properties:
«  Real symmetric, invariant under orthogonal transformations: time-
reversal 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
Dyson’s threefold way
« Conventional closed quantum systems described by
Hermitian Hamiltonians
« Three important universality classes depending on time-
reversal properties:
«  Real symmetric, invariant under orthogonal transformations: time-
reversal 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
Dyson’s threefold way
« Gaussian orthogonal/unitary/symplectic ensembles:
A† + A Amn: independently distributed normal random
variables over the real/complex numbers /
H=
2
quaternions
Dyson’s threefold way
« Gaussian orthogonal/unitary/symplectic ensembles:
A† + A Amn: independently distributed normal random
variables over the real/complex numbers /
H=
2
quaternions
« Probability distribution on space of matrices:
8
>
< e
P(H) /
e
>
: e
2
1
2 Tr(H )
, GOE
Tr(H 2 )
, GUE
2Tr(H 2 )
, GSE
« Invariant under orthogonal/unitary/symplectic
transformations
Dyson’s threefold way
« Gaussian orthogonal/unitary/symplectic ensembles:
A† + A Amn: independently distributed normal random
variables over the real/complex numbers /
H=
2
quaternions
« Probability distribution on space of matrices:
8
>
< e
P(H) /
e
>
: e
2
1
2 Tr(H )
, GOE
Tr(H 2 )
, GUE
2Tr(H 2 )
, GSE
« Invariant under orthogonal/unitary/symplectic
transformations
« Spectral properties analytically known for arbitrary
matrix size
2x2 Gaussian ensembles
« One-level distributions:
GOE
GUE
GSE
8
3
3
R1 ( ) = p ( 4+ 2+ )e
2
16
3 2⇡
2
2
2x2 Gaussian ensembles
« One-level distributions:
GUE
GOE
GSE
8
3
3
R1 ( ) = p ( 4+ 2+ )e
2
16
3 2⇡
« Level spacing distributions:
P (s) =
8
>
>
>
>
<
>
>
>
>
:
⇡ 2
4s
⇡
2se
32 2
⇡2 s
e
218 4
36 ⇡ 3 s
e
,
4 2
⇡s
GOE
,
64 2
9⇡ s
GUE
,
GSE
2
2
The Ginibre ensembles
« Gaussian random matrices without Hermiticity constraint
« Real Ginibre ensemble: Matrices with independently
distributed real normal random elements
« Invariant under orthogonal
transformations
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
PT-symmetric random matrix theory?
« What about PT-symmetric random
matrices?
PT-symmetric random matrix theory?
« What about PT-symmetric random
matrices?
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.)
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?
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
C. M. Bender and P. Mannheim, Phys. Lett. A 374 (2010) 1616
Outline
« Standard Random Matrix Theory:
Gaussian ensembles, spectral features,
Ginibre ensemble
« PT-symmetric as split-Hermitian systems:
Split-complex numbers, split-quaternions, splitHermitian matrices
i2 =
1
j2 = k2 = 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 42 072001 (2011)
Split-complex numbers
« Hyperbolic version of complex numbers – imaginary unit
squares to plus one
z = x + jy
« Conjugate:
x, y 2 R
z̄ = x
jy
j2 = +1
Split-complex numbers
« Hyperbolic version of complex numbers – imaginary unit
squares to plus one
z = x + jy
« Conjugate:
x, y 2 R
z̄ = x
j2 = +1
jy
« Representation as real 2x2 matrix:
z$
✓
x
y
y
x
◆
Split-complex numbers
« Hyperbolic version of complex numbers – imaginary unit
squares to plus one
z = x + jy
« Conjugate:
x, y 2 R
z̄ = x
j2 = +1
jy
✓
◆
x y
« Representation as real 2x2 matrix: z $
y x
✓
◆
x y
2
« Indefinite “norm”: |z| = z z̄ = det
= x2
y x
y2
(Split)-quaternions
z = z0 + iz1 + jz2 + kz3
zj 2 R
Sir William Rowan
Hamilton
1805 - 1865
Split-quaternions
z = z0 + iz1 + jz2 + kz3
z
2
R
j
2
i =
1
j2 = k2 = ijk = +1
Sir James Cockle
1819 - 1895
« Conjugate:
z̄ = z0
iz1
jz2
kz3
Split-quaternions
z = z0 + iz1 + jz2 + kz3
z
2
R
j
2
i =
1
j2 = k2 = ijk = +1
Sir James Cockle
1819 - 1895
« Conjugate:
« 2x2 matrix representation:
z̄ = z0
iz1
z$
z0 + iz1
z2 iz3
✓
jz2
kz3
z2 + iz3
z0 iz1
◆
Split-quaternions
z = z0 + iz1 + jz2 + kz3
z
2
R
j
2
i =
1
j2 = k2 = ijk = +1
Sir James Cockle
1819 - 1895
« Conjugate:
« 2x2 matrix representation:
« Indefinite “norm”:
z̄ = z0
iz1
z$
z0 + iz1
z2 iz3
✓
z z̄ = z02 + z12
z22
jz2
z32
kz3
z2 + iz3
z0 iz1
◆
Split-Hermitian matrices
« “Inner product” on split-quaternionic vector space:
(~u, ~v ) =
N
X
ūn vn
n=1
« Adjoint of split-quaternionic matrix:
(~u, A~v ) = (A† ~u, ~v )
= transpose and split-quaternionic conjugate
Split-Hermitian matrices
« “Inner product” on split-quaternionic vector space:
(~u, ~v ) =
N
X
ūn vn
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!
Split-Hermitian matrices
« Space of split-Hermitian N ⇥ N matrices has 2N 2
real dimensions
N
Split-Hermitian matrices
« Space of split-Hermitian N ⇥ N matrices has 2N 2
real dimensions
N
« Use 2x2 matrix representation to define eigenvalues &
eigenvectors
) Real characteristic polynomial
) Eigenvalues doubly degenerate in 2N ⇥ 2N problem
Split-Hermitian matrices
« Space of split-Hermitian N ⇥ N matrices has 2N 2
real dimensions
N
« Use 2x2 matrix representation to define eigenvalues &
eigenvectors
) Real characteristic polynomial
) Eigenvalues doubly degenerate in 2N ⇥ 2N problem
Split-Hermitian matrices can be viewed as a
representation of PT-symmetric matrices!
Split-Hermitian matrices
« Space of split-Hermitian N ⇥ N matrices has 2N 2
real dimensions
N
« Use 2x2 matrix representation to define eigenvalues &
eigenvectors
) Real characteristic polynomial
) Eigenvalues doubly degenerate in 2N ⇥ 2N problem
Split-Hermitian matrices can be viewed as a
representation of PT-symmetric matrices!
« Split-complex Hermitian $ real PT-symmetric matrices
«  Zero inner product between eigenvectors belonging to distinct
eigenvalues
Outline
« Standard Random Matrix Theory:
Motivation, Gaussian ensembles, Ginibre
ensemble
« PT-symmetric as split-Hermitian systems:
Split-complex numbers, split-quaternions, splitHermitian matrices
i2 =
1
j2 = k2 = ijk = +1
« Split-Hermitian Gaussian ensembles
Probability distribution on space of matrices,
2x2: Analytical results, relation to Ginibre
ensemble
Split-Hermitian Gaussian ensembles
« Construct split versions of Gaussian unitary and
symplectic ensembles:
A† + A
H=
2
Amn: independently distributed normal random
variables over the split-complex numbers / splitquaternions
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
Split-Hermitian Gaussian ensembles
« Construct split versions of Gaussian unitary and
symplectic ensembles:
A† + A
H=
2
Amn: 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) =
⇣ 1 ⌘ N2 ⇣ 2 ⌘ 12 N (N
⇡
2
P(H) =
⇡
N
2
⇡
2
p
⇡
2N (N
1)
e
1)
e
Tr(HH T )
Tr(HH I +H I H)
transpose & complex conjugation
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
Split-Hermitian Gaussian ensembles
« Construct split versions of Gaussian unitary and
symplectic ensembles:
A† + A
H=
2
Amn: 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) =
⇣ 1 ⌘ N2 ⇣ 2 ⌘ 12 N (N
⇡
2
P(H) =
⇡
N
2
⇡
2
p
⇡
2N (N
1)
e
1)
e
Tr(HH T )
Tr(HH I +H I H)
transpose & complex conjugation
« Invariant under orthogonal/unitary transformations!
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-complex Hermitian ensemble
« 2 ⇥ 2 split-complex Hermitian matrix:
H=
✓
⇤1
+j
j
⇤2
◆
⇤1,2 , ,
2R
« Probability distribution:
2
P(H) = 2 e
⇡
Tr(HHT )
2
= 2e
⇡
(⇤21 +⇤22 +2
2
+2
2
)
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-complex Hermitian ensemble
« 2 ⇥ 2 split-complex Hermitian matrix:
H=
✓
⇤1
+j
j
⇤2
◆
⇤1,2 , ,
2R
« Probability distribution:
2
P(H) = 2 e
⇡
Tr(HHT )
2
= 2e
⇡
(⇤21 +⇤22 +2
2
+2
2
)
« Related to Ginibre ensemble in real 4 ⇥ 4 representation:
0
⇤1
B 0
H$B
@
0
⇤1
⇤2
0
1
0
⇤2
C
B +
T
C=O B
@ 0
0A
⇤2
0
⇤1
0
0
0
0
⇤1
1
0
0 C
CO
+ A
⇤1
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-quaternionic Hermitian ensemble
« 2 ⇥ 2 split-quaternionic Hermitian matrix:
H=
✓
⇤1
✓ + iµ + j⌫ + k
✓
iµ
j⌫
⇤2
k
◆
✓, µ, ⌫,
2R
« Probability distribution:
32
P(H) = 3 e
⇡
2(⇤21 +⇤22 +2(✓ 2 +µ2 +⌫ 2 +
2
))
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-quaternionic Hermitian ensemble
« 2 ⇥ 2 split-quaternionic Hermitian matrix:
H=
✓
⇤1
✓ + iµ + j⌫ + k
✓
iµ
j⌫
⇤2
k
◆
✓, µ, ⌫,
2R
« Probability distribution:
32
P(H) = 3 e
⇡
2(⇤21 +⇤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, arxiv:1505.07810
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-complex Hermitian ensemble
2|=( )|
e
R1 ( ) = p
⇡
2(<( )2 =( )2 )
+ (=( ))
erfc(2|=( )|)
!
2
2
e
e 2
erf( ) + p
2
2 ⇡
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-complex Hermitian ensemble
2|=( )|
e
R1 ( ) = p
⇡
2(<( )2 =( )2 )
+ (=( ))
erfc(2|=( )|)
!
2
2
e
e 2
erf( ) + p
2
2 ⇡
1
« Probability that eigenvalues are real: P ( 2 R) = p
2
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-complex Hermitian ensemble
R1R (
)=
2
e
2
2
2
e
erf( ) + p
2 ⇡
1
« Probability that eigenvalues are real: P ( 2 R) = p
2
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-complex Hermitian ensemble
R1R ( ) =
2
e
2
2
e 2
erf( ) + p
2 ⇡
GOE:
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-complex Hermitian ensemble
R1I (
2|=( )|
e
)= p
⇡
2(<( )2 =( )2 )
erfc(2|=( )|)
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-quaternionic Hermitian ensemble
r
R1 ( ) =2
2
|=( )|e
⇡
+ (=( ))
4 (<( ))2 +(=( ))2
e 2 h
p + p
2
2
8
2⇡
2⇡
e
4
2
2
2
+1
8
1 i
2
!
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-quaternionic Hermitian ensemble
r
R1 ( ) =2
2
|=( )|e
⇡
+ (=( ))
4 (<( ))2 +(=( ))2
e 2 h
p + p
2
2
8
2⇡
2⇡
e
4
2
« Probability that eigenvalues are real: P (
2
2
+1
2 R) = 1
8
1 i
2
!
1
p
2 2
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-quaternionic Hermitian ensemble
R1R ( ) =
4
e
8
2
p
2
2⇡
e
+ p
2
2
2⇡
« Probability that eigenvalues are real: P (
h
2
2
+1
2 R) = 1
8
1 i
2
1
p
2 2
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-quaternionic Hermitian ensemble
R1R ( ) =
4
e
8
2
p
2
2⇡
e
+ p
2
2
2⇡
h
2
2
+1
8
1 i
2
GUE:
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-Hermitian ensembles
« Analytic expressions for spectral properties
« One-level distribution:
Split-quaternionic Hermitian ensemble
R1I (
)=2
r
2
|=( )|e
⇡
4 (<( ))2 +(=( ))2
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
2x2 split-Hermitian ensembles
Level spacing distributions for real eigenvalues
split-complex
⇡
P (s) = se
2
split-quaternionic:
P (s) =
split-quaternionic
⇡ 2
4s
p
2 a
1
1
p
2 2
as2 e
p
as
⇡
2
+
p
2
p
as eas erfc( 2a s)
p
2 2
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
!
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?
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810
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?
EMG, S Mudute-Ndumbe, M Taylor, arxiv:1505.07810