Quantum Transport and RMT
Combinatorics
All together now
Universality in chaotic quantum transport: the
concordance between random matrix and
semiclassical theories
Gregory Berkolaiko, Texas A&M University
Based on work with J.Kuipers
IMS, Singapore, June 18, 2012
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
A handwaving introduction to Quantum Mechanics
A
B
A
B
Probability: P(A → B) = Ppath1 + Ppath2 ,
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
A handwaving introduction to Quantum Mechanics
A
B
A
B
Probability: P(A → B) = Ppath1 + Ppath2 ,
QM: Ψ(A → B) = Ψpath1 + Ψpath2 , where |Ψpath |2 = Ppath .
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
A handwaving introduction to Quantum Mechanics
A
B
A
B
Probability: P(A → B) = Ppath1 + Ppath2 ,
QM: Ψ(A → B) = Ψpath1 + Ψpath2 , where |Ψpath |2 = Ppath .
But Ψ are not real positive, so interference is possible (due to
phases).
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport: an experimental setup
_
+
Prepare sample: cut through conducting layer to make a
cavity with small openings
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport: an experimental setup
_
+
Prepare sample: cut through conducting layer to make a
cavity with small openings
Apply voltage and measure current fluctuations
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport: theory (one channel per lead)
replaced by
unitary
S=
r1 t 0
t r2
i1
o1
S
i1
o1
:
7→
i2
o2
t is the quantum amplitude for scattering from i1 to o2 .
Gregory Berkolaiko
Universality in chaotic quantum transport
o2
i2
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport: theory (one channel per lead)
replaced by
unitary
S=
r1 t 0
t r2
i1
o1
S
i1
o1
:
7→
i2
o2
t is the quantum amplitude for scattering from i1 to o2 .
|t|2 =: T is the transmission probability.
Gregory Berkolaiko
Universality in chaotic quantum transport
o2
i2
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport: theory (one channel per lead)
replaced by
unitary
S=
r1 t 0
t r2
i1
o1
S
i1
o1
:
7→
i2
o2
t is the quantum amplitude for scattering from i1 to o2 .
|t|2 =: T is the transmission probability.
Landauer formula for time-averaged current (voltage V )
Ī = 2e
Gregory Berkolaiko
eV
T.
h
Universality in chaotic quantum transport
o2
i2
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport: theory (one channel per lead)
replaced by
unitary
S=
r1 t 0
t r2
i1
o1
S
i1
o1
:
7→
i2
o2
t is the quantum amplitude for scattering from i1 to o2 .
|t|2 =: T is the transmission probability.
Landauer formula for time-averaged current (voltage V )
Ī = 2e
eV
T.
h
Shot noise intensity is (variance of the binomial process)
2
eV
δI = 2e 2
T (1 − T ).
h
Gregory Berkolaiko
Universality in chaotic quantum transport
o2
i2
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport: theory (multiple channels)
Typically there are many channels (107 for a wire of radius 1 µm).
1
...
N1
S
N1+1
...
S=
N1+N2
r1 t 0
t r2
:
ileft
iright
7→
N1 channels in the left lead, N2 channels in the right.
Gregory Berkolaiko
Universality in chaotic quantum transport
oleft
oright
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport: theory (multiple channels)
Typically there are many channels (107 for a wire of radius 1 µm).
1
...
N1
S
N1+1
...
S=
N1+N2
r1 t 0
t r2
:
ileft
iright
7→
N1 channels in the left lead, N2 channels in the right.
S is N × N, where N = N1 + N2 .
Gregory Berkolaiko
Universality in chaotic quantum transport
oleft
oright
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport: theory (multiple channels)
Typically there are many channels (107 for a wire of radius 1 µm).
1
...
N1
S
N1+1
...
S=
N1+N2
r1 t 0
t r2
:
ileft
iright
7→
N1 channels in the left lead, N2 channels in the right.
S is N × N, where N = N1 + N2 .
t is N2 × N1 : left-to-right transmission.
Gregory Berkolaiko
Universality in chaotic quantum transport
oleft
oright
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport: theory (multiple channels)
Typically there are many channels (107 for a wire of radius 1 µm).
1
...
N1
S
N1+1
...
S=
N1+N2
r1 t 0
t r2
:
ileft
iright
7→
oleft
oright
N1 channels in the left lead, N2 channels in the right.
S is N × N, where N = N1 + N2 .
t is N2 × N1 : left-to-right transmission.
Tj are eigenvalues of t ∗ t and Landauer-Büttiker formula is
Ī = 2e
eV X
Tj ∼ tr(t ∗ t).
h
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport: theory (multiple channels)
Typically there are many channels (107 for a wire of radius 1 µm).
1
...
N1
S
N1+1
...
S=
N1+N2
r1 t 0
t r2
:
ileft
iright
7→
oleft
oright
N1 channels in the left lead, N2 channels in the right.
S is N × N, where N = N1 + N2 .
t is N2 × N1 : left-to-right transmission.
Tj are eigenvalues of t ∗ t and Landauer-Büttiker formula is
Ī = 2e
eV X
Tj ∼ tr(t ∗ t).
h
Shot noise intensity ∼ tr t ∗ t − (t ∗ t)2 .
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport (cont)
S=
r1 t 0
t r2
If time-reversal (TR) invariant, then
Ψ(A → B) = Ψ(B → A) =⇒ S = S T .
Verified by experiment (e.g. Chang, Baranger, Pfeiffer, West
’94): properties of S depend on classical properties of the
cavity shape:
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Quantum Transport (cont)
S=
r1 t 0
t r2
If time-reversal (TR) invariant, then
Ψ(A → B) = Ψ(B → A) =⇒ S = S T .
Verified by experiment (e.g. Chang, Baranger, Pfeiffer, West
’94): properties of S depend on classical properties of the
cavity shape:
But how to access S ?
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Random Matrix model
Blümel and Smilansky’88: take S to be a random matrix with
suitable symmetry.
Broken TR:
U unitary,
measure invariant under U 7→ WU for any unitary W ,
Circular Unitary Ensemble (CUE),
Haar measure on U(N).
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Random Matrix model
Blümel and Smilansky’88: take S to be a random matrix with
suitable symmetry.
Broken TR:
U unitary,
measure invariant under U 7→ WU for any unitary W ,
Circular Unitary Ensemble (CUE),
Haar measure on U(N).
TR:
V unitary symmetric,
measure invariant under V 7→ W T VW for any unitary W ,
V = U T U, where U ∈ CUE(N),
Circular Orthogonal Ensemble (COE),
NOT Haar measure on O(N).
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
U(N) integration
Example 1: Calculate conductance
*N
+
N
1
X
X
∗
htr(t t)i =
Uoi Uoi
i=1 o=N1 +1
Gregory Berkolaiko
= N1 N2 Uoi Uoi U(N)
U(N)
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
U(N) integration
Example 1: Calculate conductance
*N
+
N
1
X
X
∗
htr(t t)i =
Uoi Uoi
i=1 o=N1 +1
= N1 N2 Uoi Uoi U(N)
U(N)
Theorem (Samuel’80)
hUa1 b1 · ·Uas bs Uα1 β1 · ·Uαt βt iN =
δts
X
VN (σ
σ,π∈St
−1
π)
t
Y
α
k=1
where VN (ρ) is a ratio of polynomials in N.
Gregory Berkolaiko
β
δakσ(k) δbkπ(k) .
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
U(N) integration
Example 1: Calculate conductance
*N
+
N
1
X
X
∗
htr(t t)i =
Uoi Uoi
i=1 o=N1 +1
= N1 N2 Uoi Uoi U(N)
U(N)
Theorem (Samuel’80)
hUa1 b1 · ·Uas bs Uα1 β1 · ·Uαt βt iN =
δts
X
VN (σ
−1
π)
t
Y
α
k=1
σ,π∈St
where VN (ρ) is a ratio of polynomials in N.
Example 1 (cont): We need only VN (1) =
htr(t ∗ t)i = N1NN2 .
Gregory Berkolaiko
1
N,
β
δakσ(k) δbkπ(k) .
so
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
U(N) integration
Example 1: Calculate conductance
*N
+
N
1
X
X
∗
htr(t t)i =
Uoi Uoi
i=1 o=N1 +1
= N1 N2 Uoi Uoi U(N)
U(N)
Theorem (Samuel’80)
hUa1 b1 · ·Uas bs Uα1 β1 · ·Uαt βt iN =
δts
X
VN (σ
σ,π∈St
−1
π)
t
Y
α
β
δakσ(k) δbkπ(k) .
k=1
where VN (ρ) is a ratio of polynomials in N.
Example 1 (cont): We need only VN (1) = N1 , so
htr(t ∗ t)i = N1NN2 .
Example 2: For U12 U13 U13 U12 use π = (12) and σ = id or
(12).
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
U(N) integration (cont)
Theorem (Samuel’80)
hUa1 b1 · ·Uas bs Uα1 β1 · ·Uαt βt iN =
δts
X
σ,π∈St
VN (σ
−1
π)
t
Y
α
β
δakσ(k) δbkπ(k) ,
k=1
where VN are class functions that satisfy VN (∅) = 1 and
X
NV (c1 , . . . , ck ) +
V (p, q, c2 , . . . , ck )
p+q=c1
+
k
X
cj V (c1 + cj , . . . , cˆj , . . . , ck ) = δc1 ,1 V (c2 , . . . , ck ),
j=2
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
U(N) integration (cont)
Theorem (Samuel’80)
NV (c1 , . . . , ck ) +
X
V (p, q, c2 , . . . , ck )
p+q=c1
+
k
X
cj V (c1 + cj , . . . , cˆj , . . . , ck ) = δc1 ,1 V (c2 , . . . , ck ),
j=2
V (ρ) depends only on the conjugacy class of the permutation
ρ which is determined by the lengths of cycles of ρ. For
example ρ = (1 4)(3 2 5) is of class (2, 3).
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
U(N) integration (cont)
Theorem (Samuel’80)
NV (c1 , . . . , ck ) +
X
V (p, q, c2 , . . . , ck )
p+q=c1
+
k
X
cj V (c1 + cj , . . . , cˆj , . . . , ck ) = δc1 ,1 V (c2 , . . . , ck ),
j=2
V (ρ) depends only on the conjugacy class of the permutation
ρ which is determined by the lengths of cycles of ρ. For
example ρ = (1 4)(3 2 5) is of class (2, 3).
From the recursion we see V (1) = 1/N, as promised.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
U(N) integration (cont)
Theorem (Samuel’80)
NV (c1 , . . . , ck ) +
X
V (p, q, c2 , . . . , ck )
p+q=c1
+
k
X
cj V (c1 + cj , . . . , cˆj , . . . , ck ) = δc1 ,1 V (c2 , . . . , ck ),
j=2
V (ρ) depends only on the conjugacy class of the permutation
ρ which is determined by the lengths of cycles of ρ. For
example ρ = (1 4)(3 2 5) is of class (2, 3).
From the recursion we see V (1) = 1/N, as promised.
V (1, 1) = 1/(N 2 − 1) and V (2) = −1/(N 2 − 1)N.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
U(N) integration (cont)
Theorem (Samuel’80)
NV (c1 , . . . , ck ) +
X
V (p, q, c2 , . . . , ck )
p+q=c1
+
k
X
cj V (c1 + cj , . . . , cˆj , . . . , ck ) = δc1 ,1 V (c2 , . . . , ck ),
j=2
V (ρ) depends only on the conjugacy class of the permutation
ρ which is determined by the lengths of cycles of ρ. For
example ρ = (1 4)(3 2 5) is of class (2, 3).
From the recursion we see V (1) = 1/N, as promised.
V (1, 1) = 1/(N 2 − 1) and V (2) = −1/(N 2 − 1)N.
Sometimes VN is called “Weingarten function”.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Further RMT references
Review: Beenakker’97
First few moments: Baranger, Mello 94; Jalabert, Pichard,
Beenakker ’94
Diagrammatics: Brouwer and Beenakker ’94
From Selberg integrals: Savin, Sommers, Wieczorek ’06-’08;
P. Vivo, E. Vivo ’08; Novaes ’08; Livan, P.Vivo ’11; Mezzadri,
Sims ’11.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Semiclassical approximation
RM model is very successful but why? Limits of applicability?
Semiclassical approximation
X
i
Soi (E ) ∼
Aγ (E )e ~ Sγ (E ) ,
γ(i→o)
so, for example,
hSoi Soi i ∼
*
X
+
Aγ Aγ 0 e
i
(Sγ (E )−Sγ 0 (E ))
~
γ,γ 0
Sγ ≈ Sγ 0
=⇒
,
E
γ 0 must mostly follow γ.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Semiclassical approximation
RM model is very successful but why? Limits of applicability?
Semiclassical approximation
X
i
Soi (E ) ∼
Aγ (E )e ~ Sγ (E ) ,
γ(i→o)
so, for example,
hSoi Soi i ∼
*
X
+
Aγ Aγ 0 e
i
(Sγ (E )−Sγ 0 (E ))
~
γ,γ 0
Sγ ≈ Sγ 0
=⇒
,
E
γ 0 must mostly follow γ.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Semiclassical approximation
RM model is very successful but why? Limits of applicability?
Semiclassical approximation
X
i
Soi (E ) ∼
Aγ (E )e ~ Sγ (E ) ,
γ(i→o)
so, for example,
hSoi Soi i ∼
*
X
+
Aγ Aγ 0 e
i
(Sγ (E )−Sγ 0 (E ))
~
γ,γ 0
Sγ ≈ Sγ 0
=⇒
,
E
γ 0 must mostly follow γ.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Higher order correlations
A typical term in htr(t ∗ t)2 i is
hSo1 i1 So2 i2 So1 i2 So2 i1 i
that is
γ1 : i1 → o1
γ2 : i2 → o2
i1
o1
γ10 : i2 → o1
γ20 : i1 → o2 .
i2
o2
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Higher order correlations
A typical term in htr(t ∗ t)2 i is
hSo1 i1 So2 i2 So1 i2 So2 i1 i
that is
γ1 : i1 → o1
γ10 : i2 → o1
Examples:
i1
o1
i2
o2
γ2 : i2 → o2
i1
o1
γ20 : i1 → o2 .
i2
o2
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Higher order correlations
A typical term in htr(t ∗ t)2 i is
hSo1 i1 So2 i2 So1 i2 So2 i1 i
that is
γ1 : i1 → o1
γ10 : i2 → o1
Examples:
γ2 : i2 → o2
i1
o1
γ20 : i1 → o2 .
i2
o2
i1
o1 i 1
o1
i2
o2 i 2
o2
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Higher order correlations
A typical term in htr(t ∗ t)2 i is
hSo1 i1 So2 i2 So1 i2 So2 i1 i
that is
γ1 : i1 → o1
γ10 : i2 → o1
Examples:
γ2 : i2 → o2
i1
o1
γ20 : i1 → o2 .
i2
o2
i1
o1 i 1
o1
i1
o1
i2
o2 i 2
o2 i 2
o2
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Transport moments in the universal regime — history
h
i
Task: Evaluate tr (t ∗ t)n to all orders in 1/N.
(Universal regime only: τE τd )
terms
?
d
3
f
2 b
1 a
c
1
2
e
3
n
(a) Diagonal approximmation: Blümel and Smilansky ’88
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Transport moments in the universal regime — history
h
i
Task: Evaluate tr (t ∗ t)n to all orders in 1/N.
(Universal regime only: τE τd )
terms
?
d
3
f
2 b
1 a
c
1
2
e
3
n
(b) Breakthrough of Richter and Sieber ’02: first off-diagonal term
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Transport moments in the universal regime — history
h
i
Task: Evaluate tr (t ∗ t)n to all orders in 1/N.
(Universal regime only: τE τd )
terms
?
d
3
f
2 b
1 a
c
1
2
e
3
n
(c) First term for shot noise: Schanz, Puhlmann and Geisel ’03
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Transport moments in the universal regime — history
h
i
Task: Evaluate tr (t ∗ t)n to all orders in 1/N.
(Universal regime only: τE τd )
terms
?
d
3
f
2 b
tr(t ∗ t)
1 a
c
1
2
e
3
n
tr(t ∗ t)2
(d)
and
to all orders of 1/N (and diagram evaluation
rules): Müller, Heusler, Braun and Haake ’07
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Transport moments in the universal regime — history
h
i
Task: Evaluate tr (t ∗ t)n to all orders in 1/N.
(Universal regime only: τE τd )
terms
?
d
3
f
2 b
(e) first term of
1 a
c
1
2
tr(t ∗ t)n
e
3
n
for all n: GB, Harrison and Novaes ’09
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Transport moments in the universal regime — history
h
i
Task: Evaluate tr (t ∗ t)n to all orders in 1/N.
(Universal regime only: τE τd )
terms
?
d
3
f
2 b
1 a
c
1
2
(f) first several terms of
e
3
tr(t ∗ t)n
Gregory Berkolaiko
n
for all n: GB and Kuipers ’11
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Quantum Transport
Random Matrices
Semiclassical approximation
Evaluation rules
Müller, Heusler, Braun and Haake ’07:
Each vertex: ×(−N)
Each edge: × N1
Assumes chaotic system with fast equilibration and long stretches
between encounters.
Uses Hannay - Ozorio de Almeida sume rule.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Leading order diagrams: combinatorial meaning
From leading order contributions to tr(t ∗ t)4 :
i1
(a)
o1
o1
i2
i4
o2
i3
i1
(b)
o4
i2
o2
i3
o4
o3
i4
o3
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Leading order diagrams: combinatorial meaning
From leading order contributions to tr(t ∗ t)4 :
i1
(a)
o1
o1
i2
i4
o2
i3
i1
(b)
o4
i2
o2
i3
o4
o3
i4
o3
All diagrams are plane trees.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Leading order diagrams: combinatorial meaning
From leading order contributions to tr(t ∗ t)4 :
i1
(a)
o1
o1
i2
i4
o2
i3
i1
(b)
o4
i2
o2
i3
o4
o3
i4
o3
All diagrams are plane trees.
Correspond to factorizations of the permutation (1 2 3 4), e.g.
(1 2)(2 4)(2 3) and (3 4)(1 2 4).
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Leading order diagrams: combinatorial meaning
From leading order contributions to tr(t ∗ t)4 :
i1
(a)
o1
o1
i2
i4
o2
i3
i1
(b)
o4
i2
o2
i3
o4
o3
i4
o3
All diagrams are plane trees.
Correspond to factorizations of the permutation (1 2 3 4), e.g.
(1 2)(2 4)(2 3) and (3 4)(1 2 4).
Factorizations count up to the permutation of commuting
factors, e.g. (1 3)(1 2)(3 4) = (1 3)(3 4)(1 2)
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Mathematical description of diagrams
Maps or ribbon graphs: graphs embedded on a surface.
Examples:
i1
o1 i 1
o1
i1
o1
i2
o2 i 2
o2 i 2
o2
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Mathematical description of diagrams
Maps or ribbon graphs: graphs embedded on a surface.
Examples:
i1
o1 i 1
o1
i1
o1
i2
o2 i 2
o2 i 2
o2
become
o1
i1
i2
o2
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Mathematical description of diagrams
Maps or ribbon graphs: graphs embedded on a surface.
Examples:
i1
o1 i 1
o1
i1
o1
i2
o2 i 2
o2 i 2
o2
become
o2
o1
o1
i1
i2
i1
i2
o2
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Mathematical description of diagrams
Maps or ribbon graphs: graphs embedded on a surface.
Examples:
i1
o1 i 1
o1
i1
o1
i2
o2 i 2
o2 i 2
o2
become
o2
i1
i1
o1
o1
i2
i1
o2
i2
i2
o2
Gregory Berkolaiko
Universality in chaotic quantum transport
o1
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Conditions on diagrams
To be a valid diagram for tr(t ∗ t)n ,
o2
i1
o1
i1
o2
i2
i2
or
2n vertices of deg 1: i1 , . . . , in , o1 , . . . , on
Gregory Berkolaiko
Universality in chaotic quantum transport
o1
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Conditions on diagrams
To be a valid diagram for tr(t ∗ t)n ,
o2
i1
o1
i1
o2
i2
i2
or
2n vertices of deg 1: i1 , . . . , in , o1 , . . . , on
all other vertices degree 2m ≥ 4
Gregory Berkolaiko
Universality in chaotic quantum transport
o1
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Conditions on diagrams
To be a valid diagram for tr(t ∗ t)n ,
o2
i1
o1
i1
o2
i2
o1
i2
or
2n vertices of deg 1: i1 , . . . , in , o1 , . . . , on
all other vertices degree 2m ≥ 4
unique boundary walk which goes i1 , o1 , i2 , o2 , . . . unicellular
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Conditions on diagrams
To be a valid diagram for tr(t ∗ t)n ,
o2
i1
o1
i1
o2
i2
o1
i2
or
2n vertices of deg 1: i1 , . . . , in , o1 , . . . , on
all other vertices degree 2m ≥ 4
unique boundary walk which goes i1 , o1 , i2 , o2 , . . . unicellular
Each edge is traversed twice, by red and by black =⇒ each
cycle has an even number of odd vertices.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Factorizations and diagrams
(B. and Irving ’08-15) The inequivalent factorizations (of
π = (1 2 . . . n)) are enumerated by maps that have
2n vertices of deg 1: i1 , . . . , in , o1 , . . . , on
all other vertices degree 2m ≥ 4
unique boundary walk which goes i1 , o1 , i2 , o2 , . . . ⇒
unicellular
Each cycle has a nonzero even number of odd vertices.
This is a subset of semiclassical diagrams!
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Contribution of a diagram
o2
i1
o1
i1
o2
i2
o1
i2
Each diagram contributes (Müller, Heusler, Braun and Haake ’07):
(−N)V
1
N
= (−1)V 2g ,
N
NE
where genus g is
2g = 1 − V + E − F + C
Number of diagrams D2g (π, V ) (in all examples π = (1 2 . . . n)).
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Notation
Integer partitions, for example
11 = 1 + 1 + 1 + 2 + 3 + 3
For partition α of n,
α = [1α1 2α2 3α3 · · · ],
length |α| = |α1 | + |α2 | + |α3 | + . . .
depth hαi = 0 + α2 + 2α3 + 3α4 + . . ..
For the example above, α = [13 21 32 ] with length 6 and depth 5.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Factorizations
π = σ1 σ2 · · · σk ,
π ∈ Sn
Ordered factorizations:h σ1 , .ih
. . , σk ∈ Sin .
Example: (12345) = (134) (12)(45) .
Number denoted by A[π, k, d], where d =
Gregory Berkolaiko
Pk
j=1 hσj i.
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Factorizations
π = σ1 σ2 · · · σk ,
π ∈ Sn
Ordered factorizations:h σ1 , .ih
. . , σk ∈ Sin .
Example: (12345) = (134) (12)(45) .
P
Number denoted by A[π, k, d], where d = kj=1 hσj i.
Inequivalent factorizations: σ1 , . . . , σk are cycles.
Up to commuting factors: (134)(12)(45) = (134)(45)(12)
Partition α = [2α2 3α3 · · · ] counts cycles of length 2, 3, . . .
Example: [22 31 ].
Number denoted by Fα (π), where d = hαi.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Maps
Factorizations
Factorizations
π = σ1 σ2 · · · σk ,
π ∈ Sn
Ordered factorizations:h σ1 , .ih
. . , σk ∈ Sin .
Example: (12345) = (134) (12)(45) .
P
Number denoted by A[π, k, d], where d = kj=1 hσj i.
Inequivalent factorizations: σ1 , . . . , σk are cycles.
Up to commuting factors: (134)(12)(45) = (134)(45)(12)
Partition α = [2α2 3α3 · · · ] counts cycles of length 2, 3, . . .
Example: [22 31 ].
Number denoted by Fα (π), where d = hαi.
Primitive factorizations: σ1 , . . . , σk are transpositions.
σk = (sk tk ), with sk < tk and tk−1 ≤ tk .
Example: (12345) = (12)(45)(35)(25).
Number denoted by pd (π), where d = k.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Factorizations, RMT and diagrams
Theorem
The CUE coefficients
V (π) =
∞
(−1)hπi X
(−1)d
,
w
(π)
d
Nn
Nd
d=1
where
wd =
X
(−1)k−d A[π, k, d]
(Collins’03)
k
(GB-Irving)
(GB-Kuipers)
= pd (π).
Gregory Berkolaiko
(Matsumoto-Novak’10)
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Factorizations, RMT and diagrams
Theorem
The CUE coefficients
V (π) =
∞
(−1)hπi X
(−1)d
,
w
(π)
d
Nn
Nd
d=1
where
wd =
X
(−1)k−d A[π, k, d]
(Collins’03)
k
=
X
(−1)d−|α| Fα (π)
(GB-Irving)
α:hαi=d
(GB-Kuipers)
= pd (π).
Gregory Berkolaiko
(Matsumoto-Novak’10)
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Factorizations, RMT and diagrams
Theorem
The CUE coefficients
V (π) =
∞
(−1)hπi X
(−1)d
,
w
(π)
d
Nn
Nd
d=1
where
wd =
X
(−1)k−d A[π, k, d]
(Collins’03)
k
=
X
(−1)d−|α| Fα (π)
(GB-Irving)
α:hαi=d
=
X
(−1)V D2g (π, V )
(GB-Kuipers)
V :d=hπi+2g
= pd (π).
Gregory Berkolaiko
(Matsumoto-Novak’10)
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Concordance between RMT and semiclassics
Corollary
Since the RMT agrees with semiclassics on the level of the class
functions VN (π), they will also agree for all moments (including
nonlinear).
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Proof idea
Proof idea:
(a)
i2
o2
o1
(b)
(c)
i2
o2
i1
(d)
i2
o2
o1
o1
i2
o2
o1
i1
i1
1
Untie omax (while we can).
2
If cannot, insert / remove edge.
3
Tie things back.
If step 2 was performed we have two diagrams that cancel.
If not, we have a primitive factorization.
Gregory Berkolaiko
Universality in chaotic quantum transport
i1
Quantum Transport and RMT
Combinatorics
All together now
Summary
Taking the evaluation rule for granted, complete equivalence
of semiclassical answers and RMT predictions
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Summary
Taking the evaluation rule for granted, complete equivalence
of semiclassical answers and RMT predictions
Works for all moments, all orders, all classical symmetry
classes (Dyson’s three-fold way).
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Summary
Taking the evaluation rule for granted, complete equivalence
of semiclassical answers and RMT predictions
Works for all moments, all orders, all classical symmetry
classes (Dyson’s three-fold way).
Combinatorics more fun for the TR-invariant case
non-orientable maps
signed permutations instead of Sn
palindromic factorizations
but it all works out.
Gregory Berkolaiko
Universality in chaotic quantum transport
Quantum Transport and RMT
Combinatorics
All together now
Summary
Taking the evaluation rule for granted, complete equivalence
of semiclassical answers and RMT predictions
Works for all moments, all orders, all classical symmetry
classes (Dyson’s three-fold way).
Combinatorics more fun for the TR-invariant case
non-orientable maps
signed permutations instead of Sn
palindromic factorizations
but it all works out.
Semi-automatic way to calculate generating function for
low-order (up to 7) corrections to tr(t ∗ t)n .
Gregory Berkolaiko
Universality in chaotic quantum transport