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Department of Applied Mathematics
HAIT – Holon Academic Institute of TechnologyHolon, ISRAEL
Workshop on Random Matrix Theory: Condensed Matter, Statistical Physics, and Combinatorics The Abdus Salam International Centre for Theoretical Physics, Trieste, June 29, 2004
Towards exact integrabilityof replica field theories
H A I T
Eugene Kanzieper
Based on:Replica field theories, Painlevé transcendents,
and exact correlation functions, Phys. Rev. Lett. 89, 250201 (2002)
Thanks to:Craig Tracy (UCal-Davis), Peter Forrester (UMelb)
(for guiding through Painlevé literature)
Discussions with: Alex Kamenev (UMin), Ady Stern (WIS), Jac Verbaarschot
Supported by:Albert Einstein Minerva Centre for Theoretical
Physics(Weizmann Institute of Science, Rehovot, Israel)
(SU
NY)
Outline
Nonperturbative Methods in Physics of Disorder
Replica sigma models
Supersymmetric sigma model
Keldysh sigma model
Why Replicas? What Are Replicas and The Replica Limit?
Two Pitfalls: Analytic Continuation and (Un) controlled Approximations
Message: exact approach to replicas needed
Outline
Nonperturbative Methods in Physics of Disorder
Towards Exact Integrability of Replica Field
Theories in 0 Dimensions
Conclusions
Why Replicas? What Are Replicas and The Replica Limit?
Two Pitfalls: Analytic Continuation and (Un) controlled Approximations
Outline
Nonperturbative Methods in Physics of Disorder
Outline
Nonperturbative Methods in Physics of Disorder
Statistical description: Ensemble of grains
Ensemble averaged observable
Disorder as a perturbation ?..
11 FE
Disordered grain
Qu
an
tum
dot
mean scattering time
Fermi energy
IMPORTANTPHYSICS
LOST
Whole issue of strong localisation
Disordered grain
Qu
an
tum
dot
Weak disorder limit: long time particle evolution (times larger than the Heisenberg time)
11 FE
Disordered grain
Qu
an
tum
dot
Whole issue of strong localisation
F a i l u r e
!
to perturbative treatment of disorder
IMPORTANTPHYSICS
LOST
Disordered grain
Qu
an
tum
dot
NO
Replica sigma models (bosonic and fermionic)Wegner 1979Larkin, Efetov, Khmelnitskii 1980Finkelstein 1982
Supersymmetric sigma modelEfetov 1982
Keldysh sigma modelHorbach, Schön 1990 Kamenev, Andreev 1999
Q
Field Theoretic Approaches
finite dimensionalmatrix field
nn Q
symmetry
QQQ JiDdF 0J 2Tr
8)( 2r
Q DQZ )(exp JFJ
1Q 2
A Typical NonlinearNonlinearly constrained matrix field of
certain symmetries
Generating functional
Action
Model
Outline
Nonperturbative Methods in Physics of Disorder
Replica sigma models
Supersymmetric sigma model
Keldysh sigma model
Perturbative approach may become quite useless even in the weak disorder limit Nonperturbative approaches: the three formulations
Why Replicas? What Are Replicas and the
Replica Limit?
Outline
Nonperturbative Methods in Physics of Disorder
Replica sigma models (bosonic and fermionic)Wegner 1979Larkin, Efetov, Khmelnitskii 1980Finkelstein 1982
Supersymmetric sigma modelEfetov 1982
Keldysh sigma modelHorbach, Schön 1990 Kamenev, Andreev 1999
Field Theoretic Approaches
disorder
interaction
interactionout-of-equilibrium
disorderdisorder
Replica sigma models (bosonic and fermionic)Wegner 1979Larkin, Efetov, Khmelnitskii 1980Finkelstein 1982
Field Theoretic Approaches
interactiondisorder
A viable tool to treat an interplay between disorder
and interaction!
Replica sigma models (bosonic and fermionic)Wegner 1979Larkin, Efetov, Khmelnitskii 1980Finkelstein 1982
Field Theoretic Approaches
interactiondisorder
A viable tool to treat an interplay between disorder
and interaction!
Why Replicas? What Are Replicas and the
Replica Limit?
Outline
Nonperturbative Methods in Physics of Disorder
Replica trick
Mean level density out of
Looks easier (replica partition function)
Reconstruct through the replica limit
based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934
HE
EGˆ
1Tr)(
)ˆ(det)( HE EZ nn
)(1
)( lim0
EZE
EG nn n
commutativity!
Replica trick
Density-density correlation function out of
Looks easier (replica partition function)
Reconstruct through the replica limit
based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934
commutativity!
HE
HE EG EG
ˆ1
Trˆ
1Tr)()(
21
21
)ˆ(det)ˆ(det),( 2121 HE HE E EZ nnn
),(1
)()( 2121
2
20
21 lim E EZE E
EG EG nn n
Reconstruct through the replica limit
based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934
commutativity!),(1
)()( 2121
2
20
21 lim E EZE E
EG EG nn n
Replica trick
Word of caution:
for more than two decades no one could rigorously implement it in
mesoscopics !
Replica Trick: A Bit of Chronology
1979 1980 1985
Bosonic Replicas
FermionicReplicas
F. Wegner
Larkin, Efetov, Khmelnitskii
First Critique(RMT)
Verbaarschot, Zirnbauer
Reconstruct through the replica limit
based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934
commutativity!),(1
)()( 2121
2
20
21 lim E EZE E
EG EG nn n
Replica Trick: A Bit of Chronology
Reconstruct through the replica limit
based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934
commutativity!),(1
)()( 2121
2
20
21 lim E EZE E
EG EG nn n
Critique of the replica trick, J. Verbaarschot and M. Zirnbauer 1985
“… the replica trick … suffers from a serious drawback: it is mathematically ill founded.”
“… the replica trick for disordered electron systems is limited to those regions of parameter space where the nonlinear sigma model can be evaluated perturbatively.”
Another critique of the replica trick, M. Zirnbauer 1999
Replica Trick: A Bit of Chronology
1979 1980 1985
Bosonic Replicas
FermionicReplicas
F. Wegner
Larkin, Efetov, Khmelnitskii
First Critique(RMT)
Verbaarschot, Zirnbauer
Reconstruct through the replica limit
based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934
commutativity!),(1
)()( 2121
2
20
21 lim E EZE E
EG EG nn n
1999
Kamenev, Mézard
Replica SymmetryBreaking (RMT)
Second Critique(RMT)
Zirnbauer
Asymptotically Nonperturbative
Results:
?
19821983
SUSY
Efetov
Two Pitfalls: Analytic Continuation and (Un) controlled Approximations
Why Replicas? What Are Replicas and the
Replica Limit?
Outline
Nonperturbative Methods in Physics of Disorder
g0D limit:
Disordered grain
Qu
an
tum
dot
5.73 4.28 0.18 9.33 4.58 9.27 7.30 4.03 4.05 1.59 6.49 9.19
4.78 8.45 0.02 9.52 6.97 4.20 1.14 9.93 5.94 6.49 5.03 4.50
6.41 4.02 0.01 5.17 9.32 4.73 3.00 3.19 0.74 8.03 4.38 1.30
1.83 2.47 8.03 6.60 4.34 9.47 9.93 5.94 6.49 4.78 4.85 3.28
4.06 7.37 9.03 8.05 4.51 3.95 4.00 3.05 3.58 7.10 4.48 9.37
4.78 8.45 0.02 9.52 6.97 4.20 8.03 7.94 5.29 1.18 4.38 3.01
0.09 5.32 3.86 8.22 0.36 0.88 0.28 2.40 1.39 6.60 4.34 9.47
1.18 2.87 1.14 9.93 5.94 6.49 4.78 8.45 0.02 9.52 6.97 4.20
3.00 5.29 3.57 5.29 8.83 7.17 2.40 1.39 5.73 4.28 0.18 9.33
0.28 2.40 1.39 5.73 6.41 4.02 0.01 5.17 5.07 7.35 4.78 8.45
8.45 7.30 4.03 4.05 1.59 6.49 9.19 3.02 4.39 4.04 9.03 8.10
9.93 5.94 6.49 4.78 4.85 3.28 7.24 8.04 0.39 1.83 2.47 8.03
RMT
Two Pitfalls
(a) Analytic Continuation
)(1
ˆ1
Tr)( lim0
EZE
HE
EG nn n
)ˆ(det)( HE EZ nn
HEi DHE )ˆ(exp)ˆ(det
n
kkkkn
1
)ˆ(exp)(~ HEi D EZ
Original recipe
Field theoretic realisation
Rn
Zn
(a) Analytic Continuation
)ˆ(det)( HE EZ nn
n
kkkkn
1
)ˆ(exp)(~ HEi D EZ
Rn
Zn
van Hemmen and Palmer 1979Verbaarschot and Zirnbauer 1985Zirnbauer 1999
U n i q u e n e s
s
?..
(b) (Un) Controlled Approximations
made prior to analytic continuation
DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)
Q QE i
DQEZ ,Trexp)(det
)(~
2
)(
N
nnn
Saddle point evaluation for matrices of large dimensions
Rn,)ˆ(det)( HE EZ nn
Zn
),()(vol1~,0 ,
(sp) NzGN pnp pnnE E,Z
n Z
known explicitlyReplica Symmetry Breaking for “causal” saddle points
(b) (Un) Controlled Approximations
made prior to analytic continuation
DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)
Saddle point evaluation for matrices of large dimensions
),()(vol1~,0 ,
(sp) NzGN pnp pnnE E,Z
n Z
known explicitly)(~ Nn Obreaks down at
p
jpnpn jn
jG
npn
nG
1,, )1(
)()(vol,
)()(
)(
UUU
Analytic continuation…
0)(vol 1, npnG
(b) (Un) Controlled Approximations
made prior to analytic continuation
DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)
Saddle point evaluation for matrices of large dimensions
),()(vol1~,0 ,
(sp) NzGN pnp pnnE E,Z
n Z
)(~ Nn Obreaks down at
Does NOT existIs NOT unique: Re-enumerate Saddles!!
Analytic continuation…
In the vicinity n=0:
Kamenev, Mézard 1999
Zirnbauer 1999Kanzieper
2004 (unpublished
)
diverges for Zn
(b) (Un) Controlled Approximations
made prior to analytic continuation
DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)
Saddle point evaluation for matrices of large dimensions
),()(vol1~,0 ,
(sp) NzGN pnp pnnE E,Z
n Z
)(~ Nn Obreaks down at
Analytic continuation…..?
diverges for Zn
What’s the reason(s) for the failure ?
(b) (Un) Controlled Approximations
made prior to analytic continuation
DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)
Saddle point evaluation for matrices of large dimensions
),()(vol1~,0 ,
(sp) NzGN pnp pnnE E,Z
n Z
)(~ Nn Obreaks down at
Analytic continuation…..?
diverges for Zn
1D 2D
(b) (Un) Controlled Approximations
made prior to analytic continuation
DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)
Saddle point evaluation for matrices of large dimensions
),()(vol1~,0 ,
(sp) NzGN pnp pnnE E,Z
n Z
)(~ Nn Obreaks down at
Analytic continuation…..?
diverges for Zn
(b) (Un) Controlled Approximations
made prior to analytic continuation
DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)
Saddle point evaluation for matrices of large dimensions
),()(vol1~,0 ,
(sp) NzGN pnp pnnE E,Z
n Z
)(~ Nn Obreaks down at
Analytic continuation…..?
diverges for Zn
It’s a bit too dangerous to make analyticcontinuation based on an approximate result !
Two Pitfalls: Analytic Continuation and (Un) controlled Approximations
Why Replicas? What Are Replicas and the
Replica Limit?
Outline
Nonperturbative Methods in Physics of Disorder
Towards Exact Integrability of Replica Field
Theories in 0 Dimensions
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
),()(vol1~,0 ,
(sp) NzGN pnp pnnE E,Z
n Z
Saddle point evaluation for large matrices (KM, 1999)
duality
Q QE i
DQEZ ,Trexp)(det
)(~
2
)(
N
nnn
Rn,)ˆ(det)(ˆ
N
nn GUE
H
HEEZ
Zn
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
Let’s do everything exactly !!
duality
Q QE i
DQEZ ,Trexp)(det
)(~
2
)(
N
nnn
Rn,)ˆ(det)(ˆ
N
nn GUE
H
HEEZ
Zn
“There is of course little hope that the multi-dimensional integrals appearing in the replica formalism … can ever be evaluated non-perturbatively … ”Critique of the replica trick, J. Verbaarschot and M. Zirnbauer 1985
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
Let’s do everything exactly !!
duality
Q QE i
DQEZ ,Trexp)(det
)(~
2
)(
N
nnn
Rn,)ˆ(det)(ˆ
N
nn GUE
H
HEEZ
Zn
Exact evaluation is possible as there exists an exact link between 0D replica field theories and the theory of nonlinear integrable hierarchies. EK: PRL 89, 250201 (2002)
“There is of course little hope that the multi-dimensional integrals appearing in the replica formalism … can ever be evaluated non-perturbatively … ”Critique of the replica trick, J. Verbaarschot and M. Zirnbauer 1985
Paul Painlevé(1863-1933)
Gaston Darboux (1842-1917)
No PhotoYet
Morikazu Toda
born 1917
French Prime Minister
September-November 1917
April-November 1925
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
Let’s do everything exactly !!
Q QE i
DQEZ ,Trexp)(det
)(~
2
)(
N
nnn
Rn,)ˆ(det)(ˆ
N
nn GUE
H
HEEZ
Zn
n
kk
Nkn
n
kkn d
1
22
1
exp)(~ E i EZ
1det j
knn
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
Let’s do everything exactly !!
iE kk
112
1
22
)(~
j
kj,kj
kj,kiN
k
n
kk
nn
kkede det det EEEZ
2222
)(~ kjNij,k
nn ede EEEZ det
n
kk
Nkn
n
kkn d
1
22
1
exp)(~ E i EZ
1det j
knn
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
iE kk
112
1
22
)(~
j
kj,kj
kj,kiN
k
n
kk
nn
kkede det det EEEZ
2222
)(~ kjNij,k
nn ede EEEZ det
nn
Nikj
kjn
n ede
det EE
EEZ 2
,
22
)(~
n
kk
Nkn
n
kkn d
1
22
1
exp)(~ E i EZ
1det j
knn
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
iE kk
112
1
22
)(~
j
kj,kj
kj,kiN
k
n
kk
nn
kkede det det EEEZ
2222
)(~ kjNij,k
nn ede EEEZ det
nn
Nikj
kjn
n ede
det EE
EEZ 2
,
22
)(~
n
kk
Nkn
n
kkn d
1
22
1
exp)(~ E i EZ
1det j
knn
Q QE i
DQEZ ,Trexp)(det
)(~
2
)(
N
nnn
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
nn
Nikj
kjn
n ede
det EE
EEZ 2
,
22
)(~
Q QE i
DQEZ ,Trexp)(det
)(~
2
)(
N
nnn
Hankel determinant !!Darboux Theorem !!
),(~ Nn E
1),(~0
NE
),(~1 NE
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
nnkj
kjn
nn
n NeNe det ),(~),(~)(~
1,
22
EEEZ EEE
Q QE i
DQEZ ,Trexp)(det
)(~
2
)(
N
nnn
(Positive) Toda Lattice Equation
112 ~~~~~
nnnnn '''
Toda Lattice Hierarchy for replica partition functions
is a fingerprint of exact integrability hidden in replica field theories !
S Y M M
E T R Y
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
)(~),(~,1),(~110
2
EZE E E eNN
),(~)(exp)(~ 2 Nn nn E E EZ
112 ~~~~~
nnnnn '''
S Y M M
E T R Y
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
),(~)(exp)(~ 2 Nn nn E E EZ
112 ~~~~~
nnnnn '''
Does it help us perform analytic continuation
Zn Rnfrom to ?
Zn
Kyoto School’s Formalism
Okamoto, Noumi, Yamada
discovered a link between Toda Lattices and Painlevé equations
No PhotoYet
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
Toda Lattice can be reduced to Painlevé equationPainlevé equation contains the replica index as a parameter in its coefficients
A “simple minded” analytic continuation of so- obtained replica partition function away from integers leads to a correct replica limit
Loosely speaking:
Correctness of a such an analytic continuation can independently be proven (details in the papers), but uniqueness …
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
112 ~~~~~
nnnnn '''
),(~)(exp)(~ 2 Nn nn E E EZ Zn
E i
ZEZ0
IV )(exp)0(~)(~ tdtnn
02244 22 N'n'''t'' IVIVIVIVIVIV
+ boundary conditions
Rn
How it looks in the very end…
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
)(~1)( lim
0
EZE
EG nn n
get exact answer !
How it looks in the very end…
Exact Integrability in 0 Dimensions
DoS in GUEN from fermionic replicas (EK, 2002)
E i
ZEZ0
IV )(exp)0(~)(~ tdtnn
02244 22 N'n'''t'' IVIVIVIVIVIV
+ boundary conditions
Rn
How it looks in the very end…
Exact Integrability in 0 Dimensions
DoS-DoS in GUEN from fermionic replicas (EK, 2002)
)()2()2(log)(~log 321
2 nEEnnin OZ
042 n'''t't''t VVVVVVV
i
nn ein
G sin21)(~1
lim)( 2220 Z
+ boundary conditions
Rn
22/)(
exp)0(~)(~
0
2V
t
ntntdti nn ZZ
How it looks in the very end…
Exact Integrability in 0 Dimensions
DoS in chGUE from fermionic replicas (EK, 2002)
)()()()()(log)(~log 211
0
ntItKtItKtdtsnss
n OZ
0))(14(2222 't'''n''t IIIIIIIIIIIIIIIIII
)()()(2
)()( 112 sJsJsJ
sss
+ boundary conditions
Rn
2
0
4/2/)()(exp)(~
s
t
tntdtss IIIn
n
Z
Towards Exact Integrability of Replica Field
Theories in 0 Dimensions
Conclusions
Why Replicas? What Are Replicas and The Replica Limit?
Two Pitfalls: Analytic Continuation and (Un) controlled Approximations
Outline
Nonperturbative Methods in Physics of Disorder
Established:
Conclusions
Shown:
Derived:
Link between fermionic replica field theories in 0D and the theory of nonlinear integrable hierarchies Toda hierarchy of replica partition functions intimately related to the unitary symmetry of the replica field theory Exact nonperturbative correlation functions in the GUE, Ginibre’s ensemble of complex random matrices, and in chiral GUE (Dustermaat-Heckman theorem is violated) – by reduction of Toda Lattices to Painlevé equations combined with a “simple minded” analytic continuation
symmetry classes (Pfaff Lattices)
41 ,
Puzzle of bosonic replicas
Questions left unanswered
What is about the uniqueness of analytic continuation?
Is there exact integrability of 0D replica field theories
with interaction?
Workshop on Random Matrix Theory: Condensed Matter, Statistical Physics, and Combinatorics The Abdus Salam International Centre for Theoretical Physics, Trieste, June 29, 2004
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