Transcript Atomistic Picture: Chains - ZMP-HH

From Random Matrices to Supermanifolds
ZMP Opening Colloquium (Hamburg, Oct 22, 2005)
• Why random matrices? What random matrices?
• Which supermanifolds?
• How do random matrix problems lead to questions
about supermanifolds and supersymmetric field
theories?
Total cross section versus c.m. energy for scattering of neutrons on
The resonances all have the same spin 1/2 and positive parity.
Wigner ´55
Poisson
232
Th.
NDE
1726 spacings
Nearest-neighbor spacing distribution for the ``Nuclear Data Ensemble´´ comprising 1726 spacings. For
comparison, the RMT prediction labelled GOE and the result for a Poisson distribution are also shown.
Universality of spectral fluctuations
In the spectrum of the Schrödinger, wave, or Dirac operator
for a large variety of physical systems, such as
•
atomic nuclei (neutron resonances),
•
disordered metallic grains,
•
chaotic billiards (Sinai, Bunimovich),
•
microwaves in a cavity,
•
acoustic modes of a vibrating solid,
•
quarks in a nonabelian gauge field,
•
zeroes of the Riemann zeta function,
one observes fluctuations that obey the laws given by random matrix
theory for the appropriate Wigner-Dyson class and in the ergodic limit.
Spacing distribution of the Riemann zeroes
GUE
from A. Odlyzko (1987)
Wigner-Dyson universality
Wigner-Dyson symmetry classes:
•
A : complex Hermitian matrices (‘unitary class’, GUE)
•
AI : real symmetric matrices (‘orthogonal class’, GOE)
•
AII : quaternion self-dual matrices (‘symplectic class’, GSE)
Dyson (1962, The 3-fold way): ``The most general kind of matrix
ensemble, defined with a symmetry group which may be
completely arbitrary, reduces to a direct product of independent
irreducible ensembles each of which belongs to one of three
known types.’’
This classification has proved fundamental to various areas of
theoretical physics, including the statistical theory of complex
many-body systems, mesoscopic physics, disordered electron
systems, and the field of quantum chaos.
Outline
•
Motivation: universality of disordered spectra
•
Symmetry classes of disordered fermions:
from Dyson‘s threefold way to the 10-way classification
•
Riemannian symmetric superspaces as target spaces of
susy nonlinear sigma models
•
Howe duality: ratios of random characteristic polynomials
•
Spontaneous symmetry breaking of a hyperbolic sigma
model
Symmetry classes of disordered fermions
Consider one-particle Hamiltonians (fermions):
H  W (c c  c c )  (Z c c  Z c c )
Canonical anti-commutation relations: c c  c c  
Applications/examples:
•
Hartree-Fock-Bogoliubov theory of superconductors
•
Dirac equation for relativistic spin ½ particles
Classify such Hamiltonians according to their symmetries! What
are the irreducible blocks that occur?
Theorem [Heinzner, Huckleberry & MRZ, CMP 257 (2005) 725]:
Every irreducible block that occurs in this setting corresponds to
one of a large family of irreducible symmetric spaces.
Ten large families of symmetric spaces
W
H   
Z
Z 

 W t 
family
symmetric space
form of
A
U( N )
complex Hermitian
AI
U( N ) / O(N)
real symmetric
A II
U(2 N ) / USp (2 N )
quaternion self-adjoint
C
USp (2 N )
Z complex symmetric, W=W*
CI
USp (2 N ) / U( N )
Z complex symmetric, W=0
D
SO (2 N )
Z complex skew, W=W*
DIII
SO (2 N ) / U( N )
Z complex skew, W=0
AIII
U( p  q) / U( p)  U(q)
Z complex pxq, W=0
BD I
SO ( p  q) / SO ( p)  SO (q)
Z real pxq, W=0
C II
USp (2 p  2q) / USp (2 p)  USp (2q)
Z quaternion 2px2q, W=0
Physical realizations
AI : electrons in a disordered metal with conserved spin and
with time reversal invariance
A : same as AI, but with time reversal broken by a magnetic
field or magnetic impurities
AII: same as AI, but with spin-orbit scatterers
CI : quasi-particle excitations in a disordered spin-singlet
superconductor in the Meissner phase
C : same as CI but in the mixed phase with magnetic vortices
DIII: disordered spin-triplet superconductor
D : spin-triplet superconductor in the vortex phase, or with
magnetic impurities
AIII: massless Dirac fermions in SU(N) gauge field background
(N > 2)
BDI: same as AIII but with gauge group SU(2) or Sp(2N)
CII : same as AIII but with adjoint fermions, or gauge group
SO(N)
Altland, Simons & MRZ: Phys. Rep. 359 (2002) 283
Open Mathematical Problems
Conjecture 1:
All states (at arbitrarily weak disorder) are localized in
two space dimensions for symm. classes A, AI, C, CI
Conjecture 2:
In dimension d  3 metallic behavior is stable, i.e.
states remain extended under perturbation by weak
disorder for any symmetry class
Conjecture 3:
In the so-called ergodic regime the level-correlation
functions are universal. The universal laws are given
by the invariant Gaussian random-matrix ensemble
of the appropriate symmetry class
Random matrix methods
Methods based on the joint probability density for the eigenvalues
of a random matrix:
•
•
Orthogonal polynomials + Riemann-Hilbert techniques
(Scaling limit) reduction to integrable PDE’s (Painleve-type)
In contrast, superanalytic methods apply to band random matrices,
granular models, random Schrödinger operators etc.
•
•
Hermitian (or Hamiltonian) disorder:
Schäfer-Wegner method (1980) , Fyodorov’s method (2001)
see MRZ, arXiv:math-ph/0404057 (EMP, Elsevier, 2006)
Unitary (scattering, time evolution) disorder:
color-flavor transformation (1996), Howe duality (2004)
Wegner’s N-orbital model
(class A)
Hermitian random matrices H for a
lattice  with N orbitals per site i  
Hilbert space V  iVi ,
Vi  N
Orthogonal projectors i : V  Vi
Fourier transform of probability measure d (H ):
 exp (i Tr HK ) d ( H )  exp  
ij
i  j : Jii  O( N1 ) ,
J ij Tr K  i K  j
i  j : Jij  O( N 0 )
Local gauge invariance U(V1 )  U(V2 ) ... U(V|| )

Symmetric supermanifolds: an example
pq
Unitary vector space U  
The space of all orthogonal decompositions
U  U   U    p  q
is a Grassmann manifold U pq / U p  Uq  M1
pq
Pseudo-unitary vector space V  
of signature (p,q).
The pseudo-orthogonal decompositions
V  V   V    p  q
form a non-compact Grassmannian U p,q / U p  Uq  M 0
Globally symmetric Riemannian manifold M  M1  M 0
(type AIII)
Example (cont’d)

 M
Vector bundle F 
A point m  M determines U  U  U  , V  V  V 
Fibre  1 (m)  Hom(U  ,V  )  Hom(U  ,V  )
 Hom(V  ,U  )  Hom(V  ,U  )
Minimal case:
 1 (m)  4
S2


H2
The algebra of sections (M ,  F  ) carries a canonical
(U V )  
action of the Lie superalgebra  
p  q| p  q
)
Riemannian symmetric superspace (M ,  F  , 
Universal construction of symmetric superspaces
Complex Lie superalgebra
(with Cartan involution)
 
0 
1
( 2 -grading)
 (
0 
0 )  (
1 
1)
Pick such real Lie groups H 0  G0 that G0 / H 0  M
is Riemannian symmetric space in the geometry
induced by the Cartan-Killing form.
H0
H 0 acts on 1 by Ad.
Form the associated vector bundle F  G0 H0 
1 M
 canonically acts on ‘superfunctions’ (M ,  F  )
Supersymmetric nonlinear sigma models
Graded-commutative algebra of sections   (M ,  F  )
action on  determines metric tensor
Invariance w.r.t. to 
g : Der  Der  
Susy sigma model is functional integral of maps
 : d    F
Action functional is given by the metric tensor in the usual way.
Riemannian structure is important for stability!
The 10-Way Table
CI
Correspondence between random matrix models
and supersymmetric nonlinear sigma models:
RME
A
AI
AII
noncomp. AIII BDI CII
C
CI
D
DIII AIII BDI CII
CI
C
A
AI
AII
DIII D
A
AII
AI
DIII D
susy NLsM
compact
AIII CII
BDI CI
C
MRZ, J. Math. Phys. 37 (1996) 4986
Open problems – in sigma model language
Conjecture 1:
All states (at arbitrarily weak disorder) are localized in
two space dimensions for symm. classes A, AI, C, CI.
Nonlinear sigma model has mass gap.
Conjecture 2:
In dimension d  3 metallic behavior is stable, i.e.
states remain extended under perturbation by weak
disorder for any symmetry class.
Noncompact symmetry is spontaneously broken.
Conjecture 3:
In the so-called ergodic regime the level-correlation
functions are universal. The universal laws are given
by the invariant Gaussian random-matrix ensemble of
the appropriate symmetry class.
RG flow takes nonlinear sigma model to Gaussian fixed point.
Supersymmetric Howe pairs: motivation
Riemann zeta function on critical line:
f  (2 N (t ))   (1/ 2  it )
Smooth counting function N (t )  (t / 2 ) log(t / 2 )  (t / 2 )
Farmer’s conjecture for the autocorrelation function of ratios:
1
lim 

 ds
0

f  ( s  1 ) f  ( s  2 )
f  ( s  1 ) f  ( s  2 )
(1  2 )( 1  2 )
(  1 )( 2  2 )
 ei ( 1  2 ) 1
( 1  2 )(1  2 )
( 2  1 )(1  2 )

1  0  
2
Det(1  ei 1 / N u ) Det(1  e i 2 / N u )
 lim N   du
i1 / N
i2 / N
Det
(
1

e
u
)
Det
(
1

e
u)
UN
Conrey, Farmer & MRZ, math-ph/0511024
Autocorrelation of ratios as a character
2  graded vector space V  V1 V0
GL (V1 ) representation   ( g ) :  (V1 )   (V1 )
dimV
has character k 0 (t ) k Tr  k ( g )  Det V (1  tg )
1
1
GL(V0 )  representation  ( g ) : S (V0 ) S (V0 )

k
k
1
has character k 0 t Tr  ( g )  Det V0 (1  tg )
Super Fock space 
V
 (V1 )  S(V0 )
     : GL(V1 )  GL(V0 )  GL( V )
ST r  ( g1 , g 0 ) 
Det(1  g1 )
 SDet V (1  g ) 1
Det(1  g 0 )
g  diag ( g1 , g0 )
STr  ( g )  STr  ( g )
Notice advantage: linearization!
Set V  V1 V0  2|2  N
acts there

2|2
UN
acts here
 V  (N )  (N )  S(N )  S(N )
The actions of U N and 
2|2 on 
V
commute.
is susy dual pair in the sense of R. Howe

2|2 , U N
[Howe (1976/89): Remarks on classical invariant theory]
t  diag (ei1 , ei 2 , ei1 , ei2 ); u  UN
Propn.  (t ) 

UN
SDet V (1  t  u) 1 du
is character of highest-weight irreducible representation
of 
2|2 on the U N invariants in  V .
Determining the character
Main idea: the function t   (t ) uniquely extends to radial
analytic section of symmetric superspace (M ,  F  , 
2|2 )
Properties of  :
i)  lies in the kernel of the full ring of 
2|2-invariant
differential operators for (M ,  F  )
ii)  has convergent weight expansion
 (t )   sdim (W ) e (logt ) where   ik (mk k  nkk )
and n1  0  m1 , m2  N  n2
Details in: Conrey, Farmer & MRZ
Heuristic picture: naive transcription of Weyl character formula
to this situation gives the correct answer!
Noncompact nonlinear sigma models
space(time)

Energy (action) function: S 

target M
symmetric space
(noncompact)


(M )
| D |2   d d x g ( )   a   b g ab
Regularization: lattice   
Targets
M : SO 2,1 / SO 2 , U p,q / U p  Uq , ...
d = 1: M. Niedermaier, E. Seiler, arXiv:hep/th-0312293
d = 2: Duncan, Niedermaier & Seiler, Nucl. Phys. B 720
(2005) 235
d = 3: Spencer & MRZ, Commun. Math. Phys. 252 (2004) 167
Spontaneous Symmetry Breaking
Consider the simplest case of M  SO 2,1 / SO 2
Discrete field i   ( xi ) ( xi  )
Define action via geodesic distance of M :
S   ij/ cosh  dist (i , j )   k cosh  dist (k , o)
Gibbs measure:
eS k dvol(k )
Theorem (Spencer & MRZ):

1d chain
cosh  dist( 0 , o)
2
if   vol  1 and if d  3 and  is not too small.
S



 M
o
 const
Proof: Use Iwasawa decomposition G  NAK for SO 2,1 .
Integrate out nilpotent degrees of freedom, resulting in convex
action for torus variables. Apply Brascamp-Lieb inequality.