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?
Wigner ´55
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.
Total cross section versus c.m. energy for scattering of neutrons on Th.The resonances all have the same spin 1/2 and positive parity.
232
Poisson NDE1726 spacings
Universality of spectral fluctuations
In the spectrum of the Schrödinger, wave, or Dirac operatorfor 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
from A. Odlyzko (1987)
GUE
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)
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.
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.’’
Wigner-Dyson universality
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):
Canonical anti-commutation relations:Applications/examples:• Hartree-Fock-Bogoliubov theory of superconductors • Dirac equation for relativistic spin ½ particles
cccc
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.
)()( ccZccZccccWH
Ten large families of symmetric spaces
A
IA
IIA
C
IC
D
IIID
IIIA
IBD
IIC
complex Hermitian
real symmetric
quaternion self-adjoint
Z complex symmetric, W=W*
Z complex symmetric, W=0
Z complex skew, W=W*
Z complex skew, W=0
Z complex pxq, W=0
Z real pxq, W=0
Z quaternion 2px2q, W=0
tWZZW
Hform ofsymmetric spacefamily
)2(SO N
)(U N
O(N)/)(U N
)2(USp/)2(U NN
)2(USp N
)(U/)2(USp NN
)(SO)(SO/)(SO qpqp
)2(USp)2(USp/)22(USp qpqp
)(U)(U/)(U qpqp
)(U/)2(SO NN
Physical realizationsAI : 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 impuritiesAII: same as AI, but with spin-orbit scatterersCI : quasi-particle excitations in a disordered spin-singlet
superconductor in the Meissner phase
C : same as CI but in the mixed phase with magnetic vorticesDIII: disordered spin-triplet superconductor D : spin-triplet superconductor in the vortex phase, or with
magnetic impuritiesAIII: 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 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
Conjecture 1: All states (at arbitrarily weak disorder) are localized in two space dimensions
Conjecture 2: In dimension metallic behavior is stable, i.e. states remain extended under perturbation by weak disorder
3d
for symm. classes A, AI, C, CI
for any symmetry class
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
)(U...)(U)(U ||21 VVVLocal gauge invariance
Fourier transform of probability measure :)(Hd
jiij ij KKJHdHK Tr exp)()Tr i(exp
)(:,)(: 01 NOJjiNOJji ijii
Hermitian random matrices for a lattice with orbitals per site
Hilbert space
Orthogonal projectors
Hi
Niii VVV ,
ii VV :
N
(class A)
Symmetric supermanifolds: an example
Globally symmetric Riemannian manifold 01 MMM
Unitary vector spaceThe space of all orthogonal decompositions
is a Grassmann manifold 1UU/U Mqpqp
qpU
qpUUU
Pseudo-unitary vector space of signature (p,q). The pseudo-orthogonal decompositions
form a non-compact Grassmannian 0, UU/U Mqpqp
qpV
qpVVV
(type AIII)
Example (cont’d)
Vector bundleA point determinesFibre
MF
VVVUUU ,
),(Hom),(Hom UVUV),(Hom),(Hom)(1 VUVUm
Mm
2H2S
Minimal case:
41 )( m
The algebra of sections carries a canonical action of the Lie superalgebra
),( FM
Riemannian symmetric superspace ),,( FMqpqpVU |)(
Universal construction of symmetric superspaces
0H
),( FMcanonically acts on ‘superfunctions’
Form the associated vector bundle MGF H 10 0
Pick such real Lie groups that is Riemannian symmetric space in the geometry induced by the Cartan-Killing form.
00 GH MHG 00 /
Complex Lie superalgebra ( -grading)
(with Cartan involution) 2)()( 1100
10
acts on by Ad. 0H 1
Supersymmetric nonlinear sigma models
),( FMGraded-commutative algebra of sections
Fd :
Susy sigma model is functional integral of maps
DerDer:g
Invariance w.r.t. to action on determines metric tensor
Riemannian structure is important for stability!
Action functional is given by the metric tensor in the usual way.
The 10-Way TableIC
RME A AI AII C CI D DIII AIII BDI CII
noncomp. AIII BDI CII DIII D CI C A AI AII
compact AIII CII BDI CI C DIII D A AII AI
Correspondence between random matrix models and supersymmetric nonlinear sigma models:
MRZ, J. Math. Phys. 37 (1996) 4986
susy NLsM
Open problems – in sigma model language
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.
Conjecture 2: In dimension metallic behavior is stable, i.e. states remain extended under perturbation by weak disorder for any symmetry class.
3d
Noncompact symmetry is spontaneously broken.
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.
Supersymmetric Howe pairs: motivation
Farmer’s conjecture for the autocorrelation function of ratios:
Riemann zeta function on critical line:)i2/1())(2( ttNf
)2/()2/log()2/()( ttttN Smooth counting function
0 21
211
)()()()(lim
sfsfsfsfds
))(())((e
))(())((
2112
2211)(i
2121
2121 21
21 0
N
uuuudu NN
NN
NU
/i/i
/i/i
)e1(Det)e1(Det)e1(Det)e1(Detlim
21
21
Conrey, Farmer & MRZ, math-ph/0511024
Autocorrelation of ratios as a character
01 VVV graded vector space2
representationhas character
)(GL 1V )()(:)( 11 VVg
1
1
dim
0)1(Det)(Tr)(V
k Vkk tggt
Super Fock space )S()( 01 VVV
)GL()GL()GL(: 01 VVV
1
0
101 )1(SDet
)1(Det)1(Det),(STr
ggggg V
),(diag 01 ggg
)(S)(S:)( 00 VVg representationhas character
01)1(Det)(Tr
0k Vkk tggt
)(GL 0V
Notice advantage: linearization! )(STr)(STr gg
Set
Nut U);e,e,e,(ediag 2121 iiii
N
duutt VU
1)1(SDet)(
is character of highest-weight irreducible representation of on the invariants in .2|2 NU V
Propn.
NVVV 2|201
2|2 acts there NU acts here
NU,2|2 is susy dual pair in the sense of R. Howe[Howe (1976/89): Remarks on classical invariant theory]
)(S)(S)()( NNNNV
The actions of and on commute.2|2NU V
Determining the character
Details in: Conrey, Farmer & MRZ
Main idea: the function uniquely extends to radial analytic section of symmetric superspace ),,( 2|2 FM
)(tt
Heuristic picture: naive transcription of Weyl character formula to this situation gives the correct answer!
Properties of : i) lies in the kernel of the full ring of -invariant
differential operators forii) has convergent weight expansion
2|2
),( FM
)(loge)(sdim)( tWt where
2211 ,0 nNmmn and
k kkkk nm )(i
Noncompact nonlinear sigma models
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
space(time)
Mtarget symmetric space
(noncompact)
Regularization: lattice
)()(
2|| Mab
bad ggxdDS
Energy (action) function:
Targets ...,UU/U,SO/SO: ,21,2 qpqpM
Spontaneous Symmetry Breaking
Proof: Use Iwasawa decomposition for . Integrate out nilpotent degrees of freedom, resulting in convex action for torus variables. Apply Brascamp-Lieb inequality.
NAKG 2,1SO
Theorem (Spencer & MRZ):
if and if and is not too small.
const),dist(cosh 02
So
1vol 3d
)(dvole kkS Gibbs measure:
21,2 SO/SOMConsider the simplest case of
)()( iii xxDiscrete field
),(distcosh),(distcosh/ oS kkjiij
Define action via geodesic distance of :M
o
1d chain M