From Random Matrices to Supermanifolds ZMP Opening Colloquium (Hamburg, Oct 22, 2005) Why random matrices? What random matrices? Which supermanifolds?

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 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

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

tWZ

ZWHform 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

H

i

Niii VVV ,

ii VV :

N

(class A)

Symmetric supermanifolds: an example

Globally symmetric Riemannian manifold 01 MMM

Unitary vector space

The 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 bundle

A point determines

Fibre

MF

VVVUUU ,

),(Hom),(Hom UVUV

),(Hom),(Hom)(1 VUVUmMm

2H2S

Minimal case:

41 )( m

The algebra of sections carries a canonical

action of the Lie superalgebra

),( FM

Riemannian symmetric superspace ),,( FM

qpqpVU |)(

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

sfsf

sfsfds

))((

))((e

))((

))((

2112

2211)(i

2121

2121 21

21 0

N

uu

uudu

NN

NN

N

U/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

representation

has 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

gg

ggg V

),(diag 01 ggg

)(S)(S:)( 00 VVg representation

has character

0

1)1(Det)(Tr0k V

kk 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 thereNU 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 for

ii) 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