Plateau transition in Hall effect: Matrix model for this class of

problems

A. Sedrakyan

Yerevan Physics Institute

Cologne-2015

1. Random Networks and their connection to gravity

a) Plateau transition in Hall effect b) Integrable models on random surfaces: Non-critical strings c) Sign-factor in 3D Ising model

3. Matrix Models for random networks

For any random network consisting of different n-channel S-matrices (n-incoming and n-outgoing waves)

Definen

Khachatryan, Sedrakyan, Sorba- Nucl.Phys. B 825 (2009) 444Khachatryan , Schrader, Sedrakyan -J. Phys. A 42 (2009)304019

and plug into network

S1

1

nn

,

Rψ1 ψ2

'

ψ1

'ψ2=e

ψi

' S ij ψj+ψ2

'ψ1

' S0 ψ1 ψ2

Rψ1 ψ2

'

ψ1

'ψ2 =

ψ1'ψ2

'

ψ2ψ1

R R

R

RRR

R R

R

RR

RRR

R

R

R

R

R

R

R

R

R

R

ML

Feynman diagrams of the gauge field

)(Nsuaa

A

Z=∫∏rdψr ∏

plaquetts

Rψ1ψ2

'

ψ1

'ψ2

∫∏rdψr e

∑r ,μ

ψr

' S r ,r+μψr+μ+ ∑

plaquettes

ψ2

'ψr+μ x

' S0 ψr+μ x+μ yψr+μ y=

This can be done for all quantum spin chain models with fixed R-matrix. All chain models with local Hamiltonian have R-matrix.

What we will have if network is random?

Exp{-S( , , )}

Putting into partition function Z we get full action of the fermionic part

=

=

=

e

n

nnn

niα

nnniα

nn ψψ+)e,ψ,S(ψ=e,,ψS }{

eiα

XXR

Example: XX-model with phases, equivalent to CC model

The coordinate systems on randomML and regular ML can be connectedvia tetrads

S [ Aa ,F ]=i2

ψσa [∂a−∂a+Aa ]−ψ(m+F σ3) ψ

aadxedx

curved regular

aa e

S [ ψ , Aa ,F ]=∫d ξ ei2

ψσa eαa [∂α−∂α+Aα ]−ψ(e m+F σ3)ψ

We got fermions interacting with U(1) gauge fieldand gravity

For the averaging over U(1) disorder in supersymmetricapproach (Efetov) we need to introduce two fermionic fields,and interacting with U(1) gauge field with opposite charges together with two complex bosonic fields , which have the same action as fermions

ψ↑

ψ↓

ϕ↓ ,↑

S [ ψ↓ ,↑ ,ϕ↓ ,↑ , Aα , F ]total S [ ψ↑ ,−Aα ,−F ]S [ ψ↓ , Aα ,F ]

S [ϕ↑ ,−Aα ,−F ]S [ϕ↓ , Aα , F ]

=

++

+

W. Nudding, I. Gruzberg, A. Kluemper, A. S.-article in preparation

3D Ising model, fermionic string representation. Polyakov-1979

Sign factor exhibits Pauli principle for strings. Fermionic string

Z=∑x⃗(ξ )eα Area

Φ[ x⃗ (ξ)]

Sign-factor in 3D Ising model Kavalov, Sedrakyan—Preprint ERPHI-695(10)-1984 Nucl. Phys. B285[FS19](1987)

S=∑n,μ

ψ nΩn,n+μψ n+μ tn,n+μ

Ωn,n+μ∈SU (2 )

Φ[ x⃗ (ξ) ,ψ]=∫ dψe−S=∏CΩn ,n+μ=(−1)N N= # of fluxes

Μatrix Model for Random Network

Action

)2( NSUMconsiderj

''';'

;

'';''

;

j

j

i

i

ji

ji RRmatrixR

][][][;'';''

;'';'' MMMRMn

n

n

n

jiji

jijiTrTrMS

eMS

DMZ

FunctionPartition][

J. Ambjorn, A. Sedrakyan - Nucl.Phys.B874, 877(2013)J. Ambjorn, Sh.Khachatryan, A.Sedrakyan - PRD92, 026002(2015)

M=U mU+.

Consider adjoined representation

τa∈su((2 s+1)N )

Action become

S [m ,Λ]=mbΛb

aRacΛc

umu−∑aV (ma)

Λ=Tr [ τaU τbU ]

Action is quadratic. Gaussian integration may be used if is parametrized byfree coordinates

Λba

DM=DU dm=DΛ dm

Λ∈F (1,1,...1)=U ((2 s+1)N )

U (1)(2 s+1)N ≃∏1

(2 s+1)NCPi

Flag Manifold

DΛ=∏k=1

(2 s+1 )N−1D [CPk

]=∏k=1

(2 s+1)N−1D [

Sk

S1 ]

.≃∏k=1

(2 s+1 )N−1D [ Sk

]

¿

D [ Sk]=δ(∑1

k|z i|

2−1)∏ dz i

Z=∫∏ dmad λa e−W (ma , λa )

Summary

Random network problems and strings in real space have the same physical background.They contain matter and gauge fields interacting with gravity

Random network