The instanton vacuum of generalized models

The instanton vacuum of generalized models

I.S. Burmistrov

and

A.M.M. PruiskenInstitute for Theoretical Physics, University of Amsterdam

cond-mat/0407776 accepted in Annals of Physics

L.D. Landau Institute for Theoretical Physics

The instanton vacuum of generalized modelsIn

trodu

ction-1

I. Burmistrov and A.M.M. Pruisken

Nonlinear sigma model with topological term

is defined on coset

Introduction Lan

dau

ITPPruisken ‘84

Wegner ‘79

Efetov, Larkin, Khmelnitzkii ‘80

Dynamical variable – unitary matrix field

mean-field longitudinal DC conductivity

mean-field Hall DC conductivity

The instanton vacuum of generalized modelsL

and

au IT

PI. Burmistrov and A.M.M. Pruisken

Introduction

Introd

uction

-2

2D disordered electron gas in magnetic fieldO(3) model

model

n=m=0

n=m=1

n=1, m=N-1

It contains

The instanton vacuum of generalized modelsIn

trodu

ction-3

I. Burmistrov and A.M.M. Pruisken

Independently on (m, n)

Introduction Lan

dau

ITP

1. Massless chiral edge exictations

2. Quantum Hall effect, i.e. robust quantization of

3. Divergent correlation length at = k+1/2

Dependent on (m, n)

1. Order of plateau-plateau transitions

2. Critical exponents for plateau-plateau transitions


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Mass terms (linear and bilinear in Q operators)

where

with

Introd

uction

-4

Introduction

Wegner ’79Pruisken ’85


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Non-perturbative renormalization group equations

where

Euler constant

Results

Resu

lts-1

Perturbative resutls byE. Brezin, S. Hikami and

J. Zinn-Justin ‘80


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Nature of the plateau-plateau transition for different (m,n)

Results

FP at zero

FP at nonzero

Resu

lts-2

O(3)


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Nature of the plateau-plateau transition for different (m,n)

Results

Large m, n

Small m, n

Resu

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Quantum Hall Effect for m,n < 1Renormalization group flow diagram

Results

Resu

lts-4

Khmelnitzkii ’83Pruisken ’83


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QHE in 2DEG (n=m=0)

Fixed point at

Results

Resu

lts-5

Non-perturbative renormalization group equations

Pruisken ’87(4 times larger coefficient)


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Linear environment of FP

Divergent localization length

Critical exponents

relevant

irrelevant

Resu

lts-6

Results Plateau-plateau transition

Pruisken ‘88


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Inverse participation ratio (IPR)

It can be related with antisymmetric operator as

Critical exponent for IPR

Extended -- zero

Localized – finite

Resu

lts-7

Results Multifractality

=2

Wegner ‘79


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Generalized inverse participation ratio

It can be related with higher order antisymmetric operators and written as

Critical exponent

All exponents are different!

Results

Resu

lts-8

Multifractality


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New variable

Singularity function

Maximum at

The result (from NPRGEqs) is parabola

Legendre transform!

Results Multifractality

Resu

lts-9


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Results

Resu

lts-10

Comparison with numerics


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Localization length exponent

Resu

lts-11

Quantum Hall effect for n=m >0Results


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Irrelevant exponent

Resu

lts-12



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Anomalous dimensions

Resu

lts-13



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NL M action

Mass terms

where

Derivation

-1

Derivation Action


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Topological charge

If at the boundary

then C[Q] is integer valued

Why should it be?

Derivation

-2

Boundary conditionsDerivation


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Change of variables

where at the boundary

Derivation

-3



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split

Derivation

-4



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where

Effective action for the edge

we can write

where physical observables

Derivation

-5

Effective action for the edgeDerivation

Background fields


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In the case of finite localization length

then

No renormalization of k!Skoric, Pruisken, Baranov ‘98

Robust quantization of Hall conductance

Derivation

-6

Effective action for the edgeDerivation


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Derivation

-7

Bulk actionDerivation


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conductances

masses

Derivation

-8

Physical observablesDerivation

Pruisken ’87


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Derivation

-9


Specific choice of t Generators of U(m+n)

Effective action


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Derivation

-10


Generators

Fiertz identity


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Instanton solution

Action on the instanton solution

Finite

Derivation

-11

InstantonsDerivation

O(3) instantonBelavin Polyakov ‘75


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For then

where

Logarithmic divergences in mass terms on the instanton solutions!?

Derivation

-12

InstantonsDerivation


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where

Stereographic projection

Derivation

-13

Quantum fluctuationsDerivation


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Spectrum

Eigenfunctions

Jacobi polynomials

Derivation

-14

Quantum fluctuationsDerivation

Zero modes

sizepositionrotations


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Spatially varying masses

Derivation

-15

Mass termsDerivation

Transformation preserves logarithms!

Linear terms


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Derivation

-16

Mass termsDerivation


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Set of parameters

Such that

Derivation

-17

Pauli-Villars regularizationDerivation

Replacement

‘t Hooft ‘76


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Derivation

-18

Thermodynamic potentialDerivation

where

Quantities

are exactly the same as one can obtain in perturbative renormalization !!


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Transformation from curved space to flat space

Derivation

-19

TransformationDerivation


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Derivation

-20


where

How is related with ?


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Derivation

-21


Local counterterms (‘t Hooft)

The action becomes


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Derivation

-22

TransformationsDerivation

}Local counterterms

(‘t Hooft)

Renormalization by fluctuations


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Derivation

-23


where

Prescription

Similarly


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Non-perturbative results for conductivities

Derivation

-24

ConductivitiesDerivation

Hence (there is no dependence on !)


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Derivation

-25


Similarly

where


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Derivation

-26

MassesDerivation

Non-perturbative results for masses ( <0)

where

“Magnetization”


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Derivation

-27

MassesDerivation

Perturbative results only are needed

Hence ( <0)

and


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Derivation

-28

MassesDerivation

Non-perturbative results for masses ( >0)

then


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Derivation

-29

MassesDerivation

Non-perturbative results for masses

Hence


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Con

clusion

s-1

Conclusions

Non-perturbative (one instanton) results for beta and gamma (anomalous dimension) functions in generalized models

QHE in free electron gas (m=n=0) is not the special case of replica limit

Instanton analysis provides estimation for critical exponents for plateau-plateau transitions

The method lays the foundation for a non-perturbative analysis of the electron gas that includes the effects of electron-electron interaction

Instanton analysis produces the main features of the QHE

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The instanton vacuum of generalized models