The instanton vacuum of generalized models
-
Upload
gavin-johnson -
Category
Documents
-
view
34 -
download
4
description
Transcript of 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
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Mass terms (linear and bilinear in Q operators)
where
with
Introd
uction
-4
Introduction
Wegner ’79Pruisken ’85
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Non-perturbative renormalization group equations
where
Euler constant
Results
Resu
lts-1
Perturbative resutls byE. Brezin, S. Hikami and
J. Zinn-Justin ‘80
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Nature of the plateau-plateau transition for different (m,n)
Results
FP at zero
FP at nonzero
Resu
lts-2
O(3)
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Nature of the plateau-plateau transition for different (m,n)
Results
Large m, n
Small m, n
Resu
lts-3
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Quantum Hall Effect for m,n < 1Renormalization group flow diagram
Results
Resu
lts-4
Khmelnitzkii ’83Pruisken ’83
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
QHE in 2DEG (n=m=0)
Fixed point at
Results
Resu
lts-5
Non-perturbative renormalization group equations
Pruisken ’87(4 times larger coefficient)
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Linear environment of FP
Divergent localization length
Critical exponents
relevant
irrelevant
Resu
lts-6
Results Plateau-plateau transition
Pruisken ‘88
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
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
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
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
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
New variable
Singularity function
Maximum at
The result (from NPRGEqs) is parabola
Legendre transform!
Results Multifractality
Resu
lts-9
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Results
Resu
lts-10
Comparison with numerics
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Localization length exponent
Resu
lts-11
Quantum Hall effect for n=m >0Results
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Irrelevant exponent
Resu
lts-12
Quantum Hall effect for n=m >0Results
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Anomalous dimensions
Resu
lts-13
Quantum Hall effect for n=m >0Results
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
NL M action
Mass terms
where
Derivation
-1
Derivation Action
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Topological charge
If at the boundary
then C[Q] is integer valued
Why should it be?
Derivation
-2
Boundary conditionsDerivation
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Change of variables
where at the boundary
Derivation
-3
Boundary conditionsDerivation
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
split
Derivation
-4
Boundary conditionsDerivation
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
where
Effective action for the edge
we can write
where physical observables
Derivation
-5
Effective action for the edgeDerivation
Background fields
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
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
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-7
Bulk actionDerivation
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
conductances
masses
Derivation
-8
Physical observablesDerivation
Pruisken ’87
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-9
Physical observablesDerivation
Specific choice of t Generators of U(m+n)
Effective action
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-10
Physical observablesDerivation
Generators
Fiertz identity
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Instanton solution
Action on the instanton solution
Finite
Derivation
-11
InstantonsDerivation
O(3) instantonBelavin Polyakov ‘75
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
For then
where
Logarithmic divergences in mass terms on the instanton solutions!?
Derivation
-12
InstantonsDerivation
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
where
Stereographic projection
Derivation
-13
Quantum fluctuationsDerivation
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Spectrum
Eigenfunctions
Jacobi polynomials
Derivation
-14
Quantum fluctuationsDerivation
Zero modes
sizepositionrotations
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Spatially varying masses
Derivation
-15
Mass termsDerivation
Transformation preserves logarithms!
Linear terms
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-16
Mass termsDerivation
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Set of parameters
Such that
Derivation
-17
Pauli-Villars regularizationDerivation
Replacement
‘t Hooft ‘76
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-18
Thermodynamic potentialDerivation
where
Quantities
are exactly the same as one can obtain in perturbative renormalization !!
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Transformation from curved space to flat space
Derivation
-19
TransformationDerivation
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-20
Physical observablesDerivation
where
How is related with ?
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-21
TransformationDerivation
Local counterterms (‘t Hooft)
The action becomes
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-22
TransformationsDerivation
}Local counterterms
(‘t Hooft)
Renormalization by fluctuations
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-23
TransformationDerivation
where
Prescription
Similarly
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Non-perturbative results for conductivities
Derivation
-24
ConductivitiesDerivation
Hence (there is no dependence on !)
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-25
TransformationDerivation
Similarly
where
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-26
MassesDerivation
Non-perturbative results for masses ( <0)
where
“Magnetization”
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-27
MassesDerivation
Perturbative results only are needed
Hence ( <0)
and
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-28
MassesDerivation
Non-perturbative results for masses ( >0)
then
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
Derivation
-29
MassesDerivation
Non-perturbative results for masses
Hence
The instanton vacuum of generalized modelsL
and
au IT
PI. Burmistrov and A.M.M. Pruisken
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