Spectral Properties of the Renormalization Group › ~meiyin › RG.pdfSpectral Properties of the Renormalization Group Mei Yin Department of Mathematics University of Arizona July

AcknowledgmentsIntroduction

Decimation TransformationMajority Rule Transformation

Spectral Properties of the Renormalization Group

Mei Yin

Department of MathematicsUniversity of Arizona

July 7, 2009

Mei Yin Spectral Properties of the Renormalization Group



The renormalization group (RG) approach is largely responsible forthe considerable success which has been achieved in developing aquantitative theory of phase transitions.Physical properties emerge from considering the spectral propertiesof the linearization of the RG map at a fixed point. We considerreal-space RG for classical lattice systems. The linearization actson an infinite-dimensional Banach space of interactions. At atrivial fixed point (zero interaction), the spectral properties of theRG linearization can be worked out explicitly, without anyapproximation.




In this talk, I will talk about results obtained for the RG mapscorresponding to decimation and majority rule. They indicatespectrum of an unusual kind: dense point spectrum for which theadjoint operators have no point spectrum at all, but only residualspectrum. This may serve as a lesson in what one might expect inmore general situations.Part of the talk is inspired by R.B. Israel’s paper: Banach algebrasand Kadanoff transformations. In particular, some of the theoremsto be presented here were stated in this paper without proof.




I got this work started when I was working with my advisorProf. William Faris in the 2008 program in Combinatorics andStatistical Mechanics at the Isaac Newton Institute in Cambridge,organized by Prof. Alan Sokal.I greatly benefited from many useful discussions with my advisorand I especially thank him for his patience with me.I learnt a lot about math physics from talking to the professors inmy department, especially Prof. Tom Kennedy. I gained an insightinto renormalization group from his carefully prepared mathphysics lectures.




We need some basic setup first.

Our original and image configuration spaces are Zd , where dis any positive integer for decimation and d = 1 for majorityrule.

σ : Zd → {−1, 1}, the spin function for Ising-type systems.

σx , commonly referred to as the spin at site x , where x ∈ Zd

is a d-dimensional vector.

σX =∏

x∈X σx , and we require X to be nonempty.

J and J ′ are defined on nonempty finite subsets of Zd andtake values in the complex numbers for the convenience ofspectral theory.

A constant pure magnetic field is such that J(X ) = 0 exceptfor one-point sets {x}, where J({x}) = m, a constant.









σX =∏












σX =∏












σX =∏












σX =∏












σX =∏












σX =∏







We consider RG transformations which are formally defined by,

e∑

Y J′(Y )σ′Y =

∑σ

T (σ, σ′)e∑

X J(X )σX (1)

where T is a probability kernel from the original configurationspace to the image configuration space, with

∑σ′ T (σ, σ′) = 1 for

every σ.




We work in two types of Banach spaces, B (with pair space B∗)and Br (with pair space B∗r ).B (Br ) is technically not the dual space of B∗ (B∗r ) and B∗ (B∗r ) istechnically not the dual space of B (Br ). B and B∗ (Br and B∗r )are paired in the sense that each one is part of the dual space ofanother, or in other words, each one consists of continuous linearfunctions defined on another.




The norms are respectively defined by,

||J|| = supx∈L

∑x∈X

|J(X )| (2)

||J||∗ =∑x∈L

supx∈X

1

|X ||J(X )| (3)

||J||r = supx∈L

∑x∈X

|J(X )|er |X | (4)

||J||∗r =∑x∈L

supx∈X

1

|X ||J(X )|e−r |X | (5)

where the constant r > 0 and |X | is the cardinality of the set X .It is not hard to verify that these are suitable definitions.




Note: Here the ||J||r and the ||J||∗r norms are somewhat differentfrom the corresponding norms defined in Israel’s paper. Withoutmuch difficulty, we can verify that same results to be presentedhere would also hold if we adopt his definitions.




The Banach Space BThe Banach Space BrThe Banach Space B∗The Banach Space B∗rFurther Thought

For decimation transformation in a d-dimensional space withblocking factor bd , the probability kernel T (σ, σ′) is defined by

T (σ, σ′) =

{1 if σ′x = σbdx for all x ;0 otherwise.

where the slightly strange notation bdx denotes the d-dimensionalvector obtained by multiplying every component of thed-dimensional vector x by b.





At the trivial fixed point (zero interaction), with the help of Fourier

theory on the group {−1, 1}Zd, we can work out the explicit

infinite volume expression of the linearization of the RG map (‘L’stands for linearization),

∂J ′(X )

∂J(Y )= δ(Y , bdX ) (6)

LJ(X ) = J(bdX ) (7)

where bdX = ∪x∈X{bdx}.





By the usual correspondence between adjoint operators,∑X

J1(X )LJ2(X ) =∑Y

J2(Y )L∗J1(Y )

The explicit expression of the adjoint operator L∗ is also obtained,

L∗J(X ) =

{J( 1

bd X ) if X is a block;

0 otherwise.(8)

where X is a block means that X = bdY for some set Y , and wedefine 1

bd X = Y .





Theorem (Israel)

In the Banach Space B, the spectrum of L is all point spectrum,|λ| ≤ 1.

The proof of this theorem will follow from several propositions.





Theorem (Israel)

In the Banach Space B, the spectrum of L is all point spectrum,|λ| ≤ 1.






Proposition

||L|| = 1.

Proof.

We check that for each fixed x ∈ L,∑

x∈X |LJ(X )| ≤ ||J||, whichwould imply ||L|| ≤ 1.Recall that for the eigenvector J, LJ(X ) = J(bdX ), so naturally,

∑x∈X

|LJ(X )| =∑x∈X

|J(bdX )| =∑

bdx∈bdX

|J(bdX )|

≤∑

bdx∈X

|J(X )| ≤ ||J||

Realizing that a constant pure magnetic field is an eigenvectorwith eigenvalue 1, we conclude ||L|| = 1.





Corollary

Every eigenvalue λ of L satisfies |λ| ≤ 1.





Proposition

Every |λ| ≤ 1 is an eigenvalue.

Proof.

For a generic λ, we display one eigenvector here. In fact, withsome further thought, it is not hard to show that there areinfinitely many eigenvectors for each λ.The eigenvector J is defined by,

J({(bn, 0, ..., 0)}) = λnJ({(1, 0, ..., 0)}) = λn

and for all the other subsets X , J(X ) is set to zero.





Theorem (Israel)

In the Banach Space Br , the spectrum of L is all point spectrum,|λ| ≤ 1.

The proof of this theorem is quite analogous to the previous one,with minor revisions.





Theorem (Israel)

In the Banach Space Br , the spectrum of L is all point spectrum,|λ| ≤ 1.






Theorem

In the Banach Space B∗, the spectrum of L∗ is all residualspectrum, |λ| ≤ 1.






Theorem

In the Banach Space B∗, the spectrum of L∗ is all residualspectrum, |λ| ≤ 1.






Proposition

||L∗|| = 1.

Proof.

Standard analysis fact for Banach spaces, as ||L|| = 1.





Proposition

For every eigenvalue λ 6= 0, there is no nontrivial eigenvector.

Proof.

Fix an arbitrary finite subset X of the infinite lattice, after a finitenumber of iterations of L∗ (say n times), X will not take on theshape of a block.Thus (L∗)n+1J(X ) = 0 = λn+1J(X ), which implies J(X ) = 0.





Proposition

For λ = 0, there is no nontrivial eigenvector.

Proof.

Suppose the nontrivial eigenvector J(X ) = m 6= 0 for some finitesubset X , then the crucial fact that we can always find Y , withL∗J(Y ) = J(X ) will do the job.L∗J(Y ) = λJ(Y ) = 0, we reach a contradiction.





Corollary

The point spectrum of L∗ is empty.

In order to verify our theorem, the only thing left to show now isthat for |λ| ≤ 1, Range(λI − L∗) 6= B∗.We divide into two cases: |λ| < 1 and |λ| = 1.





Corollary

The point spectrum of L∗ is empty.

In order to verify our theorem, the only thing left to show now isthat for |λ| ≤ 1, Range(λI − L∗) 6= B∗.We divide into two cases: |λ| < 1 and |λ| = 1.





• |λ| < 1.Define J({(bn, 0, ..., 0)}) = λ

n, and J(X ) = 0 for all other subsets

X .Recall the norm definition in B∗, ||J||∗ =

∑x∈L supx∈X

1|X | |J(X )|.

In our current case,

||J||∗ =∞∑

n=0

|λ|n =1

1− |λ|< ∞

This says that J lies in B∗.However, J can not be approximated by any J ′ in Range(λI − L∗).





• |λ| < 1.Define J({(bn, 0, ..., 0)}) = λ



∑x∈L supx∈X

1|X | |J(X )|.


||J||∗ =∞∑

n=0

|λ|n =1

1− |λ|< ∞






• |λ| < 1.Define J({(bn, 0, ..., 0)}) = λ



∑x∈L supx∈X

1|X | |J(X )|.


||J||∗ =∞∑

n=0

|λ|n =1

1− |λ|< ∞






To see this, note that

J ′(X ) =

{λK (X )− K ( 1

bd X ), if X is a block

λK (X ), otherwise

for some K that lies in B∗.Also

||J − J ′||∗ =∑x∈L

supx∈X

1

|X ||J(X )− J ′(X )|

≥∑

x=(bn,0,...,0)

supx∈X

1

|X ||J(X )− J ′(X )|

≥∞∑

n=0

|J({(bn, 0, ..., 0)})− J ′({(bn, 0, ..., 0)})|





However,

∞∑n=0

J({(bn, 0, ..., 0)})J ′({(bn, 0, ..., 0)}) =∞∑

n=0

λnJ ′({(bn, 0, ..., 0)})

= λK ({(1, 0, ..., 0)}) + λ(λK ({(bn, 0, ..., 0)})− K ({(1, 0, ..., 0)})) + · · ·

= 0

∞∑n=0

|J({(bn, 0, ..., 0)})− J ′({(bn, 0, ..., 0)})|

≥√∑∞

n=0 |J({(bn, 0, ..., 0)})− J ′({(bn, 0, ..., 0)})|2

≥

√√√√ ∞∑n=0

|λn|2 =

√1

1− |λ|2Mei Yin Spectral Properties of the Renormalization Group




• |λ| = 1.Define J({(1, 0, ..., 0)}) = 1, and J(X ) = 0 for all other subsets X .Suppose there exists a J ′ approximating J such that

1

2≥ ||J − J ′||∗ =

∑x∈L

supx∈X

1

|X ||J(X )− J ′(X )|

≥ |λK ({(1, 0, ..., 0)})− 1|+ |λK ({(bn, 0, ..., 0)})− K ({(1, 0, ..., 0)})|+ · · ·

where K is defined as before.





• |λ| = 1.Define J({(1, 0, ..., 0)}) = 1, and J(X ) = 0 for all other subsets X .Suppose there exists a J ′ approximating J such that

1

2≥ ||J − J ′||∗ =

∑x∈L

supx∈X

1

|X ||J(X )− J ′(X )|

≥ |λK ({(1, 0, ..., 0)})− 1|+ |λK ({(bn, 0, ..., 0)})− K ({(1, 0, ..., 0)})|+ · · ·

where K is defined as before.





Then, as |λ| = 1, we would have, for any n ≥ 0.

|λn+1K ({(bn, 0, ..., 0)})−1| ≤ 1

2|K ({(bn, 0, ..., 0)})| ≥ 1

2

But then,

||K ||∗ =∑x∈L

supx∈X

1

|X ||K (X )| ≥

∞∑n=0

|K ({(bn, 0, ..., 0)})| = ∞

contradiction.





Theorem

In the Banach Space B∗r , the spectrum of L∗ is all residualspectrum, |λ| ≤ 1.






Theorem

In the Banach Space B∗r , the spectrum of L∗ is all residualspectrum, |λ| ≤ 1.






These theorems might not come as a surprise once we realize thesimilarity between L/L∗ and left translation/right translation: Lacts like left translation and L∗ acts like right translation onsequences (X , bdX , ...) for all possible subsets X .Moreover, ignoring multiplicity of the eigenvalues, the spectrum ofL is the same as that of left translation in l∞, and the spectrum ofL∗ is the same as that of right translation in l1, which might be aresult of the norms in our Banach spaces being a combination ofl∞ and l1 norms.




The Banach Space BThe Banach Space BrThe Banach Space B∗The Banach Space B∗r

For majority rule transformation in one-dimension with blockingfactor 3, the probability kernel T (σ, σ′) is defined by

T (σ, σ′) =

{1 if σ′x = 1

2(σ3x−1 + σ3x + σ3x+1 − σ3x−1σ3xσ3x+1), ∀x ;0 otherwise.

At the trivial fixed point (zero interaction), with the help of Fouriertheory on {−1, 1}Z, we can work out the explicit infinite volumeexpression of the linearization of the RG map (‘L’ stands forlinearization),

∂J ′(X )

∂J(Y )=

{ ∏x∈X χ(Y ∩ Rx) if Y ⊂ RX ;

0 otherwise.(9)

LJ(X ) =∑

Y :Y⊂RX

J(Y )∏x∈X

χ(Y ∩ Rx) (10)





where χ is defined by,

χ(X ) =

12 if |X | = 1;−1

2 if |X | = 3;0 otherwise.

Rx denotes the set {3x − 1, 3x , 3x + 1} and RX denotes the set∪x∈X{3x − 1, 3x , 3x + 1}. (‘R’ stands for rescaling)





By the usual correspondence between adjoint operators,∑X

J1(X )LJ2(X ) =∑Y

J2(Y )L∗J1(Y )

The explicit expression of the adjoint operator L∗ is also obtained,

L∗J(X ) =

{± 1

2n J(⋃

n{m}) if X =⋃

n Wm;0 otherwise.

(11)

where Wm is either one of the following four sets,{3m − 1}, {3m}, {3m + 1} or {3m − 1, 3m, 3m + 1} (m integer).⋃

n Wm is a finite union of n of these disjoint Wm’s. And “+” istaken when |X | − n is divisible by 4; and “−” is taken otherwise.





Theorem

In the Banach Space B, the spectrum of L is all point spectrum,|λ| ≤ 3

2 .






Theorem

In the Banach Space B, the spectrum of L is all point spectrum,|λ| ≤ 3

2 .






Proposition

||L|| = 32 .

Proof.

We check that for each fixed x ∈ L,∑

x∈X |LJ(X )| ≤ 32 ||J||, which

would imply ||L|| ≤ 32 .

As x ∈ X , LJ(X ) is a linear combination of J(Y )’s, |Y ∩Rx | beingeither 1 or 3. Due to the definition of χ, the coefficients of J(Y )’sare bounded above by 1

2 .We group the sum LJ(X ) into three parts:

Y ∩ Rx = {3x − 1}Y ∩ Rx = {3x} or Y ∩ Rx = {3x − 1, 3x , 3x + 1}Y ∩ Rx = {3x + 1}





Proof.

Ignoring the coefficients of J(Y )’s, these three parts are less thanor equal to

∑3x−1∈X |J(X )|,

∑3x∈X |J(X )|, and

∑3x+1∈X |J(X )|

respectively, therefore bounded above by ||J|| by definition.Realizing that a constant pure magnetic field is an eigenvectorwith eigenvalue 3

2 , we conclude ||L|| = 32 .





Corollary

Every eigenvalue λ of L satisfies |λ| ≤ 32 .





Proposition

Every |λ| ≤ 32 is an eigenvalue.

Proof.

For a generic λ, we display one eigenvector here. In fact, withsome further thought, it is not hard to show that there areinfinitely many eigenvectors for each λ.The eigenvector J is defined by,

J({0}) = J({1})

J({−1}) = 2(λ− 1)J({0})

J({3n − 1}) = J({3n}) = J({3n + 1}) =2

3λJ({n}) for n 6= 0

and for all the other subsets X , J(X ) is set to zero.





Theorem

In the Banach Space Br , the spectrum of L is all point spectrum,|λ| ≤ 3

2 .






Theorem

In the Banach Space Br , the spectrum of L is all point spectrum,|λ| ≤ 3

2 .






Theorem

In the Banach Space B∗, the point spectrum of L∗ is empty.






Theorem

In the Banach Space B∗, the point spectrum of L∗ is empty.






Proposition

For every eigenvalue λ 6= 0 and λ 6= 12 , there is no nontrivial

eigenvector.

Proof.

Fix an arbitrary finite subset X of the infinite lattice. For λ 6= 0,J(X ) is either zero or a nonzero constant multiple of J({0}) as aresult of the action of L∗.For λ 6= 1

2 , λJ({0}) = L∗J({0}) = 12J({0}), which implies that

J({0}) = 0.





Proposition

For λ = 0, there is no nontrivial eigenvector.

Proof.

Suppose the nontrivial eigenvector J(X ) = m 6= 0 for some finitesubset X , then the crucial fact that we can always find Y , withL∗J(Y ) a nonzero constant multiple of J(X ) will do the job.L∗J(Y ) = λJ(Y ) = 0, we reach a contradiction.





Proposition

For λ = 12 , every nontrivial eigenvector has norm infinity.

Proof.

We must have J({0}) = m 6= 0 in order for J to be nontrivial.As 1

2J({−1}) = L∗J({−1}) = 12J({0}), we see that J({−1}) = m

also.Following similar fashion, J({n}) = m for arbitrary n.But then,

||J||∗ =∑x∈L

supx∈X

1

|X ||J(X )| = ∞

contradiction.





Theorem

In the Banach Space B∗r , the point spectrum of L∗ is empty.

The proof of this theorem is quite similar to the previous one, withminor revisions.





Theorem

In the Banach Space B∗r , the point spectrum of L∗ is empty.

The proof of this theorem is quite similar to the previous one, withminor revisions.





Thank You!


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Spectral Properties of the Renormalization Group › ~meiyin › RG.pdfSpectral Properties of the Renormalization Group Mei Yin Department of Mathematics University of Arizona July