Fixed points & renormalization flow diagrams

Taking the finite scaling parameter to be infinitesimally small, we can define a flow equation for the couplings. It is customary to “invert” the exponential map and defineℓ“lnc. Then (2.119) can be written as

g¹pcq “ lim

nÑ8 ℓÑ0

”

Rcrgsın

« lim

nÑ8 ℓÑ0

r1`nRℓs rgs “g`Rpgqdℓ ñ dg

dℓ “Rpgq (2.120) which defines the change in the coupling under a smooth zooming out process. We can compare this expression to the Callan-Symanzik equation (2.142) arising from correlation function renor-malization³² and identify the flow equation (2.120) as the generalized (coupling dependent) β-function.

i

Find all scale invariant theories with the additional symmetries imposed by the given uni-versality class,i.e. determine all possible scale covariant³⁴ operators and their scaling di-mension.

ii

Analyze the local behavior of all scale invariant effective theories by deformations close to their fixed points. This amounts to the analog of higher loop calculation in perturbative QFT, which can be viewed as quantum corrections to the naïve or engineering dimension of the scale covariant operators. Map out the local RG flows by solving the linearized RG flow equations near the fixed points.

iii

Connect all fixed points by extending the local RG flows to integrated global RG flows.

The last point is usually intractable and only possible for weak coupling fixed points. This is the arena where the Gauge/Gravity Duality will step in and (may) provide invaluable insight, add more strongly coupled scale invariant theories and new calculational tools for tackling strongly coupled RG flows (see Chapter3and in particular Section 3.4). The stability analysis close to the fixed pointsg^‹follows the stability analysis of first order differential equations. Linearizing the generalizedβ-function

Rpgq “Rpg^‹q `∇Rpgqˇˇ

g“g^‹pg´g^‹q ñ R^a_b “ BRpg^aq Bg^b

ˇˇ ˇˇ

g“g^‹

, (2.122)

the stability is controlled by the eigensystem of the in general non-symmetric linear mapping R^a_b. In RG parlance the components of the left-eigenvector of R^a_b are (dangerously) called scaling fields and obey the simple flow equation with the (trivial) exponential solution

dv^a

dℓ “y^av^a ñ v^a “Ce^ℓya. (2.123)

Note also that the corresponding operator scales with ∆_O “ D´y^a. The solution of (2.123) allows for three different types of scaling fields:

• Repy^aq ą0(relevant):

For increasingℓ(zooming out) the system is driven away from the fixed point and thus the corresponding scaling field is called relevant. Starting with a small relevant deformation, the system flows to a different fixed point with a possibly different effective theory/univer-sality class.

• y^a “0(marginal):

Invariant scaling fields under the RG flow (2.123) are called marginal and do not contribute to the flow. However, the second order derivatives along marginal directions allow for a more subtle classification, i.e. positive values are marginally relevant whereas negative values are marginally irrelevant. In any case on approaching the fixed point the flow of a marginal field will decrease until it is stationary at the fixed point.

• Repy^aq ă0(irrelevant):

For increasingℓ the system is attracted to the fixed point and the corresponding scaling field is termed irrelevant. Under a small irrelevant deformation close to the fixed point, the theory will flow back to the fixed point.³⁵

34Operators that transform under the scale transformation,i.e. respect the scaling symmetry of the theory.

35There are also dangerously irrelevant scaling fields inducing singularities in the free energy which usually leads to violations of hyperscaling.

Figure 2.4.

A typical yet intricate renormalization flow diagram connecting stable fixed points (‚) to unstable (‚) and mixed fixed points (˝), where the flow direction is indicated by the arrows. The mixed or critical fixed point defines a critical surface ( ) spanned by the irrelevant directions. Tuning the cou-pling of the physical system is represented by the blue dashed line ( ) and the crit-ical point (‚) is given by the intersection with the critical surface. Crossing the crit-ical surface allows the system to flow to a different stable fixed point (stable phase of matter), whereas right at the intersection point the system flows to the critical fixed point, describing a phase transition with universal features.

The local behavior of the RG flow close to the fixed point leads to the following stability classifi-cation:

• Stable fixed points:

A stable fixed point possesses no relevant scaling fields. Thus, small microscopic differences in the initial theory cannot deform the effective theory under rescaling and so we can identify stable fixed points with stable states of matter.

• Unstable fixed points:

Unstable fixed points possess only relevant scaling fields. In principle, these fixed points are not accessible by any RG flow and are unphysical yet important artifacts that control the global RG flow behavior.

• Mixed fixed points:

A mixed fixed point has relevant and irrelevant directions. The irrelevant scaling fields define a tangent space denoted as critical surface. Any set of coupling constants describing a point on the critical surface will be attracted to the mixed fixed point. A small deviation away from the critical surface will drive the system to other fixed points in the respective directions of the relevant scaling fields. In this sense, a mixed fixed point is related to a phase transition and the existence of such a fixed point implies the existence of a stable fixed point (where the relevant scaling fields will be irrelevant). However, the critical point of a second order phase transition is in general not identical to a mixed or critical fixed point, it is sufficient to cross the critical surface in the vicinity of the actual critical point, c.f. Figure2.4

In order to elucidate the power of the renormalization group method, let us apply it to the simple scalar field theory describing the vectorOpnquniversality class we already investigated in Section2.3.4 with mean field methods. Starting with the total effective theory involving all possible terms respecting theOpnqsymmetry

Srϕ;Js “

ż

d^Dx

«ÿ

j

g^pjq`

∇^jϕ˘2

`ÿ

k

g^pkqϕ^k´ϕJ ff

, (2.124)

we can easily determine the scaling behavior of the coupling and the field by a simple dimen-sional analysis. First the action must be dimensionless under rescaling by construction, so rSsc “ 0 and the scaling transformation (2.115) implies rxsc “ ´1. Furthermore, we fix the scaling dimension of the fieldϕby identifying the leading gradient term as the free field dimen-sionless contribution

rSrϕ;Jssc“0 ñ

„ż

d^Dxp∇ϕq²



c

“0 ^ rxsc“ ´1 ñ r∇sc “1, (2.125) to obtain

rϕsc “D´2

2 ñ ”

g^pjqı

c“2pj´1q, ” g^pkqı

c“k´ ˆk

2 ´1

˙

D, rJsc“ D`2

2 . (2.126) The corresponding RG flow equations are found by linearizing the scaling transformationg^aÝÑ c^rgasg^a. The trivial fixed point is g^‹ “ 0 which is called Gaussian fixed point for reasons that will become apparent in a moment. Let us determine the irrelevant operators of the Gaussian fixed point. The gradient terms are independent of the dimensionality and the leading free field gradient is the only marginal gradient term of the action (2.124) since

dg^pjq

dℓ “2p1´jqg^pjq ÝÑ 2p1´jq ă0, ją1 (2.127) whereas the power series terms in the fields yields

dg^pkq dℓ “k´

ˆk 2 ´1

˙

Dg^pkq ÝÑ ką4^

$&

%

4ăD 2k

k´2 ăDă4. (2.128) We immediately rediscover theGinzburg Criterionon page52and obtain the Gaussian/free field critical exponents. ForDą4all higher interaction terms are irrelevant and the system is flowing to the Gaussian fixed point described by the Gaussian model

Srϕ;Js “

ż

d^Dx

„b

2p∇ϕq²`r 2ϕ²



. (2.129)

Note that the ϕ² term is always relevant (the mass term introduces an energy scale breaking conformal invariance) as well as the source term which naturally must be a relevant deforma-tion as an external force preparing the system in a particular state. The RG flow diagram and generalizedβ-function are shown in Figure2.5. We see that the Gaussian fixed point inD ą4 dimensions is actually a mixed fixed point denoting a phase transition at the critical pointr^‹“0 andu^‹ “0. If we consider for example the Ising universality class then the coupling rcan be identified with the reduced temperature (2.109) and the stable r^‹ “ 8fixed point describes the paramagnetic (high-temperature) phase whereas forră0 we are flowing to ther^‹ “ ´8 fixed point describing the (low-temperature) ferromagnetic phase. However, forDă4we see in (2.128) that the interaction terms are becoming more and more relevant whereϕ⁴is the leading relevant term. The effective theory is therefore given by the Landauϕ⁴-theory (2.109)

Srϕ;Js “

ż

d^Dx

„b

2p∇ϕq²`r 2ϕ<sup>2</sup>` u

4!ϕ⁴´ϕJ



. (2.130)

Relevant deformations/perturbations in urenders the Gaussian fixed point unstable inD ă 4 and the system is flowing to a new fixed pointr^‹ ‰0andu^‹ ‰0, known as the Wilson-Fisher

r

0 u

0 β

g βprq

βpuq

r

0 u

0 β

g βprq

βpuq

Figure 2.5. In the left panel we show the renormalization group flow forD ą4. In this case we find a critical fixed point atr^‹“0andu^‹“0where the scaling fieldris rele-vant and the scaling fielduis irrelevant. The Gaussian fixed pointpr^‹“0, u^‹“0q describes a phase transition with mean field critical exponents between the sta-ble disordered high-temperature fixed point r Ñ 8 (paramagnetic phase) and the stable ordered low-temperature fixed point r Ñ ´8 (ferromagnetic phase).

In the right panel we show the situation in the case of D ă 4. Here the Gaus-sian fixed point destabilizes and a new critical fixed point emerges, the so-called Wilson-Fisher fixed point. Due to the importance of the fluctuations, the mean field solution is modified and forră0a flow to the disordered phase is possible. The insets describe the functional dependence of theβ-function in (2.131).

fixed point,c.f. Figure 2.5. For the Landau-Wilson effective theory the fast fluctuation modes and the slow fluctuating modes are not decoupled as in the Gaussian model, so the local RG flow equations cannot be determined from the scaling behavior alone. Corrections to the leading RG behavior can be computed by various schemese.g. momentum shell renormalization or the operator product expansion of the local operators close to the critical point. The renormalized effective action (at the one loop level regularized by dimensional regularization) leads to the following RG flow equations³⁶

dr

dℓ “2r`pn`2qC 6

u r`λ², du

dℓ “ p4´Dqu´pn`8qC 6

u² pr`λ²q²,

(2.131)

wherendenotes the number of components of theOpnqvector model andCdescribes the phase space factor of the momentum integration. The Wilson-Fisher fixed point is determined by

r^‹“ ´pn`1qC 12

u^‹ r^‹`λ<sup>2</sup>, u<sup>‹</sup>“6pr<sup>‹</sup>`λ²q²

pn`8qC p4´Dq,

(2.132)

36The details are outlined in almost all books about QFT and renormalization, a particular nice exposition can be found in [6,68]

which can be solved by an expansion inǫ“4´D³⁷ r^‹“ ´ n`2

2pn`8qǫ`Opǫ²q, u^‹“ 6

Cpn`8qǫ`Opǫ²q,

(2.133)

and expressing all length in units ofλ^´1 for convenience. The linearized RG equations for the scaling fields close to the Wilson-Fisher fixed point read

dv^

dℓ “2´n`2

n`8ǫ`Opǫ²q, dv^

dℓ “ ´ǫ`Opǫ²q, (2.134) wherev^a are linear combination ofranduaccording to the transformation matrix

¨

˝v¹ v²

˛

‚“

¨

˝

12pn`3ǫ`8q pn`2qp6ǫn`n`12ǫ`8q 1

0 1

˛

‚

¨

˝r u

˛

‚. (2.135)

We see that the Wilson-Fisher fixed point is the critical/mixed fixed point, where the new scaling fieldv¹ is now a non-trivial combination of the old couplingsrandu. Again the Wilson-Fisher fixed point resides atv¹ “v^‹“0and we may regard the couplingv¹as the reduced “tempera-ture” describing deviations away from the critical point. Note that in term of the “old” couplings, for sufficiently strong interactionsu, the phase transition from the ferromagnetic to paramag-netic phase might occur for r ă 0. Due to the strong fluctuations, the ferromagnetic phase is disfavored and thus even forră0the system will flow to the disordered fixed point atr^‹“ 8. The punchline of this example is to show that once we determine the right coupling in the vicin-ity of the critical fixed point, we automatically include the effects of fluctuations. Applying the scaling theory, which is briefly explained in the following paragraph, we are able to determine all critical exponent including all “anomalous” contributions.

Scaling & Critical Phenomena

To make contact with experiments we need to connect the RG flow results to the measurable correlation functions. The correlators must be independent of the purely theoretical renormal-ization steps which allows us to write

Cpx;vq “c^n∆ϕCpc^´1x;tc^yav^auq “ e^{n{2pD´2`ηqℓ}Cpc^´1x;tc^yav^auq, (2.136) where we used the general scaling behavior of n fieldsϕ(2.126). Note that x stands for all coordinates whereϕis inserted. Taking the most relevant scaling field to be the leading defor-mation,³⁸ we can rescale the correlation function by fixing the rescaling parameter of the most relevant scaling fieldv¹ “ v to be c^y1v¹ “ c^yv “ 1 since (2.136) must be true for arbitrarily chosenℓand hence

Cpx;v, v^aq “v^´2∆ϕ{yC´

v^1{yx,1, v^´ya{yv^a¯

vÑ0

“ v^´2∆ϕ{yF

´ v^1{yx

¯

“ξ^D´2`ηF ˆx

ξ

˙

(2.137)

37Note that dimensional regularization introduced a real valued dimensionality.

38Usually for most fixed points after proper diagonalization there is only one relevant scaling field left.

The functionF is an universal scaling function encoding the relevant contribution close to the critical point. Approaching the critical pointv Ñ 0 all other scaling field contributions vanish and the intrinsic characteristic length scale diverges. Therefore, we may assign to the relevant scaling field v the intrinsic characteristic length scale of correlation functions, the correlation length ξ. Comparing with Table 2.5 the relevant RG eigenvalue can be related to the critical exponentν via

ξ„v^´1{y „|T´Tc|^´ν ñ ν “ 1

y. (2.138)

This relation is commonly known as a hyperscaling relation. Strictly speaking this relation is only true forone relevant scaling field. If there are more competing relevant scaling fields the physical description is more complex and there are competing length/energy scales related to the behavior near the critical point. Indeed for thermodynamic or global transport properties at zero momentum (static equilibrium) the second most relevant deformation will play the role of the reference scale. In this case (2.137) is modified to

Cpv¹, v²q “`

v¹˘´np^∆ϕ{^y1q

F

„`

v¹˘´^y2{^y1

v²



. (2.139)

All other critical exponents can be related to the leading relevant scaling fields. The derivation follows the same steps as above, where we apply the RG method to the reduced free energy densityf “Srϕ;Js{L^dof the Landau-Ginzburg action (2.112). According to (2.126), the exter-nal source field and the reduced temperature are the most relevant terms, whereas all irrelevant scaling fields withy^aă0are denoted byv^a

fpr, J, v^aq “c^´Df´

c^yrr, c^yJJ, c^´|ya|v^a¯

“r^D{yrf´

˘1, r^´yJ{yrJ, r^|ya|{yrv^a¯

“r^D{yJfsing

´

r^´yJ{yrJ¯

, (2.140)

withfsingbeing the singular part of the reduced free energy density at the critical pointrÑ0.

Taking the definitions from Table2.5one can show that the following scaling relations hold true α“2´D

y_r, β“ D´y_J

y_r , γ“2yJ´D

y_r , δ“ y_J

D´y_J. (2.141) Note that Fisher’s scaling lawγ “ p2´ηqν can be directly derived by integrating (2.137) and comparing to the definition of the susceptibility in Table2.5. Using the reduced free energy, Fisher’s scaling law takes the formη “2`D´2yJ. Eliminating the eigenvalues of the scaling fields from (2.141) yields the thermodynamic scaling relations betweenα,β, γandδ. Expres-sions involving the critical exponents related to the correlation functions η and ν are called hyperscaling relations, which may be violated by dangerously irrelevant scaling fields impacting on correlation functions and Euclidean actions/free energies, differently. As an aside the scaling behavior of the correlation function is instrumental in deriving the Callan-Symanzik equations, usually defined not in terms of length scales but an energy scaleµ

d

dµCpx;gq “0 ñ ˆ

µ B

Bµ`βpgq B

Bg`nγpgq

˙

Cpx;gq “0. (2.142) where theβ andγ-function are defined as

βpgq “µBg

Bµ, γpgq “µBlna Z_ϕ

Bµ . (2.143)

Of course the physical content of (2.142) and the combination of the RG flow equations with the scaling functions is equivalent.

dx

dy dw

ÝÑ dv

zÑz² ÝÝÝÝÝÝÝÝÑ

Figure 2.6. Holomorphic complex functions describe conformal maps between their domain complex planez“x`iyand the image complex planefpzq “Refpzq`i Imfpzq “ w`iv. In the above figure the analytic functionfpzq “ z² is shown. As one can see the angles of the intersection of the coordinate functionsxandyare preserved, but not the area enclosed by contour lines. Moreover, the plots indicate that the transformation of an infinitesimal area elementdxdy Ñdwdvcan be composed of a scaling and a rotation as shown in the inset.

Im Dokument Gauge/Gravity duality (Seite 76-83)