Renormalization, and the renormalization group, on the lattice

. (16.101)

The Fourier transform of this (in the continuum limit) is G(k˜ ²)∝(

k²+ξ⁻²(T))−1

, (16.102)

as we learned in section 1.3.3. Comparing (16.100) with (16.87), it is clear that (16.100) is proportional to the propagator (or Green function) for the field s(x); (16.102) then shows that ξ⁻¹(T) is playing the role of a mass termm.

Now, near a critical point for a statistical system, correlations exist over very large scalesξcompared to the inter-atomic spacing a; in fact, at the critical point ξ(Tc) ∼ L, where L is the size of the system. In the quantum field theory, as indicated earlier, we may regarda⁻¹ as playing a role analogous to a momentum cut-off Λ, so the regimeξ>>ais equivalent to m<<Λ, as was indeed always our assumption. Thus studying a quantum field theory this way is analogous to studying a four-dimensional statistical system near a critical point. This shows rather clearly why it is not going to be easy: correlations over all scales will have to be included. At this point, we are naturally led to the consideration ofrenormalizationin the lattice formulation.

16.4 Renormalization, and the renormalization group, on the lattice

16.4.1 Introduction

In the continuum formulation which we have used elsewhere in this book, fluctuations over short distances of order Λ⁻¹ generally lead to divergences

16.4. Renormalization, and the renormalization group, on the lattice 173 in the limit Λ → ∞, which are controlled (in a renormalizable theory) by the procedure of renormalization. Such divergent fluctuations turn out, in fact, to affect a renormalizable theory only through the values of some of its parameters and, if these parameters are taken from experiment, all other quantities become finite, even as Λ → ∞. This latter assertion is not easy to prove, and indeed is quite surprising. However, this is by no means all there is to renormalization theory: we have seen the power of ‘renormal-ization group’ ideas in making testable predictions for QCD. Nevertheless, the methods of chapter 15 were rather formal, and the reader may well feel the need of a more physical picture of what is going on. Such a pic-ture was provided by Wilson (1971a) (see also Wilson and Kogut 1974), us-ing the ‘lattice + path integral’ approach. Another important advantage of this formalism is, therefore, precisely the way in which, thanks to Wil-son’s work, it provides access to a more intuitive way of understanding renor-malization theory. The aim of this section is to give a brief introduction to Wilson’s ideas, so as to illuminate the formal treatment of the previous chapter.

In the ‘lattice + path integral’ approach to quantum field theory, the degrees of freedom involved are the values of the field(s) at each lattice site, as we have seen. Quantum amplitudes are formed by integrating suitable quantities over all values of these degrees of freedom, as in (16.87) for example.

From this point of view, it should be possible to examine specifically how the

‘short distance’ or ‘high momentum’ degrees of freedom affect the result. In fact, the idea suggests itself that we might be able to perform explicitly the integration (or summation) over those degrees of freedom located near the cutoff Λ in momentum space, or separated by only a lattice site or two in co-ordinate space. If we can do this, the result may be compared with the theory as originally formulated, to see how this ‘integration over short-distance degrees of freedom’ affects the physical predictions of the theory. Having done this once, we can imagine doing it again – and indeed iterating the process, until eventually we arrive at some kind of ‘effective theory’ describing physics in terms of ‘long-distance’ degrees of freedom.

There are several aspects of such a programme which invite comment.

First, the process of ‘integrating out’ short-distance degrees of freedom will obviously reduce the number of effective degrees of freedom, which is neces-sarily very large in the case ξ >> a, as envisaged above. Thus it must be a step in the right direction. Secondly, the above sketch of the ‘integrating out’ procedure suggests that, at any given stage of the integration, we shall be considering the system as described by parameters (including masses and couplings) appropriate to that scale, which is of course strongly reminiscent of RGE ideas. And thirdly, we may perhaps anticipate that the result of this ‘integrating out’ will be not only to render the parameters of the theory scale-dependent, but also, in general, to introduce new kinds of effective in-teractions into the theory. We now consider some simple examples which we hope will illustrate these points.

16.4. Renormalization, and the renormalization group, on the lattice 175

FIGURE 16.5

A ‘coarsening’ transformation applied to the lattice portion shown in figure 16.4. The new (primed) spin variables are situated twice as far apart as the original (unprimed) ones.

where we have used (s^'₀s1)²= 1. It follows that

exp(Ks^'₀s1) = coshK(1 +s^'₀s1tanhK), (16.107) and similarly

exp(Ks1s^'₁) = coshK(1 +s1s^'₁tanhK). (16.108) Thus the sum overs1is

E

s1=±1

cosh²K(

1 +s^'₀s1tanhK+s1s^'₁tanhK+s^'₀s^'₁tanh²K)

. (16.109) Clearly, the terms linear ins1vanish after summing, and thes1 sum becomes just

2 cosh²K(

1 +s^'₀s^'₁tanh²K)

. (16.110)

Remarkably, (16.110) contains a new ‘nearest-neighbour’ interaction, s^'₀s^'₁, just like the original one in (16.103), but with analtered coupling (and a dif-ferent spin-independent piece). In fact, we can write (16.110) in the standard form

exp [g1(K) +K^'s^'₀s^'₁] (16.111) and then use (16.107) to set

tanhK^' = tanh²K (16.112)

and identify

g1(K) = ln

(2 cosh²K coshK^'

)

. (16.113)

Exactly the same steps can be followed through for the sum ons3in (16.105), and indeed for all the sums over the ‘integrated out’ spins. The upshot is that, apart from the accumulated spin-independent part, the new partition

| |

' ' '

' ' ' ' '

' ' '

'

174 16. Lattice Field Theory, and the Renormalization Group Revisited

FIGURE 16.4

A portion of the one-dimensional lattice of spins in the Ising model.

16.4.2 Two one-dimensional examples

Consider first a simple one-dimensional Ising model with Hamiltonian (16.99) and partition function

N−1

Z = E

exp K E

snsn+1 , (16.103)

{sn} n=0

where K = J/(kBT) > 0. In (16.103) all the s_n variables take the values

±1 and the ‘sum over {sn}’ means that all possible configurations of the N variables s0, s1, s2, . . . , sN−1 are to be included. The spin sn is located at the lattice site na, and we shall (implicitly) be assuming the periodic boundary condition sn = sN+n. Figure 16.4 shows a portion of the one-dimensional lattice with the spins on the sites, each site being separated by the lattice constant a. Thus, for the portion {sN−1, s0, . . . s4} we are evaluating

E exp[K(s_N−1s₀ + s0s₁ + s1s₂ + s2s₃ + s3s4)]. (16.104)

sN −1,s0,s1,s2,s3,s4

Now suppose we want to describe the system in terms of a ‘coarser’ lattice, with lattice spacing 2a, and corresponding new spin variables s^' _n. There are many ways we could choose to describe the s^' _n, but here we shall only consider a very simple one (Kadanoff 1977) in which each s^' _n is simply identified with the s_n at the corresponding site (see figure 16.5). For the portion of the lattice under consideration, then, (16.104) becomes

E exp [K(sN−1s₀ + s₀s1 + s1s₁ + s₁s3 + s3s₂)] . (16.105)

sN −1,s ,s₀ 1,s ,s₁ 3,s₂

If we can now perform the sums over s₁ and s₃ in (16.105), we shall end up (for this portion) with an expression involving the ‘effective’ spin variables s₀, s and s₂, situated twice as far apart as the original ones, and therefore providing a more ‘coarse grained’ description of the system. Summing over s₁ and s₃ corresponds to ‘integrating out’ two short-distance degrees of freedom as discussed earlier.

In fact, these sums are easy to do. Consider the quantity exp(Ks₀^' s1), expanded as a power series:

K² K³

exp(Ks^' ₀s1) = 1 +Ks^' ₀s₁ + + (s₀s1) + . . . (16.106) 2! 3!

1

174 16. Lattice Field Theory, and the Renormalization Group Revisited

FIGURE 16.4

A portion of the one-dimensional lattice of spins in the Ising model.

16.4.2 Two one-dimensional examples

Consider first a simple one-dimensional Ising model with Hamiltonian (16.99) and partition function

Z= E

{sn}

exp

| K

NE−1 n=0

snsn+1

|

, (16.103)

where K = J/(kBT) > 0. In (16.103) all the sn variables take the values

±1 and the ‘sum over {sn}’ means that all possible configurations of the N variabless0, s1, s2, . . . , sN−1are to be included. The spin snis located at the lattice sitena, and we shall (implicitly) be assuming the periodic boundary condition sn = sN+n. Figure 16.4 shows a portion of the one-dimensional lattice with the spins on the sites, each site being separated by the lattice constanta. Thus, for the portion{sN−1, s0, . . . s4}we are evaluating

E

s_N−1,s0,s1,s2,s3,s4

exp[K(s_N−1s0+s0s1+s1s2+s2s3+s3s4)]. (16.104) Now suppose we want to describe the system in terms of a ‘coarser’ lattice, with lattice spacing 2a, and corresponding new spin variabless^'_n. There are many ways we could choose to describe thes^'_n, but here we shall only consider a very simple one (Kadanoff 1977) in which eachs^'_n is simply identified with thesnat the corresponding site (see figure 16.5). For the portion of the lattice under consideration, then, (16.104) becomes

E

sN−1,s^'₀,s1,s^'₁,s3,s^'₂

exp [K(sN−1s^'₀+s^'₀s1+s1s^'₁+s^'₁s3+s3s^'₂)]. (16.105) If we can now perform the sums overs1 ands3 in (16.105), we shall end up (for this portion) with an expression involving the ‘effective’ spin variables s^'₀, s^'₁ and s^'₂, situated twice as far apart as the original ones, and therefore providing a more ‘coarse grained’ description of the system. Summing overs1

ands3corresponds to ‘integrating out’ two short-distance degrees of freedom as discussed earlier.

In fact, these sums are easy to do. Consider the quantity exp(Ks^'₀s1), expanded as a power series:

exp(Ks^'₀s1) = 1 +Ks^'₀s1+K² 2! +K³

3! (s^'₀s1) +. . . (16.106)

'

'

' '

' ' ' '

' '

' '

' '

16.4. Renormalization, and the renormalization group, on the lattice 175

FIGURE 16.5

A ‘coarsening’ transformation applied to the lattice portion shown in figure 16.4. The new (primed) spin variables are situated twice as far apart as the original (unprimed) ones.

where we have used (s₀s1)² = 1. It follows that

exp(Ks^'0 s1)= coshK (1 +s₀s1 tanh K), (16.107) and similarly

exp(Ks1s₁)= coshK (1 +s1s₁ tanh K). (16.108) Thus the sum over s1 is

E cosh² K (1 + s K)

0s₁ tanh K + ₁ tanh² .

s1=±1

Clearly, the terms linear in s1 vanish after summing, and the s1 sum becomes just

s1s₁ tanh K + s₀s (16.109)

K (1 + s K)

1 tanh² .

Remarkably, (16.110) contains a new ‘nearest-neighbour’ interaction, s

2 cosh² ₀s (16.110)

0s just like the original one in (16.103), but with an altered coupling (and a dif-ferent spin-independent piece). In fact, we can write (16.110) in the standard form

1,

exp [g1(K) + K^' s₀s₁] (16.111) and then use (16.107) to set

tanh K^' = tanh² K (16.112)

and identify

g1(K) = ln

(2 cosh² K cosh K^'

). (16.113)

Exactly the same steps can be followed through for the sum on s₃ in (16.105), and indeed for all the sums over the ‘integrated out’ spins. The upshot is that, apart from the accumulated spin-independent part, the new partition

16.4. Renormalization, and the renormalization group, on the lattice 177

FIGURE 16.6

‘Renormalization flow’: the arrows show the direction of flow of the coupling K as the lattice constant is increased. The starred values are fixed points.

FIGURE 16.7

The renormalization flow for the transformation (16.120).

this will correspond to a critical point at a finite temperature. A simple such example given by Kadanoff (1977) is the transformation

K^' =1

2(2K)² (16.119)

for a doubling of the effective lattice size, or K⁽ⁿ⁾=1

2(2K)ⁿ (16.120)

fornsuch iterations. The model leading to (16.120) involves fermions in one dimension, but the details are irrelevant to our purpose here. The renormal-ization transformation (16.120) has three fixed points: K^∗= 0,K^∗=∞and the finite pointK^∗= ¹₂. The renormalization flow is shown in figure 16.7.

The striking feature of this flow is that the motion is always away from the finite fixed point, under successive iterations. This may be understood by recalling that at the fixed point (which is a critical point for the statistical system) the correlation length ξ must be infinite (asL→ ∞). As we iterate away from this point, ξ decreases and we leave the fixed (or critical) point.

For this model,ξis given by Kadanoff (1977) as ξ= a

|ln 2K| (16.121)

which indeed goes to infinity atK= ¹₂.

16.4.3 Connections with particle physics

Let us now begin to think about how all this may relate to the treatment of the renormalization group in particle physics, as given in the previous chapter.

176 16. Lattice Field Theory, and the Renormalization Group Revisited function, defined on a lattice of size 2a, has the same form as the old one, but with a new coupling K^' related to the old one K by (16.112).

Equation (16.112) is an example of a renormalization transformation: the number of degrees of freedom has been halved, the lattice spacing has doubled, and the coupling K has been renormalized to K^' .

It is clear that we could apply the same procedure to the new Hamiltonian, introducing a coupling K^'' which is related to K^' , and thence to K by

tanh K^'' = (tanh K^')² = (tanh K)⁴ . (16.114) This is equivalent to iterating the renormalization transformation; after n iterations, the effective lattice constant is 2ⁿa, and the effective coupling is given by

tanh K⁽ⁿ⁾ = (tanh K)ⁿ . (16.115) The successive values K^', K^'' , . . . of the coupling under these iterations can be regarded as a ‘flow’ in the (one-dimensional) space of K-values: a renor-malization flow.

Of particular interest is a point (or points) K^∗ such that

tanh K ^∗ = tanh² K ^∗ . (16.116) This is called a fixed point of the renormalization tranformation. At such a point in K-space, changing the scale by a factor of 2 (or 2ⁿ for that matter) will make no difference, which means that the system must be in some sense ordered. Remembering that K = J/(kBT), we see that K = K^∗ when the temperature is ‘tuned’ to the value T = T^∗ = J/(kBK^∗). Such a T^∗ would be the temperature of a critical point for the thermodynamics of the system, corresponding to the onset of ordering. In the present case, the only fixed points are K^∗ = ∞ and K^∗ = 0. Thus there is no critical point at a non-zero T^∗, and hence no transition to an ordered phase. However, we may describe the behaviour as T → 0 as ‘quasi-critical’. For large K, we may use

tanh K - 1− 2e^−2K (16.117)

to write (16.115) as

K⁽ⁿ⁾ = K −1 ln n, (16.118) 2

which shows that Kⁿ changes only very slowly (logarithmically) under itera-tions when in the vicinity of a very large value of K, so that this is ‘almost’

a fixed point.

We may represent the flow of K, under the renormalization transformation (16.115), as in figure 16.6. Note that the flow is away from the quasi-fixed point at K^∗ = ∞ (T = 0) and towards the (non-interacting) fixed point at K^∗ = 0.

A renormalization transformation which has a fixed point at a finite (nei-ther zero nor infinite) value of the coupling is clearly of greater interest, since

176 16. Lattice Field Theory, and the Renormalization Group Revisited function, defined on a lattice of size 2a, has the same form as the old one, but with a new couplingK^' related to the old oneK by (16.112).

Equation (16.112) is an example of arenormalization transformation: the number of degrees of freedom has been halved, the lattice spacing has doubled, and the couplingK has been renormalized toK^'.

It is clear that we could apply the same procedure to the new Hamiltonian, introducing a couplingK^'' which is related toK^' , and thence toKby

tanhK^''= (tanhK^')²= (tanhK)⁴. (16.114) This is equivalent to iterating the renormalization transformation; after n iterations, the effective lattice constant is 2ⁿa, and the effective coupling is given by

tanhK⁽ⁿ⁾= (tanhK)ⁿ. (16.115) The successive valuesK^', K^'', . . . of the coupling under these iterations can be regarded as a ‘flow’ in the (one-dimensional) space ofK-values: a renor-malization flow.

Of particular interest is a point (or points)K^∗ such that

tanhK^∗= tanh²K^∗. (16.116) This is called afixed point of the renormalization tranformation. At such a point inK-space, changing the scale by a factor of 2 (or 2ⁿ for that matter) will make no difference, which means that the system must be in some sense ordered. Remembering that K = J/(kBT), we see that K = K^∗ when the temperature is ‘tuned’ to the value T =T^∗ =J/(kBK^∗). Such a T^∗ would be the temperature of acritical pointfor the thermodynamics of the system, corresponding to the onset of ordering. In the present case, the only fixed points areK^∗=∞andK^∗= 0. Thus there is no critical point at a non-zero T^∗, and hence no transition to an ordered phase. However, we may describe the behaviour asT →0 as ‘quasi-critical’. For largeK, we may use

tanhK-1−2e^−2K (16.117)

to write (16.115) as

K⁽ⁿ⁾=K−1

2lnn, (16.118)

which shows thatKⁿ changes only very slowly (logarithmically) under itera-tions when in the vicinity of a very large value ofK, so that this is ‘almost’

a fixed point.

We may represent the flow ofK, under the renormalization transformation (16.115), as in figure 16.6. Note that the flow is away from the quasi-fixed point at K^∗ =∞ (T = 0) and towards the (non-interacting) fixed point at K^∗= 0.

A renormalization transformation which has a fixed point at a finite (nei-ther zero nor infinite) value of the coupling is clearly of greater interest, since

16.4. Renormalization, and the renormalization group, on the lattice 177

FIGURE 16.6

‘Renormalization flow’: the arrows show the direction of flow of the coupling K as the lattice constant is increased. The starred values are fixed points.

FIGURE 16.7

The renormalization flow for the transformation (16.120).

this will correspond to a critical point at a finite temperature. A simple such example given by Kadanoff (1977) is the transformation

K^' = 1

(2K)² (16.119)

2 for a doubling of the effective lattice size, or

K⁽ⁿ⁾ = 1 (2K)ⁿ (16.120)

2

for n such iterations. The model leading to (16.120) involves fermions in one dimension, but the details are irrelevant to our purpose here. The renormal-ization transformation (16.120) has three fixed points: K^∗ = 0, K^∗ = ∞and the finite point K^∗ = 1₂ . The renormalization flow is shown in figure 16.7.

The striking feature of this flow is that the motion is always away from the finite fixed point, under successive iterations. This may be understood by recalling that at the fixed point (which is a critical point for the statistical system) the correlation length ξ must be infinite (as L → ∞). As we iterate away from this point, ξ decreases and we leave the fixed (or critical) point.

For this model, ξ is given by Kadanoff (1977) as

ξ = a (16.121)

|ln 2K| which indeed goes to infinity at K = 1₂ .

16.4.3 Connections with particle physics

Let us now begin to think about how all this may relate to the treatment of the renormalization group in particle physics, as given in the previous chapter.

16.4. Renormalization, and the renormalization group, on the lattice 179 We have emphasized that, at a critical point, and in the continuum limit, the correlation lengthξ→ ∞, or equivalently the mass parameter (cf (16.102)) m=ξ⁻¹→0. In this case, the Fourier transform of the spin-spin correlation function should behave as

G(k˜ ²)∝ 1

k². (16.125)

This is indeed thek²-dependence of the propagator of a free, massless scalar particle, but – as we learned for the fermion propagator in section 15.5 – it is no longer true in an interacting theory. In the interacting case, (16.125) generally becomes modified to

G(k˜ ²)∝ 1

(k²)^1−η2, (16.126) or equivalently

G(x)∝ 1

|x|^1+η (16.127)

in three spatial dimensions, and in the continuum limit. Thus, at a critical point, the spin-spin correlation function exhibits scaling under the transforma-tionx^' =fx, but it is not free-field scaling. Comparing (16.126) with (15.75), we see that η/2 is precisely theanomalous dimension of the field s(x), so – just as in section 15.5 – we have an example of scaling with anomalous di-mension. In the statistical mechanics case, η is a critical exponent, one of a number of such quantities characterizing the critical behaviour of a system.

In general, η will depend on the coupling constant η(K): at a non-trivial fixed point,ηwill be evaluated at the fixed point valueK^∗,η(K^∗). Enormous progress was made in the theory of critical phenomena when the powerful methods of quantum field theory were applied to calculate critical exponents (see for example Peskin & Schroeder 1995, chapter 13, and Binney et al.

1992).

In our discussion so far, we have only considered simple models with just one ‘coupling constant’, so that diagrams of renormalization flow were one-dimensional. Generally, of course, Hamiltonians will consist of several terms, and the behaviour of all their coefficients will need to be considered under a renormalization transformation. The general analysis of renormalization flow in multi-dimensional coupling space was given by Wegner (1972). In simple terms, the coefficients show one of three types of behaviour under renormal-ization transformations such thata→f a, characterized by their behaviour in the vicinity of a fixed point: (i) the difference from the fixed point value grows asf increases, so that the system moves away from the fixed point (as in the single-coupling examples considered earlier); (ii) the difference decreases asf increases, so the system moves towards the fixed point; (iii) there is no change in the value of the coupling asf changes. The corresponding coefficients are called, respectively, (i)relevant, (ii)irrelevantand (iii)marginalcouplings; the terminology is also frequently applied to the operators in the Hamiltonians 178 16. Lattice Field Theory, and the Renormalization Group Revisited

FIGURE 16.8

FIGURE 16.8

Im Dokument Non-Abelian Gauge Theories (Seite 188-197)