4 Continuous Symmetries

There are a number of reasons why it is useful to characterise states of matter in terms of their (broken) symmetry. The original idea of Landau was that, as we’ve seen with the Ising model, symmetry provides a powerful mechanism to understand when a phase transition will take place. In particular, there must be a phase transition whenever $H$ changes.

However, it turns out that this is not the only thing symmetry is good for. As we will see below, knowledge of $G$ and $H$ is often sufficient to determine many of the low energy properties of a system, both through a result known as Goldstone’s theorem (that we will describe in Section 4.2) and through various topological considerations (some of which we will see in Section 4.4).

Finally, and particularly pertinent for this course is the role that symmetry plays in the renormalisation group and specifically in universality. One can ask: when do two systems lie in the same universality class? Although the full answer to this question is not yet understood, a fairly good guess is: when they share the same symmetry $G$.

There are many different systems and choices of $G$ that we could look at. A particularly interesting class occurs when we take $G={𝐑}^d\times SO(d)$, the group of spatial translations and rotations. The pattern of symmetry breaking provides, for example, a clean distinction between a liquid/gas and a solid, with the latter breaking $G$ down to its crystal group. In this framework, there is not one solid phase of matter, but many, with each different crystal structure preserving a different $H$ and hence representing a different phase of matter. The different breaking patterns of spatial rotations also allows us to define novel phases of matter, such as liquid crystals. Viewed in this way, even soap, which can undergo a discontinuous change to become slippery, constitutes a new phase of matter. We will not discuss this form of symmetry breaking in this course, but you can learn more about it in next term’s course on Soft Matter.

Here, instead, we will be interested in phases of matter that are characterised by “internal” symmetry groups $G$ that are continuous, as opposed to the discrete symmetry of the Ising model. This includes materials like magnets, where the spin is a vector that is free to rotate. It also includes more exotic quantum materials such as Bose-Einstein condensates, superfluids and superconductors. We will see that systems with continuous symmetry groups $G$ exhibit a somewhat richer physics than we’ve seen in the Ising model.

These kind of excitations, which arise from the spontaneous breaking of continuous symmetries, are known as Goldstone bosons, or sometimes Nambu-Goldstone bosons. In the particular context of magnets, they are called spin waves.

There is a dizzying array of names for these kind of excitations, reflecting their ubiquity and importance. In general, an excitation whose energy cost vanishes as the wavelength goes to infinity is referred to as a soft mode or, alternatively, is said to be gapless. These are to be contrasted with gapped excitations whose energy remains finite in this limit. In the context of quantum field theory, “gapless” = “massless”, and “gapped” = ”massive”, with the energy gap coming from $E=m{c}^2$.

Gapless excitations often dominate the low-temperature behaviour of a system, where they are the only excitations that are not Boltzmann suppressed. In many systems, these gapless modes arise from the breaking of some symmetry. A particularly important example, that we will not discuss in these lectures, are phonons in a solid. These can be thought of as Goldstone bosons for broken translational symmetry.

For our $O(N)$ model, the $G=O(N)$ symmetry is broken by a choice of $⟨\mathit{\varphi }⟩$ to $H=O(N-1)$. (To see this, note that if $\mathit{\varphi }=(M_0,0,\mathrm{\dots },0)$ then there is a surviving $O(N-1)$ symmetry which acts on the on the string of zeros.) The space of ground states has a group theoretic interpretation as the coset space

${𝐒}^{N-1}={\displaystyle \frac{O(N)}{O(N-1)}}$

This idea generalises. If a continuous symmetry $G$ is spontaneously broken to $H$, then the manifold of ground states is given by $G/H$. We get a Goldstone boson for each broken symmetry generator, so the total number is

$\text{\# Goldstone Bosons }=\dim G-\dim H$

For the $O(N)$ model, $G=O(N)$ and $H=O(N-1)$ so the number of Goldstone modes is then $\frac{1}{2}N(N-1)-\frac{1}{2}(N-1)(N-2)=N-1$, which is indeed the dimension of the sphere ${𝐒}^{N-1}$.

The kinetic terms for the Goldstone bosons above take the form of the metric on the two-sphere ${𝐒}^2$, i.e. $d{s}^2=d{\theta }^2+{\sin }^2\theta d{\varphi }^2$. This is no coincidence: the Goldstone bosons describe fluctuations around the minima of the free energy $F[\mathit{\varphi }]$. In the present case, this set of minima is ${𝐒}^2$, and this geometry gets imprinted on the dynamics of the Goldstone modes. We will explore this more in Section 4.3.

This manifests itself in the correlation function, which decays as a power-law rather than exponential. This is simplest to see in the XY-model. (We will discuss $O(N)$ models with $N\ge 3$ in more detail in Section 4.3.) The free energy (4.116) is

$F[\theta ]={\displaystyle \int d^{d}x\frac{\gamma }{2}{M}_0^2{(\nabla \theta )}^2}+\mathrm{\dots }$

where the higher order terms are all derivatives and will not affect the discussion below. To compute the correlation function $⟨\theta (𝐱)\theta (𝐲)⟩$, we can simply import the result (2.53). (There are some subtleties in doing the path integral because $\theta (𝐱)$ is periodic, now valued in $[0,2\pi )$ rather than $𝐑$. These subtleties turn out not to be important here but we will revisit them in Section 4.4.) The long distance behaviour is

$⟨\theta (𝐱)\theta (𝐲)⟩={\displaystyle \frac{1}{\gamma M_0^2}}{\displaystyle \int \frac{d^{d}k}{{(2\pi )}^d}\frac{e^{i𝐤\cdot (𝐱-𝐲)}}{k^2}}$

(4.118)

This is similar to the behaviour of the correlation function at the critical point. Indeed, a critical point can be thought of as having gapless excitations. But there are differences.

First, the power-law decay at critical point requires some fine-tuning of a parameter; we must pick the temperature to be exactly $T=T_c$. In contrast, spontaneous symmetry breaking is more robust, and we get power-law decay for all . In other words, we have a phase with long range correlations, rather than just a point in the phase diagram. (For $T>T_c$, where there is no symmetry breaking, all modes still decay exponentially as in the Ising model.)

The second difference is that Goldstone bosons are much simpler to understand than the gapless modes at a critical point. As we have seen, at critical points the power-law decay of correlation functions suffers a correction due to integrating out short distance modes, resulting in the critical exponent $\eta \ne 0$. There are no such subtleties for Goldstone modes since all the dynamics is constrained by symmetry, and the correlation function (4.118). There are two caveats to this statement, both of which we will elaborate upon below. The first is that the simplicity only holds when ; when we sit at the critical point $T=T_c$, things become interesting once again. The second caveat is that we have to be above the lower critical dimension for the Goldstone bosons to exist.

First, in $d=2$ there are no critical points with $G=O(N)$ symmetry. We’ll see why in Section 4.2.3 and explore the physics more in Sections 4.3 and 4.4.

In $d=3$, the theories flow to a different critical point for each $N$. The critical exponents are known to be:

where the other critical exponents, $\alpha$, $\beta$, $\gamma$ and $\delta$ all follow from the scaling relations that we saw in Section 3.2.

While the values of $\eta$ and $\nu$ do not look very different from the Ising exponents, there is an important difference in the critical exponent for the heat capacity $c\sim {|T-T_c|}^{-\alpha }$, which is given by $\alpha =2-3\nu$. For both the $O(2)$ and $O(3)$ transition, $\alpha$ is negative. For example, $\alpha \approx -0.16$ for the $O(2)$ transition. This means that the heat capacity exhibits a cusp, rather than a true divergence.

For example, the superfluid transition of helium lies in the XY universality class. The heat capacity has long been known to exhibit cusp-like behaviour as shown in Figure 34⁹⁹ 9 This data is taken from Buckingham and Fairbank, “The Nature of the Lambda Transition”, in Progress in Low Temperature Physics III, 1961.. This characteristic shape means that the second order superfluid transition is sometimes referred to as the “lambda transition”. It turns out that the accuracy in these experiments is limited by the effect of the Earth’s gravitational field. In the 1990s, these measurements were made on a space shuttle flight, in broad (but not perfect) agreement with theoretical prediction of $c\sim A_{\pm }-B{t}^{-\alpha }$ for the critical exponent $\alpha \approx -0.16$ and suitable coefficients $A_{\pm }$ and $B$.

The transition to Bose-Einstein condensate also sits in the XY universality class. This is a particularly clean system which allows precision experiments. For example, the data above shows the behaviour of the correlation length as a gas of ultracold rubidium-87 atoms passes through the critical point. The critical exponent is found to be $\nu =0.67\pm 0.13$, in good agreement with the theoretical prediction (albeit with fairly large error bars)¹⁰¹⁰ 10 This data is taken from the paper “Critical behaviour of a trapped interacting Bose gas” by T. Donner et. al., arXiv:0704.1439..

It is not too difficult to repeat the RG calculations that we did in Sections 3.3 - 3.5 for the $O(N)$ model. As before, we rescale fields so that our order parameter – which we now call ${\varphi }_a(𝐱)$, with $a=1,\mathrm{\dots },N$ – has free energy

$\beta F[\varphi ]$

$=$

${\displaystyle \int d^{d}x\left[\frac{1}{2}\nabla {\varphi }_a\cdot \nabla {\varphi }_a+\frac{1}{2}{\mu }_0^2{\varphi }_a{\varphi }_a+g_0{({\varphi }_a{\varphi }_a)}^2+\mathrm{\dots }\right]}$

The study of the Gaussian fixed point, at ${\mu }_0^2=g_0=0$, goes through much as before. Indeed, a simple dimensional analysis argument tells us that

$[{\varphi }_a]={\mathrm{\Delta }}_{\varphi }={\displaystyle \frac{d-2}{2}}$

and, so

$[{\mu }_0^2]=2\mathit{\hspace{1em}\hspace{1em} }\mathrm{and}\mathit{\hspace{1em}\hspace{1em} }[g_0]=4-d$

so that ${\mu }_0^2$ is always a relevant deformation, while $g_0$ is relevant in dimensions. So far, nothing depends on $N$.

The differences arise in perturbation theory. The part of the free energy which mixes long and short wavelength modes is

$\beta F_I[\varphi ]={\displaystyle \int d^{d}xg{({\varphi }_a{\varphi }_a)}^2}$

The presence of the internal indices, $a=1,\mathrm{\dots },N$, means that the interaction has more structure than previously. To reflect this, we need to change our rules for drawing diagrams. First, each line should now be accompanied by an internal index $a$. Second, it is useful to split the interaction vertex as

$\mathit{\hspace{1em}\hspace{1em}}\to \mathit{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}}\sim g_0{\delta }_{ab}{\delta }_{cd}$

where the red ellipse splits the four legs into two pairs, each of which is a singlet under the $O(N)$ symmetry, as shown in the delta function structure. (You may have to squint in some of the following pictures to see which pairs of legs are contracted.)

The rest of the calculation proceeds as in Section 3.4.1. We find that, at this order, we have a renormalisation of the quadratic term given by

${\mu }_0^2\to {\mu }^{\prime \mathrm{\hspace{0.17em}2}}={\mu }_0^2+4(N+2)g_0{\displaystyle {\int }_{\mathrm{\Lambda }/\zeta }^{\mathrm{\Lambda }}}{\displaystyle \frac{d^{d}q}{{(2\pi )}^d}}{\displaystyle \frac{1}{q^2+{\mu }_0^2}}$

This agrees with our earlier result (3.94) when $N=1$.

The story is different when we have continuous symmetries. Now there are no domain walls because the space of ground states is continuously connected. However, there is a more prominent phenomena which means that the lower critical dimension is raised to $d_l=2$.

We see that there is a qualitative difference between $d>2$ and $d\le 2$. For $d>2$, the two point correlator $⟨\theta (𝐱)\theta (0)⟩$ decays to a constant as $r\to \mathrm{\infty }$. This constant is cancelled by the other term in (4.120), which means that the phase returns to its original value $⟨\theta ⟩=0$.

In contrast, for $d\le 2$, the fluctuation of the phase grows indefinitely as we go to larger distances. You may have thought that you had placed the system in a fixed ground state, but the thermal fluctuations of the Goldstone mode mean that it doesn’t stay there. The interpretation is that there is no ordered phase with $⟨\mathit{\varphi }⟩\ne 0$ in $d=2$ dimensions or below.

This is a general result, known as the Mermin-Wagner theorem. A continuous symmetry cannot be spontaneously broken in $d=2$ dimensions or below: there are no Goldstone bosons in $d=2$ dimensions.

This leaves us with a delicate question. The existence of the gapless Goldstone modes was predicated on the idea of spontaneous symmetry breaking. But for $d\le 2$ dimensions, no such symmetry breaking happens. What, then, is the resulting physics?

In $d=1$, the physics is straightforward: there are no gapless modes. As before, this can also be understood in the language of quantum mechanics, where the spectrum of a particle moving on ${𝐒}^{N-1}$ is discrete and gapped. For $d=2$, the physics is more interesting. It turns out that the answer is somewhat different for $O(N)$ models with $N\ge 3$ and for the XY-model with $N=2$. We will discuss the fate of the Goldstone modes for each of these in Sections 4.3 and 4.4 respectively.

Instead, there is a better approach, first introduced in this context by Polyakov, called the background field method. First, suppose that $𝐧(𝐱)$ takes some profile which varies slowly in space,

$n^a(𝐱)={\tilde{n}}^a(𝐱)$

This profile must obey $\widetilde{𝐧}\cdot \widetilde{𝐧}=1$. This will play the role of our long wavelength modes.

On top of this background, we want to introduce short wavelength modes which change rapidly in space. To parameterise these modes, we first introduce frame fields. These are a basis of $N-1$ unit vectors $e_{\alpha }^a(𝐱)$, with $a=1,\mathrm{\dots },N$ and $\alpha =1,\mathrm{\dots }N-1$, which are orthogonal to ${\tilde{n}}^a(𝐱)$,

${\tilde{n}}^a(𝐱)e_{\alpha }^a(𝐱)=0\mathit{\hspace{1em}\hspace{1em} }\forall \alpha \mathit{\hspace{1em}\hspace{1em}}\mathrm{and}\mathit{\hspace{1em}\hspace{1em}}{e}_{\alpha }^a(𝐱)e_{\beta }^a(𝐱)={\delta }_{\alpha \beta }$

(4.127)

The frame fields are, like ${\tilde{n}}^a$, slowly varying in space. There is an ambiguity in the definition of these frame fields; we can always rotate them by a local $O(N-1)$ transformation and we will still have a good set of frame fields.

The short wavelength modes sit on top of our original field $\widetilde{𝐧}(𝐱)$ and fluctuate in the direction of the frame fields. We call these ${\chi }_{\alpha }(𝐱)$, and write the full configuration as

$n^a(𝐱)={\tilde{n}}^a(𝐱){(1-\chi {(𝐱)}^2)}^{1/2}+{\displaystyle \sum _{\alpha =1}^{N-1}}{\chi }_{\alpha }(𝐱)e_{\alpha }^a(𝐱)$

(4.128)

By construction, this configuration still satisfies the constraint (4.122). This is morally equivalent to our previous Fourier space decomposition $\varphi ={\varphi }_{-}+{\varphi }_{+}$, but now in real space. We will integrate out the short wavelength modes $\chi$ and determine their effect on the long wavelength mode $\widetilde{𝐧}$.

First, we have

	$\nabla n^a$	$=$	$\nabla {\tilde{n}}^a{(1-{\chi }^2)}^{1/2}+{\tilde{n}}^a\nabla {(1-{\chi }^2)}^{1/2}+\nabla ({\chi }_{\alpha }{e}_{\alpha }^a)$
		$=$	$\nabla {\tilde{n}}^a(1-{\displaystyle \frac{1}{2}}{\chi }^2)-{\tilde{n}}^a{\chi }_{\alpha }\nabla {\chi }_{\alpha }+\nabla ({\chi }_{\alpha }{e}_{\alpha }^a)+𝒪({\chi }^2)$

The gradient term then becomes

${(\nabla n^a)}^2={(\nabla {\tilde{n}}^a)}^2(1-{\chi }^2)+{(\nabla {\chi }_{\alpha })}^2+{\chi }_{\alpha }{\chi }_{\beta }\nabla e_{\alpha }^a\nabla e_{\beta }^a+2\nabla {\tilde{n}}^a\nabla ({\chi }_{\alpha }{e}_{\alpha }^a)+𝒪({\chi }^3)$

where we have used the identities (4.127). One of the cross-terms vanishes by dint of the fact that ${\tilde{n}}^a{\tilde{n}}^a=1$ so that ${\tilde{n}}^a\nabla {\tilde{n}}^a=0$.

Our partition function is now

$Z={\displaystyle \int 𝒟\tilde{n}\delta ({\tilde{n}}^2-1)e^{-\frac{1}{2e^2}\int d^{d}x{(\nabla \tilde{n})}^2}\int 𝒟\chi e^{-\frac{1}{2e^2}\int d^{d}x{(\nabla \chi )}^2}{e}^{-F_I[\tilde{n},\chi ]}}$

where the interaction between ${\tilde{n}}^a$ and ${\chi }_{\alpha }$ are captured in

$F_I[{\tilde{n}}^a,{\chi }_{\alpha }]={\displaystyle \frac{1}{2e^2}}{\displaystyle \int d^{d}x\left[-{\chi }^2{(\nabla {\tilde{n}}^a)}^2+{\chi }_{\alpha }{\chi }_{\beta }\nabla e_{\alpha }^a\nabla e_{\beta }^a+2\nabla {\tilde{n}}^a\nabla ({\chi }_{\alpha }{e}_{\alpha }^a)\right]}$

As previously, we interpret the functional integral over $\chi$ as computing the expectation value of $e^{-F_I}$ using the probability distribution $\exp (-\frac{1}{2e^2}\int d^{d}x{(\nabla \chi )}^2)$. In other words,

The expectation value can be Taylor expanded,

(4.129)

As usual, the renormalisation group will generate many terms when we integrate out $\chi$. Our interest is in how the leading order kinetic terms ${(\nabla \tilde{n})}^2$ are affected; all other terms will turn out to be irrelevant. At leading order, it is sufficient to focus on $⟨F_I⟩$. (Given the term linear in $\chi$, one might wonder if such a term can be generated from $⟨F_I^2⟩$; a closer inspection shows that this in fact gives rise only to terms like ${(\nabla \tilde{n})}^4$.) We have

$⟨F_I⟩={\displaystyle \frac{1}{2e^2}}{\displaystyle \int d^{d}x\left(-{\delta }_{\alpha \beta }{(\nabla {\tilde{n}}^a)}^2+\nabla e_{\alpha }^a\nabla e_{\beta }^a\right)⟨{\chi }_{\alpha }(𝐱){\chi }_{\beta }(𝐱)⟩}$

(4.130)

where we’ve used the fact that $⟨\chi (𝐱)⟩=0$ to lose the linear term. The correlator that we want takes the same form that we calculated in previous sections. (See, for example, (2.54).)

$⟨{\chi }_{\alpha }(𝐱){\chi }_{\beta }(𝐱)⟩=e^2{\delta }_{\alpha \beta }{I}_d$

where

$I_d={\displaystyle {\int }_{\mathrm{\Lambda }/\zeta }^{\mathrm{\Lambda }}}{\displaystyle \frac{d^{d}q}{{(2\pi )}^d}}{\displaystyle \frac{1}{q^2}}$

Here we’ve introduced the limits on the integral to reflect the fact that, as in our earlier RG analysis, the short wavelength modes – which are here ${\chi }_{\alpha }(𝐱)$ – have support only in . The integral is simple to perform; we have

where ${\mathrm{\Omega }}_{d-1}$ is the area of the unit sphere ${𝐒}^{d-1}$. Substituting this into (4.130), it is clear that the first term corrects the kinetic term in the sigma model. But what of the second term? Using the fact that the correlator is proportional to ${\delta }_{\alpha \beta }$, it takes the form $\nabla e_{\alpha }^a\nabla e_{\alpha }^a$. We can massage this into the form we need using some identities. Between then, the fields $({\tilde{n}}^a,e_{\alpha }^a)$ provide an orthonormal basis of ${𝐑}^N$. Inverting this, we have

${\tilde{n}}^a{\tilde{n}}^b+e_{\alpha }^a{e}_{\alpha }^b={\delta }^{ab}$

(4.131)

Using this, we can write

$\nabla e_{\alpha }^a\nabla e_{\alpha }^a=\nabla e_{\alpha }^a\nabla e_{\alpha }^b({\tilde{n}}^a{\tilde{n}}^b+e_{\beta }^a{e}_{\beta }^b)$

But since ${\tilde{n}}^a{e}_{\alpha }^a=0$, we have ${\tilde{n}}^a\nabla e_{\alpha }^a=-(\nabla {\tilde{n}}^a)e_{\alpha }^a$ so

	$\nabla e_{\alpha }^a\nabla e_{\alpha }^a$	$=$	$e_{\alpha }^a{e}_{\alpha }^b(\nabla {\tilde{n}}^a)(\nabla {\tilde{n}}^b)+(e_{\beta }^a\nabla e_{\alpha }^a)(e_{\beta }^b\nabla e_{\alpha }^b)$
		$=$	$\nabla {\tilde{n}}^a\nabla {\tilde{n}}^a+(e_{\beta }^a\nabla e_{\alpha }^a)(e_{\beta }^b\nabla e_{\alpha }^b)$

where, in going to the second line, we’ve used (4.131) again, together with the fact that ${\tilde{n}}^a\nabla {\tilde{n}}^a=0$. The first term is just what we want; the second term is a new term that we can add to the sigma model and will be generated by RG. It is related to the geometric concept of torsion; it turns out to be irrelevant and we will not discuss it further here.

Both terms in (4.130) now give a contribution to ${(\nabla {\tilde{n}}^a)}^2$; the first is $-{\delta }_{\alpha \beta }{\delta }^{\alpha \beta }=-(N-1)$; the second is simply $+1$. The upshot of this is that the $⟨F_I⟩$ includes the term

$⟨F_I⟩=(2-N)I_d{\displaystyle \int d^{d}x\frac{1}{2}{(\nabla {\tilde{n}}^a)}^2}$

We now exponentiate this so that, to the order we’re working at, (4.129) becomes $⟨e^{-F_I}⟩=e^{-⟨F_I⟩}$. This gives us an effective free energy for the long wavelength field $\widetilde{𝐧}$,

$F[\widetilde{𝐧}]={\displaystyle \int d^{d}x\frac{1}{2e^{\prime \mathrm{\hspace{0.17em}2}}}\nabla \widetilde{𝐧}\cdot \nabla \widetilde{𝐧}}$

with

${\displaystyle \frac{1}{e^{\prime \mathrm{\hspace{0.17em}2}}}}={\displaystyle \frac{1}{e_0^2}}+(2-N)I_d$

Usually there are two further steps in the RG programme. First, we need to rescale our momentum cut-off back up to $\mathrm{\Lambda }$, and in doing so rescale all length by $1/\zeta$. This proceeds as before. The second step, advertised in Section 3, is to rescale the fields so that the kinetic term is canonically normalised. This step is not for us, since the normalisation of the kinetic term is the only coupling we have. Instead, the fields are normalised correctly by imposing the constraint (4.122). The upshot is that we have the running coupling constant

${\displaystyle \frac{1}{e^2(\zeta )}}={\zeta }^{d-2}\left({\displaystyle \frac{1}{e_0^2}}+(2-N)I_d\right)$

(4.132)

The first term comes from the naive dimensional analysis $[e^2]=2-d$ that we saw in (4.125); the second term is the one-loop correction from integrating out the high momentum modes.

Note that this second term vanishes when $N=2$. This reflects the fact that the Goldstone boson in the XY-model is free; this can be seen in (4.116) where there are no interaction terms. Interesting things happen in the XY-model but we will postpone discussion to Section 4.4. In contrast, for $N\ge 3$ the Goldstone bosons are interacting, as can be seen for the $O(3)$ model in (4.117), and this drives the running of the coupling constant.

Asymptotically free theories are rather rare in physics. Perhaps the best known example is QCD or, in general, Yang-Mills theory with some small number of matter fields.

The sign of the beta function is telling us that, in $d=2$ (and indeed in $d=1$), the weakly coupled ordered phase that we started with is unstable. This is a manifestation of the Mermin-Wagner theorem that we mentioned in Section 4.2.3; there are no Goldstone bosons in $d\le 2$. Unfortunately to really understand the infra-red physics in these low dimensions we will have to figure out how to deal with the strongly interacting theory. We will introduce a particularly useful approach in Section 4.3.3.

In higher dimensions, say $d=3$, the beta function is negative. This means that the sigma-model flows towards weak coupling in the infra-red, telling us that the ordered phase is stable.

However, there is now something new we can do. We can look at what happens in dimension

$d=2+ϵ$

Here the beta function takes the form

${\displaystyle \frac{de}{ds}}=-{\displaystyle \frac{ϵ}{2}}e+(N-2){\displaystyle \frac{e^3}{4\pi }}{\mathrm{\Lambda }}^{ϵ}$

This has a fixed point that lies within the remit of perturbation theory, namely

$e_{\star }^2={\displaystyle \frac{2\pi ϵ}{N-2}}{\mathrm{\Lambda }}^{-ϵ}$

(4.133)

However, in contrast to the story of Section 4.2, this is now a UV fixed-point, rather than an IR fixed point. How should we interpret this?

To understand this, let’s recap the story so far. The $O(N)$ model, with unconstrained fields $\mathit{\varphi }$, has a Wilson-Fisher fixed point in dimension $d=3$. This has one relevant deformation which is, roughly speaking, ${\mathit{\varphi }}^2$. If we turn on this relevant deformation with negative sign, we flow to the ordered phase which is described by our sigma model.

What we’ve seen above is this story in reverse. Starting from the ordered phase, described by the sigma model, we have managed to claw our way back up the RG flow to find a UV fixed point, at least in dimensions $d=2+ϵ$. It is natural to identify this with the Wilson-Fisher point, viewed through different eyes. This provides us with a different handle on the $O(N)$ Wilson-Fisher fixed points for $N\ge 3$; we can either approach them from above using the $d=4-ϵ$ expansion, or from below using the $d=2+ϵ$ expansion.

To extract the critical exponent $\nu$, we need to understand how $e^2$ is related to the temperature. From our definition of the path integral (4.124), we see that $1/e^2$ sits in the exponent where $\beta$ would sit in a usual partition function. This motivates us to identify $e^2$ with temperature $T$ and the fixed point $e_{\star }^2$ with the critical temperature $T_{\star }$. We then linearise about the fixed point by writing $e^2=e_{\star }^2+\delta e^2$ to find

${\displaystyle \frac{d(\delta e^2)}{ds}}=+ϵ\delta e^2$

This gives ${\mathrm{\Delta }}_t=ϵ$ and, correspondingly, the critical exponent

$\nu ={\displaystyle \frac{1}{ϵ}}$

(4.134)

independent of $N$. To compute the critical exponent $\eta$, one could add the interaction $\int d^{d}x𝐁\cdot 𝐧(𝐱)$ or, alternatively, extract the anomalous dimension of $𝐧$ from the calculation above. One finds that

$\eta ={\displaystyle \frac{ϵ}{N-2}}$

The $d=2+ϵ$ expansion does not give great results if we just go ahead and plug in $ϵ=1$. But then, there is little reason that it should! For example, for $N=3$, we can compare the best known results with mean field, and with the $d=4-ϵ$ and $d=2+ϵ$ expansions, where we work to first order and plug in $ϵ=1$. We have

	$\eta$	$\nu$
MF	0	$\frac{1}{2}$
$d=4-ϵ$	0	0.61
$d=2+ϵ$	1	1
Actual	0.0386	0.702

Questions like this are typically hard. As $e^2\to \mathrm{\infty }$, it naively appears that all field configurations contribute equally to the path integral, no matter how wildly they vary and how far they are from the saddle point. We have very few techniques to deal with such situations. Often we have to turn to some hidden and surprising symmetry, or to some unusual limit where the theory is soluble.

In the present case, it turns out that such a limit exists: it is $N\to \mathrm{\infty }$. To proceed, we first rewrite the delta-function in the path integral (4.124) as

	$Z$	$=$	${\displaystyle \int 𝒟n\delta ({𝐧}^2-1)\exp \left(-\frac{1}{2e_0^2}\int d^{d}x\nabla 𝐧\cdot \nabla 𝐧\right)}$		(4.135)
		$=$	${\displaystyle \int 𝒟𝐧𝒟\sigma \exp \left(-\frac{1}{2e_0^2}\int d^{d}x\nabla 𝐧\cdot \nabla 𝐧-\frac{i}{2e_0^2}\int d^{d}x\sigma (𝐧\cdot 𝐧-1)\right)}$

Here the field $\sigma (𝐱)$ plays the role of a Lagrange multiplier; integrating it out gives us back the delta-function, imposing the field constraint ${𝐧}^2=1$.