Derivation of the Ising Model

In this section we will take the $\lambda\rightarrow\infty$ limit of the $\lambda \varphi ^{4}$ model. This can be done for more general models than the one-component $\lambda \varphi ^{4}$ model we are examining here, namely the multi-component $\lambda \varphi ^{4}$ models which are invariant by the $SO(\mathfrak{N})$ groups of transformations. In general the $\lambda\rightarrow\infty$ limit of these models will take us to the corresponding $SO(\mathfrak{N})$-invariant sigma models. In our case here, however, we will deal only with the one-component $\lambda \varphi ^{4}$, which is invariant by the sign reflections. The simple discrete set of transformations given by the identity and the reflection also forms a group of transformations, a discrete group which is known by either $O(1)$ or $\mathbb{Z}_{2}$. In this case the corresponding sigma model is simply the Ising model which was mentioned in [13]. In this way we will establish that we can use the Ising model as a direct representation of the infinite-coupling limit of the $O(1)$-symmetrical $\lambda \varphi ^{4}$ model.

Figure 2.1.1: Limits leading from the polynomial models to the sigma models by means of negative-slope straight lines departing from the Gaussian point. Note that lines parallel to the $\alpha$ and $\lambda$ coordinate axes are excluded.

The Ising model can be obtained from the $\lambda\varphi^4$ polynomial model in the limit in which the coupling parameter $\lambda$ tends to positive infinity over negative-slope straight lines in the parameter plane of the $\lambda\varphi^4$ model. These lines must exist only for $\lambda>0$, because otherwise they would cross the region where the model is unstable. Also, their slopes must be strictly negative (not zero) and finite, which rules out horizontal and vertical lines in the parameter plane. Since the slopes must be strictly negative and finite, in these limits we will have $\alpha\rightarrow-\infty$ and $\lambda\rightarrow\infty$ in such a way that $-\alpha/\lambda$ is a positive constant. Figure 2.1.1 illustrates the situation for lines starting from the Gaussian point. Observe that in these limits only the slope of the lines really matters. It makes no difference whether we use $\lambda=-C_{1}\alpha$ or $\lambda=-C_{1}\alpha+C_{2}$ for some finite constant $C_{2}$, because the finite additional term becomes irrelevant in the limit. In fact, we may take the limit over any curve that approaches asymptotically a negative-slope straight line. An important example of this is the critical curve of the $\lambda\varphi^4$ model, which in the $\lambda\rightarrow\infty$ limit approaches the critical point of the corresponding Ising model.

In order to establish this connection between the two models we start by recalling that the action of the $\lambda\varphi^4$ model without external sources, as it was defined in section 1.1, is given by

$$S[\varphi]=\frac{1}{2}\sum_{\ell}(\Delta_{\ell}\varphi)^{2} ... ...um_{s}\varphi^{2}(s) +\frac{\lambda}{4}\sum_{s}\varphi^{4}(s).$$

As we already discussed in section 1.1, we may now separate the action of the model in two parts, a kinetic part $S_{K}$ containing only the derivative terms,

$$S_{K}[\varphi]=\frac{1}{2}\sum_{\ell}(\Delta_{\ell}\varphi)^{2},$$

and a potential part $S_{V}$ containing the polynomial terms,

$$S_{V}[\varphi]=\sum_{s}V(\varphi),\mbox{   where   } V(\varphi)=\frac{\alpha}{2}\varphi^{2}+\frac{\lambda}{4}\varphi^{4}.$$

The functional integrals that appear in the definition of the observables of the quantum theory may now be written as

$$\int[{\bf d}\varphi]\;e^{-S[\varphi]}=\int[{\bf d}\varphi] \;e^{-S_{V}[\varphi]}\;e^{-S_{K}[\varphi]},$$

so that, including the exponential of $S_{V}$ in the measure of the distribution of the model, we may write this distribution as

$$[{\bf d}\varphi]\;e^{-S[\varphi]}=\left[{\bf d}\varphi \;e^{-V(\varphi)}\right]e^{-S_{K}[\varphi]}.$$

Since we may multiply this quantity by any constant independent of the fields without changing the observables, we may write the measure as

$$[{\bf d}\varphi\;\varrho(\alpha,\lambda,\varphi)],$$

where $\varrho(\alpha,\lambda,\varphi)$ is the local part of the distribution, which we thus include in the measure, normalized as

$$\varrho(\alpha,\lambda,\varphi)=\frac{\displaystyle e^{-V(\v... ...playstyle \int_{0}^{\infty}{{\rm d}}\varphi\;e^{-V(\varphi)}}.$$ (2.1.1)

Observe that the integration runs only over the positive values of $\varphi$ and hence can be interpreted as an integration over the absolute value of the field. With this the following normalization condition for $\varrho$ holds,

$$\int_{0}^{\infty}{\rm d}\varphi\;\varrho(\alpha,\lambda,\varphi)=1.$$

The integration over the absolute value of the field, when generalized to the more complex models having $SO(\mathfrak{N})$ symmetry, corresponds to the separation of the integration variables into a radial part and an angular part. In this way the arguments being presented here can be easily generalized to that case. In general the radial variable will be the modulus of the $\mathfrak{N}$-dimensional field vector in the internal symmetry space, which in our case here reduces to the absolute value of $\varphi$. Also, in our discrete $O(1)$-symmetrical case the integral over the angular part reduces to a sum over only two points, corresponding to rotations by each one of the two angles $0$ and $\pi$, and hence to the two possible signs, $\cos(0)=1$ and $\cos(\pi)=-1$. Therefore, in our present case the integral over all the values of $\varphi$ at each site is being decomposed in this way into an integral over the positive values of $\varphi$ and a sum over the two possible signs. Since both $\varrho(\alpha,\lambda,\varphi)$ and $V(\varphi)$ are even functions of $\varphi$, they are in fact functions only of the absolute value of $\varphi$, and independent of its sign, so it is not necessary to make their dependence on the absolute value explicit. Given all this, we may write the functional integrals as

$$\int[{\bf d}\varphi\;\varrho(\alpha,\lambda,\varphi)]\;e^{-S... ...t\;\varrho(\alpha,\lambda,\varphi)] \;e^{-S_{K}[\psi\varphi]},$$

where $\sum[\psi=\pm 1]$ represents the sum over all the possibilities for combinations of the sign of the field at each site, over all sites. Writing this functional integral explicitly this once, for clarity, we have

$$\int[{\bf d}\varphi\;\varrho(\alpha,\lambda,\varphi)]\;e^{-S... ...[\alpha,\lambda,\varphi(s)] \right\}\;e^{-S_{K}[\psi\varphi]},$$

where $\psi(s)$ is a new variable holding the sign of the field at the site $s$, while $\varphi$ assumes only positive values, due to the limits of integration adopted.

Let us now examine the behavior of $\varrho(\alpha,\lambda,\varphi)$ when $\lambda\rightarrow\infty$ and $\alpha=-\beta\lambda$, for some positive $\beta$. Note that, since lines making angles $0$ and $\pi/2$ with the $\alpha$ coordinate axis are excluded, so are the corresponding values $\beta=\infty$ and $\beta=0$. Executing the calculation of the integral in the denominator we obtain (problem 2.1.1), in terms of the parabolic cylinder functions ${\bf D}_{\nu}$,

$$\int_{0}^{\infty}{{\rm d}}\varphi\;e^{-V(\varphi)} =\frac{\s... ...}_{-\frac{1}{2}}\!\left(\frac{\alpha}{\sqrt{2\lambda}}\right).$$ (2.1.2)

Using the asymptotic form of ${\bf D}_{-\frac{1}{2}}$ (problem 2.1.2) and substituting $\alpha$ in terms of $\lambda$, we may write the distribution $\varrho(\lambda,\varphi)=\varrho(\alpha=-\beta\lambda,\lambda,\varphi)$, for large values of $\lambda$, as

$$\varrho(\lambda,\varphi)\simeq\sqrt{\frac{\beta\lambda}{\pi}} \;e^{-\frac{\lambda}{4}(\varphi^{2}-\beta)^{2}}.$$ (2.1.3)

We see here that indeed we cannot have either $\beta=0$ or $\beta=\infty$, because in either case $\varrho$ would vanish identically and hence would cease to be a normalizable statistical distribution. Given a finite and non-zero value of $\beta$ we also see that, when $\lambda\rightarrow\infty$, $\varrho$ tends to zero for all $\varphi$ except for $\varphi=\sqrt\beta$, where it diverges as $\sqrt\lambda$. Hence, given a continuous and limited function $f(\varphi)$ and considering the normalization of $\varrho$, one can verify that, in the $\lambda\rightarrow\infty$ limit (problem 2.1.3),

$$\int_{0}^{\infty}{{\rm d}}\varphi\;f(\varphi)\;\varrho(\lamb... ...arphi \;\varrho(\lambda,\varphi)=f\left(\sqrt{\beta} \right).$$ (2.1.4)

In other words, the distribution $\varrho(\lambda,\varphi)$ tends to a Dirac delta function,

$$\lim_{\lambda\rightarrow\infty}\varrho(\lambda,\varphi) =\de... ...qrt{\beta} \right) =2\sqrt{\beta}\;\delta(\varphi^{2}-\beta).$$

The conclusion is that in this limit the expectation values of the polynomial model may be written as

$$\langle{\cal O}\rangle_{N}=\frac{\displaystyle \sum[\psi=\pm... ...i\vert\;\delta(\varphi^{2}-\beta)] \;e^{-S_{K}[\psi\varphi]}},$$

where the remaining part of the measure can be written explicitly as

$$[{\bf d}\vert\varphi\vert\;\delta(\varphi^{2}-\beta)]=\prod_{s}{\rm d}\vert\varphi(s)\vert\;\delta[\varphi^{2}(s)-\beta],$$

where the Dirac delta functions imply a condition of constraint on the fields, $\varphi^{2}=\beta$, or $\varphi=\sqrt{\beta}$, since the sign of $\varphi$ is being considered explicitly and hence $\varphi$ is positive. We may now use the Dirac delta functions to perform all the integrations over $\vert\varphi\vert$, thus obtaining for the expectation values

$$\langle{\cal O}\rangle_{N}=\frac{\displaystyle \sum[\psi=\pm... ...\displaystyle \sum[\psi=\pm 1]\;e^{-S_{K}[\sqrt{\beta}\psi]}}.$$

We may now examine the form of the action $S_{K}$ under these conditions, in order to simplify it and exhibit it in a more familiar form. In terms of the new variables $\psi$ and the parameter $\beta$ this action can be written as

$$S_{K}[\psi]=\frac{\beta}{2}\sum_{\ell}(\Delta_{\ell}\psi)^{2},$$

where the new field variables $\psi=\varphi/\sqrt{\beta}$ satisfy the constraint $\psi^{2}=1$, since they are just signs, and the parameter $\beta$ appears now multiplying the action, just like the parameter $\beta=1/(kT)$ of statistical mechanics. Although we may want to think of $\beta$ as the inverse of a fictitious temperature, in order to guide our intuition about the behavior of the models, we should remember that our $\beta$ is, in truth, related to the mass parameter $\alpha$ and the coupling constant $\lambda$, and not to any truly physical temperature related to the dynamical system we are studying. The phenomenon that something relating to a coupling constant appears multiplying the action due to a scaling redefinition of the fields is typical of the gauge theories, as we may see in future volumes.

Note that, although $S_{K}[\psi]$ is a purely quadratic functional of $\psi$, the model is not the free theory, due to the fact that the field $\psi$ satisfies an equation of constraint, and is not, therefore, a free real variable. This is a situation in which the non-linearities, instead of being introduced by a term in the action, are introduced instead by the measure of the functional integral, which is where the constraint is implemented in the quantum theory. We may now perform one more transformation of the action of the model, with the intention of showing in a clearer way its relation with the Ising model of statistical mechanics. If we expand the squares of the derivatives contained in the action, we get

$$(\Delta_{\ell}\psi)^{2} =\psi^{2}(\ell_{-})-2\psi(\ell_{-})\psi(\ell_{+})+\psi^{2}(\ell_{+}),$$

where $\psi(\ell_{-})$ and $\psi(\ell_{+})$ are the fields at the two ends of the link $\ell$. Using now the equation of constraint $\psi^{2}=1$ we see that the two terms containing the squares are constant, independent of the fields, which means that they can be neglected in the action without changing the observables. We are left, therefore, with the bilinear term, and we write the action as (problem 2.1.4)

$$S_{K}[\psi]=-\beta\sum_{\ell}\psi(\ell_{-})\psi(\ell_{+}).$$ (2.1.5)

We have here an interaction between next neighbors involving the product of unit spins, exactly like in the Ising model. Hence we see that the infinite coupling limit of the $O(1)$ polynomial model is indeed the Ising model. Therefore, the expectation values of the polynomial model can be written as expectation values in this model, by means of a simple rescaling of the variable appearing within the observable,

$$\langle{\cal O}[\varphi]\rangle_{N}=\frac{\displaystyle \sum... ...psi=\pm 1]\;e^{\beta\sum_{\ell}\psi(\ell_{-})\psi(\ell_{+})}}.$$

Observe that, once the $\lambda\rightarrow\infty$ limit is taken in the way explained here, this relationship between the two classes of models is exact and involves no approximations of any kind.

It is important to discuss here the situation regarding the introduction of external sources into the model in this limit. The Ising model inherits from the polynomial model the introduction of external sources by means of a linear term in the action,

$$-\sum_{s}j(s)\varphi(s)=-\sqrt{\beta}\sum_{s}j(s)\psi(s).$$

It follows therefore that, apart from a rescaling of the sources by $\sqrt{\beta}$, the introduction of external sources is to be done in the usual way. In order to write this term in the form which is customary in statistical mechanics, we define the external sources $\eta(s)$ for the Ising model as $j(s)=\sqrt{\beta}\eta(s)$, so that the external-source term of the action acquires the form

$$-\sum_{s}j(s)\varphi(s)=-\beta\sum_{s}\eta(s)\psi(s).$$

At first sight it might seem natural to include the source term of the polynomial model in the potential part of the action, together with the $\alpha$ and $\lambda$ terms, and then to rework the derivation of the large-coupling limit. However, this should not be done, for two reasons: first, the external source term does not change in the limit and has in fact no role to play in it; second, it is not simply a polynomial term in $\varphi$, because its coefficient $j$ is not a constant like $\alpha$ or $\lambda$, but rather an arbitrary function of the sites. The external-source term should therefore be left in the action, together with the kinetic term $S_{K}$, and should not be included in the measure with the potential term $S_{V}$.

When the external-source term is treated in this way, the derivation of the large-coupling limit proceeds exactly as before, nothing changes in the derivation because no steps in it depend on other terms that the complete action may contain, besides the potential term $S_{V}$. We may therefore introduce external sources into the resulting Ising model exactly as we would do in the original polynomial model, and the whole functional generator formalism is made available for the analysis of the Ising model, without any change. Hence the Ising model is in fact an exact and complete direct representation of the infinite-coupling limit of the corresponding $\lambda \varphi ^{4}$ polynomial model. This relationship can be generalized to the multicomponent $SO(\mathfrak{N})$-invariant polynomial models and the corresponding sigma models. It can also be generalized to models with larger powers of the fields. It is possible to show (problem 2.1.5) that the Ising model can also be obtained as the $\lambda\rightarrow\infty$ limit of the $\lambda\varphi^{2p}$ polynomial models, with $p=3,4,5,\ldots$, in a way which is completely analogous to the $p=2$ case that we examined here.

Another aspect which we must examine here is the one relating to the superposition process involved in the definition of the block variables. Once again this is inherited by the Ising model from the polynomial model, so we still have a simple linear superposition of the fields $\psi$ at the various sites within the block. Note that although these fundamental fields satisfy the constraint $\psi^{2}=1$, the same will not be true for the block variables. If we consider the process of linear superposition of the fields within a block, in order to define a block variable, it is evident (problem 2.1.6) that the sum and the average of a collection of signs $\psi=\pm 1$ will not itself have unit absolute value. If the fields are distributed is a very random way, without any appreciable alignment, their average will tend to have an absolute value much smaller than $1$. Only in the opposite case, when the fields are highly aligned, the absolute value of the average will tend to $1$. The absolute value of the sum may have any value, either larger or smaller than $1$.

We see therefore that in general the introduction of external sources will cause the block variables, which are the variables in terms of which we should interpret the theory, to deform to any value, without respecting an equation of constraint. In fact, they will behave much like the corresponding variables of the polynomial models. In short, we see that the constraint that appears for the fundamental field in the large-coupling limit does not survive the block-variable superposition process and that the Ising model we get in the limit is not fundamentally different from the $\lambda \varphi ^{4}$ polynomial model it derives from. Hence we confirm that the Ising models are not to be seen as a completely different class of models, but as a way to examine directly the behavior of the polynomial models in the $\lambda\rightarrow\infty$ limit. Since, as was mentioned before, the Ising models can also be obtained as the limits of the $\lambda\varphi^{2p}$ polynomial models for any $p\geq 2$, they become a tool for the examination of the large-coupling limit of a large class of models.

We will finish this section with some comments about the critical behavior of the models. As we shall see later, approximate calculations based on the mean-field technique show that the Ising models have well-defined critical points $\beta_{c}$ for dimensions $d\geq 3$. Since they are the $\lambda\rightarrow\infty$ limits of the polynomial models, we see that the perturbative Gaussian approximation also predicts well-defined critical points for the Ising models, although they are infinitely distant from the Gaussian point. In fact, the two predictions do not differ very much from each other, and are both confirmed qualitatively by the computer simulations. In each dimension $d\geq 3$ we have therefore the same situation, a critical line in the parameter plane of the polynomial model, connecting at one end with the Gaussian point at the origin, and connecting at the other end with the critical point of the Ising model, over the arc at infinity.

The situation for $d=2$ is very peculiar in the case of the $O(1)$ models and deserves to be mentioned here. In this case the polynomial model does not exist in the vicinity of the Gaussian point, except for the Gaussian point itself. There is, therefore, no critical line connecting the Gaussian point to the critical point of the two-dimensional Ising model, which does exist, however, as is well known. In the corresponding $\lambda \varphi ^{4}$ model there is a convergent expansion near the Gaussian point [28], which shows that the observables are analytical functions of the parameters and that there is therefore no critical behavior, as our perturbative results indicate. However, it has also been shown that there is a phase transition in the polynomial model for sufficiently large $\lambda$ [28], indicating that there should be a critical line starting somewhere within the critical diagram, away from the Gaussian point, and extending from there to the critical point of the Ising model over the arc at infinity. The details regarding this peculiar situation are currently unknown.

Going back to the cases $d\geq 3$, besides the indications that we saw here, the computer simulations also indicate that the Ising models have the same triviality behavior of the polynomial models, in the sense that $\lambda_{R}$ vanishes in the continuum limit. In all these models it seems that the kinetic term of the action completely dominates the dynamics, and that the potential terms are not sufficient to change qualitatively the behavior dictated by the kinetic term. If even making the coupling parameter $\lambda$ tend to infinity we cannot obtain truly interacting models, it becomes clear that a deeper structural change of the models is necessary. This is exactly what one does when one discusses gauge theories involving vector fields, in which the interactions are introduced precisely through the kinetic term. There still is, however, a rather long path to follow before we get to that.

Problems

1. Calculate the integral in the denominator of equation (2.1.1), obtaining the result given in equation (2.1.2); see for example [15].

2. Use the asymptotic form of the parabolic cylinder functions ${\bf D}_{\nu}$ in order to write the local distribution of the Ising models in the form given in equation (2.1.3); see for example [25].

3. Show that the equation (2.1.4) for the local distribution of the Ising models is valid in the $\lambda\rightarrow\infty$ limit. In order to do that, show that the exchange of $f(\varphi)$ for $f(\sqrt{\beta})$ in the left-hand side does not change the limit.

4. Expand the squares of the finite differences in the action $S_{K}$, use the condition of constraint and neglect field-independent constants, in order to write the action in the form given in equation (2.1.5).

5. Repeat in a qualitative way the derivation presented in the text for the case of the $\lambda \varphi ^{4}$ models, in order to show that the Ising models are obtained, in limits in which $\alpha\rightarrow-\infty$ and $\lambda\rightarrow\infty$, with constant $-\alpha/\lambda$, from the corresponding $\lambda\varphi^{2p}$ models, for any $p\geq 3$.

6. Show that the average of the field $\psi$ over any number of sites has an absolute value smaller than or equal to $1$. Show also that the absolute value of the sum of $\psi$ over a block of sites may have any positive value in the continuum limit.