Triviality Tests

The fact that the renormalized masses depend on the external sources, as we saw above, has important consequences for the design of computer simulations targeted at probing the triviality issue. One way to do this is to perform simulations on finite lattices in which one tries to measure the relation between the external sources and the expectation value of the field. The argument is based on the fact that on symmetry grounds it is reasonable to expect that the model has an effective action with the general form

$$\begin{eqnarray*} \Gamma_{N}[\vec{\varphi}_{\rm c}] & = & \sum_{n_{\mu}}^{N^{... ...^{2} - j_{0}\varphi_{\mathfrak{N},{\rm c}}(n_{\mu}) \right\}, \end{eqnarray*}$$

where $\vec{\varphi}_{\rm c}(n_{\mu})$ is the classical field, which is just another name for the expectation value of the field, given an arbitrary external source $j_{0}$. In this expression $\alpha_{0}$ and $\alpha_{\mathfrak{N}}$ are the renormalized masses, and $\lambda_{R}$ is the renormalized coupling constant. It is possible that additional terms may appear in $\Gamma_{N}[\vec{\varphi}_{\rm c}]$, but terms containing derivatives are not relevant for the argument that follows, and terms with higher powers can be easily included in the analysis if need be.

If one considers only homogeneous external sources $j_{0}$, then it is clear that the classical field must also be a constant, $\varphi_{\mathfrak{N},{\rm c}}(n_{\mu})=v_{0}$, and hence all terms containing derivatives vanish. We are left with only the part of the effective action that contains the effective potential. In addition to this, since the external source is in the direction of $\varphi_{\mathfrak{N},{\rm c}}(n_{\mu})$ in the internal $SO(\mathfrak{N})$ space, it is clear that the expectation values of all the other field components are zero, so that we are left with

$$\Gamma_{N}[\vec{\varphi}_{\rm c}] = \sum_{n_{\mu}}^{N^{d}... ...+ \frac{\lambda_{R}}{4}\, v_{0}^{4} - j_{0}v_{0} \right],$$

where we now wrote $v_{0}$ for the expectation value of the field. Since the behavior of $\varphi_{\mathfrak{N},{\rm c}}(n_{\mu})$ is ruled by the minimum of this action in the classical or long-wavelength limit, which is consistent with the use of a homogeneous external source, we may now differentiate with respect to the classical field $v_{0}$, and equate the result to zero, thus obtaining

$$j_{0} = \alpha_{\mathfrak{N}}\, v_{0} + \lambda_{R}\, v_{0}^{3}.$$

By measuring $v_{0}$ as a function of $j_{0}$ one may then determine from this equation the coefficients $\alpha_{\mathfrak{N}}$ and $\lambda_{R}$, and thus probe into the triviality of the model. For $d\geq 4$ triviality would result if it can be shown that

$$\lim_{N\to\infty}\lambda_{R} = 0.$$

In other words, a linear result for the relation between $j_{0}$ and $v_{0}$, with the renormalized mass parameter as the coefficient, indicates triviality. Any deviations from linearity imply the existence of interactions, either on finite lattices or in the continuum limit. This technique avoids the necessity for the direct measurement of the four-point function, which is generally much more difficult to do numerically than to measure the one-point and two-point functions.

However, we have shown here that $\alpha_{\mathfrak{N}}$ itself depends on $j_{0}$. Therefore, even if $\lambda_{R}$ is in fact zero on finite lattices this relation will not result linear if the simulations are performed by varying $j_{0}$ at fixed values of parameters $\alpha$ and $\lambda$. It is therefore necessary to adjust these parameters, as one varies $j_{0}$, in order to keep $\alpha_{\mathfrak{N}}$ constant. For this purpose the value of $\alpha_{\mathfrak{N}}$ can be obtained independently via the measurement of the propagator of the field component $\varphi_{\mathfrak{N}}(n_{\mu})$, of course. At the end of the day, its value can be confirmed by the value resulting for the linear coefficient from a polynomial fit to the relation between $j_{0}$ and $v_{0}$.

Given a certain chosen value for $\alpha_{\mathfrak{N}}$, for each value of $j_{0}$ one must search the parameter plane of the model looking for a point where $\alpha_{\mathfrak{N}}$ has that value, and only then measure $v_{0}$. This can become a computationally expensive search. This can be done by keeping $\lambda$ constant and varying $\alpha$, thus traversing horizontal lines on the parameter plane, or by keeping $\alpha$ constant and varying $\lambda$, thus traversing vertical lines. Given the structure of the phase transition and of the critical line, one attractive alternative is to keep $\alpha^{2}+\lambda^{2}$ constant and vary the angle $\theta$ around the position of the critical line, where

$$\frac{\lambda}{-\alpha} = \tan(\theta),$$

not forgeting that in general $\alpha$ will be negative. In any case, the formula giving $\alpha_{\mathfrak{N}}$ in terms of $\alpha$, $\lambda$, $v_{0}$ and $j_{0}$ that we derived here, shown in Equation (6), may serve to provide at least a good initial guess for this costly search in the parameter plane.