Fixed Boundary Conditions

In order to illustrate in a more familiar form the analogy between the $d=3$ classical field theory as defined here and electrostatics, it is necessary to change the boundary conditions. We will do this, defining what we will call fixed boundary conditions, first of all on finite lattices. This type of boundary condition and other similar types, derived from it, will also have a role to play in the quantum theory, as we may see in a future opportunity, but in the scope of this book we will use them only in this chapter. We will represent a finite lattice with fixed boundary conditions in the form shown in figure 2.2.1, where the sites marked with crosses are the one in the external border. These sites have a different role from the others, which we will call internal sites, being for this reason marked in a different way. Note that each one of them connects to a single internal site, independently of the dimension $d$ of the lattice.

Figure 2.2.1: A two-dimensional lattice with fixed boundary conditions.

In the scope of the classical theory we assume that the values of the fields are fixed and given a-priori at these border sites. The form of the action we considered before in equation (1.3.1) does not change, except for the fact that the sum over links includes now the links connecting the internal sites to the border. The sum over sites does not change at all, it remains running over the internal sites. Under these conditions the classical finite equation of motion in equation (2.1.3) also does not change form. It applies to all internal sites, but not to the border sites, for two reasons. For one thing, it is not necessary that it determine the values of the field at these sites, since they are given a-priori. For another, it is not possible to calculate the value of the Laplacian at these sites, because of the lack of sufficient data.

In order to see this, it is necessary to derive the equation in detail on a finite lattice. Note that the action itself changes slightly in form, because in this case we have, repeating the derivation made previously for the periodical boundary conditions, in $d=1$,

$$\begin{eqnarray*} \lefteqn{\sum_{l}\left(\Delta_{\ell}\varphi\right)^{2} =\left(... ...hi_{n}+\varphi_{n+1}) +\varphi_{N+1}(\varphi_{N+1}-\varphi_{N}), \end{eqnarray*}$$

where $\varphi_{0}$ and $\varphi_{N+1}$ are the fields at the external border, $\varphi_{1}\ldots\varphi_{N}$ are the fields in the interior, $\Delta_{\ell_{b\pm}}$ are the derivatives at the two opposite borders and we have detailed what happens at each one of the two ends. This time it results that the final form relating a sum over links with a sum over sites is

  $$\sum_{l}\left(\Delta_{\ell}\varphi\right)^{2} =-\varphi_{0}\Delta... ...=1}^{N}\varphi(n)\Delta^{2}\varphi(n) +\varphi_{N+1}\Delta_{\ell_{b+}}\varphi,$$ (2.2.1)

where we now have, unlike the previous case, surface terms to consider.

In order to derive the equation of motion it is more convenient to start from the initial form of the action given in equation (1.3.1), which we now write separating explicitly the internal links $\ell_{i}$ and the links to the border $\ell_{b\pm}$, still in one dimension, for simplicity,

$$S_{0}[\varphi]=\frac{1}{2}(\Delta_{\ell_{b-}}\varphi)^{2} +\... ...varphi)^{2} +\frac{\alpha_{0}}{2}\sum_{n=1}^{N}\varphi^{2}(n).$$

In order to find the configuration that minimizes the action we use the usual techniques of the calculus of variations, making variations $\delta\varphi$ of the field on all the internal sites. On the border sites the field remains fixed at the values given by the boundary conditions. We impose then that the variation of the action be zero for any $\delta\varphi(n)$. This variation of the action is given by

$$\delta S_{0}=\Delta_{\ell_{b-}}\varphi\delta(\Delta_{\ell_{b... ...}\varphi) +\alpha_{0}\sum_{n=1}^{N}\varphi(n)\delta\varphi(n).$$

It is easy to verify (problem 2.2.1) that, on any of the links,

$$\delta(\Delta_{\ell}\varphi)=\Delta_{\ell}\delta\varphi(s),$$

and we therefore have

$$\delta S_{0}=\Delta_{\ell_{b-}}\varphi\Delta_{\ell_{b-}}\del... ...ta\varphi +\alpha_{0}\sum_{n=1}^{N}\varphi(n)\delta\varphi(n).$$

We write now, explicitly, the part of the sum with the first terms containing derivatives close to one of the ends of the lattice, recalling that $\delta\varphi(0)=\delta\varphi(N+1)=0$,

$$\begin{eqnarray*} \lefteqn{(\varphi_{1}-\varphi_{0})\delta\varphi_{1} +(\varphi_... ...\delta\varphi_{2}(\varphi_{1}-2\varphi_{2}+\varphi_{3}) +\ldots. \end{eqnarray*}$$

With this we see that the variation of the action may be written as

$$\delta S_{0}=-\sum_{n=1}^{N}[\Delta^{2}\varphi(n)]\delta\var... ...}[-\Delta^{2}\varphi(n)+\alpha_{0}\varphi(n)]\delta\varphi(n).$$

Se now see that, for $\delta S_{0}$ to vanish for any function $\delta\varphi(n)$ it is necessary that, for all $n$,

$$-\Delta^{2}\varphi(n)+\alpha_{0}\varphi(n)=0,$$

which is the equation of motion for this case. We see that its form does not change, we have the same equation, although with different boundary conditions, of course.

Note that in this type of lattice there are $N$ internal sites in each direction, but a larger number, $N+1$, of links in each direction. In order to take the continuum limit we proceed in a way similar to the one used before, but this time the relation between the size $L$ of the box and the length $a$ of the links is $L=(N+1)a$. The volume of the integration element is the same as before, $a^{d}$, but there are now surface terms to consider in the integrals. The new representation of the lattice with all these elements is shows in figure 2.2.2.

Figure 2.2.2: The geometrical elements of a two-dimensional lattice with a fixed boundary.

In the continuum limit the equation of motion is the same differential equation we obtained before for periodic boundary conditions, shown in (2.1.1). What changes, naturally, are the boundary conditions. When $m_{0}=0$ and $d=3$ this equation reduces to the three-dimensional Laplacian and in this case we indeed have the theory of electrostatics, $\varphi$ being the electric potential within a cubic box of volume $V=L^{3}$ that contains no charges. However, this time the solution is not necessarily trivial, because there may be charges on the surfaces that are the borders of the box in the continuum limit, causing the potential not to be zero at these surfaces. In this case the solution of the equation will not be simply $\varphi\equiv 0$ within the box, but will have, instead of this, a value that will depend on the fixed values at the borders. This is the typical example of electrostatic problem that may be solved numerically, for example, by the relaxation method associated to the Laplacian (problems 2.2.2 and 2.2.3).

Observe that the sum over links in equation (2.2.1) may be written, in terms of the dimensional variables, generalizing again to $d$ dimensions, as

$$\sum_{\vec{n}}a^{d}\sum_{\mu} \frac{\left[\Delta_{\mu}\phi(\... ...ec{n}}a^{d}\phi(\vec{n})\frac{\Delta^{2}\phi(\vec{n})}{a^{2}},$$

where $\sum_{\vec{n}_{b}}$ is a sum over the external border, $\phi_{b}$ is the field at the border and $\Delta_{\ell_{be}}\phi$ is the finite external normal derivative of the field at the border. We have now the integral $\sum_{\vec{n}}a^{d}$ over the volume and the integral $\sum_{\vec{n}_{b}}a^{d-1}$ over the oriented external surface, so that in the continuum limit we obtain

$$\int_{V}{\rm d}^{d}x \sum_{\mu}\left[\partial_{\mu}\phi(\vec... ...-\int_{V}{\rm d}^{d}x\;\phi(\vec{x})\partial^{2}\phi(\vec{x}),$$

where $S=\partial V$ is the surface which is the border of the volume $V$ and $\partial_{\perp}$ is the external normal derivative to this surface. If we assume that in the continuum limit the derivatives that appear in this expression have finite values, it now becomes clear that in this limit the surface terms only vanish in the case in which the field is kept equal to zero at the external surface. Hence, we see that the form of the action changes by the addition of surface terms both on finite lattices and in the continuum limit of the classical theory, while the equation of motion does not change, the difference remaining implicitly only in the boundary conditions to be used to solve the equation.

It is important to observe here that this kind of boundary condition will not have a fundamental role to play in the quantum theory. This is due to the fact that in the quantum theory the value of the field at sites is neither an observable nor a controllable quantity that one can fix at a constant value without fluctuations. The boundary conditions discussed here will be associated, in the context of the quantum theory, to an approximation scheme, namely the so-called mean-field method. In any case we do see here, by means of these examples, that our discrete structure on the lattice reduces in fact to known cases of classical field theories in the continuum limit, when we introduce into the structure a dimensional scale which is external to the structure of our theory on the lattice.

Problems

1. If $f(s)$ is an arbitrary function of the sites, $s_{+}$ and $s_{-}$ are the sites at the two ends of a link $\ell$, $\Delta_{\ell}f=f(s_{+})-f(s_{-})$ is the finite difference of the fields at these sites, and assuming that an infinitesimal variation $\delta\varphi(s)$ at each site $s$ of the lattice is made, show that

   
   

   $$\delta(\Delta_{\ell}\varphi)=\Delta_{\ell}\delta\varphi.$$
   
   

2. Show, on finite lattices in $d$ dimensions, for an arbitrary position $\vec{n}$ in the interior of the lattice, that the equation $\Delta^{2}\varphi(\vec{n})=0$ implies that $\varphi(\vec{n})$ is equal to the average value of its $2d$ nearest neighbors.

3. ($\star$) Write a program that uses the relaxation method, which is based on the result of problem 2.2.2, to find the field $\varphi(s)$ that satisfies the equation $\Delta^{2}\varphi=0$ on a two-dimensional lattice in which $\varphi=1$ at two opposite sides of the border and $\varphi=-1$ at the other two opposite sides of the border.