Putting a Euclidean field on a lattice

A finite lattice changes the probability distribution we calculate, and a finite sample changes how accurately we know it. Keeping those two effects apart lets us improve a calculation for the right reason. More samples reduce random uncertainty. A smaller spacing tests the short-distance approximation. A larger interval tests the influence of the boundary.

Prerequisites: the classical-field and free-field quantization chapters of this course, together with the ideas of a mean and a variance. We will use a matrix as a compact way to write coupled quadratic terms; the steps below explain what its entries mean. After this chapter, you should be able to design separate tests for spacing, volume and sampling errors.

Replace an integral with neighboring sites

Consider one real field on a line. Measure distance in a reference length $\ell_0$, mass in $1/\ell_0$, and field amplitude in $\sqrt{\ell_0}$. We use natural units, so the Euclidean action is dimensionless. The numerical inputs below are expressed in these units.

Let $a$ be the site spacing and $m>0$ the mass parameter. On a periodic ring, $N$ sites fill a length $L=Na$. Neighboring values approximate a derivative: the change in field divided by the distance between them. Replacing the continuum integral by a sum gives

A configuration with large differences between neighbors has a larger action and receives less probability, according to $P(\phi)\propto e^{-S_E}$. This is why nearby sites tend to fluctuate together.

The coordinate is Euclidean. It can represent imaginary time for a quantum harmonic oscillator. A periodic interval then gives finite-temperature paths; increasing the interval approaches the ground-state limit. Fixing both endpoint values instead gives a conditioned path distribution. The equal-real-time covariance of a quantum field in one spatial dimension has a different mode weight, $1/(2\omega_k)$. Our one-dimensional Euclidean calculation uses the inverse quadratic operator.

Find the covariance before drawing samples

Expand the neighboring differences. Each interior square occurs in two bonds, giving a diagonal coefficient $2/a+am^2$. Each adjacent pair contributes an off-diagonal coefficient $-1/a$. These numbers form a matrix $A$, and the action becomes $S_E=\phi^TA\phi/2$.

The covariance matrix $C$, whose entry is $C_{ij}=\langle\phi_i\phi_j\rangle$, obeys $AC=I$. One way to see this is to integrate the derivative of $\phi_j e^{-S_E}$ with respect to $\phi_i$. The boundary term at infinite field amplitude vanishes because the positive quadratic action suppresses it. Differentiating the two factors gives $\delta_{ij}-\sum_k A_{ik}\langle\phi_k\phi_j\rangle=0$. Thus $C=A^{-1}$.

On the ring, a mode with phase change $2\pi q/N$ per site turns the neighboring-difference operator into the number $4\sin^2(\pi q/N)/a$. Summing the contribution of every mode gives

Here $n$ counts site intervals and $r=na$ is physical separation. This exact finite-lattice curve is available before sampling. The workbench draws fields by factoring $A$ and solving triangular equations, then compares them with this separately calculated spectral reference.

A numerical example with three answers

Take $N=48$, $a=0.25$, $m=0.75$ and $r=1.5$, so $L=12$ and $n=6$. The exact lattice correlation is $0.216126$ in units of $\ell_0$. The continuum Green function on a ring of the same length is

It gives $0.216715$. The second exponential describes the other way around the ring. Both values are deterministic predictions for their respective models.

The saved default record has model version gaussian-chain-1.0.0, seed $314159$ and $1024$ independent fields. Multiplying the values at the two probes in each field and averaging gives $\widehat G=0.238240$, with standard error $0.022873$. Its difference from the lattice prediction is about $0.97$ standard errors. This is an ordinary sampling fluctuation in that particular record.

The lattice-to-continuum difference is only $0.000589$. It stays the same when we change the seed. Resolving that small spacing effect calls for a much more precise sample estimate.

Why correlated samples need wider error bars

To give the sampling process a controllable memory, update the whole field by

Each $\eta_s$ is independent. The first field is also drawn from $\mathcal N(0,C)$, so the chain starts in its stationary distribution. The update index $s$ counts samples. At $\rho=0$, each update replaces the field completely.

Let $Y_s$ be the product at the two probes. Gaussian fourth moments give $\operatorname{Corr}(Y_s,Y_{s+k})=\rho^{2k}$. Expand the variance of the mean as a double sum over sample pairs. There are $M-k$ pairs at positive lag $k$, and equally many at negative lag. Dividing by $M^2$ gives

where $v$ is the variance of one product. The effective count is $M/F_M$. For $M=1024$ and $\rho=0.9$, $F_M=9.48249$ and the effective count is about $108$.

The ordinary sample variance $s^2$ itself has expectation $v(M-F_M)/(M-1)$ for this chain. Correcting that bias gives the workbench's estimate

These error bars use the known sampler correlation. The displayed gold autocorrelation curve is estimated from the saved products and consequently fluctuates around its reference.

Experiments that answer separate questions

Open Euclidean Field Correlations. Predict what will happen before making each change.

• To isolate sampling, keep the lattice fixed and change only the seed. The gold estimate moves; both reference curves stay put. Increase the retained count and compare the size of those movements over several seeds.
• To test spacing at fixed volume, compare $(N,a)=(24,0.5),(48,0.25),(96,0.125)$. Two inputs must change together to keep $L=12$. Keep $r=1.5$, which requires $n=3,6,12$. The lattice predictions are $0.214401$, $0.216126$ and $0.216567$, approaching the continuum value.
• To test volume on a periodic ring, hold $a$ and $m$ fixed and change only $N$. Keep the same probe separation. The contribution that wraps around the ring shrinks as the ring grows.

Export the record before changing settings. It contains every field vector, the seed and the model version, so another calculation can reconstruct the sample means.

Practice

You quadruple the number of independent samples. What happens to the standard error, and to the spacing error?

A colleague measures the field at one site instead of a two-point product. With $\rho=0.9$, may they reuse the product's effective count?

The correlation curve moves when you double the number of sites at fixed spacing. Design one further comparison that can tell whether the movement was caused by volume or resolution.

The Gaussian Green-function derivation follows Tong, Statistical Field Theory, section 2.2. The relation between autocorrelation and uncertainty is developed in Wolff, Monte Carlo errors with less errors, version 4, equations 21–25.