Continuum scaling and universality checks

For a continuum Euclidean scalar action with a canonical gradient term, use units in which its statistical or quantum action scale has been absorbed so the action is dimensionless. In d dimensions the physical field has length dimension one minus half d. The lattice field rescaling then makes the nearest-neighbor difference term dimensionless. The quadratic and quartic bare coefficients contain powers of lattice spacing shown below. Their renormalized counterparts need not equal these tree-level bare expressions in an interacting theory.

A continuum limit requires the correlation length in lattice spacings to grow while the physical reference length stays fixed. Near an interacting critical point this generally means tuning the bare mass parameter toward a coupling-dependent critical value, not merely setting its classical value to zero. Set the scale using a specified observable and match appropriate renormalized couplings along the refinement sequence. If ratios of independent physical observables drift without control, the simulations may be following different physical theories instead of one theory with a smaller cutoff. Existence and nontriviality of the limiting theory are additional questions, not consequences of refining a grid.

With L sites in one direction, the physical box length is a times L. Refining at fixed physical box therefore increases L. This can define a continuum theory in a finite box, but it does not make the box infinite relative to the correlation length. A thermodynamic or infinite-volume result requires a separately controlled ratio of box size to physical correlation length. At a critical point where that length diverges, finite-size scaling replaces a simple assumption that every box is already much larger than the correlations.

For a nearest-neighbor free scalar, the lattice momentum kernel differs from the continuum squared momentum by terms quadratic in spacing at fixed low momentum. This gives a useful analytic cutoff benchmark, with the small parameter being momentum times spacing. An interacting observable can have logarithms, other correction exponents, or action-dependent leading powers, so an arbitrary straight line in spacing squared is not universally justified. At fixed physical volume the number of lattice variables grows as inverse spacing to the d, and critical slowing can add further cost.

Compare distinct lattice discretizations only after matching their symmetries, dimensionality, relevant couplings, and physical scales. In an appropriate common scaling regime, critical exponents and suitably normalized dimensionless relations can agree while bare critical couplings and nonuniversal amplitudes differ. Binder crossing values also depend on shape and boundary conventions. Algorithmic dynamic exponents can differ even when both algorithms sample the same static universality class. Agreement of one exponent or a good-looking collapse is therefore supporting evidence rather than a complete identification.

Preserve the action and lattice geometry, update schedule, thermalization decision, measurement definitions, random generator and stream construction, code version, precision, and raw or adequately blocked data. A seed alone is insufficient when generators, parallel ordering, or analysis change. Record covariance handling, fit ranges, correction assumptions, and scale-setting correlations so the extrapolation can be repeated. Independent implementations and exact tiny-system benchmarks support this record. The planar matrix lab below illustrates one-axis cell-area bookkeeping only; it has no spacing-dependent field action or continuum solver.