The Criticality Compass: design of an instrument for finding and following multivariate criticality — design note

Defensive publication: this note is deposited as a dated, public, enabling disclosure of the mechanisms listed in its Section 8. This note describes a tool we intend to build. It is not finished, and it has not been shown to work. We are publishing the design now so that the ideas are on public record with a date. Big systems, such as magnets, fluids and materials, may change their whole character when a setting is turned. Sometimes this edge behaviour is not tied to one setting. It spreads across several settings at once, as a whole line of edge states, and such lines may meet or come to an end. In very thin magnetic films, for example, a whole range of temperatures can sit on the edge at once. The tool is for finding and following this kind of edge behaviour. The tool has two parts. The Detector is meant to look at the data from a system and answer yes, no, or “I can’t tell”. When the answer is no, it is meant to still say briefly what it saw, such as a single sharp tipping point or a sudden jump. That is a side result, not the purpose. The Navigator is meant to switch on only after a yes. It is meant to be the feedback part of a running simulation or experiment. It takes each new reading, chooses the next setting, and hands that setting back to the process. In this way it is meant to walk along the edge and, over time, build a map of where the edge runs and where its lines meet or end. Working over data already collected is its secondary mode. Early pieces are being tried on test problems whose answers are already known. This design note fixes the design of the Criticality Compass as of its date. The Compass is for multivariate criticality (MC). MC is critical behaviour spread across more than one tuning parameter: a line of critical points or of fixed points, or a point where such lines meet or end. The Detector is designed to look at data from a system and decide whether it shows MC, does not, or cannot tell. If it does, the Navigator is designed to follow the MC lines as feedback to a running simulation or experiment, and over time to map them. Targets include the Berezinskii–Kosterlitz–Thouless line, lines of continuously varying exponents, and tricritical points. For a system that is not MC, it is designed to name what it saw, as a side result. The measurement uses a Monte Carlo block-spin renormalisation-group comparison, finite-size scaling and a replicate interval. The decision uses a declared-input contract, a sign cell and a partition rulebook. Nothing is yet shown to work; the methods paper will follow.