Compression theory for inhomogeneous systems

Abstract

The physics of complex systems stands to greatly benefit from the qualitative changes in data availability and advances in data-driven computational methods. Many of these systems can be represented by interacting degrees of freedom on inhomogeneous graphs. However, the lack of translational invariance presents a fundamental challenge to theoretical tools, such as the renormalization group, which were so successful in characterizing the universal physical behaviour in critical phenomena. Here we show that compression theory allows the extraction of relevant degrees of freedom in arbitrary geometries, and the development of efficient numerical tools to build an effective theory from data. We demonstrate our method by applying it to a strongly correlated system on an Ammann-Beenker quasicrystal, where it discovers an exotic critical point with broken conformal symmetry. We also apply it to an antiferromagnetic system on non-bipartite random graphs, where any periodicity is absent.

Fig. 1. Schematic for constructing collective degrees of freedom in inhomogeneous systems.

Distinct systems like tissues (left, in green) and colloidal suspensions (right, in blue) can be abstracted into a set of vector degrees of freedom ${\mathcal{V}}^i$ (indicated by stacks of squares, i = 1, 2, 3) living on an irregular graph with local structure. The final component of each vector is shown by a coloured box to indicate potentially different types of internal degree of freedom, unique to each sub-system. To derive a compressed representation of such systems, it is essential to tailor the coarse graining transformation Λⁱ for each local neighbourhood i. This is achieved by an information theoretic variational principle, where ${\mathbf{\Lambda }}^i:{\mathcal{V}}^i\mapsto {\mathcal{H}}^i$ maximises the mutual information $I\left({\mathcal{H}}^i:{\mathcal{E}}^i\right)$. This allows the compressed variables ${\mathcal{H}}^i$ to capture the emergent long-range physics according to the statistics of the surrounding distant environment ${\mathcal{E}}^i$. Local optimisation can produce compressed variables with varying cardinality across the system, here illustrated by vectors ${\mathcal{H}}^i$ with varying numbers of components. The connectivity of the emergent supergraph is determined through the correlations of the new variables.

Fig. 2. Self-similarity of the Ammann-Beenker tiling, and the coarse graining blocks.

a A microscopic dimer configuration on the AB tiling’s edges, with an overlaid AB super-quasilattice, self-similar to the microscopic one. The effective degree of freedom at a supervertex with valence n will be obtained by coarse graining the dimer configuration in the surrounding polygon tile Vⁿ. In total there are 4 classes of such polygons, here shown in green, blue, red and yellow for n = 8, 3, 4, 5, respectively. The shape of the block tile is dictated by the valence n of the central supervertex in matching colour. b The inflation (deflation) σ²⁽⁻²⁾ of the elementary rhombi and squares generating the tiling, with parts of the polygonal domains indicated in colour. Coarse graining all such polygonal patches executes a deflation σ⁻² of the original AB quasilattice, yielding the super-quasilattice shown.

Fig. 3. Finding collective clock variables.

a Coarse graining transformation Λ compressing Monte Carlo configurations $\mathcal{V}$ into bitstrings $\mathcal{H}$ on supervertices of the σ⁻² deflated AB tiling. Each bit ${\mathcal{H}}_k$ is decided by the sign of the linear transformation ${\mathrm{\Lambda }}_k\cdot \mathcal{V}$. b, f The length of the bitstring ${\mathcal{H}}^{8\left(3\right)}$ is determined by the saturation point (shown in green) of mutual information at 4 (2) bits at 8- (3-)supervertices. c, g The respective optimal filters Λ⁸ and Λ³ carry a representation of the local spatial symmetries of corresponding supervertices, namely ${\mathsf{C}}_8$ and mirror. d, h The probability distributions $P\left({\mathcal{H}}^{8\left(3\right)}\right)$ occupy the space of codes sparsely, and form abstract ${\mathbb{Z}}_{8\left(3\right)}$ clock variables. e In particular, ${\mathcal{H}}^8$ forms a closed 8-loop, where each state has exactly two neighbours with Hamming-distance 1. i The representations of the local symmetries on filters induce transitions between adjacent clock-states, enabling the identification of abstract clock-states with spatial directions along the links of the quasiperiodic lattice.

Fig. 4. Emergent dimer exclusion rule and self-similar dimer-dimer correlations across scales.

a The probability distribution of microscopic (i.e. δ⁰) dimers (in greyscale) on an AB patch, conditioned on one of the links (in orange) hosting a dimer. b, c First two columns: the probabilities $P\left({\mathcal{H}}^8\mid {\mathcal{H}}^3\right)$ of the emergent clock variables on the δ² and δ⁴ super-quasilattice (in greyscale), conditioned on two distinct states of one of the 3-clocks (in orange). The third column shows distributions $P\left({\mathcal{H}}^3\mid {\mathcal{H}}^8\right)$ conditioned on a state of the central 8-clock. Binding of adjacent clock variables into super-dimers obeying dimer exclusion constraints is revealed by sharply peaked conditional distributions. The effective super-dimers reproduce also longer-range dimer-dimer correlations at both δ² and δ⁴ scales. d, e Examples of optimal coarse-graining filters producing the central 8-state clock variable at scales δ² and δ⁴. The latter comprises 2760 microscopic links.