A chaotic lattice field theory in two dimensions

TL;DR

This work reframes chaotic turbulence in extended lattice field theories as a deterministic Euclidean field theory where spatiotemporally periodic states serve as the fundamental building blocks of the partition function. By introducing orbit Jacobians and exploiting Bravais-lattice geometry via Floquet-Bloch analysis, it derives a spatiotemporal zeta function that encodes weights of prime orbits and enables exact (up to cycle-expansion truncation) predictions for observables. The approach unifies field theory, solid-state methods, and chaos theory, with explicit demonstrations on spatiotemporal cat and nonlinear φ^3/φ^4 models, illustrating multiplicativity of weights across repeats and shadowing that accelerates convergence. This framework promises a systematic, geometry-aware path to understanding turbulence in spatially extended systems and opens questions about continuum limits, symmetry reductions, and computational scalability.

Abstract

We describe spatiotemporally chaotic (or turbulent) field theories discretized over d-dimensional lattices in terms of sums over their multi-periodic orbits. `Chaos theory' is here recast in the language of statistical mechanics, field theory, and solid state physics, with the traditional periodic orbits theory of low-dimensional, temporally chaotic dynamics a special, one-dimensional case. In the field-theoretical formulation, there is no time evolution. Instead, treating the temporal and spatial directions on equal footing, one determines the spatiotemporally periodic orbits that contribute to the partition sum of the theory, each a solution of the system's defining deterministic equations, with sums over time-periodic orbits of dynamical systems theory replaced here by sums of d-periodic orbits over d-dimensional spacetime geometries, the weight of each orbit given by the Jacobian of its spatiotemporal orbit Jacobian operator. The weights, evaluated by application of the Bloch theorem to the spectrum of periodic orbit's Jacobian operator, are multiplicative for spacetime orbit repeats, leading to a spatiotemporal zeta function formulation of the theory in terms of prime orbits.

Figures (14)

• Figure 1: A bird's eye view of the quantum action landscape over the primitive cell state space $\mathcal{M}_\mathbb{A}$, Eq. (\ref{['torusStatesp']}). White ellipses indicate the stationary points Eq. (\ref{['eqMotionLagrang']}), i.e., the set of all deterministic solutions $\{\mathsf{\Phi}_a,\mathsf{\Phi}_b,\mathsf{\Phi}_c,\cdots,\mathsf{\Phi}_g\}$. They form the skeleton on which the partition sums of both quantum chaos and deterministic chaos / turbulence are evaluated, with the common deterministic backbone, but different weights. (a) For a quantum theory, the semiclassical partition sum Eq. (\ref{['n-pt-corr1']}) is an approximation, with quantum probability amplitude phases given by deterministic solutions' actions, and stability weights given by square roots of the deterministic ones. (b) For a det-er-min-is-tic field theory the probabilities that form the partition sum Eq. (\ref{['detPartSum']}) are exact, a Dirac porcupine of delta function quills, a quill for each solution of defining equations.
• Figure 2: (Color online) The intersections of the light grey lines -lattice sites $z \in \mathbb{Z}^2$- form the integer square lattice Eq. (\ref{['LattField']}). (a) Translations of the primitive cell $\mathbb{A}={[3\!\times\!2]_{1}}$ spanned by primitive vectors $\mathbf{a}_1=(3,0)$ and $\mathbf{a}_2=(1,2)$ define the Bravais lattice $\mathcal{L}_\mathbb{A}$. (b) The primitive vectors $\mathbf{a}_1=(2,-2)$ and $\mathbf{a}_2=(-1,4)$ form a primitive cell ${\mathbb{A}'}$ equivalent to (a) by a unimodular transformation. The intersections (red points) of either set of dashed lines form the same Bravais lattice $\mathcal{L}_\mathbb{A}=\mathcal{L}_{\mathbb{A}'}$. The volume Eq. (\ref{['lattVol']}) of either primitive cell is ${N}_\mathcal{L}=6$, the number of integer lattice sites within the cell, with the tips of primitive vectors and tiles' outer boundaries belonging to the neighboring tiles. Continued in Fig. \ref{['f:2x1rpo']}.
• Figure 3: Examples of ${[{L_{}}\!\times\!{T_{}}]_{S_{}}}$ field configurations Eq. (\ref{['2DHermiteBas']}) or 'bricks', together with their spa-tio-temp-or-al Bravais lattice tilings, visualized as brick walls. (a) ${[2\!\times\!1]_{1}}$, primitive vectors $\mathbf{a}_1=(2,0)$, $\mathbf{a}_2=(1,1)$; (b) ${[3\!\times\!2]_{1}}$ of Fig. \ref{['f:BravLatt']} (a), primitive vectors $\mathbf{a}_1=(3,0)$, $\mathbf{a}_2=(1,2)$. Rectangles enclose the primitive cell and its Bravais lattice translations. Continued in Fig. \ref{['f:pcellTiling']}.
• Figure 4: (Color online) (a) Bravais lattice $\mathbb{A}={[6\!\times\!4]_{2}}$, blue dots, is a sublattice of Bravais lattice $\mathbb{A}_p = {[3\!\times\!2]_{1}}$, blue and red dots. Its primitive cell ${\mathbb{A}}$ (green parallelogram spanned by primitive vectors (6,0) and (2,4)) is tiled by repeats of the primitive cell $\mathbb{A}_p$ (gray parallelogram spanned by primitive vectors (3,0) and (1,2)). The primitive vectors of the 2 Bravais lattices are related by $\mathbb{A} = \mathbb{A}_p \mathbb{R}$ where $\mathbb{R} = {[2\!\times\!2]_{0}}$. (b) Transform the primitive cell $\mathbb{A}_p$ to the unit square of a new square lattice, where each unit square supports a multiplet of 6 fields belonging to a prime $\mathcal{L}_{\mathbb{A}_p}$-periodic state. In this new square lattice, the prime periodic state is a steady state whose primitive cell is a $[1\!\times\!1]_{0}$ unit square (gray square), while the repeat of the prime is a $\mathcal{L}_\mathbb{R}$-periodic state, whose primitive cell is $\mathbb{R} = {[2\!\times\!2]_{0}}$ (green square).
• Figure 5: (Color online) (a) Bravais lattice $\mathbb{A}={[3\!\times\!2]_{1}}$ of Fig. \ref{['f:BravLatt']}, red dots, is a sublattice of Bravais lattice $\mathbb{A}'={[3\!\times\!1]_{2}}$, blue and red dots, even though the primitive cell ${\mathbb{A}}$ (green parallelogram spanned by primitive vectors (3,0) and (1,2)) does not appear to be tiled by a repeat of the primitive cell ${\mathbb{A}'}$ (blue parallelogram spanned by primitive vectors (3,0) and (2,1)). (b) If we shift the top edge of primitive cell ${\mathbb{A}}$ by 3 lattice units, to ${[3\!\times\!2]_{4}}={[3\!\times\!2]_{1}}$ (green parallelogram spanned by primitive vectors (3,0) and (4,2)), the tiling is clear.