[Paper Review] Dynamics in Systems with Modulated Symmetries

Pablo Sala, Julius Lehmann+2 more|arXiv (Cornell University)|Oct 15, 2021

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper introduces generalized spatially modulated symmetries—exponential and (quasi)-periodic—extending multipole and subsystem symmetries. It demonstrates that such symmetries lead to exotic non-equilibrium dynamics: in 1D, they induce diffusive correlation scaling modulated by a finite momentum; in higher dimensions, they generate sub-diffusive behavior with lattice-scale spatial oscillations and conserved momentum lines/surfaces, while exponential modulation yields infinitely long-lived boundary correlations.

ABSTRACT

We extend the notions of multipole and subsystem symmetries to more general {\it spatially modulated} symmetries. We uncover two instances with exponential and (quasi)-periodic modulations, and provide simple microscopic models in one, two and three dimensions. Seeking to understand their effect in the long-time dynamics, we numerically study a stochastic cellular automaton evolution that obeys such symmetries. We prove that in one dimension, the periodically modulated symmetries lead to a diffusive scaling of correlations modulated by a finite microscopic momentum. In higher dimensions, these symmetries take the form of lines and surfaces of conserved momenta. These give rise to exotic forms of sub-diffusive behavior with a rich spatial structure influenced by lattice-scale features. Exponential modulation, on the other hand, can lead to correlations that are infinitely long-lived at the boundary, while decaying exponentially in the bulk.

Motivation & Objective

• To generalize multipole and subsystem symmetries to broader classes of spatially modulated symmetries, including exponential and (quasi)-periodic forms.
• To construct explicit microscopic models in 1D, 2D, and 3D that realize these symmetries through locally interacting stochastic cellular automata.
• To investigate the long-time dynamics of such systems, particularly the role of modulated symmetries in shaping correlation functions and transport behavior.
• To identify how lattice-scale features influence macroscopic hydrodynamic behavior, especially in systems with non-commuting spatial symmetries.
• To establish a connection between conserved momentum hypersurfaces in momentum space and anomalous spatial decay and oscillations in real space.

Proposed method

• Formalize modulated symmetries via conserved charges of the form $\mathcal{Q}_{\{\alpha_{\bm{r}}\}} = \sum_{\bm{r}} \alpha_{\bm{r}} q_{\bm{r}}$, where $\alpha_{\bm{r}}$ is spatially modulated.
• Use a stochastic cellular automaton framework with local gates $G_x = \{n_{-1}, n_0, n_{+1}\}$ to simulate dynamics and enforce symmetries.
• Derive recurrence relations for $\alpha_j$ such that $\alpha_{j+2} = \frac{p}{q}\alpha_{j+1} - \alpha_j$, ensuring conservation under gate dynamics.
• Analyze real-space correlation decay via inverse Fourier transforms of conserved momentum manifolds in momentum space.
• Apply stationary phase approximation to asymptotic integrals over closed loops in momentum space, yielding $r^{-1/2}$ decay for generic directions.
• Numerically evaluate real-space modulations $\alpha(\bm{r})$ using parametrized integrals over momentum-space hypersurfaces, such as hyperbolic arcs in 2D.

Experimental results

Research questions

• RQ1 How do spatially modulated symmetries—beyond standard multipole or subsystem symmetries—affect the long-time dynamics of quantum systems?
• RQ2 What is the real-space structure and decay behavior of conserved quantities associated with non-uniform momentum-space manifolds?
• RQ3 How does the presence of conserved momentum lines or surfaces in 2D and 3D lead to sub-diffusive or oscillatory correlation dynamics?
• RQ4 Can exponential modulation of symmetry charges lead to infinitely long-lived correlations at system boundaries?
• RQ5 To what extent do lattice-scale features influence the macroscopic hydrodynamics in systems with non-translationally invariant conserved quantities?

Key findings

• In one dimension, periodically modulated symmetries lead to diffusive scaling of correlations modulated by a finite microscopic momentum, with oscillations on microscopic length scales.
• In two and three dimensions, conserved momentum manifolds take the form of closed curves or surfaces, leading to sub-diffusive dynamics with rich spatial oscillations influenced by lattice-scale features.
• For exponentially modulated symmetries, real-space conserved quantities decay as $r^{-1/2}$ in generic directions, with long-lived boundary correlations and exponential decay in the bulk.
• The asymptotic decay of real-space modulations from momentum-space loops follows $r^{-1/2}$ due to stationary phase approximation at points where the direction vector is normal to the loop.
• Even in non-generic cases or when integrating over square regions in momentum space, decay remains slow, with $r^{-1}$ enhancement in some configurations.
• Numerical evaluation of inverse Fourier transforms over parametrized momentum-space curves (e.g., hyperbolic arcs) reveals complex spatial oscillations and slow decay, confirming the analytical predictions.