The presence of global conserved quantities in interacting systems generically leads to diffusive transport at late times. Here, we show that systems conserving the dipole moment of an associated global charge, or even higher-moment generalizations thereof, escape this scenario, displaying subdiffusive decay instead. Modeling the time evolution as cellular automata for specific cases of dipole- and quadrupole conservation, we numerically find distinct anomalous exponents of the late time relaxation. We explain these findings by analytically constructing a general hydrodynamic model that results in a series of exponents depending on the number of conserved moments, yielding an accurate description of the scaling form of charge correlation functions. We analyze the spatial profile of the correlations and discuss potential experimentally relevant signatures of higher-moment conservation.
The presence of global conserved quantities in interacting systems generically leads to diffusive transport at late times. Here, we show that systems conserving the dipole moment of an associated global charge, or even higher-moment generalizations thereof, escape this scenario, displaying subdiffusive decay instead. Modeling the time evolution as cellular automata for specific cases of dipole- and quadrupole conservation, we numerically find distinct anomalous exponents of the late time relaxation. We explain these findings by analytically constructing a general hydrodynamic model that results in a series of exponents depending on the number of conserved moments, yielding an accurate description of the scaling form of charge correlation functions. We analyze the spatial profile of the correlations and discuss potential experimentally relevant signatures of higher-moment conservation.
This work investigates the out-of-equilibrium dynamics of dipole and higher-moment-conserving systems with long-range interactions, drawing inspiration from trapped ion experiments in strongly tilted potentials. We introduce a hierarchical sequence of multipole-conserving models characterized by power-law decaying couplings. Although the moments are always globally conserved, adjusting the power-law exponents of the couplings induces various regimes in which only a subset of multipole moments are effectively locally conserved. We examine the late-time hydrodynamics analytically and numerically using an effective classical framework, uncovering a rich dynamical phase diagram that includes subdiffusion, conventional diffusion, and L\'evy flights. Our results are unified in an analytic reciprocal rule that captures the nested hierarchy of hydrodynamics in multipole conserving systems where only a subset of the moments are locally conserved. Moreover, we extend our findings to higher dimensions and explore the emergence of long timescales, reminiscent of prethermal regimes, in systems with low charge density. Lastly, we corroborate our results through state-of-the-art numerical simulations of a fully quantum long-range dipole-conserving system and discuss their relevance to trapped-ion experimental setups.
We show that the simplest universality classes of fracton hydrodynamics in more than one spatial dimension, including isotropic theories of charge and dipole conservation, can exhibit hidden quasiconservation laws, in which certain higher multipole moments can only decay due to dangerously irrelevant corrections to hydrodynamics. We present two simple examples of this phenomenon. First, an isotropic dipole-conserving fluid in the infinite plane conserves an infinite number of harmonic multipole charges within linear response; we calculate the decay or growth of these charges due to dangerously irrelevant nonlinearities. Second, we consider a model with xy and x^{2}-y^{2} quadrupole conservation, in addition to dipole conservation, which is described by isotropic fourth-order subdiffusion, yet has dangerously irrelevant sixth-order corrections necessary to relax the harmonic multipole charges. We confirm our predictions for the anomalously slow decay of the harmonic conserved charges in each setting by using numerical simulations, both of the nonlinear hydrodynamic differential equations, and in quantum automaton circuits on a square lattice.
In systems with a conserved density, the additional conservation of the center of mass (dipole moment) has been shown to slow down the associated hydrodynamics. At the same time, long-range interactions generally lead to faster transport and information propagation. Here, we explore the competition of these two effects and develop a hydrodynamic theory for long-range center-of-mass-conserving systems. We demonstrate that these systems can exhibit a rich dynamical phase diagram containing subdiffusive, diffusive, and superdiffusive behaviors, with continuously varying dynamical exponents. We corroborate our theory by studying quantum lattice models whose emergent hydrodynamics exhibit these phenomena.
We investigate the coupled dynamics of charge and energy in interacting lattice models with dipole conservation. We formulate a generic hydrodynamic theory for this combination of fractonic constraints and numerically verify its applicability to the late-time dynamics of a specific bosonic quantum system by developing a microscopic nonequilibrium quantum field theory. Employing a self-consistent 1/N approximation in the number of field components, we extract all entries of a generalized diffusion matrix and determine their dependence on microscopic model parameters. We discuss the relation of our results to experiments in ultracold atom quantum simulators.
The scrambling of quantum information in closed many-body systems, as measured by out-of-timeordered correlation functions (OTOCs), has received considerable attention lately.Recently, a hydrodynamical description of OTOCs has emerged from considering random local circuits.Numerical work suggests that aspects of this description are universal to ergodic many-body systems, even without randomness; a conjectured explanation for this is that while the random circuits have noise built into them, deterministic quantum systems, much like classically chaotic ones, "generate their own noise" and look effectively random on sufficient length scales and timescales.In this paper, we extend this approach to systems with locally conserved quantities (e.g., energy).We do this by considering local random unitary circuits with a conserved U(1) charge and argue, with numerical and analytical evidence, that the presence of a conservation law slows relaxation in both time-ordered and out-of-time-ordered correlation functions; both can have a diffusively relaxing component or "hydrodynamic tail" at late times.We verify the presence of such tails also in a deterministic, periodically driven system.We show that for OTOCs, the combination of diffusive and ballistic components leads to a wave front with a specific asymmetric shape, decaying as a power law behind the front.These results also explain existing numerical investigations in non-noisy ergodic systems with energy conservation.Moreover, we consider OTOCs in Gibbs states, parametrized by a chemical potential μ, and apply perturbative arguments to show that for μ ≫ 1 the ballistic front of information spreading can only develop at times exponentially large in μ-with the information traveling diffusively at earlier times.We also develop a new formalism for describing OTOCs and operator spreading, which allows us to interpret the saturation of OTOCs as a form of thermalization on the Hilbert space of operators.
We show how the tracer motion of tagged, distinguishable particles can effectively describe transport in various homogeneous quantum many-body systems with constraints. We consider systems of spinful particles on a one-dimensional lattice subjected to constrained spin interactions, such that some or even all multipole moments of the effective spin pattern formed by the particles are conserved. On the one hand, when all moments---and thus the entire spin pattern---are conserved, dynamical spin correlations reduce to tracer motion identically, generically yielding a subdiffusive dynamical exponent z=4. This provides a common framework to understand the dynamics of several constrained lattice models, including models with XNOR or tJz constraints. We consider random unitary circuit dynamics with such a conserved spin pattern and use the tracer picture to obtain asymptotically exact expressions for their late-time dynamical correlations. Our results can also be extended to integrable quantum many-body systems that feature a conserved spin pattern but whose dynamics is insensitive to the pattern, which includes for example the folded XXZ spin chain. On the other hand, when only a finite number of moments of the pattern are conserved, the dynamics is described by a convolution of the internal hydrodynamics of the spin pattern with a tracer distribution function. As a consequence, we find that the tracer universality is robust in generic systems if at least three multipole moments of the spin pattern (its total charge, dipole moment and quadrupole moment) remain conserved. In cases where only total magnetization and dipole moment of the pattern are constant, we uncover an intriguing coexistence of two processes with an equal dynamical exponent but different scaling functions, which we relate to phase coexistence at a first-order transition.