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arXiv:2208.07480 · Physical review. B./Physical review. B

Distinction between transport and Rényi entropy growth in kinetically constrained models

Zhi-Cheng Yang

Published Dec 7, 2022• 7 citations
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Abstract

Conservation laws and the associated hydrodynamic modes have important consequences on the growth of higher R\'enyi entropies in isolated quantum systems. It has been shown in various random unitary circuits and Hamiltonian systems that the dynamics of the R\'enyi entropies in the presence of a U(1) symmetry obey ${S}^{(n\ensuremath{\ge}2)}(t)\ensuremath{\propto}{t}^{1/z}$, where $z$ is identified as the dynamical exponent characterizing transport of the conserved charges. Here, however, we demonstrate that this simple identification may not hold in certain quantum systems with kinetic constraints. In particular, we study two types of U(1)-symmetric quantum automaton circuits with XNOR (exclusive NOR) and Fredkin constraints, respectively. We find numerically that while spin transport in both models is subdiffusive, the second R\'enyi entropy grows diffusively in the XNOR model, and superdiffusively in the Fredkin model. For systems with the XNOR constraint, this distinction arises since the spin correlation function can be attributed to an emergent tracer dynamics of tagged particles, whereas the R\'enyi entropies are constrained by collective transport of the particles. Our results suggest that care must be taken when relating transport and entanglement entropy dynamics in generic quantum systems with conservation laws.

DOI
10.1103/physrevb.106.l220303

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Conservation laws and the associated hydrodynamic modes have important consequences on the growth of higher R'enyi entropies in isolated quantum systems. It has been shown in various random unitary circuits and Hamiltonian systems that the dynamics of the R'enyi entropies in the presence of a U(1) symmetry obey S(n\ensuremath≥2)(t)\ensuremath∝t1/z{S}^{(n\ensuremath{\ge}2)}(t)\ensuremath{\propto}{t}^{1/z}S(n\ensuremath≥2)(t)\ensuremath∝t1/z, where zzz is identified as the dynamical exponent characterizing transport of the conserved charges. Here, however, we demonstrate that this simple identification may not hold in certain quantum systems with kinetic constraints. In particular, we study two types of U(1)-symmetric quantum automaton circuits with XNOR (exclusive NOR) and Fredkin constraints, respectively. We find numerically that while spin transport in both models is subdiffusive, the second R'enyi entropy grows diffusively in the XNOR model, and superdiffusively in the Fredkin model. For systems with the XNOR constraint, this distinction arises since the spin correlation function can be attributed to an emergent tracer dynamics of tagged particles, whereas the R'enyi entropies are constrained by collective transport of the particles. Our results suggest that care must be taken when relating transport and entanglement entropy dynamics in generic quantum systems with conservation laws.
We investigate the dynamics of quantum entanglement after a global quench and uncover a qualitative difference between the behavior of the von Neumann entropy and higher Rényi entropies. We argue that the latter generically grow sub-ballistically, as ∝sqrt[t], in systems with diffusive transport. We provide strong evidence for this in both a U(1) symmetric random circuit model and in a paradigmatic nonintegrable spin chain, where energy is the sole conserved quantity. We interpret our results as a consequence of local quantum fluctuations in conserved densities, whose behavior is controlled by diffusion, and use the random circuit model to derive an effective description. We also discuss the late-time behavior of the second Rényi entropy and show that it exhibits hydrodynamic tails with three distinct power laws occurring for different classes of initial states.
We study the entanglement dynamics of quantum automaton (QA) circuits in the presence of U(1) symmetry. We find that the second Rényi entropy grows diffusively with a logarithmic correction as tln⁡t , saturating the bound established by Huang \cite{Huang_2020}. Thanks to the special feature of QA circuits, we understand the entanglement dynamics in terms of a classical bit string model. Specifically, we argue that the diffusive dynamics stems from the rare slow modes containing extensively long domains of spin 0s or 1s. Additionally, we investigate the entanglement dynamics of monitored QA circuits by introducing a composite measurement that preserves both the U(1) symmetry and properties of QA circuits. We find that as the measurement rate increases, there is a transition from a volume-law phase where the second Rényi entropy persists the diffusive growth (up to a logarithmic correction) to a critical phase where it grows logarithmically in time. This interesting phenomenon distinguishes QA circuits from non-automaton circuits such as U(1)-symmetric Haar random circuits, where a volume-law to an area-law phase transition exists, and any non-zero rate of projective measurements in the volume-law phase leads to a ballistic growth of the Rényi entropy.
We investigate how transport properties of U(1)U(1)U(1)-conserving dynamics impact the growth of quantum resources characterizing the complexity of many-body states. We quantify wave-function delocalization using participation entropy (PE), a measure rooted in the coherence theory of pure states, and assess nonstabilizerness through stabilizer Rényi entropy (SRE). Focusing on the XXZ spin chain initialized in domain-wall state, we demonstrate universal power-law growth of both PE and SRE, with scaling exponents explicitly reflecting the underlying transport regimes, ballistic, diffusive, or KPZ-type superdiffusive. Our results establish a solid connection between quantum resources and transport, providing insights into the dynamics of complexity within symmetry-constrained quantum systems.
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=4z=4z=4. This provides a common framework to understand the dynamics of several constrained lattice models, including models with XNOR or tJzt{J}_{z}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.
We study how conservation laws shape the spreading of quantum coherence in many-body dynamics. Focusing on U(1)U(1)U(1)-symmetric random circuits, charge-and-dipole conserving circuits, as well as ergodic Hamiltonian dynamics, we probe coherences both globally, via the participation entropy, and locally, via the relative entropy of coherence. Combining exact vector evolution, matrix product state simulations, and replica tensor networks methods, we find that conservation laws replace the logarithmic saturation of unconstrained circuits with slow hydrodynamic relaxation of the global coherence measures. Locally, symmetry-constrained circuits show a clean rise-peak-fall structure whose peak time grows algebraically with subsystem size. In contrast, ergodic Hamiltonians broaden the peak into an extended plateau at larger subsystems, highlighting a qualitatively distinct mechanism. Coherence thus emerges as a sensitive probe of symmetry-constrained thermalization, linking quantum resource dynamics to many-body transport.
We investigate the spectral and transport properties of many-body quantum systems with conserved charges and kinetic constraints. Using random unitary circuits, we compute ensemble-averaged spectral form factors and linear-response correlation functions, and find that their characteristic timescales are given by the inverse gap of an effective Hamiltonian-or equivalently, a transfer matrix describing a classical Markov process. Our approach allows us to connect directly the Thouless time, t_{Th}, determined by the spectral form factor, to transport properties and linear-response correlators. Using tensor network methods, we determine the dynamical exponent z for a number of constrained, conserving models. We find universality classes with diffusive, subdiffusive, quasilocalized, and localized dynamics, depending on the severity of the constraints. In particular, we show that quantum systems with "Fredkin" constraints exhibit anomalous transport with dynamical exponent z≃8/3.