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arXiv:2205.07901 · Physical review. B./Physical review. B

Emergent tracer dynamics in constrained quantum systems

Johannes Feldmeier, William Witczak‐Krempa, Michael Knap

Published Sep 12, 2022• 31 citations
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Abstract

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 $t{J}_{z}$ 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.

DOI
10.1103/physrevb.106.094303

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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.
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 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 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.
Understanding the non-equilibrium dynamics of quantum many-body systems remains one of the grand challenges of modern physics. In particular, increasing attention has been devoted to the emergence of non-equilibrium universality classes that have no equilibrium counterparts. A prominent example is the Kardar-Parisi-Zhang universality class realized in dissipative Bose-Einstein condensates. In this Letter, motivated by recent experimental advances, we investigate the universal dynamics of dissipative quantum systems with dipole moment conservation. We develop an effective field theory description, supported by a concrete quantum spin model, to capture the resulting universal behaviors. Our analysis unveils a novel strongly interacting non-equilibrium fixed point that governs the equal-time phase fluctuations in systems with either strong or weak dipole symmetries. Moreover, charge transport becomes subdiffusive in the presence of strong dipole symmetry, while it remains diffusive in the weakly symmetric case. Our results reveal the intricate interplay between kinetic constraints and dissipation in quantum many-body systems.
The spin dynamics of strongly interacting one-dimensional quantum magnets at high-temperature displays an emergent coarse-grained hydrodynamic behavior, understood as the result of conserved quantities. Different kinds of emergent hydrodynamics can be characterized by the dynamical exponent zzz governing the length-time scaling $x\ensuremath{\sim}{t}^{1/z}$. Relying on extensive numerical simulations, the authors extract the exponent zzz for various microscopic quantum spin-SSS models. They identify three universal regimes for the spin transport (superdiffusive with z=3/2z=3/2z=3/2, ballistic with z=1z=1z=1, and diffusive with z=2z=2z=2), which depend only on the isotropy and integrability of the system, but are independent of its microscopic details.