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arXiv:2108.02205 · Physical Review Letters

Subdiffusion and Many-Body Quantum Chaos with Kinetic Constraints

Hansveer Singh, Brayden Ware, Romain Vasseur, Aaron J. Friedman

Published Dec 1, 2021• 73 citations
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Abstract

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.

DOI
10.1103/physrevlett.127.230602

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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.
We investigate spectral statistics in spatially extended, chaotic many-body quantum systems with a conserved charge. We compute the spectral form factor K(t) analytically for a minimal Floquet circuit model that has a U(1) symmetry encoded via spin-1/2 degrees of freedom. Averaging over an ensemble of realizations, we relate K(t) to a partition function for the spins, given by a Trotterization of the spin-1/2 Heisenberg ferromagnet. Using Bethe ansatz techniques, we extract the "Thouless time" t_{Th} demarcating the extent of random matrix behavior, and find scaling behavior governed by diffusion for K(t) at t≲t_{Th}. We also report numerical results for K(t) in a generic Floquet spin model, which are consistent with these analytic predictions.
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.
Fredkin-type kinetic constraints appear in systems ranging from classical interfaces to th e Fredkin and Motzkin quantum spin chains. Here, we identify a pseudolocal conserved charge associated with Fredkin-constrained dynamics that may be leveraged to subdiffusively bound the dynamical exponent $z\ensuremath{\ge}5/2$. Our results establish that transport in Fredkin-constrained systems belongs to a distinct dynamical universality class that is separate from ordinary diffusion. This universality class is stable to local perturbations that conserve the pseudolocal charge.
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 numerically study the spectral statistics of open quantum many-body systems (OQMBS) as signatures of quantum chaos (or the lack thereof), using the dissipative spectral form factor (DSFF), a generalization of the spectral form factor to complex spectra. We show that the DSFF of chaotic OQMBS generically displays the quadratic\textit{quadratic}quadratic ramp-plateau behaviour of the Ginibre ensemble from random matrix theory, in contrast to the linear ramp-plateau behaviour of the Gaussian ensemble in closed quantum systems. Furthermore, in the presence of many-body interactions, such RMT behaviour emerges only after a time scale τdevτ_{\mathrm{dev}}τdev​, which generally increases with system size for sufficiently large system size, and can be identified as the non-Hermitian analogue of the many-body Thouless time\textit{many-body Thouless time}many-body Thouless time. The universality of the random matrix theory behavior is demonstrated by surveying twelve models of OQMBS, including random Kraus circuits (quantum channels) and random Lindbladians (Liouvillians) in several symmetry classes, as well as Lindbladians of paradigmatic models such as the Sachdev-Ye-Kitaev (SYK), XXZ, and the transverse field Ising models. We devise an unfolding and filtering procedure to remove variations of the averaged density of states which would otherwise hide the universal RMT-like signatures in the DSFF for chaotic OQMBS. Beyond chaotic OQMBS, we study the spectral statistics of non-chaotic OQMBS, specifically the integrable XX model and a system in the many-body localized (MBL) regime in the presence of dissipation, which exhibit DSFF behaviours distinct from the ramp-plateau behaviour of random matrix theory. Lastly, we study the DSFF of Lindbladians with the Hamiltonian term set to zero, i.e. only the jump operators are present, and demonstrate that the results of RMT universality and scaling of many-body Thouless time survive even without coherent evolution.