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Addendum to "Dynamics of Renyi entanglement entropy in local quantum circuits with charge conservation''

Yichen Huang

Published Jul 27, 2020• 1 citation
OpenAlex

Abstract

In this addendum to [Huang, arXiv:1902.00977], I explicitly present results for qudit systems, i.e., spin systems with constant local dimension $d\ge2$.

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In this addendum to [Huang, arXiv:1902.00977], I explicitly present results for qudit systems, i.e., spin systems with constant local dimension d≥2d\ge2d≥2.
Abstract My previous work [arXiv: 1902.00977 ] studied the dynamics of Rényi entanglement entropy R α in local quantum circuits with charge conservation. Initializing the system in a random product state, it was proved that R α with Rényi index α > 1 grows no faster than ‘diffusively’ (up to a sublogarithmic correction) if charge transport is not faster than diffusive. The proof was given only for qubit or spin-1/2 systems. In this note, I extend the proof to qudit systems, i.e., spin systems with local dimension d ≥ 2 .
In local quantum circuits with charge conservation, we initialize the system in random product states and study the dynamics of the Renyi entanglement entropy RαR_αRα​. We rigorously prove that RαR_αRα​ with Renyi index α>1α>1α>1 at time ttt is ≤O(tln⁡t)\le O(\sqrt{t\ln t})≤O(tlnt​) if the transport of charges is diffusive. Very recent numerical results of Rakovszky et al. show that this upper bound is saturated (up to the sub-logarithmic correction) in random local quantum circuits with charge conservation.
Abstract Entanglement helps in understanding diverse phenomena, going from quantifying complexity to classifying phases of matter. Here we study the influence of conservation laws on entanglement growth. Focusing on systems with U(1) symmetry, i.e., conservation of charge or magnetization, that exhibits diffusive dynamics, we theoretically predict the growth of entanglement, as quantified by the Rényi entropy, in lattice systems in any spatial dimension d and for any local Hilbert space dimension q (qudits). We find that the growth depends both on d and q, and is in generic case first linear in time, similarly as for systems without any conservation laws. Exception to this rule are chains of 2-level systems where the dependence is a square-root of time at all times. Predictions are numerically verified by simulations of diffusive Clifford circuits with upto ~ 105 qubits. Such efficiently simulable circuits should be a useful tool for other many-body problems.