Signatures of a quantum stabilized fluctuating phase and critical dynamics in a kinetically constrained open many-body system with two absorbing states

Federico Carollo, Markus Gnann, Gabriele Perfetto, and Igor Lesanovsky

Phys. Rev. B 106, 094315 – Published 29 September 2022

Abstract

We introduce and investigate an open many-body quantum system in which kinetically constrained coherent and dissipative processes compete. The form of the incoherent dissipative dynamics is inspired by that of epidemic spreading or cellular-automaton-based computation related to the density-classification problem. It features two nonfluctuating absorbing states as well as a $Z_{2}$ -symmetric point in parameter space. The coherent evolution is governed by a kinetically constrained $Z_{2}$ -symmetric many-body Hamiltonian which is related to the quantum XOR-Fredrickson-Andersen model. We show that the quantum coherent dynamics can stabilize a fluctuating state and we characterize the transition between this active phase and the absorbing states. We also identify a rather peculiar behavior at the $Z_{2}$ -symmetric point. Here the system approaches the absorbing-state manifold with a dynamics that follows a power law whose exponent continuously varies with the relative strength of the coherent dynamics. Our work shows how the interplay between coherent and dissipative processes as well as symmetry constraints may lead to a highly intricate nonequilibrium evolution and may stabilize phases that are absent in related classical problems.

Received 5 May 2022
Revised 22 August 2022
Accepted 20 September 2022

DOI:https://doi.org/10.1103/PhysRevB.106.094315

Physics Subject Headings (PhySH)

Irreversible processes Nonequilibrium statistical mechanics Open quantum systems Phase transitions

Nonequilibrium lattice models Quantum spin models

Mean field theory Tensor network methods

Statistical Physics & ThermodynamicsGeneral Physics

Authors & Affiliations

Federico Carollo ¹, Markus Gnann ¹, Gabriele Perfetto ¹, and Igor Lesanovsky ^1,2

¹Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
²School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, The University of Nottingham, Nottingham, NG7 2RD, United Kingdom

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Issue

Vol. 106, Iss. 9 — 1 September 2022

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Images

Figure 1
Open quantum system with two absorbing states. (a) Quantum chain made of two-level sites, which can assume the states $•$ and $\circ$ . Each site can undergo a (classical) incoherent state change $\circ / • ↭ • / \circ$ , occurring at a rate $γ_{• / \circ} = γ (1 \pm κ)$ per neighbor in the state $• / \circ$ [see Eq. (4)]. A site can further coherently change its state with rate (“Rabi frequency”) $Ω (1 - α)$ , if its neighbors are both in the same state, or $Ω (1 + α)$ otherwise [see Eq. (2)]. (b) This system is dual to a domain-wall model, with kink ( $\circ •$ ) domain-wall particles A and antikink ( $• \circ$ ) domain-wall particles B. For $α = 1$ (absorbing-state regime), these can only hop, both coherently and incoherently, and annihilate with rates $2 γ_{• / \circ}$ . (c) For large (in modulus) values of the parameter $Γ = κ γ^{2} / Ω^{2}$ and $α = 1$ the stationary state is one of two absorbing states, according to the sign of $Γ$ . For small $| Γ |$ , instead, a stationary fluctuating phase emerges, as predicted by Eq. (5). (d) The density of domain-wall particles $n_{DW}$ [cf. panel (b)] decays with time $t$ as a power law for $α = 1$ and $κ = 0$ . The decay law varies from a diffusive behavior for $Ω / γ = 0$ to a superdiffusive one, $t^{- δ}$ with $1 / 2 < δ < 1$ , for $Ω / γ \neq 0$ [see Fig. 4 below].
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Figure 2
Mean-field phase diagram. (a) Magnetization $Z_{SS}$ in stationary state I as a function of $α$ and $Γ$ . (b) Same as in panel (a) for stationary state II, which also provides the fluctuating phase shown in Fig. 1 for $α = 1$ . Highlighted in the plot are the $α = 1$ line, the absorbing-state phase transition points [cf. Fig. 1], the two Ising-like critical points, and the associated direction manifesting Ising criticality. (c) Stationary magnetization $Z_{SS}$ along the dashed line crossing the Ising critical point at $Γ > 0$ in panel (b).
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Figure 3
Single-cluster dynamics. (a) Probability $P_{•} (t)$ of finding a cluster of length $L$ as a function of time and of $Ω / γ$ , for $κ = 0.5$ and $L = 50$ . (b) Stationary behavior of $P_{•}$ for a system of $L = 50$ sites, as a function of $Ω / γ$ and $κ$ . (c) Stationary probability $1 - P_{•}$ as a function of $L$ for $Ω / γ = 1.5, 2, 2.5$ and $κ = 0.5$ . (d) Behavior of the single-seed survival probability $S$ as a function of $t$ for $Ω / γ = 1$ , $κ = 0$ , and different values of $L$ . Upon increasing $L$ , a power-law behavior $\approx t^{- 1 / 2}$ is approached.
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Figure 4
Many-body tensor-network simulations. (a) Magnetization $Z$ , starting from the Néel state, for $α = 1$ and $κ = 0.1$ . The different sets of curves are for $Ω / γ = 0, 0.5, 2, 3$ . For each $Ω / γ$ , we display results for $L = 4, 6, \dots, 14$ . For sufficiently large values of $Ω / γ$ , upon increasing the system size $L$ , a fluctuating phase in which the magnetization remains small emerges. The inset shows an estimated phase diagram. (b) Estimate—fit of the numerical data for $γ t \in [1, 2]$ —of the dynamical critical exponent $δ$ as a function of $Ω / γ$ , for $κ = 0$ and $L = 14$ . The insets show, in a log-log plot, the domain-wall density $n_{DW}$ as a function of time for $L = 4, 6, \dots, 14$ , with $Ω / γ = 0$ (left) and $Ω / γ = 1$ (right). The results presented are for a bond dimension $χ = 64$ .
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