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arXiv:2209.12949 · Physical Review Letters

Sufficient Condition for Gapless Spin-Boson Lindbladians, and Its Connection to Dissipative Time Crystals

Leonardo da Silva Souza, Luis Fernando dos Prazeres, Fernando Iemini

Published May 3, 2023• 20 citations
arXivHTMLPDF

Abstract

We discuss a sufficient condition for gapless excitations in the Lindbladian master equation for collective spin-boson systems and permutationally invariant systems. The condition relates a nonzero macroscopic cumulant correlation in the steady state to the presence of gapless modes in the Lindbladian. In phases arising from competing coherent and dissipative Lindbladian terms, we argue that such gapless modes, concomitant with angular momentum conservation, can lead to persistent dynamics in the spin observables with the possible formation of dissipative time crystals. We study different models within this perspective, from Lindbladians with Hermitian jump operators, to non-Hermitian ones composed by collective spins and Floquet spin-boson systems. We also provide a simple analytical proof for the exactness of the mean-field semiclassical approach in such systems based on a cumulant expansion.

DOI
10.1103/physrevlett.130.180401

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We discuss a sufficient condition for gapless excitations in the Lindbladian master equation for collective spin-boson systems and permutationally invariant systems. The condition relates a nonzero macroscopic cumulant correlation in the steady state to the presence of gapless modes in the Lindbladian. In phases arising from competing coherent and dissipative Lindbladian terms, we argue that such gapless modes, concomitant with angular momentum conservation, can lead to persistent dynamics in the spin observables with the possible formation of dissipative time crystals. We study different models within this perspective, from Lindbladians with Hermitian jump operators, to non-Hermitian ones composed by collective spins and Floquet spin-boson systems. We also provide a simple analytical proof for the exactness of the mean-field semiclassical approach in such systems based on a cumulant expansion.
Boundary time crystals (BTCs) are paradigmatic examples of continuous time crystals that emerge in collective, dissipative quantum systems. Their microscopic origin, however, remains obscured by the complexity of Lindbladian many-body dynamics and has largely been investigated through semiclassical and numerical approaches. Here, we develop an exact operator space framework for collective spin Lindbladians based on irreducible tensor representations. We map the Lindbladian dynamics onto a local non-Hermitian hopping problem, where weak symmetries impose selection rules on operator transport, while the hopping amplitudes and their relative strengths determine the resulting dynamical regimes. We identify the microscopic ingredients that distinguish models supporting conventional collective precession from those supporting BTC dynamics. Our approach further reveals how non-unital dissipation can provide a source for the traceless operator subspace, providing a microscopic mechanism for the reduced sensitivity of BTC dynamics to initial conditions beyond mean-field descriptions. More broadly, our results establish operator space transport as a general perspective for understanding dissipative many-body dynamics and its connections to non-Hermitian phenomena.
Strong dynamical symmetries provide an algebraic description of persistent oscillations in open quantum systems governed by the Lindblad equation. For finite-dimensional systems with a full-rank stationary state, such symmetries are necessary and sufficient for non-decaying oscillatory modes to appear in the Liouvillian's spectrum. These modes can be directly constructed by acting on stationary states with the identified symmetry operators. Here we establish an asymptotic counterpart of this algebra for collective spins. Taking a single spin SSS in a magnetic field, subject to dissipation through linear jump operators, we identify the exact conditions on the field and dissipative couplings under which non-decaying oscillatory modes emerge as S→∞S \rightarrow \inftyS→∞. We then write down a simple, closed form for the corresponding `symmetry' operator which can be applied to the steady state to construct these modes, which are eigenvectors of the Liouvillian, with imaginary eigenvalues, up to O(1/S)O(1/S)O(1/S) corrections. A second operator, an emergent conserved quantity, completes the algebra, and its slow relaxation determines how quantum noise selects the stationary state. The construction can be directly extended to a pair of interacting spins sharing a common reservoir, with the resulting quantum algebra making the entanglement during a quench analytically accessible.
Dissipative time crystals can appear in spin systems, when the Z2{Z}_{2}Z2​ symmetry of the Hamiltonian is broken by the environment, and the square of total spin operator S2{S}^{2}S2 is conserved. In this paper, we relax the latter condition and show that time-translation-symmetry-breaking collective oscillations persist, in the thermodynamic limit, even in the absence of spin symmetry. We engineer an ad hoc Lindbladian using power-law-decaying spin operators and show that time-translation-symmetry breaking appears when the decay exponent obeys $0<\ensuremath{\eta}\ensuremath{\le}1$. This model shows a surprisingly rich phase diagram, including the time-crystal phase as well as first-order, second-order, and continuous transitions of the fixed points. We study the phase diagram and the magnetization dynamics in the mean-field approximation. We prove that this approximation is quantitatively accurate, when $0<\ensuremath{\eta}<1$ and the thermodynamic limit is taken, because the system does not develop sizable quantum fluctuations, if the Gaussian approximation is considered.
We investigate how symmetries and conserved quantities relate to the occurrence of the boundary time crystal (BTC) phase in a generalized spin model with Lindblad dissipation. BTCs are a nonequilibrium phase of matter in which the system, coupled to an external environment, breaks the continuous time translational invariance. We perform a detailed mean-field study aided by a finite-size analysis of the quantum model of a p,qp,qp,q-spin-interaction system, a generalized ppp-spin-interaction system, which can be implemented in fully connected spin-1/2 ensembles. We find the following conditions for the observation of the BTC phase. First, the BTC appears when the discrete symmetry held by the Hamiltonian, Z2{\mathbb{Z}}_{2}Z2​ in the considered models, is explicitly broken by the Lindblad jump operators. Second, the system must be coupled uniformly to the same bath in order to preserve the total angular momentum during the time evolution. If these conditions are not satisfied, any oscillatory behavior appears only as a transient in the dynamics and a time-independent stationary state is eventually reached. Our results suggest that these two elements may be general requirements for the observation of a stable BTC phase relating symmetries and conserved quantities in arbitrary spin models.
We characterize nonequilibrium phases in long-range dissipative spin systems through the statistical properties of quantum jump trajectories. While the average dynamics governed by the Lindblad master equation provides access to steady-state expectation values of order parameters, the quantum trajectory framework reveals features encoded in the spatial and temporal correlations of detection events. Focusing on a model exhibiting a paramagnetic-to-ferromagnetic phase transition, we investigate the full counting statistics of quantum jumps using a tilted Lindbladian approach. We combine this with cluster mean-field and cumulant expansion techniques, which allow us to capture, respectively, the short- and long-range structure of jump correlations. In addition, we study the waiting-time distributions of detection events. We show that quantum jump correlations display clear signatures of the underlying phases and reveal distinct dynamical features across the transition. Our results highlight the potential of trajectory-resolved observables as probes of collective behavior in open quantum many-body systems and provide new insights into the role of long-range interactions in shaping nonequilibrium dynamics.
We investigate the emergence of a time crystal (TC) in a driven dissipative many-body spin array. In this system the interplay between incoherent spin pumping and collective emission stabilizes a synchronized non-equilibrium steady state which in the thermodynamic limit features a self-generated time-periodic pattern imposed by collective elastic interactions. In contrast to prior realizations where the time symmetry is already broken by an external drive, here it is only spontaneously broken by the elastic exchange interactions and manifest in the two-time correlation spectrum. Employing a combination of exact numerical calculations and a second-order cumulant expansion, we investigate the impact of many-body correlations on the TC formation and establish a connection between the regime where it is stable and where the system features a slow growth rate of the mutual information. This observation allows us to conclude that the TC studied here is an emergent semi-classical out-of-equilibrium state of matter. We also confirm the rigidity of the TC to single-particle dephasing. Finally, we discuss an experimental implementation using long lived dipoles in an optical cavity.