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arXiv:2101.05710 · Physical review. B./Physical review. B

Symmetries and conserved quantities of boundary time crystals in generalized spin models

G. Piccitto, Matteo M. Wauters, Franco Nori, Nathan Shammah

Published Jul 19, 2021• 58 citations
arXivHTMLPDF

Abstract

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 $p,q$-spin-interaction system, a generalized $p$-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, ${\mathbb{Z}}_{2}$ 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.

DOI
10.1103/physrevb.104.014307

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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.
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.
Boundary time crystals (BTC's) are nonequilibrium phases of matter occurring in quantum systems in contact to an environment, for which a macroscopic fraction of the many-body system breaks time translation symmetry. We study BTC's in collective ddd-level systems, focusing on the cases with d=2d=2d=2, 3, and 4. We find that BTC's appear in different forms for the different cases. We first consider the model with collective d=2d=2d=2-level systems [Phys. Rev. Lett. 121, 035301 (2018)], whose dynamics is described by a Gorini-Kossakowski-Sudarshan-Lindblad master equation, and perform a throughout analysis of its phase diagram and Jacobian stability for different interacting terms in the coherent Hamiltonian. In particular, using perturbation theory for general (non-Hermitian) matrices, we obtain analytically how a specific Z2{\mathbb{Z}}_{2}Z2​ symmetry-breaking Hamiltonian term destroys the BTC phase in the model. Based on these results we define a d=4d=4d=4 model composed of a pair of collective two-level systems interacting with each other. We show that this model support richer dynamical phases, ranging from limit cycles, period-doubling bifurcations, and a route to chaotic dynamics. The BTC phase is more robust in this case, not annihilated by the former symmetry-breaking Hamiltonian terms. The model with collective d=3d=3d=3-level systems is defined similarly, as competing pairs of levels, but sharing a common collective level. The dynamics can deviate significantly from the previous cases, supporting phases with the coexistence of multiple limit cycles, closed orbits and a full degeneracy of zero Lyapunov exponents.
We demonstrate the existence of a dynamical quantum phase transition (DQPT) in a dissipative collective-spin model that exhibits the boundary time crystal (BTC) phase. We initialize the system in the ground state of the Hamiltonian in either the BTC or the non-BTC phase, and drive it across the BTC transition. The driving is done by an abrupt quench or by a finite-time linear ramp of a Hamiltonian control parameter under Markovian Lindblad dynamics. We diagnose DQPTs through zeros of the fidelity-based Loschmidt echo between the initial state and the evolving mixed state, which induce nonanalytic cusp-like features in the associated rate function. For quenches into the BTC phase, the Loschmidt echo exhibits repeated zeros due to the emergent time-periodic steady state, whereas for quenches into the non-BTC phase, the overlap vanishes and remains zero once the dynamics relaxes to a stationary state. We further show that the DQPT persists under the ramp protocol followed by unitary evolution with the final Hamiltonian. Finally, we analyze the finite-size scaling of the first critical time and find convergence to a constant in the thermodynamic limit, with distinct power-law approaches for the quench and the ramp protocols.
Abstract Open many-body quantum systems can exhibit intriguing nonequilibrium phases of matter, such as time crystals. In these phases, the state of the system spontaneously breaks the time-translation symmetry of the dynamical generator, which typically manifests through persistent oscillations of an order parameter. A paradigmatic model displaying such a symmetry breaking is the boundary time crystal (BTC), which has been extensively analyzed experimentally and theoretically. Despite the broad interest in these nonequilibrium phases, their thermodynamics and their fluctuating behavior remain largely unexplored, in particular for the case of coupled time crystals. In this work, we consider two interacting BTCs and derive a consistent interpretation of their thermodynamic behavior. We fully characterize their average dynamics and the behavior of their quantum fluctuations, which allows us to demonstrate the presence of quantum and classical correlations in both the stationary and the time-crystal phases displayed by the system. We furthermore exploit our theoretical derivation to explore possible applications of time crystals as quantum batteries, demonstrating their ability to efficiently store energy.
Driven-dissipative many-body systems support nontrivial quantum phases absent in equilibrium. As a prominent example, the interplay between coherent driving and collective dissipation can lead to a dynamical quantum phase that spontaneously breaks time-translation symmetry. This so-called boundary time crystal (BTC) is fragile in the presence of local dissipation, which can easily relax the system to a stationary state. In this Letter, we demonstrate a robust BTC that is intrinsically induced by local dissipation. We provide extensive numerical evidence to support the existence of the BTC and study its behaviors in different regimes. In particular, with decreasing interaction range, we identify a transition from mean-field limit cycles to correlated BTCs featuring sizable quantum correlations. Our studies significantly broaden the scope of nonequilibrium phases and shed new light on the experimental search for dynamical quantum matter.
Boundary time crystals (BTCs) break time-translation symmetry and exhibit long-lived, robust oscillations insensitive to initial conditions. We show that collective spin BTCs can admit emergent topological winding numbers in operator space. Expanding the density operator in a spherical tensor basis, we map the Lindblad dynamics onto an effective local hopping problem, where collective degrees of freedom label sites of an emergent two-dimensional operator space lattice and identify topological obstructions that enforce the delocalization of operator modes on the lattice. The resulting spectral delocalization provides a natural explanation for the robust oscillatory dynamics observed in BTCs. When combined with non-reciprocal transport of operator weight across operator space, this mechanism moreover also leads to the universality of long-time dynamics across a broad class of initial states. Our results frame BTC dynamics as a form of topologically constrained operator space transport and suggest a close connection to non-Hermitian skin-effects.