Classical approaches to prethermal discrete time crystals in one, two, and three dimensions

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Classical approaches to prethermal discrete time crystals in one, two, and three dimensions

Andrea Pizzi, Andreas Nunnenkamp, and Johannes Knolle

Phys. Rev. B 104, 094308 – Published 27 September 2021

See Viewpoint: A Classical View of Quantum Time Crystals

Abstract

We provide a comprehensive account of prethermal discrete time crystals within classical Hamiltonian dynamics, complementing and extending our recent work [A. Pizzi, A. Nunnenkamp, and J. Knolle, Phys. Rev. Lett. 127, 140602 (2021)]. Considering power-law interacting spins on one-, two-, and three-dimensional hypercubic lattices, we investigate the interplay between dimensionality and interaction range in the stabilization of these nonequilibrium phases of matter that break the discrete time-translational symmetry of a periodic drive.

Received 15 August 2021
Accepted 18 August 2021

DOI:https://doi.org/10.1103/PhysRevB.104.094308

Physics Subject Headings (PhySH)

Chaos Classical mechanics Dynamical phase transitions Exotic phases of matter Nonequilibrium statistical mechanics

Collective dynamics Floquet systems

Condensed Matter, Materials & Applied PhysicsNonlinear DynamicsStatistical Physics & Thermodynamics

Viewpoint

A Classical View of Quantum Time Crystals

Published 27 September 2021

Numerical studies indicate that certain types of time crystals might be described using classical physics—a result that could vastly simplify the theoretical description of these systems.

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Authors & Affiliations

Andrea Pizzi ¹, Andreas Nunnenkamp ², and Johannes Knolle ^3,4,5

¹Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, England, United Kingdom
²School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, England, United Kingdom
³Department of Physics, Technische Universität München, James-Franck-Straße 1, D-85748 Garching, Germany
⁴Munich Center for Quantum Science and Technology, 80799 Munich, Germany
⁵Blackett Laboratory, Imperial College London, London SW7 2AZ, England, United Kingdom

Classical Prethermal Phases of Matter

Andrea Pizzi, Andreas Nunnenkamp, and Johannes Knolle

Phys. Rev. Lett. 127, 140602 (2021)

Floquet Phases of Matter via Classical Prethermalization

Bingtian Ye, Francisco Machado, and Norman Y. Yao

Phys. Rev. Lett. 127, 140603 (2021)

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Issue

Vol. 104, Iss. 9 — 1 September 2021

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Figure 1
Prethermal time crystals of (classical) spins on a hypercubic lattice. (a) Schematic of the system for dimensionality $D = 2$ . The spins are accommodated on a hypercubic lattice and subject to intermitting power-law interactions with characteristic exponent $α$ and longitudinal and transverse fields. (b) The Hamiltonian ruling the system dynamics is binary: the $z z$ interaction and longitudinal $z$ field (red) are alternated with the longitudinal $x$ field (blue). (c) Initially ( $t = 0$ ), each spin is moved away from the north pole by a random Gaussian polar angle $θ$ with zero mean and standard deviation $2 π W$ . The perturbed copy of the system ${S^{'}}$ differs from ${S}$ of an amount $\approx Δ ≪ W, 1$ . (d) Phenomenology of a prethermal $4 − DTC$ . With a one-to-one sphere-to-color mapping, we unambiguously represent the state of a two-dimensional grid of $N = 200^{2}$ spins at some representative times. With a period four-tupled response, the spins are mostly polarized along $+ z, - y, - z, y, + z, \dots$ at times $t / T = 4 k, 4 k + 1, 4 k + 2, 4 k + 3, 4 k + 4, \dots$ , respectively. This is visualized by alternating predominant colorations red, yellow, green, blue, red, etc. This subharmonic response holds across a long prethermal regime that extends beyond the timescale characterizing chaos, $τ_{preth} \sim \frac{1}{λ}$ , with $λ$ the Lyapunov exponent. Only at a very long time $t = 10^{5} \sim τ_{th}$ does the system order break, with the nucleation and proliferation of domains of opposite magnetizations. The infinite-temperature state with random spin orientations is reached at later times $t = 10^{7} T ≫ τ_{th}$ . Here, $g = 0.255, h = 0.1, ω = 3.14, α = \infty, R = 1,$ and $W = 0.1$ .
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Figure 2
Quantum vs classical: phenomenology of a prethermal discrete time crystal. We show that the phenomenology of a prethermal DTC is essentially the same in classical and quantum systems by drawing a close analogy with Fig. 1 in Ref. [22]. As observables, we consider (a) the energy $H_{T}$ averaged over one period, (b) the decorrelator $D$ measuring the distance between two initially very similar copies of the system (renormalized by its infinite-temperature value $D_{\infty} = \sqrt{2}$ ), and (c) the magnetization $m = {〈 s_{i}^{z} 〉}_{i, runs}$ . By making the association decorrelator $\leftrightarrow$ entanglement entropy, the full phenomenology of the quantum prethermal $2 − DTC$ described in Ref. [22] is recovered: (i) for a short-range model ( $α = \infty$ , left column) heating occurs—over an exponentially long (in frequency) timescale—the signature of “standard” prethermalization; (ii) for a long-range model and “cold” initial condition ( $α = 1.5$ and $W = 0.1$ , central column), prethermalization comes with the realization of a nontrivial time-crystalline (subharmonic) response of the magnetization $m$ ; (iii) with long-range interactions, standard prethermalization is recovered for a “hot” initial condition ( $α = 1.5$ and $W = 0.2$ , right column). Here, we used $N = 100, R = 100, h = 0.1, g = 0.515, Δ = 0.01$ .
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Figure 3
Interplay of dimensionality and interaction range in prethermal discrete time crystals.. We characterize the prethermal $4 − DTC$ by looking at (a) the energy $H_{1}$ (at stroboscopic times $t = 4 n T$ ), (b) the magnetization $m = {〈 s_{i}^{z} 〉}_{i, runs}$ , and (c) the decorrelator $d$ . In one dimension (left column), a prethermal response emerges for sufficiently long-ranged interactions ( $α ⪅ 2$ ). The thermalization as well as the prethermalization times increase with decreasing $α$ (d). The phenomenology is similar in two dimensions, where the separation of timescales between prethermalization and thermalization can be well appreciated for a broader range of $α ⪅ 6$ . In three dimensions, crucially, the separation of timescales extends all the way to $α = \infty$ . Here, we used $N = 200, 400, 343$ in $D = 1, 2,$ and 3, respectively, and $ω = 2.2, h = 0.1, g = 0.26$ , and $Δ = 0.01$ . Solid lines and errorbars are obtained as averages and standard deviations over $R = 50$ independent runs.
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Figure 4
Scaling with frequency. (a–c) We investigate the dependency of the prethermal $4 − DTC$ on the frequency $ω$ by means of the same diagnostics as in Fig. 3. For an interaction range $α = 1.5, 4,$ and $\infty$ in $D = 1, 2$ , and 3, respectively, the observed phenomenology is the same: prethermalization occurs over a timescale $τ_{preth} \approx 1 / λ$ that barely depends on the frequency $ω$ , whereas the full thermalization to an infinite-temperature state occurs at a much later time $τ_{th} \sim e^{c ω}$ . (d) The two timescales $τ_{preth}$ and $τ_{th}$ are extracted as the times at which the decorrelator crosses the 10 and $90 %$ of its maximum value $D_{\infty}$ , that is, the times at which its prethermal plateau can be considered to start and end. For this figure we used the same parameters as in Fig. 3.
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Figure 5
Phase diagrams. We investigate the stability of the prethermal $4 − DTC$ for dimensionality $D = 1$ (left), 2 (center), and 3 (right). As a diagnosis for the $4 − DTC$ we use the subharmonic spectral function, that is, the Fourier transform $\tilde{m}$ of the magnetization at frequency $ω / 4$ , computed over the first $10^{4}$ periods. (a–c) Phase diagrams in the plane of the transverse field strength $g$ and power-law exponent $α$ . In one and two dimensions (a, b) the $4 − DTC$ is stable for interactions that are sufficiently long range, that is, for $α < α_{c}$ , with $α_{c} \approx 2$ and 6 for $D = 1$ and 2, respectively. In striking contrast, in $D = 3$ dimensions (c) the $4 − DTC$ extends all the way up to $α = \infty$ , that is, to the short-range (nearest-neighbor) limit. (d, e) Phase diagrams in the plane of transverse field $g$ and drive period $T = 2 π / ω$ . The phase diagram looks qualitatively the same in all dimensions: the frequency should be large enough for the $4 − DTC$ to be stable (in a prethermal fashion), but the larger the frequency and the smaller the range of $g$ over which the $4 − DTC$ is stable. Here, we used $N = 150, 144,$ and 216 in $D = 1, 2,$ and 3, respectively; $T = 2.5$ in (a)–(c); $α = 1.5, 4,$ and $inf$ in (d)–(f), respectively; $R = 50$ ; $h = 0.1$ ; and $W = 0.1$ .
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Figure 6
Stability against perturbation of the initial condition. We investigate the stability of the prethermal $4 − DTC$ with respect to perturbations of the initial condition, for dimensionality $D = 1, 2,$ and 3 and for power-law exponent $α = 1.5, 4$ , and $\infty$ , respectively. As in Fig. 5, we compute the subharmonic spectral function at frequency $ω / 4$ over the first 5000 periods, and use it as an order parameter for the $4 − DTC$ . We observe a phase transition at a certain finite critical value of the noise strength $W_{c} \sim 0.18$ , above which the prethermal DTC disappears in favor of trivial thermalization. Note, the observed fluctuations are a finite-size effect. Here, we used $N = 2000, 2500,$ and 8000 in $D = 1, 2,$ and 3, respectively, whereas $T = 2.5$ , $R = 20$ , $g = 0.25$ , and $h = 0.1$ .
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