Time-crystalline behavior in an engineered spin chain

Robin Schäfer, Götz S. Uhrig, and Joachim Stolze

Phys. Rev. B 100, 184301 – Published 5 November 2019

Abstract

Time crystals break the discrete time translational invariance of an external periodic drive by oscillating at an integer multiple of the driving period. In addition to this fundamental property, other aspects are often considered to be essential characteristics of a time crystal, such as the presence of disorder or interactions, robustness against small variations of system parameters, and the free choice of the initial quantum state. We study a finite-length polarized XX spin chain engineered to display a spectrum of equidistant energy levels without drive and show that it keeps a spectrum of equidistant Floquet quasienergies when subjected to a large variety of periodic driving schemes. Arbitrary multiples of the driving period can then be reached by adjusting parameters of the drive, for arbitrary initial states. This behavior is understood by mapping the XX spin chain with $N + 1$ sites to a single large spin with $S = N / 2$ invoking the closure of the group SU(2). Our simple model is neither intrinsically disordered nor is it an interacting many-body system (after suitable mapping), and it does not have a thermodynamic limit in the conventional sense. It does, however, show controllable discrete time translational symmetry breaking for arbitrary initial states and a degree of robustness against perturbations, thereby carrying some characteristic traits of a discrete time crystal.

Received 28 April 2019
Revised 16 October 2019

DOI:https://doi.org/10.1103/PhysRevB.100.184301

Physics Subject Headings (PhySH)

1-dimensional spin chains Classical spin models Floquet systems Quantum spin models

Condensed Matter, Materials & Applied PhysicsNonlinear DynamicsStatistical Physics & Thermodynamics

Authors & Affiliations

Robin Schäfer^1,2,*, Götz S. Uhrig^1,†, and Joachim Stolze ^1,‡

¹Fakultät Physik, Technische Universität Dortmund, D-44221 Dortmund, Germany
²Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany

^*robin.schaefer@tu-dortmund.de, schaefer@pks.mpg.de
^†goetz.uhrig@tu-dortmund.de
^‡joachim.stolze@tu-dortmund.de

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Issue

Vol. 100, Iss. 18 — 1 November 2019

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Images

Figure 1
Floquet multipliers (left) and quasienergies (right) for an integer spectrum $ɛ_{α} = ɛ_{0} α mod ω$ of 25 quasienergies. ( $α = - 24, - 22, ..., 24, ɛ_{0} = 0.13, ω = 1$ .) The incommensurate structure of the Floquet multipliers prevents periodic behavior at any reasonable multiple of the driving period.
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Figure 2
Same as Fig. 1, for $α = - 24, - 22, ..., 24, ɛ_{0} = 1 / 6, ω = 1$ . Both Floquet multipliers and reduced quasienergies show commensurate structures leading to a time-translation symmetry breaking period $T_{S} = 3 T$ .
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Figure 3
Absolute value $| ɛ_{0} | < ω / 2$ of the characteristic quasienergy scale (38), reduced to the first Brillouin zone in time, for binary driving with driving frequency $ω = 1$ . The parameter $λ$ denotes the strength of the nearest-neighbor couplings and $h$ is the amplitude of time-dependent external field in the Hamiltonian (20) with coefficients (21).
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Figure 4
Stability of the time-translation symmetry breaking against spatial randomness in the system. Shown is the absolute value of the return amplitude $f_{0, 0} = 〈 0 | U (m T_{S}) | 0 〉$ in the system with multiplicative Gaussian disorder (68) in the Hamiltonian, as a function of the standard deviations $σ_{h}$ and $σ_{λ}$ . Time-translation symmetry is broken with $T_{S} = 2 T$ in panels (a) and (b), and with $T_{S} = 3 T$ in panels (c) and (d), respectively. Panel (d) shows results for harmonic driving; all other results are for binary driving. In all cases, $N = 29$ and $ω = 1$ ; all data were averaged over 640 measurements. A Gaussian smoothing was applied to the fluctuating raw data in order to avoid wildly fluctuating contour lines. Parameters used were $λ = 0.3, h = 0.273316$ in panels (a) and (b), $λ = 0.2, h = 0.304237$ in panel (c), and $λ = 1.2, h = 1.279452$ in panel (d).
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Figure 5
Spectral weight $w (\frac{ω}{2})$ of the fundamental time crystal mode, for different kinds of disorder, as a function of system size. The time correlation $〈 f_{0 N} (t) 〉$ was calculated for 4000 $t$ values between zero and $8 T_{S}$ , averaging over 500 disordered system configurations. The modulus squared of the (numerical) Fourier transform of $〈 f_{0 N} (t) 〉$ defines the spectral weight. The height of the peak at $\frac{ω}{2}$ (where $ω$ is the driving frequency) of that quantity is divided by the same quantity, calculated without disorder, to eliminate the intrinsic (disorder-independent) size dependence. All data shown are for Gaussian disorder. Types of disorder are multiplicative, for $σ = 0.01$ (open red circles) and $σ = 0.02$ (open red squares), as well as additive, for $σ = 0.1$ (filled blue circles) and $σ = 0.2$ (filled blue squares).
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