Absolute stability and spatiotemporal long-range order in Floquet systems

C. W. von Keyserlingk, Vedika Khemani, and S. L. Sondhi

Phys. Rev. B 94, 085112 – Published 8 August 2016

Abstract

Recent work has shown that a variety of novel phases of matter arise in periodically driven Floquet systems. Among these are many-body localized phases which spontaneously break global symmetries and exhibit novel multiplets of Floquet eigenstates separated by quantized quasienergies. Here we show that these properties are stable to all weak local deformations of the underlying Floquet drives—including those that explicitly break the defining symmetries—and that the models considered until now occupy submanifolds within these larger “absolutely stable” phases. While these absolutely stable phases have no explicit global symmetries, they spontaneously break Hamiltonian-dependent emergent symmetries, and thus continue to exhibit the novel multiplet structure. The multiplet structure in turn encodes characteristic oscillations of the emergent order parameter at multiples of the fundamental period. Altogether these phases exhibit a form of simultaneous long-range order in space and time which is new to quantum systems. We describe how this spatiotemporal order can be detected in experiments involving quenches from a broad class of initial states.

Received 20 May 2016

DOI:https://doi.org/10.1103/PhysRevB.94.085112

Physics Subject Headings (PhySH)

Magnetic order Many-body localization Order parameters

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

C. W. von Keyserlingk, Vedika Khemani, and S. L. Sondhi

Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

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Vol. 94, Iss. 8 — 15 August 2016

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Images

Figure 1
(a) Schematic depiction of the manifold of Floquet unitaries that are absolutely stable and characterized by Hamiltonian-dependent emergent symmetries (gray area). Special submanifolds (colored lines) within the absolutely stable manifold are characterized by Hamiltonian-independent unitary $(U_{i})$ and antiunitary $(T_{i})$ symmetries. Special models (black stars) can lie at the intersection of several submanifolds with exact symmetries. As an example, the $π SG$ model defined in Refs. [25, 26] is absolutely stable and possesses the Ising unitary symmetry $P$ and an antiunitary symmetry $T = K P$ , where $K$ is complex conjugation. (b) Schematic depiction of the spatiotemporal long-range order found in absolutely stable phases–the order looks “antiferromagnetic” in time and glassy in space.
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Figure 2
Disorder and eigenstate averaged spectral gaps for the generically perturbed model (6) without any $P$ and $T$ symmetries plotted as a function of the perturbation strength $λ$ and system size $L$ . The nearest-neighbor quasienergy gap $Δ_{0}$ shows no $λ$ dependence but decreases exponentially with $L$ . On the other hand $Δ_{π}$ which measures the spectral pairing of even-odd parity states scales as $λ^{L}$ (fits to this form superimposed). Thus, there is a window of $λ s$ for which $Δ_{π} ≪ Δ_{0}$ and the system exhibits robust spectral pairing in the $L \to \infty$ limit. Gaps smaller than $\sim 10^{- 14}$ are below numerical precision, thus the initial $λ$ -independent trend in the $Δ_{π}$ data for larger $L$ . Inset: Cartoon of the quasienergy spectrum illustrating the definitions of $Δ_{0}$ and $Δ_{π}$ .
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Figure 3
Disorder and eigenstate averaged end-to-end connected correlation functions for $σ^{x, z}$ in the “generic” model (6) with no $P, T$ symmetries (blue squares, red circles) and a model [52] with $T$ symmetry obtained by setting $h^{y} = 0$ in (6) (black diamonds, green triangles). As discussed in the text, the generic model shows long-range order for both operators which is signaled here by correlations scaling as $λ^{2}$ independently of system size. On the other hand, in the model with $T$ symmetry, only $σ^{z}$ shows long-range order while the $σ^{x}$ correlator scales as $λ^{f (L)}$ , where $f (L) \sim 0.9 L - 1.4$ (fits shown) and thus vanishes in the $L \to \infty$ limit. This is to be expected from symmetry constraints. The $σ^{y}$ correlators (not shown here) also display long-range order in the generic model but not in the $T$ -symmetric model.
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Figure 4
Fourier transform over time window $Δ t = 500 T$ of one-point time-dependent expectation values $〈 ψ_{0} | σ^{{x, y, z}} (n T) | ψ_{0} 〉$ in the “generically” perturbed model (6). The initial state $| ψ_{0} 〉$ is a product state with physical spins $σ^{α}$ randomly pointing on the Bloch sphere and uncorrelated from site to site. As discussed in the text, the response looks “glassy” with several incommensurate Fourier peaks in addition to the peak at $π / T$ , although we expect these to decay away in the $L \to \infty, T \to \infty$ limit. Data are shown for a single disorder realization in a system of length $L = 10$ .
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Figure 5
Left: Phase diagram for the MBL Ising-symmetric drives presented in Refs. [25, 26] showing the $0 SG$ and $π SG$ phases which are long-range ordered and spontaneously break Ising symmetry, as well as the $0 π PM$ and trivial paramagnetic phases which have no LRO. The $0 π PM$ is an SPT with nontrivial edge modes and can spontaneously break time translation symmetry on its edges. Right: On perturbing with generic Ising-symmetry-breaking fields $h^{gen}$ , only the $π SG$ is absolutely stable and continues into a phase with LRO and an emergent symmetry. The other three phases can be continuously connected to the trivial MBL paramagnet in the presence of $h^{gen}$ .
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