Nonequilibrium quantum order at infinite temperature: Spatiotemporal correlations and their generating functions

Sthitadhi Roy and Achilleas Lazarides

Phys. Rev. B 98, 064208 – Published 28 August 2018

Abstract

Localization-protected quantum order extends the idea of symmetry breaking and order in ground states to individual eigenstates at arbitrary energy. Examples include many-body localized static and $π$ spin glasses in Floquet systems. Such order is inherently dynamical and difficult to detect as the order parameter typically varies randomly between different eigenstates, requiring specific superpositions of eigenstates to be targeted by the initial state. We show that two-time correlators overcome this, reflecting the presence or absence of eigenstate order even in fully mixed, infinite temperature states. We show how spatiotemporal correlators are generated by the recently introduced dynamical potentials, demonstrating this explicitly using an Ising spin glass and a Floquet $π$ spin glass and focusing on features mirroring those of equilibrium statistical mechanics such as bimodal potentials in the symmetry-broken phase.

Received 16 May 2018

DOI:https://doi.org/10.1103/PhysRevB.98.064208

Physics Subject Headings (PhySH)

Eigenstate thermalization Many-body localization Nonequilibrium statistical mechanics Quantum criticality

Disordered systems Floquet systems Nonequilibrium systems

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Sthitadhi Roy^1,2,* and Achilleas Lazarides^3,†

¹Physical and Theoretical Chemistry, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom
²Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Oxford University, Parks Road, Oxford OX1 3PU, United Kingdom
³Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany

^*sthitadhi.roy@chem.ox.ac.uk
^†acl@pks.mpg.de

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Vol. 98, Iss. 6 — 1 August 2018

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Images

Figure 1
The dynamical potential $Θ_{M_{z}}$ as a function of $s$ and $t$ for different system sizes, $L$ , in the spin glass (a) and paramagnet (b) phases, respectively. Note that they have opposite trends with increasing $L$ with regard to their curvature at $s = 0$ . The second derivative of $Θ_{M_{z}}$ with respect to $s$ at $s = 0$ as a function of $t$ scaled with $L$ and $t$ suggests that $Θ_{M_{z}} \sim s^{2} t^{2} L$ in the spin glass phase (c) and vanishes in the thermodynamic limit in the paramagnet phase (d). The distribution $P_{M_{z}} (m / t)$ shows a Gaussian and exponential behavior in the spin glass phase (e) and the paramagnet phase (f), respectively. Other parameters are $J_{z} = 0$ , $J = 5$ , $J_{x} = 0.3$ , and $h = 0.3$ .
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Figure 2
(a) The phase diagram of the model in Eq. (7) for $J_{x} = 0$ . The markers show the parameter values of $J_{z}$ and $h_{x}$ , corresponding to which the numerical results are shown in the rest of the panels. (b) $\partial_{s}^{2} Θ_{M_{z}} (n^{*}) |_{s = 0} / n^{*} L$ as a function of the frequency $ω$ at fixed time $n^{*} = 100$ for the three different $(J_{z}, h_{x})$ values marked in panel (a) for different $L$ . The collapse of the data for different $L$ suggests that the response survives in the thermodynamic limit. (c)–(e) The behavior of $\partial_{s}^{2} Θ_{M_{z}} |_{s = 0} / n L$ as a function of $ω$ and $n$ together shows that the integrated two-time correlator grows quadratically with time, if probed at $ω = 0$ and $π$ in the 0 and $π$ spin glass phases, respectively. Other parameters are $J = 5$ , $J_{x} = 0.1$ , $h = 0.3$ , and $L = 14$ for panels (c)–(e).
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Figure 3
The potentials $Θ_{G_{z}}$ as a function of $s$ and $t$ in the spin glass phase (a) and the paramagnet phase (b) for various $L$ . The behavior of the corresponding $\partial_{s}^{2} Θ_{G_{z}} |_{s = 0}^{/} t L^{2}$ suggests the form $Θ_{G_{z}} \sim s^{2} t^{2} L^{2}$ in the spin glass phase (c) and $Θ_{G_{z}} \sim s^{2} t^{2} L^{0}$ in the paramagnet phase (d) (see inset for collapse). Other parameters are $J_{z} = 0$ , $J = 5$ , $J_{x} = 0.3$ , and $h = 0.3$ .
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Figure 4
The branches $Θ_{G_{z}}^{\pm}$ for the spin glass and the paramagnet phase are superposed on $Θ$ for two different system sizes for the spin glass phase (a) and the paramagnet phase (b). The first derivative of the branches with respect to $s$ at $s = 0$ as a function of $t$ and various system sizes suggests that their leading behavior is of the form $Θ_{G_{z}}^{\pm} \sim \pm s t L$ in the spin glass phase (c) and $Θ_{G_{z}}^{\pm} \sim s t L^{- 1}$ in the paramagnet phase (d).
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Figure 5
The distribution $P_{G_{z}}$ showing a bimodal nature in the spin glass phase (a), which seems persistent with increasing $L$ , and a unimodal distribution in the paramagnet phase (b). The inset shows that, if the branch construction is nevertheless used in the paramagnetic phase, one again obtains a bimodal distribution, but crucially the bimodality systematically goes away with increasing $L$ eventually converging to the unimodal distribution in the thermodynamic limit.
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