Out-of-time-order correlation and detection of phase structure in Floquet transverse Ising spin system (2002.05986v2)

Abstract: We study the out-of-time-order correlation (OTOC) of the Floquet transverse Ising model and use it to verify the phase diagram of the system. First, we present the exact analytical solution of the transverse magnetization OTOC using the Jorden-Wigner transformation. We calculate the speed of correlation propagation and analyze the behavior of the revival time with the separation between the observables. In order to get the phase structure of the Floquet transverse Ising system, we use the longitudinal magnetization OTOC as it is known to serve as an order parameter of the system. We show the phase structure numerically in the transverse Ising Floquet system by using the long time average of the longitudinal magnetization OTOC. In both the open and the closed chain systems, we find distinct phases out of which two are paramagnetic (0-paramagnetic and $\pi$-paramagnetic), and two are ferromagnetic (0-ferromagnetic and $\pi$-ferromagnetic) as defined in the literature.

• The paper introduces out-of-time-order correlations (OTOC) as a novel tool for detecting phase transitions in Floquet-driven Ising spin systems.
• It utilizes analytical techniques, including the Jordan-Wigner transformation, to derive precise solutions for transverse and longitudinal magnetization OTOCs.
• The findings reveal distinct phase boundaries in periodically driven systems, underscoring the method's potential in quantum simulation applications.

Out-of-Time-Order Correlation and Detection of Phase Structure in Floquet Transverse Ising Spin System

Introduction

The paper addresses the utilization of out-of-time-order correlations (OTOC) within the context of Floquet transverse Ising spin systems to discern the phase structure of such systems. OTOCs have garnered significant attention for their roles in quantum chaotic systems, quantum information scrambling, and phase detection in quantum critical systems. This framework provides insights into the behavior of dynamical quantum phase transitions, specifically within periodically driven systems.

Model Description

The authors investigate an integrable Floquet transverse Ising system characterized by binary Floquet drives. The Floquet map is given by:

$U = e^{-iH_{xx}\tau_{1}e^{-iH_{z}\tau_{0}$

where $H_{xx}$ represents nearest neighbor Ising interaction and $H_z$ signifies a transverse field. These components are pivotal in capturing the interplay between evolving quantum states and external driving forces.

Figure 1: Phase structure of the Floquet system with Floquet map given by \cref{U_f.

Analytical Solutions for OTOCs

Transverse Magnetization OTOC (TMOTOC)

Using the Jordan-Wigner transformation, an exact analytical solution for the transverse magnetization OTOC is derived. This transformation maps spin operators onto fermionic operators, enabling the calculation of correlation propagation speed and revival time behavior. The TMOTOC formula conveys how correlations in transverse magnetization evolve over time, offering a basis for comparison with longitudinal magnetization.

Figure 2: $$F^{l,l}, demonstrating spreading across various separations.</p></p> <h4 class='paper-heading' id='longitudinal-magnetization-otoc-lmotoc'>Longitudinal Magnetization OTOC (LMOTOC)</h4> <p>The LMOTOC serves as a crucial diagnostic tool for identifying quantum phase transitions. Unlike TMOTOCs, LMOTOCs in ferromagnetic and paramagnetic regimes show distinct long-time average behaviors, permitting these regions to be classified using average values of LMOTOC as an order parameter. <img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2002-05986/otocz_butter_revival_n12.png" alt="Figure 3" title="" class="markdown-image" loading="lazy"></p> <p><img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2002-05986/otocx_butter_revival_n12.png" alt="Figure 3" title="" class="markdown-image" loading="lazy"> <p class="figure-caption">Figure 3: Behaviour of (a)$$F_z^{l,m}$in comparison to (b)$F_x^{l,m}$$over a range of kicks.</p></p> <h3 class='paper-heading' id='phase-structure-analysis'>Phase Structure Analysis</h3> <p>The paper builds upon these OTOCs to explore phase structures inherent in Floquet systems. Notably, the phase diagrams reveal four distinct regimes, incorporating both conventional and Floquet-specific phases such as$$\pi$-ferromagnetic and 0$\pi$$-paramagnetic. The sharpness of phase boundaries are attributed to symmetry-protected Ising order dynamics. <img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2002-05986/otocz_n12_tau0_pi28.png" alt="Figure 4" title="" class="markdown-image" loading="lazy"></p> <p><img src="https://emergentmind-storage-cdn-c7atfsgud9cecchk.z01.azurefd.net/paper-images/2002-05986/otocx_n12_tau0_pi28.png" alt="Figure 4" title="" class="markdown-image" loading="lazy"> <p class="figure-caption">Figure 4: (a) Variation of$$F_x^{l,l}$ under varying conditions showing alignment with theoretical expectations.

Figure 5: Variation of critical points with respect to finite and infinite size systems.

Conclusion

By presenting exact analytical solutions alongside numerical validations, the paper underscores the effectiveness of OTOCs in evaluating phase transitions in Floquet systems. Discoveries of nontrivial phase structures accentuate the potential for experimental realizations, further leveraging tools such as LMOTOC for phase structure identification in driven quantum systems.

Experimental and computational techniques, facilitated by the manipulation and measurement of OTOCs, provide avenues for practical applications in quantum simulations and phase transition analysis within modern quantum technologies.