Universal defect density scaling in an oscillating dynamic phase transition

Wei-Can Yang, Makoto Tsubota, Adolfo del Campo, and Hua-Bi Zeng

Phys. Rev. B 108, 174518 – Published 30 November 2023

Abstract

Universal scaling laws govern the density of topological defects generated while crossing an equilibrium continuous phase transition. The Kibble-Zurek mechanism (KZM) predicts the dependence on the quench time for slow quenches. By contrast, for fast quenches, the defect density scales universally with the amplitude of the quench. We show that universal scaling laws apply to dynamic phase transitions driven by an oscillating external field. The difference in the energy response of the system to a periodic potential field leads to energy absorption, spontaneous breaking of symmetry, and its restoration. We verify the associated universal scaling laws, providing evidence that the critical behavior of nonequilibrium phase transitions can be described by time-average critical exponents combined with the KZM. Our results demonstrate that the universality of critical dynamics extends beyond equilibrium criticality, facilitating the understanding of complex nonequilibrium systems.

Received 16 July 2023
Revised 7 November 2023
Accepted 14 November 2023
Corrected 30 July 2024

DOI:https://doi.org/10.1103/PhysRevB.108.174518

Physics Subject Headings (PhySH)

Continuous phase transition Critical phenomena Dynamical phase transitions Nonequilibrium statistical mechanics Superconducting phase transition Topological defects

Nonequilibrium systems

Condensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Corrections

30 July 2024

Correction: The omission of the “Contact author” label in the footnote for the last author has been fixed.

Authors & Affiliations

Wei-Can Yang ¹, Makoto Tsubota ^1,2,*, Adolfo del Campo ^3,4,†, and Hua-Bi Zeng ^5,6,‡

¹Department of Physics, Osaka Metropolitan University, 3-3-138 Sugimoto, 558-8585 Osaka, Japan
²Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
³Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg
⁴Donostia International Physics Center, E-20018 San Sebastián, Spain
⁵Center for Theoretical Physics, Hainan University, Haikou 570228, People's Republic of China
⁶Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China

^*tsubota@omu.ac.jp
^†adolfo.delcampo@uni.lu
^‡Contact author: hbzeng@yzu.edu.cn

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Issue

Vol. 108, Iss. 17 — 1 November 2023

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Images

Figure 1
Universal breakdown of KZM in a fast nonequilibrium phase transition. The effective relaxation time $\tilde{τ たう} (λ らむだ (t))$ should be the time average of the real relaxation time $τ たう (λ らむだ (t))$ (red line). The freeze-out time determined by equating the time elapsed after crossing the phase transition to average relaxation time (large black dot). In fast quench regions with finite depth ${λ らむだ}_{f}$ , the quench is saturated at a fixed equilibrium relaxation time $\tilde{τ たう} ({λ らむだ}_{f})$ , which we will explain later.
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Figure 2
(a) Evolution of order parameters in an external oscillating field. With the addition of the oscillating field, the order parameters gradually decrease and finally reach a nonequilibrium steady state. The red line has an amplitude lower than the critical field $E / E_{c} = 0.83$ and corresponds to an oscillatory steady state. The black line has an amplitude equal to the critical field $E / E_{c} = 1$ and shows the decay of the order parameter in the steady state. (b) In the case of $E / E_{c} = 0.83$ , the red line represents the rate at which the order parameters fall in response to a large electric field, while the black line represents the rate at which the small electric field order parameters recover to rise. (c) Phase diagram of the final average order parameter as a function of the amplitude of the field. When the amplitude of the field is greater than the critical value, the system is completely disordered and enters the normal state.
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Figure 3
Dynamic evolution of the nonequilibrium system with topological defects generated after a finite time quench with an oscillating field. The solid black line shows the number of topological defects with noise. The red line shows the evolution of the order parameter. The insets show the profile of order parameters before and after the freeze-out time, manifesting a distribution of vortices in the latter stage.
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Figure 4
The number of vortices $N_{vortices}$ as a function of the quench time ${τ たう}_{Q}$ in which the the electric field decays. For each ${τ たう}_{Q}$ point, the number of topological defects is averaged over 100 trajectories. For slow quenches with ${τ たう}_{Q} > 200$ , the vortex number is consistent with the KZM scaling $N \sim {τ たう}_{Q}^{- 0.4832 \pm 0.0182}$ . For fast quenches with ${τ たう}_{Q} < 200$ , the KZM is no longer satisfied, and the number of vortices does not vary with the quench rate. However, the value at the plateau scales universally with the amplitude of the quench, as shown from top to bottom, for the values of $E_{f} = 0 E_{c}, 0.4 E_{c}, 0.7 E c$ , with the symbols $◯, ⋄, □$ , respectively.
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Figure 5
Universal dynamics beyond KZM after a fast quench of the electric field with an oscillating dynamic phase transition. Panel (a) shows the linear scaling of the vortex number as a function of the final field intensity, i.e., $N \sim {ε いぷしろん}_{f}^{1.003 \pm 0.0351}$ . Panel (b) shows the dependence of the freeze-out time on ${ε いぷしろん}_{f}$ , with data fitted to $\hat{t} \sim {ε いぷしろん}_{f}^{- 1.042 \pm 0.014}$ . Panel (c) shows the dependence of the critical quench time on ${ε いぷしろん}_{f}$ , with data fitted to ${τ たう}_{Q}^{c} \sim {ε いぷしろん}_{f}^{- 1.985 \pm 0.031}$ .
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