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arXiv:1710.05256 · New Journal of Physics

Quench dynamics in the Aubry–André–Harper model with p-wave superconductivity

Qi-Bo Zeng, Shu Chen, Rong Lü

Published Apr 16, 2018• 13 citations
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Abstract

The Anderson localization phase transition in the Aubry–André–Harper (AAH) model with p -wave superconducting (SC) pairing is numerically investigated by suddenly changing the on-site potential from zero to various finite values which fall into the extended, critical and localized phase regimes shown in this model. The time evolution of entanglement entropy (EE), mean width of wave packets and Loschmidt echo of the system exhibit distinct but consistent dynamical signatures in those three phases. Specifically, the EE grows as a power function of time with the exponent of which varies in the extended phase but keeps almost unchanged in the critical phase for different quench parameters. However, if the system is in the localized phase after a quench, the EE grows much slower and will soon get saturated. The time-dependent width of wave packets in the system shows similar behaviors as the EE. In addition, from the perspective of the dynamical phase transition, we find that the Loschmidt echo oscillates and always keeps finite when the system is quenched in the extended phase. In contrast, in the critical or localized phase, the echo will reach to zero at some time intervals or will decay almost to zero after a long-time evolution. The universal features of these quantities in the critical phase of the system with various SC pairing amplitudes are also observed.

DOI
10.1088/1367-2630/aabe39

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The Anderson localization phase transition in the Aubry–André–Harper (AAH) model with p -wave superconducting (SC) pairing is numerically investigated by suddenly changing the on-site potential from zero to various finite values which fall into the extended, critical and localized phase regimes shown in this model. The time evolution of entanglement entropy (EE), mean width of wave packets and Loschmidt echo of the system exhibit distinct but consistent dynamical signatures in those three phases. Specifically, the EE grows as a power function of time with the exponent of which varies in the extended phase but keeps almost unchanged in the critical phase for different quench parameters. However, if the system is in the localized phase after a quench, the EE grows much slower and will soon get saturated. The time-dependent width of wave packets in the system shows similar behaviors as the EE. In addition, from the perspective of the dynamical phase transition, we find that the Loschmidt echo oscillates and always keeps finite when the system is quenched in the extended phase. In contrast, in the critical or localized phase, the echo will reach to zero at some time intervals or will decay almost to zero after a long-time evolution. The universal features of these quantities in the critical phase of the system with various SC pairing amplitudes are also observed.
We investigate the nonequilibrium dynamics of the one-dimension Aubry-Andr\'e-Harper model with ppp-wave superconductivity by changing the potential strength with slow and sudden quench. Firstly, we study the slow quench dynamics from the localized phase to the critical phase by linearly decreasing the potential strength VVV. The localization length is finite and its scaling obeys the Kibble-Zurek mechanism. The results show that the second-order phase transition line shares the same critical exponent $z\ensuremath{\nu}$, giving the correlation length $\ensuremath{\nu}=1$ and dynamical exponent $z=1.373\ifmmode\pm\else\textpm\fi{}0.023$, which are different from the Aubry-Andr\'e model. Secondly, we also study the sudden quench dynamics between three different phases: localized phase, critical phase, and extended phase. In the limit of V=0V=0V=0 and $V=\ensuremath{\infty}$, we analytically study the sudden quench dynamics via the Loschmidt echo. The results suggest that, if the initial state and the post-quench Hamiltonian are in different phases, the Loschmidt echo vanishes at some time intervals. Furthermore, we found that, if the initial value is in the critical phase, the direction of quenching is the same as one of the two limits mentioned before, and similar behaviors will occur.
We investigate the quantum criticality and universality in Aubry-Andr\'e-Harper (AAH) model with ppp-wave superconducting pairing $\mathrm{\ensuremath{\Delta}}$ in terms of the generalized fidelity susceptibility (GFS). We show that the higher order GFS is more efficient in spotlighting the critical points than lower order ones, and thus the enhanced sensitivity is propitious for extracting the associated universal information from the finite-size scaling in quasiperiodic systems. The GFS obeys power-law scaling for localization transitions and thus scaling properties of the GFS provide compelling values of critical exponents. Specifically, we demonstrate that the fixed modulation phase $\ensuremath{\phi}=\ensuremath{\pi}$ alleviates the odd-even effect of scaling functions across the Aubry-Andr\'e transition with $\mathrm{\ensuremath{\Delta}}=0$, while the scaling functions for odd and even numbers of system sizes with a finite $\mathrm{\ensuremath{\Delta}}$ cannot coincide irrespective of the value of $\ensuremath{\phi}$. A thorough numerical analysis with odd number of system sizes reveals the correlation-length exponent $\ensuremath{\nu}\ensuremath{\simeq}$ 1.000 and the dynamical exponent $z\phantom{\rule{4pt}{0ex}}\ensuremath{\simeq}$ 1.388 for transitions from the critical phase to the localized phase, suggesting the unusual universality class of localization transitions in the AAH model with a finite ppp-wave superconducting pairing lies in a different universality class from the Aubry-Andr\'e transition. The results may be testified in near term state-of-the-art experimental settings.
In this work, we focus on the critical dynamics of the one-dimensional quasiperiodic p-wave-paired Aubry-André-Harper model, which exhibits a transition point between the gapped critical and the gapless localized phases. First, we disclose that the dynamical exponent features two distinct plateaus in the gapless localized phase. Besides the plateau with dynamical exponent z=1.388z=1.388z=1.388 near the transition point, there is a second one with z=1z=1z=1 away from the transition point. Then we demonstrate that these two plateaus intrinsically affect the critical dynamics with moderate quench rate by employing both one-way and round-trip quench protocols. In the one-way quench protocol, we clarify that the Kibble-Zurek (KZ) regime consists of two sub-regimes with different KZ exponents, which is a direct consequence of the two-plateau structure of the dynamical exponent. We also diagnose the KZ, pre-saturated, and saturated regimes by varying the quench rate from slow to fast limits. While, in the round-trip quench protocol, we confirm the interference effect of two critical dynamics and show a narrow neck in the oscillatory density of defects. It turns out that the position of the neck depends on the ratio of to-and-fro quench rates and the p-wave superconductivity pairing amplitude. We show how the position of the neck can be used to quantitatively determine the turning point of the two KZ sub-regimes.
Anderson localization, where quenched disorder leads to the localization of single-particle wave functions, has been studied extensively. However, the effect of interaction and non-random potentials on many-body localization (MBL) has gained attention recently. We investigate the dynamics of the interacting Aubry-André model to assess the stability of the many-body localized (MBL) phase. Our motivation to consider the Aubry–André potential as the disorder potential is to eliminate the avalanche mechanism, which often destabilizes MBL in systems with quenched disorder. We analyze the time evolution and the system-size scaling across a range of disorder strengths for the entanglement entropy, number entropy, and the Hartley number entropy. In the non-interacting case, our results confirm an Anderson localization transition at a critical disorder strength Dc = 2, which is consistent with earlier studies. However, when interactions are introduced, the number entropy appears to grow unboundedly as SN ~ ln ln t in the thermodynamic limit, even for large disorder. Meanwhile, the entanglement entropy grows logarithmically, S ~ ln t, and both entropies saturate in a correlated manner at finite times td ~ e^(L·D). These findings suggest that using a quasiperiodic potential to remove the avalanche mechanism does not stabilize true MBL behavior. Instead, the long-time dynamics are qualitatively similar to those found in systems with random disorder. In this thesis, we first introduce the fundamental concepts of thermalization, quantum quench, and symmetry-resolved entropies, then provide a study of localization, specifically in the Aubry-André model. Finally, we place our numerical results in the broader context of the current debate on the stability of many-body localized phases.
We numerically study the evolution of spectral correlations in the entanglement Hamiltonian (EH) of noninteracting fermions in the Aubry-Andr\'e-Harper (AAH) model. We analyze the time evolution of the EH spectrum in a nonequilibrium setting by studying several quantities: spectral distribution, level statistics, entanglement entropy, and spectral form factor (SFF) in the context of the delocalization-localization transition in the AAH model. It is observed that the SFF of the entanglement spectrum in the delocalized phase and at the phase-transition point evolves in three time intervals. We make a systematic study of the emergence of these three timescales for various initial states and find that the number of time intervals remains 3 unless the Hamiltonian is tuned in the localized phase or when the initial state is maximally entangled, when there is a featureless time evolution. We find a broad direct correlation between the entanglement entropy and the length of the ramp of the SFF. We also find that in the delocalized phase the spectral correlations are stronger in the center of the spectrum and grow progressively weaker as more and more of the spectrum is considered.
We have investigated scaling properties near the quantum critical point between the extended phase and the critical phase in the Aubry-Andr\'e-Harper model with ppp-wave pairing, which have rarely been exploited as most investigations focus on the localization transition from the critical phase to the localized phase. We find that the spectrum-averaged entanglement entropy and the generalized fidelity susceptibility act as eminent universal order parameters of the corresponding critical point without gap closing. We introduce a Widom scaling Ansatz for these criticality probes to develop a unified theory of critical exponents and scaling functions. We thus extract the correlation-length critical exponent $\ensuremath{\nu}$ and the dynamical exponent zzz through the finite-size scaling given the system size increase in the Fibonacci sequence. The retrieved values of $\ensuremath{\nu}\ensuremath{\simeq}1.000$ and $z\ensuremath{\simeq}3.610$ indicate that the transition from the extended phase to the critical phase belongs to a different universality class from the localization transition. Our approach sets the stage for exploring the unconventional quantum criticality and the associated universal information of quasiperiodic systems in state-of-the-art quantum simulation experiments.