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arXiv:2012.07001 · Physical review. B./Physical review. B

Dynamics of a quantum phase transition in the Aubry-André-Harper model with p -wave superconductivity

Xianqi Tong, Ye-Ming Meng, Xunda Jiang, Chaohong Lee, G. D. de Moraes Neto, Gao Xianlong

Published Mar 10, 2021• 35 citations
arXivHTMLPDF

Abstract

We investigate the nonequilibrium dynamics of the one-dimension Aubry-Andr\'e-Harper model with $p$-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 $V$. 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 $V=0$ 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.

DOI
10.1103/physrevb.103.104202

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arXiv:1710.05256•New Journal of Physics•Apr 16, 2018•13 citations

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We investigate the quantum criticality and universality in Aubry-Andr\'e-Harper (AAH) model with $p$-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 $p$-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.

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Ting Lv, Yubin Liu, Tian-Cheng Yi +3 more

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 $p$-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 $z$ 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.

arXiv:2210.06740•Physical review. B./Physical review. B•Oct 18, 2022•14 citations
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\ensuremathνz\ensuremath{\nu}z\ensuremathν, giving the correlation length \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1 and dynamical exponent z=1.373\ifmmode±\else\textpm\fi0.023z=1.373\ifmmode\pm\else\textpm\fi{}0.023z=1.373\ifmmode±\else\textpm\fi0.023, 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∞V=\ensuremath{\infty}V=\ensuremath∞, 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.
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.
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.
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.
We explore quantum criticality and Kibble-Zurek scaling (KZS) in the Aubry-Andr\'e-Stark (AAS) model, where the Stark field of strength $\ensuremath{\varepsilon}$ is added onto the one-dimensional quasiperiodic lattice. We perform scaling analysis and numerical calculations of the localization length, the inverse participation ratio (IPR), and the energy gap between the ground and first excited states to characterize critical properties of the delocalization-localization transition. Remarkably, our scaling analysis shows that, near the critical point, the localization length $\ensuremath{\xi}$ scales with $\ensuremath{\varepsilon}$ as $\ensuremath{\xi}\ensuremath{\propto}{\ensuremath{\varepsilon}}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\approx}0.3$, a new critical exponent for the AAS model, which is different from the counterparts for both the pure Aubry-Andr\'e (AA) model and the pure Stark model. The IPR I\mathcal{I}I scales as $\mathcal{I}\ensuremath{\propto}{\ensuremath{\varepsilon}}^{s}$ with the critical exponent $s\ensuremath{\approx}0.098$, which is also different from both of the pure models. The energy gap $\mathrm{\ensuremath{\Delta}}E$ scales as $\mathrm{\ensuremath{\Delta}}E\ensuremath{\propto}{\ensuremath{\varepsilon}}^{\ensuremath{\nu}z}$ with the same critical exponent $z\ensuremath{\approx}2.374$ as that for the pure AA model. We further reveal hybrid scaling functions in the overlap between the critical regions of the Anderson and Stark localizations. Moreover, we investigate the driven dynamics of the localization transitions in the AAS model. By linearly changing the Stark (quasiperiodic) potential, we calculate the evolution of the localization length and the IPR, and we study their dependence on the driving rate. We find that the driven dynamics from the ground state is well described by the KZS with the critical exponents obtained from the static scaling analysis. When both the Stark and the quasiperiodic potentials are relevant, the KZS form includes the two scaling variables. This work extends our understanding of critical phenomena on localization transitions and generalizes the application of the KZS to hybrid models.
The critical phase of a non-Hermitian quasicrystal can support stronger transport than its surrounding delocalized phase. We demonstrate this anomalous behavior in the one-dimensional nonreciprocal Aubry-André-Harper model through a combined study of dynamical quantum phase transitions (DQPTs) and wavepacket diffusion. Using a parity-sorted energy-spectrum classification that directly encodes the generalized PT\mathcal{PT}PT symmetry, we find that DQPTs in this system are energy-resolved, in contrast to the energy-independent DQPTs of Hermitian quasicrystals. The energy-resolved features are most pronounced when the initial and final Hamiltonians belong to different phases (localized or extended), and they are tied to the even-odd index structure of the spectrum, which we exploit to organize the quench-dynamical landscape. For wavepacket dynamics after a single-site quench, the diffusion exponent βββ, extracted from the long-time power-law scaling of the root-mean-square displacement σ(τ)σ(τ)σ(τ), partitions the phase diagram into four distinct regimes. In the Hermitian limit the extended phase is ballistic (β=1β=1β=1), the critical phase is normally diffusive (β=0.5β=0.5β=0.5), and the localized phase yields β→0β\to 0β→0. Nonreciprocity reverses this hierarchy: the extended phase becomes normally diffusive, while the critical phase turns ballistic. We trace the anomalous β=1β=1β=1 at criticality to the self-similar multifractal structure of the critical eigenstates, whose nodal positions are organized by the golden ratio. A finite-size scaling ansatz built on the wave-front propagation yields σ(τ)∝τσ(τ)\proptoτσ(τ)∝τ. The parity-resolved DQPTs and the βββ-phase diagram establish two complementary dynamical diagnostics of nonreciprocal quasicrystals, in which nonreciprocity promotes transport at the critical point and suppresses it in the delocalized phase.
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.