Dynamical signature of the moir\'e pattern in a non-Hermitian ladder

Dynamical signature of the moiré pattern in a non-Hermitian ladder

X. M. Yang, X. Z. Zhang, C. Li, and Z. Song

Phys. Rev. B 98, 085306 – Published 17 August 2018

Abstract

We study the dynamical behavior of a non-Hermitian moire superlattice system, which consists of two-coupled Su-Schrieffer-Heeger (SSH) chains with staggered imaginary on-site potentials. The slight difference, which indicates that the in-lattice spacing of nearest-neighbor lattice sites at the top is different from that in the bottom chain, can result in periodical occurrence of the two main spatial regions. We show that two such distinct regions are dimerized and tetramerized, which determine the distinct dynamical behaviors. The Dirac probability can oscillate periodically, increase quadratically, and increase exponentially, which correspond to the unbroken phase, exceptional points, and the broken phase of the tetramerized region. In comparison, the Dirac probability can exhibit high-frequency oscillation in the dimerized region. These phenomena demonstrate the dynamical signature and provide insightful information of the moire pattern in the non-Hermitian regime.

Received 6 June 2018
Revised 24 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.085306

Physics Subject Headings (PhySH)

PT-symmetric quantum mechanics Quantum phase transitions

1-dimensional systems

Numerical approximation & analysis PT-symmetry Tight-binding model

Condensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

X. M. Yang¹, X. Z. Zhang^2,*, C. Li¹, and Z. Song^1,†

¹School of Physics, Nankai University, Tianjin 300071, China
²College of Physics and Materials Science, Tianjin Normal University, Tianjin 300387, China

^*zhangxz@tjnu.edu.cn
^†songtc@nankai.edu.cn

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Vol. 98, Iss. 8 — 15 August 2018

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Figure 1
Schematic illustration of the modified non-Hermitian two-leg ladder system. It consists of two SSH chains with staggered imaginary potentials. The irregular structure arises from the slight difference of lattice constants between two legs. There are three types of approximate regular ladder structures (circled by the red, blue, and green dotted lines, respectively), which appear periodically in a large scale. The interleg hopping rates are $κ, κ^{'}, κ_{1}$ , and $κ_{2}$ , in various regions, respectively. Note that $κ_{1}$ and $κ_{2}$ describe the interaction strength in the crossover between two dashed rectangles, which depends on the location of the sites in the leg. (b1), (c1), and (d1) are drafts of three typical structures in different regions. (b2), (c2), and (d2) are drafts of the reductions of the structures in (b1), (c1), and (d1), respectively, in the limit case $κ, κ^{'} ≫ w ≫ ν$ limit. (b2)–(d2) capture the main features of (b1)–(d1). Systems (b2) and (d2) have fully real spectra for small enough $γ$ but distinguishable dynamical behaviors (see text). (c2) can also have full real spectrum when $γ$ is small enough, which can be seen from Fig. 3. These phenomena indicate that three regions should have distinguishable dynamical behaviors, which are the signature of moire patterns.
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Figure 2
Schematic illustration of three types of regular ladder systems. (a) and (c) have $PT$ symmetry, while (b) is irrelevant to the symmetry involving the bilinear operator T. The red dot lines indicate the axis of reflection operation.
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Figure 3
Phase diagrams of the systems $H_{T}, H_{C}$ , and $H_{D}$ , which are presented by the distributions of critical $γ_{c}$ on the $v w$ plane (in unit of $κ$ ). Light regions correspond to small or zero $γ_{c}$ , at which the reality of the spectrum is fragile. Beyond these regions, there always exist common parameters $(w, v, γ)$ to maintain the full real spectrum for three types of non-Hermitian systems. This fact ensures the probable existence of stable dynamical signature of moire pattern.
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Figure 4
Profiles of Dirac probabilities of time evolution for four typical dynamical behaviors. Plots of $P_{T}^{R} (t), P_{T}^{I} (t), P_{T}^{EP} (t)$ , and $P_{D}^{R} (t)$ are obtained from Eq. (33) with $w = 0.5, γ = 0.395,$ Eq. (35) with $w = 0.5, γ = 0.505,$ Eq. (37) with $w = γ = 0.5$ , and Eq. (43) with $γ = 0.395$ , respectively. We take $κ = 1$ for all cases. Panels (a)–(c) show the same plots in different scales. We can see the four types of systems exhibit distinguishable dynamical behaviors, which are building blocks of moire pattern in Fig. 5.
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Figure 5
Propagation of the initial state being distributed on each site with equal probability. The system parameters are (a) $κ = 1, w = 0.5, v = 0.1, γ = 0.1, Δ = 1 / 301$ , (b) $κ = 0.6, w = 0.5, v = 0.1, γ = 0.1, Δ = 1 / 301$ , (c) $κ = 1, w = 0.5, v = 0.1, γ = 0.645, Δ = 1 / 101$ , and (d) $κ = 1, w = 0.5, v = 0.1, γ = 0.655, Δ = 1 / 101$ , respectively. The dashed red and black lines in each panel denote the two distinct dynamical behaviors which are determined by the tetramerized and dimerized substructures. The panels (a) and (b) show that probability $P (j, t)$ varied periodically with different period as time $t$ increases. In panels (c) and (d), one can see that the Dirac probability increases in the tetramerized phase while it exhibits periodical oscillation in the dimerized phase. Note that $P (t)$ in the tetramerized phase increases quadratically with time $t$ in panel (c), which is associated with the EP as given in Eq. (37). In comparison, $P (t)$ increases exponentially in the broken phase of the tetramerized region, which is in agreement with Eq. (35). These four panels show apparently that the slight mismatched lattice constants lead to the long period moire patterns.
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Physical Review B

covering condensed matter and materials physics

Dynamical signature of the moiré pattern in a non-Hermitian ladder

X. M. Yang, X. Z. Zhang, C. Li, and Z. Song

Phys. Rev. B 98, 085306 – Published 17 August 2018

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Physical Review B

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Dynamical signature of the moiré pattern in a non-Hermitian ladder

X. M. Yang, X. Z. Zhang, C. Li, and Z. Song

Phys. Rev. B 98, 085306 – Published 17 August 2018

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