Emergent Long-Lived Zitterbewegung in Su–Schrieffer–Heeger Lattice with Third-Nearest-Neighbor Hopping

We investigate the wavepacket dynamics of quasiparticles in a Su–Schrieffer–Heeger lattice with third-nearest-neighbor hopping. The results reveal that the life-span of Zitterbewegung can be prolonged. To better understand the mechanism, we discuss the band structure and the long-time average of inverse participation rate. The results show that the band structure can be effectively manipulated as a quasi-flat band by introducing the third-nearest-neighbor hopping. This, as a unique advantage over the standard Su–Schrieffer–Heeger model, will bring about restrained diffusion of the wavepacket as well as dramatically stretched life-span of Zitterbewegung, thus will promise wide applications in condensed matter physics.


INTRODUCTION
With the Dirac equation as its core, relativistic quantum mechanics predicts that particles show a high-frequency Zitterbewegung (ZB) even in free motion [1,2]. For a free electron, the Dirac equation predicts the ZB is so high in frequency ( 10 21 Hz) and so small in amplitude ( m) that it is impossible to observe it directly with current experimental techniques [3][4][5]. In addition, recent studies have pointed out that the ZB of a free particle is unobservable, which is only a mathematical property of the single-particle Dirac equation rather than a physical observable of a real particle [6][7][8][9][10][11]. The reason is that the real physical picture is distorted in all representations except for the Foldy-Wouthuysen one [12]. In the latter representation, ZB of a free particle does not appear and ZB of an interacting particle can exist. Though early studies on ZB were mainly focused on electrons, current research has proved that ZB phenomenon is actually not unique to electrons, but widely exists in condensed matter systems with linear dispersion such as graphene [13][14][15], Weyl semimetal [16], topological insulator [17], and other systems [18][19][20][21]. A valid Dirac equation is shared by all these systems to describe the dynamical properties. With the progress of quantum simulation technology, the simulation of ZB has been realized on various quantum simulation platforms, such as phononic/photonic crystals [22][23][24][25], trapped ions [26], and ultracold atoms [27][28][29][30][31][32][33][34][35][36].
Furthermore, J.A. Lock predicted in 1979 that the finite-sized initial wavepacket will lead to the damping of ZB, i.e., the corresponding amplitude will decay very fast over time [37]. His prediction has been verified by theories [38,39] and experiments [26,30] in succession. The fleeting existence of ZB makes it difficult to observe in experiments [26,30]. Besides, recent studies have found that the introduction of the third-nearest-neighbor (N3) hopping can change the energy band structure, leading to Dirac point metamorphosis, a growing number of research concern relative systems with N3 hopping [40][41][42][43][44][45]. It is natural to ask whether a quasi-flat band structure can be produced by Dirac point metamorphosis, thereby reducing the diffusion rate of the quasi-particles and ultimately producing the long-lived ZB phenomenon? To answer this question, we propose a N3 Su-Schrieffer-Heeger (SSH) model which features not only the nearest-neighbor hopping, but also the N3 hopping.
The rest of this paper is organized as follows. In Section 2, we introduce the N3 SSH model. In Section 3, the visualized dynamics of wavepacket are studied by numerical methods. In Section 4, we theoretically analyze on how the effect of the N3 hopping pro-

MISCELLANEOUS
longs the life-span of ZB effect. In Section 5, an experimental scheme of cold atomic lattice system is proposed. We summary the results in Section 6.

MODEL
The standard SSH model [46,47] is one of the simplest topological models, which can be implemented in experiments by using phononic/photonic crystals [48,49], cold atoms [50] or superconducting circuit [51]. As for cold atomic lattice system, the standard SSH can be implemented by loading rubidium atoms, where the primitive cell is made of AB sublattice [50]. By introducing the auxiliary laser, one can effectively control the N3 hopping (see Fig. 1a). The corresponding tight-binding Hamiltonian is given by the form where and denote the creation and annihilation operators of particles on A (B) sublattice of the nth primitive cell, respectively, and N is the total number of the primitive cells. There are two types of the nearest-neighbor hopping, i.e., intracell and intercell w. and are the N3 hopping amplitudes  which link the long-range AB sublattice. By introducing Fourier transform (hereinafter we set the lattice spacing ), one can obtain the Bloch Hamiltonian (without loss of generality, hereinafter we take the natural unit system, namely ) in the form (4) where and denote the Pauli matrices. Since the dynamical properties of the quasiparticles are dominated by the band structure near the high symmetry point , by considering , where is the displacement from the high symmetry point K, one can obtain the low-energy effective Hamiltonian as (5) where denotes the effective light velocity in x direction.
is the gap parameter and is the effective mass. Then, the corresponding energy-momentum relations can be obtained as (6) The energy-momentum relation with different gap parameter Δ is plotted in Figs. 1b-1f. , The corresponding dynamical properties of quasiparticles in a N3 SSH lattice can be described by the following equation of motion, i.e., 3. NUMERICAL WAVEPACKET DYNAMICS In order to study the wavepacket dynamics of quasiparticles in the N3 SSH lattice, we directly solve the equation of motion Eq. (7) with Hamiltonian (5). Throughout the calculation, a universal Gaussian wavepacket without initial velocity is considered as the initial state, i.e., (8) The above initial wave function Eq. (8) can be seen as of two parts. The first part is the spatial distribution of the wavepacket, which is the Gaussian function. L is the initial width of the wavepacket. The second part represents the spin direction of the wavepacket, commonly known as spinor. Without loss of generality, we then select as the initial spinor, where T stands for the matrix transposition. Visualized evolution of wavepacket is plotted in Figs. 2a-2f.
As shown in the Fig. 2, the centroid of the wavepacket exhibits periodic oscillations whenever , which is the evidence of ZB effect. To be specific, effect decreases first and then increases with the decreasing . Notably, one can see a stable ZB effect when (see Fig. 2b), where the width (amplitude) of wavepacket barely expands (decreases). On the other hand, when , no matter how the parameter Δ changes, the attenuation of ZB effect is still very strong and its life-span is very short. To further grasp the mechanism behind the longer-lived ZB effect, we provide a theoretical analysis of the dynamical evolution in the next section.

Centroid of Wavepacket
First, we calculate the analytical expression of the centroid motion. In Heisenberg picture, the timedependent position operator reads (9) After plugging the Hamiltonian (5) into the above expression and doing some operator gymnastics, one can obtain the expression as (10) where It can be seen clearly that is a 2 × 2 matrix, i.e., . The first two terms of Eq. (10) (12) where (13) Both analytical and numerical results are plotted in Figs. 2g and 2h, respectively. The analytical result (solid lines) shows good agreement with the former numerical one (symbols). Physical quantities such as the drift velocity of the wavepacket, the amplitude and frequency of ZB can be handily deduced from Eq. (12). To be specific, v dr ~ A , ω ZB = 2|E| = and . As shown in Fig. 3, the analytical results show that when , the velocity reaches its peak, and when Δ goes to infinity, the drift velocity is rapidly reduced to zero. In addition, the oscillation frequency increases linearly with the absolute value of the gap parameter, while the amplitude becomes inversely proportional accordingly. Furthermore, from the expression of drift velocity, one can find that when the parameter , drift velocity of the system remains almost zero, which allows the positive and negative part of the wave function to fully interfere, resulting in a longer-lived stable ZB phenomenon.

Corresponding Inverse Participation Rate
We know that ZB is caused by interference between the positive and negative part of the wave function. The reason for the rapid decay of the traditional ZB phenomenon is that the positive and negative part of the wave function cease to be overlapped after they drift fast away from the center with the opposite direction of velocity, hence the vanished ZB effect. It can also be seen that the quasi-flat band induced by N3 hopping makes group velocity of the central wavepacket tend to be zero. In this case, the positive and negative part of the wave function will stay in the central interference area for a long time to enhance stable life-span of ZB. To be exact, it is a relatively stable localized wavepacket that helps prolong the existence of ZB. For the sake of clarity, we focus on the system's inverse participation rate (IPR). It is a physical quantity to describe localization of the wave function, and the bigger the IPR, the more expanded the quasiparticle's wavepacket (vice versa) [52][53][54][55]. The expression reads (14) The evolutionary trend of the system's IPR over time is plotted in Fig. 4.
From Fig. 4a, we can see that the IPR of the system oscillates when , indicating the oscillation of the wave function derived from ZB with time. Furthermore, when the energy gap parameter is set to be around -0.6, will be long confined in a very small range. This shows that under certain circumstances, the positive and negative part of the wave function remain overlapping for a long time, and the life-span of ZB is prolonged. To find the set of parameters that can produce the longest-lived ZB, we calculate the long-time average IPR of the system, i.e., (15) The long-time average IPR is plotted in Fig. 4b, where the red and blue lines correspond to the N3 SSH model and the standard SSH model, respectively. It is obvious in the Fig. 4b that the N3 SSH model has a minimum value of , i.e., . In addition, when , in N3 SSH model tends to be much smaller than that in standard SSH model, showing that the introduction of N3 hopping effectively reduces the diffusion of wavepacket and thus produces longer-lived ZB effect.
On the one hand, for the standard SSH model, the Dirac equation can be used to describe the dynamical properties of relativistic quasiparticles in most cases.

However, when
, the Bloch band of the system tends to be flat, leaving the one-dimensional Dirac equation unable to well depict the system's dynamical properties. In this case, since hopping between the AB sublattice in the primitive cell is switched off, the system exhibits dynamical properties in the double well, i.e., Josephson-like oscillation rather than the ZB effect, which is the reason why in the Fig. 4b the case of is marked by dashed lines. On the other hand, for the N3 SSH model, when , one can see that increases first and then decreases with Δ, and when , increases first and then decreases with , but will increase again beyond the minimum value . It is obvious that of wavepacket near -0.6 is relatively small, resulting in prolonged life-span of ZB in this range. This well explains why ZB in Fig. 2b features dramatically longer life-span.

Momentum Distribution
As a very effective tool in studying ZB phenomena, the momentum analysis can reveal the underlying reason in the spectrum structure [56]. All square-integrable solution of the Dirac equation can be written in the form (16) The functions and can be determined by the Fourier transform of the initial function by a projection onto the positive or negative energy subspace.
Numerical integral of Eq. (16) provides a method to compute the momentum analysis of free time evolution for an arbitrary initial wavepacket. We calculate the momentum distribution of positive-and negativeenergy branches for , , , respectively (see Fig. 5). For the wavepacket Eq. (8), Fig. 5 shows the momentum distributions of the parts with positive and negative energy, i.e., the functions and . We see that the positive energy part has its momentum distribution slightly shifted towards positive average momentum, whereas the negative energy part has a negative average momentum. As we know, the slope of the energy spectrum reflects the group velocity of the wavepacket. has a positive drift velocity (see Fig. 2h). However, for the energy spectrum shown in the inset of Fig. 5a, the energy spectrum structure is the direct opposite of the above cases near the , which give a negative drift velocity (see Fig. 2g).
Furthermore, the size of the overlapping area between positive and negative energy parts represents the intensity of ZB. As shown in Fig. 5a, the overlapping area of positive and negative energy branches is large, and the energy spectrum is tending to be flat near . Therefore, the slope of the energy spectrum tends to be 0, i.e., the group velocity of the wavepacket will approach 0. The wavepacket hardly spreads away in this case and can remain localized for a long time, so the ZB oscillation can maintain a stable existence. For the cases of Δ = -0.2 and +0.6, we see in Figs. 5b and 5c that although the positive and negative energy branches overlap, the slope of their energy spectrum near is much larger than the slope of the energy spectrum in the case of Δ = -0.6, so the corresponding group velocity is significantly larger = 0 x q = 0 x q = 0 x q and the wavepacket will spread out, leaving ZB to disappear rapidly.

EXPERIMENTAL PROPOSAL
The N3 SSH model studied in this paper, is very possible to implement in various systems inclusive of superconducting circuit lattice [57,58], ion trap [59], ultracold atomic ensemble [50,60] and optical waveguide array [61][62][63], which may contain ZB effect with longer and more stable existence as predicted in this paper.
One can load rubidium atomic cloud into a SSH optical lattice, and then realize and manipulate the N3 hopping with the help of auxiliary lasers, as exemplified in cold atom experiments conducted by I. Bloch's group at LMU [64,65] and I. Spielman's group at NIST [30]. Then shut down the magneto-optical trap, which is used as the laser cooling source, and let the atomic ensemble evolve in the N3 SSH lattice. Finally, a time-of-flight process is carried out to amplify the wavepacket for detection. The intensity of hopping parameter can be controlled between 0.17 and 2.0 kHz in the above platforms [66,67] and the corresponding time unit is , so in this paper is equal to 4.8-56.2 ms, which means the stabilization time of ZB effect is significantly prolonged. The long-lived ZB phenomenon makes it possible to have more sufficient time for experimental manipulation and detection.

CONCLUSIONS
In summary, the wavepacket dynamics of quasiparticles in a SSH lattice with N3 hopping are investigated through analytical and numerical methods, respectively. The results reveal that the introduction of N3 hopping changes the band structure, forming a quasiflat band in the vicinity of high symmetry points, reducing the expansion of the quasiparticle wavepacket, prolonging the interference time of the positive and negative branches, so an emergent long-lived ZB phenomenon occurs. Furthermore, we propose an experimental scheme in cold atomic system. Due to its easy accessibility and high controllability, we hope the long-lived ZB effect predicted here can be verified experimentally in the near future.

CONFLICT OF INTEREST
The authors declare that they have no conflicts of interest. /w = 60 t This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.