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Recent experiments performed the nonreciprocal magneotransport in ZrTe$_5$ and obtained a giant magnetochiral anisotropy (MCA) coefficient $\gamma'$. The existing theoretical analysis was based on the semiclassical Boltzmann equation. In this paper, we develop a full quantum theory to calculate $\gamma'$ and further explore the underlying physics. We reveal that the $xz$-mirror symmetry breaking term also breaks the parity symmetry of the system and leads to mixed selection rules and nonvanishing second-order conductivity $\sigma_{xxx}$. The calculations show that $\gamma'$ decreases with the magnetic field, survives only to weak impurity scatterings, and exhibits a nonmonotonous dependence on the strength of the $xz$-mirror symmetry breaking. Our paper can provide a deeper insight into the intrinsic nonreciprocal magnetotransport phenomena in the topological semimetal material., Comment: 8 pages, 4 figures

Recent experiments reported that the magnetic field can drive the Lifshitz transition and one-dimensional (1D) Weyl nodes in the quantum limit of three-dimensional pentatellurides, as they own low carrier densities and can achieve the extreme quantum limit at a low magnetic field. In this paper, we will investigate the conditions for the existence of the 1D Weyl nodes and their dc transport properties. We find that in the strong topological insulator (TI) phase of ZrTe5, the formation of the Weyl nodes depends heavily on the carrier density; while in the weak TI phase of HfTe5, the Weyl nodes are more likely to appear. These behaviors are attributed to the fact that in the strong and weak TI phases, the zeroth Landau levels exhibit opposite evolutions with the magnetic field. Moreover, the signatures of the critical fields that characterize the distinct behaviors of the system can be directly captured in the conductivities., Comment: 10 pages, 7 figures

Apparent diffusion coefficient (ADC) is a measure of the magnitude of diffusion of water molecules within tissues. We argue that ADC value contains information of both diffusion and T2 relaxation. In this letter, we list literature evidence to support our argument. Firstly, we list uterine myometrium tumors as examples. Myometrium has a T2 relaxation time of 79 ms at 3T. Literature shows, when the myometrium tumors are T2 weighted signal intensity hypertensive (relative to myometrium), these tumors likely show diffusion restriction. This is similar to that, while the spleen is T2 weighted signal intensity hypertensive relative to the liver, the spleen demonstrates diffusion restriction relative to the liver. On the other hand, when the myometrium tumors are T2 weighted signal intensity hypotensive (relative to myometrium), these tumors likely do not show diffusion restriction. However, when the myometrium tumors are very highly hypertensive such as the cases of leiomyoma cystic degeneration and myxoid degeneration, the relationship between T2 weighted signal intensity and diffusion is similar to that of a normal gallbladder, i.e., T2 weighted signal highly hypertensive without diffusion restriction. For most of the tumors, much longer T2 may suggest a tumor is waterier. On b=0 images, the signal of tumor (relative to other tissues) can be dominated by T2 effect. On a high b-value images such as b=1000 images, it is possible that tumor regions are dominated by noises. In the end, the ADCs, which are determined by the slope between the signal at b=0 and the signal at b=1000, are also dominated by the T2 effect, with longer T2 being associated with higher ADC measure. In fact, since there is no diffusion gradient with b=0 images, and there are only noises on b=1000 (or b=800), ADC map is devoid of diffusion information.

Considering liver as the reference, that both fast diffusion (PF) and slow diffusion (Dslow) of the spleen are much underestimated is likely due to the MRI properties of the spleen such as the much longer T2 relaxation time. It is possible that longer T2 relaxation time partially mitigates the signal decay effect of various gradients on diffusion weighted image. This phenomenon will not be limited to the spleen. Most liver tumors have a longer T2 relaxation time than their native normal tissue and this is considered to be associated with oedema. On the other hand, most tumors are measured with lower MRI diffusion (despite being oedematous). The reason why malignant tumors have lower diffusion value [apparent diffusion coefficient (ADC) and Dslow] are poorly understood but has been proposed to be related to a combination of higher cellularity, tissue disorganization, and increased extracellular space tortuosity. These explanations may be true, but it is also possible to that many tumors have MRI properties similar to the spleen such as longer T2 (relative to the liver) and these MRI properties may also contribute to the lower MRI measured ADC and Dslow . In other words, if we could hypothetically plant a piece of spleen tissue in the liver, MRI would recognize this planted spleen tissue as being similar to a tumor and measure it to have lower diffusion than the liver.

Recent experiments have reported a lot of spectacular transport properties in topological materials, such as quantum oscillations and three-dimensional (3D) quantum Hall effect (QHE) in ZrTe$_5$. In this paper, by using a strong topological insulator model to describe ZrTe$_5$, we study the magnetotransport property of the 3D system. With fixed carrier density, we find that there exists a deferring effect in the chemical potential, which favors distinguishing the saddle points of the inverted LLs. On the other hand, with fixed chemical potential, the features of 3D QHE are demonstrated and we attribute the underlying mechanisms to the interplay between Dirac fermions, magnetic field and impurity scatterings., Comment: 11 pages, 5 figures

Recently, dynamical characterization of bulk topology has been experimentally realized under nonadiabatic sudden quench dynamics. However, it has been shown that only the topology of final phase can be characterized when the system is quenched from initial topologically trivial phase. In this paper, taking the two-dimensional Chern insulator as an example, we make a thorough investigation of different types of quenching processes under nonadiabatic slow quench dynamics, and study not only the processes between nontrivial phase and trivial phase, but also between the phases with different topological invariants. We find that, under slow quench dynamics, both the initial and final topological phase can be characterized and the topological invariant can be captured by time-averaged spin polarization. Moreover, different types of processes can be distinguished from the special regions where the time-averaged spin polarization vanishes. All the dynamical characterization schemes are entirely based on the experimentally measurable quantity time-averaged spin polarization, and thus one can expect our findings may provide reference for future experiments., Comment: 11 pages,10 figures

We use the Dirac cone model to explore the high Chern number (C) phases that are realized in the magnetic-doped topological insulator (TI) multilayer structures by Zhao et al. [Nature 588, 419 (2020)]. The Chern number is calculated by capturing the evolution of the phase boundaries with the parameters and then the Chern number phase diagrams of the TI multilayer structures are obtained. The high-C behavior is attributed to the band inversion of the renormalized Dirac cones, along with which the spin polarization at the \Gamma point will get increased. Moreover, another two TI multilayer structures as well as the TI superlattice structures are also studied., Comment: 8 pages, 8 figures

The optical selection rules of an electron system under a magnetic field play key roles in determining its optical properties, from which the band structures and underlying symmetries can be derived. In this Letter, based on a three-dimensional strong topological insulator model describing ZrTe5,we study the Landau levels (LLs) and magneto-optical conductivity under an in-plane magnetic field. We reveal that in the transverse conductivity Re(\sigma_{zz}), the unconventional optical selection rules n\righatarrow n\pm 2 dominate, with n being the LL index. We attribute the unconventional selection rules to the peculiar distribution of parity carried by the LLs, resulting from the chiral symmetry of the sub-Hamiltonians. Moreover, we predict that, if the strong anisotropic system is tuned to be nearly isotropic, the LLs would redistribute and the conventional selection rules n\rightarrow n\pm 1 can be recovered., Comment: 6 pages, 4 figures

Dynamical characterization of equilibrium topological phases has attracted considerable attention in recent years. In this paper, we make a thorough exploration of the non-adiabatic characterization of topological phases under slow quench protocol. We first propose an exactly solvable multi-state Landau-Zener model that can be directly applied to the non-adiabatic slow quench dynamics of topological systems. Then we present two different schemes to characterize the bulk topology of the system based on the so called spin inversion surface. The first one needs least number of quenching processes, but requires to measure the gradients of time-averaged spin-polarization on the SIS. The second one only needs to measure the value of time-averaged spin-polarization on the SIS, thus makes it possible to directly characterize the topological phases by introducing an extra quenching process. Moreover, high-order SIS or band inversion surface (BIS) relying on the dimension reduction approach, is also generalized to the above two different characterization schemes. One can extract the topological invariant from pairs of points with opposite signs both on the 0D highest order BIS and on the 0D highest order SIS, which greatly simplifies the measurement strategy and characterization process. In a word, superior to the sudden quench protocol, we demonstrate that the topological invariant can be captured not only by the topological information on BIS, but also on the SIS. In particular, direct characterization of topological phases based on BIS and SIS can be realized., Comment: 11 pages, 11 figures

In this paper, we give a systematic theoretical study on the Landau levels (LLs) and magneto-optical conductivity Re$(\sigma_{\alpha\alpha})$ in a gapped Dirac semimetal model with mixed linear and parabolic dispersions under a magnetic field, which was recently proposed by Jiang \textit{et al.} [Phys. Rev. Lett. {\bf125}, 046403 (2020)] to explain the experimental magnetoinfrared spectroscopy in the three-dimensional ZrTe$_5$ crystal. We find that the strong magnetic field can drive the LLs become noninverted and thus the strong topological insulator phase in ZrTe$_5$ turns to be a trivial insulator. In the different magnetic field regions, the density of states and Re$(\sigma_{\alpha\alpha})$ can exhibit distinct signatures. Moreover, when the magnetic field is weak, a qualitative relation in Re$(\sigma_{zz})$ between the peaks at the saddle points is revealed as Re$(\sigma_{zz}^{\zeta_n})>$Re$(\sigma_{zz}^\Gamma)$, which is in good agreement with the experiment., Comment: 8 page2, 3 figures

In this paper, we make a comprehensive study of the properties of a gapped Dirac semimetal model, which was originally proposed in the magnetoinfrared spectroscopy measurement of ZeTe$_5$, and includes both the linear and parabolic dispersions in all three directions. We find that, depending on the band inversion parameters, $\zeta'$ and $\zeta_z'$, the model can support three different phases: the single Dirac point (DP) phase, the double DPs phase and the Dirac ring phase. The three different phases can be distinguished by their low-energy features in the density of states (DOS) and optical conductivity. At high energy, both the DOS and optical conductivity exhibit power-law like behaviors, with the asymptotic exponents depending heavily on the signs of $\zeta'$ and $\zeta_z'$. Moreover, the thumb-of-rule formula between the DOS and optical conductivity is satisfied only when $(\zeta',\zeta_z')>0$. The implications of our results for experiments are discussed., Comment: 8 pages, 5 figures

The high-Chern number phases with a Chern number C>1 have been observed in a recent experiment that performed on the topological insulator (TI) multilayer structures, consisting of the alternating magnetic-doped and undoped TI layers. In this paper, we develop an effective method to determine the Chern numbers in the TI multilayer structures and then make a systematic study on the Chern number phase diagrams that are modulated by the magnetic doping and the middle layer thickness. We point out that in the multilayer structure, the high-C behavior can be attributed to the band inversion mechanisms. Moreover, we find that the lowest bands may be multifold degenerate around the Gamma point, and when they are inverted, the Chern number change will be larger than one. Besides the TI multilayer structures implemented in the experiment, we also explore the high-C phase realizations in two other kinds of the TI multilayer structures. The implications of our results for experiments are discussed., Comment: 10 pages, 8 figures

Compared with twisted bilayer graphene, twisted double bilayer graphene (TDBG) provides another important platform to realize the moir\'e flat bands. In this paper, we first calculate the valley Chern number phase diagram of TDBG in the parameter space spanned by the twist angle and the interlayer electric potential. To include the effects of interactions, we then phenomenologically introduce the spin-splitting and valley-splitting. We find that when the valley splitting is larger than the bandwidth of the first conduction band so that a gap is opened and the spin splitting is relatively weak, the orbital Chern insulator emerges at half-filling, associated with a large orbital magnetization (OM). Further calculations suggest that there is no sign reversal of the OM when the Fermi energy goes from the bottom to the top of the half-filling gap, as the OM remains negative in both AB-AB stacking and AB-BA stacking. The implications of our results for the ongoing experiments are also discussed., Comment: 12 pages, 9 figures

Understanding the low-energy excitation state in three-dimensional layered compound ZrTe5 remains a challenging problem in the study of novel topological materials. Recently, a two-dimensional conical model was proposed to explain the experimental optical spectroscopy in the 3D ZrTe5, Martino et al. [Phys. Rev. Lett. 122, 217402 (2019)]. Motivated by this work, in this paper, we perform a systematic theoretical study on the optical conductivity of this model in both cases without and with an external magnetic field to further demonstrate the validity of this model and to recover new physics.We find that there exist completely different characteristics for optical conductivity along different directions, due to anisotropic low-energy excitations in this two-dimensional conical model. Specifically, for the interband optical conductivity, we find asymptotic dependence on the optical frequency as Re(\sigma_x)\sim\omega^{1/2} and Re(\sigma_z)\sim{\omega}^{3/2}, which are universal both in the gapped insulator phase and Weyl semimetal phase. For the magneto-optical conductivity, on the contrary, Re(\sigma^B_{x/z}) shows distinct signatures in the gapped insulator phase and Weyl semimetal phase, which can help distinguish the two phases. Our results, to be verified in future experiments, could provide more insights into the understanding of the topological nature of ZrTe5., Comment: Phys.Rev.B 101, 195201 (2020)

Gapless Weyl semimetals (WSMs) are a novel class of topological materials that host massless Weyl fermions as their low-energy excitations. When the Weyl cone is tilted, the Lorentz invariance is broken and the Lifshitz transition drives the system from type-I WSMs into type-II WSMs. Here, based on the lattice model, we perform a systematic numerical study of the effect of on-site disorder on the diagonal magnetoconductivity in tilted WSMs.We use the self-consistent Born approximation to calculate the disorder-induced self-energy and then apply the Kubo-Streda formula to calculate the diagonal magnetoconductivity. For the transverse magnetoconductivity \sigma_xx, the disorder is found to exhibit distinct effects in the quantum limit regime and in the quantum oscillation regime of type-I WSMs, which could be understood from Einstein's relationship.With strong disorder, \sigma_xx shows a linear relation with the inverse magnetic field 1/B , which exhibits certain robustness to both the Fermi energy and the Weyl cone tilting. For the longitudinal magnetoconductivity \sigma_zz , the strong disorder can break the positive magnetoconductivity as well as the Shubnikov-de Haas oscillations.By analyzing the spectral function, we find that the chiral anomaly is still preserved at strong disorder even when the Weyl cone is overtilted, as there is no gap opening around the Weyl nodes. The implications of our results for experiments are discussed., Comment: 10 pages, 6 figures

The study of Weyl semimetals (WSMs) lies at the forefront of the nontrivial topological phenomena in condensed-matter physics. In this work, we study the effect of on-site repulsive Hubbard interaction on the WSM system with a nonzero tilt at half filling. Within the Hartree-Fock mean-field approximation, we treat the Hubbard interaction self-consistently and find that the Fock exchange field vanishes, while the Hartree field can renormalize the topological mass, the tilt, and the Fermi velocity of the Weyl cones. When the renormalized tilt is larger than the renormalized Fermi velocity, the Hubbard interaction will induce the quantum phase transition from a type-I WSM to a type-II WSM. We then provide the interaction-induced phase diagrams of WSMs in different parametric spaces, in which the antiferromagnetic order at strong interaction is also considered. In addition, we analyze another model hosting two pairs of Weyl nodes, and similar results are obtained. The implications of these results are discussed.

The interplay between topology and correlation lies at the forefront of modern condensed matter physics. In this paper, we study the extended fermion-Hubbard model, including the on-site as well as the nearest-neighbor repulsive interactions, on a topological square lattice that supports the Chern insulator. Within the mean-field method, we find that the spontaneous symmetry breaking (SSB) charge-density wave or antiferromagnetic insulator dominates the system when the onsite or nearest-neighbor interactions are strong enough. It is interesting that the antiferromagnetic Chern insulator will appear in the phase diagram when there is an explicitly nonvanishing sublattice potential. In addition, we explore how a finite-size ribbon structure affects the phase diagram and point out that the critical interaction for SSB occurs with weaker strength than the bulk system.

The birefringent fermions possess a spectrum with two distinct Fermi velocities. Here based on the lattice model, we use the mean-field method to investigate the interaction-induced phase transitions of the birefringent fermions. We consider both the short-range nearest-neighbor (NN) and next-nearest-neighbor (NNN) repulsive interactions and calculate the phase diagrams under different conditions. We find that the NN interactions can induce the asymmetric charge density wave order, while the NNN interactions can drive the asymmetric quantum anomalous Hall order (AQAH) in the large limit of anisotropy in the hopping integrals, \beta\rightarrow 1. The AQAH order is characterized by the unequal loop currents connecting the NNN sites and by the appearance of a gap between the two conduction (valence) bands. Such asymmetric fermionic orders can be attributed to the specific lattice geometry of the birefringent fermions. The implications of our results for experiments are also discussed.

Motivated by the recent experiment realizing bidirectional spin-orbit-coupled Bose-Einstein condensates (BEC), we theoretically explore the properties of repulsive fermions in the two-dimensional (2D) optical lattice with such non-Abelian gauge fields. Within the mean-field level, we find a novel phase of topological antiferromagnetic (TAFM) order which incorporates both the non-trivial topology due to spin-flip hopping and spontaneous symmetry breaking (SSB) for the in-plane spin order. We argue that the appearance of such a phase is generic for repulsive fermions in Chern bands achieved through spin-orbit coupling. Our work paves the way for further studies of fermionic generalization of 2D non-Abelian spin-orbit-coupled quantum gases.

Introduction: This study evaluated algorithm-based qualitative (ABQ) method for vertebral fracture (VF) evaluation with reference to semi-quantitative (SQ) method and bone mineral density (BMD) measurement. Methods: Mr. OS (Hong Kong) and Ms. OS (Hong Kong) represent the first large-scale cohort studies on bone health in elderly Chinese men and women. The current study compared Genant's SQ method and ABQ method in these two cohorts. Based on quantitative measurement, the severity of ABQ method detected fractures was additionally classified into grade-1, grad-2, and grade-3 according to SQ's deformity criteria. The radiographs of 1,954 elderly Chinese men (mean: 72.3 years) and 1,953 elderly Chinese women (mean: 72.5 years) were evaluated. Results: according to ABQ, grade-1,-2,-3 VFs accounted for 1.89%, 1.74%, 2.25% in men, and 3.33%, 3.07%, and 5.53% in women. In men and women, 15.7% (35/223) and 34.5% (48/139) of vertebrae with SQ grade-1 deformity were ABQ(+, with fracture) respectively. In men and women, 89.7% (35/39) and 66.7% (48/72) of vertebrae with ABQ grade-1 fracture had SQ grade-1 deformity. For grade-1 change, SQ (-, negative without fracture) & ABQ (+, positive with vertebral cortex line fracture) subjects tend to have a lower BMD than the SQ(+)& ABQ(-) subjects. In subjects with SQ grade-2 deformity, those were also ABQ(+) tended to have a lower BMD than those were ABQ(-). In all grades, SQ(-)&ABQ(-) subjects tended to have highest BMD, while SQ(+)&ABQ(+)subjects tended to have lowest BMD. Conclusion: ABQ method may be more sensitive to VF associated mild lower BMD than SQ method., Comment: 26 pages,4 figures, 6 tables

Physical therapy is the first line of treatment for adults with symptoms from degenerative spondylolisthesis. Surgical management is offered when nonoperative options have not adequately relieved symptoms. We performed PubMed literature search with the word 'spondylolisthesis', and updated till September 18, 2016. We selected original research data involving surgical treatment of degenerative spondylolisthesis, and in total there were n articles, including data reported from Japan (n=37 series), South Korea (n=11 series), mainland China (n=5 series), and ROC (Taiwan, n=3 series), America (n=20 series), Europe (n=23 series). The mean age of each study were extracted and used a single entry. We tried our best to filter out double/multiple reported data. Our results showed the median age of degenerative spondylolisthesis patients underwent surgical treatment was 66 years in Japan, 60 years in South Korea, 59 years in mainland China, 59 years in ROC (Taiwan), 65 years in USA, and 66 years in Europe. This study indicates Japan and South Korea may have different surgical practice pattern. It will be of interested to investigate whether more proportion of degenerative spondylolisthesis patients have been treated surgically in South Korea than in Japan. The cost-effectiveness of different approaches may warrant further analysis by professional spine surgeons.

Purpose: To explore the usability and normal T1rho value of liver parenchyma with a novel single breathhold black blood single shot fast spin echo acquisition based liver imaging sequence. Materials and Methods: In total 19 health subjects (10 males, 9 females; mean age: 37.4 yrs; range: 23-54 yrs) participated in the study. 11 subjects had liver scanned twice in the same session to access scan-rescan repeatability. 12 subjects had liver scanned twice in two sessions with 7-10 days' interval to access scan-rescan reproducibility. MR was performed with a 3.0 T scanner with dual transmitter. The MR sequence allows simultaneous acquisition of 4 spin lock times (TSLs: 0ms, 10 ms, 30 ms, 50ms) in 10 second. Inherent black blood effect of fast spin echo and double inversion recovery were utilized to achieve blood signal suppression. Results: The technique demonstrated good image quality and minimal artifacts. For liver parenchyma, Bland-Altman plot showed the scan-rescan repeatability mean difference was 0.025 ms (95% limits of agreement: -1.163 to 1.213 ms), and intraclass correlation coefficient (ICC) was 0.977. The scan-rescan reproducibility mean difference was -0.075 ms (95% limits of agreement: -3.280 to 3.310 ms), and ICC was 0.820 which is better than the ICC of 0.764 of a previous bright blood multi-breath hold gradient echo acquisition technique. The liver T1rho value was 39.9 +/- 2.4 ms (range: 36.1 - 44.2 ms), which is lower than the value of 42.8=/-2.1 ms acquired with the previous bright blood technique. Conclusion: This study validated the application of a single breathhold black blood single shot fast spin echo acquisition based for human liver T1rho imaging. The lower liver parenchyma T1rho value and higher scan rescan reproducibility may improve of the sensitivity of this technique., Comment: Total 30 pages, 4 figures, 1 table

Purpose: Rupture of an intracranial aneurysm is the most common cause of subarachnoid haemorrhage (SAH), which is a life-threatening acute cerebrovascular event that typically affects working-age people. This study aims to investigate the aneurysmal SAH incidence rate in elderly population than in middle aged population in China. Materials and methods: Aneurysmal SAH cases were collected retrospectively from the archives of 21 hospitals in Mainland China. All the cases collected were from September 2016 and backward consecutively for a period of time up to 8 years. SAH was initially diagnosed by brain computed tomography, and CT angiography (CTA) or digital subtraction angiography (DSA) was followed and SAH was confirmed to be due to cerebral aneurysm. When for cases multiple bleeding occurred, the age of the first SAH was used in this study. The toltal incidence from all hospital at each age were summed together for females and males; then adjusted by the total population number at each age for females and males. The total population data was from the 2010 population census of the People's Republic of China. Results: In total there were 8,144 cases, with 4,861 females and 3,283 males. Our analysis shows for both females and males the relative aneurysmal SAH rate started to decrease after around 65 years old. The males the relative aneurysmal SAH rate might have started to decrease after around 55 years old. Conclusion: In contrast to previous reports, our data demonstrated a decreased aneurysmal subarachnoid hemorrhage incidence rate in elderly population than in middle aged population. Our data therefore support the hypothesis that aneurysms do not grow progressively once they form but probably either rupture or stabilize and that very elderly patients are at a reduced risk of rupture compared with atients who are younger with the same-sized aneurysms., Comment: Total 16 pages, 3 figures

The epidemiology of lumbar degenerative spondylolisthesis (DS) remains controversial. We performed a systemic review with the aim to have a better understanding of DS's prevalence in general population. The results showed the prevalence of DS is very gender specific and age specific. Both women and men have few DS before 50 years old, after 50 years old both women and men start to develop DS, with women having a faster developing rate than men. For elderly Chinese (>=65 yrs, mean age: 72.5 yrs), large population based studies (MsOS(Hong Kong) and MrOS (Hong Kong), females n=2000 and males n=2000) showed DS prevalence was for 25.0% for women and 19.1% for men, and the prevalence F:M (women:men) ratio was 1.3:1. The published data (MsOS(USA) and MrOS(USA) studies) seem to show elderly Caucasian American has a higher DS prevalence, being approximately 60-70% higher than elderly Chinese; however the prevalence F:M ratio was similar to elderly Chinese population. Patient data showed female patients more often received treatment than men; and preliminary data show the ratio of numbers of female patients received treatment compared with men did not differ between Northeast Asians (Chinese, Japanese, and Korean) and European and American Caucasians, being around 2:1 in elderly population. The existing data also suggest that menopause may be a contributing factor for the accelerated DS development in post-menopausal

We investigate the topological phase transitions in graphene under the modulation of circularly polarized light, by analyzing the changes of edge states and its topological structures. A full phase diagram, with up to ten different topological phases, is presented in the parameter space spanned by the driving frequency and light strength. We find that the high-Chern number behavior is very common in the driven system. While the one-photon resonance can create the chiral edge states in the $\pi-$gap, the two-photon resonance will induce the counter-propagating edge modes in the zero-energy gap. When the driving light strength is strong, the number and even the chirality of the edge states may change in the $\pi-$gap. The robustness of the edge states to disorder potential are also examined. We close by discussing the feasibility of experimental proposals., Comment: 6pages, 3 figures