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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