After extensive investigation on the Floquet second-order topological insulator (FSOTI) in two dimension (2D), here we propose two driving schemes to systematically engineer the hierarchy of Floquet first-order topological insulator, FSOTI, and Floquet third-order topological insulator in three dimension (3D). Our driving protocols allow these Floquet phases to showcase regular $0$, anomalous $\pi$, and hybrid $0$-$\pi$-modes in a unified phase diagram, obtained for both 2D and 3D systems, while staring from the lower order topological or non-topological phases. Both the step drive and the mass kick protocols exhibit the analogous structure of the evolution operator around the high symmetry points. These eventually enable us to understand the Floquet phase diagrams analytically and the Floquet higher order modes numerically based on finite size systems. The number of $0$ and $\pi$-modes can be tuned irrespective of the frequency in the step drive scheme while we observe frequency driven topological phase transitions for the mass kick protocol. We topologically characterize some of these higher order Floquet phases (harboring either $0$ or anomalous $\pi$ mode) by suitable topological invariant in 2D and 3D cases.
We examine the response of the Fermi arc in the context of quasi-particle interference (QPI) with regard to a localized surface impurity on various three-dimensional Weyl semimetals (WSMs). Our study also reveals the variation of the local density of states (LDOS), obtained by Fourier transforming the QPI profile, on the two-dimensional surface. We use the $T$-matrix formalism to numerically (analytically and numerically) capture the details of the momentum space scattering in QPI (real space decay in LDOS), considering relevant tight-binding lattice and/or low-energy continuum models modeling a range of different WSMs. In particular, we consider multi-WSM (mWSM), hosting multiple Fermi arcs between two opposite chirality Weyl nodes (WNs), where we find a universal $1/r$-decay ($r$ measuring the radial distance from the impurity core) of the impurity-induced LDOS, irrespective of the topological charge. Interestingly, the inter-Fermi arc scattering is only present for triple WSMs, where we find an additional $1/r^3$-decay as compared to double and single WSMs. The untilted single (double) [triple] WSM shows a straight-line (leaf-like) [oval-shaped] QPI profile. The above QPI profiles are canted for hybrid WSMs where type-I and type-II Weyl nodes coexist, however, hybrid single WSM demonstrates strong non-uniformity, unlike the hybrid double and triple WSMs. We also show that the chirality and the positions of the Weyl nodes imprint marked signatures in the QPI profile. This allows us to distinguish between different WSMs, including the time-reversal-broken WSMs from the time-reversal-invariant WSM, even though both of the WSMs can host two pairs of Weyl nodes. Our study can thus shed light on experimentally obtainable complex QPI profiles and help differentiate different WSMs and their surface band structures.
Time reversal and inversion symmetric materials fail to yield linear and nonlinear responses since they possess net zero Berry curvature. However, higher-order Hall response can be generated in these systems upon constraining the crystalline symmetries. Motivated by the recently discovered third-order Hall (TOH) response mediated by Berry connection polarizability, namely, the variation the Berry connection with respect to an applied electric field, here we investigate the existence of such Hall effect in the surface states of hexagonal warped topological insulator (e.g., Bi$_2$Te$_3$) under the application of electric field only. Using the semiclassical Boltzmann formalism, we investigate the effect of tilt and hexagonal warping on the Berry connection polarizability tensor and consequently, the TOH effect provided the Dirac cone remains gapless. We find that the magnitude of the response increases significantly with increasing the tilt strength and warping and therefore, they can provide the tunability of this effect. In addition, we also explore the effect of chemical doping on TOH response in this system. Interestingly, we show based on the symmetry analysis, that the TOH can be the leading-order response in this system which can directly be verified in experiments.
We study the non-equilibrium dynamics of a one-dimensional system of hard core bosons (HCBs) in the presence of an onsite potential (with an alternating sign between the odd and even sites) which shows a quantum phase transition (QPT) from the superfluid (SF) phase to the so-called Mott Insulator (MI) phase. The ground state quantum fidelity shows a sharp dip at the quantum critical point (QCP) while the fidelity susceptibility shows a divergence right there with its scaling given in terms of the correlation length exponent of the QPT. We then study the evolution of this bosonic system following a quench in which the magnitude of the alternating potential is changed starting from zero (the SF phase) to a non-zero value (the MI phase) according to a half Rosen Zener (HRZ) scheme or brought back to the initial value following a full Rosen Zener (FRZ) scheme. The local von Neumann entropy density is calculated in the final MI phase (following the HRZ quench) and is found to be less than the equilibrium value ($\log 2$) due to the defects generated in the final state as a result of the quenching starting from the QCP of the system. We also briefly dwell on the FRZ quenching scheme in which the system is finally in the SF phase through the intermediate MI phase and calculate the reduction in the supercurrent and the non-zero value of the residual local entropy density in the final state. Finally, the loss of coherence of a qubit (globally and weekly coupled to the HCB system) which is initially in a pure state is investigated by calculating the time-dependence of the decoherence factor when the HCB chain evolves under a HRZ scheme starting from the SF phase. This result is compared with that of the sudden quench limit of the half Rosen-Zener scheme where an exact analytical form of the decoherence factor can be derived.
Being motivated by intriguing phenomena such as the breakdown of conventional bulk boundary correspondence and emergence of skin modes in the context of non-Hermitian (NH) topological insulators, we here propose a NH second-order topological superconductor (SOTSC) model that hosts Majorana zero modes (MZMs). Employing the non-Bloch form of NH Hamiltonian, we topologically characterize the above modes by biorthogonal nested polarization and resolve the apparent breakdown of the bulk boundary correspondence. Unlike the Hermitian SOTSC, we notice that the MZMs inhabit only one corner out of four in the two-dimensional NH SOTSC. We extend the static MZMs into the realm of Floquet drive. We find anomalous $\pi$-mode following low-frequency mass-kick in addition to the regular $0$-mode that is usually engineered in a high-frequency regime. We further characterize the regular $0$-mode with biorthogonal Floquet nested polarization. Our proposal is not limited to the $d$-wave superconductivity only and can be realized in the experiment with strongly correlated optical lattice platforms.
Much having explored on two-dimensional higher-order topological superconductors (HOTSCs) hosting Majorana corner modes (MCMs) only, we propose a simple fermionic model based on a three-dimensional topological insulator proximized with $s$-wave superconductor to realize Majorana hinge modes (MHMs) followed by MCMs under the application of appropriate Wilson-Dirac perturbations. We interestingly find that the second-order topological superconductor, hosting MHMs, appear above a threshold value of the first type perturbation while third-order topological superconducting phase, supporting MCMs, immediately arises incorporating infinitesimal perturbation of the second kind. Thus, a hierarchy of HOTSC phases can be realized in a single three-dimensional model. Additionally, the application of bulk magnetic field is found to be instrumental in manipulating the number of MHMs, leaving the number for MCMs unaltered. We analytically understand these above mentioned numerical findings by making resort to the low energy model. We further topologically characterize these phases with a distinct structure of the Wannier spectra. From the practical point of view, we manifest quantized transport signatures of these higher-order modes. Finally, we construct Floquet engineering to generate the ladder of HOTSC phases by kicking the same perturbations as considered in their static counterpart.
We investigate the interplay between the non-Hermiticity and finite temperature in the context of mixed state dynamical quantum phase transition (MSDQPT). We consider a $p$-wave superconductor model, encompassing complex hopping and non-Hermiticity that can lead to gapless phases in addition to gapped phases, to examine the MSDQPT and winding number via the intra-phase quench. We find that the MSDQPT is always present irrespective of the gap structure of the underlying phase, however, the profile of Fisher zeros changes between the above phases. Such occurrences of MSDQPT are in contrast to the zero-temperature case where DQPT does not take place for the gapped phase. Surprisingly, the half-integer jumps in winding number at zero-temperature are washed away for finite temperature in the gapless phase. We study the evolution of the minimum time required by the system to experience MSDQPT with the inverse temperature such that gapped and gapless phases can be differentiated. Our study indicates that the minimum time shows monotonic (non-monotonic) behavior for the gapped (gapless) phase.
Time reversal and inversion symmetric materials fail to yield linear and nonlinear responses since they possess net zero Berry curvature. However, higher-order Hall response can be generated in these systems upon constraining the crystalline symmetries. Motivated by the recently discovered third-order Hall (TOH) response mediated by Berry connection polarizability, namely, the variation the Berry connection with respect to an applied electric field, here we investigate the existence of such Hall effect in the surface states of hexagonal warped topological insulator (e.g., Bi$_2$Te$_3$) under the application of electric field only. Using the semiclassical Boltzmann formalism, we investigate the effect of tilt and hexagonal warping on the Berry connection polarizability tensor and consequently, the TOH effect provided the Dirac cone remains gapless. We find that the magnitude of the response increases significantly with increasing the tilt strength and warping and therefore, they can provide the tunability of this effect. In addition, we also explore the effect of chemical doping on TOH response in this system. Interestingly, we show based on the symmetry analysis, that the TOH can be the leading-order response in this system which can directly be verified in experiments.
Much have been learned about universal properties of entanglement entropy (EE) and participation ration (PR) for Anderson localization. We find a new sub-extensive scaling with system size of the above measures for algebraic localization as noticed in one-dimensional long-range hopping models in the presence of uncorrelated disorder. While the scaling exponent of EE seems to vary universally with the long distance localization exponent of single particle states (SPSs), PR does not show such university as it also depends on the short range correlations of SPSs. On the other hand, in presence of correlated disorder, an admixture of two species of SPSs (ergodic delocalized and non-ergodic multifractal or localized) are observed, which leads to extensive (sub-extensive) scaling of EE (PR). Considering typical many-body eigenstates, we obtain above results that are further corroborated with the asymptotic dynamics. Additionally, a finite time secondary slow growth in EE is witnessed only for correlated case while for uncorrelated case there exists only primary growth followed by the saturation. We believe that our findings from typical many-body eigenstate would remain unaltered even in the weakly interacting limit.
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