Topological Phase Transition Point Research Articles

Electrical, thermal and thermoelectric transport in open long-range Kitaev chain

We study electrical, thermal and thermoelectric transport in a hybrid device consisting of a long-range Kitaev (LRK) chain coupled to two metallic leads at two ends. Electrical and thermal currents are calculated in this device under both voltage and thermal bias conditions. We find that the transport characteristics of the LRK chain are distinguishably different from its short-range counterpart, which is well known for hosting zero energy Majorana edge modes under some specific range of values of the model parameters. The emergence of massive Dirac fermions, the absence of gap closing at the topological phase transition point and some special features of the energy spectrum which are unique to the LRK chain, significantly alter electrical/thermal current vs. voltage/temperature bias characteristics in comparison with that of the short-range Kitaev chain. These novel transport characteristics of the LRK model can be helpful in understanding nontrivial topological phases of the LRK chain.

Topological phase transition in the antiferromagnetic topological insulator MnBi_2Te_4 from the point of view of axion-like state realization

This work aims to study the conditions of topological phase transition (TPT) between the topological and trivial states in the antiferromagnetic topological insulator (AFM TI) MnBi_2Te_4 and propose some theory about the relationship of this TPT with the possibility of axion-like state realization in this material. Using the density functional approach we have analyzed the changes in the electronic and spin structure of topological surface states (TSSs) and the nearest conduction and valence bands (CB and VB) including the changes in the bulk band gap as well as the Dirac point (DP) gap in TSSs under variation of the spin-orbit coupling strength in the region of the TPT for infinite crystal and slab with a surface both. We have shown that in both cases the TPT occurs with inversion of the contributions of the Bi-p_z and Te-p_z states of different parity at the gap edges related to change in the gap sign. In the case of surface calculations, the Bi-p_z and Te-p_z states at the edges of the bulk band gap and their inversion at the TPT point are transformed into the TSSs with an energy gap at the DP. In this case the TPT takes place without closing the band gap, i.e. with a “jump” through zero and the formation of the nonzero gap during such a transition. Our calculations show that the TPT point is also characterized by an inversion of the out-of-plane spin polarization s_z at the Gamma point for lower and upper parts of the Dirac cone and a significant spatial redistribution of the TSSs between the surface and the bulk. We suppose that the nonzero Dirac gap can have some relationship with the formation of the axion-like state, which presumably couples nonmagnetic spin-orbit and magnetic contributions at the boundary between the topological and trivial phases for a system with parameters close to the TPT conditions. A practically realized system is proposed - the AFM TI with a stoichiometry close to that of MnBi_2Te_2Se_2 with partial (about 50%) substitution of Te atoms for Se atoms in MnBi_2Te_4 which can be an experimental platform for the implementation and experimental analysis of the TPT and the corresponding possibility of the axion-like state realization in Condensed Matter. Besides, such system could serve as a good platform for studying the dynamic axion effect, where the axion field fluctuations are maximised when a small external field is applied to the system which state is close to the TPT.

Band topology and Bloch oscillation in an extended Creutz ladder

We study an extended Creutz ladder system, which supports topological phase transition as revealed by the change of topological invariant and by distinct pseudospin textures. With a linear force along the ladder, Bloch oscillation on topological bands is studied. At the topological phase transition point, the oscillation period is doubled due to band crossings and the wave packet evolves alternatively on the two bands. The magnitude of pseudospin polarizations in Bloch oscillation provide a dynamical identification of the topological feature. The local pseudospin polarizations along y − and z − axis are directly related to measurable inter-leg current and density difference. We also show the micromotion and non-adiabatic propagations of the Bloch wave packet. A small density oscillation beyond the description of Bloch oscillation is revealed. At avoided band crossings, the transition probability of the Landau-Zener tunneling is obtained. Our results exhibit a transparent framework to understand the properties of topological bands and of Bloch wave dynamics.

Non-Hermitian topological Fermi superfluid near the p -wave unitary limit

We discuss theoretically the non-Hermitian superfluid phase transition in one-dimensional two-component Fermi gases near the $p$-wave Feshbach resonance accompanied by the two-body loss associated with dipolar relaxation. We point out that this system gives us an opportunity to explore the interplay among various nontrivial properties such as universal thermodynamics at divergent $p$-wave scattering lengths, the topological phase transition at vanishing chemical potential, and the non-Hermitian Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensate (BEC) transition, in a unified manner. In the BCS phase, the loss-induced superfluid-normal transition occurs when the exceptional point appears in the effective non-Hermitian Hamiltonian. In the BEC phase, the diffusive gapless mode can be regarded as a precursor of the instability of the superfluid state. Moreover, we show that the superfluid state is fragile against the two-body loss near the topological phase transition point.

Nonlinear Hall effects in strained twisted bilayer WSe2

Recently, it has been pointed out that the twisting of bilayer WSe2 would generate topologically non-trivial flat bands near the Fermi energy. In this work, we show that twisted bilayer WSe2 (tWSe2) with uniaxial strain exhibits a large nonlinear Hall (NLH) response due to the non-trivial Berry curvatures of the flat bands. Moreover, the NLH effect is greatly enhanced near the topological phase transition point which can be tuned by a vertical displacement field. Importantly, the nonlinear Hall signal changes sign across the topological phase transition point and provides a way to identify the topological phase transition and probe the topological properties of the flat bands. The strong enhancement and high tunability of the NLH effect near the topological phase transition point renders tWSe2 and related moire materials available platforms for rectification and second harmonic generations.

Thermal control of the topological edge flow in nonlinear photonic lattices

The chaotic evolution resulting from the interplay between topology and nonlinearity in photonic systems generally forbids the sustainability of optical currents. Here, we systematically explore the nonlinear evolution dynamics in topological photonic lattices within the framework of optical thermodynamics. By considering an archetypical two-dimensional Haldane photonic lattice, we discover several prethermal states beyond the topological phase transition point and a stable global equilibrium response, associated with a specific optical temperature and chemical potential. Along these lines, we provide a consistent thermodynamic methodology for both controlling and maximizing the unidirectional power flow in the topological edge states. This can be achieved by either employing cross-phase interactions between two subsystems or by exploiting self-heating effects in disordered or Floquet topological lattices. Our results indicate that photonic topological systems can in fact support robust photon transport processes even under the extreme complexity introduced by nonlinearity, an important feature for contemporary topological applications in photonics.

Spontaneous spin polarization at the edge of the chiral superconducting states in a spin Hall metal

We discuss the chiral superconducting state in a square lattice system with an intrinsic spin-orbit coupling causing the spin Hall effect. We estimate the spontaneous spin polarization (local moment) 〈mi 〉 at the i-th site and its sum M = ∑ i 〈mi 〉 in ribbons. In the zigzag ribbon, we see 〈mi 〉 ≠ 0 near the edge. The energy spectra of the chiral edge states show the spin splitting and then we have non-zero M. Moreover, we find that M is enhanced significantly at a topological phase transition point, where the energy gap is closing. We also have non-zero 〈mi 〉 near the edge of the straight ribbon, but in contrast, there is no spin splitting in the spectra and M vanishes.

Optical spin transport theory of spin- 12 topological Fermi superfluids

We theoretically investigate optical (frequency-dependent) bulk spin transport properties in a spin-1/2 topological Fermi superfluid. We specifically consider a one-dimensional system with an interspin {\it p}-wave interaction, which can be realized in ultracold atom experiments. Developing the BCS-Leggett theory to describe the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensate (BEC) evolution and the $\mathbb{Z}_2$ topological phase transition in this system, we show how the spin transport reflects these many-body aspects. We find that the optical spin conductivity, which is a small AC response of a spin current, shows the spin gapped spectrum in the wide parameter region and the gap closes at $\mathbb{Z}_2$ topological phase transition point. Moreover, the validity of the low-energy effective model of the Majorana zero mode is discussed along the BCS-BEC evolution in connection with the scale invariance at {\it p}-wave unitarity.

Unconventional real-complex spectral transition and Majorana zero modes in nonreciprocal quasicrystals

Non-Hermiticity from nonreciprocal hoppings has recently been shown to induce exotic non-Hermitian phenomena. Here, we study a class of nonreciprocal quasicrystal superconductors described by one-dimensional Kitaev and dimerized Kitaev chains with complex quasiperiodic potentials. We unveil that the quasiperiodic potentials lead to a localization transition and all states become localized, but the nonreciprocal hopping is favorable to the extension of these states. We also uncover the existence of correspondence between the localization transition and the unconventional real-complex transition of the energy spectrum. Moreover, the localization transition takes place accompanied by a topological phase transition, which can be determined by calculating the energy gap closing point and topological invariants. Especially in the dimerized Kitaev chain, three distinguishable regions of the wave function are observed, i.e., an extended region, a localized region, and an intermediate region with mobility edges, which originates from the interplay between the dimerized hopping and the non-Hermitian quasiperiodic potential. In this case, the topological phase transition point is consistent with the boundary between the intermediate and localized regions. Furthermore, the dimerized hopping can induce a second localization transition, where the extended bulk states become localized and then extended again. These unusual properties provide a distinctive paradigm compared with the diagonal quasiperiodic systems.

Topological phase transition of the extended non-Hermitian Su-Schrieffer-Heeger model

Herein, the topological phase transition of an extended non-Hermitian Su-Schrieffer-Heeger model with both asymmetric nearest-neighbor and next-nearest-neighbor hopping is investigated. Moreover, an analytical method to determine the topological phase transition point under the open boundary condition is developed through a combination of the conditions required for constructing the generalized Brillouin zone and the topological invariants associated with distinct topological regions. The analytical results obtained thereafter are observed to be consistent with the analysis of the numerical energy spectrum under the open boundary condition. Finally, the bulk-boundary correspondence in such a non-Hermitian system is also discussed.

Topological Phase Transitions in Disordered Electric Quadrupole Insulators.

We investigate disorder-driven topological phase transitions in quantized electric quadrupole insulators in two dimensions. We show that chiral symmetry can protect the quantization of the quadrupole moment q_{xy}, such that the higher-order topological invariant is well defined even when disorder has broken all crystalline symmetries. Moreover, nonvanishing q_{xy} and consequent corner modes can be induced from a trivial insulating phase by disorder that preserves chiral symmetry. The critical points of such topological phase transitions are marked by the occurrence of extended boundary states even in the presence of strong disorder. We provide a systematic characterization of these disorder-driven topological phase transitions from both bulk and boundary descriptions.

Theory of Surface Andreev Bound States and Odd-Frequency Pairing in Superconductor Junctions

In this review, we mention about the theoretical achievement of physics of surface Andreev bound state (SABS) and odd-frequency pairing during this quarter century based on our research. Currently, topological superconductors with Majorana edge state have become a central issue in condensed matter physics. It is noted that Majorana edge state is nothing but a zero energy surface Andreev bound state (ZESABS) and the study of ZESABS has started in the context of the pairing symmetry of high Tc cuprate. We mention our obtained conductance formula of quasiparticle tunneling and Josephson current in unconventional superconductor junctions through ZESABS have already captured the essence of the charge transport profile via the Majorana fermion. We also mention that in the presence of ZESABS, odd-frequency pairing like spin-triplet s-wave or spin-singlet p-wave is hugely enhanced near the surface or interface. We further analyze the spectroscopy of the induced odd-frequency pairings by extending bulk-boundary correspondence in ZESABS. If we consider a semi-infinite superconductor with ZESABS, the induced odd-frequency pair amplitude can be expressed by a c-number defined in bulk, which we call spectral bulk-boundary correspondence. The odd-frequency pair amplitude is classified into singular part and regular one, the former part is proportional to $\sim 1/z$ (z is a complex frequency) reflecting the presence of ZESABS, and the latter one is proportional to $\sim z$ . The coefficient of the latter one diverges at the topological phase transition point. Analyzing the spectral behavior of the odd-frequency pairing is useful to understand the critical phenomena near topological transition points.

Topologically switchable behavior induced by an elastic instability in a phononic waveguide

While topological insulators have been widely studied, they typically appear in configurations and properties that are set once a structure is fabricated. As such, there is significant interest in developing topologically tunable or switchable concepts. In this work, we demonstrate that geometric nonlinearity in the form of an elastic snap-through instability can be exploited to switch the topological properties of a Timoshenko arch beam unit cell. We first demonstrate that the phonon band structure can be tuned using geometric nonlinearity and large displacement to reveal the existence of a topological phase transition point. To make this concept fully stable under the removal of the applied force, we then demonstrate the emergence of a bistable unit cell by varying the parameters of the unit cell. In doing so, we show that the bistability of the arch beam unit cell can be harnessed to design a switch that controls the topological nature of an interface between two different 1D phononic crystals.

Topological Phase Transition in 2D 1T′‐WSTe

The topological phase transition between normal and topological insulators is essential in topological mechanism analysis and search for new topological materials. Herein, on the basis of the well‐known monolayer 2D topological insulator 1T′‐WTe2, a new 2D material 1T′‐WSTe is constructed. It is a typical central inversion asymmetric 2D topological insulator. The fascinating feature of this material is that its low‐energy electronic structure is very similar to the inversion broken topological phase transition point. The first‐principles calculations reveal that the ±4% uniaxial strain can effectively turn the 2D material into topological insulator or normal insulator phases.

Transport signatures of bulk topological phases in double Rashba nanowires probed by spin-polarized STM

We study a double-nanowire setup proximity coupled to an $s$-wave superconductor and search for the bulk signatures of the topological phase transition that can be observed experimentally, for example, with an STM tip. Three bulk quantities, namely, the charge, the spin polarization, and the pairing amplitude of intrawire superconductivity are studied in this work. The spin polarization and the pairing amplitude flip sign as the system undergoes a phase transition from the trivial to the topological phase. In order to identify promising ways to observe bulk signatures of the phase transition in transport experiments, we compute the spin current flowing between a local spin-polarized probe, such as an STM tip, and the double-nanowire system in the Keldysh formalism. We find that the spin current contains information about the sign flip of the bulk spin polarization and can be used to determine the topological phase transition point.

Measuring an interaction-induced topological phase transition via the single-particle density matrix

We discuss the topological phase transition of the spin-$\frac{1}{2}$ fermionic Haldane model with repulsive on-site interaction. We show that the Berry curvature of the topological Hamiltonian, the first Chern number, and the topological phase transition point can be extracted from the single-particle density matrix for this interacting system. Furthermore, we design a tomography scheme for the single-particle density matrix of interacting fermionic two-band models in experimental realizations with cold atoms in optical lattices.

Diagonal entropy and topological phase transitions in extended Kitaev chains

We investigate the diagonal entropy for ground states of the extended Kitaev chains with extensive pairing and hopping terms. The systems contain rich topological phases equivalently represented by topological invariant winding numbers and Majorana zero modes. Both the finite size scaling law and block scaling law of the diagonal entropy are studied, which indicates that the diagonal entropy demonstrates volume effect. The parameter of volume term is regarded as the diagonal entropy density, which can identify the critical points of symmetry-protected topological phase transitions efficiently in the studied models, even for those with higher winding numbers. The formulation of block scaling law and the capability of diagonal entropy density in detecting topological phase transitions are independent of the chosen bases. In order to manifest the advantage of diagonal entropy, we also calculate the global entanglement, which cannot show clear signatures of the topological phase transitions. This work provides a new quantum-informatic approach to characterize the feature of the topologically ordered states and may motivate a deep understanding of the quantum coherence and diagonal entropy in various condensed matter systems.

Theory of the Josephson current on a magnetically doped topological insulator

Proximity induced superconducting states in the surface of magnetically doped topological insulators can host chiral Majorana modes. We consider a Josephson junction in that system with changing the chemical potential, which drives a topological phase transition in the induced superconducting states as well as a metal-insulator transition in the surface states. The local density of states and the Josephson current are analytically calculated by McMillan's Green's function method in terms of the Andreev reflection coefficient. We show that although the magnitude of the Josephson current is greatly enhanced when the surface state changes from insulating to metallic, its temperature dependence drastically changes at the topological phase transition point, reflecting the appearance of the chiral Majorana modes

Topological edge states in the Su-Schrieffer-Heeger model

Su-Schrieffer-Heeger (SSH) model on two-dimensional square lattice exhibits a topological phase transition, which is related to the Zak phase determined by bulk band topology. The strong modulation of electron hopping causes nontrivial charge polarization even in the presence of inversion symmetry. The energy band structures and topological edge states have been calculated numerically in previous studies. Here, however, full energy spectrum and explicit form of wave functions for two-dimensional bulk and one-dimensional ribbon geometries of SSH model are analytically derived using wave mechanics approach. Explicit analytic representations of wave functions provide the information of parity for each subband, localization length and critical point of topological phase transition in SSH ribbon. It is also shown that the dimensional crossover of topological transition point for SSH model from one to two-dimension.

Odd-frequency pairs in chiral symmetric systems: Spectral bulk-boundary correspondence and topological criticality

Odd-frequency Cooper pairs with chiral symmetry emerging at the edges of topological superconductors are a useful physical quantity for characterizing the topological properties of these materials. In this work, we show that the odd-frequency Cooper pair amplitudes can be expressed by a winding number extended to a nonzero frequency, which is called a `spectral bulk-boundary correspondence,' and can be evaluated from the spectral features of the bulk. The odd-frequency Cooper pair amplitudes are classified into two categories: the amplitudes in the first category have the singular functional form $\sim 1/z$ (where $z$ is a complex frequency) that reflects the presence of a topological surface Andreev bound state, whereas the amplitudes in the second category have the regular form $\sim z$ and are regarded as non-topological. We discuss the topological phase transition by using the coefficient in the latter category, which undergoes a power-law divergence at the topological phase transition point and is used to indicate the distance to the critical point. These concepts are established based on several concrete models, including a Rashba nanowire system that is promising for realizing Majorana fermions.