In condensed matter physics, non-Abelian statistics for Majorana zero modes (or Majorana Fermions) is very important, really exotic, and completely robust. The race for searching Majorana zero modes and verifying the corresponding non-Abelian statistics becomes an important frontier in condensed matter physics. In this letter, we generalize the Majorana zero modes to non-Hermitian (NH) topological systems that show universal but quite different properties from their Hermitian counterparts. Based on the NH Majorana zero modes, the orthogonal and nonlocal Majorana qubits are well defined. In particular, due to the particle-hole-symmetry breaking, NH Majorana zero modes have irrational non-Abelian statistics with continuously tunable braiding Berry phase from pi/8 to 3pi/8. This is quite different from the usual non-Abelian statistics with fixed braiding Berry phase pi/4 and becomes an example of "irrational topological phenomenon". The one-dimensional NH Kitaev model is taken as an example to numerically verify the irrational non-Abelian statistics for two NH Majorana zero modes. The numerical results are exactly consistent with the theoretical prediction. With the help of braiding these two zero modes, the pi/8 gate can be reached and thus universal topological quantum computation becomes possible.
A single unit cell contains all the information about the bulk system, including the topological feature. The topological invariant can be extracted from a finite system, which consists of several unit cells under certain environment, such as a non-Hermitian external field. We investigate a non- Hermitian finite-size Kitaev chain with PT-symmetric chemical potentials. Exact solution at the symmetric point shows that Majorana edge modes can emerge as the coalescing states at exceptional points and PT symmetry breaking states. The coalescing zero mode is the finite-size projection of the convent…
Majorana edge modes are candidate elements of topological quantum computing. In this work, we purpose a Floquet engineering approach to generate arbitrarily many non-Hermitian Majorana zero and $π$ modes at the edges of a one-dimensional topological superconductor. Focusing on a Kitaev chain with periodically kicked superconducting pairings and gain/losses in the chemical potential or nearest neighbor hopping terms, we found rich non-Hermitian Floquet topological superconducting phases, which are originated from the interplay between drivings and non-Hermitian effects. Each of the phases is ch…
We study a 1D chain of non-interacting bosonic cavities which are subject to nearest-neighbour parametric driving. With a suitable choice of drive phases, this model is strongly analogous to the celebrated Kitaev chain model of a 1D p-wave superconductor. The system exhibits phase-dependent chirality: photons propagate and are amplified in a direction that is determined by the phase of the initial drive or excitation. Further, we find a drastic sensitivity to boundary conditions: for a range of parameters, the boundary-less system has only delocalized, dynamically unstable modes, while a finit…
Topological characterization of non-Hermitian band structures demands more than a straightforward generalization of the Hermitian cases. Even for one-dimensional tight-binding models with nonreciprocal hopping, the appearance of point gaps and the skin effect leads to the breakdown of the usual bulk-boundary correspondence. Luckily, the correspondence can be resurrected by introducing a winding number for the generalized Brillouin zone for systems with even number of bands and chiral symmetry. Here, we analyze the topological phases of a nonreciprocal hopping model on the stub lattice, where o…
For topological characterization of non-Hermitian multiband systems, Majorana's stellar representation (MSR) is applied to 1D multiband models consisting of asymmetric nearest-neighbor hopping and imaginary on-site potentials. The number of edge states isolated from the continuous bulk bands in the complex energy plane is successfully linked with a topological invariant constructed from MSR. Specifically, the number of isolated edge states can be obtained from a winding number defined for the Majorana stars, which also allows for a geometric visualization of the topology related to the isolate…
We introduce non-Hermitian generalizations of Majorana zero modes (MZMs) which appear in the topological phase of a weakly dissipative Kitaev chain coupled to a Markovian bath. Notably, the presence of MZMs ensures that the steady state in the absence of decoherence events is two-fold degenerate. Within a stochastic wavefunction approach, the effective Hamiltonian governing the coherent, non-unitary dynamics retains BDI classification of the closed limit, but belongs to one of four non-Hermitian "flavors" of the ten-fold way. We argue for the stability of MZMs due to a generalization of partic…
Time-quasiperiodic Majoranas are generalizations of Floquet Majoranas in time-quasiperiodic superconducting systems. We show that in a system driven at $d$ mutually irrational frequencies, there are up to $2^d$ types of such Majoranas, coexisting despite spatial overlap and lack of time-translational invariance. Although the quasienergy spectrum is dense in such systems, the time-quasiperiodic Majoranas can be stable and robust against resonances due to localization in the periodic-drives induced synthetic dimensions. This is demonstrated in a time-quasiperiodic Kitaev chain driven at two freq…
Quantum phase transitions are intriguing and fundamental cooperative phenomena in physics. Analyzing a superconducting nanowire with spin-dependent non-Hermitian hopping, we discover a topological quantum phase transition driven by infinitesimal cascade instability. The anomalous phase transition is complementary to the universal non-Bloch wave behavior of non-Hermitian systems. We show that an infinite small magnetic field drastically suppresses the non-Hermitian skin effect, inducing a topological phase with Majorana boundary states. Furthermore, by identifying the bulk topological invariant…
We propose to realize Majorana edge and corner states in electric circuits. First, we simulate the Kitaev model by an LC electric circuit and the $p_{x}+ip_{y}$ model by an LC circuit together with operational amplifiers. Zero-energy edge states emerge in the topological phase, which are detectable by measuring impedance. Next, we simulate the Bernevig-Hughes-Zhang model by including an effective magnetic field without breaking the particle-hole symmetry, where zero-energy corner states emerge in the topological phase. It is demonstrated that they are Ising anyons subject to the braiding. Name…
Andreev bound states (ABSs) in hybrid semiconductor-superconductor nanowires can have near-zero energy in parameter regions where band topology predicts trivial phases. This surprising fact has been used to challenge the interpretation of a number of transport experiments in terms of non-trivial topology with Majorana zero modes (MZMs). We show that this ongoing ABS versus MZM controversy is fully clarified when framed in the language of non-Hermitian topology, the natural description for open quantum systems. This change of paradigm allows us to understand topological transitions and the emer…
Coupled Majorana zero modes with nonzero energies are generally detrimental to the non-Abelian statistics due to the additional dynamic phase. Nevertheless, we show that a well-connected lead can introduce a local non-Hermitian dissipation term to shift the energies of the both coupled Majorana modes to zero, and surprisingly turn the coupled Majorana mode far from the lead into a dark Majorana mode with exponentially small dissipation. This dark Majorana mode can conquer the drawback of the partially overlapped Majorana zero modes and possess the properties of true Majorana zero mode such as …
Non-Hermitian topological superconductivity with symmetry-enriched spectral and eigenstate features
Institute of Physics, Academia Sinica（中央研究院物理研究所）；Department of Physics, National Taiwan Normal University（台湾师范大学物理系）；Department of Applied Physics, Tokyo University of Science（东京理科大学应用物理系）；Department of Basic Sciences, Kyushu Institute of Technology（九州工业大学基础科学系）；Physics Division, National Center for Theoretical Sciences（国家理论科学中心物理组）
Higher-winding phases in one-dimensional non-Hermitian topological superconductors
Institute of Physics, Academia Sinica（中央研究院物理研究所）；Department of Physics, National Taiwan University（国立台湾大学物理系）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Physics Division, National Center for Theoretical Sciences（国家理论科学中心物理组）
We study theoretically a junction of a $\mathcal{PT}$-symmetric non-Hermitian superconductor (PTS) placed between two conventional superconductors. We show that due to non-Hermitian electron-electron interaction in the PTS region and the combination of symmetries, only discrete values of phases of the conventional superconductors yield solutions for Andreev bound states. Remarkably, in the case of $0$ and $π$, we obtain growing and decaying in time Andreev bound states. For $π/2$ and $3π/2$, there is a Majorana zero mode penetrating through the junction in only one direction forming a quasipar…
Parafermions are a natural generalization of Majorana fermions. We consider a breathing Kagome lattice with complex hoppings by imposing $\mathbb{Z}_{3}$ clock symmetry in the complex energy plane. It is a non-Hermitian generalization of the second-order topological insulator characterized by the emergence of topological corner states. We demonstrate that the topological corner states are parafermions in the present $\mathbb{Z}_{3}$ clock-symmetric model. It is also shown that the model is realized in electric circuits properly designed, where the parafermion corner states are observed by impe…
We investigate the roles of non-Hermitian topology in spectral properties and entanglement structures of open systems. In terms of spectral theory, we give a unified understanding of two interpretations of non-Hermitian topology: quantum anomaly and non-Hermitian skin effects, in which the bulk spectra extremely depend on the boundary conditions. In this context, the fact that the intrinsic higher-dimensional skin effects under the full open boundary condition need the presence of the topological defects is understood in terms of the anomalous fermion production such as the Rubakov-Callan effe…
We systematically study a Kitaev chain with imbalanced pair creation and annihilation, which is introduced by non-Hermitian pairing terms. Exact phase diagram shows that the topological phase is still robust under the influence of the conditional imbalance. The gapped phases are characterized by a topological invariant, the extended Zak phase, which is defined by the biorthonormal inner product. Such phases are destroyed at the points where the coalescence of groundstates occur, associating with the time-reversal symmetry breaking. We find that the Majorana edge modes also exist for the open c…
In recent research, it has been shown that non-Hermitian systems exhibit sensitivity to boundaries, and it is caused by the non-Hermitian skin effect. In this work, we construct the non-Bloch band theory in bosonic Bogoliubov--de Gennes (BdG) systems. From our theory, we can calculate the generalized Brillouin zone and the energy spectrum in such systems with open boundary conditions in the thermodynamic limit, and we can thus discuss its non-Hermitian nature, despite Hermiticity of an original Hamiltonian. In fact, we find that the bosonic Kitaev-Majorana chain exhibits rich aspects of the no…
Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians. While several properties unique to non-Hermitian topological systems were uncovered, the fundamental role of symmetry in non-Hermitian physics has yet to be fully understood, and it has remained unclear how symmetry protects non-Hermitian topological phases. Here we show that two fundamental anti-unitary symmetries, time-reversal and particle-hole symmetries, are topologically equivalent in the complex energy plane and hence unified in non-Hermitian physics. A striking consequen…
We investigate the localization and topological transitions in a one-dimensional (interacting) non-Hermitian quasiperiodic lattice, which is described by a generalized Aubry-André-Harper model with irrational modulations in the off-diagonal hopping and on-site potential and with non-Hermiticities from the nonreciprocal hopping and complex potential phase. For noninteracting cases, we reveal that the nonreciprocal hopping (the complex potential phase) can enlarge the delocalization (localization) region in the phase diagrams spanned by two quasiperiodical modulation strengths. We show that the …
Recently, it was established that there exists a direct relation between the non-Hermitian skin effects, -strong dependence of spectra on boundary conditions for non-Hermitian Hamiltonians-, and boundary zero modes for Hermitian topological insulators. On the other hand, in terms of the spectral theory, the skin effects can also be interpreted as instability of spectra for nonnormal (non-Hermitian) Hamiltonians. Applying the latter interpretation to the former relation, we develop a theory of zero modes with quantum anomaly for general Hermitian lattice systems. Our theory is applicable to a w…
We show that topologically protected defect states can exist in open (leaky or lossy) systems even when these systems are topologically trivial in the closed limit. The states appear from within the continuum, thus in absence of a band gap, and are generated via exceptional points (a spectral transition that occurs in open wave and quantum systems with a generalized time-reversal symmetry), or via a degeneracy induced by charge-conjugation-symmetry (which is related to the pole transition of Majorana zero modes). We demonstrate these findings for a leaking passive coupled-resonator optical wav…
We propose a scheme to detect the Majorana bound states (MBSs) by a thermodynamically stable D.C. Josephson current with $4π$-periodicity in the superconducting phase difference, which is distinct from the previous A.C. $4π$-periodicity found in topological superconducting Josephson junctions. The scheme, consisting of a quantum dot coupled to two s-wave superconducting leads and a floating topological superconductor supporting two MBSs at its ends, only exploits the interplay of a local Zeeman field and the exotic helical and self-Hermitian properties of MBSs, without requiring the conservati…
In many of the experimental systems that may host Majorana zero modes, a so-called chiral symmetry exists that protects overlapping zero modes from splitting up. This symmetry is operative in a superconducting nanowire that is narrower than the spin-orbit scattering length, and at the Dirac point of a superconductor/topological insulator heterostructure. Here we show that chiral symmetry strongly modifies the dynamical and spectral properties of a chaotic scatterer, even if it binds only a single zero mode. These properties are quantified by the Wigner-Smith time-delay matrix $Q=-i\hbar S^\dag…
We investigate a topological superconducting wire with balanced gain and loss that is effectively described by the non-Hermitian Kitaev/Majorana chain with parity-time symmetry. This system is shown to possess two distinct types of unconventional edge modes, those with complex energies and nonorthogonal Majorana zero modes. The latter edge modes cause nonlocal particle transport with currents that are localized at the edges and absent in the bulk. This anomalous particle transport results from the interplay between parity-time symmetry (non-Hermiticity) and topological superconductivity.
We develop a complete theory of symmetry and topology in non-Hermitian physics. We demonstrate that non-Hermiticity ramifies the celebrated Altland-Zirnbauer symmetry classification for insulators and superconductors. In particular, charge conjugation is defined in terms of transposition rather than complex conjugation due to the lack of Hermiticity, and hence chiral symmetry becomes distinct from sublattice symmetry. It is also shown that non-Hermiticity enables a Hermitian-conjugate counterpart of the Altland-Zirnbauer symmetry. Taking into account sublattice symmetry or pseudo-Hermiticity a…
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 …
A beam splitter consisting of two normal leads coupled to one end of a topological superconducting nanowire via double quantum dot is investigated. In this geometry, the linear current cross-correlations at zero temperature change signs versus the overlap between the two Majorana bound states hosted by the nanowire. Under symmetric bias voltages the net current flowing through the nanowire is noiseless. These two features highlight the fermionic nature of such exotic Majorana excitations though they are based on the superconductivity. Moreover, there exists a unique local particle-hole symmetr…
We study a non-Hermitian Kitaev chain model that contains three sources of non-Hermiticity: a constant imaginary potential, asymmetry between hopping amplitudes $t_{\rm R}$ and $t_{\rm L}$ in the right and left directions, and imbalance in pair potentials $Δ_{\rm c}$ and $Δ_{\rm a}$ for pair creation and annihilation, respectively. We show that bulk--boundary correspondence holds in this system; two topological invariants defined in bulk geometry under a modified periodic boundary condition correctly describe the presence or absence of a pair of boundary zero modes in boundary geometry under a…
Non-Hermitian (NH) crystals, quasicrystals and amorphous network display an accumulation of a macroscopic number of states near one of its specific interfaces with vacuum, such as edge, surface, hinge or corner. This phenomenon is known as the NH skin effect, which can only be observed with open boundary condition. In this regard self-similar fractals, manifesting inner boundaries in the interior of the system, harbor a novel phenomenon, the \emph{inner skin effect} (ISE). Then the NH skin effect appears at the inner boundaries of the fractal lattice with periodic boundary condition. We showca…
A topological superconductor is characterized by having a pairing gap in the bulk and gapless self-hermitian Majorana modes at its boundary. In one dimension, these are zero-energy modes bound to the ends, while in two dimensions these are chiral gapless modes traveling along the edge. Majorana modes have attracted a lot of interest due to their exotic properties, which include non-abelian exchange statistics. Progress in realizing topological superconductivity has been made by combining spin-orbit coupling, conventional superconductivity, and magnetism. The existence of protected Majorana mod…
We provide a detailed theoretical analysis of the adiabatic evolution of degenerate open quantum systems, where the dynamics is induced by time-dependent fluctuating loop paths in control parameter space. For weak system-bath coupling, the fluctuations around a deterministic base path obey Gaussian statistics, where we assume that the quantum adiabatic theorem is also satisfied for fluctuating paths. We show that universal non-Abelian geometric dephasing (NAGD) contributions are contained in the fluctuation-averaged evolution operator. This operator plays a key role in all experimental protoco…
