Exact solution of non-Hermitian systems with generalized boundary conditions: size-dependent boundary effect and fragility of skin effect

Systems with non-Hermitian skin effects are very sensitive to the imposed boundary conditions and lattice size, and thus an important question is whether non-Hermitian skin effects can survive when deviating from the open boundary condition. To unveil the origin of boundary sensitivity, we present exact solutions for one-dimensional non-Hermitian models with generalized boundary conditions and study rigorously the interplay effect of lattice size and boundary terms. Besides the open boundary condition, we identify the existence of non-Hermitian skin effect when one of the boundary hopping terms vanishes. Apart from this critical line on the boundary parameter space, we find that the skin effect is fragile under any tiny boundary perturbation in the thermodynamic limit, although it can survive in a finite size system. Moreover, we demonstrate that the non-Hermitian Su-Schreieffer-Heeger model exhibits a new phase diagram in the boundary critical line, which is different from either open or periodical boundary case.

Defective Majorana zero modes in non-Hermitian Kitaev chain

We investigate the number-anomalous of the Majorana zero modes in the non-Hermitian Kitaev chain, whose hopping and superconductor paring strength are both imbalanced. We find that the combination of two imbalanced non-Hermitian terms can induce defective Majorana edge states, which means one of the two localized edge states will disappear due to the non-Hermitian suppression effect. As a result, the conventional bulk-boundary correspondence is broken down. Besides, the defective edge states are mapped to the ground states of non-Hermitian transverse field Ising model, and the global phase diagrams of ferromagnetic-antiferromagnetic crossover for ground states are given. Our work, for the first time, reveal the break of topological robustness for the Majorana zero modes, which predict more novel effects both in topological material and in non-Hermitian physics.

Anomoulas Non-Abelian Statistics for Non-Hermitian Majorana Zero Modes: Alternative Approach to Universal Topological Quantum Computation

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, the non-Abelian statistics for these NH Majorana zero modes become anomalous, which is different from the usual non-Abelian statistics. The usual Ivanovs braiding operator for two Majorana modes is generalized to a non-Hermitian Ivanovs braiding perator. The one-dimensional NH Kitaev model is taken as an example to numerically verify the anomalous 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.

Anomalous Spontaneous Symmetry Breaking in non-Hermitian Systems with Biorthogonal Z2-symmetry

Landaus spontaneous symmetry breaking theory is a fundamental theory that describes the collective behaviors in many-body systems. It was well known that for usual spontaneous symmetry breaking in Hermitian systems, the order-disorder phase transition with gap closing and spontaneous symmetry breaking occur at the same critical point. In this paper, we generalized the Landaus spontaneous symmetry breaking theory to the cases in non-Hermitian (NH) many-body systems with biorthogonal Z2 symmetry and tried to discover certain universal features. We were surprised to find that the effect of the NH terms splits the spontaneous biorthogonal Z2 symmetry breaking from a (biorthogonal) order-disorder phase transition with gap closing. The sudden change of similarity for two degenerate ground states indicates a new type of quantum phase transition without gap closing accompanied by spontaneous biorthogonal Z2 symmetry breaking. We will take the NH transverse Ising model as an example to investigate the anomalous spontaneous symmetry breaking. The numerical results were consistent with the theoretical predictions.

State-dependent Topological Invariants and Anomalous Bulk-Boundary Correspondence in non-Hermitian Topological Systems

The breakdown of the bulk-boundary correspondence in non-Hermitian (NH) topological systems is an open, controversial issue. In this paper, to resolve this issue, we ask the following question: Can a (global) topological invariant completely describe the topological properties of a NH system as its Hermitian counterpart? Our answer is no. One cannot use a global topological invariant (including non-Bloch topological invariant) to accurately characterize the topological properties of the NH systems. Instead, there exist a new type of topological invariants that are absence in its Hermitian counterpart -- the state dependent topological invariants. With the help of the state-dependent topological invariants, we develop a new topological theory for NH topological system beyond the general knowledge for usual Hermitian systems and obtain an exact formulation of the bulk-boundary correspondence, including state-dependent phase diagram, state-dependent phase transition and anomalous transport properties (spontaneous topological current). Therefore, these results will help people to understand the exotic topological properties of various non-Hermitian systems.

Effective non-Hermitian physics for degenerate ground states of a nonHermitian Ising model with $mathcal{RT}$ symmetry

In this paper, based on a one-dimensional non-Hermitian spin model with $mathcal{RT}$-invariant term, we study the non-Hermitian physics for the two (nearly) degenerate ground states. By using the high-order perturbation method, an effective pseudo-spin model is obtained to describe non-Hermitian physics for the two (nearly) degenerate ground states, which are precisely consistent with the numerical calculations. We found that there may exist effective (anti) $mathcal{PT}$ symmetry for the effective pseudo-spin model of the two (nearly) degenerate ground states. In particular, there exists spontaneous (anti) $mathcal{PT}$ -symmetry breaking for the topological degenerate ground states with tunable parameters in external fields. We also found that even a very tiny imaginary external field applied will drive $mathcal{PT}$ phase transition.

Fermi Arcs in Tilted Weyl Semimetals: Classification, Evolution and Transport Properties

The Weyl semimetal is a new quantum state of topological semimetal, of which topological surface states -- the Fermi arcs exist. In this paper, the Fermi arcs in Weyl semimetals are classified into two classes -- class-1 and class-2. Based on a tight-binding model, the evolution and transport properties of class-1/2 Fermi arcs are studied via the tilting strength of the bulk Weyl cones. The (residual) anomalous Hall conductivity of topological surface states is a physical consequence of class-1 Fermi arc and thus class-1 Fermi arc becomes a nontrivial topological property for hybrid or type-II Weyl semimetal. Therefore, this work provides an intuitive method to learn topological properties of Weyl semimetal.