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We study many-body phase transitions in a one-dimensional ferromagnetic transversed field Ising model with an imaginary field and show that the system exhibits three phase transitions: one second-order phase transition and two $\mathcal{PT}$ phase transitions. The second-order phase transition occurring in the ground state is investigated via biorthogonal and self-normal entanglement entropy, for which we develop an approach to perform finite-size scaling theory to extract the central charge for small systems. Compared with the second-order phase transition, the first $\mathcal{PT}$ transition is characterized by the appearance of an exceptional point in the full energy spectrum, while the second $\mathcal{PT}$ transition only occurs in specific excited states. Furthermore, we interestingly show that both of exceptional points are second-order in terms of scalings of imaginary parts of the energy. This work provides an exact solution for unconventional many-body phase transitions in non-Hermitian systems., Comment: 7 pages, 5 figures

Till now, the foundation of quantum mechanics is still mysterious. To explore the mysteries in the foundation of quantum mechanics, people always take it for granted that quantum processes must be some types of fields/objects on a rigid space. In this paper, we give a new idea -- the space is no more rigid and matter is really the certain changing of "space" itself rather than extra things on it. According to the new idea, the dynamics for "space" leads to quantum mechanics. Based on this starting point, we develop a new framework on the foundation of quantum mechanics that may have a far-reaching impact on modern physics in future.

Novel topological properties that arose in the periodically driven system are unique, in which there are two kinds of quasienergy gaps, the zero quasienergy gap and the $\pi$ quasienergy gap. The corresponding edge modes would traverse either the zero quasienergy gap or the $\pi$ quasienergy gap, or traverse both two quasienergy gaps. And the characterization of these two kinds of edge modes might not be the same. However, in this paper, we find that both the zero edge modes and the $\pi$ edge modes in the Floquet Chern insulators can be characterized by the same topological invariant, where the corresponding Dirac mass term is periodically kicked. Particularly, we take the periodically kicked Qi-Wu-Zhang model as an illustrative example. In this model, the topology is characterized by the Floquet Chern number $C_F$, and there are six different topological phases in total, denoted as $C_F=\{-1_0,-2,-1_\pi,1_\pi,2,1_0\}$. Furthermore, we find that the Floquet operator associated with the periodically kicked Qi-Wu-Zhang model reduces to a Dirac Hamiltonian in the low-energy limit. Then, the phase diagram is uncovered by examining the topology of this Dirac Hamiltonian. Additionally, we explore the orders of topological phase transitions in the context of Floquet stationary states by analyzing the von Neumann entropy of these states. Our work provides further insights into the topological phases in periodically driven systems., Comment: 10 pages, 7 figures

We show that many-body fermionic non-Hermitian systems require two distinct sets of topological invariants to describe the topology of energy bands and quantum states respectively, with the latter yet to be explored. We identify 10 symmetry classes -- determined by particle-hole, linearized time-reversal, and linearized chiral symmetries. Each class has topological invariant associated with each dimension, dictating the topology of quantum states. These findings pave the way for deeper understanding of the topological phases of many-body non-Hermitian systems.

The non-Hermitian extension of quasicrystals (QC) are highly tunable system for exploring novel material phases. While extended-localized phase transitions have been observed in one dimension, quantum phase transition in higher dimensions and various system sizes remain unexplored. Here, we show the discovery of a new critical phase and imaginary zeros induced first-order quantum phase transition within the two-dimensional (2D) Haldane model with a quasicrystal potential on the upper boundary. Initially, we illustrate a phase diagram that evolves with the amplitude and phase of the quasiperiodic potential, which is divided into three distinct phases by two critical boundaries: phase (I) with extended wave functions, PT-restore phase (II) with localized wave functions, and a critical phase (III) with multifunctional wave functions. To describe the wavefunctions in these distinct phases, we introduce a low-energy approximation theory and an effective two-chain model. Additionally, we uncover a first-order structural phase transition induced (FOSPT) by imaginary zeros. As we increase the size of the potential boundary, we observe the critical phase splitting into regions in proportion to the growing number of potential zeros. Importantly, these observations are consistent with groundstate fidelity and energy gap calculations. Our research enhances the comprehension of phase diagrams associated with high-dimensional quasicrystal potentials, offering valuable contributions to the exploration of unique phases and quantum phase transition., Comment: 11 pages, 7 figures

Recently, much attention has been paid to uncovering the influence of dissipation on a quantum system, particularly on how the non-Hermitian (NH) terms modify the band topology of topological materials and reshape the profile of the wavefunctions of a system (or the NH skin effect). In this paper, a specific NH skin effect that induced by local dissipation is studied for chiral edge modes, in which the NH term corresponds to the local imaginary Fermi velocity of the chiral edge modes. By solving the NH Schr\"{o}dinger equation of the non-Hermitian chiral edge modes (nhCEs) with complex Fermi velocity, we uncovered the remarkable complex spectra and the wavefunctions of the nhCEs. We find that the complex spectra of these modes is a straight line in the topological materials, and its chirality can separates the modes with positive energy from those with negative energy, in which they are localized at different positions. We also studied the nhCEs at the boundary of 2-dimensional (2D) topological materials, the 2D $p$-wave superconductor and the Qi-Wu-Zhang model, in which the general law of nhCEs was verified. We expect that our findings will pave the way for researching the transport properties of the chiral edge modes in the non-equilibrium context., Comment: 7 pages, 6 figures

In the paper, we study the non-Hermitian system under dissipation and give the effective 2*2 Hamiltonian in the k-space by reducing the N*N Hamiltonian in the real space for them. It is discovered that the energy band shows an imaginary line gap. To describe these phenomena, we propose the theory of "non-Hermitian tearing", in which the tearability we define reveals a continuous phase transition at the exceptional point. The non-Hermitian tearing manifests in two forms -- separation of bulk state and decoupling of boundary state. In addition, we also explore the one-dimensional Su-Schrieffer-Heeger model and the Qi-Wu-Zhang model under dissipation using the theory of non-Hermitian tearing. Our results provide a theoretical approach for exploring the controlling of non-Hermitian physics on topological quantum states.

We develop a generalized Aufbau principle for non-Hermitian systems that allows for building up the configurations of indistinguishable particles. The Aufbau rule of non-Hermitian systems is unexpectedly shown to be identical to that developed in Hermitian systems when the real parts of the complex energy levels are considered. We derive full many-body energy spectra of the fermionic and bosonic Hatano-Nelson models as examples by filling single-particle energy levels in the momentum space. For open boundary conditions, we show that many-body non-Hermitian skin effects persist in all many-body eigenstates for both fermions and bosons. Furthermore, we find surprisingly that the ground state of bosons is an anomalous Bose-Einstein condensation with all of the particles simultaneously localizing in both the real and momentum space beyond the Heisenberg uncertainty principle. For periodic boundary conditions, we show that hard-core bosons cannot be mapped to fermions. This work establishes a general framework for understanding the many-body physics of non-Hermitian systems, revealing rich unique non-Hermitian many-body physics., Comment: 6 pages, 4 figures

We present a systematic study of statistical mechanics for non-Hermitian quantum systems. Our work reveals that the stability of a non-Hermitian system necessitates the existence of a single path-dependent conserved quantity, which, in conjunction with the system's Hamiltonian, dictates the equilibrium state. By elucidating the relationship between the Hamiltonian and the supported conserved quantity, we propose criteria for discerning equilibrium states with finite relaxation times. Although our findings indicate that only non-Hermitian systems with real energy spectrum precisely possess such conserved quantities, we also demonstrate that an effective conserved quantity can manifest in certain systems with complex energy spectra. The effective conserved quantity, alongside the effective transitions within their associated subspace, collectively determines the system's equilibrium state. Our results provide valuable insights into non-Hermitian systems across more realistic contexts and hold potential for applications in a diverse range of physical systems.

We classify the dissipative topological insulators (TIs) with edge dark states (EDS) by using the 38-fold way of non-Hermitian systems in this paper. The dissipative dynamics of these quadratic open fermionic systems is captured by a non-Hermitian single-particle matrix which contains both the internal dynamics and the dissipation, refereed to as damping matrix $X$. And the dark states in these systems are the eigenmodes of $X$ which the eigenvalues' imaginary part vanishes. However, there is a constraint on $X$, namely that the modes in which the eigenvalues' imaginary parts are positive are forbidden. In other words, the imaginary line-gap of $X$ is ill-defined, so the topological band theory classifying the dark states can not be applied to $X$. To reveal the topological protection of EDS, we propose the double damping matrix $\tilde{X} = \text{diag}\left( X, X^* \right)$, where the imaginary line-gap is well defined. Thus, the 38-fold way can be applied to $\tilde{X}$, and the topological protection of the EDS is uncovered. Different from previous studies of EDS in purely dissipative dynamics, the EDS in the dissipative TIs are robust against the inclusion of Hamiltonians. Furthermore, the topological classification of $\tilde{X}$ not only reflects the topological protection of EDS in dissipative TIs but also provides a paradigm to predict the appearance of EDS in other open free fermionic systems., Comment: 8 pages, 4 figures

We investigate the transition from PT-symmetry to PT-symmetry breaking and vice versa in the non-Hermitian Landau-Zener (LZ) models. The energy is generally complex, so the relaxation rate of the system is set by the absolute value of the gap. To illustrate the dynamics of phase transitions, the relative population is introduced to calculate the defect density in nonequilibrium phase transitions instead of the excitations in the Hermitian systems. The result shows that the adiabatic-impulse (AI) approximation, which is the key concept of the Kibble-Zurek (KZ) mechanism in the Hermitian systems, can be generalized to the PT-symmetric non-Hermitian LZ models to study the dynamics in the vicinity of a critical point. Therefore, the KZ mechanism in the simplest non-Hermitian two-level models is presented. Finally, an exact solution to the non-Hermitian LZ-like problem is also shown., Comment: 11 pages, 3 figures

In this study, we investigate the many-body non-Hermitian skin effect at finite temperatures in the thermodynamic limit. Our findings indicate an interesting correspondence between the non-Hermitian skin effect and a linear electric potential effect in this case. This correspondence leads to a unique distribution in non-Hermitian systems; particles in these many-body non-Hermitian systems do not inhabit the energy eigenstates of their single-body counterparts. As a result, the many-body non-Hermitian skin effect is significantly different from the single-body non-Hermitian skin effect. Specifically, for fermionic systems, the non-Hermitian skin effect disrupts the original phase, leading to a real-space Fermi surface. For bosonic systems, it can direct bosons to condense in corners at a decay rate that surpasses exponential, even at high temperatures. It also triggers a remarkable phase transition, resulting in spontaneous U(1) symmetry breaking. Uniquely, this does not generate a Goldstone mode, presenting a deviation from traditional expectations as per the Goldstone theorem.

We study parity-time-symmetric non-Hermitian quantum systems at finite temperature, where the Boltzmann distribution law fails to hold. To characterize their abnormal physical properties, a new quantum statistics theory (the so-called quantum Liouvillian statistics theory) was developed, in which the Boltzmann distribution law was replaced by the Liouvillian-Boltzmann distribution law. Using it, we derived analytical results of thermodynamic properties for thermal PT systems and found that a "continuous" thermodynamic phase transition occurs at the exceptional point, where a zero-temperature anomaly exists.

We show that non-Hermitian biorthogonal many-body phase transitions can be characterized by the enhanced decay of Loschmidt echo. The quantum criticality is numerically investigated in a non-Hermitian transverse field Ising model by performing the finite-size dynamical scaling of Loschmidt echo. We determine the equilibrium correlation length critical exponents that are consistent with previous results from the exact diagonalization. More importantly, we introduce a simple method to detect quantum phase transitions with the short-time average of rate function motivated by the critically enhanced decay behavior of Loschmidt echo. Our studies show how to detect equilibrium many-body phase transitions with biorthogonal Loschmidt echo that can be observed in future experiments via quantum dynamics after a quench., Comment: 6 pages, 3 figures

Topological stability is an important property for topological materials. However, the non-Hermitian effects may change this situation. Here, we investigate the robustness of edge states in the non-Hermitian Kitaev chain with imbalanced tunneling term and superconducting pairing term. By defining the similarity of Majorana zero modes (MZMs) and magnetic factor, the coalescing phase diagram of the MZMs and corresponding spin polarization phase diagram are provided. Because of the non-Hermitian coalescence effect and non-Hermitian suppression effect induced by the breakdown of sublattice symmetry and particle-hole symmetry, the system emergence very interesting phenomenons, such as defective MZMs, number-anomalous bulk-boundary correspondence, coalescing of many-body ground states, the magnetic phase crossover without gap closing. Those novel non-Hermitian effects offer fresh insights into MZMs and topological physics., Comment: 9 pages, 5 figures

We develop the perturbation theory of the fidelity susceptibility in biorthogonal bases for arbitrary interacting non-Hermitian many-body systems with real eigenvalues. The quantum criticality in the non-Hermitian transverse field Ising chain is investigated by the second derivative of ground-state energy and the ground-state fidelity susceptibility. We show that the system undergoes a second-order phase transition with the Ising universal class by numerically computing the critical points and the critical exponents from the finite-size scaling theory. Interestingly, our results indicate that the biorthogonal quantum phase transitions are described by the biorthogonal fidelity susceptibility instead of the conventional fidelity susceptibility., Comment: 7 pages, 4 figures

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., Comment: 6 pages, 2 figures

Landau's 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 Landau's 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.

The chiral QED$_3$--Gross-Neveu-Yukawa (QED$_3$-GNY) theory is a $2+1$-dimensional U(1) gauge theory with $N_f$ flavors of four-component Dirac fermions coupled to a scalar field. For $N_f=1$, the specific chiral Ising QED$_3$-GNY model has recently been conjectured to be dual to the deconfined quantum critical point that describes Neel--valence-bond-solid transition of frustrated quantum magnets on square lattice. We study the universal critical behaviors of the chiral QED$_3$-GNY model in $d=4-\epsilon$ dimensions for an arbitrary $N_f$ . We calculate the boson anomalous dimensions, inverse correlation length exponent, as well as the scaling dimensions of nonsinglet fermion bilinear in the chiral QED$_3$-GNY model. The Pad$\acute{e}$ estimates for the exponents are obtained in the chiral Ising-, XY- and Heisenberg-QED$_3$-GNY universality class respectively. We also establish the general condition of the supersymmetric criticality for the ungauged QED$_3$-GNY model. For the conjectured duality between chiral QED$_3$-GNY critical point and deconfined quantum critical point, we find the inverse correlation length exponent has a lower boundary $\nu^{-1}>0.75$, beyond which the Ising-QED$_3$-GNY--$\mathbb{C}$P$^1$ duality may hold., Comment: 15 pages,5 figures

The fluctuations-driven continuous quantum criticality has sparked tremendous interest in condensed matter physics. It has been verified that the gapless fermions fluctuations can change the nature of phase transition at criticality. In this paper, we study the fermionic quantum criticality with enlarged Ising$\times$Ising fluctuations in honeycomb lattice materials. The Gross-Neveu-Yukawa theory for the multicriticality between the semimetallic phase and two ordered phases that break Ising symmetry is investigated by employing perturbative renormalization group approach. We first determine the critical range in which the quantum fluctuations may render the phase transition continuous. We find that the Ising criticality is continuous only when the flavor numbers of four-component Dirac fermions $N_f\geq1/4$. Using the $\epsilon$ expansion in four space-time dimensions, we then study the Ising$\times$Ising multicriticality stemming from the symmetry-breaking electronic instabilities. We analyze the underlying fixed-point structure and compute the critical exponents for the Ising$\times$Ising Gross-Neveu-Yukawa universality class. Further, the correlation scaling behavior for the fermion bilinear on the honeycomb lattice at the multicritical point are also briefly discussed., Comment: 10 pages, 2 figures

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., Comment: 6 pages, 2 figures

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.

It was known that for non-Hermitian topological systems due to the non-Hermitian skin effect, the bulk-edge correspondence is broken down. In this paper, by using one-dimensional Su-SchriefferHeeger model and two-dimensional (deformed) Qi-Wu-Zhang model as examples, we focus on a special type of non-Hermitian topological system without non-Hermitian skin effect-topological systems under non-Hermitian similarity transformation. In these non-Hermitian systems, the defective edge states and the breakdown of bulk-edge correspondence are discovered. To characterize the topological properties, we introduce a new type of inversion symmetry-protected topological invariant-total Z2 topological invariant. In topological phases, defective edge states appear. With the help of the effective edge Hamiltonian, we find that the defective edge states are protected by (generalized) chiral symmetry and thus the (singular) defective edge states are unstable against the perturbation breaking the chiral symmetry. In addition, the results are generalized to nonHermitian topological insulators with inversion symmetry in higher dimensions. This work could help people to understand the defective edge states and the breakdown of bulk-edge correspondence for non-Hermitian topological systems., Comment: 13pages,8figures

In this paper, based on a non-Hermitian toric-code model, we surprisingly find that the degeneracy of ground states can be changed by a local non-Hermitian perturbation (even in thermodynamic limit). We call it non-Hermitian avalanche effect. As the physics consequences of the non-Hermitian avalanche effect, a correspondence between bulk quasi-particles and topologically protected degenerate ground states for Z2 topological order is borken down. In addition, the PT symmetry breaking transition of the topologically degenerate ground states subspace can be observed by fidelity susceptibility., Comment: 8 pages, 8 figures

Unconventional lattice fermions with high degeneracies beyond Weyl and Dirac fermions have attracted intensive attention in recent years. In this paper, attention is drawn to the pseudospin-1 Maxwell fermions and the $(2+1)$ dimensional parity anomaly, which goes beyond the scope of "fermion doubling theorem". We have derived the Hall conductivity of a single Maxwell fermion, and showcased each Maxwell fermion contributes a quantized Hall conductance $e^{2}/h$. We observe that the parity is spontaneously broken in the effective theory of lattice Maxwell fermions interacting with an (auxiliary) $U(1)$ gauge field, leading to an effective anomaly-induced Chern-Simons theory. An interesting observation from the parity anomaly is that the lattice Maxwell fermions beyond the "fermion doubling theorem", so there exists single Maxwell fermion in a lattice model. In addition, our work indicates the quantum anomaly in odd-dimensional spinor space., Comment: 8 pages,2 figures

Non-Hermitian topological systems show quite different properties as their Hermitian counterparts. An important, puzzled issue on non-Hermitian topological systems is the existence of defective edge states beyond usual bulk-boundary correspondence (BBC) that localize either on the left edge or the right edge of the one-dimensional system. In this paper, to understand the existence of the defective edge states, the theory of anomalous bulk-boundary correspondence (A-BBC) is developed that distinguishes the non-Bloch bulk-boundary correspondence (NB-BBC) from non-Hermitian skin effect. By using the one-dimensional non-Hermitian Su-Schrieffer-Heeger model as an example, the underlying physics of defective edge states is explored. The defective edge states are physics consequence of boundary exceptional points of anomalous edge Hamiltonian. In addition, with the help of a theorem, the number anomaly of the edge states in non-Hermitian topological systems become a mathematic problem under quantitative calculations by identifying the Abelian/non-Abelian non-Hermitian condition for edge Hamiltonian and verifying the deviation of the BBC ratio from 1. In the future, the theory for A-BBC can be generalized to higher dimensional non-Hermitian topological systems (for example, 2D Chern insulator).

In modern physics, a profound thought is "symmetry inducing interaction". In this paper, following this though, we pointed out that a symmetry of the quantum state of the vacuum (the so-called "intrinsic symmetry") rather than the usual symmetry of the Lagrangian or the action that plays the key role to construct a correct theory for quantum gravity. Based on a hypotheses about the intrinsic symmetry of the spacetime, we developed the theory for quantum gravity and found that the gravitational interaction naturally emerges., Comment: 2 figures

In this paper, with the help of Hermitian/non-Hermitian dynamic strings, the theory of non-Hermitian topological order is developed based on a non-Hermitian Wen-plaquette model. The effective models for bosonic topological excitations (e-particle and m-particle) are Hermitian tight-binding lattice model; the effective model for fermionic topological excitation (f-particle) becomes a non-Hermitian tight-binding lattice model. In addition, the effective pseudo-spin model for topologically degenerate ground states is derived by calculating the expectation values of Hermitian/non-Hermitian topological closed dynamic strings. For the topologically degenerate ground states of non-Hermitian Wen-plaquette model on an even-by-odd, odd-by-even and odd-by-odd lattice, anomalous topological degeneracy occurs, i.e., the number of the topologically protected ground states may be reduced from 2 to 1. Now, the effective pseudo-spin model turns into the typical PT-symmetric non-Hermitian Hamiltonian with spontaneous PT-symmetry breaking. At exceptional points, the topologically degenerate ground states merge with each other and the topological degeneracy turns into non-Hermitian degeneracy. In the end, the application of the non-Hermitian Z2 topological order and its possible physics realization are discussed.

Deconfined quantum critical point was proposed as a second-order quantum phase transition between two broken symmetry phases beyond the Landau-Ginzburg-Wilson paradigm. However, numerical studies cannot completely rule out a weakly first-order transition because of strong violations of finite-size scaling. We demonstrate that the fidelity is a simple probe to study deconfined quantum critical point. We study the ground-state fidelity susceptibility close to the deconfined quantum critical point in a spin chain using the large-scale finite-size density matrix renormalization group method. We find that the finite-size scaling of the fidelity susceptibility obeys the conventional scaling behavior for continuous phase transitions, supporting the deconfined quantum phase transition is continuous. We numerically determine the deconfined quantum critical point and the associated correlation length critical exponent from the finite-size scaling theory of the fidelity susceptibility. Our results are consistent with recent results obtained directly from the matrix product states for infinite-size lattices using others observables. Our work provides a useful probe to study critical behaviors at deconfined quantum critical point from the concept of quantum information., Comment: 6 pages, 4 figures

Using the linear local induction approximation, we investigated the self-induced motion of a vortex line that corresponds to the motion of a particle in quantum mechanics. Concerning Kelvin waves, the effective Schr\"odinger equation, physical quantities operators, and the corresponding path-integral formula are obtained. The vortex line-particle mapping may help in understanding particle motion in quantum mechanics.

In this paper we discuss the N$\acute{e}$el and Kekul$\acute{e}$ valence bond solids quantum criticality in graphene Dirac semimetal. Considering the quartic four-fermion interaction $g(\bar{\psi}_i\Gamma_{ij}\psi_j)^2$ that contains spin,valley, and sublattice degrees of freedom in the continuum field theory, we find the microscopic symmetry is spontaneously broken when the coupling $g$ is greater than a critical value $g_c$. The symmetry breaking gaps out the fermion and leads to semimetal-insulator transition. All possible quartic fermion-bilinear interactions give rise to the uniform critical coupling, which exhibits the multicritical point for various orders and the Landau-forbidden quantum critical point. We also investigate the typical critical point between N$\acute{e}$el and Kekul$\acute{e}$ valence bond solid transition when the symmetry is broken. The quantum criticality is captured by the Wess-Zumino-Witten term and there exist a mutual-duality for N$\acute{e}$el-Kekul$\acute{e}$ VBS order. We show the emergent spinon in the N$\acute{e}$el-Kekul$\acute{e}$ VBS transition , from which we conclude the phase transition is a deconfined quantum critical point. Additionally, the connection between the index theorem and zero energy mode bounded by the topological defect in the Kekul$\acute{e}$ VBS phase is studied to reveal the N$\acute{e}$el-Kekul$\acute{e}$ VBS duality., Comment: 11 pages, 0 figures

Recently, topological semimetals become hot topic in condensed matter physics, including Dirac semimetal, Weyl semimetal, and nodal line semimetal (NLSM). In this paper, a new type of node- line semimetal - type-II NLSM is proposed based on a two-band cubic lattice model. For type-II NLSM, the zero energy bulk states have a closed loop in momentum space but the (local) Weyl cones on nodal line become tilted. The effect of magnetic field and that of correlation on type-II NLSM are studied. In particular, after considering repulsive interaction and additional spin degrees of freedom, different types of long range magnetic orders appear in bulk states. In addition, the interaction-induced ferromagnetic order of surface states may exist. At critical point between type-I NLSM and type-II NLSM, arbitrary tiny interaction induces ferromagnetic order due to a flat band at Fermi surface.

In this paper, knot physics on entangled vortex-membranes are studied including classification, knot dynamics and effective theory. The physics objects in this paper are entangled vortex-membranes that are called composite knot-crystals. Under projection, a composite knot-crystal is reduced into coupled zero-lattices. In the continuum limit, the effective theories of coupled zero-lattices become quantum field theories. After considering the topological interplay between knots and different types of zero-lattices, gauge interactions emerge. Based on a particular composite knot-crystal with (N=4, M=3) (we call it standard knot-crystal), the derived effective model becomes the (one-flavor) Standard model. As a result, the knot physics may provide an alternative interpretation on quantum field theory., Comment: 85 pages, 15 figures. arXiv admin note: text overlap with arXiv:1604.07217

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., Comment: 4 pages,4 figures

In this paper, the Kelvin wave and knot dynamics are studied on three dimensional smoothly deformed entangled vortex-membranes in five dimensional space. Owing to the existence of local Lorentz invariance and diffeomorphism invariance, in continuum limit gravity becomes an emergent phenomenon on 3+1 dimensional zero-lattice (a lattice of projected zeroes): On the one hand, the deformed zero-lattice can be denoted by curved space-time for knots; on the other hand, the knots as topological defect of 3+1 dimensional zero-lattice indicates matter may curve space-time. This work would help researchers to understand the mystery in gravity., Comment: 15 pages, 4 figures. arXiv admin note: text overlap with arXiv:1604.07217

In this paper, a Bose-Hubbard extension of a Weyl semimetal is proposed that can be realized for ultracold atoms using laser assisted tunneling and Feshbach resonance technique in three dimensional optical lattices. The global phase diagram is obtained consisting of a superfluid phase and various Mott insulator phases by using Landau theory. The Bogoliubov excitation modes for the weakly interacting case have nontrivial properties (Weyl nodes, bosonic surface arc, etc.) analogs of those in Weyl semimetals of electronic systems, which are smoothly carried over to that of Bloch bands for the noninteracting case. The properties of the insulating phases for the strongly interacting case are explored by calculating both the quasiparticle and quasihole dispersion relation, which shows two quasiparticle spectra touch at Weyl nodes., Comment: 10 pages, 4 figures

Quantum mechanics is a successful theory that describes the behavior of photons, electrons, and other atomic- and molecular-scale objects. However, it is far from being well understood. In this paper, a new theory - knot physics for entangled vortex-membranes is developed to explore the underline physics of quantum mechanics. From point view of information, three dimensional quantum Dirac model is developed to effectively describe the fluctuations of local deformation of vortex-membranes: The elementary excitations are knots with information unit; The physics quality to describe local deformation is knot density that is defined by the density of zeros between two projected vortex-membranes; The Biot-Savart equation for Kelvin waves becomes Schrodinger equation for probability waves of knots; the classical functions for perturbative Kelvin waves become wave functions for knots; The probability of quantum mechanics comes from dynamic projection for inner observers owing to "fast clock" effect; The angular frequency for leapfrogging motion plays the role of the mass of knots; etc. This work would help people to understand the mystery of quantum mechanics., Comment: 66 pages, 14 figures

Weyl semimetal (WSM) is an exotic topological state in condensed matter physics. In this paper, based on a two-band cubic lattice model, we studied WSMs with a pair of tunable Weyl nodes. It is pointed out that there exist three types of WSMs with different tilt strengths: type-I WSM, type-II WSM and type-1.5 WSM. In particular, type-1.5 WSM has one type-I node and one type-II node. We studied chiral modes, surface Fermi arcs and quantum anomalous Hall effect in different types of WSMs. In addition, we give an experimental setup to realize the different types of WSMs based on timely technique., Comment: 5 pages, 4 figures

Knot physics is the theory of the universe that not only unified all the fundamental interactions but also explores the underlying physics of quantum mechanics. In knot physics, the most important physical result is the unification of everything (including matter, motion, interaction and space-time, ...) into the entangled vortex-membranes (a knot): our universe is a projection of a four three dimensional entangled vortex-membranes in five dimensional inviscid incompressible fluid, of which the low energy effective theory not only reproduces the Standard model -- an SU_{strong}(3)}*SU_{weak}(2)*U_{Y}(1) gauge theory but also leads to the physics of general relativity. In a word, it from knot., Comment: 5.2 pages, 4 figures

In this paper, we numerically study the non-Abelian statistics of the zero-energy Majorana fermions on the end of Majorana chain and show its application to quantum computing by mapping it to a spin model with special symmetry. In particular, by using transverse-field Ising model with Z2 symmetry, we verify the nontrivial non-Abelian statistics of Majorana fermions. Numerical evidence and comparison in both Majorana-representation and spin-representation are presented. The degenerate ground states of a symmetry protected spin chain therefore previde a promising platform for topological quantum computation., Comment: 5 pages,4 figures

In this paper, we study the quantum properties of a bilayer graphene with (asymmetry) line defects. The localized states are found around the line defects. Thus, the line defects on one certain layer of the bilayer graphene can lead to an electric transport channel. By adding a bias potential along the direction of the line defects, we calculate the electric conductivity of bilayer graphene with line defects using Landauer-B\"{u}ttiker theory, and show that the channel affects the electric conductivity remarkably by comparing the results with those in a perfect bilayer graphene. This one-dimensional line electric channel has the potential to be applied in the nanotechnology engineering., Comment: 5 pages, 7 figures

Manganites are famous mostly for the colossal magnetoresistive effect, which involves the phase separation between ferromagnetic phase and charge-ordered CE-type antiferromagnetic phases. Furthermore, manganites contain some typical magnetic ferroelectrics, e.g. E-type antiferromagnetic $o$-HoMnO$_3$. Here we re-examined these zigzag-winding antiferromagnetic phases (CE-type and E-type antiferromagnets) from the topological perspective. Our theoretical analysis proved that the E-type phase is a weak topological insulator belonging to the $\mathbb{Z}$ class. In momentum space, we classify the symmetries of this phase, and find the three symmetry operators for the chiral, particle-hole, and time-reversal symmetry. The CE-type phase can be described by the Duffin-Kemmer-Petiau algebra, implying that it is a new class of topological insulator and hence extends the existing classification. The corresponding topological end states are demonstrated via numerical calculations, which may implicate the experimental observed ferromagnetic edge states in manganite strips (Nat. Commun. 6, 6179 (2015)) and may play a crucial role in the colossal magnetoresistive effect., Comment: 8 pages, 4 figures

Topological insulators are new states of quantum matter with metallic edge/surface states. In this paper, we pointed out that there exists a new type of particle-hole symmetry-protected topological insulator - topological hierarchy insulator (THI), a composite topological state of a (parent) topological insulator and its defect-induced topological mid-gap states. A particular type of THI is topological fractal insulator, that is a THI with self-similar topological structure. In the end, we discuss the possible experimental realizations of THIs., Comment: 4.5 pages, 3 figures

Recently, Majorana Fermions (MFs) have attracted intensive attention due to their exotic statistics and possible applications in topological quantum computation (TQC). They are proposed to exist in various two-dimensional (2D) topological systems, such as px+ipy topological superconductor and nanowire-superconducting hybridization system. In this paper, two types of Majorana Fermions with different polygon sign rules are pointed out. A "smoking gun" numerical evidence to identify MF's classification is presented through looking for the signature of a first order topological quantum phase transition. By using it, several 2D topological superconductors are studied.