Hidden continuous quantum phase transition without gap closing in non-Hermitian transverse Ising model

Continuous phase transition in quantum matters is a significant issue in condensed matter physics. In general, the continuous quantum phase transitions in many-body systems occur with gap closing. On the other hand, non-Hermitian systems could display quite different properties as their Hermitian counterparts. In this paper, we show that a hidden, continuous quantum phase transition occurs without gap closing in non-Hermitian transverse Ising model. By using a projected Jordan–Wigner transformation, the one-dimensional (1D) non-Hermitian transverse Ising model with ferromagnetic order is mapped on to 1D non-Hermitian Kitaev model with topological superconducting order and becomes exactly solvable. A hidden, continuous quantum phase transition is really normal–abnormal transition for fermionic correlation in the 1D non-Hermitian Kitaev model. In addition, similar hidden, continuous quantum phase transition is discovered in two-dimensional non-Hermitian transverse Ising model and thus becomes a universal feature in certain non-Hermitian many-body systems.


Introduction
Landau's spontaneous symmetry breaking theory is a fundamental theory that plays an important role in condensed matter physics. According to Landau's spontaneous symmetry breaking theory, quantum phases are classified by different symmetries and different quantum phases are divided by (continuous) quantum phase transitions. Thus, continuous phase transition in quantum matters becomes a significant issue in condensed matter physics. Generally, continuous quantum phase transitions are relevant to the ground state properties of many-body systems and are accompanied by the gap closure for elementary excitations.
Recently, non-Hermitian (NH) systems attract great research interest from different fields in recent years. In certain non-Hermitian systems, when the non-Hermitian effects are considered, there exist many exotic physics properties, especially it's topology properties . As long as non-Hermitian parity-time spontaneous breaking [30], non-Hermitian skin effects [31][32][33], anomalous edge-bulk correspondence [34][35][36][37][38][39], and one way propagation [40][41][42]. In reference [43], a continuous quantum phase transition without gap closing in Kitaev's toric-code model was found due to nonorthogonality of eigenstates. This result indicates that the usual relationship between the correlation length and the energy gap could be changed in non-Hermitian systems.
In this paper, we ask the following questions: 'does there exist continuous quantum phase transitions without gap closing in other NH systems?' and 'Do there exist new universal features of these types of quantum phase transition in NH systems?' Motivated by the above questions, we investigate the non-Hermitian transverse Ising model, a hidden, continuous quantum phase transition without gap closing is studied. By using a projected Jordan-Wigner transformation, the one-dimensional (1D) non-Hermitian transverse Ising model with ferromagnetic order is mapped on to 1D non-Hermitian Kitaev model with topological superconducting order. According to exactly solvable properties of 1D non-Hermitian Kitaev model, the universal features of the hidden, continuous quantum phase transitions is explored that is really normal-abnormal transition.
The remainder of the paper is organized as follows. In section 2, we reviewed the theory of an open quantum system of a multi-spin system under post-selection sub-system and derived an effective 1D non-Hermitian transverse Ising model. In this section, we discussed the (global) similarity of non-Hermitian transverse Ising model. In section 3, we studied the quantum phase transition of spontaneous PT symmetry breaking. In section 4, we studied the quantum phase transition with gap closing in the PT symmetric phase. This is an order-disorder transition for the non-Hermitian system. In section 5, we studied the hidden quantum phase transition without gap closing in the PT symmetric phase with long range order by mapping the original 1D non-Hermitian transverse Ising model to an 1D non-Hermitian Kitaev model with the help of projected Jordan-Wigner transformation. In this section, the universal feature for a hidden quantum phase transition is discovered. In section 6, for the 1D non-Hermitian transverse Ising model, a complex, global phase diagram with four phases was obtained. In section 7, similar hidden, continuous quantum phase transition was studied in two-dimensional (2D) non-Hermitian transverse Ising model. Finally, the conclusions are given in section 8.

One-dimensional non-Hermitian transverse Ising model
In open quantum system, the quantum dynamics is effectively described by a non-Hermitian Hamiltonian by projecting out quantum jumping processes. We consider an open quantum system of a multi-spin system under post-selection sub-system. The total Hamiltonian of the whole system iŝ is the Hamiltonian of the multi-spin system,Ĥ B is the Hamiltonian of the environment andĤ BS = j σ + j ⊗B j + h.c. represents the coupling between them.B j is the operator of the environment. N is the number of lattice sites.
Within the Markovian approximation, the non-unitary dynamics of an open quantum system is generated by a Liouvillian superoperator L acting on the reduced density matrix ρ S (t) of the system [44,45], that is described by Gorini-Kossakowski-Sudarshan-Lindblad equation For the model in equation (1), the Lindblad operatorL describes loss from the coupling with the environment. Due to weak measurement, the amplitudes of quantum states are changed and the amplitude changing is described by the term of − i 2 N j=1L † jL j . The term N j=1 (L j ρ S (t)L † j ) describes the quantum jumping processes that would cause the system decoherence.
When we do post-selection on the multi-spin system, the quantum jumping terms N j=1 (L j ρ S (t)L † j ) are projected out and then an effective non-Hermitian Hamiltonian of multi-spin model is obtained aŝ withĤ NTI =Ĥ † NTI . The normalization procedure has been used in the derivation, the constant imaginary shift cancels out in the time evolution [46]. And the plus or minus sign before h z indicates the gain or loss process. Here J > 0 is ferromagnetic Ising coupling constant between two nearest neighbor spins and h y is the strength of a real transverse field along y-axis, h z is the strength of a imaginary transverse field along z-axis. In this paper, the coupling parameter J is set to be unit, J ≡ 1.

Quantum phase transition of spontaneous PT symmetry breaking
Firstly, we study the quantum phase transition of spontaneous PT symmetry breaking.
It is obvious that the non-Hermitian Hamiltonian has PT -symmetry, i.e., P,Ĥ NTI = 0 and T ,Ĥ NTI = 0, but PT ,Ĥ NTI = 0. Here the time reversal operator T is defined as T iT = −i and the parity operator P is defined by rotating each spin by π about the y-axis P = Π N j=1 (iσ y j ). A spontaneous PT symmetry breaking occurs at |h y | = |h z |. For the case |h y | > |h z |, the system has PT symmetry. Now, the energy spectra ofĤ NTI are same to the usual Hermitian transverse Ising model, Re h = 0 and Im h = 0. So, the energy spectra are real; for the case |h y | < |h z |, PT symmetry is broken. Now, the energy spectra ofĤ NTI are same to a transverse Ising model with imaginary transverse field,Ĥ 0 = j (−Jσ x j σ x j+1 + hσ y j ) with Re h = 0 and Im h = 0. So, the energy spectra become complex.
ForĤ NTI , another important property is non-Hermitian similarity. The operation of non-Hermitian Here, β denotes non-Hermiticity. In what follows, for clarity, we denote S by S(β).
We represent the original non-Hermitian transverse Ising model aŝ where (σ or h sinh β = h z , h cosh β = h y . The non-Hermiticity β is obtained as As a result, under an inverse non-Hermitian similarity transformation S −1 ,Ĥ NTI is transformed aŝ Under the inverse non-Hermitian similarity transformation S −1 , the energy levels E n (β) ofĤ NTI are same to those of the Hermitian model E n (β NH = 0) ofĤ 0 , i.e., The corresponding eigenstates ofĤ NTI become where |Ψ 0 denotes eigenstates ofĤ 0 .

Order-disorder quantum phase transition with gap closing
Secondly, we study the quantum phase transition with gap closing in the PT symmetric phase, |h y | > |h z |. Now, the energy spectra ofĤ NTI are same to the usual Hermitian transverse Ising model, with Im h = 0. In the region of |h y | > |h z |, under non-Hermitian (inverse) similarity transformation, the energy levels E n (β) ofĤ NTI are same to the Hermitian model E n (β = 0) ofĤ 0 = j (−Jσ x j σ x j+1 + hσ y j ), i.e., E n (β) = E n (−β) = E n (β = 0). As a result, the real energy gap is obtained as [47]  and the quantum phase transition with the gap closing forĤ NTI is same to that forĤ 0 that is obtained as |J| = |h|.
For non-Hermitian systems, an order parameter is defined by calculating the expectation value on ground states in the biorthogonal sets { vac R (β) and vac L (β) } [48], i.e., Under the condition S −1 σ x S = σ x , due to the similarity of ground states, |vac where A 0 is the order parameter for the Hermitian modelĤ 0 ; i.e., In figure 1, from exact diagonal numerical calculation for 1D non-Hermitian transverse Ising model with N = 16, in which the yellow dashed lines come from theoretical prediction. As a result, in the region of |J| > |h|, the ground state becomes a long range ferromagnetic order with A bi = A 0 = 0; in the region of |J| < |h|, the ground state is disordered state with A bi = A 0 = 0. In addition, near the order-disorder transition for the non-Hermitian system described byĤ NTI , the universal critical phenomenon is the same as that of the Hermitian modelĤ 0 .

Hidden, continuous quantum phase transitions without gap closing
Thirdly, we study the hidden, continuous quantum phase transition without gap closing in the PT symmetric phase with long range order, |h y | > |h z | and |J| > |h|.
By mapping the original 1D non-Hermitian transverse Ising modelĤ NTI to an 1D non-Hermitian Kitaev model with the help of projected Jordan-Wigner transformation, a hidden, continuous quantum phase transition is explored.

Projected Jordan-Wigner transformation
Before we start our study, we will introduce the projected Jordan-Wigner transformation. For 1D non-Hermitian transverse Ising modelĤ NTI , the usual Jordan-Wigner transformation is not applicable [49]. However, we found that in the region with long range ferromagnetic order for the 1D non-Hermitian transverse Ising modelĤ NTI , projected Jordan-Wigner transformation can be applied, by which we map the original spin model to a non-Hermitian Kitaev model. Here, 'projected' means that the Jordan-Wigner transformation can only available for this non-Hermitian transverse Ising model with long range ferromagnetic order. That means, we should concentrate on the non-Hermitian spin model in region of |h y | > |h z | and |J| > |h|. See the logical schematic diagram for projected Jordan-Wigner transformation in figure 2. Let us show the details for projected Jordan-Wigner transformation step by step: Step 1: under a global inverse similarity transformation S −1 (β),Ĥ NTI is transformed into a Hermitian one, i.e.,Ĥ where is the operator of a global non-Hermitian similarity transformation on spin system and S j (β) is defined as with β = ln( h y +h z h y −h z ).
Step 2: we do usual Jordan-Wigner transformation onĤ 0 by the string-like annihilation and creation operators Under the usual Jordan-Wigner transformation, the HamiltonianĤ 0 becomes that for a 1D superconducting statê The c n satisfy c N+1 = c 1 for periodic boundary condition and satisfy c N+1 = −c 1 for antiperiodic boundary condition. Where h β=0 and . It is obvious that the energy levels forĤ NTI can be exactly characterized by the Hermitian superconducting modelĤ β=0 Step 3: we then consider the projection condition in ferromagnetic ordered phase. The two degenerate ground states can be phenomenologically described by the product states of ( Due to the similarity, the basis of two degenerate ground states become Now, for the ground state with uniform distribution of spin direction, a moving Fermion (that is really a kink) will reverse the spin direction. Each reversed spin will contribute an additional factor e ±β on the weight of fermion's operators. A fermion at site j is accompanied by an 'amplitude' string with length j, i.e., e ±β · e ±β . . . · e ±β . By considering the contribution from the 'amplitude' string, there exists an additional weight e ±jβ on fermion's operators from the similarity transformation S(β), The arrow '⇒' denotes that this equation is under the ferromagnetic ordered condition (or the protected condition).
According to the result from this projected Jordan-Wigner transformation, one can see that the fermions have an additional 'imaginary' wave vector k 0 , i.e., Step 4: finally, under the order condition, the resulting 1D non-Hermitian Kitaev model corresponding to the 1D non-Hermitian transverse Ising modelĤ NTI is obtained aŝ where

Hidden quantum phase transition as normal-abnormal transition for fermionic correlation
After the projected Jordan-Wigner transformation, the non-Hermitian spin model is mapped into a 1D non-Hermitian Kitaev model, we can study the correlation between two fermions for the Kitaev model.
Here, the correlation length is obtained as 1/ξ = ln( J h ). We call the exponential decay of vac|c † l c j |vac to be normal correlation between two fermions far away, i.e., vac|c † l c j |vac → 0 in the limit of |l − j| → ∞. Next, we consider the case of 1D non-Hermitian Kitaev model, . We also denote the fermionic correlation byC lj = vac|c † lc j |vac , where |vac = |vac(β = 0) . Due to the existence of additional 'imaginary' wave vector k 0 = iβ, we havẽ The non-Hermitian effect is embodied into the creation and annihilation operators of excitations. From the fermionic correlations, there are two phases: for the case ±β + 1/ξ > 0 or |h y /J ± h z /J| < 1, we have a normal correlation between two fermions far away, i.e., where |l − j| → ∞; for the case of ±β + 1/ξ < 0 or |h y ± h z | > J, we have an abnormal correlation between two fermions far away, i.e.,C where |l − j| → ∞. As a result, a quantum phase transition may occur at since 1/ξ = ln( J h ), combined with equations (6) and (25), we find that According to the energy dispersion of quasi-particles ±2 √ (−J cos k + h) 2 + (J sin k) 2 , there indeed does not exist energy closing at |h y | ± |h z | = J. As a result, the continuous quantum phase transition at |h y | ± |h z | = J is an unusual phase transition without gap closing.
Thus, in the region of |J| > |h|, the 1D non-Hermitian Kitaev model has topological superconducting order. The ground state on a closed chain is two-fold degenerate, one is a ground state for fermions under periodic boundary condition, |vac R + = |0 ; the other is a ground state for fermions under anti-periodic boundary condition, |vac R − = |π .

Effective Hamiltonian for the two degenerate ground states
From above discussion, the ferromagnetic order (|h| < J) in non-Hermitian transverse Ising modelĤ NTI corresponds to the topological phase in non-Hermitian 1D Kitaev modelĤ β F . The ground state on a closed chain has two-fold degeneracy: |0 and |π . To observe the hidden, continuous quantum phase transition without gap closing, we derive the effective Hamiltonian for the two degenerate ground states in topological superconducting order for 1D non-Hermitian Kitaev model. To accurately characterize the physics of the degenerate ground states, we introduce an two-level effective Hamiltonian [50] where are the basis of the degenerate ground states.
It is known that when a virtual fermion (or a virtual kink) moves around the closed chain, the periodic boundary condition is switched to the anti-periodic boundary condition. Consequently, after the quantum tunneling process one ground state changes to the other, i.e., |0 → |π or |π → |0 . See the illustration of the quantum tunneling processes from virtual fermion (or a virtual kink) around the closed chain in figure 3. Thus, for the 1D non-Hermitian Kitaev model, the quantum tunneling process between two degenerate ground states comes from virtual fermion moving around the closed chain. To show the physical consequences of the hidden quantum phase transition, we obtain the effective Hamiltonian for the two degenerate ground states by calculating the contribution from quantum tunneling processes step by step.
Firstly, we calculate the transition matrix Δ ± between two degenerate ground states based on the 1D Hermitian Kitaev modelĤ β=0 F . We denote the transition matrix Δ + between two degenerate ground states from virtual fermions (or virtual kinks) by where j + N and j are the same lattice site and ( ) means the virtual fermion moving along single direction from j to j + N. For the quantum tunneling process of a virtual fermion propagating along one direction around the chain, the transition matrix is obtained as For the Hermitian model, the conjugate transition matrix Δ − is defined by Next, we consider the case of non-Hermitian Kitaev modelĤ β F . We denote the transition matrix Δ + between two degenerate ground states from virtual fermion (or a virtual kink) by vac|c † j+N ( )c j |vac where j + N and j are the same lattice site and ( ) means the virtual fermion moving along single direction from j to j + N. We have and Thirdly, we add a perturbative term slightly breaking the global symmetry, i.e., H NH →Ĥ NH =Ĥ NH + δĤ where δĤ slightly breaks the degeneracy of ground states and chooses a particular ground state with lowest energy vac R (β) . Due to δĤ, there exists energy difference between two degenerate ground states ε. This process plays a role of ι z on the quantum states. where

The observation of the hidden, continuous quantum phase transition
In this section, we show the existence of the hidden, continuous quantum phase transition and provide approaches to observe it. From above discussion, for the non-Hermitian transverse Ising model, the effective Hamiltonian for the degenerate ground states is obtained asH where β+κ) . Here κ is model-dependent parameter. As a result, in thermodynamic limit N → ∞, there exists a phase transition between a phase with normal fermionic correlation and the other with abnormal fermionic correlation at ±β = κ or |h y ± h z | = J that comes from intrinsic correlation anomaly.
On the one hand, the other phase is denoted by Δ + → 0, Δ + → 0. In this phase, we have e N(±β−κ) → 0 or ±β − κ < 0. In this region, the effective Hamiltonian for the degenerate ground states is reduced intoH Now, in the thermodynamic limit N → ∞, the state-similarity Υ of the two degenerate ground states turns to zero, i.e., On the other hand, one is denoted by Δ + → 0, Δ + → ∞ or Δ + → ∞, Δ + → 0. In the region, we have e N(±β−κ) → ∞ or ±β − κ > 0. The effective Hamiltonian for the degenerate ground states is reduced intõ Now, in the thermodynamic limit N → ∞, the state-similarity of the two degenerate ground states turns to 1, i.e., As a result, to observe the existence of the hidden, continuous quantum phase transition, one can detect the sudden change of state-similarity with increasing β. In figure 4, we numerically calculate the state-similarity of the two degenerate ground states and find the sudden change due to the hidden, continuous quantum phase transition. The numerical results are agree with the theoretical prediction. In addition, the fidelity of the ground state with lowest energy could illustrates the hidden, continuous quantum phase transition. We calculate the fidelity of the ground state with lowest energy by exactly numerical. The fidelity of ground state is defined by  where is a parameter term of the model, δ is the increment of the term. See the results in figure 5, one can see that the fidelity becomes divergent at h y = ±0.5 for h z = 0.5. The result indicates a quantum phase transition, which is consistent to our theoretical prediction.

Global phase diagram
Therefore, as shown in figure 6, for the 1D non-Hermitian transverse Ising model described byĤ NH , we get a complex phase diagram with four phases (see figure 6): I phase has PT symmetry and long range ferromagnetic order with normal fermionic correlation; II phase has PT symmetry and long range ferromagnetic order with abnormal fermionic correlation; III phase has PT symmetry and short range ferromagnetic order; IV phase has PT symmetry breaking. There exist three kinds of phase transitions: (a) Spontaneous PT symmetry breaking at |h y | = |h z |; (b) Quantum phase transition at J = |h| that corresponds to the order-disorder phase transition with gap closing; (c) Quantum phase transition at |h y | ± |h z | = J that is characterized by normal-abnormal transition for fermionic correlation in 1D non-Hermitian Kitaev model. This corresponds to the hidden, continuous quantum phase transition without gap closing.

Hidden, continuous quantum phase transitions in two-dimensional non-Hermitian transverse Ising model
At last to generalize our study from a 1D case to two dimensions, we found the hidden, continuous quantum phase transition also exists in two-dimensional non-Hermitian transverse Ising model. In the last section, we study two-dimensional non-Hermitian transverse Ising model [51][52][53][54][55][56], of which the Hamiltonian is given byĤ where J > 0 is ferromagnetic Ising coupling constant between two nearest neighbor spins and h y is the strength of a real transverse field along y-axis, h z is the strength of a imaginary transverse field along z-axis.
In this paper, the coupling parameter J is set to be unit, J ≡ 1. This Hamiltonian also shows PT -symmetry, i.e., P,Ĥ 2D NTI = 0 and T ,Ĥ 2D NTI = 0, but PT ,Ĥ 2D NTI = 0. We consider a non-Hermitian similarity transformation S(β) = j S j (β) on spin system with S j (β) as where β = ln( h y +h z h y −h z ) denotes non-Hermiticity. Under the non-Hermitian similarity transformation, the original spin model turns into where h = |h y | 2 − |h z | 2 . For the case of |h y | > |h z |, the energy spectra for excitations are all real; for the case of |h y | < |h z |, the energy spectra for the excitations become complex. At |h y | = |h z |, there exist exceptional points for excitations. In this paper, we focus on the case of |h y | > |h z |. The order parameter is defined by the expectation value for the ground states in the biorthogonal sets vac R and vac L , i.e., vac R | 1 N j σ x j |vac L = A bi . In the region of |h y | > |h z |, h < h c , A bi = 0, there exists long range ferromagnetic order. In the region of |h y | > |h z |, h > h c , A bi = 0, the ground state is a disordered state without degeneracy. At |h c | ≈ 3.04J that indicates the order-disorder phase transition occurs. For the non-Hermitian transverse Ising model, in the region of |h y | > |h z |, J > h c , the ground states have two-fold degeneracy. The effective Hamiltonian for the degenerate ground states is written as where Δ ± ∼ Δ 0 e ±Nβ and Δ 0 is an amplitude from quantum tunneling effect. Here, N is the total lattice number. Unfortunately, for the 2D non-Hermitian transverse Ising model, the projected Jordan-Wigner transformation fails. In this paper, we just estimate the tunneling amplitude Δ 0 to be proportional to ( h J ) N . As a result, we have Δ ± ∼ ( h J ) N e ±Nβ . In thermodynamic limit N → ∞, there exists the competition between the exponential decay of Δ ∼ e −κN (κ = ln( h J )) with the size of the system from quantum tunneling effect and the exponential enhancement of e ±βN with the size of the system from non-Hermitian similarity effect. Therefore, in thermodynamic limit N → ∞, there exist two phases: one is denoted by e N(±β−κ) → 0 or |β| < |κ| = | ln( h J )|, the other is denoted by e N(±β−κ) → ∞ or |β| > |κ| = | ln( h J )|. At |β| = |κ| = | ln( h J )| or |h y | ± |h z | = |h c |, a hidden quantum phase transition without gap closing occurs.
We then study the two dimensional non-Hermitian transverse Ising model by exactly diagonal numerical calculation on 4 × 4 square lattice. Figures 7 and 8 show the energy gap and order parameter from exact diagonal numerical calculations for 2D non-Hermitian transverse Ising model on 4 × 4 square lattice, respectively. The energy difference ΔE = E 1 − E 0 between the lowest two energy levels E 1 and E 0 is numerically obtained from exact diagonalization calculations for the model on N = 4 × 4 lattice. From figure 7, we can clearly find that the ground states are doubly degenerate in the ordered phase, while the ground state is unique for the disordered phase. This result indicates there is order-disorder phase   . The state similarity vac R − (β)|vac R + (β) of two degenerated ground state for 2D non-Hermitian transverse Ising model with N = 4 × 4. Where the green line represents the PT symmetry breaking happens, the blue line is |h y | 2 − |h z | 2 = |h| = 2 which the order-disorder phase transition with gap closing happens, the red line is |h y | ± |h z | = 2 that hidden phase transition without gap closing happens. Those are agree with our theoretical prediction by replacing |h c | as 2 due to the finite size effect. transition, where the order parameters vac R | 1 N j σ x j |vac L can be calculated by using the biorthogonal basis as shown in figure 8. We find that the behaviors of state-similarity vac R − (β)|vac R + (β) for the two degenerate ground states persist in 2D non-Hermitian transverse Ising model as demonstrated in figure 9. A hidden quantum phase transition occurs at |h y | ± |h z | = |h c |. We found that at |β| = |κ| = ln( J h ) (or |h y | ± |h z | = |h c |) there indeed exists sudden change of state-similarity vac R − (β)|vac R + (β) from 0 to 1, i.e., vac R − (β)|vac R + (β) = 0, |β| < |κ| ; vac R − (β)|vac R + (β) = 1, |β| > |κ| .
As a result, the global phase diagram is quite similar to that of the 1D non-Hermitian transverse Ising model, except for the positions of critical values. We note that the critical value |h c | ≈ 2 obtained from our N = 4 × 4 lattice at h z = 0 is smaller than |h c | ≈ 3.04 [57,58] for the 2D Hermitian transverse Ising model in thermodynamic limit due to the finite-size effect. Finally, we conclude the hidden, continuous quantum phase transitions without gap closing are universal in non-Hermitian transverse Ising model.

Conclusion and discussion
In this paper, we studied the 1D non-Hermitian transverse Ising model, a typical non-Hermitian many-body model with PT symmetry. For the 1D non-Hermitian transverse Ising model, we show a much more complex phase diagram than that for its Hermitian counterpart. This fact indicates the richness of non-Hermitian many-body physics. In particular, a hidden, continuous quantum phase transition without gap closing is discovered by using projected Jordan-Wigner transformation. The numerical results are all consistent with the theoretical predictions, that indicates the results in this paper are valid and universal for non-Hermitian many-body physics with spontaneous Z 2 -symmetry breaking. In the future, the quantum phases and quantum phase transitions in many-body non-Hermitian models will be studied to explore exotic, universal features of the non-Hermitian physics.