Hall conductance of a non-Hermitian Weyl semimetal

In recent years, non-Hermitian (NH) topological semimetals have garnered significant attention due to their unconventional properties. In this work, we explore one of the transport properties, namely the Hall conductance of a three-dimensional dissipative Weyl semi-metal formed as a result of the stacking of two-dimensional Chern insulators. We find that unlike Hermitian systems where the Hall conductance is quantized, in presence of non-Hermiticity, the quantized Hall conductance starts to deviate from its usual nature. We show that the non-quantized nature of the Hall conductance in such NH topological systems is intimately connected to the presence of exceptional points. We find that in the case of open boundary conditions, the transition from a topologically trivial regime to a non-trivial topological regime takes place at a different value of the momentum than that of the periodic boundary spectra. This discrepancy is solved by considering the non-Bloch case and the generalized Brillouin zone (GBZ). Finally, we present the Hall conductance evaluated over the GBZ and connect it to the separation between the Weyl nodes, within the non-Bloch theory.


Introduction
Topological materials have been of great research interest for the past few decades for their various intriguing properties.The branch of topology started entering into condensed matter physics soon after the discovery of the quantum Hall effect [1].Subsequently, new materials with unique topological properties were discovered and coined as topological insulators (TIs) [2][3][4][5][6][7], where the spin-orbit interaction plays a crucial role.In this arena of research, the classification of the topological phases arises due to different symmetries preserved or broken in the system [7].In TIs, the existence of unusual edge properties is protected by time-reversal (TR) symmetry.On the other hand, in some cases, we encounter topologically protected edge states for TR symmetry broken two-dimensional (2D) systems, which are termed as Chern insulating phases [8].Furthermore, topologically nontrivial phases also arise in gap-less materials.Graphene in 2D [9] and Weyl semi-metals (WSMs) [10][11][12][13][14][15][16][17][18][19][20][21][22][23] in 3D show a gap-less band structure.In WSMs the conduction and valence bands touch each other at some special points, i.e. the Weyl nodes, which appear in pairs with quantized Berry charge.The surface states of these systems are in the form of an open-ended arc, coined as the Fermi arc.These Weyl nodes are protected from several kinds of disorder, apart from those with broken discrete translation symmetry or broken charge conservation symmetry [10][11][12][13][14][15][16][17][18][19][20][21][22][23].Remarkably, the fact that the WSM phases can be achieved by stacking multiple layers of Chern insulators has been discussed in literature [10,24,25].
Recently, NH topological phases have drawn considerable attention of the research community ranging from photonics to condensed matter physics [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45].Dissipation in both classical and quantum mechanical systems is quite common, and this may lead to NH loss and gain.Recent experimental endeavours [46][47][48] in controlling dissipation have brought prodigious versatility in the synthesis of NH properties in open classical and quantum systems.NH systems exhibit unique features absent in their Hermitian counterparts.For instance, in such systems at some particular points of the spectra, the energy eigenvalues and eigenvectors coalesce; in other words, the Hamiltonian describing the system becomes defective.These points are known as EPs [49][50][51].Another such property is the NH skin effect (NHSE), where all the eigenstates get localized at the boundary of the system under open boundary condition (OBC).
In order to establish a bridge between the NH topological systems and their transport properties, a few efforts have been made in the recent years [52][53][54][55][56][57][58].One such transport property is the Hall conductance.In Hermitian 2D Chern insulators, the Hall conductance is quantized and is given by σ = (e 2 /h)C, which is the celebrated Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula [59] for a 2D system.Here C is the Chern number, the topological invariant of the system.However, in the presence of the non-Hermiticity the usual nature of the Hall conductance of the 2D system starts to deviate from the quantized value [52][53][54][55].But, the nature of the Hall conductance of a three-dimensional system in the presence of non-Hermiticity has not been thoroughly investigated so far.
In this paper, we study the Hall conductance of a 3D NH WSM in detail.We consider a stack of 2D layers of Chern insulators coupled to their neighboring layers with a tunneling amplitude t 3 , giving rise to a 3D WSM.Here, it is important to mention that though [60] studied topological properties of NH WSM, the physical motivation of our model building is quite different from [60], where the authors discuss a WSM phase but for a cubic lattice, whereas, in our model, we consider stacking of Chern insulators.In this unique stacked system, the introduction of imaginary intra-cell hopping makes it NH.Then, we investigate the Hall conductance in corroboration with the exceptional topology of the system along the direction of the stacking.Our study reveals that the deviation of the Hall conductance from the quantized nature follows a characteristic pattern.Next, we investigate the effect of non-Bloch topology on Hall conductance as well as to quantify the edge transport.In [40] the authors studied the NHSE and broken bulk boundary correspondence and their restoration with the help of the non-Bloch theory framework and a specially designed Chern number.We use their formalism as an essential tool to characterize the Hall conductance of a family of NH WSMs and discover the effects of exceptional topology on them.Finally, we discuss the experimental feasibility of our proposal.
The rest of the paper is organized as follows: In section 2, we start with the Hamiltonian of the stacked NH Chern insulator and discuss the complex eigenspectra.In section 3, we discuss the Hall conductance for our model and the deviation of the Hall conductance from the usual quantized pattern is analyzed.In section 4, the spectra for OBC are analyzed and it is found that the phase transition points are different for the OBC and the Bloch theory.Subsequently, the non-Bloch theory is invoked to show that the momentum phase transition values for OBC and non-Bloch theory match exactly in the generalized Brillouin zone (GBZ).Finally, we discuss the experimental feasibility in section 5 and conclude in section 6.

Non-Hermitian (NH) Weyl semimetal
We consider a stack of 2D layers of Chern insulators forming a 3D WSM [25].The Hamiltonian of our system describing the two-band model is where t and r are the inter-cell hopping amplitudes.Here m is the onsite energy, and t 3 is the inter-layer hopping amplitude.This Hamiltonian reduces to a Hamiltonian describing a 2D Chern insulator when this inter-layer hopping vanishes, i.e. t 3 = 0. We introduce the intra-cell hopping amplitude iγ in each 2D layer, which results in non-Hermiticity and thus the Hamiltonian describes an NH WSM.The schematic of our model is shown in figure 1, with the different hopping terms identified.This unique stacked NH system has not been analyzed before.
In the absence of the imaginary term iγ, the Hermitian Hamiltonian supports two distinct phases depending on the relative strength of the parameters-when m < t 3 , two pairs of Weyl points appear in the system at (k x , k y , k z ) = (0, π, ±k 0 ) and at (k x , k y , k z ) = (π, 0, ±k 0 ), where k 0 = arccos( m t3 ).In the other phase, a pair of Weyl points appear at (k x , k y , k z ) = (0, 0, ±k 0 ), where k 0 = arccos ( m−2r t3 ), when |m − 2r| < t 3 .We note that the 3D WSM has broken TR symmetry but the inversion symmetry is preserved, consequently the Weyl points are separated in momentum space but occur at the same energy.In our work, we have chosen the second phase for further analysis.We characterize the topological phases of WSM using a topological invariant, namely the Chern number.The Chern number for nth isolated band is defined as, where F n (k) is the Berry curvature and A n (k) = −i⟨u n,k |∇ k |u n,k ⟩ is the Berry connection.u n,k is the Bloch state eigenfunction.For NH systems, the left and right eigenstates are different and we need to consider a combination of left and right eigenfunctions.Now, in our system, we compute the Chern number using equation (2) in the k x -k y plane, keeping k z as a variable, i.e. the integration is over k x and k y .When k z lies between ±k 0 , the Chern number is found out to be 1, and the system supports a topologically non-trivial phase.However, when k z lies outside this range, a topologically trivial phase is obtained where the Chern number becomes zero.In most of our analysis, we choose m = 2 and t = r = t 3 = 1, so that k 0 = π 2 .The NH WSM described by equation (1) breaks the TR symmetry and admits the pseudo-Hermiticity.The Hamiltonian satisfies η −1 H † η = H where η is the product of the spatial inversion and σ z .This specific symmetry guarantees a spectrum comprising purely real eigenvalues and pairs of complex conjugated eigenvalues.As a result, the energy eigenvalues of our Hamiltonian appear in complex conjugate pair and are given by, A pair of EPs appear corresponding to each Weyl point along the ).Here k i 0 and k o 0 are the coordinates of inner and outer EPs, respectively.Thus two pairs of EPs appear when k z ∈ (−π, π).For k y ̸ = 0, ECs appear in the k x = 0 plane.We choose the WSM phase by fixing the parameters and as a result in our NH system, it is possible to adjust the position of the EPs along the k z direction as well as the sizes of the ECs by tuning the strength of the non-Hermiticity parameter γ.This has been illustrated and shown in figure 2. The sizes of the ECs formed in the NH system increases with increasing strength of the non-Hermiticity parameter.Four EPs (blue dots) are formed along k x = k y = 0 line.
There are two topological invariants, the vorticity and the winding number, which are used to characterize the spectral topology of the system.The vorticity is associated with the energy dispersion of the NH band structures.The vorticity, v mn , is defined for a pair of bands as [61] where v mn (Γ) is the vorticity associated with the bands m and n.Here Γ represents a closed loop in momentum space.Since eigenvalues for an NH Hamiltonian are in general complex, they can always be written as ϵ(k) = |ϵ(k)|e iθL , where θ L = arctan Im(ϵ) Re(ϵ) .For a periodic cycle, θ L evolves as θ L → θ L + 2π µ, with µ is an integer.When the loop contains no EPs, the parameter µ = 0.However, when the loop involves EPs, takes half-integer or integer values and is identified as a topological invariant specific to the NH systems.Thus the vorticity provides information about how many EPs are involved within the loop and cannot take arbitrary values.The vorticity results in a half-integer value when a single EP is enclosed within a loop.On the other hand, it becomes an integer (zero) when it encircles two EPs with the same (opposite) vorticity.Due to the square root singularity present in the energy eigenvalues, in the complex plane energy bands get exchanged at an EP, and it takes two loops to come back to the initial state.In figure 3, the vorticity associated with a pair of EPs (for other pair of EPs same follows) located at k x = k y = 0, k z = k i,o z in our system is presented.As demonstrated in figures 3(a) and (b), the two bands wind around each other in clockwise and anticlockwise directions, resulting in the vortices of the EPs (at Apart from the vorticity, another important topological invariant for NH systems is the winding number.The winding number is related to the Berry phase acquired by the eigenstates of the system [62], and it is protected by the chiral symmetry.We map out the complete phase diagram enabling NH topological phase transitions by evaluating the winding number.Since the system has chiral symmetry for k y = 0, and the EPs appear along the k z direction, we compute the winding number for every one-dimensional chain along the k x direction by treating k z as a parameter, which has the final form as follows [63,64] (see appendix A for a detailed analysis) Such a fractional winding number is a characteristic feature of the NH systems.The winding number has been shown to be closely related to the NH generalization of the Berry phase.Thus, this specially designed NH winding number suggests that if we encircle a single EP with a closed loop, we indeed obtain a half-quantized winding number.It is to be noted here that when the closed contour we consider in the calculation of the winding number contains two EPs of the same winding direction the resulting winding number is ±1 (with an NH Berry phase value of π) reminiscent of Hermitian topology.Else, the value of the winding number becomes trivial, i.e. 0. In NH systems, it is important to analyze the effect of these EPs on the transport properties of the system.In the next section, we present such an analysis of the Hall conductance for our system.Having discussed these two topological invariants, it is important to mention that vorticity is well-defined in absence of any symmetry, but the winding number formulation requires the system posses chiral symmetry.

Hall conductance
To study the Hall conductance of our 3D system, let us first illustrate the Hall conductance of a two-band system which is described by a generic two-band Hamiltonian of the form where d j,k = d R j,k + id I j,k (j = x, y, z) are in general complex for NH systems.It is well known that in Hermitian systems using linear response theory, and Kubo formalism, the Hall conductance can be computed.However, the natural question arises about the validity of such theories in the NH domain.In [65][66][67], the authors have proposed a Kubo-like formalism for the NH systems, where the correlation functions contain distribution functions, which are different from the Fermi distribution functions.Interestingly one can show that the long-time behavior of the NH distribution functions is similar to the Fermi distribution [65].Thus using [53,65,66], the Hall conductance in the x − y plane is written as Here K xy (ω) is the current-current correlation function and is given by with n F (ϵ) as the NH distribution function at temperature T. The J x(y) is the current operator.In NH scenario, one needs to deal with both the real and imaginary energies and this causes a departure in the definition of the current operators from the Hermitian case.In our systems, the current operators are defined only with the real part of the energy as follows The Hall conductance as a function of kz given by equation (10).The conductance for three values of γ is plotted in units of e 2 h .The red line corresponds to the Hermitian case and it is fully quantized.The green and blue curves correspond to NH cases for different strengths of non-Hermiticity, i.e. for γ = 0.3 and γ = 0.5, respectively.They considerably deviate from quantization.In this plot, we identify three different regions between the inner EPs, where the conductance value decreases from 1 but stays constant; between a pair of EPs, where the Hall conductance displays a shoulder shape; and outside the EPs, where it decreases to zero.Here, the nature of the Hall conductance is a direct consequence of the EPs.The deviation from the quantization can be thought of as the skin effect induced leakage of Weyl points.In plot (b) the Hall conductance is shown as a function of γ for three different values of kz lying in the three different regions identified in (a).The blue curve (kz = 0) and the green curve (kz = π/2) correspond to the regions where −k i 0 < kz < k i 0 and k i 0 < kz < k o 0 , respectively.In these cases, σ decreases with non-Hermiticity strength γ.The brown curve corresponds to kz = 2.0, which lies outside the outer EP for a few values of γ and for other values of γ lies in between the pair of EP.Consequently, σ remains near zero initially for small values of γ and then makes a transition near γ ≈ 0.43 to become finite.The parameters are chosen to be m = 2 and t = r = t3 = 1.For values of kz where σ was finite in the Hermitian system, σ decreases monotonically with increasing non-Hermiticity strength γ in the NH system.For other values of kz, initially, σ remains zero, then it may or may not increase to a finite value depending on the value of γ.
In NH systems, equation ( 9) may create a natural query of not using the whole complex spectra in the definition of the current operator.If we consider the wave-packet dynamics of our model, the group velocity of a wave packet, as is needed for the definition of the current operator, does not depend on the imaginary part of the Hamiltonian [68,69].Here in equation (8), is the retarded Green's function.Here, α = ± denotes the two bands of the Hamiltonian and in the definition of Green's function, we use the left and right eigenvectors of the NH system.Using equations ( 7) and ( 8), the final form of the Hall conductance at zero temperature is found to be [53] where and Here, Ω xy (k) is the Berry curvature and the term ν(k) captures the effect of non-Hermiticity on the Hall conductance.It is to be noted here that the value of ν(k) is always less than one which causes the deviation from the quantized nature of the Hall conductance [53] for NH systems.Up to now, we have given a general discussion on how to calculate the Hall conductance for a generic two-band Hamiltonian.It is worth mentioning here that we consider a 3D system, which is formed by stacking 2D Chern insulators along the z-direction.In our system thus k z is treated as a parameter.Following the same procedure as discussed above and considering d k = (t sin k x + iγ, t sin k y , (m − r cos k x − r cos k y − t 3 cos k z )), we have numerically calculated the corresponding Berry curvature of the system and using equation (10), we plot the nature of the Hall conductance for different values of the NH parameters.
In figure 4(a), the Hall conductance for various strengths of the NH parameter γ is presented.For γ = 0, i.e. in the Hermitian case, the Hall conductance remains quantized between the two Weyl nodes, which is represented by the solid red curve in figure 4(a).As we allow γ to be non-zero, σ xy starts deviating from the quantized value and causes a drop in the maximum value of the Hall conductance.The value of the conductance for a particular γ remains constant between two inner EPs.It exhibits a shoulder between a pair of EPs before finally diminishing to zero outside the EPs.Physically, the finite imaginary part of the spectra introduces a finite lifetime corresponding to each carrier.The carriers having momenta in z-direction in the range −k i 0 < k z < k i 0 contribute most in the Hall conductance.In this region, the Hall conductance gets suppressed with increasing γ value due to the decreasing lifetime of the carriers.Furthermore, the carriers having momenta k z in the region between (|k i 0 |, |k o 0 |) contribute in the Hall conductance.Interestingly, the nature of the Hall conductance in this region owes to the fact that a Weyl point in the Hermitian system gives rise to a pair of EPs in the NH system.The presence of EPs leads to short-lived low-lying excitations in this region, resulting in shoulder-like behavior in the Hall conductance.The carriers that have momenta |k z | > k o 0 do not contribute to the Hall conductance of the system.In figure 4(b), the Hall conductance is presented as a function of γ for different values of k z .We choose γ in a way that k z = 0 always lies in the region where −k i 0 < k z < k i 0 ; thus for k z = 0 the Hall conductance σ decreases with γ from the value of one.Next, we choose k z = π/2-it is the position of the Weyl point in the parent Hermitian system-which always lies between a pair of EPs in the NH system.We find that σ again decreases with γ.Finally, when we set k z = 2.0, the strength of γ determines whether Here σ remains zero in the first case and becomes finite in the second case.Now using the same prescription we analyze the Hall conductance of an NH WSM which is a stack of Chern insulators with C = 2.The motivation is to check whether our results are system-specific or general for a family of NH WSMs.The Hamiltonian of a WSM formed by stacking 2D Chern insulators of Chern number C = 2 with intra-cell hopping amplitude iγ is written as As long as we are in the parameter regime where |m − 2r| < t 3 , the parent Hermitian double WSM hosts two Weyl points at (k x = k y = 0 and k z = ± cos ( m−2r t3 ).The Berry charge for these Weyl points is ±2.Each Weyl point splits into a pair of EPs when γ ̸ = 0. Calculating in the same manner as described above, we find that the Hall conductance for this model behaves similar to NH WSM with topological charge ±1, as shown in figure 5.The value of the Hall conductance decreases with increasing strength of non-Hermiticity.The similarity between the nature of the Hall conductance of an NH double WSM (C = 2) and an NH WSM (C = 1) indicates the generality of our findings.The nature of Hall conductance of the family of NH stacked Chern insulators is due to the fact that the carriers in the system contribute to the conductivity depending on the momenta along the stacked direction.
We note that the term ν(k) in equation (10), manifesting in the NH generalization of the TKNN formula.Nonetheless, the Bloch theory dramatically fails to match the open boundary edge modes information resulting in broken bulk-boundary correspondence (BBC) [70][71][72][73].Consequently, the direct mapping between Hall conductance and the existing edge mode transport still needs to be explored.This

OBC and non-Bloch theory
Having discussed the Bloch band properties of the model using its complex eigenvalue spectrum and its Hall conductance, we next discuss the OBC and the non-Bloch theory.We find that, strikingly, our system encounters the NHSE.This dictates that a macroscopically large number of states are exponentially localized at the edge under OBC, leading to the violation of the celebrated BBC [32,60].The condition for obtaining zero modes under OBC in such systems was also discovered to be different from the periodic boundary conditions in the Bloch band theory framework.These consequences suggest a re-examination of the topological invariants in the GBZ to characterize their topology in terms of open boundary modes.Next, our analysis goes as follows.First, we numerically investigate the spectral topology under OBC.Then we employ the non-Bloch theory to find the topological zero modes enabling topological phase transitions.Finally, we characterize the edge transport and analyze the Hall conductance defined over GBZ in terms of the exceptional Weyl points and boundary states.Since in our system, the degeneracies in the band diagrams are found to be in the k x = 0 plane, therefore, we consider the system to be open along the x-direction and treat k y , k z as parameters.We then investigate the OBC spectra for N = 42 unit cells with the parameters fixed at m = 2, r = t = t 3 = 1, and γ = 0.5 in figure 6, where the absolute, real and imaginary parts of energy are plotted as a function of k z .The zero energy (both real and imaginary) edge modes are shown in red.In our system, the NHSE gives rise to hybridization between the edge mode and bulk skin modes that possess finite imaginary energy components, leading to their participation in conductance.As a result, the Hall conductance deviates from quantization.Although, purely real zero-energy edge modes (shown in red) retain a non-decaying current enabling edge transport.
Next, we study the non-Bloch band theory which restores the BBC, following the formalism derived in [74].First, we consider the low energy continuum model derived from our parent Hamiltonian in equation ( 1) by setting sin x,y 2 and treating k z as parameter We consider the wave vector to be complex-valued as where kr = (k x , k y , k z ) and ki = ( γ t , 0, 0).We obtain the non-Bloch Hamiltonian for our model to be Here m = m − 2r − rγ 2 2t 2 − t 3 cos k z .The wave vector k lies in the GBZ in the complex plane just as k lies in the conventional Brillouin zone in the real plane.The non-Bloch Chern number C can be determined from the Hamiltonian (equation ( 16)) using the usual prescription [60].C is 1 when m < 0 and 0 when m > 0. Thus, using non-Bloch theory we can determine the condition for the our system to undergo a phase transition from topologically non-trivial phase to normal insulator phase.This is given by Substituting the values of the parameter that we fixed initially, we finally have k z = k c z = arccos(− γ 2 2 ).Notably, this is different from the topological transition point derived using Bloch band theory but exactly matches with the condition obtained using the OBC spectra.This is illustrated in figure 7, where the phase diagram for our model is presented.Since the Hamiltonian is symmetric in k z , we choose only positive values of k z to plot the phase diagram.In figure 7, the shaded region denotes the topologically non-trivial phase ( C = 1), whereas the black dotted lines are the phase boundaries found from the Bloch continuum model.The overlapping red solid and blue dashed lines are the phase boundaries derived using the OBC computations and the non-Bloch continuum model, respectively.This results in the restoration of BBC in our NH system.The energy bands of the non-Bloch Hamiltonian touch at ±k c z at discrete points, leading to Weyl EPs under PBC.The GBZ allows us to evaluate the total Hall conductance, σ total xy .We find it as [24,25] where C is the NH Chern number defined over the GBZ.Therefore, numerically the total Hall conductance in WSM is proportional to the distance between the two NH Weyl points.This parallels the Hermitian case.However, one requires the formulation of the GBZ to get the true topological phase boundary that matches with OBC as well as to obtain the total Hall conductance of the family of 3D NH WSMs. ) derived using the Bloch model, where for k i z < kz < k o z the system remains gapless.The red line shows the phase boundary obtained using OBC and the blue dashed line corresponds to the non-Bloch continuum model.We note that the Bloch phase boundary significantly differs from the OBC spectra, which invalidates the conventional BBC.This mismatch suggests a non-Bloch framework to characterize their topological-trivial phase transitions.As we see in the plot, the blue dashed line obtained from the non-Bloch theory nicely captures the topological phase transitions.The edge modes will be found in the shaded region.Therefore, this region is the topologically non-trivial phase.The other parameters are chosen to be m = 2 and t = r = t3 = 1.

Experimental feasibility
NH systems have been experimentally realized in various platforms [47,75,76].Optics, offering an ideal platform for the physical implementation of gain and loss, opens avenues for investigating light dynamics in NH topological systems.The development of periodic chains of 'PT-atoms' has unveiled exotic features absent in Hermitian lattices, such as band-merging effects and exceptional lines [77].Also in photonic systems, controlled investigations into topological systems have become feasible.An experimental demonstration using angle-resolved scattering measurements has recently been performed, where a bulk Fermi arc (connecting a pair of EPs) develops from NH radiative losses in an open system of photonic crystal slabs [76].Furthermore, the authors illustrated the fundamental connections between the half-integer topological charges and the half-integer topological index of an EP, manifested as its mode-switching property.In our work, the non-Bloch Brillouin zone (or GBZ) allows us to evaluate the Hall conductance.It turns out that total Hall conductance is proportional to the distance between the two NH Weyl points connected by a bulk Fermi arc.Similarly, there are other various ingenious experiments that have been proposed and realized to probe the exceptional structures that strengthen the experimental feasibility of our results.For instance, NH WSM with Weyl exceptional ring has been recently realized in an optical lossy waveguide setup [46].Very recently, the NH Weyl phase has been proposed in three-dimensional TIs coupled to a ferromagnetic lead, and the NH physics of this Weyl phase can be tuned with the magnetization direction of the ferromagnetic lead [78].Another way of realizing our predictions could be in optically shaken cold atom systems by introducing loss through selective depopulation of cold atoms [75,79].Several proposals aim to experimentally measure the longitudinal and transverse conductance of ultracold atoms trapped in an optical lattice [80,81].In such a setup, the longitudinal conductance has been very recently measured [82].Finally, we note that the density can be related to the Hall conductivity via the Streda formula.This has recently been used to realize a fractional quantum Hall state in ultracold atoms [83].
Further exploration in this field encompasses the study of NH electronic dimer models, revealing thresholdless transitions and the ability to control prohibited waves in the linear regime.Interestingly, recent studies show that the particular class of unitary (pseudo-Hermiticity) and combination of unitary and anti-unitary (parity-time) symmetry leads to striking consequences in transport properties in various platforms ranging from waveguide systems to electrical set up.Delving into the electrical realm, the authors in [84] explore a NH electronic dimer system based on an imaginary resistor (Z) in a (N + 2) level atomic multipod configuration.NH systems, characterized by a gain/loss parameter governed by particular symmetries, exhibit a degeneracy at an EP that separates different phases of complex mode dynamics.This system's structural characterization and dispersive properties reveal a broad range of strong coupling, where the interplay between control and probe fields induces unique transport properties.Additionally, investigations into structures like NH trimers and electronic dimer systems with an imaginary resistor shed light on unique dispersive properties and multiple windows of transparency, further enriching our understanding of scattering and transport phenomena in NH waveguides and electrical setups [85][86][87].Thus, given the recent experimental advances, we believe that our findings on the Hall conductance can be realized in state-of-the-art platforms.

Conclusions
In this work, we explored the Hall conductance of a 3D NH WSM formed via stacking 2D NH Chern insulators.We discover that in such an NH system the Hall conductance deviates from the quantized value and exhibits a shoulder-like character, which we interpret as a consequence of the existence of pairs of EPs.Notably, the finite imaginary part of the energy introduces a finite lifetime of the carriers.The carriers having momenta in z-direction in the range −k i 0 < k z < k i 0 contribute the most to the Hall conductance, where the Hall conductance varies inversely with γ due to the short lifetime of the carriers.However, when the k z value of the carriers lie between (|k i 0 |, |k o 0 |), there is a finite contribution of the carriers to the Hall conductance.The OBC and the usual Bloch theory disagree for such NH systems and we show that the transition between the topologically non-trivial phase to the trivial phase occurs for different k z values for OBC and Bloch theory.This discrepancy was solved by using the GBZ and complex momentum values, i.e. the non-Bloch theory.We presented the Hall conductance evaluated over the GBZ and connected it to the separation between the Weyl nodes.We hope our findings stimulate further exploration of unusual transport properties of NH systems.where, P± k = |ϕ R ±,k ⟩⟨ϕ L ±,k |.We note here that the Hamiltonian in equation ( 1) can be written as,

Figure 1 .
Figure 1.Schematic of the NH WSM model as a stack of 2D Chern insulators.Two sublattices have onsite energy m (blue) and −m (pink).The inter-cell amplitudes are shown in black, cyan and red.Hopping between any two sites in a direction shown by the arrow and opposite to it will have a π phase difference.The intra-cell hopping amplitude iγ is shown in green.The inter-layer hopping amplitude is − t3 2 , shown in brown.

Figure 2 .
Figure 2. Illustration of exceptional contours (ECs) in NH WSM model.The real (green) and imaginary (red) parts of the exceptional surfaces of the NH WSM are shown for (a) γ = 0.3 and (b) γ = 0.5.The intersection of these two surfaces are the ECs.We note that as we increase the non-Hermitcity strength the size of the ECs increases.Also, four exceptional points (EPs) (blue dots) get formed at kx = ky = 0 along kz.The other parameters are chosen to be m = 2 and t = r = t3 = 1.

Figure 3 .
Figure 3. Vorticity around EPs. Vorticity of a pair of EPs when the loop Γ encloses them.The plots show the swapping of two energy bands by parameterizing the loop Γ using θ ∈ [0, 2π].The dashed curves are the projection of the trajectory of the energy bands in the complex plane.The two bands encircle EPs with opposing vorticities as they wrap around one another in opposite (clockwise and anticlockwise) directions in the two plots, (a) and (b).In (a) the contour encloses the EP located at kz = k i 0 , and in (b) it encloses the one located at kz = k o 0 .The parameters are chosen to be m = 2, t = r = t3 = 1, and γ = 0.5.

Figure 4 .
Figure 4. Hall conductance of an NH WSM.(a)The Hall conductance as a function of kz given by equation(10).The conductance for three values of γ is plotted in units of e 2 h .The red line corresponds to the Hermitian case and it is fully quantized.The green and blue curves correspond to NH cases for different strengths of non-Hermiticity, i.e. for γ = 0.3 and γ = 0.5, respectively.They considerably deviate from quantization.In this plot, we identify three different regions between the inner EPs, where the conductance value decreases from 1 but stays constant; between a pair of EPs, where the Hall conductance displays a shoulder shape; and outside the EPs, where it decreases to zero.Here, the nature of the Hall conductance is a direct consequence of the EPs.The deviation from the quantization can be thought of as the skin effect induced leakage of Weyl points.In plot (b) the Hall conductance is shown as a function of γ for three different values of kz lying in the three different regions identified in (a).The blue curve (kz = 0) and the green curve (kz = π/2) correspond to the regions where −k i 0 < kz < k i 0 and k i 0 < kz < k o 0 , respectively.In these cases, σ decreases with non-Hermiticity strength γ.The brown curve corresponds to kz = 2.0, which lies outside the outer EP for a few values of γ and for other values of γ lies in between the pair of EP.Consequently, σ remains near zero initially for small values of γ and then makes a transition near γ ≈ 0.43 to become finite.The parameters are chosen to be m = 2 and t = r = t3 = 1.For values of kz where σ was finite in the Hermitian system, σ decreases monotonically with increasing non-Hermiticity strength γ in the NH system.For other values of kz, initially, σ remains zero, then it may or may not increase to a finite value depending on the value of γ.

Figure 5 .
Figure 5. Hall conductance of an NH double WSM.The Hall conductance (in units of e 2 h ) of the non-Hermitian Stack of Chern insulator with C = 2 is presented as a function of kz for different values of γ.The red solid curve represents the Hermitian case when γ = 0.The green and blue solid curves represent non-Hermitian cases where the γ takes values 0.3 and 0.5, respectively.The other parameters are chosen to be m = 2 and t = r = t3 = 1.The Hall conductance for the NH double WSM varies with momenta kz in a similar manner to the NH WSM-it remains nearly constant between two inner EPs, exhibits a shoulder in between a pair of EPs and then falls to zero.Here also, the deviation from the quantization can be thought of as the skin effect induced leakage of Weyl points.The Hall conductance starts deviating from the maximum value σ = 2, indicative of an NH double WSM arising from the Chern insulator (C = 2).
New J. Phys.26 (2024) 023057 S Dey et al requires us to restructure the framework, which quantifies the behavior of edge transport over the GBZ and restores the BBC.

Figure 6 .
Figure 6.Open boundary spectra for WSM model.Numerically obtained spectra of the system as a function of kz for N = 42 unit cells, when the system has OBCs along the x-direction.(a)-(c) are the absolute, real and imaginary parts of the energy spectra.The parameters are chosen to be m = 2, t = r = t3 = 1, and γ = 0.5.Here, the zero modes are shown in red.Using OBC we find that the true topological transition point is at |kz| = arccos (− γ 22 ).

Figure 7 .
Figure 7. Topological phase diagram on the γ-kz plane.The black dotted lines are the phase boundaries (kz = k iz and kz = k o z ) derived using the Bloch model, where for k i z < kz < k o z the system remains gapless.The red line shows the phase boundary obtained using OBC and the blue dashed line corresponds to the non-Bloch continuum model.We note that the Bloch phase boundary significantly differs from the OBC spectra, which invalidates the conventional BBC.This mismatch suggests a non-Bloch framework to characterize their topological-trivial phase transitions.As we see in the plot, the blue dashed line obtained from the non-Bloch theory nicely captures the topological phase transitions.The edge modes will be found in the shaded region.Therefore, this region is the topologically non-trivial phase.The other parameters are chosen to be m = 2 and t = r = t3 = 1.

)
Then we can write P± k = 1 2 (1 ± dk .σ)and inserting it in the following expression we get, Now we consider our system is close to equilibrium, thus we approximate the distribution to the Fermi function at zero temperature.Now evaluating the integration in equation (B3) and combining the result with equation (B5), we get, kΩ xy (k) + Ω * xy (k) 2 × 2 tan −1 (Re ϵ (k) /Im ϵ (k)) /π.(B6)