2024 Berry curvature inside parity-time-symmetry protected exceptional surface

A three-dimensional non-Hermitian Hamiltonian with parity-time symmetry can exhibit a closed exceptional surface (EP surface) in momentum space, which is a non-Hermitian deformation of the degeneracy line (DL). Since the degeneracy line lacks an internal space, the distributions of Berry curvature inside the EP surface becomes particularly intriguing. This paper studies the distributions taking a torus-like EP surface as an example. In a meridian cross-section, the Berry connection exhibits a vortex-like ﬁeld with only angular components, while the Berry curvature is perpendicular to this cross-section; in a equatorial cross-section, the Berry curvature forms a closed curve surrounding the central genus. Both Berry connection and curvature converge along the coplanar axis and diverge at the surface. We ﬁnd the Berry ﬂux depends on the radius of the integration region and is not quantized inside the EP torus. Approaching the surface, the Berry ﬂux tends to inﬁnity and the dynamical phase oscillates violently. We point that the streamlines of Berry curvature can be used to estimate the zero or non-zero Berry ﬂux. We generalize the above patterns to the case of EP surfaces with complex shapes, and present a proposal of realizing the EP surface in an electrical circuit. Our research outcomes enhance the comprehension of EP surfaces and the topological characteristics of non-Hermitian systems with parity-time ( PT ) symmetry.

The high dimensional EP structure is a noteworthy problem.Exceptional rings (EP rings) have been intensively discussed theoretically and experimentally [119][120][121][122][123][124][125].An EP ring can be analogous to a vortex filament, and the curl field related to the vortex filament is equivalent to the Berry connection [122].In three-dimensional (3D) momentum space, non-Hermitian Hamiltonians with combined parity and time reversal symmetry spontaneously meet conditions for the appearance of exceptional surfaces (EP surfaces) [126,127].The EP surface is stable as long as the protecting symmetry is preserved [128].The EP surface inherits the topological properties of the degenerate line (DL); the nodal volume, which represents bulk Fermi arcs in 3D space, indicates the remarkable control of the density of states (DOS) [127].The topological properties of the EP surface can also be characterized by Z 2 topological invariants, and a stable zero-gap quasi-particle state is protected by symmetry and topology [126].In a highdimensional parameter space, a hypersurface where the system remains at an EP improves the robustness and enhances the sensitivity of EPs, and a non-Hermitian sensor can be designed on the basis of the hypersurface [129].A non-Hermitian Bardeen-Cooper-Schrieffer (BCS) Hamiltonian with a weak complex interaction possesses an EP surface in the quasi-partile Hamiltonian, and non-Hermiticity induces the breaking down of superfluidity and exhibition of reentrant behavior [130].The EP sur-face affects magnetic responses in a Hubbard model; the sharp local density of states (LDOS) at the Fermi energy for sublattices with weak correlations results in the local magnetic susceptibility of strong sublattice dependence [131].Experimentally, the EP surface can be observed on a magnon polariton platform, and the EP surface can be conveniently tuned to coalesce into an anisotropic exceptional saddle point [132].
Motivated by recent theoretical advances in non-Hermitian topological systems, we investigate the distribution of Berry curvature inside the EP surface.The Berry curvature is gauge-invariant and related to the topological properties of EP surfaces.In this paper, we investigate a two-band non-Hermitian system with parity-time (PT ) symmetry and a closed EP surface in 3D momentum space.The general expression of the Berry curvature defined under the biorthogonal basis reveals that the EP surface separates the zero and nonzero Berry curvature.A Hamiltonian with a torus-like EP surface is exemplified.The topological properties of the EP surface are encoded in the distributions of the Berry curvature in the meridians and equatorial crosssections.In the meridians cross-section, the Berry connection acts as a planar vortex field and the direction of the Berry curvature is perpendicular to this crosssection; in the equatorial cross-section, Berry curvatures form closed curves.Both Berry connection and curvature are convergent at the coplanar axis and divergent at the EP surface.The surface integral of the Berry curvature yields a non-quantized Berry flux.The numerical simulation implies that the non-quantized Berry flux is consistent with the dynamics phase accumulated in the adiabatic evolution, and both of them oscillates violently near the EP surface.The Berry flux can be evaluated by the distribution of Berry curvature.Berry flux is nonzero if the Berry curvatures have the same direction in the meridians cross-section otherwise vanishes.These patterns can be generalized to the EP surfaces with complicated geometries.Finally, rather than the realization in coupled resonators [133], we point that the EP surface can be measured in a electrical circuit.
The remainder of the paper is organized as follows: In Sec.II, we introduce the 3D PT -symmetric non-Hermitian two-band system.In Sec.III, we present the formal expression of Berry connection and curvature.In Sec.IV, we introduce a concrete model to exhibit the Berry curvature inside a torus-like EP surface.The nonquantized Berry flux is elucidated from the distribution of the Berry curvature.In Sec.V, the adiabatic evolution is implemented.In Sec.VI, a topological system that possesses more complicated EP surface is further discussed.In Sec.VII, the proposal of realizing the EP surface is given in the electrical circuits.In Sec.VIII, we summarize the results.

II. NON-HERMITIAN TWO-BAND SYSTEM
We consider a non-Hermitian two-band Hamiltonian in the momentum space k = {k x , k y , k z }, where σ ={σ x ,σ y ,σ z } is the Pauli matrix; the component {B x (k) , B y (k)} of the auxiliary field B (k) is the real and periodic function of k = {k x , k y , k z }; and the other component B z = iγ is a constant, which is introduced as the gain and loss.hk possesses the PT -symmetry [PT , h k ] = 0, where P = σ x is the parity operator and T is the time-reversal operator that T −1 iT = −i.
The eigenvalues of PT -symmetric systems are either real numbers or complex conjugate pairs respectively associated with PT -symmetry unbroken or broken eigenstates, respectively.Considering the specific form of the band in Eq. ( 1), i.e ± B 2 x + B 2 y − γ 2 , the complex conjugate pairs are reduced to purely imaginary numbers.The eigenstate expressions involve parameters defined in terms of energy, therefore the real/imaginary eigenvalues make the expressions more concise.In addition, h k is the pseudo anti-Hermitian , where h k |φ = ε |φ .This implies that σ z |φ becomes the left eigenstate corresponding to the right eigenstate |φ when ε is purely imaginary.These characteristics of h k simplify the calculations in the following text and are reflected in Sec.III.
In the Hermitian case (γ = 0), the band degeneracy is determined by the following equations B x (k) = 0 and B y (k) = 0 each represent a surface in the 3D momentum space.The intersection of two surfaces is the degeneracy line (DL).The topological properties of the DL are captured by the topological number Berry flux or winding number.The former is the integral of the Berry connection on a closed circle, while the latter is obtained by dividing the Berry flux by π.The Berry flux is quantized to π (0) if the closed circle is (not) linked with the DL [135,136].In the presence of gain and loss for γ = 0, the DL becomes an EP surface.The EP surface is the zero-energy surface in the form of We consider the case in which Eq. (3) describes a closed 2D surface in the 3D momentum space at the selected {B x (k) , B y (k)}.In this situation, the energy is real outside the closed EP surface and is purely imaginary inside the closed EP surface.We regard the purely imaginary region as the nodal volume wrapped by the EP surface.These data serve as the 3D bulk Fermi arcs [127].
PT -symmetry protects the EP surface which inherits the Berry flux of the DL [126,127].In this work, we focus on the Berry curvature distributions inside and outside the EP surface.

III. BERRY CONNECTION AND BERRY CURVATURE
This section provides the general expressions of Berry connections and curvatures inside and outside the EP surface.
We first calculate the eigenstates of the Bloch Hamiltonian under biorthogonal norm.The right eigenstates φ R ± of the Bloch Hamiltonian satisfy They are normalized under the biorthogonal norm φ L α φ R α = 1 (α = +/−).The EP surface serves as a boundary separating the real and complex energies.The Bloch Hamiltonian possesses an entirely real spectrum outside the EP surface (i.e. the PT -symmetry unbroken phase) and possesses an entirely imaginary spectrum inside the EP surface (i.e. the PT -symmetry broken phase).For the geometric features of the lower band φ R − with energy − B 2 x + B 2 y − γ 2 , in the unbroken PT -symmetry region γ 2 < B 2 x + B 2 y , the right and left eigenstates are in the form respectively, where ε = B 2 x + B 2 y − γ 2 , α and β are determined by tan α = γ/ε and tan β = B y /B x respectively, and Ω = −2ie iα sin α.In the broken region γ 2 > B 2 x + B 2 y , the right and left eigenstates are in the form respectively, where ε = and the Berry curvature is defined as F = ∇ × A, where ∇ = ∂ kx êx + ∂ ky êy + ∂ kz êz .Therefore, the formal expressions of the Berry connection and Berry curvature differ between the unbroken and broken PT -symmetric phases.The Berry connection is complex in both the broken and unbroken regions, the imaginary part amplifies the Dirac probability of the adiabatic evolved state, whereas the real part is related to the topological properties of the system [137].Therefore, we consider only the real part of the Berry connection in the definition.The detailed calculations are provided in the Appendix, and the results are presented concisely as follows.
Inside the EP surface PT -symmetry is broken.The components of Berry connection A j and Berry curvature where x − B 2 y .A j and F j converge at the DL in the Hermitian case (i.e., B x = B y = 0 or ε = 0) and are infinite at the singularity ε = 0 (i.e., the EP surface).
Outside the nodal volume the PT symmetry holds.The Berry connection and Berry curvature become Equations ( 9) and ( 11) imply that the EP surface acts as the boundary between zero and non-zero Berry curvature.To extract more explicit information on the Berry connection and curvature, we further simplify these formulas inside a torus-like EP surface.

IV. TORUS-LIKE EP SURFACE
We use a concrete model possessing an EP surface to study the distributions of Berry connection and curvature in the broken region, from which the inheritance of the Berry flux is well interpreted.The auxiliary field B (k) = {B x , B y } of the concrete Hamiltonian is in the form where The physical realization of the concrete Hamiltonian is proposed [127,138].
The general geometric property of the EP surface is determined by the components {B x (k) , B y (k)}.Equations (2), (3), and ( 12) indicate there are two identical nodal volumes located at the k z = 0 and k z = π planes, only the former is studied for convenience.With fixed parameters {m, a}, Eq. ( 3) implies that the maximum of B y is B y = γ (i.e., s sin k z = γ) in the situation B x = 0; therefore, the maximum of k z on the EP surface is k zmax = arcsin(γ/s), and the restriction γ < 1 is imposed.In fact, if γ = 1, the two EP surfaces touch at k zmax = arcsin γ.In addition, the EP surface possesses a mirror symmetry with respect to the k z = 0 plane.
The system possesses a torus-like EP surface under the appropriate parameters (see Sec. VIII B).A schematic diagram of the torus-like EP surface is shown in Fig. 1(a).The the red coplanar circular axis is DL in the Hermitian case.Two types of cross-sections are studied in this

A. Distribution in the meridional cross-section
This section discusses the distribution of Berry curvature in the cross-section S V of the concrete model in Eq. (12).
In the cross-section S V , polar coordinates are used to describe the physical quantities.As shown in Fig. 1(b), the green EP ring divides the plane into two parts: the Hamiltonian has an entirely imaginary spectrum in the yellow region inside the EP ring and an entirely real spectrum outside the EP ring.The EP ring is subcircular with radius γ.The circular dashed line is the energy contour L with radius r.The red point O(0, k y0 , 0) is center of the S V and is the degenerate point (DP) in the Hermitian case (γ = 0).The arbitrary point P (0, k y , k z ) on the contour L can be rewritten as P (0, r, θ) in the cylindrical coordinate system, where θ is the included angle between position vector P and coordinate axis k y .The three unit vectors {ê θ , êr , êx } of the cylindrical coordinate system are presented.Under the parameter settings given in Sec.VIII B, the Hamiltonian in the cross-section S V is reduced to It is straightforward to check that the reduced Hamiltonian H obeys the PT -symmetry, i.e., T σ x H(T σ x ) −1 = H.For the case with complex matrices, a numerical result can be obtained, exhibiting a deformed but similar distribution of Berry curvature, as shown in Sec.VI.The distributions of the Berry connection and Berry curvature in S V are illustrated in Fig. 1(c) and (d).Inside the EP ring, the expression of the Berry connection at the position P (0, r, θ) (r < γ) in Eq. ( 8) is reduced to where ε = γ 2 − r 2 .The expression of the Berry connection in the above equation is equal to the expression directly calculated from Eq. (13).Equation ( 14) indicates that the radial component e r vanishes, and the angular component e θ is non-zero, so A is a planar vortex field.We show the direction of A by the arrows without considering its intensity according to Eq. ( 14), and each arrow is tangent to the energy contour L.
which indicates that A converges at r = 0 (i.e., DP at γ = 0).A is divergent at ε = 0. Equation ( 9) can be reduced to inside the EP ring.The expression for the Berry curvature in the above equation is equal to the expression directly calculated from Eq. (13).Equation (16) indicates that only the axial component e x is non-zero.The Berry curvature has a divergent value at r = γ (i.e. the EP ring) and a convergent value at r = 0.In Fig. 1(d), we exhibit the direction of F .As we can see, all the arrows in S V point in the same direction, which is the normal of S V .As an analogy, these arrows can be regarded as magnetic field lines, and the total magnetic flux is the number of magnetic field lines that pass through S V .In addition, in two adjacent meridional cross-sections, these arrows are connected end to end, as shown in the inset of Fig. 2, where the green solid lines denote the top view of meridional cross-sections; and the dashed red line is the DL in the Hermitian case (γ = 0).The arrows in all the meridional cross-sections form a closed curve (see Fig. 2).The Berry flux is related to the distribution of the Berry curvature or Berry connection and can be used to capture the topological nature of the EP surface, where S is the integral surface, which is the shaded region surrounded by L presented in Fig. 1(b).By substituting Eq. ( 16) into Eq.(17a), we can obtain Φ B is divergent on the EP surface (r = γ).Therefore the geometric phase oscillates sharply when the integration path approaches the EP surface.Due to the divergence of the Berry connection and curvature on the exceptional point (EP) surface, the line integral of the Berry connection will not be equal to the surface integral of the Berry curvature when the integration path is located in the unbroken region, that is, the Stokes theorem does not hold.
In addition, the Berry flux can be regarded as the total magnetic flux.In Fig. 1(d), the uniform pointing of the arrows indicate that the same sign contributes to the Berry flux and therefore a non-zero Berry flux.

B. Distribution in the equatorial cross-section
This section investigates the distribution of the Berry curvature in S H .
In the equatorial cross-section S H , the Berry curvature in Eq. ( 9) inside the EP ring is reduced to Therefore, the orientation of the Berry curvature at an arbitrary point In addition Eq. ( 3) can be reduced to The above results hold true as long as B y is a function of only k z .The streamlines of the Berry curvature in S H according to Eq. ( 20) are shown in Fig. 2. A different closed black curve is depicted by setting different γ ′ .The red solid EP lines (γ ′ = γ) serve as the boundary separating non-zero and zero Berry curvatures; the region between the two EP lines has non-zero Berry curvature.The dashed red line (γ ′ = 0) represents the DL for the Hermitian case.The black curves (0 < γ ′ < γ) with arrows represent the orientation of the Berry curvature, and the background color indicates the intensity of the Berry curvature.The intensity values are shown in the color bar.The Berry curvature approached infinity near the EP lines.The streamlines surrounding the hole flow anticlockwise.All the streamlines of the Berry curvature are closed, which coincides with the equation ∇• F = 0, meaning that Berry curvatures act as a field without sources.In addition, Eq. ( 21) can be generalized to the other intersection between the k z = k z ′ plane and the EP surface.The distributions claim a clear physical correspondence for the Berry curvature and EP surface.The Berry curvature can be analogous to magnetic lines generated by a solenoid, and the EP surface can be connected to this solenoid.The total magnetic flux is the number of magnetic field lines passing through certain cross-section.This section examined the distribution of the Berry curvature inside the EP surface using two types of crosssections as examples, and calculates the Berry flux.Before moving on to the next section, there are three points that need to be supplemented and explained: i) The above results are obtained under the biorthogonal basis sets.Under the Dirac basis sets, the directions of the Berry connection and Berry curvature at any point inside the EP surface are the same but the magnitudes are different, and the two kinds of Berry fluxes are different.Both the Berry connection and Berry curvature under the Dirac basis sets converge on the EP surface, therefore the Stokes theorem holds.
ii) The winding number associated with the berry connection cannot be used to capture the topological nature of the EP surface.iii) There is an open question that is the topological connection between the isolated EPs and the EP surface in the context of topological number.The nontrivial topological nature of an isolated EP depends on the scalar field defined by the spectral phase Arg(E + − E − ) (see Fig. 3(a)) [44,139].The EP is regarded as a vortex of the scalar field where the spectral phase cannot be effectively defined.The topological nature of an EP can be characterized by the topological number π or 1/2, the former is the spectral phase difference accumulated when encircling the vortex while the latter is the winding number obtained through dividing this phase difference by 2π.Therefore EPs can be analogous to π-vortices, which hangs together with the totpological defect in a nematic [140][141][142][143][144] or defects in TIs [145][146][147][148][149]. The isolated EPs may merge accompanied by the algebraic addition of topological numbers [150].The subject of this study is the EP surface, which is a collection of infinite EPs. Figure 3(b) exhibits the spectral phase of EP surface.There is no obvious evidence that the topological properties of EP surface are related to the spectral phase.Corresponding to the same spectral pahse in Fig. 3(b), Figs.1(d) and 5(f) exhibit two distinct distributions of Berry curvature.The above analyses indicate that the spectral phase cannot describe the topology of the EP surface completely, and the winding number corresponding to EP surface is also not equal to ±1/2.Therefore EP surface cannot be simply understood as the merger of isolated EPs.The topological connection between the EPs and EP surfaces deserves further investigation.To verify the above results, we numerically simulate the adiabatic evolution driven by the Hamiltonian in Eq. ( 13) and compare the geometric phase obtained by numerical simulation and the analytical results in Eq. 18.

V. ADIABATIC EVOLUTION
Considering the adiabatic evolution on the circular contour L with a radius r [see Fig. 1(b)].H in Eq. ( 13) is a periodic function of θ, H(θ) = H(θ+2π).The lower band eigenstate φ R − (0) reverts to φ R − (0) if θ varies adiabatically from 0 to 2π, and the evolved state is the instantaneous lower band eigenstate φ R − (θ) .More explicitly, the adiabatic evolution of the initial state φ R − (0) under the Hamiltonian H(θ) can be expressed as where the dynamic phase α(θ) and the adiabatic phase γ(θ) have the form A (θ) is presented in Eq. ( 14) and γ(θ) is equivalent to the Berry flux in Eq. ( 18).The imaginary part of A (θ) in Eq. ( 39) vanishes due to the invariability η = (γ − ε)/ γ 2 − ε 2 on the contour L and therefore does not contribute to adiabatic evolution.However α(θ) is imaginary and the Dirac probability increases exponentially.To eliminate the exponential growth in probability induced by the imaginary dynamic phase, we add a factor i before the Hamiltonian H in the numerical simulation; consequently α(θ) becomes real and γ(θ) is unaffected.
Figure .4 numerically and analytically exhibits the geometric phase e iγ(θ) on the contour L with r ranging from 0 to γ (r = γ indicates that L is the EP ring).The numerical results correspond with the analytical calculations in Eq (18).The oscillating frequency of the real and imaginary parts of the geometric phase accelerates as the contour L approaches the EP ring (i.e., r = γ).As a sample, the inset numerically presents the fidelity when r = γ/2, which is defined as A fidelity of 1 indicates that the time evolution process is adiabatic.

VI. EP SURFACE WITH COMPLICATED GEOMETRY
The distribution of the Berry curvature for the new geometry is studied in this section.The EP surface has diverse geometries when the Hamiltonian in Eq. ( 12) is generalized to and s = 2. On the basis of the discussion of the physical realization [127,138], the generalized Hamiltonian could be realized by adding long-range perturbations in the x and y directions in a periodic metallic-mesh 3D photonic crystal with PTsymmetric non-Hermitian elements.In Sec.VII, we discuss in detail how to realize an EP surface in the electrical circuit.
Although the geometry of the EP surface changes when the parameters {m, a, b, c, d} vary, the EP still maintains the following features: (i) It possesses a mirror symmetry with respect to the k z = 0 plane.(ii) The Hamiltonian still has two layers of EP surfaces, the two EP surfaces touch each other when γ = s, and we discuss only the lower-layer EP surface near k z = 0. (iii) The meridional cross-sections become irregular circles rather than disks.(iv) The equatorial cross-section no longer has regular geometry, but the streamlines of the Berry curvature in the equatorial cross-section retain the feature discussed in the previous section: they are closed curves that can be depicted according to Eq. (20).To illustrate the distribution of the Berry curvature, the geometries of the EP surface under two sets of parameters are studied and the other cases share similar distributions.In the equatorial cross-section, Eq. ( 19) remains valid, and Figs.5(c) and (d) depict the streamlines of the Berry curvature.The two similar equatorial cross-sections in Fig. 5(c In Fig. 5(d), in addition to the three types of streamlines, there are yet another type of streamline surrounding one hole that flows clockwise.The appearance of the new type of streamline is a consequence of the separation between the two left (or right) holes in Fig. 5(d).
In the meridional cross-section, the Berry curvature has non-zero radial and angular components and may be not perpendicular to the meridional cross-section.Berry flux is nonzero if all the arrows representing the Berry The shchematic diagram of the tight-binding lattice in Eq. ( 27).The red, blue and black lines correspond to the hoppings m − a, −a/2, and s respectively.The green and orange spheres correspond to the gain and loss.An unit cell is marked in the shadow.(b) The circuit elements of an unit cell.The red, blue and black capacitors, which are denoted with C1, C2 and C3, correspond to the hoppings in (a).The orange (green) node is connected to the ground by a inductance and a potentiometers RA (negative impedance converter RB) which represents the gain (loss).
curvature point in the same direction in the meridional cross-section.There are specific meridional cross-sections that contain no DPs or several DPs.A natural question to ask is what the Berry curvature distribution is in these specific meridional cross-sections.A specific meridional cross-section containing no DPs is illustrated in Fig. 5(b) and the top view of this cross-section is shown in Fig. 5(d) (i.e., the yellow rectangle).In Fig. 5(f), the arrows indicate that the Berry curvature points in the positive x-direction on the left semicircle and in the negative xdirection on the right semicircle.The signs cancel out and the Berry flux vanishes.In accordance with this distribution, inside the yellow transparent rectangle in Fig. 5(d), these streamlines flow up on the right side and down on the left, and the total flux is zero.In Fig. 5(e), the non-zero winding number of the arrows indicating the Berry connection outside the EP ring is consistent with the above conclusion in Sec.IV A, which indicates that the winding number has no relation with the Berry flux.

VII. EXPERIMENTAL SCHEME IN ELECTRICAL CIRCUIT
The EP surface can be measured using an electrical circuit which is a powerful platform for investigating topological physics [151][152][153][154].For the sake of convenience, this section discusses the experimental scheme of EL in an electrical circuit, i.e., the intersection line of the EP surface and the S V cross-section (see Fig. 1(b)).There are two reasons for doing this.First, the topological properties of the EL are consistent with those of the EP surface.Second, the experimental setup corresponding to the EL can be smoothly generalized to that of the EP surface due to the design flexibility of the electrical circuit.
We first show the tight-binding lattice model possessing EL.In the S V cross-section where k x = 0, the auxiliary field B (k) = {B x (k), B y (k)} in Eq. ( 12) can be reduced to where f (k y ) = m − a − a cos k y , and s = 1.Substituting the Fourier transformation into the core matrix ky ,kz B (k y , k z ) • σ, we get the lattice model where r =x + y l is the position vector, , l represents the unit vectors, and the system size is N A schematic diagram of the lattice model is shown in Fig. 6(a).The hoppings −a/2, m − a and −s are represented by the blue, red and black lines, respectively.The on-site gains and losses are shown in orange and green, respectively.We can extend this lattice system in the x-direction to obtain a model possessing an EP surface.
The lattice system can be represented by an electrical circuit with N nodes.An N × N matrix J(ω, r), termed circuit Laplacian or admittance matrix, can be used to represent the Hamiltonian of a tight-binding model [151,152,155].J(ω, r) describes the voltage response V(ω, r) to an ac input current I(ω, r) according to V(ω, r) = J(ω, r)I(ω, r), (28) where ω is the AC driving frequency and r represents the nodes.The vector components of V and I correspond to the nodes or sites in the circuit.The matrix elements of J(ω, r) are determined on the admittance of circuit elements between nodes or between nodes and the ground.The onsite gain iγ or loss −iγ are realized using potentiometers or a negative impedance converter to ground.The admittance matrix has an alternative representation in momentum space, denoted as J(ω, k).J(ω, k) can be obtained by performing M linearly independent measurements in the electrical circuit [152,[155][156][157][158], where M describes the number of inequivalent nodes in the network.Each measurement consists of a local excitation of the circuit network and a global measurement of the voltage response, from which all the components of J(ω, k) can be extracted.Then EL can be obtained by diagonalizing the admittance matrix J(ω, k).

VIII. DISCUSSION
In summary, we have investigated the distribution of Berry curvature inside the EP surface of PT -symmetric 3D non-Hermitian two-band systems.The EP surface acts as the separation between the zero and non-zero Berry curvatures.Inside a torus-like EP surface, the distributions of Berry connections and curvatures in the meridional and equatorial cross-sections are discussed.In the meridional cross-section, the Berry connection serves as a planar vortex field and diverges at the DP and EP surface.The Berry curvature has only an axial component and diverges at the EP surface.In the equatorial cross-sections, the Berry curvature forms the closed curves inside the EP surface.The distributions of Berry curvature are analogous to the magnetic lines generated by the solenoid, and the EP surface can be analogous to the solenoids.On the basis of the distribution of the Berry curvature, we obtain the nonquantized Berry flux.The key to identifying the zero or non-zero Berry flux in a meridional cross-section is determining whether all the arrows indicating Berry curvatures point in the same direction.The numerical adiabatic evolution corresponds with the aforementioned analysis of the nonquantized Berry flux.We also discuss the distribution of Berry curvature in a general case in which the EP surface has more complicated geometry.In the equatorial crosssections, the Berry curvatures retain the form of closed curves.The streamlines with arrows indicating the direction of the Berry curvature are categorized by arrow orientations and the number of the holes they surround.We discuss a scheme of realizing the EP surface in an electrical circuit.Our findings deepen understanding of EP surfaces and the topological properties of PT -symmetric non-Hermitian systems.
move cos 2 α to the right-hand side of "=" and we get A similar calculation performing for tan β = B y /B x yields Substitute Eq. ( 31) and Eq. ( 32) into the Eq. ( 29), we have b. Berry curvature.Substitute Eq. ( 29) into Firstly, we prove ∂ j (∂ l α) can be obtained by swapping j with l in Eq. ( 35) and ∂ l (∂ j α) has the same expression with ∂ j (∂ l α), which means Secondly, we prove that ∂ j (∂ l β) can be obtained by swapping j with l and it is not difficult to check that ∂ So we prove that F j = 0, i.e.F = ∇ × A = 0.
The EP surface is in the form of a torus under the condition m = Replace m and a with γ in Eq. ( 46), we have a sin k y0 = ±1.Here we discuss the case of a sin k y0 = −1.In polar coordinates, (k y , k z ) = (k y0 + r cos θ, r sin θ), where r is small.Using the above parameter settings and taking the Taylor expansion, the Hamiltonian in Eq. ( 1) can be rewritten as and the Berry connection in Eq. ( 8) and Berry curvature in Eq. ( 9) can be reduced into , and By using a coordinate transformation of where the form for Berry connection and curvature in polar coordinates, can be easily obtained.

IX. BERRY CONNECTION AND BERRY CURVATURE DEFINED UNDER THE DIRAC NORM
This section gives the expressions for the Berry connection and Berry curvature under the Dirac orthonormal basis in the broken region.As the purpose of this section is to compare the results with those obtained under the definition of biorthogonal bases sets, and the details of calculation is similar to that in Sec.VIII A, therefore only the results will be presented.
The expression for the eigenstates of the Hamiltonian in Eq. ( 1) is x − B 2 y ,where Berry connection can be defined by where the component reads and Berry curvature can be defined by The denominator of A d j has one less factor of ε compared to the denominator of A j in Eq. ( 8), which leads a convergent A d j at the EP surface.

FIG. 1 .
FIG. 1.(a) Torus EP surface (green) at γ = 0.05, a ≈ 3.2, m ≈ 4.05.The dark disk is the representative cross-section SV and the red co-planar circular axis is the DL.(b) Schematic diagram of the the cross-section SV .(c) Berry connection and (d) Berry curvature in the cross-section SV .

FIG. 2 .
FIG. 2. Streamlines of Berry curvature in the equatorial crosssection.Inset: top view of Berry curvature in SV cross-section and its adjacent cross-section.
in S H where |γ ′ | |γ|.Equation (20) represents a closed curve inside the equatorial cross-section for a fixed γ ′ .This closed curve is the intersection between the k z = 0 plane and the EP surface and is determined by replacing γ with γ ′ (|γ ′ | < |γ|) in Eq. (3).If γ ′ = γ, the curve is the EP ring as well as the periphery of S H .If γ ′ changes from −γ to γ, all the curves determined by every γ ′ constitute the equatorial cross-section, and no two curves have a crossing point.The tangent of a curve at (k x , k y , 0) is dk y /dk x = −(∂ x B x )/(∂ y B x ) as a result of complete differentiation on both sides of Eq. (20).Compared with the equation F y /F x = −(∂ x B x )/(∂ y B x ), we conclude that the direction of Berry curvature at the point (k x , k y , 0) is identical to the tangent of the curve passing through this point,

FIG. 3 .
FIG. 3. (a) Plot of Arg(E+ − E−) for two isolated EPs.The two red points represent two isolated EPs (i.e., two vortices).(b) Plots of Arg(E+ − E−) for EP surface.The black line represents the EL (i.e., a cross-section of EP surface).

Figure 1 (
c) shows the direction of A denoted by arrows.The winding numbers of the arrows along the contour L outside and inside the EP ring are both non-zero.However, this non-zero winding number is not related to the non-zero Berry flux.The Berry flux in Eq. (17a) can be rewritten as the loop integral of the Berry connection, i.e.Φ B = L A•d l k , and it is not equal to the expression of the winding number for the Berry connection W = (2π) −1 L (A y ∇A x − A x ∇A y )/ |A| 2 dk.

1 FIG. 4 .
FIG. 4. Schematics of the geometric phase e iγ(θ) .(a) Real part and (b) imaginary part.The blue points are numerical results and the red solid line are analytical results according to Eq. (18).Inset: fidelity for r = γ/2.The parameters are the same as those in Fig. 1(a).

FIG. 5 .
FIG. 5. Genus at parameters a = 2, b = 2.3, c = d = 2.8, m = 0.3, (a) γ = 109/140 and (b) γ = 1.9.(c) and (d): Streamlines of Berry curvature in the equatorial cross-section for the configurations in (a) and (b), respectively.(e) and (f): Berry connection outside the EP ring and Berry curvature inside the EP ring on a specific cross-section, this specific cross-section is depicted in (b) and (d).
) and (d) have five holes, and the two left or right holes are touching (separated) in (c) [(d)].We sort the streamlines by the number and orientation of the holes they surround.Figure.5(c) exhibits three types of streamlines surrounding one hole (the center hole, and the orientation of streamlines are anticlockwise), two holes (the two left or right holes, clockwise), and five holes (clockwise).

Figure 6 (
b) shows a schematic diagram of the circuit elements corresponding to a unit cell.The lattice sites are represented by circuit nodes.The variable hoppings, m − a, −a/2 and −s can be realized by tuning the capacitors C 1 , C 2 and C 3 , respectively.