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In a warm plasma, the charged particles' macroscopic motion can be described by the fluid dynamics (Navier-Stokes equation) with parameters, such as pressure and density. In contrast to conventional fluid where only longitudinal wave exists due to the absence of tangential resilience, warm plasma's dispersion is more complicated due to the presence of electromagnetic force. In this work, MHD in combination with Maxwell equations are used to describe the fluid motions of the plasma under electromagnetic interactions. Taking into account the dissipation terms, which include the collision and viscosity, the plasma fluid dynamics equation is given as,
$$ \frac{{\partial {\boldsymbol{u}}_{\boldsymbol{e}}}}{{\partial t}} + {\boldsymbol{u}}_{\boldsymbol{e}} \cdot \nabla {\boldsymbol{u}}_{\boldsymbol{e}} = q_en_{\mathrm{e}}\left( {{\boldsymbol{E}} + \frac{1}{c}{\boldsymbol{u}}_{\boldsymbol{e}} \times {\boldsymbol{B}}} \right) - \frac{{\nabla P_{\mathrm{e}}}}{{{\it{n}}_{\it{e}}{\it{m}}_{\it{{\mathrm{e}}}}}} + \frac{{{\boldsymbol{u}}_{\boldsymbol{e}}}}{\tau } + \eta \nabla ^2{\boldsymbol{u}}_{\boldsymbol{e}} $$ (1) Where ne is the plasma density, me is the electron mass, qe is the elementary charge, Pe is the electrons thermal pressure, η is the viscosity coefficient, which arises from the frictions between counter-propagating electrons, τ is the mean free time between electrons collision. The corresponding Hamiltonian formalism is given as (Supplementary Note 1 for detail).
$$ \omega _p\left[ {\begin{array}{*{20}{c}} {} & { - {\boldsymbol{K}} \times /k_P} & {} & { - i} \\ {{\boldsymbol{K}} \times /k_P} & {} & {} & {} \\ {} & {} & {} & {\kappa {\boldsymbol{K}}/k_P} \\ i & {} & {\kappa {\boldsymbol{K}}/k_P} & {\omega _c\Delta - iI/\tau - iI\eta K^2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\boldsymbol{E}} \\ {\boldsymbol{H}} \\ n \\ {\boldsymbol{j}} \end{array}} \right] = \omega \left[ {\begin{array}{*{20}{c}} {\boldsymbol{E}} \\ {\boldsymbol{H}} \\ n \\ {\boldsymbol{j}} \end{array}} \right] $$ (2) where $\omega _p = \left({n_0e^2/m_e} \right)^{1/2}$ is the equilibrium plasma frequency, $\omega _c = \frac{{q_eB}}{{m_ec}}$ is the electron cyclotron frequency, $k_p = \omega _p/c$, Δ=$\left[{\sigma _y\, 0;0\, 0} \right]$, I is the identity matrix, and ${\rm{\kappa }}^2 = \frac{{P_0\gamma }}{{n_0mc^2}}$ with γ being the ratio of specific heats, n is the fluctuation of plasma density, j the polarized current. In the rest of the paper, the frequencies are normalized by ωp, η has unit of ${\mathrm{kg}}/\left({{\mathrm{m}} \!\cdot\! {\mathrm{s}}^2} \right)$ corresponding to actual viscosity coefficient to η/ωp whose unit is ${\mathrm{kg}}/\left({{\mathrm{m}} \!\cdot\! {\mathrm{s}}} \right)$. The traditional mathmatically complicated calculation of the nonlocal dielectric functions is thus transformed into an eigen-value problem.
We start by considering only the collision dissipation while neglecting the effect of viscosity. The dispersion of MHD plasma can be solved by diagonalizing the Hamiltonian in Eq. (1). The absolute value of the complex frequency of the non-ideal magnetized warm plasma is shown in Fig. 1a with $\omega _p = 3.5 \times 10^{12}\, {\mathrm{rad}}/{\mathrm{s}}$, $\omega _c = 2\omega _p$, κ=0.4, τ=2, corresponding to a magnetized plasma system with an electron density of $3.9 \times 10^{19}\, 1/{\mathrm{m}}^3$, an external magnetic field B of 8.01T, an electron thermal temperature of 27.3keV and an electronic collision frequency of $1.75 \times 10^{12}\, {\mathrm{rad}}/{\mathrm{s}}$. The band structure shows four bands, which, ordered from low to high frequencies, are the first right-handed circularly polarized band (R-wave), the longitudinal Langmuir (LM) wave (see detail in Supplementary Fig. 1), the left-handed circularly polarized band (L-wave) and the second band of R-wave. In the lossless limit R-wave and L-wave's dispersions are expressed as $k_{{\mathrm{R/L}}} = \frac{\omega }{c}\left[{1- \frac{{\omega _{pe}^2}}{{\omega ^2\left({1 \mp \omega _{ce}/\omega } \right)}}} \right]^{1/2}$. The LM wave's dispersion is given by $\omega ^2 = \omega _{pe}^2 + \kappa ^2c^2k^2$, which can reduce to the cold plasma's4 oscillation frequency by taking κ=0. (Supplementary Note 2 for detail and Supplementary Fig. 2). The positions of the Weyl points are caculated in Supplementary Fig. 3. To better understand these linear band crossing, we apply the k·p theory (effective Hamiltonian theory) under collisionless conditions to obtain the approximate Hamiltonian near the degeneracies. Expanding to first order in the vicinity of the outer degeneracy, we find the effective Hamiltonian as:
$$ H_1 = \frac{{A + N}}{2}k_z\sigma _z + Mk_x\sigma _x - Mk_y\sigma _y + \frac{{A - N}}{2}Ik_z $$ (3) Fig. 1 Existence of Nodal disks in magnetohydrodynamic plasma.
a Dispersion of the absolute value of eigen frequency ω with . Along the kz axis, the bands are differentiated as right/left handedness circular polarization wave (R/L-wave) and Langmuir wave whose polarization is only in the z-direction. The bands along transverse directions (x-direction) have degeneracies denoted by the red/blue dots. Owing to the rotational symmetry in the plasma system, the dots in a form the loops in b. Akin to the Weyl points, there are two species of Nodal disks in the plasma system with opposite topological charge (+/-1 charge denoted by the red/blue color) that are connect by the spatial inversion operation. c-f Real and imaginary part of the eigen frequency in the kx-kz and kx-ky plane. Note that in the kx-kz plane, real and imaginary parts are simultaneously degenerated at isolated points away from the kz axis. While in the kx-ky plane, the degeneracies form a ring$\omega _c = 2, \, \kappa = 0.4, \, \tau = 2$ Fig. 2 Phase map, the emergence, and annihilations of the exceptional loops at τ=2.
a The number of nodal disks is due to change by the strength of magnetic field ωc and the electron thermal parameter κ. Along the path crossing the border line, for example the path shown as the yellow arrow in a, the annihilation or emergence of nodal disks could be observed. Three-dimensional (3-D) band structures before and at the critical transition point are given in b and c. Difference of the real part of the eigen frequencies around the nodal disks before, at and after the annihilation of the nodal disks are given in d-f. The dashed lines represent the locations where the difference goes to zero. The positions of the nodal disks are located at the edge of the dashed lines. d Before the coalesce of the two nodal disks. e At the border between the two phases, the two nodal disks would coalesce and dissipate their topological charges. f After the coalesce, the zeros in the e would shift from zeroFig. 3 Berry curvature distribution and dissipative fermi arcs of the nodal disks with $\omega _c = 2, \, \kappa = 0.4, \, \tau = 2$.
a Berry curvature for the upper band of collision warm magnetic plasma in the kx-kz plane with. The red rings are the surrounding of the Nodal disks in kx-ky plane and the blue solid line is the projection of the bulk nodal disk in kx-kz plane. Scales of the arrow represent the intensity of Berry curvatures and orientation indicates the direction. The arrows show that the bulk nodal disks are pure source of Berry curvature. b fermi arcs connect at the larger (1.96), equals (1.86), and smaller (1.76) bulk nodal disk frequency. For the Nodal disks frequency, Fermi arc connects at center of the real dependency line while Fermi arc emits from the bulk states when the frequency shifts. The color represents the value of the corresponding imaginary frequency. c dissipating fermi arc between plasma and vacuum, which connects the two bulk nodal disk with opposite chirality, the color represents the value of the corresponding imaginary frequencywhere $A = \frac{{\sqrt {\varepsilon _{12}} }}{{2\left({\varepsilon _{12}^2\, -\, \varepsilon _{12} \, +\, 2} \right)}}$, $N = \frac{{\omega _p^2\sqrt {\omega _c} }}{{2\left({\omega _p \, +\, \omega _c} \right)\sqrt {\omega _c \, -\, \omega _p} }}\kappa ^2$, $M = \frac{{\sqrt {\varepsilon _{12}} }}{{2\sqrt {\varepsilon _{12}^2 \, -\, \varepsilon _{12} \, +\, 2} }}$, and ε12 is sum of the diagonal and off-diagonal elements of the dielectric matrix evaluated at the Weyl's frequency, which can be expresses as $\omega _{{\mathrm{weyl}}} = \omega _p + \frac{{\omega _p^2\omega _c}}{{2\left({\omega _c^2 \, -\, \omega _p^2} \right)}}\kappa ^2$ (Supplementary Note 3 for detail). These linear degeneracies are Weyl points that function as the sources and drains of Berry curvature flux lines. Owing to the electrons collision, the new degeneracy points where both the real and the imaginary part of the complex frequency spectrum are degenerate shift away from the kz axis, as denoted by the colored dots in Fig. 1a. Owing to the rotational symmetry of the system around kz axis, the degeneracy points form exceptional loops in the kx-ky plane in Fig. 1b. Figure 1c, d show the real and imaginary parts of the eigen frequency in kx-kz direction, where the degeneracy line for the real part of the eigen frequency in the kx-kz plane forms a bulk nodal disk due to the rotational symmetry. The dispersion of the real and imaginary parts of the eigen frequency on the kx-ky plane are presented in Fig. 1e, f, respectively, which show the presence of an EL. As shown in Fig. 1, the modified dispersion of the LM wave is curved due to the thermal effect, providing the possibility of forming three pairs of nodal disks between the LM Wave and the L/R-wave. In contrast the cold plasma can at most possess two pairs of Weyl points due to its dispersionless LM mode39. As shown in Fig. 1a, two types of band-crossing exist: the band crossings between the LM wave and the L-wave at small k, and the other two between the LM wave and the first R-wave at larger k's. These band crossings are guaranteed by the polarization orthogonality between the longitudinal LM waves and the transverse L and R waves.
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By tuning the thermal parameter κ and the cyclotron frequency ωc under a fixed collision coefficient τ=2, the nodal disks' location in the momentum space can be shifted along the kz direction, or even merged or annihilated. In Fig. 2a, the transition of the number of nodal disks is illustrated in a phase map in the parameter space (ωc, κ), which shows that the system acquires either three pairs or a single pair of nodal disks. Right at the phase boundary, the two nodal disks at larger kz join with each other. The difference in the real part of the frequency between the LM wave and R-wave constituting the nodal disks for different κ values is given in Fig. 2d-f. As shown in Fig. 2d, when κ=0.52, ωc=2 (below the critical transition value), the nodal disks are separate (white dash lines). While at the critical transition value κ=0.5228, ωc=2 the exceptional loops merge into each other and the nodal disks form a closed nodal surface, as shown in Fig. 2e. The corresponding 3-D band structures to Fig. 2d, e are given in Fig. 2b, c. It should be noted that even though the Chern number of the closed nodal surface is zero since the topological charges of the two nodal disks cancel each other, the Berry curvatures do not vanish because of the nonzero topological charge density on the surface. Further increase of κ beyond the critical value leads to simply shrinkage of the closed nodal surface and lifting of the degeneracy as is shown in Fig. 2f.
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Similar to Weyl points in Hermitian systems, nodal disks are sources and drains of Berry curvatures in non-Hermitian systems. In such a dissipative system, Berry curvatures can be derived from any entry of the Berry connection matrix: $A_\mu ^{I, J} = i\langle I{\mathrm{|}}\partial _\mu {\mathrm{|}}J\rangle, \, I, J \in \left\{ {L, R} \right\}$. In Supplementary Note 5, it is shown that this matrix is Hermitian, whose diagonal entries are real, and off-diagonal complex. The matrix entries satisfy the relation: $A_\mu ^{L, L} = A_\mu ^{R, R}\, \ne\, A_\mu ^{L, R} = A_\mu ^{L, R \ast }$, where the sign * stands for the complex conjugate. It could be further shown that the Berry curvature calculated from different matrix entries are locally different, however, their integral over the nodal disk, the Chern numbers, are the same (for the complex term $A_\mu ^{L, R}$ and $A_\mu ^{L, R}$, only real parts are integrated)15. In dissipative MHD plasma, Berry curvature flux distribution from the nodal disk between the LM and the R waves is shown in Fig. 3a, where the red rings represent the ELs in kx-ky plane and the blue solid line is the projection of the nodal disk in kx-kz plane. Figure 3a clearly shows that the nodal disk is the source of Berry curvature. In proximity to the EL, the derived Berry curvatures diverage is at the rate of $1/\!\sqrt r$, where r is the distance to the exceptional loop (see detail in Supplementary Note 5). This means the Berry curvature emitted by the exceptional loop has neligible value. Hence it is confirmed that the topological charge is continuously distributed across the bulk nodal disks, rather than concentrated at the ELs.
Owing to the nontrivial topological charge of the nodal disk, presence of Fermi arcs is expected. To solve for the Fermi arc, we consider an abrupt interface between the MHD plasma and vacuum, which can be found in magnetic confinement plasmas such as Tokamak configurations. Because of the nonlocality of the system, an additional boundary condition that the current density diminishes in the surface normal direction should be imposed, together with the conventional electromagnetic wave continuity conditions. Considering the continuity of bulk displacement current normal to the interface, this additional boundary condition is equivalent to continuity of Ex across the interface40. The dissipating Fermi arc near the outer the nodal disk at the real frequency parts of 1.76ωp, 1.86ωp, and 1.96ωp is given in Fig. 3b (more details are shown in Supplementary Fig. S7). Interestingly, at the EL frequency, the dissipative Fermi arc connects between the projected nodal disks right in the middle, as shown in Fig. 3c. However, away from the exception loop frequency, the Fermi arc could exist inside the bulk state continuum, as shown in Fig. 3b. Detailed derivations are provided in the Supplementary Note 4, and the surface modes at different frequencies are presented in Supplementary Fig. S4.
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In the above, we have studied the warm magnetic plasma with collision loss. However, in tokamaks and helical devices, there also exists strong anomalous viscosity14. As indicated by Eq. (2), viscosity leads to an extra quadratic term in the dispersion of the LM wave, which becomes significant at large momentum. The significantly modified dispersion of the LM wave could introduce an extra pair of nodal disks by intersecting the R-wave again. The real part of the eigen frequency spectrum around the nodal disks is given in Fig. 4a. Comparing with that in a plasma with only collisions, the nodal disk in a plasma with viscosity dissipation is highly curved due to the presence of the higher order term in the momentum. The curved nodal disks can be attributed to the $k_{x, y}^2$ terms present in the effective Hamiltonian model (Supplementary Note 7). Increasing the strength of the viscosity force could induce coalescence and annihilation of the nodal disks as shown in Fig. 4b-d. Similar to the effects shown in previous sections, at a certain viscosity force, tuning the magnetic field strength and electron thermal parameter, the 2nd and 3rd nodal disks could annihilate, leaving only the 1st and the 4th exceptional loops.
Fig. 4 Viscosity force induced extra Nodal disks and their phase transitions.
a According to different viscosity parameter range at , the magnetohydrodynamic plasma could have either four pairs of Nodal disks or two pairs by the annihilations of the 2nd and the 3rd, or the 3rd and the 4th. White dash lines represent the real parts' degeneracy line in the momentum plane of x-y. Difference of the real part of the eigen frequencies b before, c at, and d after near to the exceptional loops' annihilation by increasing the viscosity force. The 3-D band structures are give in e-g$\omega _c = 2, \, \kappa = 0.4$