form $\delta\mu_\chi = \delta\mu_A + s_V\delta\mu_H$. As we just mentioned, this property is to be expected since when the system consists only of particles (that is, there are no antiparticles present), the helicity and chirality are indistinguishable from each other and, therefore, they enter only in the combination (5.6). Due to the very same reason, at high vector density, the helical and axial vortical waves form the same hydrodynamic excitation, the Axial-Helical Vortical Wave, which inherits features from both original waves.

We now formally introduce a small parameter $\epsilon$ such that $T \to \epsilon T$ and $\mu_\chi \to \epsilon\mu_\chi$. We further consider an expansion of the velocity of the form $\omega \to \tilde{\omega} = \tilde{\omega}_0 + \tilde{\omega}_1\epsilon + \dots$. A series expansion with respect to $\epsilon$ yields:

$$ \begin{aligned} \tilde{M}_{VV} &= \frac{\tilde{\omega}_0\mu_V^2}{3\pi^2} + \frac{\epsilon(4\mu_\chi k\Omega + \tilde{\omega}_1\mu_V^2)}{3\pi^2} + O(\epsilon^2), & \tilde{M}_{V\chi} &= -\frac{2\epsilon^2 k\Omega}{\mu_V}\left(\frac{T^2}{3} + \frac{2\mu_\chi^2}{\pi^2}\right) + O(\epsilon^3), \\ \tilde{M}_{\chi V} &= -\frac{2k\Omega\mu_V}{3\pi^2} + \frac{4\epsilon\tilde{\omega}_0\mu_V\mu_\chi}{3\pi^2} + O(\epsilon^2), & \tilde{M}_{\chi\chi} &= \frac{\tilde{\omega}_0\mu_V^2}{\pi^2} + \frac{\epsilon(6\mu_\chi k\Omega + \tilde{\omega}_1\mu_V^2)}{\pi^2}, \end{aligned} \quad (5.10) $$

where $\tilde{M}_{V\chi} = \tilde{M}_{VA} = s_V\tilde{M}_{VH}$, $\tilde{M}_{\chi V} = \tilde{M}_{AV} = s_V\tilde{M}_{HV}$, and $\tilde{M}_{\chi\chi} = \tilde{M}_{AA} = s_V\tilde{M}_{AH} = s_V\tilde{M}_{HA} = \tilde{M}_{HH}$. Taking the determinant, it can be seen that at order $O(\epsilon^0)$, we have $\tilde{\omega}_0^2\mu_V^4/3\pi^4 = 0$, such that $\tilde{\omega}_0 = 0$ and $\tilde{\omega}$ becomes of first order with respect to $\epsilon$. The next non-vanishing contribution is of order $\epsilon^2$, with the corresponding equation given by:

$$ \tilde{\omega}_1^2 \mu_V^4 + 10 \tilde{\omega}_1 \mu_\chi \mu_V^2 k\Omega + 16 \mu_\chi^2 k^2 \Omega^2 - \frac{4\pi^2 T^2}{3} k^2 \Omega^2 = 0. \quad (5.11) $$

Solving the above equation, we obtain the energy dispersion relation of the vortical wave in the degenerate matter:

$$ \tilde{\omega}_{\pm} = -\frac{5k\Omega\mu_{\chi}}{\mu_{V}^{2}} \pm \frac{k\Omega}{\mu_{V}^{2}} \sqrt{\frac{4\pi^{2}T^{2}}{3} + 9\mu_{\chi}^{2}}, \quad [\text{Axial-Helical Vortical Wave}]. \quad (5.12) $$

The wave appears to couple fluctuations $\tilde{\delta}\mu_V$ in the vector chemical potential and the fluctuations in the combined axial-helical chemical potentials $\tilde{\delta}\mu_\chi$ as follows:

$$ \tilde{\delta}\mu_V^{\pm} = \frac{3\tilde{\delta}\mu_\chi^{\pm}}{2\mu_V} \left( \mu_\chi \pm \sqrt{\frac{4\pi^2 T^2}{3} + 9\mu_\chi^2} \right), \quad [\text{Axial-Helical Vortical Wave}]. \quad (5.13) $$

Both the energy dispersion relation (5.12) and the charge density content (5.13) of the Axial-Helical Vortical Wave depend on the sum $\mu_\chi$ of the axial and helical chemical potentials (5.6).

In the presence of a finite axial (or helical) charge density, the propagation of the wave becomes non-reciprocal with respect to the direction of the angular velocity $\Omega$. In general, the speed of the wave $\tilde{v}_+ = \tilde{\omega}_+/k$ parallel to the direction of the angular velocity $\Omega$ and the speed $\tilde{v}_- = \tilde{\omega}_-/k$ in the direction anti-parallel to $\Omega$ are different from each other (5.12). For example, setting high temperature $T \gg |\mu_\chi|$ (but still maintaining the degenerate limit $\mu_V \gg T$), we clearly see from Eq. (5.12) that the wave propagates in both directions with an offset in the velocities given by the first term. For a positive (negative) sign of the product
form $\delta\mu_\chi = \delta\mu_A + s_V\delta\mu_H$. As we just mentioned, this property is to be expected since when the system consists only of particles (that is, there are no antiparticles present), the helicity and chirality are indistinguishable from each other and, therefore, they enter only in the combination (5.6). Due to the very same reason, at high vector density, the helical and axial vortical waves form the same hydrodynamic excitation, the Axial-Helical Vortical Wave, which inherits features from both original waves.

We now formally introduce a small parameter $\epsilon$ such that $T \to \epsilon T$ and $\mu_\chi \to \epsilon\mu_\chi$. We further consider an expansion of the velocity of the form $\omega \to \tilde{\omega} = \tilde{\omega}_0 + \tilde{\omega}_1\epsilon + \dots$. A series expansion with respect to $\epsilon$ yields:

$$ \begin{aligned} \tilde{M}_{VV} &= \frac{\tilde{\omega}_0 \mu_V^2}{3\pi^2} + \frac{\epsilon(4\mu_\chi k\Omega + \tilde{\omega}_1 \mu_V^2)}{3\pi^2} + O(\epsilon^2), & \tilde{M}_{V\chi} &= -\frac{2\epsilon^2 k\Omega}{\mu_V} \left(\frac{T^2}{3} + \frac{2\mu_\chi^2}{\pi^2}\right) + O(\epsilon^3), \\ \tilde{M}_{\chi V} &= -\frac{2k\Omega\mu_V}{3\pi^2} + \frac{4\epsilon\tilde{\omega}_0\mu_V\mu_\chi}{3\pi^2} + O(\epsilon^2), & \tilde{M}_{\chi\chi} &= \frac{\tilde{\omega}_0\mu_V^2}{\pi^2} + \frac{\epsilon(6\mu_\chi k\Omega + \tilde{\omega}_1 \mu_V^2)}{\pi^2}, \end{aligned} \quad (5.10) $$

where $\tilde{M}_{V\chi} = \tilde{M}_{VA} = s_V\tilde{M}_{VH}$, $\tilde{M}_{\chi V} = \tilde{M}_{AV} = s_V\tilde{M}_{HV}$, and $\tilde{M}_{\chi\chi} = \tilde{M}_{AA} = s_V\tilde{M}_{AH} = s_V\tilde{M}_{HA} = \tilde{M}_{HH}$. Taking the determinant, it can be seen that at order $O(\epsilon^0)$, we have $\tilde{\omega}_0^2\mu_V^4/3\pi^4 = 0$, such that $\tilde{\omega}_0 = 0$ and $\tilde{\omega}$ becomes of first order with respect to $\epsilon$. The next non-vanishing contribution is of order $\epsilon^2$, with the corresponding equation given by:

$$ \tilde{\omega}_1^2 \mu_V^4 + 10 \tilde{\omega}_1 \mu_\chi \mu_V^2 k\Omega + 16 \mu_\chi^2 k^2 \Omega^2 - \frac{4\pi^2 T^2}{3} k^2 \Omega^2 = 0. \quad (5.11) $$

Solving the above equation, we obtain the energy dispersion relation of the vortical wave in the degenerate matter:

$$ \tilde{\omega}_{\pm} = -\frac{5k\Omega\mu_{\chi}}{\mu_{V}^{2}} \pm \frac{k\Omega}{\mu_{V}^{2}} \sqrt{\frac{4\pi^{2}T^{2}}{3} + 9\mu_{\chi}^{2}}, \quad [\text{Axial-Helical Vortical Wave}]. \quad (5.12) $$

The wave appears to couple fluctuations $\tilde{\delta}\mu_V$ in the vector chemical potential and the fluctuations in the combined axial-helical chemical potentials $\tilde{\delta}\mu_\chi$ as follows:

$$ \tilde{\delta}\mu_V^{\pm} = \frac{3\tilde{\delta}\mu_\chi^{\pm}}{2\mu_V} \left( \mu_\chi \pm \sqrt{\frac{4\pi^2 T^2}{3} + 9\mu_\chi^2} \right), \quad [\text{Axial-Helical Vortical Wave}]. \quad (5.13) $$

Both the energy dispersion relation (5.12) and the charge density content (5.13) of the Axial-Helical Vortical Wave depend on the sum $\mu_\chi$ of the axial and helical chemical potentials (5.6).

In the presence of a finite axial (or helical) charge density, the propagation of the wave becomes non-reciprocal with respect to the direction of the angular velocity $\Omega$. In general, the speed of the wave $\tilde{v}_+ = \tilde{\omega}_+/k$ parallel to the direction of the angular velocity $\Omega$ and the speed $\tilde{v}_- = \tilde{\omega}_-/k$ in the direction anti-parallel to $\Omega$ are different from each other (5.12). For example, setting high temperature $T \gg |\mu_\chi|$ (but still maintaining the degenerate limit $\mu_V \gg T$), we clearly see from Eq. (5.12) that the wave propagates in both directions with an offset in the velocities given by the first term. For a positive (negative) sign of the product
$\mu_{\chi}\Omega$, the wave propagates thus faster opposite (along) the vorticity vector $|\tilde{v}_{-}| > |\tilde{v}_{+}|$ ($|\tilde{v}_{-}| < |\tilde{v}_{+}|$).

Moreover, at the critical temperature

$$T_c^{\text{deg}} = \frac{2\sqrt{3}}{\pi} |\mu_{\chi}|, \qquad (5.14)$$

the speed of propagation of the wave along (opposite to) the vorticity vector vanishes if the product $\mu_{\chi}\Omega$ takes a positive (negative) value. Thus, at the critical temperature (5.14), one of the branches of the axial-helical wave is propagating (anti-)parallel to the vorticity while the other branch represents a static mode. If the temperature is below the critical value (5.14), then both modes propagate in the same direction in a transparent manifestation of non-reciprocity.

Thus, in the cold and dense rotating matter, the hydrodynamical waves propagate in a non-reciprocal manner with respect to the global angular velocity if the background state is axially or helically imbalanced. Moreover, the relation between the magnitudes of the vector and axial-helical components of the wave differs in the waves propagating in opposite directions (5.13). These properties represent a unique feature of hydrodynamic excitations possessing the helical degree of freedom.

Considering now a vector wave constructed as in Eq. (2.44), namely $\bar{\mu}_V(t = 0, z) = \mu_V + \delta\bar{\mu}_{V;0} \cos(kz)$ and $\bar{\mu}_{\chi}(t = 0, z) = \mu_{\chi}$, with $|\mu_V| \gg |\mu_{\chi}|$, we find

$$\delta\bar{\mu}_{\bar{V}}^{\pm} = \frac{1}{2} \left(1 \pm \frac{\mu_{\chi}}{\mathfrak{s}}\right) \delta\bar{\mu}_{V;0}, \quad \delta\mu_{\chi}^{\pm} = \pm \frac{\mu_V}{3\mathfrak{s}} \delta\bar{\mu}_{V;0}, \qquad (5.15)$$

with $\mathfrak{s} = \sqrt{4\pi^2 T^2 / 3 + 9\mu_{\chi}^2}$. This leads to

$$\begin{aligned} \bar{\mu}_V(t, z) &= \mu_V + \delta\bar{\mu}_{V;0} \cos \left( kz - \frac{5\Omega\mu_{\chi}}{\mu_V^2} kt \right) \cos \left( \frac{k\Omega t}{\mu_V^2 \mathfrak{s}} \right) \\ &\quad - \frac{\mu_{\chi} \delta\bar{\mu}_{V;0}}{\mathfrak{s}} \sin \left( kz - \frac{5\Omega\mu_{\chi}}{\mu_V^2} kt \right) \sin \left( \frac{k\Omega t}{\mu_V^2 \mathfrak{s}} \right), \\ \bar{\mu}_{\chi}(t, z) &= \mu_{\chi} - \frac{\mu_V \delta\bar{\mu}_{V;0}}{3\mathfrak{s}} \sin \left( kz - \frac{5\Omega\mu_{\chi}}{\mu_V^2} kt \right) \sin \left( \frac{k\Omega t}{\mu_V^2 \mathfrak{s}} \right). \end{aligned} \qquad (5.16)$$

In the case of the initial Gaussian distribution in Eq. (2.48) for the perturbation in the vector chemical potential, $\bar{\mu}_V(0, z) = \mu_V + \delta\bar{\mu}_{V;0} e^{-z^2/2\sigma^2}$, with constant $\bar{\mu}_{\chi}(0, z) = \mu_{\chi}$, we have

$$\begin{aligned} \bar{\mu}_V(t, z) &= \mu_V + \frac{\delta\bar{\mu}_{V;0}}{2} \left[ \left(1 + \frac{\mu_{\chi}}{\mathfrak{s}}\right) e^{-(z-\bar{v}_+ t)^2/2\sigma^2} + \left(1 - \frac{\mu_{\chi}}{\mathfrak{s}}\right) e^{-(z-\bar{v}_- t)^2/2\sigma^2} \right], \\ \bar{\mu}_{\chi}(t, z) &= \mu_{\chi} + \frac{\mu_V \delta\bar{\mu}_{V;0}}{3\mathfrak{s}} \left[ e^{-(z-\bar{v}_+ t)^2/2\sigma^2} - e^{-(z-\bar{v}_- t)^2/2\sigma^2} \right], \end{aligned} \qquad (5.17)$$

where $\tilde{v}_{\pm} = -5\Omega\mu_{\chi}/\mu_V^2 \pm \Omega\mathfrak{s}/\mu_V^2$ represent the group velocities corresponding to the two propagating modes.

The propagation of the initial Gaussian distribution is considered in Fig. 5. For definiteness, we took a small background temperature of $T = 10$ MeV and a background vector
$\mu_{\chi}\Omega$, the wave propagates thus faster opposite (along) the vorticity vector $|\tilde{v}_{-}| > |\tilde{v}_{+}|$ ($|\tilde{v}_{-}| < |\tilde{v}_{+}|$).

Moreover, at the critical temperature

$$T_c^{\text{deg}} = \frac{2\sqrt{3}}{\pi} |\mu_{\chi}|, \qquad (5.14)$$

the speed of propagation of the wave along (opposite to) the vorticity vector vanishes if the product $\mu_{\chi}\Omega$ takes a positive (negative) value. Thus, at the critical temperature (5.14), one of the branches of the axial-helical wave is propagating (anti-)parallel to the vorticity while the other branch represents a static mode. If the temperature is below the critical value (5.14), then both modes propagate in the same direction in a transparent manifestation of non-reciprocity.

Thus, in the cold and dense rotating matter, the hydrodynamical waves propagate in a non-reciprocal manner with respect to the global angular velocity if the background state is axially or helically imbalanced. Moreover, the relation between the magnitudes of the vector and axial-helical components of the wave differs in the waves propagating in opposite directions (5.13). These properties represent a unique feature of hydrodynamic excitations possessing the helical degree of freedom.

Considering now a vector wave constructed as in Eq. (2.44), namely $\bar{\mu}_V(t = 0, z) = \mu_V + \delta\bar{\mu}_{V;0} \cos(kz)$ and $\bar{\mu}_{\chi}(t = 0, z) = \mu_{\chi}$, with $|\mu_V| \gg |\mu_{\chi}|$, we find

$$\delta\bar{\mu}_{\bar{V}}^{\pm} = \frac{1}{2} \left(1 \pm \frac{\mu_{\chi}}{\mathfrak{s}}\right) \delta\bar{\mu}_{V;0}, \quad \delta\mu_{\chi}^{\pm} = \pm \frac{\mu_V}{3\mathfrak{s}} \delta\bar{\mu}_{V;0}, \qquad (5.15)$$

with $\mathfrak{s} = \sqrt{4\pi^2 T^2/3 + 9\mu_{\chi}^2}$. This leads to

$$\begin{aligned} \bar{\mu}_V(t, z) &= \mu_V + \delta\bar{\mu}_{V;0} \cos \left( kz - \frac{5\Omega\mu_{\chi}}{\mu_V^2} kt \right) \cos \left( \frac{k\Omega t}{\mu_V^2 \mathfrak{s}} \right) \\ &\quad - \frac{\mu_{\chi} \delta\bar{\mu}_{V;0}}{\mathfrak{s}} \sin \left( kz - \frac{5\Omega\mu_{\chi}}{\mu_V^2} kt \right) \sin \left( \frac{k\Omega t}{\mu_V^2 \mathfrak{s}} \right), \\ \bar{\mu}_{\chi}(t, z) &= \mu_{\chi} - \frac{\mu_V \delta\bar{\mu}_{V;0}}{3\mathfrak{s}} \sin \left( kz - \frac{5\Omega\mu_{\chi}}{\mu_V^2} kt \right) \sin \left( \frac{k\Omega t}{\mu_V^2 \mathfrak{s}} \right). \end{aligned} \qquad (5.16)$$

In the case of the initial Gaussian distribution in Eq. (2.48) for the perturbation in the vector chemical potential, $\bar{\mu}_V(0, z) = \mu_V + \delta\bar{\mu}_{V;0} e^{-z^2/2\sigma^2}$, with constant $\bar{\mu}_{\chi}(0, z) = \mu_{\chi}$, we have

$$\begin{aligned} \bar{\mu}_V(t, z) &= \mu_V + \frac{\delta\bar{\mu}_{V;0}}{2} \left[ \left(1 + \frac{\mu_{\chi}}{\mathfrak{s}}\right) e^{-(z-\bar{v}_+ t)^2/2\sigma^2} + \left(1 - \frac{\mu_{\chi}}{\mathfrak{s}}\right) e^{-(z-\bar{v}_- t)^2/2\sigma^2} \right], \\ \bar{\mu}_{\chi}(t, z) &= \mu_{\chi} + \frac{\mu_V \delta\bar{\mu}_{V;0}}{3\mathfrak{s}} \left[ e^{-(z-\bar{v}_+ t)^2/2\sigma^2} - e^{-(z-\bar{v}_- t)^2/2\sigma^2} \right], \end{aligned} \qquad (5.17)$$

where $\tilde{v}_{\pm} = -5\Omega\mu_{\chi}/\mu_V^2 \pm \Omega\mathfrak{s}/\mu_V^2$ represent the group velocities corresponding to the two propagating modes.

The propagation of the initial Gaussian distribution is considered in Fig. 5. For definiteness, we took a small background temperature of $T = 10$ MeV and a background vector
<figure><img src="image_8.png" /><figcaption>**Figure 5:** Time evolution of the charge perturbations (top left) $\delta\bar{\mu}_V(t, z)$, (top right) $\delta\bar{\mu}_\chi(t, z)$ and (bottom) $\delta\bar{\mu}_{\bar{\chi}}$, corresponding to the propagation of the axial-helical vortical wave through a charge-conserving plasma. The initial conditions are given in Eqs. (3.34). The background state has parameters $T = 10$ MeV, $\Omega = 6.6$ MeV, $\mu_V = 200$ MeV and $\mu_A = \mu_H = 0$, corresponding to the degenerate limit discussed in Sec. 5. In the label, $\tilde{v}$ is the absolute value of the corresponding speed of the wave $\tilde{v}_\pm = \tilde{\omega}_\pm/k$ defined in 5.12. The analytical solution (black) 5.17 is superimposed onto the numerical solutions (blue and pink).</figcaption></figure>

chemical potential of $\mu_V = 100$ MeV, while the background axial and helical chemical potentials were set to zero. The left and middle panels demonstrate the expected analytical solution derived in Eq. (5.17). Surprisingly, the third panel reveals that the combination $\delta\bar{\mu}_{\bar{\chi}} = \delta\bar{\mu}_A - s_V\delta\bar{\mu}_H$ grows to very high values. In the degenerate limit described by the tilde quantities (e.g., $\tilde{P}$) in Eq. (5.10), the combination $\delta\bar{\mu}_{\bar{\chi}}$ is not accessible.

To understand this behaviour described above, we move back to the unpolarized plasma considered in Sec. 4 and derive the relation between all mode amplitudes before taking the degenerate limit. Considering the exact expression for the matrices $M_\omega$ and $M_\Omega$ in Eq. (4.16), we can find the relations between the amplitudes corresponding to the trivial
<figure><img src="image_8.png" /><figcaption>**Figure 5:** Time evolution of the charge perturbations (top left) $\delta\bar{\mu}_V(t, z)$, (top right) $\delta\bar{\mu}_\chi(t, z)$ and (bottom) $\delta\bar{\mu}_{\tilde{\chi}}$, corresponding to the propagation of the axial-helical vortical wave through a charge-conserving plasma. The initial conditions are given in Eqs. (3.34). The background state has parameters $T = 10$ MeV, $\Omega = 6.6$ MeV, $\mu_V = 200$ MeV and $\mu_A = \mu_H = 0$, corresponding to the degenerate limit discussed in Sec. 5. In the label, $\tilde{v}$ is the absolute value of the corresponding speed of the wave $\tilde{v}_\pm = \tilde{\omega}_\pm/k$ defined in 5.12. The analytical solution (black) 5.17 is superimposed onto the numerical solutions (blue and pink).</figcaption></figure>

chemical potential of $\mu_V = 100$ MeV, while the background axial and helical chemical potentials were set to zero. The left and middle panels demonstrate the expected analytical solution derived in Eq. (5.17). Surprisingly, the third panel reveals that the combination $\delta\bar{\mu}_{\tilde{\chi}} = \delta\bar{\mu}_A - s_V\delta\bar{\mu}_H$ grows to very high values. In the degenerate limit described by the tilde quantities (e.g., $\tilde{P}$) in Eq. (5.10), the combination $\delta\bar{\mu}_{\tilde{\chi}}$ is not accessible.

To understand this behaviour described above, we move back to the unpolarized plasma considered in Sec. 4 and derive the relation between all mode amplitudes before taking the degenerate limit. Considering the exact expression for the matrices $M_\omega$ and $M_\Omega$ in Eq. (4.16), we can find the relations between the amplitudes corresponding to the trivial
mode $\omega_0 = 0$ as follows:

$$ \delta\mu_V^0 = 0, \qquad \delta\mu_{\tilde{\chi}}^0 = -\frac{A+s_V B}{A-s_V B} \delta\mu_{\chi}^{\text{im.}} \approx -\frac{2\pi^2}{3|\alpha_V|} e^{|\alpha_V|} \delta\mu_{\chi}^0. \qquad (5.18) $$

Clearly, exciting the $\delta\mu_{\chi}^0$ amplitude leads to an exponentially-larger amplitude $\delta\mu_{\tilde{\chi}}^0$.

In the case of the non-trivial (helical) modes, when $\omega_h^\pm$ is given by Eq. (4.20), we can take the amplitude of the vector charge fluctuation, $\delta\mu_V^{h;\pm}$, as the reference amplitude. The amplitudes $\delta\mu_{\tilde{\chi}}^{h;\pm}$ and $\delta\mu_{\tilde{\chi}}^{h;\pm}$ are then given by

$$ \begin{aligned} \delta\mu_{\chi}^{h;\pm} &= \frac{k\Omega T}{2\omega_h^{\pm}} \frac{A+s_V B}{\sigma_A^{\omega} + s_V \sigma_H^{\omega}} \delta\mu_V^{h;\pm} \approx \pm \frac{\alpha_V}{\pi\sqrt{3}} \delta\mu_V^{h;\pm}, \\ \delta\mu_{\tilde{\chi}}^{h;\pm} &= \frac{k\Omega T}{2\omega_h^{\pm}} \frac{A-s_V B}{\sigma_A^{\omega} - s_V \sigma_H^{\omega}} \delta\mu_V^{h;\pm} \approx \pm \frac{s_V \alpha_V^4 \sqrt{3}}{4\pi^3} \delta\mu_V^{h;\pm}, \end{aligned} \qquad (5.19) $$

where we displayed just the leading-order contribution in the degenerate limit. It can be seen that as $|\alpha_V| \to \infty$, a small excitation in $\delta\mu_V$ in either modes $\delta\mu_V^{h;\pm}$ will lead to excitations in $\delta\mu_{\tilde{\chi}}^{h;\pm}$ that are $\alpha_V^4$ times larger, signalling the breakdown of the linear perturbations ansatz in the degenerate matter. Coming back to Fig. 5, considering the ratio $\alpha_V = \mu_V/T = 20$, our estimate above indicates $\delta\mu_{\tilde{\chi}}^{h;\pm}/\delta\mu_{V;0} \approx 1117$ to leading order in $\alpha_V$, which is in line with the results shown in the right panel of this figure.

The above discussion indicates that degenerate matter under rotation is unstable under small perturbations. A small excitation of the vector chemical potential induces huge excitations in the vector and helical chemical potentials, invalidating the linear regime ansatz of this analysis. One may question the physical soundness of these results. In our companion paper [26], we show that in a realistic plasma, the non-conservation of the helicity charge prevents the axial and helical mode amplitudes from growing, leading to perfectly reasonable propagation properties.

# 6 Summary and Conclusions

In our work, we challenge the traditional approach to chiral systems, which presumes that the chiral fluids are completely described only by the pair of the vector and axial local charges. We show that the inclusion of the helical degree of freedom enriches the hydrodynam- ical spectrum of the system. Since we explored various regimes of the vector-axial-helical triad of charges, it is worth summarizing the main features of our findings in a compact set of plots shown in Fig. 6.

In the high-temperature limit of low-density plasmas, where the temperature signif- icantly exceeds the magnitude of any background chemical potential, we confirmed the existence of the Helical Vortical Wave, which corresponds to a gapless hydrodynamic ex- citation acting mainly in the helical and vector sectors of charge densities. In the next order in the series over the inverse temperature, this wave receives a slight admixture of the axial charge (3.22), indicating that all components of the fluid, represented by the vector, helical and axial charges, fluctuate coherently as the wave propagates as it is il- lustrated in Figs. 6(a) and 6(b). This next-order-correction gives us also a qualitatively
mode $\omega_0 = 0$ as follows:

$$ \delta\mu_V^0 = 0, \qquad \delta\mu_{\tilde{\chi}}^0 = -\frac{A+s_V B}{A-s_V B} \delta\mu_{\chi}^{\text{im.}} \approx -\frac{2\pi^2}{3|\alpha_V|} e^{|\alpha_V|} \delta\mu_{\chi}^0. \qquad (5.18) $$

Clearly, exciting the $\delta\mu_{\chi}^0$ amplitude leads to an exponentially-larger amplitude $\delta\mu_{\tilde{\chi}}^0$.

In the case of the non-trivial (helical) modes, when $\omega_h^\pm$ is given by Eq. (4.20), we can take the amplitude of the vector charge fluctuation, $\delta\mu_V^{h;\pm}$, as the reference amplitude. The amplitudes $\delta\mu_{\tilde{\chi}}^{h;\pm}$ and $\delta\mu_{\tilde{\chi}}^{h;\pm}$ are then given by

$$ \begin{aligned} \delta\mu_{\chi}^{h;\pm} &= \frac{k\Omega T}{2\omega_h^{\pm}} \frac{A+s_V B}{\sigma_A^{\omega} + s_V \sigma_H^{\omega}} \delta\mu_V^{h;\pm} \approx \pm \frac{\alpha_V}{\pi\sqrt{3}} \delta\mu_V^{h;\pm}, \\ \delta\mu_{\tilde{\chi}}^{h;\pm} &= \frac{k\Omega T}{2\omega_h^{\pm}} \frac{A-s_V B}{\sigma_A^{\omega} - s_V \sigma_H^{\omega}} \delta\mu_V^{h;\pm} \approx \pm \frac{s_V \alpha_V^4 \sqrt{3}}{4\pi^3} \delta\mu_V^{h;\pm}, \end{aligned} \qquad (5.19) $$

where we displayed just the leading-order contribution in the degenerate limit. It can be seen that as $|\alpha_V| \to \infty$, a small excitation in $\delta\mu_V$ in either modes $\delta\mu_V^{h;\pm}$ will lead to excitations in $\delta\mu_{\tilde{\chi}}^{h;\pm}$ that are $\alpha_V^4$ times larger, signalling the breakdown of the linear perturbations ansatz in the degenerate matter. Coming back to Fig. 5, considering the ratio $\alpha_V = \mu_V/T = 20$, our estimate above indicates $\delta\mu_{\tilde{\chi}}^{h;\pm}/\delta\mu_{V;0} \approx 1117$ to leading order in $\alpha_V$, which is in line with the results shown in the right panel of this figure.

The above discussion indicates that degenerate matter under rotation is unstable under small perturbations. A small excitation of the vector chemical potential induces huge excitations in the vector and helical chemical potentials, invalidating the linear regime ansatz of this analysis. One may question the physical soundness of these results. In our companion paper [26], we show that in a realistic plasma, the non-conservation of the helicity charge prevents the axial and helical mode amplitudes from growing, leading to perfectly reasonable propagation properties.

# 6 Summary and Conclusions

In our work, we challenge the traditional approach to chiral systems, which presumes that the chiral fluids are completely described only by the pair of the vector and axial local charges. We show that the inclusion of the helical degree of freedom enriches the hydrodynam- ical spectrum of the system. Since we explored various regimes of the vector-axial-helical triad of charges, it is worth summarizing the main features of our findings in a compact set of plots shown in Fig. 6.

In the high-temperature limit of low-density plasmas, where the temperature signif- icantly exceeds the magnitude of any background chemical potential, we confirmed the existence of the Helical Vortical Wave, which corresponds to a gapless hydrodynamic ex- citation acting mainly in the helical and vector sectors of charge densities. In the next order in the series over the inverse temperature, this wave receives a slight admixture of the axial charge (3.22), indicating that all components of the fluid, represented by the vector, helical and axial charges, fluctuate coherently as the wave propagates as it is il- lustrated in Figs. 6(a) and 6(b). This next-order-correction gives us also a qualitatively
<figure><img src="image_9.png" /><figcaption>**Figure 6:** A summarising illustration of four classes of gapless hydrodynamic excitations discussed in the paper for the hierarchy $\mu_V > \mu_H > 0$ and $\mu_A > 0$ of the background chemical potentials. For simplicity, a gapless sinusoidal wave is shown only. A detailed description is given in the text.</figcaption></figure>

unusual result implying the non-reciprocity of these hydrodynamic excitations: the helical vortical waves propagating along and opposite to the direction of vorticity possess different velocities (3.16).

Our approach also allows us to uncover yet another gapless wave in the system: the Axial Vortical Wave. According to Eq. (3.29), this wave propagates only in the presence of the axial chemical potential $\mu_A$, since its velocity vanishes if $\mu_A = 0$. In the case when the other two charge densities are vanishing, the wave exhibits oscillations in the axial charge density, which propagate unidirectionally along (opposite to) the vorticity vector $\Omega$ for $\mu_A < 0$ ($\mu_A > 0$). In the presence of the background of vector (helical) charges, the purely axial wave is also accompanied by coherent fluctuations in axial and helical (or vector) charges (3.28), as it is illustrated in Fig. 6(c).

It is important to stress that the appearance of unidirectional transport in this simple homogeneous system is a fascinating and unexpected phenomenon that is distinct from the processes that usually occur in high-temperature plasmas. For example, in most classical acoustics and hydrodynamics, sound waves typically exhibit reciprocal propagation, meaning that the waves can travel forward and backwards along the same path without change in their propagation characteristics. On the contrary, the unidirectionally, seen from the point of view of ordinary acoustical waves, emerges as a result of the meticulously engineered breaking of time-reversal symmetry in the medium through which sound waves propagate. For example, in systems that break the time-reversal symmetry, sound waves can be en-
<figure><img src="image_9.png" /><figcaption>**Figure 6:** A summarising illustration of four classes of gapless hydrodynamic excitations discussed in the paper for the hierarchy $\mu_V > \mu_H > 0$ and $\mu_A > 0$ of the background chemical potentials. For simplicity, a gapless sinusoidal wave is shown only. A detailed description is given in the text.</figcaption></figure>

unusual result implying the non-reciprocity of these hydrodynamic excitations: the helical vortical waves propagating along and opposite to the direction of vorticity possess different velocities (3.16).

Our approach also allows us to uncover yet another gapless wave in the system: the Axial Vortical Wave. According to Eq. (3.29), this wave propagates only in the presence of the axial chemical potential $\mu_A$, since its velocity vanishes if $\mu_A = 0$. In the case when the other two charge densities are vanishing, the wave exhibits oscillations in the axial charge density, which propagate unidirectionally along (opposite to) the vorticity vector $\Omega$ for $\mu_A < 0$ ($\mu_A > 0$). In the presence of the background of vector (helical) charges, the purely axial wave is also accompanied by coherent fluctuations in axial and helical (or vector) charges (3.28), as it is illustrated in Fig. 6(c).

It is important to stress that the appearance of unidirectional transport in this simple homogeneous system is a fascinating and unexpected phenomenon that is distinct from the processes that usually occur in high-temperature plasmas. For example, in most classical acoustics and hydrodynamics, sound waves typically exhibit reciprocal propagation, meaning that the waves can travel forward and backwards along the same path without change in their propagation characteristics. On the contrary, the unidirectionally, seen from the point of view of ordinary acoustical waves, emerges as a result of the meticulously engineered breaking of time-reversal symmetry in the medium through which sound waves propagate. For example, in systems that break the time-reversal symmetry, sound waves can be en-
gineered to propagate in one direction while being significantly attenuated in the opposite
direction [50].

The theoretical basis for unidirectional sound transport can, for example, be linked to
the concept of topological insulators that represent materials that maintain propagating
excitations only along their surface and not through their bulk [51]. Analogously, the
unidirectional transport of sound waves has been realized through specific mechanisms,
such as using exotic active materials with time-varying properties, creating asymmetric
structures that induce directional bias, or employing magnetic fields in conjunction with
magneto-acoustic materials [52].

On the practical level of acoustic (or, gapless, in a more generic sense) applications,
the unidirectional transport includes acoustic diodes or rectifiers, which allow sound to pass
through in one direction while blocking it in the opposite direction [53]. These devices could
be used in noise reduction systems, acoustic logic devices, and improved ultrasonic imaging
technologies [54]. Unidirectional sound propagation is an unusual phenomenon, and its
appearance in a chiral plasma represents a fascinating, unexpected effect of an interplay of
anomalous transport effects.

Coming back to the field-theoretical applications in chiral plasmas, we highlight our
observation that at high densities, both the helical and the axial waves merge into a common
hydrodynamic excitation, the Axial-Helical Vortical Wave. In this limit of the degenerate
Dirac fluid, the magnitudes of helical and axial charge densities of the ensemble of particles
are indistinguishable from each other since the helical and axial charges of a given particle
(or anti-particle) are the same up to a sign. The Axial-Helical Vortical Wave maintains the
property of non-reciprocity reflected in the velocities of its branches (5.12). Interestingly,
there exists a critical temperature (5.14), at which one of the branches of this mixed wave
stops propagating while the other one moves parallel to the direction of the vorticity vector.
The mixed axial-helical vortical wave is illustrated in Figs. 6(d) and 6(e).

The discussion in this paper focussed on the simplified model where both the axial
and the helical charges are conserved. As already discussed in Ref. [15], the helicity charge
dissipates in a realistic plasma due to helicity-violation pair annihilation (HVPA) processes.
Furthermore, the conservation of the axial charge is violated due to interactions via the
axial anomaly [48]. The latter is dynamically relevant for the crossover region of finite-
temperature QCD [23]. For the sake of completeness, we show in Figs. 6(f) and 6(g)
the chiral vortical wave [2] which emerges in the limit where the helical degree of freedom
is frozen due to strong relaxation [26]. We explore the consequences of the axial and
helical charge non-conservation on the excitations spectrum revealed here in the companion
paper [26].

**Acknowledgments.** This work is supported by the European Union - NextGenerationEU through grant No. 760079/23.05.2023, funded by the Romanian ministry of research, innovation and digitalization through Romania's National Recovery and Resilience Plan, call no. PNRR-III-C9-2022-I8. V.E.A. gratefully acknowledges partial support through a grant of the Ministry of Research, Innovation and Digitization, CNCS - UEFISCDI, project number PN-III-P1-1.1-TE-2021-1707, within PNCDI III.
gineered to propagate in one direction while being significantly attenuated in the opposite direction [50].

The theoretical basis for unidirectional sound transport can, for example, be linked to the concept of topological insulators that represent materials that maintain propagating excitations only along their surface and not through their bulk [51]. Analogously, the unidirectional transport of sound waves has been realized through specific mechanisms, such as using exotic active materials with time-varying properties, creating asymmetric structures that induce directional bias, or employing magnetic fields in conjunction with magneto-acoustic materials [52].

On the practical level of acoustic (or, gapless, in a more generic sense) applications, the unidirectional transport includes acoustic diodes or rectifiers, which allow sound to pass through in one direction while blocking it in the opposite direction [53]. These devices could be used in noise reduction systems, acoustic logic devices, and improved ultrasonic imaging technologies [54]. Unidirectional sound propagation is an unusual phenomenon, and its appearance in a chiral plasma represents a fascinating, unexpected effect of an interplay of anomalous transport effects.

Coming back to the field-theoretical applications in chiral plasmas, we highlight our observation that at high densities, both the helical and the axial waves merge into a common hydrodynamic excitation, the Axial-Helical Vortical Wave. In this limit of the degenerate Dirac fluid, the magnitudes of helical and axial charge densities of the ensemble of particles are indistinguishable from each other since the helical and axial charges of a given particle (or anti-particle) are the same up to a sign. The Axial-Helical Vortical Wave maintains the property of non-reciprocity reflected in the velocities of its branches (5.12). Interestingly, there exists a critical temperature (5.14), at which one of the branches of this mixed wave stops propagating while the other one moves parallel to the direction of the vorticity vector. The mixed axial-helical vortical wave is illustrated in Figs. 6(d) and 6(e).

The discussion in this paper focussed on the simplified model where both the axial and the helical charges are conserved. As already discussed in Ref. [15], the helicity charge dissipates in a realistic plasma due to helicity-violation pair annihilation (HVPA) processes. Furthermore, the conservation of the axial charge is violated due to interactions via the axial anomaly [48]. The latter is dynamically relevant for the crossover region of finite-temperature QCD [23]. For the sake of completeness, we show in Figs. 6(f) and 6(g) the chiral vortical wave [2] which emerges in the limit where the helical degree of freedom is frozen due to strong relaxation [26]. We explore the consequences of the axial and helical charge non-conservation on the excitations spectrum revealed here in the companion paper [26].

**Acknowledgments.** This work is supported by the European Union - NextGenerationEU through grant No. 760079/23.05.2023, funded by the Romanian ministry of research, innovation and digitalization through Romania's National Recovery and Resilience Plan, call no. PNRR-III-C9-2022-I8. V.E.A. gratefully acknowledges partial support through a grant of the Ministry of Research, Innovation and Digitization, CNCS - UEFISCDI, project number PN-III-P1-1.1-TE-2021-1707, within PNCDI III.
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