The displacement-current term $\omega^{2}/c^{2}$ in Eq. (142), contained in
the integrand at the denominator, is essentially responsible for the
development of the transparent regime (region \normalsizeC⃝ in Fig. 2), and it
is negligible for frequencies below the plasma edge [50, 51, 55, 53]. Hence,
if we analyze the impedance for for $\omega<\omega_{p}$, we can neglect the
displacement-current term in Eq. (142), with the result

$Z(\omega)=i\frac{Z_{0}}{\pi}\frac{\omega}{\omega_{p}}\int_{-\infty}^{+\infty}dz\frac{1}{S_{yy}(z,\omega)-z^{2}}.$
(144)

In the following, we will employ the full Eq. (142) for numerical computations
of the impedance “phase diagrams”, while we will revert to Eq. (144) without
the displacement current, to derive analytical results for the boundaries
between different skin effect regimes. In any case, the conductivity that
enters into the momentum-space integral for the impedance is given by Eq. (1)
in general. The 2D isotropic limit, where the conductivity follows Eq. (84),
is analyzed for reference in Appendix B.1. We checked that our results for the
impedance modulus $|Z(\omega)|$ are qualitatively robust with respect to the
choice of interface boundary conditions: the difference between specular and
diffusive surface scattering is negligible for all Fermi-surface geometries
here considered. Boundary conditions do significantly influence the surface
resistance (real part of the impedance, $\mathrm{Re}Z(\omega)$) in the
relaxation regime \normalsizeB⃝ and in the extreme anomalous regime
\normalsizeE⃝ indicated in Fig. 2. We postpone the detailed analysis of this
sensitivity to different boundary conditions to a subsequent work; see also
the discussion in Sec. VII.3.

In what follows, we will discuss the dependence of the surface impedance,
stemming from Eqs. (144) and (143), on frequency, scattering rates and
orientation angle $\theta$, for the anisotropic 2D geometries of hexagonal and
square Fermi surfaces, at fixed Fermi velocity $v_{F}$. As we will shortly
appreciate, all numerical and analytical results for the impedance confirm the
qualitative “phase diagrams” sketched in Figs. 3, 4, 5, and 6, and agree with
the discussion anticipated in Sec. II. In particular, let us stress that the
anomalous regime is the most sensitive to anisotropy: while all other forms of
skin effect depend quantitatively but not qualitatively on Fermi-surface
geometry, the impedance in anomalous regime changes from
$\left|Z(\omega)\right|\propto\omega^{2/3}$ in isotropic systems [48, 54], to
different power laws. For example, for the geometries of Figs. 5 and 6,
$\left|Z(\omega)\right|\propto\omega^{1/2}$ for the hexagonal and square
geometries in the presence of scattering. This remarkable difference
highlights that the $\sqrt{\omega}$ behavior of the impedance, usually
associated with normal skin effect, is also encountered in anomalous regime in
anisotropic systems. Such peculiar power-law further changes in the
scattering-less limit to a perfect-conductor frequency-independent behavior.
To identify the exponent $\eta$ such that
$\left|Z(\omega)\right|\propto\omega^{\eta}$, we will follow Eq. (4), which
shows that $\eta$ can be equivalently extracted from
$\mathrm{Arg}Z(\omega)/(-\pi/2)$; see also Sec. VII.1. This way, in the
following we will span the skin-effect parameter space as a function of
$\omega$, $\gamma_{\rm mc}$, and $\gamma_{\rm mr}$.

###  VI.1 Hexagonal Fermi surface

To obtain the surface impedance for an hexagonal Fermi surface, we use Eqs.
(144) and (143) in conjunction with the conductivity (92) or (102), for the
“parallel” configuration $\phi=0$ and for generic rotation angle $\phi$
respectively.

Numerical results for the impedance modulus $\left|Z(\omega)\right|$, divided
by the vacuum impedance $Z_{0}$ and by $(\omega/\omega_{p})^{1/2}$ for visual
clarity, are displayed as a function of $\omega/\omega_{p}$ by the solid red
curves in Fig. 9. We use the parameters $v_{F}/c=0.0025$ (corresponding to the
Fermi velocity of PdCoO2 [65, 95]), $\gamma_{\rm mr}=10^{-10}\omega_{p}$, and
$\gamma_{\rm mc}=10^{-6}\omega_{p}$. Figs. 9(a) and 9(b) show the results for
$\phi=0$ and $\phi=\pi/2$, respectively.

Figure 9:  Absolute value of the surface impedance $\left|Z(\omega)\right|$,
divided by the vacuum impedance $Z_{0}$ and by $(\omega/\omega_{p})^{1/2}$, as
a function of $\omega/\omega_{p}$, for a hexagonal Fermi surface. We employ
the parameters $v_{F}/c=0.0025$, and $\gamma_{\rm mr}=10^{-4}\gamma_{\rm
mc}=10^{-6}\omega_{p}$. Dashed curves show analytical results valid in each
regime, derived in Appendix B.2. The yellow-shaded area gives a qualitative
estimation of the parameter space accessible to experiments [55]. (a)
“Parallel” configuration with rotation angle $\phi=0$. (b) Configuration with
rotation angle $\phi=\pi/2$.  Figure 10:  Orientational dependence of skin
effect regimes for a hexagonal Fermi surface, as measured by the surface
impedance $Z(\omega)$ as a function of relaxation rate $\gamma_{\rm mr}$,
momentum-conserving collision rate $\gamma_{\rm mr}$, and frequency
$\omega/\omega_{p}$, where $\omega_{p}$ is the plasma frequency (78). The
corresponding orientation is sketched on the left-hand side of each plot,
together with the applied electric field $\bm{E}=E_{y}\hat{u}_{y}$ aligned
with the $y$ axis. Red arrows depict the local Fermi velocity vectors. (a)
“Parallel” configuration in the $\left(\gamma_{\rm mr},\omega\right)$ plane,
for $\gamma_{\rm mr}=\gamma_{\rm mc}$. (b) “Parallel” configuration in the
$\left(\gamma_{\rm mc},\omega\right)$ plane, for fixed $\gamma_{\rm
mr}=10^{-8}\omega_{p}$. (c) Fermi surface rotated by $\phi=\pi/2$ with respect
to panels (a) and (b), in the $\left(\gamma_{\rm mr},\omega\right)$ plane, for
$\gamma_{\rm mr}=\gamma_{\rm mc}$. (d) Fermi surface rotated by $\phi=\pi/2$
in the $\left(\gamma_{\rm mc},\omega\right)$ plane, for fixed $\gamma_{\rm
mr}=10^{-8}\omega_{p}$. The color palette is the density plot of
$\mathrm{Arg}Z(\omega)/(-\pi/2)$, giving the exponent $\eta$ of
$Z(\omega)\propto\omega^{\eta}$. Dashed lines are the analytical crossover
boundaries derived in Appendix B.2.5.

Analytical expressions, collected in Appendix B.2, are available in each
regime of skin effect, and are displayed as dashed lines in Fig. 9.
Specifically: dashed blue lines show normal skin effect, given by Eq. (178)
for $\phi=0$ and $\phi=\pi/2$; dashed orange lines display viscous skin
effect, given by Eq. (183) for $\phi=0$; dashed green lines show anomalous
skin effect, according to Eq. (179) (Eq. (181)) for $\phi=0$ ($\phi=\pi/2$);
light-blue dashed lines showing the perfect-conductivity result, common to
both orientations. In Fig. 9(b), we also barely distinguish an intermediate
frequency range between the anomalous and perfect-conductivity regimes, where
the scattering rates are negligible in comparison with frequency, but still
$\omega<v_{F}\left|q_{x}\right|$. In this frequency window the impedance is
$Z(\omega)=-i\mu_{0}\sqrt{3/2}\lambda_{L}\omega$, in accordance with Eq.
(177).

The intersection points of the analytical results (dashed lines in Fig. 9)
provide a natural criterion to identify the crossover frequency between
different skin effect regimes. Such intersections are highlighted by the
green, blue, and red dots in Fig. 9. The resulting respective crossover
frequencies $\omega_{nv}$, $\omega_{va}$ and $\omega_{ap}$ between
normal/viscous, viscous/anomalous, and anomalous/perfect-conductor skin
effect, are functions of $\gamma_{\rm mr}$, $\gamma_{\rm mc}$ and $v_{F}$ in
general, and their analytical expressions are reported in Appendix B.2,
together with a viscous/perfect conductor boundary $\omega_{vp}$ not visible
in Fig. 9. These functions generate the dashed curves in the “phase diagrams”
presented in Fig. 10(a)-(d).

The density plots of $\mathrm{Arg}Z(\omega)/(-\pi/2)$ in Fig. 10 stem from Eq.
(142) (including the displacement current), and hence they include the
transparent regime (light blue-shaded area) found for $\omega\geq\omega_{p}$;
see also region \normalsizeC⃝ in Fig. 2. Notice how the extension of each skin
effect region in Fig. 10 depends on orientation, and excellently agrees with
the qualitative discussion of Figs. 4 and 5 in Sec. II. In particular, for
$\phi=0$ the viscous regime extends throughout regions \smallD$_1$⃝ and
\normalsizeV⃝, and it is present even for $\gamma_{\rm mr}=\gamma_{\rm mc}$;
cfr. Figs. 10(a),(b) and 4. For $\phi=\pi/2$ and $\gamma_{\rm mr}=\gamma_{\rm
mc}$, the viscous regime (green-shaded region) shrinks and there is a direct
crossover from normal to anomalous skin effect, which share the same power law
$\left|Z(\omega)\right|\propto\omega^{1/2}$ but with different numerical
prefactors. This is fully consistent with the difference between regions
\normalsizeA⃝ and \smallD$_1$⃝ in Fig. 5.

In the absence of scattering rates, a qualitative difference in anomalous
regime appears, as shown in Appendix B.2.1: while for $\phi=0$ the impedance
still follows $\left|Z(\omega)\right|\propto\omega^{1/2}$, for $\phi=\pi/2$
the impedance is purely imaginary and
$\left|Z(\omega)\right|\propto\sqrt{3/2}\lambda_{L}\omega$. This aspect
explains the distinction between regions \smallD$_1$⃝ and \smallD$_2$⃝ in Fig.
5. Hence, the anomalous impedance is very sensitive to the presence or absence
of scattering, either momentum-conserving or relaxing. We emphasize that this
contrasts with the isotropic case: for circular, spherical, or spheroidal
Fermi surfaces, the regions \smallD$_1$⃝ and \smallD$_2$⃝ are
indistinguishable [55, 85]; see Fig. 3.

Another way to recognize the occurrence of anomalous forms of skin effect is
to plot the inverse surface resistance $1/\mathrm{Re}Z(\omega)$ as a function
of inverse momentum-relaxation rate $\omega_{p}/\gamma_{\rm mr}$, at fixed
frequency $\omega$, as traditionally performed in the analysis of skin effect
after Pippard [57, 58, 48, 54]. Since $\gamma_{\rm mr}$ is an increasing
function of temperature $T$ (e.g., $\gamma_{\rm mr}\propto T^{2}$ for Fermi-
liquid electron-electron interactions), these graphs can be interpreted as the
evolution of the impedance with decreasing $T$. Figs. 11(a),(b) show our
numerical results for $1/\mathrm{Re}Z(\omega)$ in the hexagonal Fermi-surface
geometry, for $\phi=0$ and $\phi=\pi/2$ respectively, which correspond to the
geometries shown in Fig. 7(a),(b).

Figure 11:  Inverse surface resistance $1/\mathrm{Re}Z(\omega)$ normalized to
the vacuum-impedance $Z_{0}$, as a function of inverse momentum-relaxation
rate $\omega_{p}/\gamma_{\rm mr}$, for a hexagonal Fermi surface with Fermi-
velocity modulus $v_{F}=2.5\times 10^{-3}c$, $\omega/\omega_{p}=10^{-4}$, and
$\gamma_{\rm mc}=\mathrm{max}\left\\{\gamma_{\rm mr},\bar{\gamma}\right\\}$,
with $\bar{\gamma}=\left\\{10^{-5},1,10^{3}\right\\}$ for the blue, red, and
gold curves respectively. Specular boundary conditions are assumed, according
to Eq. (144). The dashed gray lines represent the limit $\gamma_{\rm
mr}=\gamma_{\rm mc}=0$. (a) Results from Eq. (92), for tilting angle $\phi=0$
with respect to the surface; see Fig. 7(a). The scattering-less limit is given
by Eq. (173). (b) Results from Eqs. (101) and (102), for $\phi=\pi/2$; see
Fig. 7(b). The scattering-less limit stems from Eq. (176).

Since we expect $\gamma_{\rm mr}\approx\gamma_{\rm mc}$ at high $T$, while
$\gamma_{\rm mr}\ll\gamma_{\rm mc}$ at low $T$, we have taken $\gamma_{\rm
mc}=\mathrm{max}\left\\{\gamma_{\rm mr},\bar{\gamma}\right\\}$, with
$\bar{\gamma}=\left\\{10^{-5},1,10^{3}\right\\}$ for the blue, red, and gold
curves in each panel respectively. We see the characteristic saturation of the
inverse surface resistance, the value of which parametrically depends on
$\gamma_{\rm mc}$ and on Fermi-surface orientation parametrized by the angle
$\phi$. In Reuter and Sondheimer theory of isotropic systems, such saturation
is associated with anomalous skin effect. As we include momentum-conserving
collisions as well, Fig. 11 shows that the asymptotic saturation value depends
on $\gamma_{\rm mc}$, which corresponds to either viscous or anomalous skin
effect, depending on whether $\omega\ll\gamma_{\rm mc}$ or
$\omega\gg\gamma_{\rm mc}$ respectively. The specific saturation value for a
given $\gamma_{\rm mc}$ is influenced by Fermi-surface orientation, as shown
by comparing curves of the same color in Figs. 11(a),(b).

Hence, the relaxationless saturation of the surface resistance in hydrodynamic
regime $\omega\ll\gamma_{\rm mc}$ constitutes evidence for viscous skin
effect, governed by momentum-conserving collisions, and quantitatively
influenced by the orientation of the anisotropic Fermi surface. Analogous
conclusions hold for a square Fermi-surface geometry, as detailed in the next
section.

###  VI.2 Square Fermi surface

For a square Fermi surface, the surface impedance follows from Eqs. (144) and
(143), and from the conductivity (110) or (102), for the “diamond-shaped”
configuration $\phi=0$ and for generic rotation angle $\phi$ respectively.

Numerical calculations for the impedance modulus $\left|Z(\omega)\right|$,
divided by the vacuum impedance $Z_{0}$ and by $(\omega/\omega_{p})^{1/2}$,
are displayed as a function of $\omega/\omega_{p}$ by the solid red curves in
Fig. 12. All parameters are the same as the ones employed for Fig. 9. Figs.
12(a) and 12(b) display the calculations for $\phi=0$ and $\phi=\pi/4$,
respectively, and the two configurations are visually represented in Fig.
8(a),(b).

Figure 12:  Absolute value of the surface impedance $\left|Z(\omega)\right|$,
divided by the vacuum impedance $Z_{0}$ and by $(\omega/\omega_{p})^{1/2}$, as
a function of $\omega/\omega_{p}$, for a square Fermi surface. We use the
parameters $v_{F}/c=0.0025$, and $\gamma_{\rm mr}=10^{-4}\gamma_{\rm
mc}=10^{-6}\omega_{p}$. Dashed curves show analytical results valid in each
regime, derived in Appendix B.3. The yellow-shaded area represents a
qualitative estimation of the parameter space accessible to experiments [55].
(a) “Diamond-shaped” configuration with rotation angle $\phi=0$. (b)
Configuration with rotation angle $\phi=\pi/4$.  Figure 13:  Orientational
dependence of skin effect regimes for a square Fermi surface, as measured by
the surface impedance $Z(\omega)$ as a function of relaxation rate
$\gamma_{\rm mr}$, momentum-conserving collision rate $\gamma_{\rm mr}$, and
frequency $\omega/\omega_{p}$, where $\omega_{p}$ is the plasma frequency
(78). The corresponding orientation is sketched on the left-hand side of each
plot, together with the applied electric field $\bm{E}=E_{y}\hat{u}_{y}$
aligned with the $y$ axis. Red arrows depict the local Fermi velocity vectors.
(a) “Diamond-shaped” configuration in the $\left(\gamma_{\rm
mr},\omega\right)$ plane, for $\gamma_{\rm mr}=\gamma_{\rm mc}$. (b) “Diamond-
shaped” configuration in the $\left(\gamma_{\rm mc},\omega\right)$ plane, for
fixed $\gamma_{\rm mr}=10^{-8}\omega_{p}$. (c) Fermi surface rotated by
$\phi=\pi/4$ with respect to panels (a) and (b), in the $\left(\gamma_{\rm
mr},\omega\right)$ plane, for $\gamma_{\rm mr}=\gamma_{\rm mc}$. (d) Fermi
surface rotated by $\phi=\pi/4$ in the $\left(\gamma_{\rm mc},\omega\right)$
plane, for fixed $\gamma_{\rm mr}=10^{-8}\omega_{p}$. The color palette is the
density plot of $\mathrm{Arg}Z(\omega)/(-\pi/2)$, giving the exponent $\eta$
of $Z(\omega)\propto\omega^{\eta}$. Dashed lines are the analytical crossover
boundaries derived in Appendix B.3.5.

Appendix B.3 reports analytical expressions in each regime of skin effect,
which are displayed by dashed lines in Fig. 12: dashed blue lines show normal
skin effect, given by Eq. (178) for $\phi=0$ and $\phi=\pi/4$; the dashed
orange line displays viscous skin effect, given by Eq. (198) for $\phi=0$; the
dashed green line shows anomalous skin effect, according to Eq. (197) for
$\phi=0$; light-blue dashed lines showing the perfect-conductivity result,
which is equal for both orientations.

Analogously to the hexagonal case, we define the crossover frequencies between
all skin effect regimes, as the intersection points between different
analytical limits (dashed lines in Fig. 12). These intersections are marked by
the green, blue, and red dots in Fig. 12, and correspond to the crossover
frequencies $\omega_{nv}$, $\omega_{va}$ and $\omega_{ap}$ between
normal/viscous, viscous/anomalous, and anomalous/perfect-conductor skin
effect. Together with a viscous/perfect conductor crossover $\omega_{vp}$, not
visible in Fig. 12, the functions $\omega_{nv}$, $\omega_{va}$ and
$\omega_{ap}$ produce the dashed curves in the impedance “phase diagrams” of
Fig. 13.

As for Fig. 10, the density plots of $\mathrm{Arg}Z(\omega)/(-\pi/2)$ in Fig.
13 stem from Eq. (142) and include the effect of the displacement current. The
qualitative behavior of $\left|Z(\omega)\right|$ in each regime is similar for
Figs. 10(a),(b) and 13(a),(b). However, Figs. 10(c),(d) and 13(c),(d) are
radically different, as the former includes effects of non-locality due to the
momentum-dependent conductivity, while the latter is equivalent to the “phase
diagram” of the local Drude model. The lack of non-locality is due to the
large Fermi-surface portions with velocity parallel to the surface and the
electric field, as described in Sec. II. Overall, the boundaries between
different regimes in Figs. 10 and 13 are consistent with the qualitative
“phase diagrams” in Figs. 4, 5, and 6.

Figure 14:  Inverse surface resistance $1/\mathrm{Re}Z(\omega)$ normalized to
the vacuum-impedance $Z_{0}$, as a function inverse momentum-relaxation rate
$\omega_{p}/\gamma_{\rm mr}$, for a square Fermi surface with Fermi-velocity
modulus $v_{F}=2.5\times 10^{-3}c$, $\omega/\omega_{p}=10^{-4}$, and
$\gamma_{\rm mc}=\mathrm{max}\left\\{\gamma_{\rm mr},\bar{\gamma}\right\\}$,
with $\bar{\gamma}=\left\\{10^{-5},1,10^{3}\right\\}$ for the blue, red, and
gold curves respectively. Specular boundary conditions are assumed, according
to Eq. (144). The dashed gray lines represent the limit $\gamma_{\rm
mr}=\gamma_{\rm mc}=0$. (a) Results from Eq. (110), for tilting angle $\phi=0$
with respect to the surface; see Fig. 8(a). The scattering-less limit is given
by Eq. (195). (b) Results from Eqs. (109) and (102), for $\phi=\pi/4$; see
Fig. 8(b). The scattering-less limit is purely imaginary, as it follows from
Eq. (175).

In the scattering-less limit, the frequency dependence in anomalous regime is
altered, as demonstrated analytically in Appendix B.3.1: for $\phi=0$ the
impedance obeys $\left|Z(\omega)\right|\propto\omega^{1/2}$ as with
$\gamma_{\rm mc}>0$, although with a different prefactor. This shows that,
similarly to the hexagonal case of Sec. VI.1, scattering (either momentum-
conserving or relaxing) qualitatively influences the impedance in anomalous
regime. This feature reflects the crossover between regions \smallD$_1$⃝ and
\smallD$_2$⃝ in Fig. 5.

The presence of anomalous and viscous skin effect for the square Fermi-surface
shape with $\phi=0$ can also be deduced by the saturation of the inverse
surface resistance $1/\mathrm{Re}Z(\omega)$ as a function of
$\omega_{p}/\gamma_{\rm mr}$, at fixed frequency $\omega$, as previously shown
for the hexagonal case in Fig. 11. Our numerical results for
$1/\mathrm{Re}Z(\omega)$ are reported in Figs. 14(a),(b), for orientations
$\phi=0$ and $\phi=\pi/4$ of the square Fermi surface respectively: these
configurations correspond to Fig. 8(a),(b). For $\phi=0$, the results are very
similar for the square and hexagonal shapes, by comparison between Fig. 14(a)
and Fig. 11(a). The surface resistance saturates to a $\gamma_{\rm
mc}$-dependent value in the limit $\gamma_{\rm mr}\rightarrow 0$, which
signals either anomalous skin effect (for $\omega\gg\gamma_{\rm mc}$) or
viscous skin effect (for $\omega\ll\gamma_{\rm mc}$). A significative
difference emerges in Fig. 14(b), in that the surface resistance vanishes in
the limit $\gamma_{\rm mr}=0$, and it is independent from $\gamma_{\rm mc}$,
due to absence of non-local effects.

##  VII Discussion

The results of Sec. VI quantitatively describe the “phase diagrams” of skin
effect, and corroborate the qualitative interpretation of such diagrams in
terms of characteristic length scales, as discussed in Sec. II. This
consistency demonstrates that the surface impedance is an ideal probe of
spatial non-locality in the electrodynamic response of isotropic and
anisotropic systems. Our collision-operator formalism can be readily
generalized to include the effects of short- and long-ranged interactions
among conducting electrons, and to estimate the influence of different
interface boundary conditions. Below we comment on the extent to which such
generalizations interplay with the relations among flat Fermi-surface
segments, scattering, non-local conductivity, and skin effect. We also offer a
method, based on the “ineffectiveness concept”, to infer the frequency scaling
of the skin depth and impedance from how the conductivity scales with $q_{x}$
and $\omega$ in all regimes discussed in Sec. VI.

###  VII.1 A generalized “ineffectiveness concept” for anisotropic skin effect

The relations between conductivity, skin depth, and impedance in different
anisotropic regimes lend themselves to an intuitive physical interpretation,
which is based on the “ineffectiveness concept” introduced by Pippard to
estimate the crossover between normal and anomalous skin effect in isotropic
systems [57, 58, 55]. The argument is summarized as follows: when the skin
depth $\delta_{s}(\omega)$ is much shorter than the mean free path
$l_{m}=v_{F}\tau_{\rm mr}$, only those electrons that travel approximately
parallel to the sample surface are able to react to the incident electric
field, and they perceive a constant field. Assuming that those electrons lie
within a small angle $\pm\theta_{\rm eff}\approx\beta\delta_{s}(\omega)/l_{m}$
with respect to the normal to the surface, the effective density of reacting
electrons is $n_{\rm eff}=n\beta\delta_{s}(\omega)/l_{m}$, where $n$ is the
physical electron density per unit volume. This can be interpreted as an
effective Ohmic (Drude-like) conductivity $\sigma_{\rm eff}(\omega)=n_{\rm
eff}e^{2}\tau_{\rm mr}/m=n\beta e^{2}n\delta_{s}(\omega)/v_{F}$, which now
depends on the skin depth itself. Alternatively, one can think about
$\sigma_{\rm eff}(\omega)$ as containing an effective scattering time
$\tau_{\rm eff}=\beta\delta_{s}(\omega)/v_{F}$ for electrons. Inserting this
effective conductivity into the expression for normal (Ohmic) skin effect [48,
54, 55]

$\delta_{s}(\omega)=\sqrt{2/\left[\mu_{0}\omega\delta_{s}(\omega)\right]},$
(145)

we obtain a self-consistent relation for the skin depth that leads to
$\delta_{s}(\omega)=\left[2l_{m}/(\mu_{0}\beta\sigma_{0}\omega)\right]^{1/3}$,
with $\sigma_{0}=ne^{2}\tau_{\rm mr}/m$: this is indeed the skin depth in
anomalous regime for isotropic systems. The assumption on which the above line
of thought relies are that the effective conductivity $\sigma_{\rm
eff}(\omega)\propto 1/q_{x}\propto\delta_{s}(\omega)$. Besides, momentum-
conserving collisions $\gamma_{\rm mc}\neq\gamma_{\rm mr}$ are not explicitly
considered, which excludes the occurrence of viscous skin effect. Pippard also
applied his qualitative arguments to anisotropic systems in 2D and 3D [79]
333Through his argument, Pippard realized in the anisotropic case that the
surface resistance is dominated by the portions of the Fermi surface that are
parallel to the applied field [79], in agreement with the present study,
maintaining the above fundamental assumptions, and later Sondheimer
quantitatively confirmed these arguments for ellipsoidal Fermi surfaces [78].

Following the ineffectiveness concept, we can generalize the qualitative
estimations as follows. Suppose that the conductivity evolves as

$\frac{\sigma_{yy}(q_{x},\omega)}{\epsilon_{0}\omega_{p}^{2}}=\frac{\alpha_{1}}{\omega^{\beta}q_{x}^{\alpha}},$
(146)

with $\alpha_{1}$ a numerical factor independent of frequency and momentum,
and $\left\\{\eta,\zeta\right\\}\in\mathbb{Z}$. From the considerations of
Sec. V, there will eventually be a single polariton branch that is damped less
that others at the highest depths into the metal, for which we can employ Eq.
(128): hence, we can estimate $\delta_{s}(\omega)\approx\alpha_{2}/q_{x}$,
with $\alpha_{2}$ a numerical factor. This gives

$\frac{\sigma_{yy}[\delta_{s}(\omega),\omega]}{\epsilon_{0}\omega_{p}^{2}}=\alpha_{3}\frac{\delta_{s}^{\alpha}(\omega)}{\omega^{\beta}},$
(147)

with $\alpha_{3}=\alpha_{1}\alpha_{2}^{-\alpha}$. Identifying Eq. (147) with
the effective Ohmic conductivity to utilize for the ineffectiveness argument,
we equate

$\sigma_{yy}[\delta_{s}(\omega),\omega]\equiv\sigma_{\rm
eff}(\omega)=\epsilon_{0}\omega_{p}^{2}\tau_{\rm eff}(\omega),$ (148)

from which we deduce an effective scattering time $\tau_{\rm
eff}(\omega)=\alpha_{3}\omega^{-\beta}\delta_{s}^{\alpha}(\omega)$, or an
effective carrier density alternatively. Inserting the effective conductivity
(148) into the Ohmic skin-depth expression Eq. (145), we obtain a self-
consistent relation that we can solve for $\delta_{s}(\omega)$:

$\delta_{s}(\omega)=\left(\frac{2}{\alpha_{3}}\right)^{\frac{1}{2+\alpha}}\lambda_{L}^{\frac{2}{2+\alpha}}\omega^{\frac{\beta-1}{\alpha+2}},$
(149)

which links the skin depth with the dependence of the conductivity (146) on
frequency and momentum. The estimation for the impedance then follows from Eq.
(4). The latter relation is derived for specular surface scattering in
Appendix C. The qualitative estimation provided by Eq. (149) can be applied
even when the conductivity has various forms of momentum dependence, which
occurs for the anomalous and viscous regimes, and to both isotropic and
anisotropic cases in different orientation. For instance: in anomalous regime
for isotropic systems we have $\beta=0$ and $\alpha=1$, so that
$\delta_{s}(\omega)\propto\omega^{-1/3}$, consistently with Eq. (164); in the
hexagonal and square geometries with $\phi=0$ we have $\beta=-1$ and
$\alpha=2$, which yields $\delta_{s}(\omega)\propto\omega^{-1/2}$, in
agreement with Eqs. (131) and (137), and their counterparts for $\gamma_{\rm
mc}>0$; in extreme hydrodynamic regime and for isotropic and anisotropic
systems alike, we have $\beta=0$ and $\alpha=2$, so that
$\delta_{s}(\omega)\propto\omega^{-1/4}$, which agrees with Figs. 20, 10 and
13(a),(b). Therefore, the generalization of the ineffectiveness concept to
momentum-dependent conductivities, as here outlined, can help in guessing the
qualitative evolution of the skin depth and the impedance modulus with
frequency, from the expression of the conductivity itself.

###  VII.2 Quasiparticle interactions

The kinetic approach employed in this work is ultimately phenomenological, as
it does not depend on the microscopic origin of momentum-conserving and
relaxing scattering. However, by the same logic, the results here derived can
be applied regardless of the specific mechanisms available for electron
scattering, i.e., to different origins of spatial non-locality, as long as the
kinetic approach remains valid. For example, the awareness that non-local
hydrodynamic behavior can not only emerge from electron-electron interactions,
but also through the interaction of carriers with phonons [99, 100, 17] and
impurities [101, 102] has recently been raised. In all these cases, the
electrodynamic phenomenology can be investigated through the Boltzmann theory
outlined in this paper, in the presence or absence of anisotropy. A
microscopic account, on how distinct forms of electron self-energies from
different interaction channels produce visco-elastic phenomenologies in
electrodynamics, will be reported elsewhere [103].

A natural extension of the present work involves the inclusion of
quasiparticle interactions, both long-ranged (i.e., the Coulomb potential) and
short-ranged (e.g., electron-hole excitations) [104]. In the isotropic 3D
case, this procedure is at the basis of Landau-Silin theory of charged Fermi
liquids [105, 106], where short-ranged interactions are expanded into the
basis of spherical harmonics. Likewise, in isotropic 2D systems the
appropriate expansion basis is given by angular momentum states [104]; see
also Sec. I of the Supplemental Material. Schematically, the inclusion of
interactions modifies the Boltzmann equation (9) to [38, 104]

$\frac{\partial f_{\bm{k}}\left(\bm{r},t\right)}{\partial
t}+\bm{F}_{t}(\bm{r},t)\cdot\mathbf{\nabla}_{\bm{k}}f_{\bm{k}}\left(\bm{r},t\right)+\bm{v}_{\bm{k}}\cdot\mathbf{\nabla}_{\bm{r}}f_{\bm{k}}\left(\bm{r},t\right)\\\
=-\left({\cal C}f\right)_{\bm{k}},$ (150)

where now $\bm{F}_{t}(\bm{r},t)$ contains the contribution of the external
electric field, and of the gradient of the interacting quasiparticle
dispersion:

$\bm{F}_{t}(\bm{r},t)=-e\bm{E}(\bm{r},t)-\nabla_{\bm{r}}\mathscr{E}_{\bm{k}}(\bm{r},t).$
(151)

In turn, the interacting dispersion
$\mathscr{E}_{\bm{k}}(\bm{r},t)=\varepsilon_{\bm{k}}+\varepsilon_{L}\left[\delta
f_{\bm{k}}(\bm{r},t)\right]+\varepsilon_{\bm{k}}^{C}(\bm{r},t)$ contains the
Landau short-ranged interaction $\varepsilon_{L}\left[\delta
f_{\bm{k}}(\bm{r},t)\right]$ and the long-ranged Coulomb term
$\varepsilon_{\bm{k}}^{C}(\bm{r},t)$ in a self-consistent Vlasov approach.
Both interactions have to be expanded in terms of the eigenfunctions of the
collision operator (14), which are known to be
$\xi_{\bm{k},m}=c_{0}e^{im\theta}$ in the isotropic case, but differ in the
presence of anisotropy. Then, from Eq. (151) it is evident that interactions
affect the source term (25) at the right-hand side of the infinite linear
system (26), which modifies the form of collective modes [104] and generates
an effective renormalized Fermi velocity $v_{F}^{\ast}$. We defer the details
of this treatment to a subsequent work. Equally interesting is the case where
interactions stem from an external magnetic field, which is able to generate
novel forms of anomalous skin effect, such as the _anomaly-induced-non-
locality_ due to the chiral anomaly in Weyl semimetals [56].

###  VII.3 Influence of interface boundary conditions

Another crucial aspect of the surface-impedance calculations are the interface
boundary conditions [39, 15, 107, 44, 108]. In Sec. VI, we assumed specular
scattering at the vacuum-metal interface, which is physically similar to a
smooth surface with no-stress boundary conditions. Recently, this type of
boundary conditions were experimentally realized in transport measurements on
GaAs/AlGaAs accumulation-mode heterostructures [109]. To investigate the
influence of a different degree of specularity in interface scattering, one
can proceed similarly to Reuter and Sondheimer [48, 78, 54], by considering
the extreme cases of specular or diffusive surface scattering for electrons in
the metal. An interpolation between these two extremes can be achieved by
assuming a specularity factor $p\in\left[0,1\right]$ for both electric field
and current density, which means that a portion $p$ of scattering is specular,
while a part $1-p$ happens diffusively. In the diffusive-scattering case, the
surface electric field $E_{y}(0^{+},\omega)$ can then be linked to the
conductivity $\sigma_{yy}(q_{x},\omega)$ through a momentum integration over
the half-space occupied by the metal, in a similar way as for the specular-
scattering case of Eq. (141). In Sec. IV of the Supplemental Material we
checked our calculations for the surface impedance modulus
$\left|Z(\omega)\right|$ assuming diffusive scattering ($p=0$), and we found
negligible differences with respect to the specular-scattering case of Sec.
VI. Therefore, the results for the impedance modulus are robust against the
effect of boundary conditions.

However, the surface resistance $\mathrm{Re}Z(\omega)$ is dramatically
sensitive to boundary conditions in the anomalous reflection, or extreme
anomalous, regime (\normalsizeE⃝ in Fig. 2). This sensitivity appears because
the region \normalsizeE⃝ is least affected by bulk scattering, with both
relaxation and momentum-conserving collisions being negligible, so that
interface scattering at the vacuum-metal surface assumes a major role for the
conduction properties [110, 51, 53]. We reserve the detailed comparison of
boundary conditions for the surface resistance to a follow-up work.

###  VII.4 Effect of rounded corners and Fermi-surface curvature

Figure 15:  The “supercircle” Fermi-surface geometry, for different values of
the parameter $p$ which controls the curvature of individual segments. The
“diamond-shaped” Fermi surface of Fig. 8 is retrieved for $p=1$, while for
$p=2$ one retrieves a circular Fermi surface.

In Sec. VI we obtained the impedance “phase diagrams” of Figs. 10 and 13 under
the assumption that the 2D Fermi surface is a perfect regular polygon. While
such geometry allowed us to clearly identify how the different regimes for
$Z(\omega)$ are affected by the anisotropy and orientation of the polygonal
shape, realistic Fermi surfaces possess more complex shapes. In particular,
even when these shapes approximately resemble a polygon, the corresponding
faces are not perfectly flat, and the sharp angles of the idealized polygonal
cases are replaced by rounded corners. For this reason, it is interesting to
investigate the effect of small curvature of Fermi-surface segments and of
rounded corners on the shapes studied in Sec. VI. Do the effects of anisotropy
on the surface impedance persist (at least partially), when we take into
account Fermi-surface non-idealities? We will demonstrate in this section that
the answer is yes, considering specific examples of non-polygonal geometries.

Figure 16:  Skin effect regimes for a “supercircle” Fermi surface (153) with
$p=3/2$, as measured by the surface impedance $Z(\omega)$ as a function of
relaxation rate $\gamma_{\rm mr}$, momentum-conserving collision rate
$\gamma_{\rm mr}$, and frequency $\omega/\omega_{p}$, where $\omega_{p}$ is
the plasma frequency (78). The Fermi-surface geometry is sketched on the left-
hand side of the plot, together with the applied electric field
$\bm{E}=E_{y}\hat{u}_{y}$ aligned with the $y$ axis, and the local Fermi
velocity vectors shown by red arrows. (a) Regimes in the $\left(\gamma_{\rm
mr},\omega\right)$ plane, for $\gamma_{\rm mr}=\gamma_{\rm mc}$. (b) Regimes
in the $\left(\gamma_{\rm mc},\omega\right)$ plane, for fixed $\gamma_{\rm
mr}=10^{-8}\omega_{p}$. The color palette is the density plot of
$\mathrm{Arg}Z(\omega)/(-\pi/2)$, which yields the exponent $\eta$ of
$Z(\omega)\propto\omega^{\eta}$.

To generalize the treatment of Secs. IV-VI, let us consider an arbitrarily
varying Fermi wave vector $k_{F}(\theta)$ in the 2D plane, as a function of
angle $\theta$. Then, in accordance with the parametrization (66), the
components of the unit vector $\hat{n}_{\bm{k}}$ along the $x$ and $y$ axes
are

$n_{\bm{k},x}=N^{-1}\left[\cos\theta
k_{F}(\theta)+\sin\theta\frac{dk_{F}(\theta)}{d\theta}\right],$ (152a)
$n_{\bm{k},y}=N^{-1}\left[\sin\theta
k_{F}(\theta)-\cos\theta\frac{dk_{F}(\theta)}{d\theta}\right],$ (152b)
$N=n_{\bm{k}}=\sqrt{\left[k_{F}(\theta)\right]^{2}+\left[\frac{dk_{F}(\theta)}{d\theta}\right]^{2}},$
(152c)

as shown in Appendix D. Hence, given a parametrization $k_{F}(\theta)$, Eqs.
(152) allow us to compute the transverse conductivity through Eqs. (71) and
(72), as well as the surface impedance through Eqs. (142) and (143). This way,
we can compare the corresponding “phase diagrams” for $Z(\omega)$ with the
ones of the previously considered polygonal shapes. Notice that the
integration line element $dS=n_{\bm{k}}(\theta)d\theta$ varies along the Fermi
surface, in accordance with Eq. (152c). This feature holds for polygonal
geometries as well, but in the present non-ideal case we cannot split the
integration over $dS$ into discrete sums as done in the polygonal case.

As an example, in Sec. VII.4.1 we consider the “supercircle” geometry, in
which a square Fermi surface can be morphed into shapes with nonzero curvature
of segments and rounding of corners by the means of a single parameter $p$.
Our investigation culminates in Sec. VII.4.2 with the analysis of a more
realistic 2D case: the in-plane experimental Fermi surface of PdCoO2, as
parametrized through angle-resolved photoemission and quantum oscillations
measurements [33, 65, 95].

####  VII.4.1 The “supercircle” geometry

The “supercircle”, a special case of Lamé curve or “superellipse” for equal
semiaxis, satisfies the equation
$\left|x\right|^{p}+\left|y\right|^{p}=r^{p}$, $p>0$, for radius $r$. This
equation allows one to smoothly interpolate an astroid (for $p<1$), a
“diamond-shaped” geometry (for $p=1$), a circle (for $p=2$), and a square with
rounded corners (for $p>2$), as a function of a single parameter $p$ which
controls the shape curvature. In our case, the radius is the Fermi wave
vector: $r\equiv k_{F}$. Using polar coordinates $k_{x}=k_{F}\cos\theta$ and
$k_{y}=k_{F}\sin\theta$, the supercircle radius results

$k_{F}=k_{F}(\theta)=\left(\left|\cos\theta\right|^{p}+\left|\sin\theta\right|^{p}\right)^{-1/p}.$
(153)

Fig. 15 shows the Fermi surface corresponding to Eq. (153), for different
values of $p>0$. In Appendix D, we show that the results for the “diamond-
shaped” and circular geometries, presented in Secs. IV.3.1, VI.2 and in Secs.
IV.1, B.1 respectively, are indeed retrieved as special cases of the
parametrization (153).

Figure 17:  Skin effect regimes for a “supercircle” Fermi surface (153) with
$p=10$, as measured by the surface impedance $Z(\omega)$ as a function of
relaxation rate $\gamma_{\rm mr}$, momentum-conserving collision rate
$\gamma_{\rm mr}$, and frequency $\omega/\omega_{p}$, where $\omega_{p}$ is
the plasma frequency (78). The Fermi-surface geometry is sketched on the left-
hand side of the plot, together with the applied electric field
$\bm{E}=E_{y}\hat{u}_{y}$ aligned with the $y$ axis, and the local Fermi
velocity vectors displayed by red arrows. (a) Regimes in the
$\left(\gamma_{\rm mr},\omega\right)$ plane, for $\gamma_{\rm mr}=\gamma_{\rm
mc}$. (b) Regimes in the $\left(\gamma_{\rm mc},\omega\right)$ plane, for
fixed $\gamma_{\rm mr}=10^{-8}\omega_{p}$. The color palette is the density
plot of $\mathrm{Arg}Z(\omega)/(-\pi/2)$, which gives the exponent $\eta$ of
$Z(\omega)\propto\omega^{\eta}$.

Fig. 16 shows the results for the intermediate case $p=3/2$, which corresponds
to a “diamond” shape with additional curvature of the four Fermi-surface
segments. Such curvature is further enhanced for $p>3/2$, until for $p=2$ the
Fermi surface becomes perfectly circular. Hence, the case $1<p<2$ is suitable
to analyze the effect of deforming the polygonal geometries by introducing a
finite curvature into the flat segments. We notice subtle differences with
respect to the “diamond-shaped” dispersion of Fig. 13(a),(b): the exponent
$\eta$ in region \smallD$_1$⃝ becomes slightly smaller than $\eta=3/4$ (for
example, we have $\eta\approx 0.73$ around the center of region \smallD$_1$⃝
in Fig. 16(b)), while in region \smallD$_2$⃝ $\eta$ is slightly larger than
$\eta=1/2$ (specifically, $\eta\approx 0.55$ around the center of region
\smallD$_2$⃝ in Fig. 16(b)). This effect is due to the finite curvature of the
Fermi-surface segments, which also produces a visually distinguishable
difference between regions \smallD$_1$⃝ and \normalsizeV⃝ in Fig. 16(b). Such
differences are gradually enhanced for $3/2<p<2$, until for $p=2$ we retrieve
the isotropic “phase diagram” of Fig. 20, where $\eta=2/3$ in both regions
\smallD$_1$⃝ and \smallD$_2$⃝. Hence, Fig. 16 provides us a practically
relevant information: if Fermi-surface segments are not exactly flat in
practice, then the exponents $\eta$ in the anomalous regions \smallD$_1$⃝ and
\smallD$_2$⃝ quantitatively differ with respect to the ideal flat-segments
case. Still, even for a deformation of the “diamond-shaped” case as large as
$p=3/2$, the “phase diagrams” of Figs. 16 and 13(a),(b) are qualitatively
consistent, and the effect of Fermi-surface anisotropy is robust with respect
to the curvature of Fermi-surface segments. Such robustness may be
rationalized by realizing that a small curvature of segments predominantly
affects the low-frequency part of the “phase diagram”, where the response is
anyway diffusive and local; e.g., in region \normalsizeA⃝ the exponent
$\eta=1/2$ is insensitive to the change in Fermi-surface curvature because the
response is dominated by momentum relaxation at rate $\gamma_{\rm mr}$.
Therefore, we conclude that the “phase diagrams” derived for the polygonal
shapes of Secs. VI.1 and VI.2 provide qualitative guidance on the effect of
dispersion anisotropy on the surface impedance, even in the presence of a
small curvature of Fermi-surface segments, for shapes that can still be
approximated by an ideal polygonal geometry. Quantitative difference with
respect to a perfect polygon emerge at the level of the numerical value of the
exponent $\eta$, especially in the anomalous regions \smallD$_1$⃝ and
\smallD$_2$⃝ which are the most sensitive to Fermi-surface geometry.

Furthermore, to selectively analyze the effect of rounded corners, we show the
results for $p=10$ in Fig. 17: this geometry consist of a square with
additional rounding of corners, and as such it can be directly compared with
Fig. 13(c),(d). The case of a perfect square is retrieved in the limit
$p\rightarrow+\infty$. Here we appreciate a qualitative difference in Fig.
17(b) with respect to Fig. 13(d): the rounding of corners has reintroduced the
viscous regime \normalsizeV⃝ with $\eta=3/4$, which was absent in the
idealized square shape of Fig. 13(d). Moreover, the crossovers between the
viscous, anomalous and perfect-conductor regions are all altered with respect
to the square shape. In fact, the “phase diagram” of Fig. 17 is qualitatively
consistent with the hexagonal case of Fig. 13(c),(d): this is because such
hexagonal geometry shares all its relevant geometrical features with the
“supercircle” for $p=10$: two Fermi-surface segments have velocity essentially
parallel to the applied field – see also Secs. IV.4 and V.3 – while the rest
of the Fermi surface does not have a velocity identically orthogonal to the
field as is the case for a perfect square. Hence, we realize that the perfect
square is an idealized special case, for which the non-local character of the
response vanishes entirely from the “phase diagram” in Fig. 13(c),(d). In
reality, rounded corners of an approximately polygonal geometry qualitatively
impact the exponent $\eta$ of the surface impedance. Still, the “phase
diagram” of Fig. 17 qualitatively differs from the isotropic case of Fig. 20,
which demonstrates that rounded corners do not completely suppress the effect
of Fermi-surface anisotropy on the electrodynamic response.

The idealized polygons of Secs. IV.2 and IV.3 serve as an illustration of the
effect of dispersion anisotropy, as they allow for a clear-cut analysis of the
different regimes of anisotropic skin effect. However, in order to test our
kinetic theory by comparing it with optical spectroscopy data, it is
advantageous to consider more realistic Fermi-surface shapes, which properly
model the experimental dispersion of the material under investigation. For
this reason, in the next section we consider the exemplary case of the
experimental parametrization of the Fermi surface of PdCoO2, which involves
both portions with finite curvature and rounded corners.

####  VII.4.2 The 2D Fermi surface of PdCoO2

Figure 18:  Orientational dependence of skin effect regimes for the 2D
parametrization (154) of the Fermi surface of PdCoO2, as measured by the
surface impedance $Z(\omega)$ as a function of relaxation rate $\gamma_{\rm
mr}$, momentum-conserving collision rate $\gamma_{\rm mr}$, and frequency
$\omega/\omega_{p}$, where $\omega_{p}$ is the plasma frequency (78). The
corresponding orientation is sketched on the left-hand side of each plot,
together with the applied electric field $\bm{E}=E_{y}\hat{u}_{y}$ aligned
with the $y$ axis. Red arrows depict the local Fermi velocity vectors. (a)
Configuration with $\theta_{0}=\pi/2$ in the $\left(\gamma_{\rm
mr},\omega\right)$ plane, for $\gamma_{\rm mr}=\gamma_{\rm mc}$. (b)
Configuration with $\theta_{0}=\pi/2$ in the $\left(\gamma_{\rm
mc},\omega\right)$ plane, for fixed $\gamma_{\rm mr}=10^{-8}\omega_{p}$. (c)
Geometry with $\theta_{0}=0$ in the $\left(\gamma_{\rm mr},\omega\right)$
plane, for $\gamma_{\rm mr}=\gamma_{\rm mc}$. (d) Results for $\theta_{0}=0$
in the $\left(\gamma_{\rm mc},\omega\right)$ plane, for fixed $\gamma_{\rm
mr}=10^{-8}\omega_{p}$. The color palette is the density plot of
$\mathrm{Arg}Z(\omega)/(-\pi/2)$, giving the exponent $\eta$ of
$Z(\omega)\propto\omega^{\eta}$.

The experimental parametrization of the Fermi surface of the delafossite
compound PdCoO2, according to quantum oscillations experiments, shows a
quasi-2D geometry with a slight warping in the direction perpendicular to the
Pd planes [33, 94]. Here we adopt another common 2D parametrization [111]:

$\bm{k}_{F}(\theta)=\rho(\theta-\theta_{0})\left(\cos\theta\hat{u}_{x}+\sin\theta\hat{u_{y}}\right),$
(154a)
$\rho(\theta-\theta_{0})=k_{0}+k_{6}\cos\left[6(\theta-\theta_{0})\right]\\\
+k_{12}\cos\left[12(\theta-\theta_{0})\right].$ (154b)

$\theta$ characterizes the angular dependence of $k_{F}$ in the
$\left(k_{x},k_{y}\right)$ plane, while $\theta_{0}$ takes into account a
possible in-plane rotation of the Fermi surface with respect to the $x$ and
$y$ axes. The angular harmonics are $k_{0}=0.9518{\AA}^{-1}$,
$k_{6}=0.0444{\AA}^{-1}$, and $k_{12}=0.0048{\AA}^{-1}$. Eqs. (154) allows us
to illustrate the effect of a more realistic 2D parametrization on our
impedance “phase diagrams”, while still taking advantage of the simplification
(46) due to the existence of two mirror symmetry planes in reciprocal space. A
more general parametrization is employed in Ref. 94, where we directly compare
our kinetic theory with surface resistance measurements on PdCoO2. In
accordance with the fits of bandstructure calculations to angle-resolved
photoemission data in Ref. 65, we still employ Eq. (66), i.e., we assume that
the Fermi-velocity modulus has no angular dependence and equals
$v_{F}=7.5\times 10^{5}$ m/s [33].

Eqs. (154) produce the Fermi surfaces on the left of Fig. 18: $\theta=\pi/2$
and $\theta=0$ correspond to the sketch at the left of Figs. 18(a) and 18(c),
respectively. The numerical results for the corresponding impedance “phase
diagrams” are shown in Figs. 18(a)-(d), which can be directly compared with
the ideal hexagonal geometry of Fig. 10. As remarked in Sec. VII.4.1 for the
rounded-square shape, the rounded corners produced by Eq. (154) quantitatively
modify the exponent $\eta$ in anomalous regime with respect to the one for a
perfect hexagonal Fermi surface. In particular, the differences between the
orientations of Figs. 18(a),(b) and 18(c),(d) are diminished, with respect to
the sharper differences between Figs. 10(a),(b) and 10(c),(d) for a hexagonal
geometry. Therefore, relaxing the approximation of a perfect hexagonal shape
quantitatively smoothens some of the features resulting from anisotropy, since
the rounded corners and slightly curved segments decrease the overall
anisotropy of the Fermi surface. However, there are still discernible
differences between Figs. 18(a),(b) and 18(c),(d), particularly in the
anomalous regimes \smallD$_1$⃝ and \smallD$_2$⃝, which qualitatively still
resemble the idealized cases of Figs. 10(a),(b) and 10(c),(d). Hence, we
conclude that anisotropic effects in the skin effect are not completely
suppressed, once we model Fermi surfaces that are more realistic than
idealized polygonal geometries. Indeed, surface resistance measurements on
PdCoO2 using a bolometric microwave spectrometer confirm the anisotropy of
skin effect in such compound, in full agreement with our kinetic theory [94].

##  VIII Conclusions

By expanding the collision integral in the eigenbasis of the collision
operator, we solved the Boltzmann equation with an electric-field source term,
and obtained the distribution function and the transverse conductivity for
arbitrary electronic dispersion, under the minimal assumption of two (in 2D)
or three (in 3D) symmetry planes in momentum space, taking into account
momentum-preserving collisions at rate $\gamma_{\rm mc}$ as well as momentum
relaxation at rate $\gamma_{\rm mr}\leq\gamma_{\rm mc}$. The conductivity (1)
is valid for both isotropic and anisotropic Fermi surfaces, which allows us to
treat spherically symmetric and polygonal geometries within the same
formalism. After checking the isotropic case of a circular Fermi surface, we
specialized to the polygonal geometries of a hexagonal and a square Fermi-
surface shape, with isotropic Fermi-velocity modulus. We analyzed the
connection between the conductivity and the skin depth for both geometries,
and their dependence on Fermi-surface orientation with respect to the
direction of the applied field. Assuming specular surface scattering, we
numerically calculated the surface impedance in all regimes of skin effect,
for circular, hexagonal and square geometries, and for different Fermi-surface
orientations for the latter two cases. We also analyzed the impedance in each
skin-effect regime, and traced quantitative boundaries to characterize the
crossovers among the different regimes. Our main results are the impedance
“phase-diagrams” shown in Figs. 20, 10, and 13, where the crossovers between
different skin effect regimes are determined, both numerically and
analytically, as a function of frequency and ratio between momentum-conserving
and relaxing scattering rates. We also interpreted these diagrams, in the
presence or absence of anisotropy, in a unified fashion: the qualitative
boundaries between adjacent regimes are described in terms of ratios between
characteristic lengths for skin effect, collected in Table 1 and visualized in
Fig. 2. This way, we classify the orientational behaviour of anisotropic skin
effect in all regimes in the diagrams 4, 5, and 6, which highlight that skin
effect regimes must be identified within the parameter space set by
$\gamma_{\rm mr}$, $\gamma_{\rm mc}$, and $\omega$.

In classifying the dependence of $\left|Z(\omega)\right|$ on Fermi-surface
geometry, a crucial role is played by the presence of Fermi-surface segments
with velocity aligned (or nearly parallel) to the direction of the incident
electric field. This alignment is capable of altering the frequency scaling
exponent $\eta$ of the skin depth and the impedance, and even suppress
altogether the non-local character (dependence on $q_{x}$) of conduction in
the anisotropic system. In the light of such phenomenon, one can rationalize
the different extension and boundaries among the viscous and anomalous
regimes, i.e. regions \normalsizeV⃝, \smallD$_1$⃝ and \smallD$_2$⃝ in Fig. 2,
for different orientations.

Furthermore, we interpreted the frequency dependence of the skin depth and
impedance, demonstrated in our theory, in terms of a generalized version of
the “ineffectiveness concept” introduced by Pippard, which allows one to infer
the evolution of skin effect with frequency, from the dependence of the
conductivity on frequency and momentum.

The orientational dependence of skin effect, as well as the modifications of
the impedance “phase diagrams” due to anisotropy, persist even if we consider
more complex non-polygonal Fermi-surface geometries, as we demonstrated by
explicitly treating the case of a “supercircle” (in Sec. VII.4.1) and of a
simplified 2D experimental parametrization of the Fermi surface of PdCoO2 [33,
65, 112] (in Sec. VII.4.2). In a companion paper, the full 3D parametrization
for PdCoO2 is employed in our kinetic theory and found in agreement with
surface-resistance microwave measurements [94]. There, it is shown that the
nonzero dispersion perpendicular to the hexagonally coordinated Pd layers has
to be taken into account, to provide an accurate description of the non-local
electrodynamics of PdCoO2. However, our 2D calculations serve as a useful
reference, to estimate to what extent the anisotropic transport in the
intralayer dimension affect skin effect, and to locate in which regime the
optical data lies in the parameter space of Fig. 2. A detailed study of the
impact of different dimensionality on non-local optics in the collision-
operator formalism represents an interesting future development of our theory.

Thus, our theory provides a flexible and compact formalism, which makes
contact with a basic electrodynamic quantity, the surface impedance, that is a
consolidated and accurate method to investigate the spatially non-local
current response. Our methods are practically adaptable to many classes of 2D
materials with anisotropic Fermi surfaces. More generally, our work provides
new guidelines for the theoretical and experimental investigation of the
ballistic, viscous and Ohmic regimes of conduction in novel 2D and 3D
materials, and for the orientational dependence of the optical properties in
anisotropic electronic systems. The potential technological impact of such
guidelines encompasses many future applications of ultra-high-conductivity
materials and integrated circuits operating at GHz frequencies [113].

##  IX Acknowledgments

We are grateful to G. Ghiringhelli, C. Berthod, A. P. Mackenzie, R. Mössner,
G. A. Inkof for insightful discussions. D. V. acknowledges partial support by
the Swiss National Science Foundation (SNSF) through the SNSF Early
Postdoc.Mobility Grant P2GEP2$\\_$181450\. D. V. and J. S. acknowledge partial
support by the European Commission’s Horizon 2020 RISE program Hydrotronics
(Grant No. 873028). G. B. and D. A. B. acknowledge support from the Max
Planck-UBC-UTokyo Center for Quantum Materials and the Canada First Research
Excellence Fund, Quantum Materials and Future Technologies Program, as well as
the Natural Sciences and Engineering Research Council of Canada
(RGPIN-2018-04280).

##  Appendix A Collision operator and eigenmodes in two-times approximation

Given the assumptions (31) for the eigenfunctions $\chi_{\bm{k}}$ and the
eigenvalue equations (19), the collision operator $\hat{C}$ in spatial
dimensions $\left\\{i\right\\}$ can be written as

$\hat{C}=\sum_{m}\left|\chi_{\bm{k},m}\right\rangle\gamma_{m}\left\langle\chi_{\bm{k},m}\right|=\left|\chi_{\bm{k},0}\right\rangle
0\left\langle\chi_{\bm{k},0}\right|\\\ +\gamma_{\rm
mr}\sum_{i}\left|\chi_{\bm{k},i}\right\rangle\left\langle\chi_{\bm{k},i}\right|+\gamma_{\rm
mc}\sum_{\alpha\neq\left\\{0,\left\\{i\right\\}\right\\}}\left|\chi_{\bm{k},\alpha}\right\rangle\left\langle\chi_{\bm{k},\alpha}\right|.$
(155)

From the completeness property (20), we can write

$\sum_{\alpha\neq\left\\{0,\left\\{i\right\\}\right\\}}\left|\chi_{\bm{k},\alpha}\right\rangle\left\langle\chi_{\bm{k},\alpha}\right|=1-\left|\chi_{\bm{k},0}\right\rangle\left\langle\chi_{\bm{k},0}\right|\\\
-\sum_{i}\left|\chi_{\bm{k},i}\right\rangle\left\langle\chi_{\bm{k},i}\right|,$
(156)

which upon insertion in Eq. (155) yields

$\hat{C}=\gamma_{\rm
mc}\left(1-\left|\chi_{\bm{k},0}\right\rangle\left\langle\chi_{\bm{k},0}\right|\right)-\left(\gamma_{\rm
mc}-\gamma_{\rm mr}\right)\times\\\
\sum_{i}\left|\chi_{\bm{k},i}\right\rangle\left\langle\chi_{\bm{k},i}\right|.$
(157)

Acting with the collision operator (157) on a generic eigenfunction
$\left|\phi_{\bm{k}}\right\rangle$ produces

$\hat{C}\left|\phi_{\bm{k}}\right\rangle=\gamma_{\rm
mc}\left|\phi_{\bm{k}}\right\rangle-\gamma_{\rm
mc}\left|\chi_{\bm{k},0}\right\rangle\left\langle\chi_{\bm{k},0}\right|\left.\phi_{\bm{k}}\right\rangle-\left(\gamma_{\rm
mc}-\gamma_{\rm mr}\right)\times\\\
\sum_{i}\left|\chi_{\bm{k},i}\right\rangle\left\langle\chi_{\bm{k},i}\right|\left.\phi_{\bm{k}}\right\rangle.$
(158)

By virtue of the definition of the scalar product (15), Eq. (158) is
equivalent to Eq. (39).

The normalization of the eigenfunction $\chi_{\bm{k},0}$ imposes

$\left\langle\chi_{\bm{k},0}\right|\left.\chi_{\bm{k},0}\right\rangle=\int_{\bm{k}^{\prime}}w_{\bm{k}^{\prime}}\chi_{\bm{k}^{\prime},0}^{*}\chi_{\bm{k},0}\\\
=\frac{2}{(2\pi)^{d}\hbar}\int d\epsilon\left[-k_{B}T\frac{\partial
f^{\left(0\right)}\left(\epsilon\right)}{\partial\epsilon}\right]\int_{S(\epsilon)}\frac{dS}{v_{\bm{k}^{\prime}}}c_{0}^{\ast}c_{0}\\\
\approx\frac{2k_{B}T}{(2\pi)^{d}\hbar}\underbrace{\int
d\epsilon\frac{-\partial
f^{\left(0\right)}\left(\epsilon\right)}{\partial\epsilon}}_{1}\int_{S_{F}}\frac{dS}{v_{\bm{k}_{F}^{\prime}}}\left|c_{0}\right|^{2}\\\
=\frac{2k_{B}T}{(2\pi)^{d}\hbar}\int_{S_{F}}\frac{dS}{v_{\bm{k}_{F}^{\prime}}}\left|c_{0}\right|^{2}\equiv
1.$ (159)

Inverting the last step of Eq. (159), we find the coefficient (36). In the
same way, the normalization condition for $\chi_{\bm{k},i}$ demands

$\left\langle\chi_{\bm{k},i}\right|\left.\chi_{\bm{k},i}\right\rangle=\int_{\bm{k}^{\prime}}w_{\bm{k}^{\prime}}\chi_{\bm{k}^{\prime},i}^{*}\chi_{\bm{k},i}\\\
=\frac{2}{(2\pi)^{d}\hbar}\int d\epsilon\left[-k_{B}T\frac{\partial
f^{\left(0\right)}\left(\epsilon\right)}{\partial\epsilon}\right]\\\
\times\int_{S(\epsilon)}\frac{dS}{v_{\bm{k}^{\prime}}}c_{0}^{\ast}c_{0}c_{i}^{\ast}c_{i}\tilde{v}_{\bm{k}^{\prime},i}\tilde{v}_{\bm{k},i}\\\
\approx\frac{2k_{B}T}{(2\pi)^{d}\hbar}\underbrace{\int
d\epsilon\frac{-\partial
f^{\left(0\right)}\left(\epsilon\right)}{\partial\epsilon}}_{1}\\\
\times\int_{S_{F}}\frac{dS}{v_{\bm{k}_{F}^{\prime}}}\left|c_{0}\right|^{2}\left|c_{i}\right|^{2}(\tilde{v}_{\bm{k},i})^{2}\equiv
1.$ (160)

The last step of Eq. (160) leads to the coefficient (38).

##  Appendix B Analysis of the surface impedance for specular scattering

###  B.1 Isotropic (circular) Fermi surface

In this appendix, we provide analytical results for the impedance of a 2D
isotropic Fermi surface. Each of these results approximates $Z(\omega)$ in a
specific regime. The exact $\left|Z(\omega)\right|$ (divided by $Z_{0}$ and by
$(\omega/\omega_{p})^{1/2}$) calculated numerically from Eqs. (143), (144) and
(84), is reported as a red solid curve in Fig. 19 as a function of
$\omega/\omega_{p}$, together with the analytical approximations obtained as
follows.

Figure 19:  Absolute value of the surface impedance $\left|Z(\omega)\right|$,
divided by the vacuum impedance $Z_{0}$ and by $(\omega/\omega_{p})^{1/2}$, as
a function of $\omega/\omega_{p}$, for a circular Fermi surface (isotropic 2D
case). We use the parameters $v_{F}/c=0.0025$, and $\gamma_{\rm
mr}=10^{-4}\gamma_{\rm mc}=10^{-6}\omega_{p}$. Dashed curves show analytical
results valid in each regime, derived in Secs. B.1.1-B.1.3. The yellow-shaded
area gives a qualitative estimation of the parameter space accessible to
experiments [55].

####  B.1.1 Normal skin effect and perfect-conductor regime

In the limit of large momentum-relaxation rate $\gamma_{\rm mr}\gg\gamma_{\rm
mc}$ and small momentum $v_{F}\left|q_{x}\right|\ll(\omega+i\gamma_{\rm mc})$,
Eq. (84) reduces to the Ohmic result

$\frac{\sigma_{yy}(q_{x},\omega)}{\epsilon_{0}\omega_{p}^{2}}=\frac{1}{\gamma_{\rm
mr}-i\omega}.$ (161)

Inserting Eq. (161) in Eqs. (143) and (144), and performing the momentum
integrations, we achieve

$\frac{Z}{Z_{0}}=\frac{1-i}{\sqrt{2}\omega_{p}}\sqrt{\gamma_{\rm
mr}-i\omega}\sqrt{\omega}.$ (162)

Eq. (162) gives $\left|Z(\omega)\right|\propto\sqrt{\gamma_{\rm mr}\omega}$ at
low frequencies (normal skin effect), while it becomes
$\left|Z(\omega)\right|=\mu_{0}\lambda_{L}\omega$ at high frequencies
(perfect-conductor regime). Eq. (162) produces the dashed blue line in Fig.
19. The high-frequency limit of Eq. (162) also gives the dashed light-blue
line in Fig. 19, in the perfect-conductivity regime.

####  B.1.2 Anomalous skin effect regime

For $\omega\gg\left\\{\gamma_{\rm mc},\gamma_{\rm mr}\right\\}$ (ballistic
regime, with negligible scattering) and
$v_{F}\left|q_{x}\right|\gg(\omega+i\gamma_{\rm mc})$ (large momentum), Eq.
(84) becomes

$\frac{\sigma_{yy}(q_{x},\omega)}{\epsilon_{0}\omega_{p}^{2}}=\frac{2}{v_{F}q_{x}}.$
(163)

Using Eq. (163) in performing the integrations over momentum in Eqs. (143) and
(144), we obtain

$\frac{Z}{Z_{0}}=\frac{2}{3}\left(\frac{1}{\sqrt{3}}-i\right)\left(\frac{v_{F}}{2c}\right)^{\frac{1}{3}}\left(\frac{\omega}{\omega_{p}}\right)^{\frac{2}{3}}.$
(164)

Hence, Eq. (164) gives $\left|Z(\omega)\right|\propto\omega^{2/3}$,
consistently with the literature on ballistic (anomalous) skin effect in
isotropic systems [48, 54, 55]. Eq. (164) yields the dashed green line in Fig.
19.

Figure 20:  Skin effect regimes for an isotropic 2D (circular) Fermi surface,
as measured by the surface impedance $Z(\omega)$ as a function of relaxation
rate $\gamma_{\rm mr}$, momentum-conserving collision rate $\gamma_{\rm mr}$,
and frequency $\omega/\omega_{p}$, where $\omega_{p}$ is the plasma frequency
(78). The Fermi-surface geometry is sketched on the left-hand side of the
plot, together with the applied electric field $\bm{E}=E_{y}\hat{u}_{y}$
aligned with the $y$ axis, and the local Fermi velocity vectors shown by red
arrows. (a) Regimes in the $\left(\gamma_{\rm mr},\omega\right)$ plane, for
$\gamma_{\rm mr}=\gamma_{\rm mc}$. (b) Regimes in the $\left(\gamma_{\rm
mc},\omega\right)$ plane, for fixed $\gamma_{\rm mr}=10^{-8}\omega_{p}$. The
color palette is the density plot of $\mathrm{Arg}Z(\omega)/(-\pi/2)$, giving
the exponent $\eta$ of $Z(\omega)\propto\omega^{\eta}$. Dashed lines are the
analytical crossover boundaries derived in Appendix B.1.4.

####  B.1.3 Hydrodynamic regime

We analyze hydrodynamic skin effect in the isotropic limit by expanding the
denominator in the conductivity Eq. (84) for $\gamma_{\rm
mc}\rightarrow+\infty$ at first order (assuming negligible momentum
relaxation):

$\frac{\sigma_{yy}(q_{x},\omega)}{\epsilon_{0}\omega_{p}^{2}}=\frac{1}{-i\omega+(v_{F}q_{x})^{2}/(4\gamma_{\rm
mc})}.$ (165)

We then perform the integrals over momentum in Eqs. (143) and (144), using the
expansion (165), and we expand the result in the low-frequency limit
$\omega\rightarrow 0^{+}$, to obtain

$\frac{Z}{Z_{0}}=\frac{(-1+i)(-1)^{\frac{7}{8}}\sqrt{v_{F}/c}}{2\sqrt{2\omega_{p}}\gamma_{\rm
mc}^{\frac{1}{4}}}\omega^{\frac{3}{4}}.$ (166)

Eq. (166) produces the dashed orange line in Fig. 19.

####  B.1.4 Crossover frequencies

To estimate characteristic frequencies for the various crossovers between
different skin effect regimes, we consider the intersection points between the
analytical results for $\left|Z(\omega)\right|$ found in Secs. B.1.1-B.1.3.
Let us begin with the crossovers in the $\left(\gamma_{\rm mc},\omega\right)$
plane, for $\gamma_{\rm mc}\geq\gamma_{\rm mr}$; see Fig. 20(b). Using the
low-frequency expansion of Eq. (162) and Eq. (166), we deduce the frequency
$\omega=\omega_{nv}$ for the crossover between normal and viscous skin effect:

$\omega_{nv}=\frac{16\gamma_{\rm mc}\gamma_{\rm
mr}^{2}}{\omega_{p}^{2}(v_{F}/c)^{2}}.$ (167)

Eq. (167) gives the green dot in Fig. 19 and the dashed yellow line in Fig.
20(b). It is consistent with the qualitative crossover condition
\normalsizeA⃝–\normalsizeV⃝ in Table 3, up to a numerical constant. Employing
Eqs. (166) and (164), we obtain the crossover frequency $\omega=\omega_{va}$
between the viscous and anomalous regimes:

$\omega_{va}=C\gamma_{\rm mc}^{3}\left(\frac{c}{\omega_{p}v_{F}}\right)^{2},$
(168)

with $C=4294967296/387420489\approx 11.086$. Eq. (168) produces the blue dot
in Fig. 19 and the dashed light-blue line in Fig. 20(b). It agrees with the
qualitative criterion \smallD$_1$⃝–\normalsizeV⃝ in Table 3. The crossover
frequency $\omega=\omega_{ap}$ between anomalous and perfect-conductor regimes
follows from Eq. (164) and $\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$:

$\omega_{ap}=\frac{32}{81\sqrt{3}}\frac{v_{F}}{\lambda_{L}}.$ (169)

The red dot in Fig. 19 and the dashed red line in Fig. 20(b) are both given by
Eq. (169). The latter is in qualitative agreement with the length-scale
criterion \smallD$_2$⃝–\normalsizeE⃝ in Table 3. Finally, the crossover
between viscous skin effect and perfect-conductor regime is derived from Eq.
(166) and $\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$, giving

$\omega_{vp}=\frac{v_{F}^{2}}{16\gamma_{\rm mc}}\frac{1}{\lambda_{L}^{2}}.$
(170)

Eq. (170) generates the dashed purple line in Fig. 20(b), and is consistent
with the qualitative condition \normalsizeV⃝–\normalsizeF⃝ in Table 3. We now
analyze the crossovers in the $\left(\gamma_{\rm mr},\omega\right)$ plane, for
$\gamma_{\rm mc}=\gamma_{\rm mr}$; see Fig. 20(a). The boundary between normal
and anomalous skin effect follows from Eqs. (162) and (164):

$\omega_{na}^{\prime}=\frac{19683}{1024}\left(\frac{\lambda_{L}}{v_{F}}\right)^{2}\gamma_{\rm
mr}^{3}.$ (171)

Eq. (171) yields the dashed yellow line in Fig. 20(a), and is consistent with
the qualitative condition \normalsizeA⃝–\smallD$_1$⃝ in Table 2. The crossover
between anomalous and perfect-conductor regimes is independent from
$\gamma_{\rm mr}$, and it follows Eq. (169) as previously found from Eq.
(164). Therefore, Eq. (169) also gives the dashed red line in Fig. 20(a), in
agreement with the condition \smallD$_2$⃝–\normalsizeE⃝ in Table 2. Lastly, a
boundary between the normal and perfect-conductor regimes stems from the low-
frequency expansion of Eq. (162) and
$\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$. We simply obtain

$\omega_{np}^{\prime}=\gamma_{\rm mr},$ (172)

which is in full agreement with the criterion \normalsizeA⃝–\normalsizeB⃝ in
Table 2, and produces the dashed light-blue line in Fig. 20(a).

###  B.2 Hexagonal Fermi surface

This appendix contains the analysis of the surface impedance for a hexagonal
Fermi surface and rotation angles $\phi=\left\\{0,\pi/2\right\\}$.

####  B.2.1 Scattering-less regime

In the absence of scattering, i.e., $\gamma_{\rm mr}=\gamma_{\rm mc}=0$, and
for $\phi=0$, the conductivity reduces to Eq. (94) and the integral over $z$
in Eq. (144) can be evaluated analytically:

$\frac{Z}{Z_{0}}=\frac{1}{2\sqrt{2}}\frac{\omega}{\omega_{p}}\frac{v_{F}}{c}\frac{\left(\alpha_{2}-i\alpha_{3}\right)-\omega/\omega_{p}\left(\alpha_{2}-i\alpha_{3}\right)/\alpha_{1}}{\alpha_{2}\alpha_{3}},$
(173a)
$\alpha_{1}=\sqrt{\left(\frac{\omega}{\omega_{p}}\right)^{2}+\left(\frac{v_{F}}{c}\right)^{2}},$
(173b)
$\alpha_{2}=\sqrt{\frac{\omega}{\omega_{p}}\left(\alpha_{1}-\frac{\omega}{\omega_{p}}\right)},$
(173c)
$\alpha_{3}=\sqrt{\frac{\omega}{\omega_{p}}\left(\alpha_{1}+\frac{\omega}{\omega_{p}}\right)}.$
(173d)

The surface impedance (173) crosses over from
$\mathrm{Re}Z(\omega)\sim\omega^{1/2}$ and
$-\mathrm{Im}Z(\omega)\sim\omega^{1/2}$ at low frequency to
$\mathrm{Re}Z(\omega)\ll-\mathrm{Im}Z(\omega)\sim\omega$ at high $\omega$, as
shown by the solid and dashed blue curves in Fig. 21, respectively. We can
understand this crossover analytically, by combining Eq. (144) with the
conductivity (94) as done in Sec. V.1: in the regime $\omega\ll
v_{F}\left|q_{x}\right|$, we directly obtain

$Z(\omega)\approx\frac{1-i}{2}Z_{0}\sqrt{\frac{v_{F}}{c}}\sqrt{\frac{\omega}{\omega_{p}}}=\mu_{0}\omega\frac{1-i}{2}\delta_{s}(\omega),$
(174)

with the skin depth $\delta_{s}(\omega)$ given by Eq. (131), while for
$\omega\gg v_{F}\left|q_{x}\right|$ the momentum dependence of the
conductivity disappears and

$Z(\omega)\approx-iZ_{0}\frac{\omega}{\omega_{p}}=-i\mu_{0}\omega\lambda_{L},$
(175)

i.e., with the frequency-independent skin depth $\delta_{s}\equiv\lambda_{L}$
associated with a perfect conductivity. In fact, as noticed in Sec. V, the
high-frequency limit (175) of the surface impedance is general and independent
from the shape of the anisotropic 2D Fermi surface.

In the limit $\gamma_{\rm mr}=\gamma_{\rm mc}=0$ and $\phi=\pi/2$, the
conductivity is given by Eq. (104) and the integral over $z$ in Eq. (144)
yields the analytical result

$\frac{Z}{Z_{0}}=i\frac{\sqrt{3}}{2}\frac{\omega}{\omega_{p}}\frac{v_{F}}{c}\frac{1}{\beta_{1}}\left[\frac{-2(\omega/\omega_{p})^{2}-(v_{F}/c)^{2}+\beta_{1}}{-2(\omega/\omega_{p})^{2}+(v_{F}/c)^{2}-\beta_{1}}\right.\\\
\left.-\frac{2(\omega/\omega_{p})^{2}+(v_{F}/c)^{2}+\beta_{1}}{-2(\omega/\omega_{p})^{2}+(v_{F}/c)^{2}+\beta_{1}}\right],$
(176a)
$\beta_{1}=\sqrt{4\left(\frac{\omega}{\omega_{p}}\right)^{2}+8\left(\frac{\omega}{\omega_{p}}\frac{v_{F}}{c}\right)^{2}+\left(\frac{v_{F}}{c}\right)^{4}}.$
(176b)

The high-frequency limit of the surface impedance (176) is characteristic of a
perfect conductivity with no momentum dependence, as mentioned in the
preceding section. However, for $\phi=\pi/2$ we have at low frequencies
$\mathrm{Re}Z(\omega)\ll-\mathrm{Im}Z(\omega)\sim\omega$, as shown by the
solid and dashed gold curves in Fig. 21. This behavior is understood by using
Eqs. (144) and (104) in the limit $\omega\ll v_{F}\left|q_{x}\right|$, which
straightforwardly gives

$Z(\omega)\approx-
iZ_{0}\sqrt{\frac{3}{2}}\frac{\omega}{\omega_{p}}=-i\mu_{0}\omega\delta_{s},$
(177)

with the skin depth $\delta_{s}=\sqrt{3/2}\lambda_{L}$ in accordance with Eq.
(134).

Figure 21:  Surface impedance $Z(\omega)$ normalized to the vacuum-impedance
$Z_{0}$, as a function of dimensionless frequency $\omega/\omega_{p}$, for a
hexagonal Fermi surface with Fermi-velocity modulus $v_{F}=2.5\times
10^{-3}c$, in the limit without relaxation and collisions, i.e., $\gamma_{\rm
mr}=\gamma_{\rm mc}=0$. Blue (gold) curves show the results for crystals
tilted at an angle $\phi=0$ ($\phi=\pi/2$) with respect to the surface,
according to Eq. (173) (Eq. (176)). Solid and dashed lines refer to
$\mathrm{Re}Z(\omega)$ and $-\mathrm{Im}Z(\omega)$, respectively.

Therefore, under the assumption $\omega\ll v_{F}q_{x}$ valid at low
frequencies, we expect a qualitative difference in the skin depth and the
surface impedance between the orientations $\phi=0$ and $\phi=\pi/2$. An
analogous difference follows from the analysis of $Z(\omega)$ for a square
Fermi surface, as performed in Sec. B.3.

####  B.2.2 Normal and perfect-conductor regimes

For orientation angle $\phi=0$, the low-momentum conductivity follows Eq.
(96). To order $o\left[(v_{F}q_{x})^{2}\right]$, such conductivity is
momentum-independent, and equivalent to the one of the Drude model, Eq. (120).
Inserting this momentum-independent conductivity in Eqs. (143) and (144), and
performing the momentum integrations, we straightforwardly obtain

$\frac{Z(\omega)}{Z_{0}}=-i\frac{\omega}{\omega_{p}}\sqrt{1+\frac{i\gamma_{\rm
mr}}{\omega}}.$ (178)

Eq. (178) gives the dashed blue line in Fig. 9(a). At low frequencies, it
implies $\left|Z(\omega)\right|\propto\omega^{1/2}$, while for
$\omega\rightarrow+\infty$ it becomes the perfect-conductor result
$Z(\omega)=-iZ_{0}\omega/\omega_{p}=-i\mu_{0}\lambda_{L}\omega$. The latter
also gives the dashed light blue line in Fig. 9(a).

The conductivity for $\phi=\pi/2$ in the low-momentum regime is the same as
for $\phi=0$, i.e., Eq. (96), to order $o\left[(v_{F}q_{x})^{2}\right]$. This
is a consequence of the property mentioned in Sec. IV.2.2, that the
electrodynamics is independent of orientation angle $\phi$ in the
$q_{x}\rightarrow 0^{+}$ limit for a hexagonal Fermi surface. Hence, Eq. (178)
also gives the dashed blue line in Fig. 9(b), while the dashed light blue line
in the same figure follows $Z(\omega)=-i\mu_{0}\lambda_{L}\omega$.

####  B.2.3 Anomalous regime

To analyze the high-momentum anomalous regime for angle $\phi=0$, we use the
expansion of Eq. (92) to leading order in $q_{x}\rightarrow+\infty$, which
gives Eq. (95). Inserting the latter into Eqs. (143) and (144) and performing
the momentum integrations analytically, we obtain

$\frac{Z(\omega)}{Z_{0}}=\frac{1-i}{\sqrt{2\omega_{p}}}\frac{\sqrt{v_{F}/c}\omega^{\frac{3}{4}}}{2(\omega+i\gamma_{\rm
mc})^{\frac{1}{4}}}.$ (179)

In the regime $\omega\gg\gamma_{\rm mc}$, Eq. (179) gives

$\frac{Z(\omega)}{Z_{0}}=\frac{1-i}{2\sqrt{2}}\sqrt{\frac{v_{F}\omega}{c\omega_{p}}}.$
(180)

Eq. (180) gives the dashed green line in Fig. 9(a).

For orientation angle $\phi=\pi/2$, we use the high-momentum expansion (106)
of the conductivity in performing the momentum integrals of Eqs. (143)-(144).
The result is

$\frac{Z(\omega)}{Z_{0}}=\frac{-i\omega}{\sqrt{2}}\sqrt{3+\frac{i(\gamma_{\rm
mc}+2\gamma_{\rm mr})}{\omega}}.$ (181)

The low-frequency expansion of Eq. (181) at leading order gives

$\frac{Z(\omega)}{Z_{0}}=\frac{1-i}{2\omega_{p}}\sqrt{(\gamma_{\rm
mc}+2\gamma_{\rm mr})\omega},$ (182)

which yields the dashed green line in Fig. 9(b). We can extract a skin depth
from Eq. (182) in the presence of scattering, by defining
$Z(\omega)=(1-i)/2\mu_{0}\omega\delta_{s}(\omega)$. This gives Eq. (135) for
the skin depth.

####  B.2.4 Hydrodynamic regime

To obtain the hydrodynamic impedance for $\phi=0$, it is sufficient to
consider Eq. (179) in the limit $\gamma_{\rm mc}\gg\omega$ (dominant momentum-
conserving collisions). This way, we obtain

$\frac{Z(\omega)}{Z_{0}}=\frac{1-i}{2\sqrt{2\omega_{p}}}\frac{\sqrt{v_{F}/c}\omega^{\frac{3}{4}}}{(i\gamma_{\rm
mc})^{\frac{1}{4}}}.$ (183)

Eq. (183) produces the dashed orange line in Fig. 9(a).

For orientation angle $\phi=\pi/2$, the analysis of the hydrodynamic regime
requires to push the expansion (106) to the next term of order
$(v_{F}q_{x})^{-2}$. Using such term in the momentum integrals in Eqs.
(143)-(144), and expanding the result for $\omega\rightarrow 0$ at leading
order, produces

$\frac{Z(\omega)}{Z_{0}}=-\frac{(-1)^{3/4}}{2\sqrt{\omega_{p}}}\left(\frac{\omega}{\gamma_{\rm
mc}}\right)^{\frac{3}{4}}\sqrt{\frac{v_{F}}{c}(\gamma_{\rm mc}+2\gamma_{\rm
mr})}.$ (184)

Eq. (184) gives the dashed orange line in Fig. 9(b).

####  B.2.5 Crossover frequencies

For $\phi=0$, we begin by finding crossover boundaries in the
$\left(\gamma_{\rm mc},\omega\right)$ plane for $\gamma_{\rm
mc}\geq\gamma_{\rm mr}$. The crossover frequency between normal and viscous
skin effect results from equating $\left|Z(\omega)\right|$ from the low-
frequency expansion of Eq. (178) and from Eq. (183), and solving for
$\omega=\omega_{nv}$. We obtain

$\omega_{nv}=\frac{16\gamma_{\rm mc}\gamma_{\rm
mr}^{2}}{\omega_{p}^{2}(v_{F}/c)^{2}}.$ (185)

Eq. (185) gives the green dot in Fig. 9(a) and the dashed yellow curve in Fig.
10(b). It is also consistent with the qualitative criterion for the crossover
\normalsizeA⃝–\normalsizeV⃝ in Table 3. To evaluate the crossover frequency
between viscous and anomalous skin effect, we equate $\left|Z(\omega)\right|$
from Eqs. (183) and (180), solving for $\omega=\omega_{va}$. This gives simply

$\omega_{va}=\gamma_{\rm mc}.$ (186)

Eq. (186) yields the blue dot in Fig. 9(a) and the dashed light-blue curve in
Fig. 10(b). Besides, Eq. (186) fully agrees with the qualitative condition
\smallD$_1$⃝–\smallD$_2$⃝ in Table 3. The crossover frequency between
anomalous and perfect-conductor regimes results from equating
$\left|Z(\omega)\right|$ stemming from Eqs. (180) and
$\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$, solving for
$\omega=\omega_{ap}$. We have

$\omega_{ap}=\omega_{p}\frac{v_{F}}{4c}.$ (187)

Eq. (187) produces the red dot in Fig. 9(a) and the dashed red curve in Fig.
10(b). Up to the numerical prefactor $1/4$, it is consistent with the boundary
\smallD$_2$⃝–\normalsizeE⃝ in Table 3. Finally, there is a crossover between
viscous and perfect-conductor regimes, found by equating Eq. (183) and
$\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$. The corresponding frequency
is

$\omega_{vp}=\left(\frac{v_{F}}{\lambda_{L}}\right)^{2}\frac{1}{16\gamma_{\rm
mc}}.$ (188)

Eq. (188) generates the dashed purple line in Fig. 10(b), and it is in
qualitative agreement with the length-scale criterion
\normalsizeV⃝–\normalsizeF⃝ from Table 3.

For $\phi=\pi/2$, we evaluate the crossover between normal and viscous skin
effect by equating $\left|Z(\omega)\right|$ which result from the low-
frequency expansion of Eq. (178) and from the lengthy analytical expression
which produces the dashed orange line in Fig. 9(b), in hydrodynamic regime.
Here we quote the final result:

$\omega_{nv}=\frac{16\gamma_{\rm mc}^{3}\gamma_{\rm mr}^{2}}{(\gamma_{\rm
mc}+2\gamma_{\rm mr})^{2}(v_{F}/c)^{2}\omega_{p}^{2}}.$ (189)

Eq. (189) gives the green dot in Fig. 9(b) and the dashed yellow curve in Fig.
10(d). It also respects the qualitative criterion \normalsizeA⃝–\normalsizeV⃝
in Table 3, up to corrections of higher order in $\gamma_{\rm mc}$ and
$\delta\gamma$. The crossover between viscous and anomalous skin effect
results from equating $\left|Z(\omega)\right|$ stemming from the analytical
expression in hydrodynamic regime and from Eq. (182). The final result is

$\omega_{va}=\frac{4\gamma_{\rm mc}^{3}}{\omega_{p}^{2}(v_{F}/c)^{2}}.$ (190)

Eq. (190) yields the blue dot in Fig. 9(b) and the dashed light-blue curve in
Fig. 10(d). It is consistent with the condition \smallD$_1$⃝–\normalsizeV⃝ in
Table 3, up to higher-order corrections in $\gamma_{\rm mc}$ and
$\delta\gamma$. We obtain the crossover frequency between anomalous and
perfect-conductor by equating $\left|Z(\omega)\right|$ stemming from Eqs.
(182) and $\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$, solving for
$\omega=\omega_{ap}$. We have

$\omega_{ap}=\frac{\gamma_{\rm mc}}{2}+\gamma_{\rm mr}.$ (191)

Eq. (191) produces the red dot in Fig. 9(b) and the dashed red curve in Fig.
10(d). It qualitatively agrees with the criterion \smallD$_1$⃝–\smallD$_2$⃝
reported in Table 3, at leading order in $\gamma_{\rm mc}$. Lastly, the
crossover between viscous and perfect-conductor regimes stems from from the
analytical expression in hydrodynamic regime and from
$\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$. We obtain

$\omega_{vp}=\frac{(\gamma_{\rm mc}+2\gamma_{\rm
mr})^{2}(v_{F}/c)^{2}}{16\gamma_{\rm mc}^{3}}\omega_{p}^{2}.$ (192)

At leading order in $\gamma_{\rm mc}$, Eq. (192) agrees with the qualitative
condition \normalsizeV⃝–\normalsizeF⃝ in 3. The dashed purple line in Fig.
10(d) stems from Eq. (192).

We now trace the visible crossover boundaries in the $\left(\gamma_{\rm
mr},\omega\right)$ plane for $\gamma_{\rm mc}=\gamma_{\rm mr}$. For $\phi=0$,
we find a crossover between normal and viscous skin effect by equating the
leading low-frequency expansion of Eq. (178) and Eq. (183):

$\omega_{nv}^{\prime}=16\left(\frac{\lambda_{L}}{v_{F}}\right)^{2}\gamma_{\rm
mr}^{3}.$ (193)

Eq. (193) yields the dashed yellow line in Fig. 10(a), and it qualitatively
agrees with the length-scale criterion \normalsizeA⃝–\smallD$_1$⃝ in Table 2.
The crossover between viscous and anomalous regimes is found through Eqs.
(183) and (180), with the simple result of Eq. (172), as in the isotropic
case. It is in perfect agreement with the criterion \smallD$_1$⃝–\smallD$_2$⃝
in Table 2. Then, Eq. (172) produces the dashed light-blue line in Fig. 10(a).
The boundary between the anomalous and perfect-conductor regimes can be
estimated through Eq. (180) and
$\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$: it is independent from
$\gamma_{\rm mr}$ and $\gamma_{\rm mc}$, and gives Eq. (187). This result
gives the dashed red line in Fig. 10(a), in accordance with the qualitative
condition \smallD$_2$⃝–\normalsizeE⃝ in Table 2. Finally, we estimate the
boundary between the normal and perfect-conductor regimes using Eq. (178) and
$\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$. We simply retrieve Eq. (172),
in the same way as for the crossover between viscous and anomalous skin
effect. Hence, Eq. (172) corresponds also to the dashed purple line in Fig.
10(a), in agreement with the condition \normalsizeA⃝–\normalsizeB⃝ in Table 2.

In the configuration with $\phi=\pi/2$, the only visible crossover with
$\gamma_{\rm mr}=\gamma_{\rm mc}$ is the one between anomalous and perfect-
conductor regimes. It is found by equating the low-frequency expansion of Eq.
(182) with $\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$, and it simply
gives

$\omega_{ap}^{\prime}=\frac{3}{2}\gamma_{\rm mr},$ (194)

This equation provides the dashed red line in Fig. 10(c), and it qualitatively
agrees with the criteria \smallD$_1$⃝–\smallD$_2$⃝ and
\normalsizeA⃝–\normalsizeB⃝ in Table 2.

###  B.3 Square Fermi surface

In this appendix, we derive analytical expressions for the surface impedance
assuming a square Fermi surface and rotation angles
$\phi=\left\\{0,\pi/4\right\\}$.

####  B.3.1 Scattering-less regime

In the same way as in Sec. VI.1, we can analyze the scattering-less limit of
the surface impedance for a square Fermi surface. In the geometry with tilting
angle $\phi=0$, using Eq. (144) and the conductivity (112), from the integral
over $z$ we directly obtain

$\frac{Z(\omega)}{Z_{0}}=i\frac{\omega}{\omega_{p}}\frac{v_{F}}{c}\frac{\gamma_{1}(\gamma_{2}-\gamma_{3})-\omega/\omega_{p}(\gamma_{2}+\gamma_{3})}{2\gamma_{1}\gamma_{2}\gamma_{3}},$
(195a)
$\gamma_{1}=\sqrt{\left(\frac{\omega}{\omega_{p}}\right)^{2}+2\left(\frac{v_{F}}{c}\right)^{2}},$
(195b)
$\gamma_{2}=\sqrt{\frac{\omega}{\omega_{p}}\left(\gamma_{1}-\frac{\omega}{\omega_{p}}\right)},$
(195c)
$\gamma_{3}=\sqrt{-\frac{\omega}{\omega_{p}}\left(\gamma_{1}+\frac{\omega}{\omega_{p}}\right)}$
(195d)

Figure 22:  Surface impedance $Z(\omega)$ normalized to the vacuum-impedance
$Z_{0}$, as a function of dimensionless frequency $\omega/\omega_{p}$, for a
square Fermi surface with Fermi-velocity modulus $v_{F}=2.5\times 10^{-3}c$,
in the limit without relaxation and collisions, i.e., $\gamma_{\rm
mr}=\gamma_{\rm mc}=0$. Blue (gold) curves show the results for crystals
tilted at an angle $\phi=0$ ($\phi=\pi/4$) with respect to the surface,
according to Eq. (195) (Eq. (175)). Solid and dashed lines refer to
$\mathrm{Re}Z(\omega)$ and $-\mathrm{Im}Z(\omega)$, respectively. Notice that
the impedance is entirely imaginary for the $\phi=\pi/4$ case.

Fig. 22 shows the real and (minus the) imaginary parts of the surface
impedance (195a) as a function of $\omega/\omega_{p}$, as blue solid and
dashed curves respectively. The behavior of $Z(\omega)$ is very similar to the
one for a hexagonal Fermi surface with $\phi=0$; see Fig. 21. We can trace
back this analogy to the similar form of the conductivities (94) and (112) for
the hexagonal and square case respectively, which also give the same
qualitative behavior of the skin depths (131) and (137). Consequently, in the
regime $\omega\ll v_{F}\left|q_{x}\right|$ the conductivity (112) and Eq.
(144) formally yield Eq. (174), but with Eq. (137) for the skin depth in
square geometry. In the high-frequency limit $\omega\gg
v_{F}\left|q_{x}\right|$ we retrieve the smallest possible skin depth
$\lambda_{L}$ and the impedance follows Eq. (175).

As the skin depth analyzed in Sec. V.2, the surface impedance of the square
Fermi surface is qualitatively modified by the presence of two large portions
of the Fermi surface parallel to the applied field for $\phi=\pi/4$; see Fig.
8. In fact, the disappearance of the momentum dependence of the conductivity
generates the London skin depth $\delta\equiv\lambda_{L}$, so that the surface
impedance is simply given by Eq. (175): it is purely imaginary and linear in
frequency. Such evolution with $\omega$ is shown by the dashed gold line in
Fig. 22 and is valid at any frequency in the absence of scattering. Hence, we
expect a qualitative change in $Z(\omega)$ between the orientations $\phi=0$
and $\phi=\pi/4$, with the disappearance of $\mathrm{Re}Z(\omega)$ in the
latter case for $\gamma_{\rm mr}=\gamma_{\rm mc}=0$. Notice that such
disappearance does not occur for $\gamma_{\rm mc}>0$, as shown by the
asymptotic limits for $\gamma_{\rm mr}\rightarrow 0$ shown in Fig. 14: in that
case, the asymptotic value of the surface resistance is entirely due to
momentum-conserving collisions.

####  B.3.2 Normal and perfect-conductor regimes

Assuming an orientation $\phi=0$, the low-momentum conductivity is given by
Eq. (113). To order $o\left[(v_{F}q_{x})^{2}\right]$, such conductivity does
not depend on momentum, and is equivalent to the one of the Drude model.
Inserting the momentum-independent term in Eqs. (143) and (144), and
performing the momentum integrations, we retrieve Eq. (178). Hence, Eq. (178)
also gives the dashed blue line in Fig. 12(a). At low frequencies, it implies
$Z(\omega)\propto\omega^{1/2}$, while in the high-frequency limit it turns
into the perfect-conductor result $Z(\omega)=-i\mu_{0}\lambda_{L}\omega$,
which is also shown by the dashed light blue line in Fig. 12(a).

For angle $\phi=\pi/4$, the conductivity is (120), identical to the one of the
Drude local conductor for any momentum. Therefore, in the regime of normal
skin effect we obtain Eq. (178) for both $\phi=0$ and $\phi=\pi/4$, and this
result gives the dashed blue line in Fig. 12(b) as well. The dashed light-blue
line in Fig. 12(b) shows the perfect-conductor high-frequency limit
$Z(\omega)=-i\mu_{0}\lambda_{L}\omega$.

####  B.3.3 Anomalous regime

For angle $\phi=0$, we use the expansion of Eq. (110) to leading order in
$q_{x}\rightarrow+\infty$, which gives Eq. (115). Using Eqs. (143) and (144),
we obtain

$\frac{Z(\omega)}{Z_{0}}=\frac{1-i}{2\sqrt{\omega_{p}}}\frac{\sqrt{v_{F}/c}i\omega}{\left[2\omega(\omega+i\gamma_{\rm
mc})\right]^{\frac{1}{4}}}.$ (196)

In the limit $\omega\gg\gamma_{\rm mc}$, Eq. (196) yields

$\frac{Z(\omega)}{Z_{0}}=\frac{1-i}{2^{5/4}}\sqrt{\frac{v_{F}}{c}}\sqrt{\frac{\omega}{\omega_{p}}},$
(197)

which gives the dashed green line in Fig. 12(a).

Rotating the Fermi surface by an angle $\phi=\pi/4$, the conductivity (120) is
momentum-independent, therefore the anomalous regime is absent.

####  B.3.4 Hydrodynamic regime

The hydrodynamic limit of the impedance for $\phi=0$ follows directly from Eq.
(196) in the limit $\omega\ll\gamma_{\rm mc}$:

$\frac{Z(\omega)}{Z_{0}}=\frac{1-i}{2^{5/4}\sqrt{\omega_{p}}}\frac{\sqrt{v_{F}/c}\omega^{\frac{3}{4}}}{(i\gamma_{\rm
mc})^{\frac{1}{4}}}.$ (198)

Eq. (198) produces the dashed orange line in Fig. 12(a).

For $\phi=\pi/4$ the momentum dependence of the conductivity vanishes, and
there is no viscous regime.

####  B.3.5 Crossover frequencies

Let us first define criteria to distinguish the skin effect regimes in the
$\left(\gamma_{\rm mc},\omega\right)$ plane for $\gamma_{\rm
mc}\geq\gamma_{\rm mr}$. For $\phi=0$, we obtain the crossover frequency
between normal and viscous skin effect from equating $\left|Z(\omega)\right|$
that stem from the low-frequency expansion of Eq. (178) and from Eq. (198),
and we solve for $\omega=\omega_{nv}$. We have

$\omega_{nv}=\frac{8\gamma_{\rm mc}\gamma_{\rm
mr}^{2}}{\omega_{p}^{2}(v_{F}/c)^{2}}.$ (199)

Eq. (199) gives the green dot in Fig. 12(a) and the dashed yellow curve in
Fig. 13(b). As for the other Fermi-surface geometries, we can qualitatively
interpret Eq. (199) as the crossover line \normalsizeA⃝–\normalsizeV⃝ in Table
3. The crossover frequency between viscous and anomalous skin effect stems
from equating $\left|Z(\omega)\right|$ from Eqs. (198) and (197). The solution
is $\omega=\omega_{va}=\gamma_{\rm mc}$, as for the hexagonal case Eq. (186).
Therefore, Eq. (186) also yields the blue dot in Fig. 12(a) and the dashed
light-blue curve in Fig. 13(b). It coincides with the lenght-scale criterion
\smallD$_1$⃝–\smallD$_2$⃝ in Table 3. The crossover between anomalous and
perfect-conductor regimes results from equating $\left|Z(\omega)\right|$
according to Eqs. (197) and $\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$,
solving for $\omega=\omega_{ap}$. We have

$\omega_{ap}=\omega_{p}\frac{v_{F}}{2\sqrt{2}c}.$ (200)

Eq. (200) generates the red dot in Fig. 12(a) and the dashed red curve in Fig.
13(b). Up to the prefactor $1/(2\sqrt{2})$, it agrees with the condition
\smallD$_1$⃝–\normalsizeV⃝ in Table 3. Finally, we find a boundary between
viscous and perfect-conductor regimes by equating Eq. (198) and
$\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$. We obtain

$\omega_{vp}=\frac{1}{8}\left(\frac{v_{F}}{\lambda_{L}}\right)^{2}\frac{1}{\gamma_{\rm
mc}}.$ (201)

Eq. (201) is in qualitative agreement with the criterion
\normalsizeV⃝–\normalsizeF⃝ in Table 3, and it produces the dashed purple line
in Fig. 13(b).

For $\phi=\pi/4$, the only crossover occurs between the normal and perfect-
conductor regimes. We achieve $\omega_{np}=\gamma_{\rm mr}$, in agreement with
Eq. (172) and independently from $\gamma_{\rm mc}$. Then, Eq. (172) gives the
red dot in Fig. 12(b) and the dashed red line in Fig. 13(d). It is perfectly
consistent with the qualitative criteria \normalsizeA⃝–\normalsizeB⃝ and
\smallD$_1$⃝–\smallD$_2$⃝ in Table 2.

We now define crossover boundaries for the square geometry, in the
$\left(\gamma_{\rm mr},\omega\right)$ plane for $\gamma_{\rm mc}=\gamma_{\rm
mr}$. For $\phi=0$, there is a crossover between normal and viscous skin
effect, found by equating the low-frequency expansion of Eq. (178) and Eq.
(198):

$\omega_{nv}^{\prime}=8\left(\frac{\lambda_{L}}{v_{F}}\right)^{2}\gamma_{\rm
mr}^{3}.$ (202)

Eq. (202) is in qualitative agreement with the criterion
\normalsizeA⃝–\smallD$_1$⃝ in Table 2, and it generates the dashed yellow line
in Fig. 13(a). A boundary between the viscous and anomalous regimes is
identified by the frequency at which Eqs. (198) and (197)coincide. As for the
hexagonal and circular geometries, this gives Eq. (172), which is equivalent
to the condition \smallD$_1$⃝–\smallD$_2$⃝ in Table 2, and it produces the
dashed light-blue line in Fig. 13(a). The crossover between the anomalous and
perfect-conductor regimes is estimated by equating Eq. (197) and
$\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$. One retrieves Eq. (200),
consistently with the previous analysis in the $\left(\gamma_{\rm
mc},\omega\right)$ plane. Therefore, Eq. (200) is also qualitatively
consistent with the condition \smallD$_2$⃝–\normalsizeE⃝ in Table 2, and it
yields the dashed red line in Fig. 13(a). Besides, there is a crossover
between normal and perfect-conductor regimes, which is identified through the
equality between the low-frequency expansion of Eq. (178) and
$\left|Z(\omega)\right|=Z_{0}\omega/\omega_{p}$. In this case, we retrieve Eq.
(172), in perfect agreement with the length-scale criterion
\normalsizeA⃝–\normalsizeB⃝ in Table 2. Hence, Eq. (172) also produces the
dashed purple line in Fig. 13(a).

In the configuration with $\phi=\pi/4$, there is but one crossover with
$\gamma_{\rm mr}=\gamma_{\rm mc}$: the one between normal and perfect-
conductor regimes. It is given by $\omega_{np}=\gamma_{\rm mr}$, which agrees
with Eq. (172) as in the $\left(\gamma_{\rm mc},\omega\right)$ plane. This
equation gives the dashed red line in Fig. 10(c), and it is equivalent to the
qualitative criteria \smallD$_1$⃝–\smallD$_2$⃝ and \normalsizeA⃝–\normalsizeB⃝
in Table 2.

##  Appendix C Frequency scaling of the surface impedance for specular
interface scattering

In this section, we derive the scaling relation (4) for the frequency
dependence of the impedance $Z(\omega)$ analytically for specular interface
scattering. We start from Eq. (142) and neglect the displacement current,
leading to Eq. (144), that is

$Z(\omega)\approx\frac{i\omega\mu_{0}}{\pi}\int_{-\infty}^{+\infty}dq_{x}\frac{1}{\mu_{0}i\omega\sigma_{yy}(q_{x},\omega)-q_{x}^{2}}.$
(203)

Assuming the scaling relation (146) for the conductivity, we can write
$\mu_{0}i\omega\sigma_{yy}(q_{x},\omega)=-\alpha_{1}(-i\omega)^{-\beta+1}q_{x}^{-\alpha}$.
Inserting the latter equation into Eq. (203), and performing the momentum
integration analytically, we obtain

$Z(\omega)=\frac{2\mu_{0}}{\pi}\Gamma\left(\frac{1+\alpha}{2+\alpha}\right)\Gamma\left(\frac{3+\alpha}{2+\alpha}\right)\alpha_{1}^{-\frac{1}{2+\alpha}}(-i\omega)^{\frac{\beta+\alpha+1}{2+\alpha}}\\\
\propto(-i\omega)^{\eta},$ (204)

where $\Gamma(z)=\int_{0}^{+\infty}dte^{-t}t^{z-1}$ is the Euler Gamma
function, and $\eta=(1+\alpha+\beta)/(2+\alpha)$. Using
$(-i)^{\eta}=e^{-i\eta\pi/2}$, we retrieve the scaling relation (4).

##  Appendix D Parametrization of arbitrarily shaped 2D Fermi surfaces

This appendix details the methodology employed to calculate the surface-
impedance “phase diagrams” for non-polygonal 2D Fermi surfaces, including the
effects of an arbitrary curvature of Fermi-surface segments. For simplicity,
here we retain the assumption of a constant Fermi velocity modulus for any
orientation in reciprocal space, in accordance with Eq. (66), however the
direction of $\bm{v}_{\bm{k}_{F}}$ varies with orientation and is locally
orthogonal to the Fermi surface by definition [92]. A straightforward
generalization to cases where even the Fermi-velocity modulus is anisotropic
may be performed by referring to the more general parametrization (56). In
this section, we also retain the hypothesis that two mirror symmetry planes
exist in reciprocal space, according to Eq. (46) and employing $\beta=y$ and
$\alpha=x$ for the space directions.

Consider the 2D parametrization $k_{F}(\theta)$ for the variation of the Fermi
wave vector as a function of angle $\theta$. Then, a vector $\bm{n}_{\bm{k}}$
locally orthogonal to the curve $k_{F}(\theta)$ corresponds to

$\bm{n}_{\bm{k}}=\frac{d\bm{k}_{F}(\theta)}{d\theta}\times\hat{u}_{z},$ (205)

where $\hat{u}_{z}$ is the unit vector normal to the 2D plane where
$k_{F}(\theta)$ lies. The differential line element $dS$, which enters into
Fermi-surface integrations, results

$dS=n_{\bm{k}}d\theta,$ (206)

where

$n_{\bm{k}}=\left|\bm{n}_{\bm{k}}\right|=\sqrt{\left[k_{F}(\theta)\right]^{2}+\left[\frac{dk_{F}(\theta)}{d\theta}\right]^{2}}.$
(207)

Eq. (208) corresponds to the rotation by $\pi/2$ of the unit vector locally
tangent to the curve $k_{F}(\theta)$ at a given angle $\theta$. The unit
vector normal to $k_{F}(\theta)$ is then

$\hat{n}_{\bm{k}}=\frac{\bm{n}_{\bm{k}}}{n_{\bm{k}}}=\frac{1}{\sqrt{\left[k_{F}(\theta)\right]^{2}+\left[\frac{dk_{F}(\theta)}{d\theta}\right]^{2}}}\\\
\times\left[\hat{u}_{k_{F}}k_{F}(\theta)-\hat{u}_{\theta}\frac{dk_{F}(\theta)}{d\theta}\right],$
(208)

where $\hat{u}_{k_{F}}$ and $\hat{u}_{\theta}$ are the unit vectors in the
radial and angular directions, respectively. Then, remembering the
transformation of unit vectors from polar to Cartesian coordinates,

$\hat{u}_{k_{F}}=\cos\theta\hat{u}_{k_{x}}+\sin\theta\hat{u}_{k_{y}},$ (209a)
$\hat{u}_{\theta}=-\sin\theta\hat{u}_{k_{x}}+\cos\theta\hat{u}_{k_{y}},$
(209b)

combining Eqs. (208) and (209) we have

$\hat{n}_{\bm{k}}=N^{-1}\left\\{\left[\cos\theta
k_{F}(\theta)+\sin\theta\frac{dk_{F}(\theta)}{d\theta}\right]\hat{u}_{k_{x}}\right.\\\
\left.+\left[\sin\theta
k_{F}(\theta)-\cos\theta\frac{dk_{F}(\theta)}{d\theta}\right]\hat{u}_{k_{y}}\right\\},$
(210)

where $N\equiv n_{\bm{k}}$ obeys Eq. (207). Further decomposing Eq. (210) into
its components along $k_{x}$ and $k_{y}$ yields Eq. (152).

As an exemplary application, in the next section we calculate the components
(152) of $\hat{n}_{\bm{k}}$ for the “supercircle” geometry (153) of Sec.
VII.4.1, using selected values of the parameter $p$.

###  D.1 Parametrization of the “supercircle” Fermi surface

For the geometry (153), the derivatives $dk_{F}(\theta)/d\theta$ have to be
separately evaluated in the four quadrants of the 2D plane. For generic $p$,
we have

$\frac{dk_{F}(\theta)}{d\theta}=\left[\left|\cos\theta\right|^{p-1}\mathrm{sign}\left\\{\cos\theta\right\\}\sin\theta-\left|\sin\theta\right|^{p-1}\mathrm{sign}\left\\{\sin\theta\right\\}\cos\theta\right]\left[\left|\cos\theta\right|^{p}+\left|\sin\theta\right|^{p}\right]^{-\frac{1+p}{p}}.$
(211)

For generic values of $p$, the integrals (72) which contribute to the
transverse conductivity (71) have to be calculated numerically. Therefore, we
do not have a general closed expression for the conductivity for any momentum
and frequency. In the following, we separately analyze specific values of $p$:
we exactly retrieve the results of Secs. IV.3.1 and VI.2 for $p=1$ (“diamond”
shape) and of Secs. IV.1 and B.1 for $p=2$ (circular shape).

####  D.1.1 “Diamond” shape ($p=1$)

The derivatives (211) for $p=1$ are

$\frac{dk_{F}(\theta)}{d\theta}=\frac{-\cos\theta+\sin\theta}{(\cos\theta+\sin\theta)^{2}},\,0\leq\theta<\frac{\pi}{2},$
(212a)
$\frac{dk_{F}(\theta)}{d\theta}=\frac{-\cos\theta+\sin\theta}{-1+\sin(2\theta)},\,\frac{\pi}{2}\leq\theta<\pi,$
(212b)
$\frac{dk_{F}(\theta)}{d\theta}=\frac{\cos\theta-\sin\theta}{(\cos\theta+\sin\theta)^{2}},\,\pi\leq\theta<\frac{3\pi}{2},$
(212c)
$\frac{dk_{F}(\theta)}{d\theta}=\frac{\cos\theta+\sin\theta}{(\cos\theta-\sin\theta)^{2}},\,\frac{3\pi}{2}\leq\theta<2\pi.$
(212d)

The normalization factors (207) are

$n_{\bm{k}}=\frac{1}{\sqrt{2}}\left[(\cos\theta+\sin\theta)^{4}\right]^{-\frac{1}{2}},\,0\leq\theta<\frac{\pi}{2}\,v\,\pi\leq\theta<\frac{3\pi}{2},$
(213a)
$n_{\bm{k}}=\frac{1}{\sqrt{2}}\left[(\cos\theta-\sin\theta)^{4}\right]^{-\frac{1}{2}},\,\frac{\pi}{2}\leq\theta<\pi\,v\,\frac{3\pi}{2}\leq\theta<2\pi.$
(213b)

Using Eqs. (212) and (214) into Eqs. (152), we identically retrieve

$\hat{n}_{\bm{k}}=\frac{1}{\sqrt{2}}\hat{u}_{k_{x}}+\frac{1}{\sqrt{2}}\hat{u}_{k_{y}},\,0\leq\theta<\frac{\pi}{2}$
(214a)
$\hat{n}_{\bm{k}}=-\frac{1}{\sqrt{2}}\hat{u}_{k_{x}}+\frac{1}{\sqrt{2}}\hat{u}_{k_{y}},\,\frac{\pi}{2}\leq\theta<\pi,$
(214b)
$\hat{n}_{\bm{k}}=-\frac{1}{\sqrt{2}}\hat{u}_{k_{x}}-\frac{1}{\sqrt{2}}\hat{u}_{k_{y}},\,\pi\leq\theta<\frac{3\pi}{2},$
(214c)
$\hat{n}_{\bm{k}}=\frac{1}{\sqrt{2}}\hat{u}_{k_{x}}-\frac{1}{\sqrt{2}}\hat{u}_{k_{y}},\,\frac{3\pi}{2}\leq\theta<2\pi,$
(214d)

which is fully consistent with the piecewise-constant parametrization of Tab.
5 in Sec. IV.3.1.

####  D.1.2 Circular shape ($p=2$)

For a circular shape, the derivatives (211) are null in all quadrants:

$\frac{dk_{F}(\theta)}{d\theta}=0.$ (215)

Using Eq. (215) into Eq. (208), we retrieve

$\hat{n}_{\bm{k}}\equiv\hat{u}_{k_{F}},$ (216)

as it should be: the direction normal to the Fermi surface is the radial
direction for a circular shape. Hence, in this special case we identically
retrieve the results of Sec. IV.1.

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