<figure><img src="image_28.png" /><figcaption>Figure 25. Bosonic spectrum for Model I (left panel) and Model II (right panel), as a function of $x_F$ and $\Delta_\phi$, respectively. In Model I, we used $\Delta = 2.5$, $g_5 = 8$, $r_1 = 10^{-10}$, and $r_2 = 15$. In Model II, we used $x_F = 1$, $\Delta = 3$, $g_5 = 8$, $\Delta_\phi = 0.2$, $r_1 = 10^{-12}$, and $r_2 = 15$. Both plots are normalised in units of the decay constant $\tilde{f}$. The colour coding for the spectrum is: singlet scalar (blue), non-singlet scalar (blue diamonds), tensor (red), pseudoscalar (black), vector (green), axial-vector (green diamonds); for further details see [1]. In Model I, since $\Delta > 2$, there is also a massless dilaton in the spectrum. In both panels, it is understood that heavier states are present in the right- and left-upper corners.</figcaption></figure>

where $\mu$ is a constant, $\kappa_0 = -2\gamma_E$, $\kappa_n = (n-1)!(n \ge 1)$, and the coefficients $h_k^{(n)}$ are defined through the expansion

$$ \frac{Y_{n-1}(z) - \frac{2}{\pi} \log\left(\frac{z}{2}\right) J_{n-1}(z)}{Y_n(z) - \frac{2}{\pi} \log\left(\frac{z}{2}\right) J_n(z)} \equiv \sum_{k \ge 0} h_k^{(n)} z^{2k+1}, \qquad (B.41) $$

one obtains the two-point functions

$$ \langle \mathcal{O}_R(q) \bar{\mathcal{O}}_R(-q) \rangle^{\pm} = \frac{2}{\not{q}} \left(\frac{Q}{2}\right)^{2n} \frac{1}{\kappa_n(n-1)!} \left[ \pi \frac{Y_{n-\frac{1}{2} \mp \frac{1}{2}}(e^{-r_1}Q)}{J_{n-\frac{1}{2} \mp \frac{1}{2}}(e^{-r_1}Q)} - 2 \log \left(\frac{Q}{\mu}\right) \right]. \qquad (B.42) $$

As can be seen, we have traded the UV scale corresponding to a finite $r_2$ for the renormalisation scale $\mu$.

## C Comparison with bosonic spectrum

In order to make the comparison between the fermionic and bosonic spectra of Models I and II, we here present two additional plots shown in figure 25, complementing the results previously obtained in [1]. We remind the Reader that what we call Model II in the current
paper is referred to as Model IIB in [1]. Furthermore, the spectrum is computed using the invariant $\mathcal{I} = \mathcal{I}_2$, which is a choice that only affects the results for the non-singlet scalar (for details, see [1]). The left panel of figure 25 shows the bosonic spectrum for Model I, as one approaches the upper bound on the number of flavours $x_F \le 2\Delta$. Conversely, the right panel shows the bosonic spectrum for Model II, as one approaches the upper bound on $\phi_c < \sqrt{3/\Delta_\phi}$ for a value of $\Delta_\phi < \frac{\Delta}{2\Delta-x_F}$. In both limits, the spectrum begins to form a continuum, i.e. the spacing between the heavy states approaches zero in units of $\tilde{f}$. However, there is an important difference in that for the former case the mass gap remains of the same order as $x_F$ is varied, resulting in a gapped continuum, while for the latter case, the continuum that forms for large values of $\phi_c$ starts at zero. We also note the presence of light states in both cases, namely the NGBs and the dilaton, which are both massless for Model I, while in Model II, increasing the source $\phi_c$ results in lifting the mass of the dilaton. The latter effect is small due to the choice of $\Delta_\phi = 0.2$ being close to marginal.

## D Holographic Wilsonian RG

For completeness, we give here a derivation of the flow equation given by eq. (4.14), following the arguments outlined in [42] for bosons, and generalised to fermions in [76, 77]. We start by observing that eq. (4.9) is invariant under change of the cutoff surface parametrized by $\mathbf{r}$. Recalling eq. (4.2), the actions appearing in eq. (4.9) are

$$ S_{\Psi}[r_1, \mathbf{r}; s_1, -1] = - \int_{r_1}^{\mathbf{r}} dr \int d^4 x \sqrt{-g} \left[ \frac{1}{2} (\overline{\Psi} \Gamma^M D_M \Psi - \overline{D_M} \Psi \Gamma^M \Psi) + H_{\Psi} \overline{\Psi} \Psi \right] \\ - \frac{s_1}{2} \int d^4 x \sqrt{-\tilde{g}} \overline{\Psi} \Psi \Big|_{r_1} + \frac{1}{2} \int d^4 x \sqrt{-\tilde{g}} \overline{\Psi} \Psi \Big|_{\mathbf{r}} \qquad (D.1) $$

and

$$ S_{UV}[\mathbf{r}, r_{UV}] = - \int d^4 q \sqrt{-\tilde{g}} \overline{\Psi}_L(-q, \mathbf{r}) f(q, \mathbf{r}) \Psi_L(q, \mathbf{r}), \qquad (D.2) $$

where, for ease of presentation, we have defined (see eq. (4.11))

$$ f(q, \mathbf{r}) \equiv \frac{i\not q}{e^{4A} N_L^2 \xi^2}, \qquad (D.3) $$

and we omitted the factor $N_\Psi$ which only appears as an overall constant, irrelevant in the following. In the large-$N_C$ limit, these actions are to be evaluated on the classical solutions, and the RG invariance of eq. (4.9) hence implies that

$$ \partial_{\mathbf{r}}(S_{\Psi}[r_1, \mathbf{r}; s_1, -1] + S_{UV}[\mathbf{r}, r_{UV}]) = 0. \qquad (D.4) $$

In addition, the variational problem demands that

$$ \begin{aligned} 0 &= \delta S_{\Psi}[r_1, \mathbf{r}; s_1, -1] + \delta S_{UV}[\mathbf{r}, r_{UV}] \\ &= \int d^4 q \sqrt{-\tilde{g}} \left\{ \left[ \overline{\Psi}_R(-q, \mathbf{r}) - \overline{\Psi}_L(-q, \mathbf{r}) f(q, \mathbf{r}) \right] \delta \Psi_L(q, \mathbf{r}) \right. \\ & \qquad \left. + \delta \overline{\Psi}_L(-q, \mathbf{r}) \left[ \Psi_R(q, \mathbf{r}) - f(q, \mathbf{r}) \Psi_L(q, \mathbf{r}) \right] \right\}, \end{aligned} \qquad (D.5) $$
leading to the boundary conditions relating $\Psi_R(\overline{\Psi}_R)$ to $\Psi_L(\overline{\Psi}_L)$:

$$ \begin{aligned} \overline{\Psi}_R(-q, \mathbf{r}) &= \overline{\Psi}_L(-q, \mathbf{r}) f(q, \mathbf{r}) \\ \Psi_R(q, \mathbf{r}) &= f(q, \mathbf{r}) \Psi_L(q, \mathbf{r}). \end{aligned} \qquad (D.6) $$

Together, eqs. (D.4) and (D.5) imply a flow equation for $f$, as follows. We have that

$$ \begin{aligned} \partial_{\mathbf{r}} \mathcal{S}_{\Psi}[r_1, \mathbf{r}; s_1, -1] &= \int d^4 x \left\{ \frac{1}{2} \partial_r (\sqrt{-\tilde{g}} \overline{\Psi} \Psi) - \sqrt{-g} \left[ \frac{1}{2} (\overline{\Psi} \Gamma^M D_M \Psi - \overline{D_M} \overline{\Psi} \Gamma^M \Psi) + H_{\Psi} \overline{\Psi} \Psi \right] \right\} \Big|_{\mathbf{r}} \\ &= \int d^4 q e^{4A} \left\{ (2 \partial_{\mathbf{r}} A - H_{\Psi}) [\overline{\Psi}_R(-q, \mathbf{r}) \Psi_L(q, \mathbf{r}) + \overline{\Psi}_L(-q, \mathbf{r}) \Psi_R(q, \mathbf{r})] \right. \\ & \qquad \left. + \overline{\Psi}_R(-q, \mathbf{r}) \partial_{\mathbf{r}} \Psi_L(q, \mathbf{r}) + \partial_{\mathbf{r}} \overline{\Psi}_L(-q, \mathbf{r}) \Psi_R(q, \mathbf{r}) \right. \\ & \qquad \left. - e^{-A} [\overline{\Psi}_L(-q, \mathbf{r})(i\not{q})\Psi_L(q, \mathbf{r}) + \overline{\Psi}_R(-q, \mathbf{r})(i\not{q})\Psi_R(q, \mathbf{r})] \right\} \qquad (D.7) \\ &= \int d^4 q e^{4A} \left\{ (4 \partial_{\mathbf{r}} A - 2H_{\Psi}) \overline{\Psi}_L(-q, \mathbf{r}) f(q, \mathbf{r}) \Psi_L(q, \mathbf{r}) \right. \\ & \qquad \left. + \overline{\Psi}_L(-q, \mathbf{r}) f(q, \mathbf{r}) \partial_{\mathbf{r}} \Psi_L(q, \mathbf{r}) + \partial_{\mathbf{r}} \overline{\Psi}_L(-q, \mathbf{r}) f(q, \mathbf{r}) \Psi_L(q, \mathbf{r}) \right. \\ & \qquad \left. - e^{-A} [\overline{\Psi}_L(-q, \mathbf{r})(i\not{q})\Psi_L(q, \mathbf{r}) + \overline{\Psi}_L(-q, \mathbf{r}) f(q, \mathbf{r})(i\not{q}) f(q, \mathbf{r}) \Psi_L(q, \mathbf{r})] \right\}, \end{aligned} $$

where we used eq. (D.5) to obtain an expression in terms of $\Psi_L$ and $\overline{\Psi}_L$. Furthermore,

$$ \partial_{\mathbf{r}} \mathcal{S}_{\text{UV}}[\mathbf{r}, r_{\text{UV}}] = - \int d^4 q e^{4A} \left\{ \begin{aligned} & 4 \partial_{\mathbf{r}} A \overline{\Psi}_L(-q, \mathbf{r}) f(q, \mathbf{r}) \Psi_L(q, \mathbf{r}) + \overline{\Psi}_L(-q, \mathbf{r}) \partial_{\mathbf{r}} f(q, \mathbf{r}) \Psi_L(q, \mathbf{r}) \\ & + \partial_{\mathbf{r}} \overline{\Psi}_L(-q, \mathbf{r}) f(q, \mathbf{r}) \Psi_L(q, \mathbf{r}) + \overline{\Psi}_L(-q, \mathbf{r}) f(q, \mathbf{r}) \partial_{\mathbf{r}} \Psi_L(q, \mathbf{r}) \end{aligned} \right\}, \qquad (\text{D.8}) $$

so that eq. (D.4) becomes

$$ \begin{aligned} 0 &= \partial_{\mathbf{r}}(\mathcal{S}_{\Psi}[r_1, \mathbf{r}; s_1, -1] + \mathcal{S}_{\text{UV}}[\mathbf{r}, r_{\text{UV}}]) \\ &= - \int d^4 q e^{4A} \overline{\Psi}_L(-q, \mathbf{r}) \left\{ \begin{aligned} & \partial_{\mathbf{r}} f(q, \mathbf{r}) + 2H_{\Psi} f(q, \mathbf{r}) \\ & + e^{-A} i \not{q} + f(q, \mathbf{r}) (e^{-A} i \not{q}) f(q, \mathbf{r}) \end{aligned} \right\} \Psi_L(q, \mathbf{r}). \end{aligned} \qquad (D.9) $$

One now observes that eq. (D.9) needs to hold for any classical solution. In particular, after imposing an IR boundary condition on $\Psi_L$, one can think of the space of such solutions as being parametrized by the value that $\Psi_L$ takes at the coordinate $r = \mathbf{r}$, which implies that

$$ \partial_{\mathbf{r}} f = -2H_{\Psi} f - e^{-A} i \not{q} - f (e^{-A} i \not{q}) f. \qquad (\text{D.10}) $$

Using the definition of $f(q, \mathbf{r})$ given in eq. (D.3), one finally obtains the RG flow equation for $\xi$:

$$ \partial_{\mathbf{r}} \xi^2 = -N_L^{-2} e^{-5A} q^2 + N_L^2 e^{3A} \xi^4. \qquad (\text{D.11}) $$
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