*6.1. Heat Current*

The behavior of the photo-assisted charge current as a function of the AC and DC voltage contribution has been already discussed both in the non-interacting [16] and in the strongly interacting regime [17,53], the latter case showing divergencies at zero temperature signature of the limitations of the perturbative approach in this regime. In Figure 2, we report the density plots of its heat counterpart whose functional form has been derived in Equation (54). The first row represents the case of a sinusoidal drive with

$$V\_{\rm ac}^{(\rm cos)}(t) = -\frac{\omega \alpha}{\varepsilon} \cos\left(\omega t\right) \tag{73}$$

both in the free fermion case *ν* = 1 (top left panel) and at *ν* = 1/3 (top right panel). Here, all curves are mirror symmetric with respect to *α* = 0 ( J*BS*(*<sup>α</sup>*, *q*) = J*BS*(−*α*, *q*)) and *q* = 0 ( J*BS*(*<sup>α</sup>*, *q*) = J*BS*(*<sup>α</sup>*, −*q*)) and consequently satisfy

$$
\overline{\mathcal{J}\_{\rm BS}}(\mathfrak{a},q) = \overline{\mathcal{J}\_{\rm BS}}(-\mathfrak{a},-q). \tag{74}
$$

Both of these remarkable features are a direct consequence of the symmetries of the photo-assisted tunneling amplitudes. Indeed, for this kind of drive, one has [16,61]

$$p\_l^{(\cos)}(a) = j\_l(-a) \, , \tag{75}$$

with *Jl* Bessel function of order *l*. These amplitudes satisfy

$$p\_l^{(\cos)}(\alpha) = p\_l^{(\cos)}(-\alpha) \tag{76}$$

and

$$p\_l^{(\text{cos})}(\mathfrak{a}) = p\_{-l}^{(\text{cos})}(\mathfrak{a}). \tag{77}$$

Considering the case of a periodic train of Lorentzian pulses of the form

$$V\_{ac}^{(\text{Lor})}(t) = \frac{\omega}{\pi e} \sum\_{l=-\infty}^{+\infty} \frac{\eta}{\eta^2 + \left(\frac{t}{T} - l\right)^2} - \frac{\omega \alpha}{e},\tag{78}$$

with width at half height given by *η* and where the photo-assisted tunneling amplitudes are given by

$$p\_l^{(\text{Lor})}(a) = a \sum\_{s=0}^{+\infty} \frac{\Gamma(\alpha + l + s)}{\Gamma(\alpha + 1 - s)} \frac{(-1)^s e^{-2\pi \eta (2s + l)}}{\Gamma(l + s + 1)\Gamma(s + 1)}.\tag{79}$$

In this case, one observes that, at *ν* = 1 (bottom left panel of Figure 2), the heat current is still highly symmetric as in the case of the sinusoidal drive. This is an accidental consequence of the fact that, at this value of the filling factor for every periodic voltage drive, one has [50]

$$\sqrt{\mathcal{J}\_{\rm BS}}(a,q) \propto \sum\_{l=-\infty}^{+\infty} |p\_l(a)|^2 \left(q+l\right)^2 = \frac{e}{\hbar\omega} \int\_0^{\top} \frac{dt}{\mathcal{T}} V^2(t) = \frac{e}{\hbar\omega} V\_{dc}^2 + \frac{e}{\hbar\omega} \int\_0^{\top} \frac{dt}{\mathcal{T}} V\_{ac}^2(t),\tag{80}$$

which is manifestly insensitive to the overall sign of both the DC and the AC contribution to the voltage. This is no more true for what concerns the filling factors in the Laughlin sequence (bottom right panel of Figure 2) where

$$
\overline{\mathcal{J}\_{\rm BS}}(\mathfrak{a},q) \quad \neq \quad \overline{\mathcal{J}\_{\rm BS}}(-\mathfrak{a},q) , \tag{81}
$$

$$
\overline{\mathcal{J}\_{BS}}(\mathfrak{a},q) \quad \neq \quad \overline{\mathcal{J}\_{BS}}(\mathfrak{a},-q). \tag{82}
$$

However, the condition

$$p\_l^{(\text{Lor})}(a) = p\_{-l}^{(\text{Lor})}(-a) \tag{83}$$

leads to the residual symmetry

$$
\widetilde{\mathcal{T}\_{BS}}(\mathfrak{a},\mathfrak{q}) = \overline{\mathcal{T}\_{BS}}(-\mathfrak{a},-\mathfrak{q}).\tag{84}
$$

**Figure 2.** Density plot of the averaged heat current J*BS* (in units of |*λ*|2) as a function of the DC voltage amplitude *q* (*x*ˆ-axis) and the AC voltage amplitude *α* (*y*ˆ-axis). Other parameters are: *η* = 0.1, *θ* = 0 and *ω*c = 10*<sup>ω</sup>*.
