**4. Heat Current**

In order to extend the previous analysis to heat transport, we need to properly define the heat current operator. For a system described by a Hamiltonian density H and with particle number density N , the heat density Q is given by

$$
\mathcal{Q} = \mathcal{H} - \mu \mathcal{N},
\tag{37}
$$

with *μ* the chemical potential of the lead where the heat density is measured. This definition is motivated by the fact that the energy density depends on an arbitrary energy reference and, thus, it is not a well-defined experimental observable. On the contrary, heat is defined as the energy carried by particles with respect to a given chemical potential, thus motivating the expression in Equation (37).

According to this and considering again the chiral propagation of the bosonic modes along the edges [66], in the chiral Luttinger description (see Equation (1)), one defines the heat densities as

$$\mathcal{Q}\_{\mathbb{R}/L}(\mathbf{x}, t) = \frac{v}{4\pi t} \left(\partial\_{\mathbf{x}} \Phi\_{\mathbb{R}/L}(\mathbf{x}, t)\right)^2,\tag{38}$$

where all the contributions proportional to the chemical potential *μ* = *vkF* are automatically taken into account [57,67]. Notice that the above equation provides the proper value of the thermal Hall conductance in agreemen<sup>t</sup> with the Wiedemann–Franz law [67,68]

The corresponding heat current operators in the terminals 2 and 3 can be expressed in terms of heat density operators [57] as

$$\mathcal{J}\_{2/3}(t) = \pm v \,\,\mathcal{Q}\_{R/L}(\pm d, t) \tag{39}$$

due to the chirality of Laughlin edge states.

Proceeding as for the charge current, the heat current operators are represented in powers of the tunneling amplitude Λ,

$$\mathcal{J}\_{2/3}(t) = \mathcal{J}\_{2/3}^{(0)}(t) + \mathcal{J}\_{2/3}^{(1)}(t) + \mathcal{J}\_{2/3}^{(2)}(t) + \mathcal{O}\left(\left|\Lambda\right|^3\right),\tag{40}$$

with

$$\mathcal{J}\_{2/3}^{(0)}(t) = \pm v \mathcal{Q}\_{R/L}^{(0)}(\pm d, t), \tag{41}$$

$$\mathcal{J}\_{2/3}^{(1)}(t) = \pm i v \int\_{-\infty}^{t} dt' \left[ H\_T(t'), \mathcal{Q}\_{\mathbb{R}/L}^{(0)}(\pm d\_\prime t) \right],\tag{42}$$

$$\mathcal{J}\_{2/3}^{(2)}(t) = \pm i^2 v \int\_{-\infty}^t dt' \int\_{-\infty}^{t'} dt'' \left[ H\_T(t''), \left[ H\_t(t'), \mathbb{Q}\_{R/L}^{(0)}(\pm d, t) \right] \right]. \tag{43}$$

In the above equations, we have denoted with Q(0) *<sup>R</sup>*/*L*(*<sup>x</sup>*, *t*) the time evolution of heat densities in the absence of tunneling, which can be obtained from the time evolution of bosonic fields in Equation (5) and reads

$$\mathcal{Q}\_{\mathbb{R}}^{(0)}(\mathbf{x},t) = \frac{\upsilon}{4\pi} \left[ \left( \partial\_{\mathbf{x}} \phi\_{\mathbb{R}}(\mathbf{x},t) \right)^{2} + 2\varepsilon \sqrt{\upsilon} \partial\_{\mathbf{x}} \phi\_{\mathbb{R}}(\mathbf{x},t) V\_{\mathbb{R}/\mathbb{L}} \left( t \mp \frac{\mathbf{x}}{\upsilon} \right) + \frac{e^{2}\nu}{\upsilon} V\_{\mathbb{R}}^{2} \left( t \mp \frac{\mathbf{x}}{\upsilon} \right) \right], \tag{44}$$

$$Q\_L^{(0)}(\mathbf{x}, t) \quad = \ \frac{\upsilon}{4\pi} \left(\partial\_\mathbf{x} \phi\_{\mathbb{R}/L}(\mathbf{x}, t)\right)^2. \tag{45}$$

The commutators involving Q(0) *<sup>R</sup>*/*L*(*<sup>x</sup>*, *t*) in Equations (42) and (43) are

$$
\mathcal{J}\_{2/3}^{(1)}(t) = \pm \mathcal{Q}\_{\mathbb{R}/L} \left( \pm d, t \right), \tag{46}
$$

$$\mathcal{J}\_{2/3}^{(2)}(t) = \pm i \int\_{-\infty}^{t - \frac{d}{v}} dt'' \left[ H\_l(t''), \dot{\mathcal{Q}}\_{\mathbb{R}/L}(\pm d, t) \right],\tag{47}$$

where

$$
\dot{Q}\_{\mathbb{R}}(\mathbf{x},t) = \upsilon \Lambda \left(\partial\_{\mathbf{x}} + ik\_{\mathbb{F}}\right) \Psi\_{\mathbb{R}}^{\dagger}(\mathbf{x},t) \Psi\_{L}(\mathbf{x},t) + \text{H.c.},\tag{48}
$$

$$\dot{Q}\_{\rm L}(\mathbf{x},t) = -v\Lambda \Psi\_{\rm R}^{\dagger}(\mathbf{x},t) \left(\partial\_{\mathbf{x}} + ik\mathbf{r}\right) \Psi\_{\rm L}(\mathbf{x},t) + \text{H.c.} \tag{49}$$

According to this, the average of heat current operators in Equation (40) reads

$$
\langle \mathcal{J}\_{2/3}(t) \rangle = \langle \mathcal{J}\_{2/3}^{(0)} \rangle + \langle \mathcal{J}\_{2/3}^{(2)} \rangle. \tag{50}
$$

Analogously with the case of the charge current, one has that J (1) 2/3 is zero due to the unbalance between annihilation and creation field operators of each chirality. Focusing on the heat current which is backscattered by the QPC into reservoir 3, one can define the backscattering heat current as

$$
\mathcal{J}\_{\rm BS}(t) = \langle \mathcal{J}\_{\rm 3}^{(2)}(t) \rangle. \tag{51}
$$

It can be expressed in terms of Green's functions in Equations (26) and (27), by exploiting the explicit expression of *Q*˙ *<sup>L</sup>*(*<sup>x</sup>*, *t*) in Equations (48) and (49), thus finding

$$\begin{split} \mathcal{J}\_{RS}(t) &= i|\Lambda|^2 \int\_{-\infty}^{t-\frac{d}{\sigma}} d\tau \Big[ G\_R^{<}\left(t', t - \frac{d}{\upsilon}\right) \left(\partial\_t - ik\_F \upsilon\right) G\_L^{>}\left(t', t - \frac{d}{\upsilon}\right) + \\ &+ G\_R^{>}\left(t', t - \frac{d}{\upsilon}\right) \left(\partial\_t + ik\_F \upsilon\right) G\_L^{<}\left(t', t - \frac{d}{\upsilon}\right) + \\ &- G\_R^{<}\left(t - \frac{d}{\upsilon}, t'\right) \left(\partial\_t - ik\_F \upsilon\right) G\_L^{>}\left(t - \frac{d}{\upsilon}, t'\right) + \\ &- G\_R^{>}\left(t - \frac{d}{\upsilon}, t'\right) \left(\partial\_t + ik\_F \upsilon\right) G\_L^{<}\left(t - \frac{d}{\upsilon}, t'\right) \Big]. \end{split} \tag{52}$$

By recalling the link between Green's functions and the function P*ν*(*t*), the heat backscattering current becomes

$$\mathcal{J}\_{\rm BS}(t) = i|\lambda|^2 \int\_0^{+\infty} d\tau \cos\left[\nu e \int\_t^{t-\tau} dt' V\_{\rm R}(t')\right] \partial\_{\rm \tau} \left[\mathcal{P}\_{2\nu}(\tau) - \mathcal{P}\_{2\nu}(-\tau)\right],\tag{53}$$

which, averaged over one period, reduces to

$$\overline{\mathcal{J}\_{BS}(t)}(a,q) = |\lambda|^2 \frac{\omega \nu}{2} \sum\_{l} |p\_l(a)|^2 (q+l) \left\{ \vec{\mathcal{P}}\_{2\nu} \left[ (q+l)\omega \right] - \vec{\mathcal{P}}\_{2\nu} \left[ -(q+l)\omega \right] \right\}.\tag{54}$$

### **5. Mixed Noise and Heat Noise**

Concerning the fluctuations, together with the conventional current noise [69], it is possible to define both the autocorrelated mixed noise and heat noise [45,46,50]. For the sake of simplicity, we will focus exclusively on the signal detected in reservoir 3 considering the quantities

$$\mathcal{S}\_X = \int\_0^\mathcal{T} \frac{dt}{\mathcal{T}} \int\_{-\infty}^{+\infty} dt' \left[ \left< j\_3(t') \mathcal{J}\_3(t) \right> - \left< j\_3(t') \right> \left< \mathcal{J}\_3(t) \right> \right],\tag{55}$$

$$\mathcal{S}\_{Q} = \int\_{0}^{\mathcal{T}} \frac{dt}{\mathcal{T}} \int\_{-\infty}^{+\infty} dt' \left[ \left< \mathcal{J}\_{3}(t')\mathcal{J}\_{3}(t) \right> - \left< \mathcal{J}\_{3}(t') \right> \left< \mathcal{J}\_{3}(t) \right> \right]. \tag{56}$$

Notice that analogous expressions can be derived focusing on the reservoir 2.

The perturbative expansion of charge and heat current operators in Equations (13) and (40) allows for also expressing these quantities perturbatively in Λ, namely

$$\mathcal{S}\_X = \mathcal{S}\_X^{(02)} + \mathcal{S}\_X^{(20)} + \mathcal{S}\_X^{(11)} + \mathcal{O}\left(|\Lambda|^3\right),\tag{57}$$

$$\mathcal{S}\_{\mathcal{Q}} = \mathcal{S}\_{\mathcal{Q}}^{(02)} + \mathcal{S}\_{\mathcal{Q}}^{(20)} + \mathcal{S}\_{\mathcal{Q}}^{(11)} + \mathcal{O}\left(|\Lambda|^3\right),\tag{58}$$

where

$$\mathcal{S}\_{X}^{(ij)} = \int\_{0}^{T} \frac{dt}{T} \int\_{-\infty}^{+\infty} dt' \left\{ \langle l\_{3}^{(i)}(t') \mathcal{J}\_{3}^{(j)}(t) \rangle - \langle l\_{3}^{(i)}(t') \rangle \langle \mathcal{J}\_{3}^{(j)}(t) \rangle \right\},\tag{59}$$

$$\mathcal{S}\_{Q}^{(ij)} = \int\_{0}^{\mathcal{T}} \frac{dt}{\mathcal{T}} \int\_{-\infty}^{+\infty} dt' \left\{ \langle \mathcal{J}\_{3}^{(i)}(t') \mathcal{J}\_{3}^{(j)}(t) \rangle - \langle \mathcal{J}\_{3}^{(i)}(t') \rangle \langle \mathcal{J}\_{3}^{(j)}(t) \rangle \right\}. \tag{60}$$

In the perturbative expansions of Equations (57) and (58), the only surviving terms are S(11) *X* and S(11) *Q* , since one can show that [50,66]

$$\mathcal{S}\_X^{(02)} = \mathcal{S}\_X^{(20)} = 0,\tag{61}$$

$$\mathcal{S}\_Q^{(02)} = \mathcal{S}\_Q^{(20)} = 0.\tag{62}$$

Mixed and heat noises are obtained in terms of Green's functions as

$$\mathcal{S}\mathbf{x} = i\left|\Lambda\right|^2 \int\_0^T \frac{dt}{T} \int\_{-\infty}^{+\infty} dt' \left[ \mathbf{G}\_R^{<}(t',t)(\partial\_t - i\mathbf{k}\cdot\mathbf{r})\mathbf{G}\_L^{<}(t',t) - \mathbf{G}\_R^{<}(t',t)(\partial\_t + i\mathbf{k}\cdot\mathbf{r})\mathbf{G}\_L^{<}(t',t) \right], \tag{63}$$
 
$$\mathcal{L}\mathbf{x} = \lim\_{\Delta t \to 0} \mathbf{A}^T \int\_0^T dt \int\_{-\infty}^{+\infty} \mathbf{r} \mathbf{G}\_{JL}^{<}(\mathbf{r} \le (\mathbf{r} \le \mathbf{r})\odot\mathbf{r}, \dots, \mathbf{r} \le \mathbf{r} \le \mathbf{r}' \le \mathbf{r}', \mathbf{r}),$$

$$\mathcal{S}\_{\mathbb{Q}} = |\Lambda|^2 \int\_0^l \frac{dt}{\mathcal{T}} \int\_{-\infty}^{+\infty} dt' \Big[ G\_{\mathbb{R}}^{<}(t', t) (\partial\_{t'} + ik\_F v) (\partial\_{t'} - ik\_F v) G\_L^{<}(t', t) + \\\\ + G\_{\mathbb{R}}^{<}(t', t) (\partial\_{t'} - ik\_F v) G\_L^{<}(t', t) \Big] \tag{64}$$
 
$$+ G\_{\mathbb{R}}^{<}(t', t) (\partial\_{t'} - ik\_F v) (\partial\_{t'} + ik\_F v) G\_L^{<}(t', t) \Big] \tag{65}$$

and, in terms of the function P*ν*(*t*) in Equation (28), they can be rewritten as

$$\mathcal{S}\_X = 2\nu e |\lambda|^2 \int\_0^\mathcal{T} \frac{dt}{\mathcal{T}} \int\_{-\infty}^{+\infty} dt' \sin\left[\nu \varepsilon \int\_{t'}^t dt'' V(t'')\right] \mathcal{P}\_\nu(t'-t) \partial\_{t'} \mathcal{P}\_\nu(t'-t),\tag{65}$$

$$\mathcal{S}\_{\mathbb{Q}} = 2|\lambda|^2 \int\_0^{\mathcal{T}} \frac{dt}{\mathcal{T}} \int\_{-\infty}^{+\infty} dt' \cos \left[ \nu \epsilon \int\_{t'}^t dt' V(t'') \right] \mathcal{P}\_{\mathbb{v}}(t'-t) \partial\_t \partial\_{t'} \mathcal{P}\_{\mathbb{v}}(t'-t). \tag{66}$$

Using again the series expansion of the voltage dependent phase factor and the Fourier transform for P*ν*(*t* − *t*), one is left with

$$\mathcal{S}\_{X}(\mathfrak{a},q) = \frac{\mathsf{V}\mathfrak{C}\omega}{2} |\lambda|^{2} \sum\_{l} |p\_{l}(\mathfrak{a})|^{2} (q+l) \left\{ \vec{\mathcal{P}}\_{2\nu} \left[ (q+l)\omega \right] + \vec{\mathcal{P}}\_{2\nu} \left[ -(q+l)\omega \right] \right\},\tag{67}$$

$$\mathcal{S}\_{\mathbb{Q}}(a,q) = \frac{|\lambda|^2}{2\pi} \sum\_{l} |p\_l(a)|^2 \int\_{-\infty}^{+\infty} dE E^2 \bar{\mathcal{P}}\_{\mathbb{V}}(\mathbb{E}) \left\{ \bar{\mathcal{P}}\_{\mathbb{V}} \left[ (q+l)\omega - E \right] + \bar{\mathcal{P}}\_{\mathbb{V}} \left[ -(q+l)\omega - E \right] \right\},\tag{68}$$

where we have explicitly indicated the dependence on the AC (*α*) and DC (*q*) voltage amplitude.

To perform the integral in the equation for S*Q*, we exploit the identity

$$\int\_{-\infty}^{+\infty} \frac{dY}{2\pi} Y^2 \tilde{\mathcal{P}}\_{\xi 1}(Y) \tilde{\mathcal{P}}\_{\xi 2}(X - Y) = \frac{\tilde{\mathcal{P}}\_{\xi 1 + \xi 2}(X)}{1 + \mathcal{g}\_1 + \mathcal{g}\_2} \left[ \mathcal{g}\_1 \mathcal{g}\_2 \pi^2 \theta^2 + \frac{\mathcal{g}\_1 (1 + \mathcal{g}\_1)}{\mathcal{g}\_1 + \mathcal{g}\_2} \omega^2 \right] \tag{69}$$

obtaining the expression

$$\mathcal{S}\_{\mathbb{Q}}(\mathfrak{a},q) = |\lambda|^2 \sum\_{l} |p\_l(\mathfrak{a})|^2 \left[ \frac{2\pi^2 \nu^2}{1+2\nu} \theta^2 + \frac{1+\nu}{1+2\nu} (q+l)^2 \omega^2 \right] \left\{ \vec{\mathcal{P}}\_{2\mathbb{P}} \left[ (q+l)\omega \right] + \vec{\mathcal{P}}\_{2\mathbb{P}} \left[ -(q+l)\omega \right] \right\}. \tag{70}$$

At temperature zero, the above expressions reduce to

$$\mathcal{S}\_X(a,q) = \nu \epsilon |\lambda|^2 \frac{\pi}{\Gamma(2\nu)} \left(\frac{\omega}{\omega\_0}\right)^{2\nu} \sum\_l |p\_l(a)|^2 |q+l|^{2\nu} \text{sign}(q+l),\tag{71}$$

$$\mathcal{S}\_{\mathbb{Q}}(a,q) = \omega |\lambda|^2 \frac{\pi(1+\nu)}{\Gamma(2\nu)(1+2\nu)} \left(\frac{\omega}{\omega\_{\mathbb{C}}}\right)^{2\nu} \sum\_{l} |p\_l(a)|^2 |q+l|^{2\nu+1}.\tag{72}$$

Here, the chiral free fermion case is recovered directly by inserting *ν* = 1. It is worth noticing that this peculiar state can be described exactly (to all order in the tunneling amplitude) in terms of the scattering theory [69,70]. Moreover, at this value of the filling factor, the dependence on *a* (finite length cut-off) disappears.

These noises will be investigated in the following with the aim of carrying out their spectroscopic analysis. For sake of simplicity, we will focused on the zero temperature limit, the finite temperature correction being negligible as far as *θ ω*.
