An impurity in a Luttinger Liquid coupled to Ohmic-class environments

We investigate the impact of noisy environments on transport through a one dimensional interacting quantum wire containing a scattering impurity. The impurity is known to induce a metalinsulator quantum phase transition as a function of the interaction strength in the wire. By considering Ohmic-class environments (ranging from Ohmic, to sub-Ohmic and super-Ohmic cases), we elucidate the influence of fluctuations on this phase transition. Within a perturbative renormalization group approach, we show that Ohmic environments keep the phase transition intact, while suband super-Ohmic environments, modify it into a smooth crossover at a scale that depends on the interaction strength in the wire. The system, however, still undergoes a metal-to-insulator-like transition when moving from sub-Ohmic to super-Ohmic environment noise. We further consider realistic experimental conditions due to finite length and finite temperatures of the wire, and explore the signatures of the phase transition in the temperature dependence of the impurity-induced back-scattering electrical conductance.

Quantum transport through a 1D wire described by a spinless TLL theory has been the subject of intense theoretical [15,16,[24][25][26][27][28][29] and experimental studies [30,31]. It was shown that special care needs to be taken when considering the coupling of such wires to external electronic leads [25,26,32,33]. When a clean wire is adiabatically coupled to leads [25,26], the conductance is independent of the electron-electron interactions inside the wire, and is given by the unit of conductance e 2 h. However, placing an impurity amidst a wire with repulsive many-body interactions, the conductance through it becomes suppressed due to back-scattering processes, and it exhibits a power-law temperature-dependence with exponents entirely determined by the many-body interactions [25,26,31]. This is reminiscent of the Kane-Fisher quantum phase transition between a metallic and an insulating state at the non-interacting critical point [15,16]. Transport across an impurity in a TLL is, therefore, of particular interest, as it reveals the interaction strength within the wire, and crucially-depends on the resulting 1D strongly-correlated state.
Motivated by technological advances in cold atom experiments [34] and circuit quantum simulations [17], there is a surge of interest in the nonequilibrium dynamics of open quantum 1D systems. Correspondingly, the problem of an impurity in a TLL is being revisited with various studies on the impact of, e.g., the induced dissipation by a lossy impurity [35][36][37] or a non-local two-body loss [38], as well as on their attenuation of critical phenomena [39]. Similarly, the impact of outof-equilibrium bias across the wire has been extensively explored [40][41][42][43]. Note that to simplify the theoretical treatment, the electronic leads are routinely modelled as one-dimensional non-interacting electrons (Ohmic environment) [25,26,43], rather than more realistic higherdimensional systems [44,45]. In an attempt to consider a more realistic environment, some of us introduced a lowpass filter (finite capacitance) to the leads to account for charging effects inside them [46]. This produces a characteristic charging time to the Ohmic environment that competes with the time of flight of the TLL plasmons, altering the high-energy properties of the wire.
In this work, we go beyond the Ohmic-environment assumption and reveal how the low-energy Kane-Fisher's physics becomes dominated by the environment fluctuations. Specifically, we consider sub-Ohmic and super-Ohmic current noise fluctuations in the environment that compete with the wire's many-body effects. We show that while super-Ohmic dissipation localizes the particles akin to a Zeno effect, the sub-Ohmic fluctuations overtake the low-energy properties of the TLL. The impurity, then, engenders a wire-environment competition with a non-monotonous renormalization group (RG) flow, leading ultimately to a metal-to-insulator-like transition in the wire as a function of the noise statistics. Furthermore, considering realistic finite wires, we predict that the non-monotonous flow implies unusual temperaturedependent scaling of the conductance coming out from arXiv:2012.04628v1 [cond-mat.str-el] 8 Dec 2020 The phase diagram of a two-level system (φ 4 -theory) coupled to an Ohmic environment for varying dissipation strength α. For small dissipation α < 1, the effective tunneling, Γ, between two potential wells diverges, i.e., the particle is delocalized, but for stronger dissipation α > 1 the particle is localized Γ → 0 [47]. (d) Phase diagram of the TLL hosting an impurity coupled to Ohmic leads [cf. action (4), with effective scattering potential V mapped to a Sine-Gordon potential] as a function of the interaction strength in the wire Kw [cf. Eq. (2)].
the wire-environment competition.
Setup and microscopic model -We consider a system of interacting spinless electrons confined in a singlechannel 1D wire of length L that contains a single impurity. The wire is additionally adiabatically connected to metallic leads, see Fig. 1(a). The Hamiltonian of the wire reads where the first term represents the kinetic energy of electrons with linearized dispersion k = α η v F k, v F the electron velocity, and α L = 1 (α R = −1) corresponds to the left-(right-)moving electrons with fermionic field operators Ψ L (Ψ R ). The second term describes local electronelectron interactions inside the wire via (normal-ordered) density operators ρ η (x) =∶ Ψ † η (x)Ψ η (x) ∶ with a constant magnitude U (x) = U ≠ 0 only inside the wire (x ∈ [−L 2, L 2]). The third term corresponds to a backscattering impurity of strength V 0 at position x 0 .
The wire is adiabatically connected to electronic leads, which we introduce by imposing appropriate boundary conditions ∂ t φ η (x = ±L 2) = 2πJ η , where J η (ω) is the current operator in the leads. The effect of the boundaries enter the correlation functions of the wire via the noise power spectrum S(ω) = ⟨J η (ω)J η (−ω)⟩, where ⟨⋯⟩ denotes thermal averaging with respect to the Fermi liquid state of the leads [46][47][48][49]. We consider an Ohmicclass noise power spectrum where ω c is the characteristic energy scale of the environment, indicating the exponential suppression of currentcurrent correlations for ω ≫ ω c . The parameter s ∈ (0, 2) distinguishes between different cases, i.e., s = 1 describes an Ohmic lead, whereas s < 1 (s > 1) corresponds to the sub-(super-)Ohmic case. The noise power exhibits a bosonic distribution n b (βω) = 1 [exp(βω) − 1] at inverse temperature β.
To realize an Ohmic environment, it suffices to consider free fermions with a well-defined Fermi-Dirac distribution. On the other hand, non-Ohmic environments with s ≠ 1 can be realized, for example, by electron-phonon coupling in the leads (s > 1) [50], or by complex RC circuit architectures (s < 1) [51]. In Fig. 1(b), we plot the frequency dependence of the noise spectrum for these three cases. Comparing to the Ohmic case, the currentcurrent fluctuations in the sub-(super)-Ohmic leads are more dominant at lower (higher) frequencies, i.e., environmental fluctuations are slower (faster). Ohmic-class environments have been extensively studied in the framework of the spin-boson model [47,[51][52][53][54], revealing the profound influence of the environment fluctuations on the nature of the ground state, as well as on the dynamics of the system. In particular, it was shown that in the Ohmic case, there exist a critical dissipation that distinguishes between a localized phase and a delocalized one, see Fig. 1(c). In contrast, in the sub-(super)-Ohmic case, the system is argued to be localized (delocalized) independent of dissipation strength [47]. Analogously, in this work, we investigate the impact of such current fluctuations in the leads on the transport through a disordered interacting wire.
We formulate the impurity scattering in imaginarytime path-integral formalism.
The action of the bosonized system at all positions x ≠ x 0 is quadratic and can be therefore integrated out, resulting in the following local (extended Sine-Gordon) action Green's function of the clean wire at x = x 0 . Without loss of generality, we assume x 0 = 0 [55]. The wire's plasmonic greater and lesser Green's functions read respectively, with ⟨⋯⟩ 0 referring here to the thermal average with respect to the clean wire. Such local Green's functions are entirely determined by the Hamiltonian (2) and the noise power spectrum at the boundaries (3) [46,49]. Note that the action (4) and the resulting correlation functions describes physics found in a variety of systems, including Brownian motion of a quantum mechanical particle in a periodic potential [56], as well as in the dissipative two-level system [47,52,57,58]; the wire-environment competition manifests through the specific functional form of G 0 ϕϕ . Infinitely long wire at zero temperature -First, we consider the limit of L → ∞ at T = 0, and investigate the impact of the noise-spectrum on the renormalization of the backscattering impurity potential in Eq. (4). In this limit, the plasmonic greater Green's function reads G >,0 ϕϕ (ω) = −iK w S(ω) ω 2 , and detailedbalance holds G <,0 ϕϕ (ω) = e −βω G >,0 ϕϕ (ω) as expected for bosons in thermal equilibrium [59]. The noise spectrum of Ohmic leads at low energies scales linearly with energy, S(ω) ∼ ω, and commonly in this limit, we observe G 0 ϕϕ (iω) = K w 2ω . More generally, we can always define a similar structure G 0 ϕϕ (iω) = K(ω) 2ω with an energydependent Luttinger parameter K(ω) incorporating also the fluctuations from the leads, see Fig. 2(a). At low frequencies ω ≪ ω c , it can be approximated as Unlike the Ohmic case, the frequency-dependence of the effective Luttinger parameter with non-Ohmic leads modifies the physics of the interacting wire. In particular, for the sub-Ohmic case (slow environmental fluctuations), at sufficiently low-frequencies, the effective interaction in the wire appears attractive [K(ω) > 1], while for the super-Ohmic case, the interaction becomes effectively We employ a perturbative RG approach (due to the presence of infrared divergences [51]), in which we integrate out high energy fields and map the system (4) to itself, but with a smaller ultraviolet cutoff Λ ′ = Λ(1 − dl), i.e., dl = dΛ Λ [15]. As a result, a renormalized scattering potential V (Λ) (up to dl 2 ) obeys the flow equation Note that due to the noisy leads, the flow involves also the renormalization of K(ω). The numerical solution of the flow equation is shown in Fig. 2. For the Ohmic case s = 1 [ Fig. 2(b)], standard Kane-Fisher physics [15] is observed, where the wire's Luttinger-Liquid parameter plays a decisive role, i.e., for The quantum critical point is at K w = 1, corresponding to non-interacting electrons, see Fig. 1(d).
In the case of a sub-ohmic noise spectrum, see Fig. 2(c), the low-frequency noise induces an effective K(ω) that increases with Λ. Therefore, starting from repulsive interactions in the wire, the renormalized scattering potential exhibits a non-monotonic behaviour. Specifically, defining Λ * such that K(Λ * ) = 1, we observe that for Λ > Λ * , the back-scattering potential increases and the transport through the wire will be suppressed, while for Λ < Λ * , it decreases and transport ignores the impurity. The transition point Λ * strongly depends on the wire's bare Luttinger Liquid parameter K w , cf. Eq. (5). Crucially, regardless of the specific initial microscopic parameters of the wire, the fixed point of the flow is K → ∞, V → 0. We compare the sub-ohmic behaviour with the super-ohmic case, where the flow lines are reversed as shown in Fig. 2(d). In this case, the effective K(ω) is reduced and the fixed point is realized at K → 0, V → ∞. We conclude that the quantum phase transition that occurs for the Ohmic environment from the insulating state to the metallic one with quantum critical point at K w = 1, for non-Ohmic environment is replaced by a smooth cross-over with characteristic energy scale Λ * that depends on interaction strength in the wire.
Conductance of a finite wire at finite temperature -At finite temperatures and for a finite length of the wire, the local plasmonic Green's function of a clean wire is the structure function of a many-body Fabry-Pérot interferometer that is formed due to the presence of the leads reflecting the plasmons at the boundaries [46,49]. The finite length of the wire introduces a characteristic time scale to the system, namely, the time of flight for the plasmons τ L = LK w v F to cross the wire. At high frequencies, ωτ L ≫ 1, the system acts similarly to the infinite wire [60], whereas at small frequencies, ωτ L ≪ 1, F (x, ω) ≈ 1 and thereby the physics of the interacting wire is washed out and the behaviour of the system is dominated by the environment. In order to study ac-conductance through the wire, we consider an external probe in the form of an electric potential U (x, t) = U (x) cos(ωt), leading to the additional term in the Hamiltonian, δH = ∑ η ∫ dxρ η (x)U (x). The ac-conductance within linear response theory [25,26,60,61] where the self-energy can be obtained by expanding the partition function corresponding to the action (4) [60], which up to V 2 0 reads with Thereby the ac-conductance can be written as ϕϕ (ω) corresponds to the conductance through a clean wire, and G b ≡ −2iωG R,0 ϕϕ Σ R (ω)G R,0 ϕϕ represents the correction to the conductance due to the presence of the impurity. In the dclimit, lim ω→0 G 0 (ω) = (ω ω c ) s−1 , which is independent of temperature and interaction strength inside the wire. For the Ohmic case, lim ω→0 G 0 (ω) = 1, for sub-Ohmic this limit diverges, and in the super-Ohmic case it vanishes.
In the following, we focus on the temperaturedependence of the correction G b , see Fig. 3. The temperature mimics the RG flow of the renormalized scattering potential (cf. Fig. 2): (i) In the Ohmic case [ Fig. 3(a)], for 1 τ L ≪ T ≪ ω c , we obtain a power-law temperature dependence of the form G b ∝ (T ω c ) 2Kw−2 . Such a power-law is characteristic for critical scaling close to a quantum phase transition [60]. Specifically, for repulsive interaction, G b grows with decreasing temperature, while it gets suppressed for attractive interactions. For a noninteracting wire, G b is independent of temperature. Interestingly, at small temperatures (T τ L ≪ 1), we observe a temperature-independent behaviour for all values of K w , corresponding to the cutoff of the critical scaling by the finite length of the wire; (ii) For the sub-and super-Ohmic case [ Figs. 3(b) and (c), respectively], we observe a characteristic energy-scale Λ * above which the system qualitatively behaves as in the Ohmic case, i.e., with a power-law decrease governed by the noise scaling with ω c . However, at T ≪ Λ * , for the sub-Ohmic case, we obtain an exponential suppression with exponents depending on the interaction strength as In contrast, in the super-Ohmic case for T ≪ Λ * , the G b grows with decreasing temperature in a power-law fashion with an exponent that is entirely independent of the interaction strength in the wire T 2s−4 . The finite length effect at low temperatures is washed away by the noisy environment.
Conclusion The noise statistics of the charge fluctuation at the boundaries of an interacting wire can modify the transport through a dirty Luttinger liquid beyond the Kane-Fisher description. Within a perturbative RG analysis, we showed how an adiabatic coupling to an Ohmic-class environment can turn a repulsivelyinteracting wire to behave as an attractively-interacting one [and vice versa] at certain energy scales. We further outline the physical implications of the wire-environment competition for transport measurements within linear response. Specifically, at low temperatures, the impurityinduced correction to conductance (i) follows the result of Kane-Fisher [15], and critically-scales with an interaction-dependent power-law up to a finite length cutoff in the Ohmic case, (ii) get washed out in the sub-Ohmic case due to the dominant role of slow (viscous) fluctuations in the environment, and (iii) is effectively amplified as the fast charge-fluctuations (super-Ohmic) at the boundary of the wire acts similarly to a Zeno effect [47]. Future work can extend our analysis beyond the linear response paradigm and employ, e.g., non-equilibrium bosonization [42], as well as explore more sophisticated impurities that allow for incoherent scattering processes [36].
We thank A. Chiocchetta for fruitful discussions. We acknowledge financial support from the Swiss National Science Foundation.

Supplemental Material for
An impurity in a Luttinger Liquid coupled to Ohmic-class environments Andisheh Khedri, AntonioŠtrkalj, and Oded Zilberberg Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland

PLASMONIC GREEN'S FUNCTION OF A CLEAN WIRE
In this section, we outline the calculation details of the plasmonic Green's function for the interacting wire. We elaborate on its crucial dependence on the environment through the boundary conditions. As mentioned in the main text, we restrict the bosonization treatment to the interacting part of the system (wire), i.e., x ∈ [−L 2, L 2], and account for the presence of the environment (leads) through the following boundary conditions (continuity equation): where the boundary operators J L R are the current operator in the non-interacting leads L R,ω the fermionic (creation) annihilation operators for right/left-moving electrons. We use the boundary condition (I.1) to solve the equations of motions for the bosonic fields ϕ(x, t), θ(x, t) = (1 . We obtain a solution in the form of the original right-(left-)mover fields that reads [1,2] , (I.5) with τ L = LK w ν F the time-of-flight required for the plasmons to cross the wire. Note that from the particlehole symmetry of the currents at the boundaries J L R (−ω) = J † L R (ω), the following symmetry for the fields holds Using the current-current correlations at the boundaries, we can define the environment noise spectrum S(ω) as Thus, we can fully determine the correlation functions of the plasmonic fields as with the structure function , (I. 8) which encodes all the information about the interacting wire, i.e., its length L, and the interaction strength K w . Note that, due to the presence of the environment and the correspondingly-imposed boundary, a Fabry-Pérot cavity . (I.9)

Analytic properties of the local structure function
It is insightful to examine the analytic properties of the local structure function (I.9). For this purpose, we generalise the structure function for any complex z ∈ C, and find its poles as with n ∈ N. For K w ∈ (0, 1), the poles are shown in Fig. 1. Note that the imaginary part of the poles grows when the repulsive interaction is decreased, and as we approach the non-interacting limit K w → 1, it becomes infinitely large. For the purpose of performing complex integration over the structure factor (I.9), it is useful to rewrite it as a sum over two function where the first/second term in the parenthesis is analytic in the upper/lower-half of the complex plane, i.e., its poles are in the lower/upper-half. Furthermore, defining y = cosh −1 [(1 + K 2 w ) (1 − K 2 w )], we have (1 ± K w ) (1 ∓ K w ) = exp(±y), and therefore we can rewrite the structure function in terms of the bosonic distribution function (I.12)

OHMIC-CLASS ENVIRONMENT
We consider a generic Ohmic-class environment with noise power spectrum S(ω) defined in Eq. (3) in the main text. In the following, we will use the retarded bosonic Green's function for the field ϕ of the clean wire, with Θ the Heaviside function. It can be expressed in terms of lesser and greater Green's function G <,0 Note that Ohmic-class environments hold detailed balance, and we have which follows from Eq. (I.7) [we use the fact that F (x, x ′ , ω) is a real and even function of ω, see Eq. (I.8)], as we expect for bosons in thermal equilibrium. Thereby, the retarded Green's function Eq. (II.1) simplifies to

Ohmic-environment
In the case of an Ohmic environment, s = 1, we extend the integral in Eq. (II.3) to the complex plane, and employ the analytic properties of the structure function [see discussion leading to Eq. (I.12)], to obtain the local retarded Green's function for Analytically continuing the obtained retarded/advanced Green's function to the imaginary axis (Matsubara space) [3] by substitution ω ± iη → iω for ω ≷ 0, we can rewrite the Matsubara Green's function as G 0 At large frequencies, ωτ L ≫ 1, K(ω) approaches the Luttinger Liquid parameter inside the wire K w , whereas for low frequencies, ωτ L ≪ 1, K(ω) goes to 1, resembling the TLL parameter of the noninteracting (Fermi liquid) leads. We note the the same results can be obtained by solving the equation of motion for Matsubara Green's function where k(x) = K w for x ∈ [−L 2, L 2], and k(x) = 1 elsewhere. This is accomplished by imposing the continuity of the Matsubara Green's function and of its derivative v F k 2 (x)∂ x G 0 ϕϕ (x, x ′ , iω) at the wire's ends x = ±L 2, as well as at the discontinuity at x = x ′ , namely In conclusion, the finite length of the wire connected to non-interacting electrons introduces an infrared cutoff 1 τ L , below which the wire acts as non-interacting electrons.
Infinite wire connected to an Ohmic-class environment We next consider a generic Ohmic-class environment with s ∈ [0, 2], for which the plasmonic spectral function reads as is shown in Fig. 2. In the super-Ohmic case, the spectral function at small frequencies, ωτ L ≪ 1, diverges slower than the ohmic case [i.e., slower than ∼ K W (2ω)], while the sub-Ohmic case is diverging faster. For the interacting wire at frequencies ωτ L > 1, the oscillations are present due to the formation of a Fabry-Pérot cavity.
Having the plasmonic spectral function, we can obtain the plasmonic Matsubara Green's function, at Matsubara frequencies ω n = 2nπ β, n ∈ N, Csc[(πs) 2], (II.9) where we used the fact that in the limit of L → ∞, F (iω) → K w , and that we are interested in ω n ≪ ω c . We can rewrite Eq. (II.9), as G 0 ϕϕ (iω) = K(ω) (2 ω ), with K(ω) = K w Csc[(πs) 2] ω ω c s−1 . In the Ohmic case, this reduces to K(ω) = K w as expected. Finite wire connected to an Ohmic-class environment In the case of a finite wire, similar to the infinite length case, we can define the effective TLL parameter as In Fig. 3, we show the numerical evaluation of the integral above, and compare it with the approximate formulã The approximation agrees well for ω ≪ ω c , irrespective of the value of s. Note that from Eq. (II.11) it follows that at high frequencies, ωτ L ≫ 1,K(ω, L) = K(ω) with K(ω) being the effective Luttinger parameter for L → ∞, cf. Eq. (5) in the main text. At small frequencies, ωτ L ≪ 1, on the other hand, the curves corresponding to different lengths merge together as is shown in Fig. 3, exhibiting that at such small frequencies, the physics of the interacting wire is washed out and the system's behaviour is dominated by the environment.

RENORMALIZATION OF THE SCATTERING POTENTIAL
In this section, we present the standard perturbative RG analysis for the problem of a single impurity immersed in a TLL. After integrating-out the leads, the action of the system in imaginary-time reads For all x ≠ x 0 , the action is quadratic, and we can integrate out all the corresponding fields to obtain the local action [Eq. (4) in the main text], which can be decomposed as where we have assumed x 0 = 0. We define an ultraviolet cut-off Λ and the corresponding scale-dependent field as ϕ Λ (τ ) = ∫ Λ −Λ dω e iωτ ϕ(ω). For any Λ ′ ∈ [0, Λ], we can decompose the bosonic fields into low-and high-frequency fields, ϕ Λ (τ ) = ϕ Λ ′ (τ ) + h(τ ). Now, we turn to the functional-integral formulation of the partition function, and integrate over the high-frequency field h(τ ) to obtain Performing the Gaussian integral over high-frequency fields we obtain We proceed with the RG procedure and consider an infinitesimal change of the ultraviolet cutoff Λ ′ = Λ − dΛ = Λ(1 − dl), dl = dΛ Λ. In order to compare the physics governed by fields ϕ Λ ′ with the one governed by the fields ϕ Λ , we re-scale the frequency ω to ω ′ = ω Λ Λ ′ = (1 + dl)ω, and the corresponding imaginary-time τ to τ ′ = τ (1 − dl), such that τ ω = τ ′ ω ′ [4]. Thereby, we have Now the high-frequency degrees of freedom can be integrated out while keeping the partition function invariant provided that the scattering potential is renormalized accordingly, i.e., Using the plasmonic Matsubara Green's function for an infinitely long-wire coupled to the Ohmic-class environment that we obtained in Eq. (II.9), the integral in the exponent becomes where Γ is the incomplete Gamma function. Therefore the flow equation up to the first order in dl is Taking the limit ω c → ∞, in the Ohmic-case (s = 1), the flow equation boils down to dV dl = V 0 (1 − K w ), which is the known result from Kane and Fisher [5]. For a finite-length wire, we can obtain the flow equation for the scattering potential in an analogous manner. Figure  4 depicts the beta function dV dl = β(Λ) as a function of scale parameter Λ for the Ohmic, sub-Ohmic, and the super-Ohmic cases, and for varying interaction strength K w . As it is shown for Λτ L ≪ 1, the beta function is independent of the interaction strength in the wire, which is consistent to our previous discussion in relation to Eq. (II.11). The flow for Λτ L ≫ 1 is similar to our analysis of the infinitely long wire in the main text.

CONDUCTANCE
In this section, we consider an external electrical potential U (x, t), and calculate the resulting conductance through the interacting wire containing an impurity. The action of the system in imaginary time reads where A s is given in Eq. (III.1), and E(x, t) = −∂ x U (x, t) is the external electric field. Similar to the previous section, for all x ≠ x 0 , the action is quadratic, and we can integrate out all the fields at x ≠ x 0 [6], and obtain the following local action In the limit ω n → 0, the structure function (I.8) and hence the Matsubara Green's function G 0 (iω n ) become independent of position (x, x ′ ), and hence the local action can be approximated as where we have assumed x 0 = 0, and ∫ L 2 −L 2 dx E(x, ω n ) = U cos(ωt). We are interested in calculating the resulting ac-current, which reads [7] I ac 2(e 2 h) with G ϕϕ being the plasmonic Matsubara Green's function in the presence of the impurity. As the exact form of the G ϕϕ is unattainable, we perform a perturbative expansion [5] in terms of the scattering potential V 0 , and obtain (IV.6) is the impurity-induced self energy. In the following, we try to simplify the above expression by using the analytic properties of Matsubara Green's functions. First, we define (IV.14) The first term in Eq. (IV.14) is the conductance of a clean wire [G 0 in the main text], and the second term corresponds to the back-scatterings from the impurity [G b in the main text]. We conclude this section by emphasising that the calculation of the conductance (IV.14) boils down to the evaluation of the two integrals in Eqs. (IV.12) and (IV.8).

Ohmic environment
In this subsection, we outline the analysis of the temperature dependence of the conductance through the wire which is in contact with Ohmic leads with noise power spectrum S(ω) = ω [1 + n b (βω)].

Infinitely long wire
For an infinitely long-wire, the local retarded Green's function is G 0,R ϕϕ (iω) = iKw 2ω+i0 + e − ω ωc . Hence, Eq. (IV.8) becomes (IV.15) Concentrating on the low temperature behaviour, ω c β ≫ 1, we obtain which makes the evaluation of Eq. (IV.12) feasible, resulting in (IV.17) However, note that this situation is unphysical, namely in order to generate a current through the interacting wire, we have to connect the system to the electronic leads, which forces us to take into account the finite-length of the wire and the resulting frequency structure of the Luttinger liquid parameter as we shall see in the following.

Finite wire connected to Ohmic leads
We now consider a finite-length wire, where the Luttinger Liquid parameter acquires a frequency dependence, as shown in the discussion related to Eq. (II.5). We evaluate the integrals [Eqs. (IV.12) and (IV.8)] numerically and show the results in Fig. 5. At high temperatures T τ L > 1, the conductance can be approximated as