Luttinger liquid coupled to Ohmic-class environments

We investigate the impact of an Ohmic-class environment on the conduction and correlation properties of one-dimensional interacting systems. Interestingly, we reveal that interparticle interactions can be engineered by the environment's noise statistics. Introducing a backscattering impurity to the system, we address Kane-Fisher's metal-to-insulator quantum phase transition in this noisy and realistic setting. Within a perturbative renormalization group approach, we show that the Ohmic environments keep the phase transition intact, while sub- and super-Ohmic environments modify it into a smooth crossover at a scale that depends on the interaction strength within the wire. The system still undergoes a metal-to-insulator-like transition when moving from sub-Ohmic to super-Ohmic environment noise. We cover a broad range of realistic experimental conditions, by exploring the impact of a finite wire length and temperature on transport through the system.

Figure 1

(a) Sketch of a quantum wire containing a single impurity, that backscatters electrons, coupled to leads. (b) The noise-power spectrum of the electronic leads as a function of frequency, where ${\omega }_c$ marks the environment bandwidth. (c) The phase diagram of a two-level system (${\varphi }^4$ theory) coupled to an Ohmic environment for varying dissipation strength $\alpha$. For small dissipation $\alpha <1$, the effective tunneling $\mathrm{\Gamma }$ between two potential wells diverges, i.e., the particle is delocalized, but for stronger dissipation $\alpha >1$ the particle is localized $\mathrm{\Gamma }\to 0$ [83]. (d) Phase diagram of the TLL hosting an impurity coupled to Ohmic leads [cf. action (4), with an effective scattering potential $V$ mapped to a sine-Gordon potential] as a function of the interaction strength in the wire $K_{\mathrm{w}}$ [cf. Eq. (4)].

Figure 2

Scaling and RG flow of the scattering potential $V\left(\omega \right)$ and effective Luttinger parameter $K\left(\omega \right)$ at zero temperature [cf. Eqs. (4, 5, 6)]. (a) The frequency dependence of $K\left(\omega \right)$ for $\omega \gg 1/{\tau }_L$ for a range of interaction strengths in the wire $K_{\mathrm{w}}\in [0.6,1.4]$ and different Ohmic-class cases, $s=1.2,1.0,0.8$. The solid lines depict the noninteracting case $K_{\mathrm{w}}=1$, and the lighter (darker) shades mark $K_{\mathrm{w}}<1$ ($K_{\mathrm{w}}>1$). (b)–(d) The flow diagram (along the direction of the arrows) of the scattering potential vs effective interaction as we change the cutoff energy scale $\mathrm{\Lambda }$ from $[\infty ,{\mathrm{\Lambda }}_f]$, with ${\mathrm{\Lambda }}_f>1/{\tau }_L$ for (b) Ohmic ($s=1.0$), (c) sub-Ohmic ($s=0.8$), and (d) super-Ohmic ($s=1.2$) cases.

Figure 3

Temperature dependence of the impurity-induced correction to conductance through a finite-length wire $[L=100{\nu }_{\mathrm{F}}/\left(K_{\mathrm{w}}{\omega }_c\right)]$ for different interaction strengths ($K_{\mathrm{w}}$) [cf. Eq. (7)]. (a) Ohmic, (b) sub-Ohmic ($s=0.8$), and (c) super-Ohmic case ($s=1.2$). The dashed lines show in (a) the expected power-law dependence $G_b\propto {(T/{\omega }_c)}^{2K_{\mathrm{w}}-2}$, (b) the exponential dependence $G_b\propto exp[-{\alpha }_s{(T/{\omega }_c)}^{s-1}]$, and (c) $G_b\propto {(T/{\omega }_c)}^{2s-4}$ [90]. The arrows in (b) and (c) mark ${\mathrm{\Lambda }}^{*}$, where $K\left({\mathrm{\Lambda }}^{*}\right)=1$.