Electron lifetime in Luttinger liquids

We investigate the decoherence of the electron wave packet in purely ballistic one-dimensional systems described through the Luttinger liquid (LL). At a finite temperature $T$ and long times $t$, we show that the electron Green’s function for a fixed wave vector close to one Fermi point decays as $\exp (-t∕{\tau }_F)$ —as opposed to the power-law behavior occurring at short times—and the emerging electron lifetime obeys ${\tau }_F^{-1}\propto T$ for spinful as well as spinless electrons. For strong interactions, $\left(T{\tau }_F\right)\ll 1$, reflecting that the electron is not a good Landau quasiparticle in LL’s. We justify that fractionalization is the main source of electron decoherence for spinful as well as spinless electrons clarifying the peculiar electron mass renormalization close to the Fermi points. For spinless electrons and weak interactions, our intuition can be enriched through a diagrammatic approach or Fermi golden rule and through a Johnson-Nyquist noise picture. We stress that the electron lifetime (and the fractional quasiparticles) can be revealed from Aharonov-Bohm experiments or momentum resolved tunneling. We aim to compare the results with those of spin-incoherent and chiral LL’s.

Figure 1

(Color online) Non-trivial self-energy diagrams to second order in $(Ua∕v_F)$.

Figure 2

(Color online) Behavior of ${\sinh }^{-\gamma }\left(x\right)$ (in the text $x=2\pi t\beta$) with the two asymptotes $x^{-\gamma }$ at short times and $2^{\gamma }\exp (-\gamma x)$ at long times. For finite $\gamma$ and large times such that $t>{\tau }_F\gamma$, the Green function $G_{+}(p,t)$ falls off exponentially with time.

Figure 3

(Color online) For $2\pi t\ll \beta$, although the overlap between the fractional wave packets is finite, the inherent fractionalization process clarifies the orthogonality catastrophe factor $t^{-\gamma }$ in ${\mathcal{G}}_{+}(x\to ut,t)$ whereas at long times the electron fractionalization leads to ${\mathcal{G}}_{+}(x\to ut,t)\propto \exp (-t∕{\tau }_F)$ with ${\tau }_F^{-1}=2\pi \gamma ∕\beta$. For very weak interactions, we obtain ${\tau }_F^{-1}\propto (Ua∕\pi v_F{)}^{2}T$.

Figure 4

(Color online) Schematic geometry of electrons tunneling between the wire and the source. A magnetic field $B$ is applied perpendicular to the plane of the wire to allow a field-dependent momentum boost $p_B=Bd$ where $d$ is the distance between the wire and the source (Ref. 26, 27). Electrons can be made “spinless” through another magnetic field $\mathrm{\Delta }B$ applied along the wire.

Figure 5

(Color online) Broadening of the momentum-resolved tunneling density of states by electron-electron interactions for an infinite and spin-polarized wire; $D=1$ and $T=0.05$. The width of the peak gives an access to $\gamma ∕u=(Q_{-}{)}^2∕(gu)=(Q_{-}{)}^2∕v_F$.

Figure 6

(Color online) The setup for electron-wave interferences. The electrons can tunnel from one wire to the other at $x=0$ and $x=d$. The lengths of the wires are much larger than $d$ such that fractionalization cannot be hindered by boundary effects.