Kinetic Processes in Fermi–Luttinger Liquids

Abstract

In this work we discuss extensions of the pioneering analysis by Dzyaloshinskii and Larkin [Sov. Phys. JETP 38, 202 (1974)] of correlation functions for one-dimensional Fermi systems, focusing on the effects of quasiparticle relaxation enabled by a nonlinear dispersion. Throughout the work we employ both, the weakly interacting Fermi gas picture and nonlinear Luttinger liquid theory to describe attenuation of excitations and explore the fermion-boson duality between both approaches. Special attention is devoted to the role of spin-exchange processes, effects of interaction screening, and integrability. Thermalization rates for electron- and hole-like quasiparticles, as well as the decay rate of collective plasmon excitations and the momentum space mobility of spin excitations are calculated for various temperature regimes. The phenomenon of spin–charge drag is considered and the corresponding momentum transfer rate is determined. We further discuss how momentum relaxation due to several competing mechanisms, viz. triple electron collisions, electron–phonon scattering, and long-range inhomogeneities affect transport properties, and highlight energy transfer facilitated by plasmons from the perspective of the inhomogeneous Luttinger liquid model. Finally, we derive the full matrix of thermoelectric coefficients at the quantum critical point of the first conductance plateau transition, and address magnetoconductance in ballistic semiconductor nanowires with strong Rashba spin–orbit coupling.

Appendices

APPENDIX A

NOTES ON BOSONIZATION

This section is prepared as a supplementary material to the main text of the paper. Here we concentrate on the derivation of an effective Hamiltonian for nonlinear Luttinger liquids and construction of the corresponding kinetic theory of spin–charge scattering processes detailed in Section 2.6. To the large extend we follow here Giamarchi textbook [14] for the bosonization procedure and notations, plus the Haldane review article [13] to include band curvature effects. An additional element presented here is a more detailed bosonization of the interaction part of the fermionic Hamiltonian. It will be shown that similar to the band-curvature terms, interaction also generates anharmonic couplings between the spin and charge modes. As discussed in the main text of the paper these nuances have important consequences for the relaxation in 1D including spin–charge drag and energy transport.

The starting point is the usual form of fermionic Hamiltonian:

$$\begin{gathered} H\, = \, - {\kern 1pt} i{{{v}}_{F}} \\ \times \sum\limits_s^{} {\int {dx[\psi _{{Rs}}^{\dag }(x){{\partial }_{x}}{{\psi }_{{Rs}}}(x)\, - \,\psi _{{Ls}}^{\dag }(x){{\partial }_{x}}{{\psi }_{{Ls}}}(x)]} } \\ - \frac{1}{{2m}}\sum\limits_s^{} {\int {dx[\psi _{{Rs}}^{\dag }(x)\partial _{x}^{2}{{\psi }_{{Rs}}}(x) + \psi _{{Ls}}^{\dag }(x)\partial _{x}^{2}{{\psi }_{{Ls}}}(x)]} } \\ + \frac{1}{2}\sum\limits_{ss'}^{} {\int {dxdx{\kern 1pt} '{\kern 1pt} V(x - x{\kern 1pt} ')\psi _{s}^{\dag }(x)\psi _{{s'}}^{\dag }(x){{\psi }_{{s'}}}(x{\kern 1pt} '){{\psi }_{s}}(x).} } \\ \end{gathered} $$

(131)

Here index s = ↑↓ stands for the spin projection, ψ_Rs and ψ_Ls are the annihilation operators for right- and left-moving spin-s electrons, while ψ_s = ψ_Rs + ψ_Ls is full operator in the interaction part of the Hamiltonian. The standard approximation is that low energy excitations take place near the Fermi points, such that electron operator is decomposed as follows

$${{\psi }_{s}}(x)\, = \,{{\psi }_{{Rs}}}(x)\, + \,{{\psi }_{{Ls}}}(x)\, = \,{{e}^{{i{{k}_{F}}x}}}{{R}_{s}}(x)\, + \,{{e}^{{ - {{k}_{F}}x}}}{{L}_{s}}(x),$$

(132)

where new fields R(L)_s(x) are assumed to vary slowly on the scale of the Fermi wavelength. In the bosonization description these fields can be expressed in terms of dosonic displacement φ_s(x) and conjugated phase ϑ_s(x)

$$\begin{gathered} {{R}_{s}}(x) = \frac{{{{\kappa }_{s}}}}{{\sqrt {2\pi a} }}\exp [i{{\vartheta }_{s}}(x) - i{{\varphi }_{s}}(x)], \\ {{L}_{s}}(x) = \frac{{{{\kappa }_{s}}}}{{\sqrt {2\pi a} }}\exp [i{{\vartheta }_{s}}(x) + i{{\varphi }_{s}}(x)], \\ \end{gathered} $$

(133)

where a is the short distance cut-off ~$k_{F}^{{ - 1}}$ and κ_s are the Klein factors that ensure proper anticommutation relation between original fermionic operators. They obey {κ_s, κ_s'} = 2δ_ss' and satisfy $\kappa _{s}^{\dag }$ = κ_s. The bosonic fields obey commutation

$$[{{\varphi }_{s}}(x),{{\vartheta }_{{s'}}}(x)] = \frac{{i\pi }}{2}\operatorname{sgn} (x - x'){{\delta }_{{ss'}}}\,.$$

(134)

With these notations at hand fermionic densities for right- and left-moving electrons become

$$\begin{gathered} {{\rho }_{{Rs}}}(x) = R_{s}^{\dag }(x){{R}_{s}}(x) = - \frac{1}{{2\pi }}{{\partial }_{x}}[{{\varphi }_{s}}(x) - {{\vartheta }_{s}}(x)], \\ {{\rho }_{{Ls}}}(x) = L_{s}^{\dag }(x){{L}_{s}}(x) = - \frac{1}{{2\pi }}{{\partial }_{x}}[{{\varphi }_{s}}(x) + {{\vartheta }_{s}}(x)]. \\ \end{gathered} $$

(135)

The total density operator per spin ρ_s(x) = $\psi _{s}^{\dag }$(x)ψ_s(x) contains the sum of the long-wavelenght part, $\rho _{s}^{{(0)}}$(x), and oscillatory part $\rho _{s}^{{(2{{k}_{F}})}}$(x):

$$\begin{gathered} {{\rho }_{s}}(x) = \rho _{s}^{{(0)}}(x) + \rho _{s}^{{(2{{k}_{F}})}}(x) \\ = - \frac{1}{\pi }{{\partial }_{x}}{{\varphi }_{s}}(x) + \frac{1}{{\pi a}}\cos [2{{\varphi }_{s}}(x) - 2{{k}_{F}}x]. \\ \end{gathered} $$

(136)

For the first two terms of Eq. (131) in the bosonization dictionary we have

$$\begin{gathered} \psi _{{Rs}}^{\dag }(x){{\partial }_{x}}{{\psi }_{{Rs}}}(x) - \psi _{{Ls}}^{\dag }(x){{\partial }_{x}}{{\psi }_{{Ls}}}(x) \\ = i\pi [\rho _{{Rs}}^{2}(x) + \rho _{{Ls}}^{2}(x)], \\ \psi _{{Rs}}^{\dag }(x)\partial _{x}^{2}{{\psi }_{{Rs}}}(x) + \psi _{{Ls}}^{\dag }(x)\partial _{x}^{2}{{\psi }_{{Ls}}}(x) \\ = - \frac{{4{{\pi }^{2}}}}{3}[\rho _{{Rs}}^{2}(x) + \rho _{{Ls}}^{2}(x)], \\ \end{gathered} $$

(137)

such that kinetic part of the Hamiltonian transforms into

$$\begin{gathered} {{H}_{{{\text{kin}}}}} = \frac{{{{{v}}_{F}}}}{{2\pi }}\sum\limits_s^{} {\int {dx[{{{({{\partial }_{x}}{{\varphi }_{s}})}}^{2}} + {{{({{\partial }_{x}}{{\vartheta }_{s}})}}^{2}}]} } \\ - \frac{1}{{6\pi m}}\sum\limits_s^{} {\int {dx[{{{({{\partial }_{x}}{{\varphi }_{s}})}}^{3}} + 3({{\partial }_{x}}{{\varphi }_{s}}){{{({{\partial }_{x}}{{\vartheta }_{s}})}}^{2}}].} } \\ \end{gathered} $$

(138)

For the interaction part of the Hamiltonian in Eq. (131) we proceed as follows. Up to an additive constant it can be rewritten as

$$\begin{gathered} {{H}_{{\operatorname{int} }}} = \frac{1}{2}\sum\limits_{ss'}^{} {\int {dxdx{\kern 1pt} '{\kern 1pt} V(x - x{\kern 1pt} ')} } \\ \times \,[\underbrace {\psi _{{Rs}}^{\dag }(x){{\psi }_{{Rs}}}(x) + \psi _{{Ls}}^{\dag }(x){{\psi }_{{Ls}}}(x)}_{q\sim 0} + \underbrace {\psi _{{Ls}}^{\dag }(x){{\psi }_{{Rs}}}(x)}_{q\sim 2{{k}_{F}}}] \\ \times \,[\underbrace {\psi _{{Rs'}}^{\dag }(x{\kern 1pt} '){{\psi }_{{Rs'}}}(x{\kern 1pt} ')\, + \,\psi _{{Ls'}}^{\dag }(x{\kern 1pt} '){{\psi }_{{Ls'}}}(x{\kern 1pt} ')}_{q\sim 0} \\ + \,\underbrace {\psi _{{Ls'}}^{\dag }(x{\kern 1pt} '){{\psi }_{{Rs'}}}(x{\kern 1pt} ')}_{q\sim 2{{k}_{F}}}] \\ \end{gathered} $$

(139)

thus separating explicitly different scattering channels, where q labels characteristic momenta transferred in the collision. From here one can read out forward and backward scattering parts of the interaction term, namely H_int = $H_{{\operatorname{int} }}^{{{\text{fs}}}}$ + $H_{{\operatorname{int} }}^{{{\text{bs}}}}$. The formed one may be easily rewritten in the bosonization dictionary

$$\begin{gathered} H_{{\operatorname{int} }}^{{{\text{fs}}}} = \frac{1}{2}\sum\limits_{ss'}^{} {\int {dxdx'V(x - x')\rho _{s}^{{(0)}}(x)\rho _{{s'}}^{{(0)}}(x')} } \\ = \frac{{{{V}_{0}}}}{{2{{\pi }^{2}}}}\sum\limits_{ss'}^{} {\int {dx({{\partial }_{x}}{{\varphi }_{s}})({{\partial }_{x}}{{\varphi }_{{s'}}}).} } \\ \end{gathered} $$

(140)

Here we expanded the density $\rho _{{s'}}^{{(0)}}$(x') around x using the fact that interaction potential V(x – x') is short-ranged and that fields φ_s(x) are slowly varying on the scale where interaction appreciably decays. By V₀ we denote zero momentum Fourier transform of the interaction potential and higher order gradients were ignored. We concentrate now on the backward scattering part of the Hamiltonian. In the bosonization language it can be written as

$$\begin{gathered} H_{{\operatorname{int} }}^{{{\text{bs}}}} = \frac{1}{{8{{\pi }^{2}}{{a}^{2}}}}\sum\limits_{ss'}^{} {\int {dxdyV(y)} } \\ \, \times [{{e}^{{ - 2i{{k}_{F}}y}}}{{e}^{{2i{{\varphi }_{s}}(x + y/2) - 2i{{\varphi }_{{s'}}}(x - y/2)}}} + h.c.]. \\ \end{gathered} $$

(141)

For the case of s ≠ s' we have sine-Gordon part of the Hamiltonian

$$H_{{\operatorname{int} }}^{{{\text{bs}}|ss'}} = \frac{{{{V}_{{2{{k}_{F}}}}}}}{{2{{\pi }^{2}}{{a}^{2}}}}\int {dx\cos [2{{\varphi }_{ \uparrow }}(x) - 2{{\varphi }_{ \downarrow }}(x)]} $$

(142)

A little more careful consideration is required for the case when s = s'. In this case one should expand the fields, which are in fact non-commuting operators. To do the expansion procedure safely the operator has to be normal-ordered:

$$\begin{gathered} \exp [i{{\varphi }_{s}}(x)] = \langle \exp [i{{\varphi }_{s}}(x)]\rangle :\exp [i{{\varphi }_{s}}(x)]: \\ = \exp [ - \langle \varphi _{s}^{2}(x)\rangle {\text{/}}2]:\exp [i{{\varphi }_{s}}(x)]: \\ \end{gathered} $$

(143)

such that one may apply usual Taylor series for the operator under the normal-ordered sign : ( ) :. The brackets 〈…〉 imply quantum averaging and to the lowest order in interaction

$$\langle {{({{\varphi }_{s}}(x) - {{\varphi }_{s}}(x{\kern 1pt} '))}^{2}}\rangle = \ln \left[ {\frac{{{\text{|}}x - x{\kern 1pt} '{\text{|}}}}{a}} \right].$$

(144)

With this formalism at hand we have

$$\begin{gathered} H_{{\operatorname{int} }}^{{{\text{bs}}|ss}} \approx \frac{1}{{8{{\pi }^{2}}{{a}^{2}}}}\sum\limits_s^{} {\int {dxdyV(y)\left[ {{{e}^{{ - 2i{{k}_{F}}y}}}{{e}^{{ - 2\ln \frac{{|y|}}{a}}}}} \right.} } \\ \times \left( {1 + 2i{{\Phi }_{s}}(x,y) + \frac{{{{{(2i)}}^{2}}}}{2}\Phi _{s}^{2}(x,y)} \right. \\ \left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}}}}}}}}}\left. { + \frac{{{{{(2i)}}^{3}}}}{6}\Phi _{s}^{3}(x,y) + \frac{{{{{(2i)}}^{4}}}}{{24}}\Phi _{s}^{4}(x,y)} \right) + h.c.} \right], \\ \end{gathered} $$

(145)

where Φ_s(x, y) = φ_s(x + y/2) – φ_s(x – y/2) ≈ yϑ_xφ_s(x) and we carried gradient expansion to the lowest order. By neglecting now constant and full derivative terms and noticing that exp[–2ln(|y|/a)] = (a/y)², that cancels cut-off dependent prefactor, one finds

$$\begin{gathered} H_{{\operatorname{int} }}^{{{\text{bs}}|ss}} \approx \frac{1}{{8{{\pi }^{2}}}} \\ \times \,\sum\limits_s^{} {\int {dxdyV(y)\left[ { - 4\cos (2{{k}_{F}}y){{{({{\partial }_{x}}{{\varphi }_{s}})}}^{2}}_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}} \right.} } \\ \left. { - \frac{8}{3}y\sin (2{{k}_{F}}y){{{({{\partial }_{x}}{{\varphi }_{s}})}}^{3}} + \frac{4}{3}{{y}^{2}}\cos (2{{k}_{F}}y){{{({{\partial }_{x}}{{\varphi }_{s}})}}^{4}}} \right]. \\ \end{gathered} $$

(146)

After the integration by parts above expression reduces to the form

$$\begin{gathered} H_{{\operatorname{int} }}^{{{\text{bs}}|ss}} = - \frac{{{{V}_{{2{{k}_{F}}}}}}}{{2{{\pi }^{2}}}}\sum\limits_s^{} {\int {dx{{{({{\partial }_{x}}{{\varphi }_{s}})}}^{2}}} } \\ + \frac{{V_{{2{{k}_{F}}}}^{'}}}{{3{{\pi }^{2}}}}\sum\limits_s^{} {\int {dx{{{({{\partial }_{x}}{{\varphi }_{s}})}}^{3}} - \frac{{V_{{2{{k}_{F}}}}^{{''}}}}{{6{{\pi }^{2}}}}\sum\limits_s^{} {\int {dx{{{({{\partial }_{x}}{{\varphi }_{s}})}}^{4}}.} } } } \\ \end{gathered} $$

(147)

The first term is conventional for the bosonization technique while the last two are new additions responsible for the interaction of bosons. At this point we perform transformation to the spin–charge representation for the boson fields:

$${{\varphi }_{\rho }} = \frac{1}{{\sqrt 2 }}({{\varphi }_{ \uparrow }} + {{\varphi }_{ \downarrow }}),\quad {{\varphi }_{\sigma }} = \frac{1}{{\sqrt 2 }}({{\varphi }_{ \uparrow }} - {{\varphi }_{ \downarrow }}),$$

(148)

and similar for the ϑ-field. The final result we split into five parts

$$H = {{H}_{2}} + H_{3}^{{{\text{bc}}}} + H_{3}^{{{\text{bs}}}} + {{H}_{4}} + {{H}_{{{\text{sg}}}}}.$$

(149)

The quadratic part

$$\begin{gathered} {{H}_{2}} = \frac{{{{{v}}_{\rho }}}}{{2\pi }}\int {dx\left[ {\frac{1}{{{{K}_{\rho }}}}{{{({{\partial }_{x}}{{\varphi }_{\rho }})}}^{2}} + {{K}_{\rho }}{{{({{\partial }_{x}}{{\vartheta }_{\rho }})}}^{2}}} \right]} \\ + \frac{{{{{v}}_{\sigma }}}}{{2\pi }}\int {dx\left[ {\frac{1}{{{{K}_{\sigma }}}}{{{({{\partial }_{x}}{{\varphi }_{\sigma }})}}^{2}} + {{K}_{\sigma }}{{{({{\partial }_{x}}{{\vartheta }_{\sigma }})}}^{2}}} \right]} \\ \end{gathered} $$

(150)

corresponds to the usual linear Luttinger liquid model of spin–charge separation explicit. The boson velocities and Luttinger liquid interaction constants are locked by relations ${{{v}}_{\rho }}$K_ρ = ${{{v}}_{\sigma }}$K_σ = ${{{v}}_{F}}$, and at the level of perturbation theory K_ρ = 1 – (2V₀ – ${{V}_{{2{{k}_{F}}}}}$)/2π${{{v}}_{F}}$ and K_σ = 1 + ${{V}_{{2{{k}_{F}}}}}$/2π${{{v}}_{F}}$. The sine-Gordon term is also belongs to the linear Luttinger liquid theory

$${{H}_{{{\text{sg}}}}} = \frac{{{{V}_{{2{{k}_{F}}}}}}}{{2{{\pi }^{2}}{{a}^{2}}}}\int {dx\cos [2\sqrt 2 {{\varphi }_{\sigma }}].} $$

(151)

The cubic order coupling terms can be split into two groups. First group is due to band curvature

$$\begin{gathered} H_{3}^{{{\text{bc}}}} = - \frac{1}{{6\sqrt 2 \pi m}}\int {dx[{{{({{\partial }_{x}}{{\varphi }_{\rho }})}}^{2}}} \\ + 3({{\partial }_{x}}{{\varphi }_{\rho }}){{({{\partial }_{x}}{{\varphi }_{\sigma }})}^{2}} + 3({{\partial }_{x}}{{\varphi }_{\rho }}){{({{\partial }_{x}}{{\vartheta }_{\rho }})}^{2}} \\ + 3({{\partial }_{x}}{{\varphi }_{\rho }}){{({{\partial }_{x}}{{\vartheta }_{\sigma }})}^{2}} + 6({{\partial }_{x}}{{\vartheta }_{\rho }})({{\partial }_{x}}{{\varphi }_{\sigma }})({{\partial }_{x}}{{\vartheta }_{\sigma }})]. \\ \end{gathered} $$

(152)

The second group is due to backscattering

$$H_{3}^{{{\text{bs}}}} = \frac{{V_{{2{{k}_{F}}}}^{'}}}{{3\sqrt 2 {{\pi }^{2}}}}\int {dx[{{{({{\partial }_{x}}{{\varphi }_{\rho }})}}^{3}} + 3({{\partial }_{x}}{{\varphi }_{\rho }}){{{({{\partial }_{x}}{{\varphi }_{\sigma }})}}^{2}}].} $$

(153)

These terms couple spin and charge excitations and lead to ρ → σσ decay processes that we discussed in the context of spin–charge drag equilibration rates. The quartic order terms

$$\begin{gathered} {{H}_{4}} = - \frac{{V_{{2{{k}_{F}}}}^{{''}}}}{{12{{\pi }^{2}}}} \\ \times \,\int {dx[{{{({{\partial }_{x}}{{\varphi }_{\rho }})}}^{4}} + 6{{{({{\partial }_{x}}{{\varphi }_{\rho }})}}^{2}}{{{({{\partial }_{x}}{{\varphi }_{\sigma }})}}^{2}} + {{{({{\partial }_{x}}{{\varphi }_{\sigma }})}}^{4}}]} \\ \end{gathered} $$

(154)

lead to ρρ, ρσ, and σσ type boson scattering. Finally, to obtain kinetic equations for bosons we use canonical oscillator representation in normal modes

$$\begin{gathered} {{\partial }_{x}}{{\varphi }_{{\rho /\sigma }}} = - \sum\limits_q^{} {\sqrt {\frac{{\pi {\text{|}}q{\text{|}}}}{{2L}}} {{e}^{{ - iqx}}}[b_{{\rho /\sigma }}^{\dag }(q) + {{b}_{{\rho /\sigma }}}( - q)],} \\ {{\partial }_{x}}{{\vartheta }_{{\rho /\sigma }}} = \sum\limits_q^{} {\sqrt {\frac{{\pi {\text{|}}q{\text{|}}}}{{2L}}} \operatorname{sgn} (q){{e}^{{ - iqx}}}[b_{{\rho /\sigma }}^{\dag }(q) - {{b}_{{\rho /\sigma }}}( - q)],} \\ \end{gathered} $$

(155)

written in terms of creation and annihilation operators. It should be also borne in mind that fields φ_ρ/σ(x) and ϑ_ρ/σ(x) contain topological terms, N^R ± N^L, which are important in defining the momentum operator [13].

APPENDIX B

MOBILE IMPURITY MODEL

The purpose of this section is to illustrate a connection between calculation of quasiparticle decay rates in fermions via three-particle collisions and in bosons via nonlinear Luttinger liquid approach. For simplicity we condense this discussion to the spinless case. Here we essentially follow the framework developed in [65] with an extension to include an additional interaction term known from the context of impurity dynamics in Luttinger liquid [64] that enables a decay processes.

The technical essence of the method can be summarized as follows. Starting from the initial fermionic model one introduces not only conventional low-energy subbands ψ_R(L) at ±k_F for right-movers and left-movers, but also the sub-band modes d around the momentum k of the high-energy particle (or hole) whose energy defines the threshold. In this approach, the fermion operator is split as ψ(x) ~ ${{e}^{{i{{k}_{F}}x}}}$ψ_R(x) + ${{e}^{{ - {{k}_{F}}x}}}$ψ_L(x) + e^ikxd(x) in which the high-energy particle acts as a mobile impurity coupled to the Luttiger liquid modes. The Hamiltonian for this model reads [65]

$$H = {{H}_{0}} + {{H}_{d}} + {{H}_{{\operatorname{int} }}},$$

(156a)

$${{H}_{0}} = \frac{{v}}{{2\pi }}\int {dx[K{{{({{\partial }_{x}}\vartheta )}}^{2}} + {{K}^{{ - 1}}}{{{({{\partial }_{x}}\varphi )}}^{2}}],} $$

(156b)

$${{H}_{d}} = \int {dx{{d}^{\dag }}(x)[\varepsilon (k) - i{{{v}}_{d}}{{\partial }_{x}}]d(x),} $$

(156c)

$${{H}_{{\operatorname{int} }}} = \frac{1}{{2\pi }}\int {dx[({{V}_{R}} - {{V}_{L}}){{\partial }_{x}}\vartheta - ({{V}_{R}} + {{V}_{L}}){{\partial }_{x}}\varphi ]{{\rho }_{d}}.} $$

(156d)

Here operator d(x) creates a mobile particle of momentum k and velocity ${{{v}}_{d}}$ = ∂ε/∂k, and ρ_d(x) = d^†(x)d(x) is the fermion density operator. In this treatment the curvature of the fermion dispersion was kept explicit. When applied to the calculation of the spectral function, this model captures power-law threshold singularities beyond the limit of linear Luttinger liquid theory and yields the universal description. However, this model does not yet capture relaxation processes as the interaction term couples mobile particle either to the left-movers or to the right-movers separately. At the level of fermionic description of the problem, we saw that finite decay rate is generated by RRL process that involves particle and two particle-hole pairs. This suggests that we need another coupling term of d-particle with both left- and right-movers. We thus add

$$H_{{\operatorname{int} }}^{'} = \gamma \int {dx{{\rho }_{d}}({{\partial }_{x}}{{\varphi }_{R}})({{\partial }_{x}}{{\varphi }_{L}})} $$

(157)

which is inspired by [64] where friction of a heavy particle moving through the Luttinger liquid was considered. An estimate for the coupling constant was given γ ~ V²/ε_F which is qualitatively consistent with three-particle scattering process.

Let us return now to the effective Hamiltonian Eq. (156) and look for a single high-energy particle. From H_d the time-ordered (retarded) free propagator of d-electron is $G_{0}^{{{\text{ret}}}}$(x, t) = 〈Td^†(x, t)d(0, 0)〉 = θ(t)e^iεtδ(x – ${{{v}}_{d}}$t) which simply describes ballistically propagating particle. The Fourier transform of the latter is $G_{0}^{{{\text{ret}}}}$(k, ω) = (ω – ε – k${{{v}}_{d}}$ + iα)^–1. The idea now is to apply perturbation theory in $H_{{\operatorname{int} }}^{'}$ to determine the self-energy for the d-particle induced by collisions with right-movers and left-movers.

The Dyson equation for the d-electron gives dressed propagator

$${{G}^{{{\text{ret}}}}}(k,\omega ) = \frac{1}{{\omega - \varepsilon - k{{{v}}_{d}} - {{\Sigma }^{{{\text{ret}}}}}(k,\omega )}},$$

(158)

where self-energy appears to the second order in γ

$${{\Sigma }^{{{\text{ret}}}}} = - i{{\gamma }^{2}}\int {dxdt{{e}^{{ - ikx + i\omega t}}}{{\Pi }_{R}}(x,t){{\Pi }_{L}}(x,t)G_{0}^{{{\text{ret}}}}(x,t)} ,$$

(159)

where free propagator for bosonic fields is

$$\begin{gathered} {{\Pi }_{{R(L)}}}(x,t) = \langle {{\partial }_{x}}{{\varphi }_{{R(L)}}}(x,t){{\partial }_{x}}{{\varphi }_{{R(L)}}}(0,0)\rangle \\ = - \frac{1}{{2\pi {{{(x \mp {v}t \pm i\alpha )}}^{2}}}}. \\ \end{gathered} $$

(160)

A slight comment here that following [65] it is a good idea to rescale bosonic fields first φ → $\sqrt K$φ and ϑ → ϑ/$\sqrt K$, which removes LL interaction parameter K from H₀ but renormalizes coefficients in $H_{{\operatorname{int} }}^{'}$. We reabsorbed all factors in the redefinition of γ and so Π_R(L)(x, t) is written above for the rescaled fields. The particle life-time is determined by the imaginary part of the self-energy, so that we look at the latter

$${\text{Im}}{{\Sigma }^{{{\text{ret}}}}}\, = \, - {\kern 1pt} \frac{{{{\gamma }^{2}}}}{{4{{\pi }^{2}}}}\int\limits_{ - \infty }^{ + \infty } {dt\frac{{{{e}^{{i(\omega - \varepsilon - k{{{v}}_{d}})t}}}}}{{{{{[({{{v}}_{d}}\, - \,{v})t\, + \,i\alpha ]}}^{2}}{{{[({{{v}}_{d}}\, + \,{v})t\, - \,i\alpha ]}}^{2}}}}} .$$

(161)

Notice here that by assumption ${{{v}}_{d}}$ > ${v}$ so that poles of the integrand are in the different parts of the complex plane. At the mass-shell we find d-particle relaxation rate

$$\tau _{d}^{{ - 1}} = \operatorname{Im} {{\Sigma }^{{{\text{red}}}}}(k,\omega = \varepsilon + {{{v}}_{d}}k) = \frac{{{{\gamma }^{2}}({v}_{d}^{2} - {{{v}}^{2}})}}{{8\pi {v}_{d}^{3}{{\alpha }^{3}}}}.$$

(162)

We arrive at the finite result which, however, depends on the cut-off parameter α. To resolve this issue we appeal to the fact that for the model to be well-defined there has to be a clear energy separation between the sub-bands of d-particle and bosonized right- and left-movers. One can argue from the dimensional analysis that cut-off should scale at low energies, k → k_F, as α^–1 ~ k – k_F. Furthermore, one can take ${v}_{d}^{2}$ – ${v}_{{}}^{2}$ ≈ 2${v}_{F}^{{}}$(k – k_F)/m*, and also γ ∝ ${{V}_{{2{{k}_{F}}}}}$(V₀ – ${{V}_{{2{{k}_{F}}}}}$)/(m*${v}_{F}^{2}$), where parametrically the latter is known from the perturbation theory in fermions of Section 2.2. Finally, combining everything together we get

$$\tau _{d}^{{ - 1}}\sim {{\varepsilon }_{F}}\frac{{V_{{2{{k}_{F}}}}^{2}{{{({{V}_{0}} - {{V}_{{2{{k}_{F}}}}})}}^{2}}}}{{{v}_{F}^{4}}}{{\left( {\frac{{k - {{k}_{F}}}}{{{{k}_{F}}}}} \right)}^{4}}.$$

(163)

This estimate matches with the zero-temperature quasiparticles relaxation rate from Eq. (16).