# Long lifetimes of ultrahot particles in interacting Fermi systems

M. Bard  Institut für Nanotechnologie, Karlsruhe Institute of Technology,
76021 Karlsruhe, Germany  I. V. Protopopov  Department of Theoretical Physics,
University of Geneva, 1211 Geneva, Switzerland  Landau Institute for
Theoretical Physics, 119334 Moscow, Russia  A. D. Mirlin  Institut für
Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
Institut für Theorie der Kondensierten Materie, Karlsruhe Institute of
Technology, 76128 Karlsruhe, Germany  Petersburg Nuclear Physics Institute,
188350 St. Petersburg, Russia

###### Abstract

The energy dependence of the relaxation rate of hot electrons due to
interaction with the Fermi sea is studied. We consider 2D and 3D systems,
quasi-1D quantum wires with multiple transverse bands, as well as single-
channel 1D wires. Our analysis includes both spinful and spin-polarized
setups, with short-range and Coulomb interactions. We show that, quite
generally, the relaxation rate is a non-monotonic function of the electron
energy and decays as a power-law at high energies. In other words, ultra-hot
electrons regain their coherence with increasing energy. Such a behavior was
observed in a recent experiment on multi-band quantum wires, J. Reiner et al,
Phys. Rev. X 7, 021016 (2017).

###### pacs:

##  I Introduction

Relaxation of excitations due to interaction with the surrounding system is a
fundamental phenomenon that governs the behavior of complex many-body systems
at sufficiently large time scales and is in the core of thermalization and
ergodization. Further, relaxation breaks the quantum coherence and curtains
quantum laws of nature, thus giving rise to the largely classical world that
surrounds us.

Already in the early days of the quantum many-body theory it was realized,
however, that the relaxation times associated with the low-energy excitations
in the many-body systems can be long (compared to other microscopic time
scales). The paradigmatic example of such a situation is the Landau Fermi-
liquid theory that rests on the fact that, in an interacting electronic
system, a Landau quasiparticle with excitation energy $\epsilon$ (counted from
the Fermi energy $\epsilon_{\mathrm{F}}$) has the decay rate (inverse
lifetime)
$1/\tau_{\epsilon}\propto\epsilon^{2}/\epsilon_{\mathrm{F}}\ll\epsilon$. The
long lifetimes of quasiparticles at low energies give rise to a plethora of
quantum phenomena in the low-temperature properties of electronic systems
observed in a variety of experiments, including, e.g. quantum corrections to
conductivity WeakLocalization  and quantum Hall interferometry
QHInterferometry .

In view of the physical importance of relaxation processes, they were
extensively studied over decades in various condensed matter systems and for
various types of excitations, including electrons in normal
metalsEERelaxationNormal , Bogolyubov quasiparticles in
superconductorsEERelaxationSuperconductors  and Bose gasesRelaxationColdAtoms
; Beliaev ; Tan2010 ; Ristivojevic2014 ; Ristivojevic2016 , electrons in one-
dimensional (1D) quantum wiresRelaxation1DExp  and quantum Hall edge
channelsRelaxationHallExp , to name a few. Most of these studies focused on
the low-energy excitations that are usually expected to relax slowly and whose
properties in many cases show remarkable degree of universality.

The fact that not only at low energies can fermionic excitations exhibit long
relaxation times was emphasized recently in Ref. Beidenkopf2017, . This work
studied the interaction-induced decoherence of hot electrons in a
semiconducting nanowire by means of scanning tunneling microscopy. Remarkably,
it was observed that, while at low energies the electron relaxation time
reduces as its energy grows (the behavior familiar from the conventional
Fermi-liquid theory), it becomes long again at excitation energies larger than
the Fermi energy. In other words, electrons regain their coherence at high
energies. This surprising behavior was explained in Ref. Beidenkopf2017,  by
an analysis of the relaxation in processes that involve only electrons from
the lowest band of the transverse quantization in the nanowire.

The results of Ref. Beidenkopf2017,  pose natural questions. How universal is
the non-monotonic behavior of the relaxation rate and the regain of coherence
at high energies? In particular, do they persist in quasi-1D systems with
several transverse bands involved in relaxation? Are they also relevant to
two-dimensional (2D) and three-dimensional (3D) systems?

The purpose of this work is to answer these questions. We show that the non-
monotonicity of the relaxation rate and the regain of quantum coherence is a
very general and robust phenomenon. We do this by studying several models of
interacting fermions, ranging from fermions with isotropic spectrum in $D\geq
2$, through multi-band quantum wires, to strictly 1D interacting fermions. We
show that in all these models the electron relaxation rate decays as a power
law with the momentum $p_{1}$ of the hot fermion, provided that the
interparticle interaction decays sufficiently fast as a function of the
momentum transfer. In 2D and 3D as well as in quasi-1D the relaxation rate
scales as $1/p_{1}$ while in 1D the rate vanishes as $1/p_{1}^{5}$. In this
analysis, we focus on models with the interaction potential in momentum
domain, $V(q)$, characterized by a single momentum scale $q_{0}$ such that the
interaction can be expanded, $V(q)=V_{0}(1-q^{2}/q_{0}^{2})$, for $q\ll q_{0}$
and gets suppressed sufficiently strongly at $q\gg q_{0}$. We also explore the
case of Coulomb interaction and demonstrate that the non-monotonic behavior of
the relaxation rate applies in this situation as well.

A related problem of decay of high-energy _bosonic_ quasiparticles was
recently studied in Refs. Tan2010, ; Ristivojevic2016,  in 1D geometry. The
authors of these works found a saturation of the decay rate at high energy,
which is in contrast to the $1/p_{1}^{5}$-decay that we predict for fermionic
quasiparticles. We will return to the origin of this difference in Sec. V.

The structure of the paper is as follows. We start in Sec. II with the
simplest possible setting: weakly interacting fermions with an isotropic
parabolic dispersion in $D\geq 2$ spatial dimensions. We analyze two-particle
collision processes and show that the corresponding relaxation rate is a non-
monotonic function of the quasiparticle energy. section III is devoted to the
discussion of the relaxation in multi-band metallic wires: a situation that
can be viewed as a strongly anisotropic limit of 2D and 3D models of Sec. II.
In Sec. IV we consider fermions in a 1D parabolic band where two-particle
collisions are forbidden by energy and momentum conservation laws and the
analysis of three-fermion collisions is required. Our results are summarized
in Sec. V.

##  II Isotropic 3D and 2D cases

The simplest setting one can assume to study the relaxation in a condensed
matter system at high energies is that of (spinless or spinful) particles with
parabolic dispersion in $(D\geq 2)$-dimensional space. Clearly, the cases
$D=2$ and $D=3$ are of particular interest from the physical point of view. We
thus consider in this section an isotropic $D$-dimensional Fermi sea with
Fermi momentum $p_{\mathrm{F}}$ and, on top of it, an electron with momentum
$p_{1}\gg p_{\mathrm{F}}$. We assume that the particles interact via an
interaction $V(q)$ characterized by a single momentum scale $q_{0}$, so that
$V(q)$ can be expanded, $V(q)\sim V_{0}(1-q^{2}/q_{0}^{2})$, for $q\lesssim
q_{0}$ and is sufficiently strongly suppressed at $q\gg q_{0}$. As a guiding
example, one can think about a model interaction with an exponential decay,
e.g., $V(q)=V_{0}e^{-(q/q_{0})^{2}}$. As we discuss in the end of this
section, our results remain applicable also in the case of screened Coulomb
interaction in 2D and 3D. We assume $q_{0}\gtrsim p_{\mathrm{F}}$, while the
relation between $q_{0}$ and $p_{1}$ can be arbitrary.

We are interested in the relaxation rate for our hot particle which is given
by the Fermi golden rule,

$\frac{1}{\tau_{p_{1}}}=\frac{1}{2!}\int
d\mathbf{p}_{2}d\mathbf{p}_{1}^{\prime}d\mathbf{p}_{2}^{\prime}\,\delta\left(E_{\rm
i}-E_{\rm f}\right)\delta\left({\bf P}_{\rm i}-{\bf P}_{\rm
f}\right)n_{\mathrm{F}}(\epsilon_{2})[1-n_{\mathrm{F}}(\epsilon_{1}^{\prime})][1-n_{\mathrm{F}}(\epsilon_{2}^{\prime})]\left|M_{\mathbf{p}_{1},\mathbf{p}_{2}}^{\mathbf{p}_{1}^{\prime},\mathbf{p}_{2}^{\prime}}\right|^{2}.$
(1)

This expression contains only the out-scattering rate since by assumption the
initial distribution consists of a filled Fermi sea and an additional high-
energy fermion. Here $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$
($\mathbf{p}_{1}^{\prime}$ and $\mathbf{p}_{2}^{\prime}$) are the particle
momenta before (after) the scattering, $E_{\rm i}$, ${\bf P}_{\rm i}$ and
$E_{\rm f}$, ${\bf P}_{\rm f}$ are the total energy and momentum of the two
particles before and after the collision, $\delta$ functions express the
energy and momentum conservation, and we use the notation
$\epsilon_{i}\equiv\epsilon_{\mathbf{p}_{i}}=\mathbf{p}_{i}^{2}/2m$. The
matrix element
$M_{\mathbf{p}_{1},\mathbf{p}_{2}}^{\mathbf{p}_{1}^{\prime},\mathbf{p}_{2}^{\prime}}$
consists of the direct and exchange terms. In the spin-polarized case we have

$\left|M_{\mathbf{p}_{1},\mathbf{p}_{2}}^{\mathbf{p}_{1}^{\prime},\mathbf{p}_{2}^{\prime}}\right|^{2}=\left[V(\left|\mathbf{p}_{1}-\mathbf{p}_{1}^{\prime}\right|)-V(\left|\mathbf{p}_{1}-\mathbf{p}_{2}^{\prime}\right|)\right]^{2}.$
(2)

In the case of fermions with spin we find, after the summation over the spin
polarization of the second electron:

$\begin{split}\left|M_{\mathbf{p}_{1},\mathbf{p}_{2}}^{\mathbf{p}_{1}^{\prime},\mathbf{p}_{2}^{\prime}}\right|^{2}=\left[V(\left|\mathbf{p}_{1}-\mathbf{p}_{1}^{\prime}\right|)\right]^{2}+\left[V(\left|\mathbf{p}_{1}-\mathbf{p}_{2}^{\prime}\right|)\right]^{2}\\\
-V(\left|\mathbf{p}_{1}-\mathbf{p}_{1}^{\prime}\right|)V(\left|\mathbf{p}_{1}-\mathbf{p}_{2}^{\prime}\right).\end{split}$
(3)

Exploiting rotational invariance of the problem, one can recast Eq. (1) into a
convenient form (see Appendix A):

$\begin{split}\frac{1}{\tau_{p_{1}}}=\frac{S_{D-2}}{2p_{1}^{D-2}}\int_{0}^{\infty}dPP^{D-1}\int_{\left|P/2-p_{1}\right|}^{P/2+p_{1}}q^{2D-3}dq\int_{0}^{\pi}d\phi\,d\phi^{\prime}\,\left(\sin\phi\sin\phi^{\prime}\right)^{D-2}&\delta\left(\epsilon_{1}-\frac{p_{1}^{2}}{2m}\right)\\\
\times&n_{\mathrm{F}}(\epsilon_{2})[1-n_{\mathrm{F}}(\epsilon_{1}^{\prime})][1-n_{\mathrm{F}}(\epsilon_{2}^{\prime})]w_{q}(\phi,\phi^{\prime}).\end{split}$
(4)

Here $S_{d}$ is the volume of a $d$-dimensional sphere ($S_{d=0}=2$).
Physically, the integration variables $P$ and $q$ in Eq. (4) are the center-
of-mass and relative momentum of the particles in the collision process,
$P=\left|{\bf p}_{1}+{\bf p}_{2}\right|=\left|{\bf p}_{1}^{\prime}+{\bf
p}_{2}^{\prime}\right|$ and $2q=\left|{\bf p}_{1}-{\bf
p}_{2}\right|=\left|{\bf p}_{1}^{\prime}-{\bf p}_{2}^{\prime}\right|$; angle
$\phi$ ($\phi^{\prime}$) is the angle between center-of-mass momentum and
relative momentum before (after) the collision. The energies $\epsilon_{i}$
and $\epsilon_{i}^{\prime}$ are functions of integration variables given by

$\displaystyle 2m\epsilon_{1,2}=\frac{P^{2}}{4}+q^{2}\pm Pq\cos\phi,$ (5)
$\displaystyle 2m\epsilon^{\prime}_{1,2}=\frac{P^{2}}{4}+q^{2}\pm
Pq\cos\phi^{\prime}.$ (6)

The $\delta$ function in Eq. (4) ensures that the momentum of one of the
incoming particles equals $p_{1}$ and fixes the angle $\phi$ to

$\phi_{0}=\arccos\frac{p_{1}^{2}-q^{2}-P^{2}/4}{Pq}.$ (7)

The limits of the $q$-integration guarantee that $0<\phi_{0}<\pi$. Finally,
the function $w_{q}(\phi,\phi^{\prime})$ represents the (properly angle-
averaged) matrix element squared. It is given by

$w_{q}(\phi,\phi^{\prime})=S_{D-3}\int_{0}^{\pi}d\gamma\left(\sin\gamma\right)^{D-3}\left[V_{+}-V_{-}\right]^{2}$
(8)

and

$w_{q}(\phi,\phi^{\prime})=S_{D-3}\int_{0}^{\pi}d\gamma\left(\sin\gamma\right)^{D-3}\left[V_{+}^{2}+V_{-}^{2}-V_{+}V_{-}\right]$
(9)

in the spinless and spinful cases, respectively, with

$V_{\pm}=V\left[q\sqrt{2\left(1\pm\cos\phi\cos\phi^{\prime}\pm\cos\gamma\sin\phi\sin\phi^{\prime}\right)}\right].$
(10)

For the special case of $D=2$, the integration over $\gamma$ in Eqs. (8) and
(9) should be understood according to

$S_{-1}\int\frac{d\gamma}{\sin\gamma}\longrightarrow\sum_{\gamma=0,\pi}.$ (11)

Figure 1: High-energy scattering process ($p_{1}\gg p_{\mathrm{F}}$) in terms
of the center-of-mass and relative momentum of the colliding particles. The
Fermi distribution $n_{\mathrm{F}}(p_{2})$ restricts the allowed values of $q$
and $P$ to a small vicinity of the point $(p_{1}/2,p_{1})$, which is
represented by a shaded region between the line $P/2+q=p_{1}$ and the ellipse
$P^{2}/2+2q^{2}-p_{1}^{2}-p_{\mathrm{F}}^{2}=0$.

Equation (4) is fully general and applies to arbitrary temperature and
momentum $p_{1}$. It simplifies considerably in the case of $T=0$ and
$p_{1}\gg p_{\mathrm{F}}$. Under these conditions, the Fermi function
$n_{\mathrm{F}}(\epsilon_{2})$ restricts the integration over $p$ and $q$ to a
small vicinity of the point $P=p_{1}$, $q=p_{1}/2$, see Fig. 1. The angle
$\phi_{0}$ defined by Eq. (7) is then small in the whole range of integration,

$\phi_{0}\simeq
2\frac{\sqrt{P+2q-2p_{1}}}{\sqrt{p_{1}}}\lesssim\frac{p_{\mathrm{F}}}{p_{1}}\ll
1.$ (12)

In contrast to the case of low-energy scattering processes familiar from the
Fermi-liquid theory, the Fermi factors associated to the outgoing momenta,
$(1-n_{\mathrm{F}}(\epsilon_{i}^{\prime}))$ do not play a major role here, as
the typical momenta after the scattering are large. Under the above
assumptions, the effect of these factors is only to exclude almost perfect
forward scattering processes characterized by

$\min(\phi^{\prime},\pi-\phi^{\prime})<\phi_{0}^{\prime}=\arccos\frac{p^{2}/4+q^{2}-p_{\mathrm{F}}^{2}}{pq}\lesssim\frac{p_{\mathrm{F}}}{p_{1}}.$
(13)

Under the assumption that the characteristic momentum transfer $q_{0}$ is
larger than (or of the order of) the Fermi momentum, $q_{0}\gtrsim
p_{\mathrm{F}}$, the smallness of the angles $\phi_{0}$ and
$\phi_{0}^{\prime}$ allows us to fully decouple the $\phi^{\prime}$
integration in Eq. (4). This yields

$\frac{1}{\tau_{p_{1}}}\sim
mp_{1}^{D-2}p_{\mathrm{F}}^{D}\int_{0}^{\pi}d\phi^{\prime}\left(\sin\phi^{\prime}\right)^{D-2}w_{p_{1}/2}(0,\phi^{\prime}).$
(14)

Here we have taken into account the characteristic value of the angle
$\phi_{0}\sim p_{\mathrm{F}}/p_{1}$ as well as the available area in the
$(q,P)$-plane, $p_{\mathrm{F}}^{3}/p_{1}$.

In Eq. (14) and in analogous formulas below, the symbol “$\sim$” means “equal
up to a numerical coefficient of order unity”. This coefficient depends on the
specific model of the interaction.

The resulting scaling of the relaxation time (14) depends on the presence of
spin as well as on the relation between $p_{1}$ and $q_{0}$. At
$p_{\mathrm{F}}\ll p_{1}\ll q_{0}$, the function $w_{p_{1}/2}$ in Eq. (14) can
be expanded in powers of $p_{1}^{-1}$. For fermions with spin, the function
$w_{p_{1}/2}$ is given by Eq. (9), which leads to the estimate
$w_{p_{1}/2}\sim V_{0}^{2}$ and results in the collision rate

$\frac{1}{\tau_{p_{1}}}\sim mV_{0}^{2}p_{\mathrm{F}}^{D}p_{1}^{D-2}\,,\qquad
p_{\mathrm{F}}\ll p_{1}\ll q_{0}.$ (15)

The result (15) is determined merely by the phase space available for the
collision process. On the other hand, for spinless (or spin-polarized)
fermions in the same parameter regime, the leading contribution to
$w_{p_{1}/2}$ vanishes due to Hartree-Fock cancellation, see Eq. (8). This
results in the suppression of the relaxation rate (compared to the spinful
case), yielding

$\frac{1}{\tau_{p_{1}}}\sim
mV_{0}^{2}\frac{p_{\mathrm{F}}^{D}p_{1}^{D+2}}{q_{0}^{4}}\,,\qquad
p_{\mathrm{F}}\ll p_{1}\ll q_{0}.$ (16)

Figure 2: Schematic plot (on log-log scale) of the momentum dependence of
relaxation rate in $D=3$, as predicted by Eqs. (15), (16), (17), (18) and
(19). At the largest energies, $p_{1}\gg q_{0}$, the relaxation rate is given
by Eq. (17) and decays as $1/p_{1}$, irrespectively of the presence of spin.
In the intermediate momentum range, $p_{\mathrm{F}}\ll p_{1}\ll q_{0}$, the
relaxation rate for spinful (spinless) fermions grows as $p_{1}$ ($p_{1}^{5}$)
[see Eqs. (15) and (16), respectively]. Finally, in the Fermi-liquid regime,
$p_{1}-p_{\mathrm{F}}\ll p_{\mathrm{F}}$, the relaxation rate exhibits
universal $(p_{1}-p_{\mathrm{F}})^{2}$ scaling, with a prefactor that is
smaller for spinless fermions due to Hartree-Fock cancellation, see Eqs. (18)
and (19).

The presence of spin becomes unimportant in the parameter regime
$p_{\mathrm{F}}\lesssim q_{0}\ll p_{1}$ (including the physically most
relevant case $q_{0}\sim p_{\mathrm{F}}$). In this situation, the interaction
$w_{p_{1}/2}(0,\phi^{\prime})$ is of the order of $V_{0}$ at $\phi^{\prime}=0$
but decays quickly beyond $\phi^{\prime}\sim q_{0}/p_{1}$, thus effectively
limiting the available region of $\phi^{\prime}$. Thus, we get

$\frac{1}{\tau_{p_{1}}}\sim
mV_{0}^{2}\frac{p_{\mathrm{F}}^{D}q_{0}^{D-1}}{p_{1}}\,,\qquad
p_{\mathrm{F}}\lesssim q_{0}\ll p_{1}.$ (17)

Equation (17) constitutes the main result of this section. It shows that the
relaxation rate of an ultra-hot particle exhibits a universal $1/p_{1}$
scaling in any spatial dimension $D\geq 2$ and irrespectively of the presence
of spin.

At low momenta, the relaxation rate $1/\tau_{p_{1}}$ follows the
characteristic Fermi-liquid scaling,
$1/\tau_{p_{1}}\propto(p_{\mathrm{F}}-p_{1})^{2}$ (up to logarithmic factors
in $D=2$). Explicitly, for spinful particles

$\frac{1}{\tau_{p_{1}}}\sim
mV_{0}^{2}p_{\mathrm{F}}^{2D-4}(p_{1}-p_{\mathrm{F}})^{2},\qquad
p_{1}-p_{\mathrm{F}}\ll p_{\mathrm{F}}\lesssim q_{0},$ (18)

while for electrons without spin the prefactor is smaller because of the
Hartree-Fock cancellation,

$\frac{1}{\tau_{p_{1}}}\sim
mV_{0}^{2}\frac{p_{\mathrm{F}}^{2D}}{q_{0}^{4}}(p_{1}-p_{\mathrm{F}})^{2},\qquad
p_{1}-p_{\mathrm{F}}\ll p_{\mathrm{F}}\lesssim q_{0}.$ (19)

Combining Eqs. (15) - (19), we conclude that the relaxation rate generically
exhibits a non-monotonic behavior, as illustrated in Fig. 2 for the case
$D=3$. In this figure, dashed and solid lines correspond to spinless and
spinful fermions, respectively. After an initial increase at relatively low
momenta, $p_{1}-p_{\mathrm{F}}\ll p_{\mathrm{F}}$ (which follows the universal
Fermi-liquid scaling but with different prefactors for spinful and spinless
cases) and an intermediate scaling regime at $p_{\mathrm{F}}\ll p_{1}\ll
q_{0}$, the relaxation rate starts to decrease as $1/p_{1}$ independently of
the presence of spin and of dimensionality of the system.

Our analytic predictions for the relaxation rate are in excellent agreement
with the direct numerical evaluation of the integral (4), as illustrated by
Fig. 3. A model interaction $V(q)=V_{0}e^{-q^{2}/2p_{\mathrm{F}}^{2}}$ was
used to generate this plot.

Figure 3: Dependence of the relaxation rate on momentum $p_{1}$ [as given by
Eq. (4)]. Model interaction $V(q)=V_{0}e^{-q^{2}/2p_{\mathrm{F}}^{2}}$ and
$D=3$ were used to generate the plot. The relaxation rate exhibits a non-
monotonic behavior with a maximum at $p_{1}\simeq 4p_{\mathrm{F}}$ and was
normalized by its maximal value $\tau_{\rm max}^{-1}$. Inset: the same
dependence in the log-log scale. Dashed lines show the analytic predictions
$1/\tau_{p}\propto(p-p_{\mathrm{F}})^{2}$ and $1/\tau_{p}\propto 1/p$ at
$p-p_{\mathrm{F}}\ll p_{\mathrm{F}}$ and $p\gg p_{\mathrm{F}}$, respectively.

Concluding this section, let us discuss the applicability of our results to
the case of Coulomb interaction between particles. To this end, we recall that
the large-energy asymptotic behavior (17) relies essentially on (i) the
kinematics of the collision process and (ii) the inability of the interaction
to transfer too large momenta [see discussion before Eq. (17)]. A careful
analysis shows that despite a relatively slow decay of the screened Coulomb
interaction,

$V_{\rm 2D}(q)=\frac{2\pi e^{2}}{\kappa+q}\,,\qquad V_{\rm 3D}(q)=\frac{4\pi
e^{2}}{\kappa^{2}+q^{2}}$ (20)

(with $\kappa$ being the inverse screening radius), it is still fast enough
for the scaling law (17) to apply. More generally, the range of applicability
of (17) extends to interaction potentials $V(q)$ that decay at large momenta
as $1/q^{\alpha}$ with $\alpha>(D-1)/2$. Indeed, inspecting Eqs. (9) and (10),
we see that $w_{q}(0,\phi^{\prime})\sim
q^{-2\alpha}\sin^{-2\alpha}(\phi^{\prime}/2)$. If $\alpha>(D-1)/2$, the
integration in Eq. (14) is dominated by small $\phi^{\prime}\sim q_{0}/p_{1}$
and we obtain the scaling (17). Using $V_{0}\sim e^{2}/\kappa$ in 2D,
$V_{0}\sim e^{2}/\kappa^{2}$ in 3D, and $q_{0}\sim\kappa$, we get the
relaxation rate of ultra-hot particles interacting via screened Coulomb
interaction (20) with $\kappa\gtrsim p_{\mathrm{F}}$:

$\frac{1}{\tau_{p_{1}}}\sim\frac{me^{4}}{p_{1}}\times\left\\{\begin{array}[]{cc}\displaystyle\frac{p_{\mathrm{F}}^{2}}{\kappa},&\quad
D=2,\\\\[11.38092pt] \displaystyle\frac{p_{\mathrm{F}}^{3}}{\kappa^{2}},&\quad
D=3.\end{array}\right.$ (21)

The origin of the $1/p_{1}$ scaling (17), (21) of the relaxation rate with the
inverse momentum of the hot particle can be explained in the following way. In
view of the suppression of the interaction at momentum transfers exceeding
$q_{0}$, the energy transferred to a particle emerging form the Fermi sea can
be at most of the order of $q_{0}^{2}/2m$. Such a change of the energy of a
hot particle corresponds to a momentum transfer in the direction of $\bf
p_{1}$ of the order of $q_{0}^{2}/p_{1}$, where we used the value $p_{1}/m$ of
the hot-particle velocity. This implies a reduction of the phase space by a
factor $q_{0}/p_{1}\ll 1$, explaining the $1/p_{1}$ scaling of $1/\tau$. The
specific $1/p_{1}$ form is thus related to the parabolicity of the spectrum.
On the other hand, the decay of the relaxation rate with $p_{1}$ is more
generic and will take place for any dispersion law with velocity increasing as
a function of momentum.

##  III Multichannel quantum wires

In the preceding section, we have presented a comprehensive analysis of the
high-energy relaxation process for the particle with isotropic quadratic
energy spectrum in $D=2$ and $3$ spatial dimensions. Let us now turn to the
analysis of the relaxation in multichannel quantum wires and demonstrate that
the non-monotonic behavior found in Sec. II persists also in this case.

###  III.1 Setup

We begin by formulating the model. A quasi-1D wire that we will consider hosts
1D energy bands with quadratic dispersion enumerated by an index ${\bf n}$
that corresponds to the transverse quantization,

$\epsilon_{\bf n}(p)=\frac{p^{2}}{2m}+\Delta_{\bf n}.$ (22)

Here, $p$ is the momentum in the direction along the wire (which we choose as
$z$-axis) and $\Delta_{\bf n}$ sets the bottom of the ${\bf n}$-th band. The
Fermi see resides in one or more of the low-lying bands. We denote by
$p_{\mathrm{F}}$ the Fermi momentum in the lowest band, see Fig. 4.

The electrons populating the wire interact via an interaction $V(|{\bf
r}|)=V(|{\bf r}^{\perp}|,z)$ that translates, in the band picture, into

$\begin{split}V_{{\bf n}_{1},{\bf n}_{2};{\bf n}_{1}^{\prime}{\bf
n}_{2}^{\prime}}(q)\equiv&\\\ \int d{\bf r}_{1}^{\perp}d{\bf
r}_{2}^{\perp}V(|{\bf r}^{\perp}_{1}-&{\bf r}^{\perp}_{2}|,q)\psi_{{\bf
n}_{1}}^{*}({\bf r}^{\perp}_{1})\psi_{{\bf n}_{2}}^{*}({\bf
r}^{\perp}_{2})\psi_{{\bf n}_{1}^{\prime}}({\bf r}^{\perp}_{1})\psi_{{\bf
n}_{2}^{\prime}}({\bf r}^{\perp}_{2}).\end{split}$ (23)

Here the integration runs over the cross-section of the wire and $\psi_{\bf
n}({\bf r}^{\perp})$ are wave functions of transversal quantization.

Our goal is to study the relaxation of a hot electron injected at some
momentum $p_{1}$ into one of the bands ${\bf n}_{1}$ such that its energy
$\epsilon_{{\bf n}_{1}}(p_{1})\gg\epsilon_{F}$. In order to accomplish this
task, one needs to know the band positions $\Delta_{\bf n}$ and the
transverse-quantization wave functions $\psi_{\bf n}({\bf r}^{\perp})$. As we
discuss below, the high-energy scattering processes involve excitations of
electrons into high energy bands. We thus expect that the corresponding
transversal wave functions are largely independent of microscopic details and
can be approximated by plane waves (e.g., with periodic boundary conditions).
In such a setting, the index $\bf n$ can be identified with the transversal
momentum. Its precise nature depends on the transversal dimensionality of the
wire. In this work, we consider two physically relevant cases (see Fig. 5):
(i) a wire defined as a stripe of width $d$ in a 2D electron gas, in which
case $\bf{n}$ is just an integer number, ${\bf n}=0,\pm 1,\ldots$, and (ii) a
wire with 2D cross-section (of characteristic size $d$), in which case ${\bf
n}=(n_{x},n_{y})$ becomes a 2D integer vector.

Figure 4: 1D bands of a quantum wire: energy $\epsilon$ as a function of the
momentum $p$ along the wire. We model a multi-channel quantum wire by a
collection of parabolic bands labeled by the index ${\bf n}$ of transverse
quantization. The Fermi sea (bold sections of lines) occupies one or more low-
lying bands; $p_{\mathrm{F}}$ denotes the Fermi momentum in the lowest band.
Figure 5: Quasi-1D geometry. (a) Two finite dimensions of size $d$ to study
the crossover to 3D. (b) One finite dimension to study the crossover to 2D.

In accordance with our identification of the band index with the transversal
momentum, we choose

$\Delta_{\bf n}=\Delta_{0}|{\bf
n}|^{2}\,,\qquad\Delta_{0}\sim\frac{1}{md^{2}},$ (24)

and impose the momentum conservation condition on the matrix elements of the
interaction

$V_{{\bf n}_{1},{\bf n}_{2};{\bf n}_{1}^{\prime}{\bf
n}_{2}^{\prime}}(q)=\frac{1}{d^{D_{\perp}}}V_{{\bf n}_{1}-{\bf
n}^{\prime}_{1}}(q)\delta_{{\bf n}_{1}+{\bf n}_{2}-{\bf n}^{\prime}_{1}-{\bf
n}^{\prime}_{2}}.$ (25)

Here, $D_{\perp}=1,2$ is the transversal dimensionality of the wire. In
analogy with Sec. II, we assume the interaction to be approximately isotropic,
$V_{{\bf n}_{1}-{\bf n}^{\prime}_{1}}(q)\simeq
V\left(Q\equiv\sqrt{q^{2}+4\pi^{2}|{\bf n}|^{2}/d^{2}}\right)$, and
characterized by a single momentum scale $q_{0}$. Specifically, the
interaction $V(Q)$ can be expanded at not too large momenta,

$V(Q)\simeq V_{0}\left(1-Q^{2}/q_{0}^{2}\right)\,,\qquad Q\ll q_{0},$ (26)

and decays fast enough at $Q\gg q_{0}$. Physically, this means the relaxation
rate is determined by momentum transfers less than or of the order of $q_{0}$.

###  III.2 Q1D setup with one lateral dimension

We start our analysis of quantum wires by considering the case of a wire
realized as a stripe of width $d$ in a 2D electron gas. Such a model can be
viewed as describing the crossover from one to two spatial dimensions.
Assuming a weak interaction, we employ the golden rule to calculate the
relaxation rate:

$\displaystyle\frac{1}{\tau}$ $\displaystyle=$ $\displaystyle\frac{1}{2!}\int
dp_{2}\,dp_{1}^{\prime}\,dp_{2}^{\prime}\\!\\!\\!\\!\sum_{n_{2},n_{1^{\prime}},n_{2^{\prime}}}\delta(p_{1}+p_{2}-p_{1}^{\prime}-p_{2}^{\prime})$
(27) $\displaystyle\times$
$\displaystyle\delta_{n_{1}+n_{2},n_{1^{\prime}}+n_{2^{\prime}}}\delta(\epsilon_{1}+\epsilon_{2}-\epsilon_{1^{\prime}}-\epsilon_{2^{\prime}})$
$\displaystyle\times$ $\displaystyle
F(\lambda_{1},\lambda_{2};\lambda_{1^{\prime}},\lambda_{2^{\prime}}).$

Here we have introduced a shorthand notation $\lambda_{i}=(n_{i},p_{i})$, and
$\epsilon_{i}\equiv\epsilon_{n_{i}}(p_{i})$ is given by Eq. (22). The
summation over transversal momentum (that is just a scalar integer in the
present case) is restricted by the conservation law assumed in our model. The
function $F$ in Eq. (27) reads

$\displaystyle
F(\lambda_{1},\lambda_{2};\lambda_{1^{\prime}},\lambda_{2^{\prime}})$
$\displaystyle=$ $\displaystyle
n_{\mathrm{F}}(\epsilon_{2})[1-n_{\mathrm{F}}(\epsilon_{1^{\prime}})][1-n_{\mathrm{F}}(\epsilon_{2^{\prime}})]$
(28) $\displaystyle\times$
$\displaystyle\left|M_{\lambda_{1},\lambda_{2}}^{\lambda_{1^{\prime}},\lambda_{2^{\prime}}}\right|^{2},$

with the modulus squared of the matrix element given by

$\left|M_{\lambda_{1},\lambda_{2}}^{\lambda_{1^{\prime}},\lambda_{2^{\prime}}}\right|^{2}=\frac{1}{d^{2}}\left[V_{n_{1}-n_{1^{\prime}}}(p_{1}-p_{1}^{\prime})-V_{n_{1}-n_{2^{\prime}}}(p_{1}-p_{2}^{\prime})\right]^{2}$
(29)

or

$\displaystyle\left|M_{\lambda_{1},\lambda_{2}}^{\lambda_{1^{\prime}},\lambda_{2^{\prime}}}\right|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{d^{2}}\Bigl{[}\left[V_{n_{1}-n_{1^{\prime}}}(p_{1}-p_{1}^{\prime})\right]^{2}$
(30) $\displaystyle+$
$\displaystyle\left[V_{n_{1}-n_{2^{\prime}}}(p_{1}-p_{2}^{\prime})\right]^{2}$
$\displaystyle-$ $\displaystyle
V_{n_{1}-n_{1^{\prime}}}(p_{1}-p_{1}^{\prime})V_{n_{1}-n_{2^{\prime}}}(p_{1}-p_{2}^{\prime})\Bigr{]}$

in the absence and presence of spin, respectively. It proves convenient to
introduce the (longitudinal) momentum transfer $q=p_{1}-p_{1}^{\prime}$ as one
of the integration variables. The integrations over $p_{2}$ and
$p_{1}^{\prime}$ can be performed with the help of the $\delta$ functions
describing the conservation of energy and longitudinal momentum, yielding

$\displaystyle\frac{1}{\tau}$ $\displaystyle=$
$\displaystyle\frac{m}{2}\sum_{\\{n_{i}\\}}\int\frac{dq}{|q|}$
$\displaystyle\times$ $\displaystyle
F_{\\{n_{i}\\}}\left(p_{1},p_{1}-q+\frac{m\Delta_{\mathrm{eff}}}{q};p_{1}-q,p_{1}+\frac{m\Delta_{\mathrm{eff}}}{q}\right),$

where we introduced the energy

$\Delta_{\mathrm{eff}}=\Delta_{0}(n_{1}^{2}+n_{2}^{2}-n_{1^{\prime}}^{2}-n_{2^{\prime}}^{2}).$
(32)

A detailed analysis of this expression can be found in Appendix B.1. We find
the following results for the relaxation rate of an ultra-hot electron with
the total momentum $p_{1}^{\rm tot}=\sqrt{p_{1}^{2}+p_{1\perp}^{2}}\gg
p_{\mathrm{F}}$ in a quasi-1D wire constituting a 2D strip:

$\frac{1}{\tau_{\epsilon}}\sim\left\\{\begin{array}[]{ll}mV_{0}^{2}p_{\mathrm{F}}^{2},&\quad
p_{\mathrm{F}}\ll p_{1}^{\rm tot}\ll q_{0},\ \ \text{spinful},\\\\[8.5359pt]
\displaystyle
m^{3}V_{0}^{2}\frac{p_{\mathrm{F}}^{2}}{q_{0}^{4}}\epsilon^{2},&\quad
p_{\mathrm{F}}\ll p_{1}^{\rm tot}\ll q_{0},\ \ \text{spinless},\\\\[8.5359pt]
\displaystyle\sqrt{m}V_{0}^{2}p_{\mathrm{F}}^{2}\frac{q_{0}}{\sqrt{\epsilon}},&\quad
p_{\mathrm{F}}\lesssim q_{0}\ll p_{1}^{\rm tot};\,(p_{1\perp},p_{1})\not\in
A,\end{array}\right.$ (33)

where we introduced the transversal momentum $p_{1\perp}=n_{1}/d$. Equation
(33) largely coincides with the corresponding results for an isotropic 2D
system, Sec. II. The difference to the 2D situation occurs when the momentum
is high and is almost in the transversal direction. More specifically, if the
momentum $(p_{1\perp},p_{1})$ belongs to the region defined by two equations
(see Fig. 6),

$\text{region A:}\qquad
p_{1\perp}>q_{0}^{2}d\quad\mathrm{and}\quad\frac{p_{1}}{p_{1\perp}}<\frac{1}{q_{0}d},$
(34)

the relaxation rate is additionally suppressed in comparison with the 2D
result due to transverse energy quantization. To understand the reason for
this, let us consider the case of an initial momentum pointing out exactly in
the transverse direction, $p_{1}=0$, which corresponds to the bottom of a
certain high energy band. A decay process then necessarily involves a
transition to a lower energy band, which means a momentum transfer $\gtrsim
1/d$ in the direction of the initial momentum. If this momentum transfer is
larger than $q_{0}^{2}/p_{1}$, such a process will be parametrically
suppressed, as explained in the end of Sec. II. This yields the first of the
conditions (34). A more detailed analysis shows that this suppression happens
not only at $p_{1}=0$ but also in a range of $p_{1}$ given by the second
condition (34). The relaxation rate in the regime (34) is determined by the
large-momentum tail of the interaction, which leads to a faster decay of the
relaxation rate than $1/\sqrt{\epsilon}$. The actual form of the decay is non-
universal as it depends on the large momentum behavior of $V(q)$. In the
continuum limit (fixed momenta $p_{\mathrm{F}}$, $q_{0}$, $p_{1}$,
$p_{1\perp}$ and $d\to\infty$), the regime (34) disappears and we recover the
usual isotropic 2D result.

Figure 6: Non-universal regime (34) in the plane spanned by the longitudinal
$p_{1}$ and transversal $p_{1\perp}$ momentum in quasi-1D wires. In this
regime, the relaxation rate decays as a function of energy $\epsilon$ faster
than $1/\sqrt{\epsilon}$. The actual dependence is non-universal since it
depends on the large-momentum tail of $V(q)$.

As in an isotropic 2D system, the above results apply also for interaction
potentials that decay at large $q$ as a power law $1/q^{\alpha}$ with
$\alpha>1/2$. This includes, in particular, the case of a (screened) Coulomb
interaction $V_{\rm 2D}(q)$, Eq. (20).

###  III.3 Q1D setup with two lateral dimensions

After the discussion of the setup with one finite transversal dimension, we
now turn to the case of a 3D wire, i.e., of a quasi-1D setup with two lateral
dimensions. In comparison to the discussion of Sec. III.2, we need to
associate two discrete indices with each electron state:
$n_{i}\to\mathbf{n}_{i}=(n_{i,x},n_{i,y})$, and replace in the matrix element
$V^{(2)}_{n}(p)/d$ by $V^{(3)}_{\mathbf{n}}(p)/d^{2}$, see Eq. (25).

In comparison to the situation with only one lateral dimension, there is an
additional subtlety here. If the integer vectors
$(\mathbf{n}_{1^{\prime}}-\mathbf{n}_{1})$ and
$(\mathbf{n}_{1^{\prime}}-\mathbf{n}_{2})$ are perpendicular to each other,
the longitudinal momentum transfer $q=0$ is allowed (“vertical relaxation”).
According to Eq. (LABEL:Eq:relaxation-Q1D-1-lateral-integration-over-q), the
contributions from those processes to the relaxation rate are formally
divergent in a logarithmic fashion. At zero temperature, such processes are
only allowed if the longitudinal momentum $p_{1}$ is smaller than the Fermi
momentum $p_{\mathrm{F}}$. The logarithmic singularity gets regularized due to
broadening of the $\delta$ function related to energy conservation by other
processes. As a result, the contribution of such processes gets an additional
logarithmic factor $\sim\ln(p_{1\perp}/p_{\mathrm{F}})$ in comparison with
“non-singular” processes. On the other hand, because of the above
orthogonality condition of the vectors
$(\mathbf{n}_{1^{\prime}}-\mathbf{n}_{1})$ and
$(\mathbf{n}_{1^{\prime}}-\mathbf{n}_{2})$, only a small fraction of all
processes belong to the “singular” class. This results in a suppression of
“singular” processes by a factor $(p_{\mathrm{F}}q_{0}d^{2})^{-1}\ll 1$. In a
parametrically broad range of momenta $p_{1\perp}$ of the hot electron, this
power-law suppression will be more important than the logarithmic enhancement,
so that the “singular” processes will yield a subleading contribution. Only at
very high $p_{1\perp}$ will the singular processes give a dominant
contribution (assuming one can still apply the model at such energies). This
will, however, modify our conclusion on $1/p^{\rm tot}_{1}$ decay of the
relaxation rate only by an additional logarithmic factor. Furthermore,
“singular” processes are absent in more realistic models in which the energy
dependence deviates from perfect parabolicity. In the following we thus
discard this subclass of processes.

Analyzing “non-singular” processes, we find for the relaxation rate of an
ultra-hot electron with the total momentum $p_{1}^{\rm
tot}=\sqrt{p_{1}^{2}+p_{1\perp}^{2}}\gg p_{\mathrm{F}}$ in a 3D wire (see
Appendix B.2 for detail)

$\frac{1}{\tau_{\epsilon}}\sim\left\\{\begin{array}[]{ll}m^{3/2}V_{0}^{2}p_{\mathrm{F}}^{3}\sqrt{\epsilon},&\qquad
p_{\mathrm{F}}\ll p_{1}^{\rm tot}\ll q_{0},\ \ \text{spinful},\\\\[8.5359pt]
\displaystyle
m^{7/2}V_{0}^{2}\frac{p_{\mathrm{F}}^{3}}{q_{0}^{4}}\epsilon^{5/2},&\qquad
p_{\mathrm{F}}\ll p_{1}^{\rm tot}\ll q_{0},\ \ \text{spinless},\\\\[8.5359pt]
\displaystyle\sqrt{m}V_{0}^{2}p_{\mathrm{F}}^{3}\frac{q_{0}^{2}}{\sqrt{\epsilon}},&\qquad
p_{\mathrm{F}}\lesssim q_{0}\ll p_{1}^{\rm tot},\\\
&\qquad(\mathbf{p}_{1\perp},p_{1})\not\in A,\end{array}\right.$ (35)

in consistency with the corresponding results for an isotropic 3D system, Sec.
II. In full analogy to the case of a wire constituting a 2D strip, there is
the regime A, Eq. (34) where the discreteness of the transverse spectrum leads
to an additional suppression of the relaxation rate.

As in an isotropic 3D system, these results apply also for interaction
potentials that decay at large $q$ as a power law $1/q^{\alpha}$ with
$\alpha>1$. This includes, in particular, the case of a (screened) Coulomb
interaction $V_{\rm 3D}(q)$, Eq. (20).

##  IV 1D wires and triple collisions

In the previous sections, we have shown that non-monotonicity is a generic
feature of the energy dependence of the relaxation rate in $D\geq 2$ spatial
dimensions as well in quasi-1D wires hosting many subbands. In the present
section, we consider the case of a single-channel 1D wire with parabolic
dispersion.

In one dimension, energy and momentum conservation restrict the two-particle
collisions studied in Sec. II to permutations of the momenta of colliding
particles. Thus, two-particle collisions can not lead to relaxation and one
has to study three-particle collision processes. In the “low-energy” domain
(energy much smaller than the Fermi energy), such a study was accomplished in
a number of worksLunde07 ; Khodas2007 ; imambekov09 ; imambekov11 ; micklitz11
; ristivojevic13 ; apostolov13 ; Lin2013 ; MatveevFurusaki2013 ; PGM2013 ;
Matveev2013 ; Dmitriev2012 ; Protopopov2014 ; Gangardt2014 ; protopopov13 .
Little is known, however, about the ultra-high-energy case
$\epsilon\gg\epsilon_{F}$. This gap is filled in the present section.

In the case of triple collisions, the Fermi golden rule reads [cf. Eq. (1) and
notations therein]:

$\frac{1}{\tau_{p_{1}}}=\frac{1}{2!3!}\int
dp_{2}dp_{3}dp_{1}^{\prime}dp_{2}^{\prime}dp_{3}^{\prime}\,\delta\left(E_{\rm
i}-E_{\rm
f}\right)\delta\left(P_{i}-P_{f}\right)n_{\mathrm{F}}(\epsilon_{2})n_{\mathrm{F}}(\epsilon_{3})[1-n_{\mathrm{F}}(\epsilon_{1}^{\prime})][1-n_{\mathrm{F}}(\epsilon_{2}^{\prime})][1-n_{\mathrm{F}}(\epsilon_{3}^{\prime})]\left|M_{p_{1},p_{2},p_{3}}^{p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime}}\right|^{2}.$
(36)

Here, $M_{p_{1},p_{2},p_{3}}^{p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime}}$
is the matrix element for the triple collisions whose precise form will be
discussed below. If the fermions have spin, the symbol
$\left|M_{p_{1},p_{2},p_{3}}^{p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime}}\right|^{2}$
implicitly incorporates the summation over spin indices.

The energy and momentum conservation constraints are conveniently resolved by
parametrizing the momenta of colliding particles according toristivojevic13 ;
Protopopov2014

$\displaystyle p_{k}=\frac{P}{3}+q\cos\left[\phi+\frac{2\pi(k-1)}{3}\right]$
$\displaystyle\,,\quad k=1,2,3,\;$ (37) $\displaystyle
p_{k}^{\prime}=\frac{P}{3}+q\cos\left[\phi^{\prime}+\frac{2\pi(k-1)}{3}\right]$
$\displaystyle\,,\quad k=1,2,3.\;$ (38)

The variable $-\infty<P<\infty$ is nothing but the total momentum of the
particles while $q\geq 0$ is analogous to the relative momenta in the two-
particle collision processes and fixes the total energy

$E_{i}=E_{f}=\frac{P^{2}}{6m}+\frac{3q^{2}}{4m}.$ (39)

One can now rewrite the collision rate (36) in the form [cf. Eq. (4) of Sec.
II]

$\begin{split}\frac{1}{\tau_{p_{1}}}=\frac{m}{2}\int_{-\infty}^{\infty}dP\int^{\infty}_{\left|P/3-p_{1}\right|}q\,dq\int_{-\pi}^{\pi}d\phi\,d\phi^{\prime}\;\delta\left[p_{1}-p_{1}(P,q,\phi)\right]n_{\mathrm{F}}(\epsilon_{2})n_{\mathrm{F}}(\epsilon_{3})[1-n_{\mathrm{F}}(\epsilon_{1}^{\prime})][1-n_{\mathrm{F}}(\epsilon_{2}^{\prime})][1-n_{\mathrm{F}}(\epsilon_{3}^{\prime})]w_{q}(\phi,\phi^{\prime}).\end{split}$
(40)

Here, $p_{1}(P,q,\phi)$ is given by Eq. (37) and $w_{q}(\phi,\phi^{\prime})$
is the modulus squared of the triple-collision matrix element,
$\left|M_{p_{1},p_{2},p_{3}}^{p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime}}\right|^{2}$,
evaluated on the mass shell.

Figure 7: Kinematics of 1D three-particle scattering process in the $(q,P)$
plane. The Fermi distributions $n_{\mathrm{F}}(\epsilon_{2})$ and
$n_{\mathrm{F}}(\epsilon_{3})$ restrict the allowed values of $q$ and $P$ to a
vicinity of the point $(2p_{1}/3,p_{1})$ represented by the shaded region
between the line $P/3+q=p_{1}$ and two hyperbolas.

The analysis of Eq. (40) proceeds to large extent along the same lines as that
of Sec. II. First of all, for $p_{1}\gg p_{\mathrm{F}}$ the Fermi functions
$n_{\mathrm{F}}(\epsilon_{2})$ and $n_{\mathrm{F}}(\epsilon_{3})$ restrict the
integration in the $(q,P)$ plane to a small neighborhood of the point
$(q=2p_{1}/3,P=p_{1})$, see Fig. 7. The $\delta$ function fixes then the angle
$\phi$ to

$\phi_{0}=\pm\arccos\frac{p_{1}-P/3}{q},\qquad|\phi_{0}|\lesssim\frac{p_{\mathrm{F}}}{p_{1}}.$
(41)

On the other hand, just as in the higher-dimensional situations and in
contrast to the low-energy scattering, the Fermi factors associated to the
outgoing particles are not important (provided that the interaction is capable
of transferring momenta of the order of $p_{\mathrm{F}}$, which is what we
have assumed). As a result, Eq. (40) reduces to

$\frac{1}{\tau_{p_{1}}}\sim
mp_{\mathrm{F}}^{2}\int_{-\pi}^{\pi}d\phi^{\prime}w_{2p_{1}/3}\left(\phi\sim\frac{p_{\mathrm{F}}}{p_{1}},\phi^{\prime}\right).$
(42)

Equation (42) relies essentially only on the kinematics of the three-particle
collision process. To complete our analysis, a dynamical input is necessary.
The matrix element
$M_{p_{1},p_{2},p_{3}}^{p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime}}$
governing the dynamics arises in the second order of the perturbation theory
in the two-body interactionKhodas2007 ; ristivojevic13 ; MatveevFurusaki2013 ;
Dmitriev2012 . In the spinless case it is given by the vacuum expectation
value

$M_{p_{1},p_{2},p_{3}}^{p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime}}=\left\langle
a_{p_{3}}a_{p_{2}}a_{p_{1}}\right|\hat{V}\frac{1}{E-\hat{H}_{0}+i0}\hat{V}\left|a^{\dagger}_{p_{1}}a^{\dagger}_{p_{2}}a^{\dagger}_{p_{3}}\right\rangle,$
(43)

where $\hat{H}_{0}$ and $\hat{V}$ are free and interaction parts of the
Hamiltonian, respectively. A straightforward albeit somewhat lengthy algebra
allows one to rewrite Eq. (43) as (see Appendix C)

$M_{p_{1},p_{2},p_{3}}^{p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime}}=M_{q}(\phi,\phi^{\prime})=\frac{4m}{3q^{2}}\frac{\sum_{j,k=0}^{2}\Gamma_{q}\left(\phi+\frac{2\pi
k}{3},\phi^{\prime}+\frac{2\pi j}{3}\right)\left[\cos\left(\phi+\frac{2\pi
k}{3}\right)-\cos\left(\phi^{\prime}+\frac{2\pi j}{3}\right)\right]}{\cos
3\phi-\cos 3\phi^{\prime}},$ (44)

with

$\begin{split}\Gamma_{q}(\phi,\phi^{\prime})=[V(q_{10})-V(q_{20})][V(q_{01})-V(q_{02})]\\\
-2V(q_{12})V(q_{21})+2V(q_{11})V(q_{22}),\end{split}$ (45)

where we have introduced shorthand notations for various momentum transfers
involved in the process:

$q_{kj}=q\left[\cos\left(\phi+\frac{2\pi
k}{3}\right)-\cos\left(\phi^{\prime}+\frac{2\pi j}{3}\right)\right].$ (46)

The matrix element $M_{q}(\phi,\phi^{\prime})$ is an odd periodic function of
each of the angles with the period $2\pi/3$ reflecting the
indistinguishability of the particles. The factor $1/q^{2}$ in Eq. (44) stems
from the typical scaling of the energy denominators in Eq. (43), while the
denominator $\cos 3\phi-\cos 3\phi^{\prime}$ takes into account the
possibility for them to become small for particular configurations of angles
$\phi=\phi^{\prime}\mod 2\pi/3$. Using the symmetry of the function
$\Gamma_{q}(\phi,\phi^{\prime})$ with respect to the exchange of $\phi$ and
$\phi^{\prime}$, it is easy to see, however, that the vanishing of the
denominator in Eq. (44) does not lead to a pole and the matrix element
$M_{q}(\phi,\phi^{\prime})$ is in fact a smooth function of angles.

A further examination of Eq. (44) shows that, for $q\gg q_{0}\gtrsim
p_{\mathrm{F}}$, the function $M_{q}(\phi\sim p_{\mathrm{F}}/q,\phi^{\prime})$
is of the order of $mp_{\mathrm{F}}V_{0}^{2}/q_{0}q^{2}$ for
$\phi^{\prime}\lesssim q_{0}/q$ and is strongly suppressed beyond this range
of angles (up to aforementioned periodicity). More specifically, at fixed
$\tilde{\phi}$ and $\tilde{\phi}^{\prime}$ and $q\gg q_{0}$ the matrix element
possesses the scaling

$\displaystyle
M_{q}\left(q_{0}\tilde{\phi}/q,q_{0}\tilde{\phi}^{\prime}/q\right)$ (47)
$\displaystyle=$
$\displaystyle\frac{4m}{9q^{2}}\left[V^{2}(q_{-})+V(0)(V(q_{-})-q_{-}V^{\prime}(q_{-}))\right]$
$\displaystyle-$ $\displaystyle\langle q_{-}\rightarrow q_{+}\rangle,$

where $q_{\pm}=\sqrt{3}q_{0}(\tilde{\phi}\pm\tilde{\phi}^{\prime})/2$ and
$V^{\prime}$ stands for the derivative of interaction with respect to
momentum. In deriving Eq. (47), we have neglected all the terms with momentum
transfer of order $q$. According to Eq. (47), the $1/q^{2}$ scaling of the
matrix element readily seen in Eq. (44) for generic values of angles $\phi$
and $\phi^{\prime}$ survives also in the regime $\phi,\phi^{\prime}\lesssim
q_{0}/q$ where the energy denominators in Eq. (44) are of the order of
$q_{0}^{2}/m$ only. The reason for this is an intricate cancellation between
various interaction processes in Eq. (44).

For $\tilde{\phi},\tilde{\phi}^{\prime}\lesssim 1$, Eq.(47) takes the form

$M_{q}\left(q_{0}\tilde{\phi}/q,q_{0}\tilde{\phi}^{\prime}/q\right)\sim\frac{mV_{0}^{2}}{q^{2}}\tilde{\phi}\tilde{\phi}^{\prime}=\frac{mV_{0}^{2}}{q_{0}^{2}}\phi\phi^{\prime}.$
(48)

Setting $\tilde{\phi}\sim p_{\mathrm{F}}/q_{0}\lesssim 1$ in Eq. (48) and
combining the resulting estimate with Eq. (42), we arrive at

$\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}p_{\mathrm{F}}^{4}V_{0}^{4}}{q_{0}p_{1}^{5}},\qquad
p_{1}\gg q_{0}\gtrsim p_{\mathrm{F}}\quad\text{(spinless)}.$ (49)

Equation (49) constitutes one of the main results of this section. It shows
that, for spinless fermions interacting via an interaction with characteristic
momentum transfer larger than $p_{\mathrm{F}}$, the relaxation rate at
ultrahigh energies, $p_{1}\gg q_{0}$, is suppressed in a power-law fashion as
$1/\tau_{p}\propto p^{-5}$. Let us note a qualitative similarity of this
result with the higher dimensional analog, Eq. (17). In both cases the
suppression of the available phase space by the maximal momentum that can be
transferred by the interaction brings the factor $1/p_{1}$. In addition, in
the case of triple collisions that govern the relaxation in 1D systems a
factor $1/p_{1}^{4}$ arises from the partial cancellations of direct and
exchange terms in the three-particle matrix element.

In the intermediate regime where the external momentum $p_{1}$ is smaller than
the momentum scale $q_{0}$ but still much larger than the Fermi momentum, we
expand the interaction potential $V(q)$ to fourth order. We obtain for the
squared matrix element

$w_{q}(\phi,\phi^{\prime})\sim\frac{m^{2}V_{0}^{4}q^{12}\sin^{2}(3\phi)\sin^{2}(3\phi^{\prime})}{q_{0}^{16}}.$
(50)

After the integration over $\phi^{\prime}$ and making use of the estimates
$q\sim 2p_{1}/3$ and $\phi\sim p_{\mathrm{F}}/p_{1}$, we find the scaling

$\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}p_{\mathrm{F}}^{4}V_{0}^{4}}{q_{0}^{16}}p_{1}^{10},\qquad
p_{\mathrm{F}}\ll p_{1}\ll q_{0}\quad\text{(spinless)}.$ (51)

In the low-energy regime we find the scaling (see Appendix D for details)

$\begin{split}\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}p_{\mathrm{F}}^{6}V_{0}^{4}}{q_{0}^{16}}(p_{1}-p_{\mathrm{F}})^{8},\quad
p_{1}-p_{\mathrm{F}}\ll&p_{\mathrm{F}}\ll q_{0},\\\
&(\mathrm{spinless}),\end{split}$ (52)

that coincides with the behavior found in Ref. Khodas2007,  if the Fermi
momentum is of the same order as the momentum scale of the interaction
$q_{0}$.

For fermions with spin, the matrix element is given by an analog of Eq. (43),
see Appendix C. After the spin summation and the parametrization according to
Eqs. (37) and (38), the square of the matrix element assumes a form that is
similar to Eq. (44). In contrast to the spinless case, non-integrable poles
appear in the squared matrix element for $\phi^{\prime}=\phi\mod 2\pi/3$.
However, at zero temperature the Fermi functions for final particles restrict
the integration over the angle $\phi^{\prime}$ to the region that excludes
those poles with the typical width of the excluded region of the order
$p_{\mathrm{F}}/p_{1}$. At non-zero temperature, the expression for the matrix
element has to be regularized. As shown in Ref. Dmitriev2012, , the divergence
is related to two consecutive two-particle scattering events separated by an
infinite time. It is shown there that a proper regularization subtracts these
double-counted two-particle processes and leads to a finite result. Since we
focus on the zero-temperature limit, such a regularization procedure is not
necessary.

At $p_{1}\gg q_{0}$, we can neglect all the terms in the transition
probability $w_{q}$ involving interactions with momentum transfer $q$ of the
order of $p_{1}$. For small angles $\phi<p_{\mathrm{F}}/p_{1}$ and
$\phi^{\prime}<q_{0}/p_{1}$, the square of the matrix element can then be
approximated by

$w_{q}(\phi,\phi^{\prime})\simeq
2m^{2}V_{0}^{2}\left(V_{0}^{\prime\prime}\right)^{2}\left[\phi^{4}+14\phi^{2}\left(\phi^{\prime}\right)^{2}+\left(\phi^{\prime}\right)^{4}\right].$
(53)

The divergence at $\phi=\phi^{\prime}$ does not manifest itself here because
the singular terms contain the interaction potential at a large momentum
transfer. Since the integration region over $\phi^{\prime}$ does not contain
the singular points (see above) and the potential is assumed to decay
sufficiently fast, such terms would only give a small correction. It is thus
indeed legitimate to drop them in Eq. (53). For values of $\phi^{\prime}$ much
larger than $q_{0}/p_{1}$ (modulo $2\pi/3$), the squared matrix element is
strongly suppressed. Comparing Eq. (53) to Eq. (48), we thus see that the
presence of spin merely enhances the transition probability by a factor of
$(\phi^{\prime}/\phi)^{2}\sim(q_{0}/p_{\mathrm{F}})^{2}$ without affecting its
scaling with momentum $p_{1}$. Accordingly, for the relaxation rate we find
the same scaling with $p_{1}$ as in the spinless case,

$\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}p_{\mathrm{F}}^{2}q_{0}V_{0}^{4}}{p_{1}^{5}},\qquad
p_{1}\gg q_{0}\gg p_{\mathrm{F}}\quad\text{(spinful)}.$ (54)

For intermediate momenta $p_{1}$ smaller than the characteristic scale of the
interaction $q_{0}$ but much larger than the Fermi momentum $p_{\mathrm{F}}$,
all momentum transfers are small. The interaction potential $V(q)$ can thus be
expanded to second order. For the square of the matrix element, we obtain

$w_{q}(\phi,\phi^{\prime})\sim\frac{m^{2}V_{0}^{4}}{q_{0}^{4}}f(\phi,\phi^{\prime}),$
(55)

where $f$ is a function of the angles $\phi$ and $\phi^{\prime}$ but is
independent of $q$. This function has second order poles at
$\phi=\phi^{\prime}\mod 2\pi/3$. However, at zero temperature, those poles are
excluded from the integration region of $\phi^{\prime}$ by the Fermi functions
of the final states [see also discussion below Eq. (52)]. The typical width of
the excluded region is of the order of $p_{\mathrm{F}}/p_{1}$. This leads to
the scaling

$\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}p_{\mathrm{F}}V_{0}^{4}}{q_{0}^{4}}p_{1},\qquad
p_{\mathrm{F}}\ll p_{1}\ll q_{0}\quad\text{(spinful)}.$ (56)

The mismatch between the scalings (54) and (56) at $p_{1}=q_{0}$ is due the
neglect of terms containing large momentum transfers and poles in
$\phi^{\prime}$ in the derivation of (54) [see discussion below Eq. (53)].
This means that there is in fact an additional intermediate regime whose
boundaries are however non-universal since they depend on the large-$q$
behavior of $V$.

In the limit of low energies we obtain for fermions with spin

$\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}V_{0}^{4}}{q_{0}^{4}}(p_{1}-p_{\mathrm{F}})^{2},\quad
p_{1}-p_{\mathrm{F}}\ll p_{\mathrm{F}}\ll q_{0}\quad(\mathrm{spinful}).$ (57)

Details of the derivation are presented in Appendix D. Apart from logarithmic
factors, this result agrees with the one obtained in Ref.
KarzigGlazmanOppen10,  for the case of unscreened Coulomb interaction.
Interestingly, the scaling with energy is the same as for the usual Fermi
liquid in 3D. However, for a weak interaction $V_{0}$, the prefactor is much
smaller than in the Fermi-liquid result, since the leading contribution in 1D
comes only from three-particle processes.

The energy dependence of the relaxation rate in 1D is schematically depicted
in Fig. 8 for both cases of spinless and spinful fermions. We see that the
non-monotonic behavior of the relaxation rate and the revival of the coherence
at high energies observed in higher dimensions persists also in the 1D case.

Figure 8: Schematic behavior of the relaxation rate in 1D on the log-log scale
for spinful and spinless fermions. At low energies, the relaxation rate is
given by Eq. (57) for spinful fermions and by Eq. (52) for spin-polarized
fermions, with a strong suppression in the latter case due to the Hartree-Fock
cancellation. In the intermediate regime, $p_{\mathrm{F}}\ll p_{1}\ll q_{0}$,
the relaxation rate grows as $p_{1}$ [see Eq. (56)] for spinful and as
$p_{1}^{10}$ [see Eq. (51)] for spinless fermions. At high energies, $p_{1}\gg
q_{0}$, the relaxation rate decays as $1/p_{1}^{5}$ in both cases, with a
prefactor for spinless fermions that is smaller by the factor
$p_{\mathrm{F}}^{2}/q_{0}^{2}$ as compared to that for fermions with spin, see
Eqs. (49) and (54), respectively. In the spinful case there is a narrow non-
universal crossover regime near $p_{1}\sim q_{0}$ marked in the plot by a
dashed line.

In Fig. 9, we show the results of a numerical evaluation of the relaxation
rate $1/\tau_{p}$ of 1D spinless fermions as given by Eq. (40). We have used a
model interaction $V(q)=\exp(-q^{2}/p_{\mathrm{F}}^{2})$ with a characteristic
scale $q_{0}$ equal to $p_{\mathrm{F}}$. In this case the above analysis
predicts the $(p-p_{\mathrm{F}})^{8}$ scaling of the relaxation rate for
$p-p_{\mathrm{F}}\ll p_{\mathrm{F}}$ and $p^{-5}$ scaling for $p\gg
p_{\mathrm{F}}$ [Eqs. (52) and (49), respectively]. The numerical results are
in very good agreement with these analytical predictions. Interestingly, the
rate $1/\tau_{p}$ shows also a local minimum in the crossover regime (at
$p\simeq 3p_{\mathrm{F}}$).

Figure 9: Log-log plot of the relaxation rate $1/\tau_{p}$ for 1D spinless
fermions as given by numerical evaluation of Eq. (40) with a model interaction
$V(q)=\exp(-q^{2}/p_{\mathrm{F}}^{2})$ corresponding to
$q_{0}=p_{\mathrm{F}}$. A small temperature was used, $T/\epsilon_{F}=0.01$,
to smear the Fermi-function steps. Straight lines correspond to the small-
momentum and high-momentum asymptotics, Eqs. (52) and (49).

Before closing this section, let us discuss the generalizations of our results
to the case of Coulomb interaction. Interestingly the pure 1D Coulomb
interaction $e^{2}\ln 1/a|q|$ nullifies identicallyristivojevic13  the three-
particle matrix element (44) although spinless fermions with Coulomb
interaction do not belong to the list of integrable models. Therefore, in our
model we take into account the short-distance cutoff for 1D Coulomb
interaction $d\equiv 1/q_{0}$ (finite width of the 1D channel) and approximate
the Coulomb interaction in momentum space by

$V_{\rm 1D}(q)=e^{2}\int
dx\,e^{iqx}\frac{1}{\sqrt{x^{2}+1/q_{0}^{2}}}=2e^{2}K_{0}({|q|}/q_{0})$ (58)

where $K_{0}$ is the modified Bessel function.

At $q\gg q_{0}$, the interaction (58) decays exponentially and in this sense
does not differ from a generic model interaction $V(q)$ studied in the rest of
this section. In particular, for spinless fermions, the scaling (47) of the
three-particle matrix element remains valid also in this case up to the
replacement of the diverging constant $V(0)$ by

$V\left[\frac{q_{0}^{2}}{2q}\left(\tilde{\phi}^{2}-(\tilde{\phi}^{\prime})^{2}\right)\right]\simeq-2e^{2}\ln\frac{q_{0}\left|\tilde{\phi}^{2}-(\tilde{\phi}^{\prime})^{2}\right|}{q}.$
(59)

Focusing now on the regime $\tilde{\phi},\tilde{\phi}^{\prime}\lesssim 1$, we
find

$M_{q}\left(q_{0}\tilde{\phi}/q,q_{0}\tilde{\phi}^{\prime}/q\right)\sim\frac{me^{4}}{q^{2}}\ln\frac{q}{q_{0}}\ln\left|\frac{\tilde{\phi}-\tilde{\phi}^{\prime}}{\tilde{\phi}+\tilde{\phi}^{\prime}}\right|,\quad
q\gg q_{0}.$ (60)

Substituting this estimate for the matrix element in Eq. (42), we obtain the
following result for the relaxation rate at highest momenta:

$\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}p_{\mathrm{F}}^{3}e^{8}\ln^{2}p_{1}/q_{0}}{p_{1}^{5}},\qquad
p_{1}\gg q_{0}\gtrsim p_{\mathrm{F}}\quad\text{(spinless)}.$ (61)

We thus see that the singular nature of the Coulomb interaction potential
leads in this regime to the enhancement of relaxation rate compared to the
regular case, Eq. (49), by a factor $(q_{0}/p_{\mathrm{F}})\ln^{2}p_{1}/q_{0}$
but does not alter its $1/p_{1}^{5}$ scaling .

At lower momenta of the incident particle, $p_{1}\lesssim q_{0}$, one can
approximate the Coulomb potential (58) by

$V_{\rm 1D}(q)\simeq
e^{2}\left(1+\frac{q^{2}}{4q_{0}^{2}}\right)\ln\frac{q_{0}}{|q|}.$ (62)

The leading behavior of the three-particle matrix element, Eq. (44), then
reads ristivojevic13

$\displaystyle M_{q}(\phi,\phi^{\prime})$ $\displaystyle\sim$
$\displaystyle\frac{me^{4}\ln
q/q_{0}}{q_{0}^{2}}\sum_{k=1}^{3}\frac{\ln|\sin\frac{\phi-\phi^{\prime}+2\pi
k/3}{2}|}{1+2\cos\left(\phi-\phi^{\prime}+2\pi k/3\right)}$ (63)
$\displaystyle-$
$\displaystyle\left\langle\phi^{\prime}\rightarrow-\phi^{\prime}\right\rangle.$

It turns out that the dominant contribution to the relaxation rate at
$p_{1}\ll q_{0}$ still comes from the region of small angles $\phi^{\prime}$
in the integral (42) where Eq. (63) reduces to [cf. Eq. (60)]

$M_{q}\left(\phi,\phi^{\prime}\right)\sim\frac{me^{4}}{q_{0}^{2}}\ln\frac{q}{q_{0}}\ln\left|\frac{\phi-\phi^{\prime}}{\phi+\phi^{\prime}}\right|,\quad
q\ll q_{0}.$ (64)

This yields the following behavior of the relaxation rate

$\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}p_{\mathrm{F}}^{3}e^{8}\ln^{2}p_{1}/q_{0}}{q_{0}^{4}p_{1}},\quad
p_{\mathrm{F}}\ll p_{1}\ll q_{0}\quad\text{(spinless)}.$ (65)

Finally, at low energies we can use the result of Ref.ristivojevic13, , which
reads (in our notations):

$\begin{split}\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}e^{8}\ln^{2}p_{\mathrm{F}}/q_{0}}{q_{0}^{4}p_{\mathrm{F}}^{2}}(p_{1}-p_{\mathrm{F}})^{4},&\quad
p_{1}-p_{\mathrm{F}}\ll p_{\mathrm{F}}\ll q_{0}.\\\ &\hskip
8.5359pt\mathrm{(spinless)}.\end{split}$ (66)

One can observe a parametric enhancement in comparison with the corresponding
result for a short-ranged interaction, Eq. (52).

Let us now turn to the analysis of the spin-unpolarized fermions with Coulomb
interaction. In full analogy with the case of a short-range interaction
potential, in the regime $p_{1}\gg q_{0}\gg p_{\mathrm{F}}$ one can focus on
the region $\phi,\phi^{\prime}\lesssim q_{0}/q$ and neglect all the terms in
the transition probability $w_{q}(\phi,\phi^{\prime})$ that contain the
interaction potential at a momentum transfer $q\sim p_{1}$. The leading
behavior of the transition probability is then given by

$w_{q}\left(q_{0}\tilde{\phi}/q,q_{0}\tilde{\phi}^{\prime}/q\right)\sim\frac{m^{2}e^{8}\ln^{2}q/q_{0}}{q^{4}}\left\\{\left[V_{\rm
1D}(q_{+})-q_{+}V^{\prime}_{\rm 1D}(q_{+})-V_{\rm
1D}(q_{-})+q_{-}V^{\prime}_{\rm 1D}(q_{-})\right]^{2}\right.\\\ +\left[V_{\rm
1D}(q_{+})-q_{+}V^{\prime}_{\rm 1D}(q_{+})\right]\left[V_{\rm
1D}(q_{-})-q_{-}V^{\prime}_{\rm 1D}(q_{-})\right]\Big{\\}}.$ (67)

Here $V_{1D}$ is given by Eq. (58) and the momenta
$q_{\pm}={\sqrt{3}}q_{0}(\tilde{\phi}\pm\tilde{\phi}^{\prime})/{2}$. As in the
case of short-range interaction potential, the transition probability (67)
does not have any poles. Setting $\tilde{\phi}\sim p_{\mathrm{F}}/q_{0}\ll 1$
in Eq. (67) and using Eq. (42), we find

$\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}p_{\mathrm{F}}^{2}q_{0}e^{8}\ln^{2}p_{1}/q_{0}}{p_{1}^{5}},\qquad
p_{1}\gg q_{0}\gtrsim p_{\mathrm{F}}\quad\text{(spinful)}.$ (68)

At smaller values of $p_{1}$, $p_{\mathrm{F}}\ll p_{1}\ll q_{0}$, we can
approximate the Coulomb interaction (58) by

$V_{\rm 1D}(q)\simeq e^{2}\ln\frac{q_{0}}{|q|},$ (69)

which results in

$w_{q}(\phi,\phi^{\prime})\sim\frac{m^{2}e^{8}\ln^{2}q/q_{0}}{q^{4}(\cos
3\phi-\cos
3\phi^{\prime})^{2}}\left[\sum_{j=0}^{2}\sin\left(\frac{\phi+\phi^{\prime}}{2}+\frac{2\pi
j}{3}\right)\ln\left|\sin\left(\frac{\phi+\phi^{\prime}}{2}+\frac{2\pi
j}{3}\right)\right|\right]^{2}+\left\langle\phi^{\prime}\rightarrow-\phi^{\prime}\right\rangle.$
(70)

Here we retain the leading behavior of the transition probability in the large
factor $\ln q_{0}/q$. At small $\phi$ and $\phi^{\prime}$, Eq. (70) reduces to

$w_{q}(\phi,\phi^{\prime})\sim\frac{m^{2}e^{8}\log^{2}q/q_{0}}{q^{4}}\left[\frac{\ln^{2}|\phi-\phi^{\prime}|}{(\phi+\phi^{\prime})^{2}}+\left\langle\phi^{\prime}\rightarrow-\phi^{\prime}\right\rangle\right].$
(71)

Substituting Eq. (71) into Eq. (42) and cutting off the divergence near
$\phi^{\prime}=\phi$ at $\phi^{\prime}-\phi\sim p_{\mathrm{F}}/p_{1}$, we get

$\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}e^{8}p_{\mathrm{F}}}{p_{1}^{3}}\ln^{2}\frac{p_{1}}{q_{0}}\ln^{2}\frac{p_{\mathrm{F}}}{p_{1}},\quad
p_{\mathrm{F}}\ll p_{1}\ll q_{0},\quad(\mathrm{spinful}).$ (72)

The low-energy regime in the case of unscreened Coulomb interaction [Eq. (69)]
for spinful fermions is discussed in Ref. KarzigGlazmanOppen10, . The result
reads (in our notations):

$\begin{split}\frac{1}{\tau_{p_{1}}}\sim&\frac{m^{3}e^{8}\ln^{2}\frac{q_{0}}{p_{\mathrm{F}}}\ln^{2}\frac{p_{1}-p_{\mathrm{F}}}{p_{\mathrm{F}}}}{p_{\mathrm{F}}^{4}}(p_{1}-p_{\mathrm{F}})^{2},\\\
&\quad p_{1}-p_{\mathrm{F}}\ll p_{\mathrm{F}}\ll
q_{0}\quad\text{(spinful)}.\end{split}$ (73)

The leading behavior $\sim(p_{1}-p_{\mathrm{F}})^{2}$ is the same as for a
short-range interaction potential [cf. Eq. (57)]. This dependence results from
a phase-space contribution $(p_{1}-p_{\mathrm{F}})^{4}$ and a strongly
enhanced squared matrix element that contributes a factor
$(p_{1}-p_{\mathrm{F}})^{-2}$ (see Appendix D).

As is seen from Eqs. (61), (65), (66), (68), (72) and (73), the behavior of
the relaxation rate is strongly non-monotonic also for the case of Coulomb
interaction (58) in 1D systems. Specifically, for the spin-polarized
electrons, $1/\tau$ increases as $(p-p_{\mathrm{F}})^{4}$ for low momenta but
then decays as $1/p$ and eventually as $1/p^{5}$ at high momenta. In the
spinful case the initial increase of relaxation rate
$1/\tau\propto(p-p_{\mathrm{F}})^{2}$ is followed up by $1/p^{3}$ and,
eventually, by $1/p^{5}$ decay. Clearly, in a physical wire, the purely 1D
analysis is valid at high momenta only as long as the energy of the hot
particle is below the bottom of the second band of transverse quantization.

Just as in the case of higher dimensions (see a comment in the end of Sec.
II), the precise $1/p^{5}$ form of the decay of $1/\tau(p)$ relies on the
parabolicity of the spectrum. However, on the qualitative level, the decay of
the relaxation rate with $p$ is a much more generic feature and remains
applicable as long as the velocity is increasing with momentum.

##  V Summary and discussion

To summarize, we have studied the energy dependence of the relaxation rate of
hot particles in an interacting fermionic system. We have shown that, quite
generally, the relaxation rate $1/\tau$ decays according to a power law with
increasing energy $\epsilon$ of a fermion in the high-energy regime. In
combination with the increase of the relaxation rate with $\epsilon$ at low
energies, this implies a non-monotonic dependence $1/\tau(\epsilon)$. In other
words, ultra-hot electrons (whose energies are much higher than the Fermi
energy) recover their coherence with increasing energy.

More specifically, we have found that, for systems of spatial dimensionality
$D\geq 2$, the relaxation rate $1/\tau$ scales as $p^{-1}$ at high momenta
$p$, see Eq. (17). This result holds under the assumptions that the spectrum
is parabolic and the interaction decays sufficiently fast at high momenta.
Importantly, the Coulomb interaction belongs to this category, so that the
results apply in this case as well, see Eq. (21). The origin of the $1/p$
scaling is related to the corresponding increase of the velocity of the hot
particle in comparison with those from the Fermi sea. Because of the inability
of the interaction to transfer momenta larger than a characteristic scale
$q_{0}$, the momentum transfer turns out to be nearly perpendicular to the
hot-particle momentum, with a deviation of the order of $q_{0}/p_{1}$. The
behavior of the relaxation rate in a 3D system in the whole range of momenta,
from the Fermi-liquid behavior to the ultra-hot regime, is shown schematically
in Fig. 2, both for spinful and spinless (spin-polarized) particles. The non-
monotonic behavior of $1/\tau$ is also visualized in Fig. 3 where a
numerically evaluated relaxation rate is shown.

Motivated by the experiment Ref. Beidenkopf2017, , we have further studied the
hot-electron relaxation in quasi-1D systems, i.e. in wires with multiple bands
of transverse quantization. We have found essentially the same results as in
bulk systems, see Eqs. (33) and (35) for quantum wires with one and two
transverse dimensions, respectively. There is, however, a non-universal regime
at very high energies, for the case of an almost transversal direction of the
hot-particle momentum [see Fig. 6]. In this regime, the discreteness of the
transversal energy spectrum is essential, resulting in a decay of the
relaxation rate that is faster than $1/p$.

We have demonstrated that the non-monotonic behavior of $1/\tau$ applies also
to single-channel wires where the relaxation is controlled by triple
collisions. Furthermore, in this case, the decay of relaxation rate at high
momenta turns out to be particularly fast, $1/\tau\propto 1/p^{5}$, see Eq.
(49). This fast decay originates from (i) a factor $1/p$ related to the
velocity mismatch and the limited possible momentum transfer of the
interaction and (ii) a factor $1/p^{4}$ resulting from partial cancellation in
the three-particle matrix element. The overall behavior of $1/\tau(p)$ for
spinful and spinless fermions in a 1D system is shown schematically in Fig. 8.
Numerical evaluation of $1/\tau$ confirms our analytical findings, see Fig. 9.
We have also analyzed the case of Coulomb interaction in single-channel 1D
geometry and demonstrated that the decay of the relaxation rate at high
energies remains applicable in this situation as well.

Our findings for the 1D geometry should be contrasted to the results of Ref.
Tan2010,  (see also Ref. Ristivojevic2016, ) where the relaxation of high-
energy quasiparticles in a 1D Bose gas was studied. It was found that the
relaxation rate saturates at a finite value in the limit of high energy. We
offer the following explanation for the difference between this result and our
$1/p^{5}$ prediction for the fermionic relaxation rate. First, the factor
$1/p^{4}$ that resulted from matrix element cancellations in our analysis is
associated with the Fermi statistics and thus does not apply in the bosonic
case. Second, the factor $1/p$ in our analysis was due to the limited momentum
transfer by the interaction. On the other hand, Ref. Tan2010,  assumed a
contact interaction, yielding a constant relaxation rate originating from
processes with energy transfer of the order of the energy of the hot particle.
This explanation is supported by the fact that the second contribution to the
relaxation rate identified in Ref. Tan2010, —the one originating from
processes with limited energy transfer—does decay as $1/p$.

Clearly, our assumption of the parabolic spectrum does not need to hold very
accurately at high energies. This may determine deviations from the predicted
laws of the behavior of the relaxation rate at high momenta.

Our results are in good agreement with the experiment of Ref. Beidenkopf2017,
where the energy dependence of the coherence length $L_{\phi}$ of electrons in
a multi-band quantum wire was measured by means of scanning tunneling
microscopy (STM). It was found (see Fig. 2 of Ref. Beidenkopf2017, ) that
$1/L_{\phi}$ is $\simeq 70$ nm near the Fermi energy, then rapidly drops down
with increasing energy, shows a minimum $L_{\phi}\simeq 20$ nm around the
energy $E\simeq 80$ meV (as counted from the Fermi energy), and rapidly
increases with further increase of energy. In particular, at the highest
energy $E\simeq 80$ meV shown in Fig. 2b of Ref. Beidenkopf2017, , the
coherent oscillations do not show any essential decay on the length scale of
$40$ nm, thus implying $L_{\phi}\gtrsim 100$ nm. (This data point is outside
the range of the $L_{\phi}$ axis in Fig. 2c of Ref. Beidenkopf2017, .)

Our theory predicts a $1/p$ behavior of the relaxation rate $1/\tau$ at high
momenta $p$ in multi-channel quantum wires with parabolic dispersion. Since
the corresponding mean free path is $\ell=v\tau$, with the velocity $v=p/m$,
we get the scaling $\ell(\epsilon)\propto\epsilon$ in this regime. Thus, our
model predicts that the mean-free path of hot electrons in a multimode quantum
wire, as determined by electron-electron collisions, increases linearly with
energy $\epsilon$. Assuming that the dominant factor controlling the
oscillation decay in an STM experiment as carried out in Ref. Beidenkopf2017,
is the relaxation due to electron-electron interaction, we obtain the same
prediction for $L_{\phi}$. This prediction is in a good agreement with
observations of Ref. Beidenkopf2017, . Indeed, as we have just discussed,
$L_{\phi}$ shows there an increase from $\simeq 20$ nm to $\simeq 100$ nm when
the energy increases from $80$ meV to $330$ meV.

Our predictions of non-monotonic behavior of the relaxation rate $1/\tau$ are
in agreement with theoretical considerations of Ref. Beidenkopf2017, . It is
worth pointing out, however, that our theory goes beyond the analysis of Ref.
Beidenkopf2017, . Specifically, Ref. Beidenkopf2017,  focused on the quasi-1D
geometry and attributed the non-monotonic behavior to the lowest transverse
subband of a quantum wire. We have explored systems of various geometry and
demonstrated that the non-monotonicity of $1/\tau$ is a rather general
phenomenon. We have also shown that in a quantum wire the non-monotonic
behavior is valid independently of the transverse subband in which the hot
electron resides, at least for the case of a parabolic spectrum. We also note
that, in our analysis, the relaxation of hot electrons in a multichannel
quantum wire is controlled by two-electron collisions independently of the
subband, at variance with Ref. Beidenkopf2017,  where the relaxation in the
lowest subband was attributed to three-particle collisions. It should be
emphasized, however, that while we have worked within the approximation of a
parabolic spectrum, the theoretical analysis of Ref. Beidenkopf2017,  aimed at
incorporating more accurately the band structure of InAs nanowires.

On a more general note, our results corroborate and extend the proposal of
Ref. Beidenkopf2017,  that the extended phase coherence at ultra-high energies
might be utilized in various quantum-technology applications. We have shown
that, while the regain of coherence is particularly strong in single-channel
wires, it also applies to quasi-1D, 2D, and 3D geometries. We hope that our
work will trigger experimental investigations of this effect in various
setups.

We close the paper with the following comment. In the present work we assumed
that the system is clean, i.e. the effect of disorder on scattering can be
neglected. One may wonder how these results are modified for the case of
stronger disorder. This is in fact a sufficiently complex question, and one
should distinguish two cases. The first possibility is that electrons in the
Fermi sea are diffusive, at least on time scales relevant for the considered
inelastic scattering process. It is known that in such situation the inelastic
scattering (decoherence) rate of low-energy electrons is strongly enhanced by
diffusive motion AltshulerAronov . On the other hand, for high-energy
electrons the relevant time scales (set by a typical energy transfer) will be
shorter, so that they can remain ballistic. We thus expect that the
enhancement of scattering will be less efficient for high-energy electrons,
implying that the non-monotonic dependence of the decay rate should hold also
for such a disordered problem. The second possibility is that the disorder is
strong enough to ensure localization, which may be the case even at a finite
temperature GMP2005 ; BAA2006 . In this case, the inelastic scattering rate at
low energies will be essentially zero. On the other hand, electrons with
sufficiently high energies will have a finite relaxation rate gornyi17 .
Again, one can ask how this rate depends on energy. A detailed analysis of the
energy dependence of the relaxation rate in a disordered system (in either
diffusive or localized regime) constitutes an interesting direction for future
research.

###### Acknowledgements.

We thank H. Beidenkopf for providing us with results of Ref. Beidenkopf2017,
prior to publication and for stimulating discussions of experimental data. We
are also thankful to I. V. Gornyi, N. Kainaris, and D. G. Polyakov for useful
discussions. This work was supported by the Swiss National Science Foundation
(Project number 200021_163005) and by the Deutsche Forschungsgemeinschaft
(Project number MI 658/9-1).

##  Appendix A Relaxation at ultra high energies: isotropic case

In this appendix, we consider scattering of a high-energy particle by an
isotropic Fermi see in $D\geq 2$ spatial dimensions and derive Eqs. (4), (8),
and (10) of Sec. II of the main text.

Our starting point is the Fermi golden rule expression, Eq. (1). Making use of
the rotational symmetry we first rewrite Eq. (1) as

$\frac{1}{\tau_{k}}=\frac{1}{mk^{D-2}S_{D-1}}\int
d\mathbf{p}_{1}d\mathbf{p}_{2}d\mathbf{p}_{1}^{\prime}d\mathbf{p}_{2}^{\prime}\;\delta\left(\frac{p_{1}^{2}}{2m}-\frac{k^{2}}{2m}\right)\delta\left(E_{\rm
i}-E_{\rm f}\right)\cdot\delta\left({\bf P}_{\rm i}-{\bf P}_{\rm
f}\right)n_{\mathrm{F}}(\epsilon_{2})[1-n_{\mathrm{F}}(\epsilon_{1}^{\prime})][1-n_{\mathrm{F}}(\epsilon_{2}^{\prime})]\left|M_{\mathbf{p}_{1},\mathbf{p}_{2}}^{\mathbf{p}_{1}^{\prime},\mathbf{p}_{2}^{\prime}}\right|^{2}.$
(74)

It is now convenient to switch to the integration over center-of-mass momentum
${\bf P}$ and the relative momenta ${\bf q}$ and ${\bf q}^{\prime}$ before and
after the scattering

${\bf
P}=\mathbf{p}_{1}+\mathbf{p}_{2}=\mathbf{p}_{1}^{\prime}+\mathbf{p}_{2}^{\prime}\,,\qquad{\bf
q}=\frac{\mathbf{p}_{1}-\mathbf{p}_{2}}{2}\,,\qquad{\bf
q}^{\prime}=\frac{\mathbf{p}_{1}^{\prime}-\mathbf{p}_{2}^{\prime}}{2}.$ (75)

After the change of integration variables we get

$\frac{1}{\tau_{k}}=\frac{1}{2k^{D-2}S_{D-1}}\int\frac{d\mathbf{P}d{\bf
q}d{\bf
q}^{\prime}}{q}\;\delta\left(\epsilon_{1}-\frac{k^{2}}{2m}\right)\delta\left(q-q^{\prime}\right)n_{\mathrm{F}}(\epsilon_{2})[1-n_{\mathrm{F}}(\epsilon_{1}^{\prime})][1-n_{\mathrm{F}}(\epsilon_{2}^{\prime})]\left|M_{\mathbf{p}_{1},\mathbf{p}_{2}}^{\mathbf{p}_{1}^{\prime},\mathbf{p}_{2}^{\prime}}\right|^{2}.$
(76)

The momenta ${\bf p}_{i}$ and ${\bf p}_{i}^{\prime}$ in Eq. (76) are assumed
to be expressed in terms of ${\bf P}$, ${\bf q}$ and ${\bf q}^{\prime}$
according to (75). In particular,

$\left|M_{\mathbf{p}_{1},\mathbf{p}_{2}}^{\mathbf{p}_{1}^{\prime},\mathbf{p}_{2}^{\prime}}\right|^{2}=\left[V(|{\bf
q}-{\bf q}^{\prime}|)-V(|{\bf q}+{\bf q}^{\prime}|)\right]^{2}$ (77)

for spinless fermions. Due to rotational invariance ${\bf P}$ can be fixed to
point in $x$-direction with the integration over direction of $P$ providing
only a factor of $S_{D-1}$. It is then convenient to parametrize ${\bf q}$ and
${\bf q}^{\prime}$ according to

$\displaystyle{\bf q}$ $\displaystyle=q\left(\cos\phi,{\bf n}\sin\phi\right),$
(78) $\displaystyle{\bf q}^{\prime}$
$\displaystyle=q\left(\cos\phi^{\prime},{\bf
n}^{\prime}\sin\phi^{\prime}\right)$ (79)

where ${\bf n}$ and ${\bf n}^{\prime}$ are $(D-1)$-dimensional unit vectors
(perpendicular to ${\bf P}$). For the scattering rate we now get

$\begin{split}\frac{1}{\tau_{k}}=\frac{1}{2k^{D-2}}\int_{0}^{\infty}dPdq\;P^{D-1}q^{2D-3}\int_{0}^{\pi}d\phi
d\phi^{\prime}\left(\sin\phi\sin\phi^{\prime}\right)^{D-2}\delta\left(\epsilon_{1}-\frac{k^{2}}{2m}\right)n_{\mathrm{F}}(\epsilon_{2})[1-n_{\mathrm{F}}(\epsilon_{1}^{\prime})][1-n_{\mathrm{F}}(\epsilon_{2}^{\prime})]\\\
\times\int d{\bf n}d{\bf
n}^{\prime}\left|M_{\mathbf{p}_{1},\mathbf{p}_{2}}^{\mathbf{p}_{1}^{\prime},\mathbf{p}_{2}^{\prime}}\right|^{2},\end{split}$
(80)

where in terms of the new integration variables

$\int d{\bf n}d{\bf
n}^{\prime}\left|M_{\mathbf{p}_{1},\mathbf{p}_{2}}^{\mathbf{p}_{1}^{\prime},\mathbf{p}_{2}^{\prime}}\right|^{2}=\int
d{\bf n}d{\bf n}^{\prime}\left(V_{+}-V_{-}\right)^{2}$ (81)

with

$V_{\pm}=V\left(q\sqrt{2(1\pm\cos\phi\cos\phi^{\prime}\pm{\bf n}\cdot{\bf
n}^{\prime}\sin\phi\sin\phi^{\prime})}\right).$ (82)

Equations (80), (81) and (82) are readily seen to be equivalent to Eqs. (4),
(8), and (10) of the main text.

##  Appendix B Relaxation rate in quasi-1D wires

In this appendix, we present details of the calculation of the relaxation rate
in multichannel quantum wires with one lateral dimension (Appendix B.1) and
two lateral dimensions (Appendix B.2) discussed in Sec. III of the paper.

###  B.1 Quasi-1D: one lateral dimension

Here we present the analysis leading to the results summarized in Eq. (33) in
Sec. III.2. We start by considering a contribution of a specific process
(fixed band indices $n_{i}$) to the relaxation rate given by Eq.
(LABEL:Eq:relaxation-Q1D-1-lateral-integration-over-q). An example of a
particular process is depicted in Fig. 10.

Figure 10: Example of a process $12\to 1^{\prime}2^{\prime}$ in a quasi-1D
setup that contributes to the relaxation of the electron 1.

Since by our assumptions there are many bands available in the energy window
between the Fermi sea and the energy of the injected particle, the sum in Eq.
(27) is dominated by the terms where the particles $1^{\prime}$ and
$2^{\prime}$ created in the collision process reside in otherwise empty bands.
We thus have to deal with only one Fermi function requiring (at zero
temperature) that the initial cold particle $2$ is within the Fermi sea and
thus limiting the range of integration over the momentum transfer $q$ to the
domain defined by the inequality

$\frac{1}{2m}\left(p_{1}-q+\frac{m\Delta_{\mathrm{eff}}}{q}\right)^{2}+\Delta_{0}n_{2}^{2}<\epsilon_{\mathrm{F}},$
(83)

where $\Delta_{\rm eff}$ is defined by Eq. (32). We note that Eq. (83)
restricting the possible values of $q$ simultaneously limits the “transversal
energy transfer” $\Delta_{\rm eff}$ by
$|\Delta_{\mathrm{eff}}|\lesssim(p_{1}^{2}-p_{\mathrm{F}}^{2})/2m$.

Figure 11: Kinematically allowed phase space (in the plane spanned by the
initial longitudinal momentum $p_{1}$ and the longitudinal momentum transfer
$q=p_{1}-p_{1}^{\prime}$) for the relaxation rate of a hot electron in a
quasi-1D setup. A process with a hole created in the lowest band of transverse
quantization ($n_{2}=0$) and with the “transverse energy transfer”
$\Delta_{\mathrm{eff}}=2\epsilon_{\mathrm{F}}$ is considered. At large
$p_{1}\gg p_{\mathrm{F}}$, contributions come from two distinct regions: (i)
$q\simeq p_{1}$, Eq. (84), and (ii) small negative $q$, Eq.(85).

In view of the anisotropy of our model, it is not clear _a priori_ whether the
relaxation rate of an ultra-hot electron with an energy
$\epsilon\gg\epsilon_{F}$ depends essentially only on the energy $\epsilon$ or
also on the direction of the momentum. To explore this point, we investigate
two distinct limiting cases. First, we consider the case of an injected
particle $1$ residing in one of the lowest bands, so that the energy
$\epsilon$ is dominated by that of the longitudinal motion. In this case, the
longitudinal momentum $p_{1}\gg|n_{1}|/d$, which also implies that $p_{1}\gg
p_{\mathrm{F}}$. The opposite limit is that of a particle in one of the highly
excited bands of the transversal motion, with $p_{1}\ll|n_{1}|/d$. We will
show that the dependence of the relaxation rate of an ultra-hot electron on
its energy is in fact almost the same in both cases. The only difference
arises at very high energies, where for electrons moving in the transversal
direction the discreteness of the spectrum becomes important.

Let us first assume that the relaxing particle is in one of the lowest bands
and $p_{1}\gg p_{\mathrm{F}}$. In this case, momentum transfers $q$ close to
$p_{1}$ and close to zero contribute to the relaxation process, see Fig. 11.
More specifically, the allowed ranges of the momentum transfer are given by

$p_{1}-\tilde{p}_{\mathrm{F}}(n_{2})<q<p_{1}+\tilde{p}_{\mathrm{F}}(n_{2})$
(84)

and

$-\frac{m\Delta_{\mathrm{eff}}}{p_{1}}(1+\delta)<q<-\frac{m\Delta_{\mathrm{eff}}}{p_{1}}(1-\delta),$
(85)

where

$\delta=\frac{\mathrm{sign}(\Delta_{\mathrm{eff}})}{p_{1}}\tilde{p}_{\mathrm{F}}(n_{2})$
(86)

and

$\tilde{p}_{\mathrm{F}}(n_{2})=\sqrt{p_{\mathrm{F}}^{2}-2m\Delta_{0}n_{2}^{2}}$
(87)

is the (positive) momentum at which the Fermi energy intersects with the band
$n_{2}$.

We now start with the simplest situation in which the momentum scale of the
interaction is the largest momentum, $p_{1}\ll q_{0}$. If we consider spinful
fermions, we can set the matrix element constant and obtain for both branches
($q\simeq-m\Delta_{\mathrm{eff}}/p_{1}$ and $q\simeq p_{1}$) the scaling

$\frac{1}{\tau_{\\{n_{i}\\}}}\sim
m|M|^{2}\frac{\sqrt{p_{\mathrm{F}}^{2}-2m\Delta_{0}n_{2}^{2}}}{p_{1}},\qquad
p_{\mathrm{F}}\ll p_{1}\ll q_{0}.$ (88)

This is the contribution from one particular process (i.e, for given band
indices). To obtain the total relaxation rate, we have to sum over all
possible processes:

$\frac{1}{\tau}=\sum_{n_{2},n_{1^{\prime}},n_{2^{\prime}}}\frac{1}{\tau_{\\{n_{i}\\}}}\delta_{n_{1}+n_{2},n_{1^{\prime}}+n_{2^{\prime}}}.$
(89)

The sum over $n_{2}$ runs over all occupied bands,
$|n_{2}|<\sqrt{\epsilon_{\mathrm{F}}/\Delta_{0}}=p_{\mathrm{F}}d/2\pi$. It can
be estimated by

$\sum_{n_{2}=0}^{p_{\mathrm{F}}d/2\pi}\sqrt{p_{\mathrm{F}}^{2}-\frac{4\pi^{2}}{d^{2}}n_{2}^{2}}\simeq\frac{p_{\mathrm{F}}^{2}d}{8}.$
(90)

for $p_{\mathrm{F}}d\gg 1$. The constraint related to the conservation of the
transversal momentum fixes the band index $n_{2^{\prime}}$. The maximum value
of the remaining band index $n_{1}^{\prime}$ can be deduced from energy
considerations: $n_{1^{\prime}}^{\mathrm{max}}\simeq
n_{1}/2+\sqrt{n_{1}^{2}/4+p_{1}^{2}/4m\Delta_{0}}$. Recalling further that we
assume that the longitudinal momentum of the initial hot electron $1$
dominates over the transversal one, $p_{1}\gg\sqrt{2m\Delta_{0}}n_{1}=2\pi
n_{1}/d$, we obtain the relaxation rate

$\frac{1}{\tau}\sim mp_{\mathrm{F}}^{2}d^{2}|M|^{2}\sim
mV_{0}^{2}p_{\mathrm{F}}^{2},\qquad p_{\mathrm{F}}\ll p_{1}\ll q_{0}.$ (91)

This rate is independent of the momentum $p_{1}$ of the hot electron and is
identical to the result (15) (with $D=2$) obtained in the same range of
$p_{1}$ for the isotropic situation in two dimensions.

For fermions without spin, the Hartree-Fock cancellation makes it necessary to
analyze the specific form (26) of the interaction. The momentum transfers of
the direct and exchange terms are

$\displaystyle
q_{\mathrm{dir}}^{2}=q^{2}+\frac{4\pi^{2}}{d^{2}}(n_{1}-n_{1^{\prime}})^{2},$
(92) $\displaystyle
q_{\mathrm{ex}}^{2}=\frac{16\pi^{4}}{q^{2}d^{4}}[(n_{1}-n_{1^{\prime}})(n_{1^{\prime}}-n_{2})]^{2}+\frac{4\pi^{2}}{d^{2}}(n_{1^{\prime}}-n_{2})^{2},$
(93)

respectively. The momentum $q$ is integrated over the regions given by Eqs.
(84) and (85). The maximum band index $n_{1^{\prime}}$ is given by
$n_{1^{\prime}}^{\mathrm{max}}\approx p_{1}d/2\sqrt{2}\pi$. For most processes
$n_{1^{\prime}}\gg n_{1},n_{2}$. Estimating the summation over all processes
yields the scaling

$\frac{1}{\tau_{\epsilon}}\sim
mV_{0}^{2}p_{\mathrm{F}}^{2}\frac{p_{1}^{4}}{q_{0}^{4}}\sim
m^{3}V_{0}^{2}\frac{p_{\mathrm{F}}^{2}}{q_{0}^{4}}\epsilon^{2},\qquad
p_{\mathrm{F}}\ll p_{1}\ll q_{0},$ (94)

which again agrees with the scaling (16) in the corresponding regime for the
isotropic 2D situation. Here, we introduced the energy $\epsilon$ of the
particle with momentum $p_{1}$ in order to facilitate the comparison between
the two situations in which the incident momentum of the particle $1$ is in
the longitudinal and transversal direction, respectively. Both regions of the
integration over $q$ [Eqs. (84) and (85)] yield contributions to the
scattering rate of the order of Eq. (94). We briefly discuss the origin of the
individual factors using the example of the region given by Eq. (84). We
estimate the integral in Eq. (LABEL:Eq:relaxation-Q1D-1-lateral-integration-
over-q) by setting $q=p_{1}$ and multiplying by the width
$\tilde{p}_{\mathrm{F}}(n_{2})$ of the integration domain. We can neglect the
$n_{2}$-dependence of the matrix element for most processes. According to Eq.
(90), the summation over $n_{2}$ in (89) yields a factor
$p_{\mathrm{F}}^{2}d$. The summation of the squared matrix element over
$n_{1^{\prime}}$ leads to the factor $(V_{0}^{2}/q_{0}^{4}d^{2})\cdot
p_{1}^{5}d$. Combining these results, we get the scaling (94).

When the longitudinal momentum exceeds the momentum scale of the interaction,
$p_{1}\gg q_{0}\gtrsim p_{\mathrm{F}}$, one of the terms in the square
brackets of Eq. (29) is much larger than the other. In this case the Hartree-
Fock cancellation is inefficient, implying that the result is essentially the
same for models with and without spin. Further, we find that only band indices
$n_{1^{\prime}}$ up to $n_{1^{\prime}}^{\mathrm{max}}\approx q_{0}d/2\pi$
contribute. For higher $n_{1^{\prime}}$, the transversal momentum transfer is
larger than $q_{0}$, so that their contributions are suppressed. The resulting
scaling of the inverse lifetime is given by

$\frac{1}{\tau_{\epsilon}}\sim
mV_{0}^{2}p_{\mathrm{F}}^{2}\frac{q_{0}}{p_{1}}\sim\sqrt{m}V_{0}^{2}p_{\mathrm{F}}^{2}\frac{q_{0}}{\sqrt{\epsilon}},\qquad
p_{\mathrm{F}}\lesssim q_{0}\ll p_{1},$ (95)

which is the same behavior as in the corresponding regime for the isotropic 2D
scaling, Eq. (17). The result (95) is obtained from (88) and (89). The sum
over $n_{2}$ contributes a factor $p_{\mathrm{F}}^{2}d$ and the summation of
the squared matrix element over $n_{1^{\prime}}$ yields a factor
$(V_{0}^{2}/d^{2})\cdot q_{0}d$.

Now we turn to the situation when the transversal momentum of the initial hot
electron dominates over the longitudinal one. This is the case when the hot
electron resides close to the bottom of a high band: $p_{1}\simeq 0$,
$\Delta_{0}n_{1}^{2}\gg\epsilon_{\mathrm{F}}$. To analyze this limit, we use
the general result Eq. (LABEL:Eq:relaxation-Q1D-1-lateral-integration-over-q)
and set there $p_{1}=0$. As has been pointed above, the dominant contribution
originates from processes in which the final particles $1^{\prime}$ and
$2^{\prime}$ reside in otherwise empty bands,
$n_{1}^{\prime},n_{2}^{\prime}>n_{2}^{\rm max}$. In this case, there is only
one Fermi function restricting the initial cold electron to be a part of the
Fermi sea. At $T=0$, this function restricts the integration over $q$ to the
following two regions:

$\begin{split}-\frac{\tilde{p}_{\mathrm{F}}}{2}-\sqrt{\frac{\tilde{p}_{\mathrm{F}}^{2}}{4}+m\Delta_{\mathrm{eff}}}<&q<\frac{\tilde{p}_{\mathrm{F}}}{2}-\sqrt{\frac{\tilde{p}_{\mathrm{F}}^{2}}{4}+m\Delta_{\mathrm{eff}}},\\\
-\frac{\tilde{p}_{\mathrm{F}}}{2}+\sqrt{\frac{\tilde{p}_{\mathrm{F}}^{2}}{4}+m\Delta_{\mathrm{eff}}}<&q<\frac{\tilde{p}_{\mathrm{F}}}{2}+\sqrt{\frac{\tilde{p}_{\mathrm{F}}^{2}}{4}+m\Delta_{\mathrm{eff}}},\end{split}$
(96)

where $\tilde{p}_{\mathrm{F}}$ stands for $\tilde{p}_{\mathrm{F}}(n_{2})$ as
defined by Eq. (87). For dominant processes for a relaxation of a hot particle
we have $m\Delta_{\mathrm{eff}}\gtrsim\tilde{p}_{\mathrm{F}}^{2}$, so that the
two integration regions (96) can be approximated by

$-\sqrt{m\Delta_{\mathrm{eff}}}-\tilde{p}_{\mathrm{F}}(n_{2})/2<q<-\sqrt{m\Delta_{\mathrm{eff}}}+\tilde{p}_{\mathrm{F}}(n_{2})/2$

and

$\sqrt{m\Delta_{\mathrm{eff}}}-\tilde{p}_{\mathrm{F}}(n_{2})/2<q<\sqrt{m\Delta_{\mathrm{eff}}}+\tilde{p}_{\mathrm{F}}(n_{2})/2.$

We start again by considering the case when the momentum scale $q_{0}$ is
large, $p_{\mathrm{F}}\ll 2\pi n_{1}/d\ll q_{0}$. In the presence of spin, the
matrix element can be replaced by a constant. A process with given values of
band indices gives the contribution

$\begin{split}\frac{1}{\tau_{\\{n_{i}\\}}}&\simeq|M|^{2}\tilde{p}_{\mathrm{F}}\sqrt{\frac{m}{\Delta_{\mathrm{eff}}}}\\\
&=|M|^{2}\tilde{p}_{\mathrm{F}}\sqrt{\frac{m}{2\Delta_{0}(n_{1}-n_{1^{\prime}})(n_{1^{\prime}}-n_{2})}}\end{split}$
(97)

to the total relaxation rate. Since $n_{1},n_{1}^{\prime}\gg n_{2}$ for
dominant processes, we can neglect the $n_{2}$-dependence in
$\Delta_{\mathrm{eff}}$. The summation over $n_{2}$ leads then to a factor
$p_{\mathrm{F}}^{2}d$ [cf. Eq. (90)]. We can now sum over $n_{1^{\prime}}$
using the Euler-Maclaurin formula:

$\displaystyle\frac{1}{\tau}$
$\displaystyle=p_{\mathrm{F}}^{2}d|M|^{2}\sqrt{\frac{m}{2\Delta_{0}}}\sum_{n_{1^{\prime}}=1}^{n_{1}-1}\frac{1}{\sqrt{n_{1^{\prime}}(n_{1}-n_{1^{\prime}})}}$
$\displaystyle\simeq
p_{\mathrm{F}}^{2}d|M|^{2}\sqrt{\frac{m}{2\Delta_{0}}}\int_{1}^{n_{1}-1}\frac{dx}{\sqrt{x(n_{1}-x)}}+\mathcal{O}(n_{1}^{-1/2})$
$\displaystyle\sim mp_{\mathrm{F}}^{2}d^{2}|M|^{2}\sim
mV_{0}^{2}p_{\mathrm{F}}^{2},\qquad p_{\mathrm{F}}\ll\frac{2\pi n_{1}}{d}\ll
q_{0}.$ (98)

The obtained relaxation rate ((98)) is identical to the result (91) for the
corresponding regime in the case of the dominant longitudinal momentum of the
initial hot particle. We thus see that the relaxation rate does not
essentially dependent on the direction of the momentum of the hot particle in
this regime. As we are going to show, this applies also to other regimes.

In the absence of spin, the above leading contribution (with the matrix
element approximated by a constant) vanishes due to Hartree-Fock cancellation.
We thus need to take into account the momentum dependence of the interaction
matrix element at the relevant momentum transfers that are given by (92) and
(93). In the considered case of the initial momentum smaller than $q_{0}$, the
momentum transfer is automatically smaller than $q_{0}$, so that the
interaction range does not impose further restrictions on the relevant phase
space. The total relaxation rate can be computed via

$\frac{1}{\tau}\sim
mp_{\mathrm{F}}^{2}d\frac{V_{0}^{2}}{q_{0}^{4}d^{5}}n_{1}^{2}\sum_{n_{1^{\prime}}=1}^{n_{1}-1}\frac{(n_{1}-2n_{1^{\prime}})^{2}}{\sqrt{n_{1^{\prime}}(n_{1}-n_{1^{\prime}})}}.$
(99)

Estimating the sum leads to

$\begin{split}\frac{1}{\tau_{\epsilon}}\sim
mV_{0}^{2}\frac{p_{\mathrm{F}}^{2}}{q_{0}^{4}}\left(\frac{2\pi
n_{1}}{d}\right)^{4}\sim
m^{3}&V_{0}^{2}\frac{p_{\mathrm{F}}^{2}}{q_{0}^{4}}\cdot\epsilon^{2},\\\
&p_{\mathrm{F}}\ll\frac{2\pi n_{1}}{d}\ll q_{0},\end{split}$ (100)

in full consistency with Eq. (94) for the situation when the incident momentum
$\mathbf{p}_{1}$ is along the longitudinal direction.

If the transversal momentum is larger than the momentum scale of the
interaction, $2\pi n_{1}/d\gg q_{0}\gtrsim p_{\mathrm{F}}$, the matrix element
suppresses some processes. Specifically, the direct term contributes if
$n_{1^{\prime}}>n_{1}-q_{0}^{2}d^{2}/4\pi^{2}n_{1}$, while the exchange term
yields an essential contribution for
$n_{1^{\prime}}<q_{0}^{2}d^{2}/4\pi^{2}n_{1}$. Accordingly, the Hartree-Fock
cancellation in the spinless situation does not occur, so that we obtain the
same scaling law as in the spinful case. We neglect again the
$n_{2}$-dependence, since for most processes $n_{1^{\prime}}\gg n_{2}$. There
is a constraint $q_{0}^{2}d^{2}/4\pi^{2}n_{1}>1$ for the sum over
$n_{1^{\prime}}$ to be non-zero. Introducing the transversal momentum of the
inserted particle $p_{1\perp}=2\pi n_{1}/d$, we can rewrite this condition as
$p_{1\perp}<q_{0}^{2}d/2\pi$. This condition ensures that the typical momentum
transfer in the transversal direction $q_{0}^{2}/p_{1\perp}$ (see discussion
in the end of Sec. II) exceeds the quantization step $2\pi/d$. If this
condition is not fulfilled, the relaxation rate is strongly suppressed since
only the large-momentum tail of $V(q)$ contributes. Therefore, in the regime
$p_{1\perp}>q_{0}^{2}d$, the effect of the discreteness of the spectrum leads
to an additional suppression for a quasi-1D system in comparison with the
isotropic 2D situation. The precise form of this suppression is non-universal,
and we do not discuss it here.

Assuming that the condition $p_{1\perp}<q_{0}^{2}d$ is fulfilled, we estimate
the relaxation rate. The summation over $n_{2}$ yields a factor $\sim
p_{\mathrm{F}}^{2}d$, see Eq. (90). The direct and exchange terms yield the
same contributions. It is thus sufficient to estimate the exchange term, which
reads

$\frac{1}{\tau}\sim
mp_{\mathrm{F}}^{2}d\frac{V_{0}^{2}}{d^{2}}\sum_{n_{1^{\prime}}=1}^{\frac{q_{0}^{2}d}{2\pi
p_{1\perp}}}\frac{d}{\sqrt{n_{1^{\prime}}(n_{1}-n_{1^{\prime}})}}.$ (101)

Replacing the sum by an integral, we get

$\begin{split}\frac{1}{\tau_{\epsilon}}\sim
mV_{0}^{2}p_{\mathrm{F}}^{2}&\frac{q_{0}}{p_{1\perp}}\sim\sqrt{m}V_{0}^{2}p_{\mathrm{F}}^{2}\frac{q_{0}}{\sqrt{\epsilon}},\\\
&\qquad p_{\mathrm{F}}\lesssim q_{0}\ll p_{1\perp}\ll q_{0}^{2}d,\end{split}$
(102)

in consistency with the scaling (95) of the relaxation rate of electrons that
move in the longitudinal direction.

As explained above, in the case when the momentum of the hot electron points
in the transversal direction, the relaxation rate is additionally suppressed
at $p_{1\perp}\gg q_{0}^{2}d$ due to discreteness of the spectrum. On the
other hand, when the momentum is in the longitudinal direction, the only
condition to recover the isotropic 2D result at high momenta $p_{1}$ is
$q_{0}d>1$. In order to analyze the crossover between both limits, we consider
the minimal momentum transfer in the situation when the hot particle is in a
high band $n_{1}\gg p_{\mathrm{F}}^{2}d^{2}$ but has also a finite momentum
component $p_{1}$ in the longitudinal direction. The longitudinal momentum
transfer $q=p_{1}-p^{\prime}_{1}$ can be estimated in this regime as

$-p_{1}q+q^{2}=\frac{n_{1^{\prime}}(n_{1}-n_{1^{\prime}})}{d^{2}}.$ (103)

For $p_{1}=0$, we obtain for the minimal longitudinal momentum transfer
$q_{\mathrm{min}}=\sqrt{n_{1}}/d=\sqrt{p_{1\perp}/d}$. The condition for the
universal regime of the relaxation rate ($1/p_{1\perp}^{-1}$ decay) is thus
given by $q_{\mathrm{min}}<q_{0}$, which reproduces the last condition in the
last line of Eq. (102) that limits $p_{1\perp}$ from above. On the other hand,
if the longitudinal momentum component $p_{1}$ becomes larger than $q_{0}$,
the condition for this regime is determined by the first term of the left-hand
side of Eq. (103). We thus find the following condition of the universal
regime in this situation: $p_{1}/p_{1\perp}>1/q_{0}d$. To summarize, a non-
universal regime with a decay of the relaxation rate that is faster than
$1/\sqrt{\epsilon}$ occurs if the longitudinal ($p_{1}$) and transversal
($p_{1\perp}$) components of momentum of particle $1$ satisfy

$p_{1\perp}>q_{0}^{2}d\quad\mathrm{and}\quad\frac{p_{1}}{p_{1\perp}}<\frac{1}{q_{0}d}.$
(104)

All results for quasi-1D wires with one lateral dimension are summarized in
Eq. (33).

###  B.2 Quasi-1D: two lateral dimensions

Here we present details of the analysis leading to the results summarized in
Eq. (35) in Sec. III.3. The calculations are to large extent analogous to
those for one lateral dimension presented in Appendix B.1 above.

We use Eq. (LABEL:Eq:relaxation-Q1D-1-lateral-integration-over-q) and consider
first the situation in which the longitudinal momentum dominates over the
transversal one, $p_{1}\gg 2\pi|\mathbf{n}_{1}|/d$. We start with the case
when the momentum scale of the interaction is large, $p_{1}<q_{0}$. In the
spinful case, the contribution of a particular process with given band indices
is given by Eq. (88) with the replacement $n_{2}\to\mathbf{n}_{2}$. The
summation of $\mathbf{n}_{2}$ over the occupied bands leads now to a factor
$\sim p_{\mathrm{F}}(p_{\mathrm{F}}d)^{2}$. The summation over
$\mathbf{n}_{1^{\prime}}$ is limited by
$|\mathbf{n}_{1^{\prime}}|<p_{1}d/2\sqrt{2}\pi$ in view of the restriction
imposed by the energy conservation. The sum over $\mathbf{n}_{1^{\prime}}$
thus leads to a factor $\sim(p_{1}d)^{2}$. Altogether, we find in the presence
of spin the following behavior of the relaxation rate:

$\frac{1}{\tau_{\epsilon}}\sim mV_{0}^{2}p_{\mathrm{F}}^{3}p_{1}\sim
m^{3/2}V_{0}^{2}p_{\mathrm{F}}^{3}\sqrt{\epsilon},\qquad p_{\mathrm{F}}\ll
p_{1}\ll q_{0}.$ (105)

In analogy with Sec. III.2, we find a full agreement of this result with the
behavior of the relaxation rate (15) in the corresponding regime of a bulk 3D
system.

In the spinless case and for large $q_{0}$, we need to analyze the precise
form (26) of the interaction at the relevant momentum transfers because of the
Hartree-Fock cancellation. The momentum transfers of the direct and exchange
term are given by Eqs. (92) and (93) with a replacement
$n_{i}\to\mathbf{n}_{i}$. For dominant processes, we can neglect the
$\mathbf{n}_{1}$ and $\mathbf{n}_{2}$ dependencies of the matrix element. The
$q$ integration over the regions (84) and (85) yields a factor $\sim
m/p_{1}\cdot\sqrt{p_{\mathrm{F}}^{2}-2m\Delta_{0}\mathbf{n}_{2}^{2}}$. Summing
over $\mathbf{n}_{2}$ results in a factor $\sim
p_{\mathrm{F}}(p_{\mathrm{F}}d)^{2}$. The summation of the squared matrix
element over $\mathbf{n}_{1^{\prime}}$ yields a factor $\sim
V_{0}^{2}/(q_{0}^{4}d^{4})\cdot p_{1}^{6}d^{2}$. The inverse lifetime is thus
given by

$\frac{1}{\tau_{\epsilon}}\sim
mV_{0}^{2}p_{\mathrm{F}}^{3}\frac{p_{1}^{5}}{q_{0}^{4}}\sim
m^{7/2}V_{0}^{2}\frac{p_{\mathrm{F}}^{3}}{q_{0}^{4}}\epsilon^{5/2},\qquad
p_{\mathrm{F}}\ll p_{1}\ll q_{0},$ (106)

which agrees with the isotropic 3D situation, Eq.(16).

In the case $p_{1}\gg q_{0}\gtrsim p_{\mathrm{F}}$, the matrix element
restricts the summation over $\mathbf{n}_{1^{\prime}}$ to the disk with radius
$q_{0}d/2\pi$. The result for the relaxation rate does not depend on the
presence of spin:

$\frac{1}{\tau_{\epsilon}}\sim
mV_{0}^{2}p_{\mathrm{F}}^{3}\frac{q_{0}^{2}}{p_{1}}\sim\sqrt{m}V_{0}^{2}p_{\mathrm{F}}^{3}\frac{q_{0}^{2}}{\sqrt{\epsilon}},\qquad
p_{\mathrm{F}}\lesssim q_{0}\ll p_{1}.$ (107)

Here the factor $m/p_{1}\cdot p_{\mathrm{F}}(p_{\mathrm{F}}d)^{2}$ stems from
the integration over $q$ and the summation over $\mathbf{n}_{2}$, while the
summation of the squared matrix element over $\mathbf{n}_{1^{\prime}}$ yields
a factor $V_{0}^{2}/d^{4}\cdot(q_{0}d)^{2}$. This result is in accordance with
the isotropic three dimensional scaling law, Eq. (17).

We discuss now the situation when the momentum of the hot electron points in a
transversal direction, i.e., the electron resides at the bottom of a high
band, $\Delta_{0}\mathbf{n}_{1}^{2}\gg\epsilon_{\mathrm{F}}$. We start again
with the spinful case. Assuming first that the transversal momenta satisfy
$p_{\mathrm{F}}\ll 2\pi|\mathbf{n}_{1}|/d\ll q_{0}$, we use Eq. (97) with a
replacement $n_{i}\to\mathbf{n}_{i}$ for the contribution of a process with
given band indices. As before, we focus on dominant processes characterized by
$\Delta_{\mathrm{eff}}>0$.

It is worth mentioning a subtlety in the case of a wire with two transverse
direction that did not exist for one transverse direction. Specifically,
besides the trivial zeros $\mathbf{n}_{1^{\prime}}=\mathbf{n}_{1}$ and
$\mathbf{n}_{1^{\prime}}=\mathbf{n}_{2}$ of $\Delta_{\mathrm{eff}}$, there are
additional zeros that occur whenever the vector
$\mathbf{n}_{1}-\mathbf{n}_{1^{\prime}}$ is orthogonal to
$\mathbf{n}_{1^{\prime}}-\mathbf{n}_{2}$. These zeros lead to singularities of
the relaxation rate. Within Eq. (LABEL:Eq:relaxation-Q1D-1-lateral-
integration-over-q), such zeros lead to logarithmic singularities. These
singularities, however, get regularized, and the corresponding terms turn out
to be of minor importance for our analysis, for a discussion of this point see
Sec. III.3. We thus drop these terms below.

The summation over $\mathbf{n}_{2}$ leads again to a factor $\sim
p_{\mathrm{F}}(p_{\mathrm{F}}d)^{2}$. Performing the summation over
$\mathbf{n}_{1^{\prime}}$, we neglect the $\mathbf{n}_{2}$-dependence since
for most processes $|\mathbf{n}_{1^{\prime}}|\gg|\mathbf{n}_{2}|$. Energetic
considerations restrict the summation over $\mathbf{n}_{1^{\prime}}$ to the
disk with a radius $|\mathbf{n}_{1}|$. The summation over
$\mathbf{n}_{1^{\prime}}$ can be approximated by the corresponding
integration. We parametrize the denominator appearing in Eq. (97):

$\frac{1}{\sqrt{(\mathbf{n}_{1}-\mathbf{n}_{1^{\prime}})\mathbf{n}_{1^{\prime}}}}=\frac{1}{\sqrt{|\mathbf{n}_{1}||\mathbf{n}_{1^{\prime}}|\cos\phi-|\mathbf{n}_{1^{\prime}}|^{2}}},$
(108)

where $\cos\phi>|\mathbf{n}_{1^{\prime}}|/|\mathbf{n}_{1}|$ to ensure the
condition $\Delta_{\mathrm{eff}}>0$. The integration over
$|\mathbf{n}_{1^{\prime}}|$ and the angle $\phi$ leads to the factor
$|\mathbf{n}_{1}|$. Collecting everything, we find

$\begin{split}\frac{1}{\tau_{\epsilon}}\sim
mV_{0}^{2}p_{\mathrm{F}}^{3}\frac{2\pi|\mathbf{n}_{1}|}{d}\sim&\,m^{3/2}V_{0}^{2}p_{\mathrm{F}}^{3}\sqrt{\epsilon}\>,\\\
&\qquad p_{\mathrm{F}}\ll\frac{2\pi|\mathbf{n}_{1}|}{d}\ll q_{0}.\end{split}$
(109)

The formula (109) for the momentum $\mathbf{p}_{1}$ pointing perpendicular to
the wire axis is in agreement with the result (105) for the momentum
$\mathbf{p}_{1}$ pointing in the longitudinal direction.

For the same conditions on the momentum scales but in the absence of spin, the
momentum dependence of the matrix element is important in view of a partial
cancellation of direct and exchange term. The momentum transfers of the direct
and exchange terms for a process with given band indices are given by Eqs.
(92) and (93), respectively (with a replacement $n_{i}\to\mathbf{n}_{i}$), and
the $q$-integration goes over the two regions of Eq. (96). As before, we
neglect the $\mathbf{n}_{2}$-dependence everywhere except for the width of the
region of $q$-integration. The summation over $\mathbf{n}_{2}$ yields a factor
$\sim p_{\mathrm{F}}(p_{\mathrm{F}}d)^{2}$. Since we consider the regime
$2\pi|\mathbf{n}_{1}|/d<q_{0}$, all momentum transfers allowed by kinematic
restrictions are smaller than $q_{0}$. The sum over $\mathbf{n}_{1^{\prime}}$
is constrained by $|\mathbf{n}_{1^{\prime}}|<|\mathbf{n}_{1}|$ as well as by
$\mathbf{n}_{1^{\prime}}(\mathbf{n}_{1}-\mathbf{n}_{1^{\prime}})>0$, which
ensures $\Delta_{\mathrm{eff}}>0$, the condition valid for dominant processes.
This sum scales as $|\mathbf{n}_{1}|^{5}$. Collecting all factors, we obtain

$\begin{split}\frac{1}{\tau_{\epsilon}}\sim
mV_{0}^{2}\frac{p_{\mathrm{F}}^{3}}{q_{0}^{4}}\left(\frac{2\pi|\mathbf{n}_{1}|}{d}\right)^{5}\sim&\,m^{7/2}V_{0}^{2}\frac{p_{\mathrm{F}}^{3}}{q_{0}^{4}}\epsilon^{5/2},\\\
&\,\,p_{\mathrm{F}}\ll\frac{2\pi|\mathbf{n}_{1}|}{d}\ll q_{0},\end{split}$
(110)

which is in accordance with Eq. (106) for the case of the hot-particle
momentum directed along the wire axis.

Finally, in the case when the hot-electron transverse momentum is the largest
scale, $2\pi|\mathbf{n}_{1}|/d\gg q_{0}\gtrsim p_{\mathrm{F}}$, the direct
term contributes if
$(2\pi/d)^{2}(\mathbf{n}_{1}^{2}-\mathbf{n}_{1}\mathbf{n}_{1^{\prime}})<q_{0}^{2}$
and the exchange term if
$(2\pi/d)^{2}\mathbf{n}_{1}\mathbf{n}_{1^{\prime}}<q_{0}^{2}$. Both regions
that constrain the summation over $\mathbf{n}_{1^{\prime}}$ do not overlap in
the considered regime. Thus, in this regime the presence or absence of spin is
immaterial: spinful and spinless fermions are characterized by the same
scaling of the relaxation rate. There is an additional constraint
$\mathbf{n}_{1^{\prime}}(\mathbf{n}_{1}-\mathbf{n}_{1^{\prime}})>0$ selecting
the most important processes ($\Delta_{\mathrm{eff}}>0$). Replacing the sum
over $\mathbf{n}_{1^{\prime}}$ by an integral, we get

$\begin{split}\frac{1}{\tau_{\epsilon}}\sim
mV_{0}^{2}p_{\mathrm{F}}^{3}q_{0}^{2}&\left(\frac{2\pi|\mathbf{n}_{1}|}{d}\right)^{-1}\sim\,\sqrt{m}V_{0}^{2}p_{\mathrm{F}}^{3}\frac{q_{0}}{\sqrt{\epsilon}},\\\
&\,\,p_{\mathrm{F}}\lesssim q_{0}\ll\frac{2\pi|\mathbf{n}_{1}|}{d}\ll
q_{0}^{2}d,\end{split}$ (111)

in full consistency with Eq. (107) for the case when $\mathbf{p}_{1}$ points
in the longitudinal direction. As in the case of one lateral dimension, the
isotropic 3D result is reproduced in Eq. (111) under the condition
$q_{0}^{2}d^{2}/|\mathbf{n}_{1}|>1$. If this condition is not met, the
relaxation rate decays even faster since only the large-momentum tail of the
interaction potential contributes. This non-universal regime exists if the
direction of the momentum of the hot particle is almost in transversal
direction. More specifically, in full analogy with the analysis in the end of
Appendix B.1, this non-universal regime is determined by the following
conditions:

$\frac{|\mathbf{n}_{1}|}{d}>q_{0}^{2}d\quad\mathrm{and}\quad\frac{p_{1}}{|\mathbf{n}_{1}|/d}<\frac{1}{q_{0}d},$
(112)

where $p_{1}$ is the longitudinal component of the momentum of particle $1$.

All the results for quasi-1D wires with two lateral dimensions are summarized
in Eq. (35).

##  Appendix C Triple collisions in 1D wires with quadratic dispersion
relation

In this appendix we derive a convenient representation for the on-shell three-
particle matrix element valid for particles with quadratic spectrum.

The general form of the three-particle matrix element for spinful fermions is
given by the vacuum expectation valueLunde07

$M_{123}^{1^{\prime}2^{\prime}3^{\prime}}=\langle
a_{1^{\prime}}a_{2^{\prime}}a_{3^{\prime}}|\hat{V}\frac{1}{E-\hat{H}_{0}+i0}\hat{V}|a^{\dagger}_{1}a^{\dagger}_{2}a^{\dagger}_{3}\rangle,$
(113)

where $\hat{H}_{0}$ and $\hat{V}$ are the free and interaction part of the
Hamiltonian, $E$ is the total energy, and $a^{(\dagger)}_{i}$ annihilates
(creates) a fermion with momentum $p_{i}$ and spin-projection $\sigma_{i}$. It
can be recast in the formLunde07

$\begin{split}M_{123}^{1^{\prime}2^{\prime}3^{\prime}}=&\sum_{(abc)\in\mathcal{P}(123)}\mathrm{sgn}(123)\sum_{(a^{\prime}b^{\prime}c^{\prime})\in\mathcal{P}(1^{\prime}2^{\prime}3^{\prime})}\mathrm{sgn}(1^{\prime}2^{\prime}3^{\prime})\\\
&\times\frac{V(p^{\prime}_{a}-p_{a})V(p^{\prime}_{c}-p_{c})\delta_{p_{a}+p_{b}+p_{c},p^{\prime}_{a}+p^{\prime}_{b}+p^{\prime}_{c}}}{\epsilon_{b}+\epsilon_{c}-\epsilon_{c^{\prime}}-\epsilon_{b+c-c^{\prime}}+i0}\\\
&\times\delta_{\sigma_{a},\sigma_{a^{\prime}}}\delta_{\sigma_{b},\sigma_{b^{\prime}}}\delta_{\sigma_{c},\sigma_{c^{\prime}}},\end{split}$
(114)

where

$\begin{split}\mathcal{P}(123)&=\\\
&\\{(123)^{+},(231)^{+},(312)^{+},(132)^{-},(213)^{-},(321)^{-}\\}\end{split}$

denotes the set of permutations and the superscript of each element denotes
its sign, $\mathrm{sgn}(123)$. In the case of a parabolic band, the energy
denominator in Eq. (114) can be rewritten as follows:

$\begin{split}\epsilon_{b}+\epsilon_{c}-\epsilon_{c^{\prime}}-\epsilon_{b+c-c^{\prime}}=&-\frac{1}{m}(p_{b}-p^{\prime}_{c})(p_{c}-p^{\prime}_{c})\\\
&=\frac{(p_{a}-p^{\prime}_{c})(p_{b}-p^{\prime}_{c})(p_{c}-p^{\prime}_{c})}{m(p^{\prime}_{c}-p_{a})}.\end{split}$
(115)

The crucial simplification comes from the observation that the numerator in
Eq. (115), when evaluated on the mass shell, is actually invariant with
respect to all the permutations of particles in the initial and final state:

$(p_{a}-p^{\prime}_{c})(p_{b}-p^{\prime}_{c})(p_{c}-p^{\prime}_{c})=\frac{q^{3}}{4}(\cos
3\phi-\cos 3\phi^{\prime}).$ (116)

To derive Eq. (116) we have used the parametrizations (37) and (38). Equation
(116) leads to

$M_{123}^{1^{\prime}2^{\prime}3^{\prime}}=\frac{4m}{q^{3}(\cos 3\phi-\cos
3\phi^{\prime})}\sum_{\begin{subarray}{c}(abc)\in\mathcal{P}(123)\\\
(a^{\prime}b^{\prime}c^{\prime})\in\mathcal{P}(1^{\prime}2^{\prime}3^{\prime})\end{subarray}}\mathrm{sgn}(123)\;\mathrm{sgn}(1^{\prime}2^{\prime}3^{\prime})\;q_{c^{\prime},a}V(q_{a^{\prime},a})V(q_{c^{\prime},c})\delta_{p_{a}+p_{b}+p_{c},p^{\prime}_{a}+p^{\prime}_{b}+p^{\prime}_{c}}\delta_{\sigma_{a},\sigma_{a^{\prime}}}\delta_{\sigma_{b},\sigma_{b^{\prime}}}\delta_{\sigma_{c},\sigma_{c^{\prime}}}.$
(117)

Here we have introduced the shorthand notation
$q_{a^{\prime},b}=p_{a}^{\prime}-p_{b}$.

Equation (117) constitutes the basis for our discussion of the transition
probability $w_{q}(\phi,\phi^{\prime})$ in Sec. IV. In the spinpolarized case
it can be further simplified by omitting the spin indices (and adjusting the
combinatorial prefactor). This leads to Eq. (44) of the main text.

##  Appendix D Low-energy regime in 1D

In this appendix we present a derivation of Eqs. (52) and (57) for the low-
momentum limit, $p_{1}-p_{\mathrm{F}}\ll p_{\mathrm{F}}$, of the relaxation
rate for fermions interacting via a short-range potential in one spatial
dimension. Our starting point is the expression (40) for the relaxation rate.
We begin by considering the kinematic constraints dictated by the Fermi
functions $n_{\mathrm{F}}(\epsilon_{2})$ and $n_{\mathrm{F}}(\epsilon_{3})$.
The area in the $q$-$P$-plane compatible with these constraints is depicted in
Fig. 12.

Figure 12: Integration region in the $q$-$P$-plane for low-energy triple
collisions in 1D, as constrained by the Fermi functions associated with
incoming particles.

Unlike the situation at high energies, $p_{1}\gg p_{\mathrm{F}}$, where
energies of the particles after a scattering event ($1^{\prime}$,
$2^{\prime}$, and $3^{\prime}$) are typically high above the Fermi sea, at low
energies the Fermi functions associated with these particles further restrict
the integration region in the $q$-$P$-plane. Because of the symmetry
$\phi^{\prime}\to\phi^{\prime}+2\pi/3$ (interchange of particles) of the
conditions resulting from the Fermi functions and of the squared matrix
element, we can restrict ourselves to the interval $0<\phi^{\prime}<2\pi/3$.
In the low-energy regime, $\delta=(p_{1}-p_{\mathrm{F}})/p_{\mathrm{F}}\ll 1$,
only a narrow interval

$\frac{\pi}{3}-\frac{\sqrt{3}}{4}\delta<\phi^{\prime}<\frac{\pi}{3}+\frac{\sqrt{3}}{4}\delta$
(118)

contributes. Furthermore, the integration domain in the $q$-$P$-plane is
strongly reduced. The integration region incorporating all kinematic
constraints is shown in Fig. 13. As shown in the figure, only a vicinity of
$q=4p_{\mathrm{F}}/3$ and $P=p_{1}\simeq p_{\mathrm{F}}$ contributes. The area
of the domain in the $q$-$P$ plane scales as
$(p_{1}-p_{\mathrm{F}})^{3}/p_{\mathrm{F}}$. The angle $\phi$ is fixed by Eq.
41 to $\phi_{0}=\pm\pi/3+\mathcal{O}(\delta)$. The involved particles reside
close to the Fermi points, with one left mover and two right movers.

Figure 13: Kinematic constraints in the $q$-$P$-plane for triple collisions in
1D at low energy. Only the upper right corner of the domain in Fig. 12
actually contributes.

Besides these phase-space considerations, we have to analyze the squared
matrix element. As before, we assume that the momentum scale of the
interaction is larger than the Fermi momentum, $q_{0}\gg p_{\mathrm{F}}$. For
spinless fermions we can use the form (50) and find for the relaxation rate

$\begin{split}\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}p_{\mathrm{F}}^{6}V_{0}^{4}}{q_{0}^{16}}(p_{1}-p_{\mathrm{F}})^{8},\quad
p_{1}-p_{\mathrm{F}}\ll&p_{\mathrm{F}}\ll q_{0},\\\
&(\mathrm{spinless})\end{split}$ (119)

which is Eq. (52) of the main text.

In the case of fermions with spin the behavior of the squared matrix element
for $\phi\approx\pi/3$ and $\phi^{\prime}\approx\pi/3$, with a typical
distance between $\phi$ and $\phi^{\prime}$ of the order of
$\delta=(p_{1}-p_{\mathrm{F}})/p_{\mathrm{F}}$, is given by

$w_{q}(\phi,\phi^{\prime})\sim\frac{m^{2}V_{0}^{4}}{q_{0}^{4}\delta^{2}}\sim\frac{m^{2}V_{0}^{4}p_{\mathrm{F}}^{2}}{q_{0}^{4}(p_{1}-p_{\mathrm{F}})^{2}}.$
(120)

Because of the energy denominators and of the absence of the Hartree-Fock
cancellation, the matrix element is strongly enhanced at low energy. Combining
all the estimates of the phase space and of the matrix element, we arrive at
the following result for the relaxation rate of spinful fermions at low energy
in 1D:

$\frac{1}{\tau_{p_{1}}}\sim\frac{m^{3}V_{0}^{4}}{q_{0}^{4}}(p_{1}-p_{\mathrm{F}})^{2},\quad
p_{1}-p_{\mathrm{F}}\ll p_{\mathrm{F}}\ll q_{0}\quad(\mathrm{spinful}),$ (121)

which is Eq. (57) of the main text.

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