Plugging in (368) into the above unitarity equations we get the following
large spin conditions:494949$|1+i\epsilon f(s)|\leq 1$ for $\epsilon\ll 1$
implies that $\text{Im}f(s)\geq 0$. In our case the small parameter $\epsilon$
is $\frac{1}{\ell^{3}}$.

$\boxed{\sum_{abc}\left(\alpha^{(1)}_{abc}\text{Im}(\rho^{a}(s)\chi^{2}(s))+\alpha^{(2)}_{abc}\text{Im}(\rho^{a}(s))\right)b\sqrt{s}\,\,\rho^{c}(-s)\geq
0,}$ (371)

$\boxed{\sum_{abc}\left(\alpha^{(1)}_{abc}\text{Im}(\rho^{a}(s)\chi^{2}(s))-\alpha^{(2)}_{abc}\text{Im}(\rho^{a}(s))\right)b\sqrt{s}\,\,\rho^{c}(-s)\geq
0,}$ (372)

$\boxed{\sum_{abc}\alpha^{(1)}_{cba}\text{Im}(\rho^{a}(s))\,b\sqrt{s}\,\,\rho^{c}(-s)\chi^{2}(-s)\geq
0,}$ (373)

valid for $s\geq 0$.

###  H.2 Large energy

We begin by considering the $s\rightarrow\infty$ expansion of a monomial term
in the ansatz:

$\displaystyle\rho^{a}(s)\rho^{b}(t)\rho^{c}(u)\approx(-1)^{a+b+c}\Bigg{[}1$
$\displaystyle-\frac{2}{\sqrt{s}}\left(ia+\frac{\sqrt{2}b}{\sqrt{1-z}}+\frac{\sqrt{2}c}{\sqrt{1+z}}\right)+i\frac{4\sqrt{2}a}{s}\left(\frac{b}{\sqrt{1-z}}+\frac{c}{\sqrt{1+z}}\right)$
(374)
$\displaystyle+\frac{1}{s}\left(-2a^{2}+\frac{8bc}{\sqrt{1-z}\sqrt{1+z}}+\frac{4b^{2}}{1-z}+\frac{4c^{2}}{1+z}\right)+\ldots$

In addition to the large energy expansion of the $\rho$ series, we also need
the large energy expansion of $\chi^{2}(s)$ , $\chi^{2}(u)$ and
$\chi(t)\chi(u)$:

$\displaystyle\chi^{2}(s)$ $\displaystyle\approx
9-\frac{48i}{\sqrt{s}}+\ldots$ (375) $\displaystyle\chi^{2}(u)$
$\displaystyle\approx 9-\frac{48i}{\sqrt{s}}\frac{1}{\sqrt{1+z}}+\ldots$
$\displaystyle\chi(s)\chi(t)\chi(u)$ $\displaystyle\approx
27-\frac{72i}{\sqrt{s}}\left(1+\frac{\sqrt{2}}{\sqrt{1+z}}+\frac{\sqrt{2}}{\sqrt{1-z}}\right)+\ldots$

This leads to the following expansions of the ansatze at large energy:

$\displaystyle\Phi_{1}$
$\displaystyle\approx\sum_{abc}\alpha^{(1)}_{abc}(-1)^{a+b+c}\Bigg{[}9-\frac{6}{\sqrt{s}}\left((8+3a)i+\frac{3\sqrt{2}b}{\sqrt{1-z}}+\frac{3\sqrt{2}c}{\sqrt{1+z}}\right)+\ldots\Bigg{]},$
(376) $\displaystyle\Phi_{2}$
$\displaystyle\approx\sum_{abc}\alpha^{(2)}_{abc}(-1)^{a+b+c}\Bigg{[}1-\frac{2}{\sqrt{s}}\left(ia+\frac{\sqrt{2}b}{\sqrt{1-z}}+\frac{\sqrt{2}c}{\sqrt{1+z}}\right)+\ldots\Bigg{]},$
$\displaystyle\Phi_{3}$
$\displaystyle\approx\sum_{abc}\alpha^{(1)}_{cba}(-1)^{a+b+c}\Bigg{[}9-\frac{6}{\sqrt{s}}\left(3ia+\frac{3\sqrt{2}b}{\sqrt{1-z}}+\frac{(8+3c)\sqrt{2}}{\sqrt{1+z}}\right)+\ldots\Bigg{]},$
$\displaystyle\Phi_{5}$
$\displaystyle\approx\sum_{abc}\alpha^{(5)}_{abc}(-1)^{a+b+c}\Bigg{[}27-\frac{18}{\sqrt{s}}\left((4+3a)i+\frac{(4+3b)\sqrt{2}}{\sqrt{1-z}}+\frac{(4+3c)\sqrt{2}}{\sqrt{1+z}}\right)+\ldots\Bigg{]}~{}.$

We now work order by order in $\frac{1}{s}$. We begin with the leading order
which is

#### O(1):

The unitarity condition on spin $0$ partial waves
$|1+i(\Phi^{0}_{1}\pm\Phi^{0}_{2})|\leq 1$ implies that

$\boxed{\sum_{abc}\alpha^{(1)}_{abc}(-1)^{a+b+c}=0}~{},$ (377)

and

$\boxed{\sum_{abc}\alpha^{(2)}_{abc}(-1)^{a+b+c}=0}~{}.$ (378)

Similarly, the unitarity condition on odd spin $\ell$ partial waves
$|1+2i(\Phi^{\ell}_{3})|\leq 1$ implies that

$\boxed{\sum_{abc}\alpha^{(1)}_{cba}(-1)^{a+b+c}=0}~{},$ (379)

which turns out to be equivalent to the condition (377). Finally the unitarity
condition (139) simplifies at leading order due to the results we just derived
above, and leads to

$\boxed{\sum_{abc}\alpha^{(5)}_{abc}(-1)^{a+b+c}=0}~{}.$ (380)

Effectively, the constraints above say that constant terms in the amplitudes
should go to 0 as $s\rightarrow\infty$. We now consider the next order which
is

#### O($s^{-1/2}$):

From the unitarity condition on spin 0 partial waves
$|1+i(\Phi^{0}_{1}\pm\Phi^{0}_{2})|\leq 1$, we get the two conditions:

$\boxed{\sum_{abc}(-1)^{a+b+c}\left((24+9a)\,\alpha_{abc}^{(1)}+a\,\alpha_{abc}^{(2)}\right)\leq
0}~{},$ (381)

and

$\boxed{\sum_{abc}(-1)^{a+b+c}\left((24+9a)\,\alpha_{abc}^{(1)}-a\,\alpha_{abc}^{(2)}\right)\leq
0}~{}.$ (382)

Consider now the unitarity condition (139) for non-zero even spin $\ell$
partial waves. The imaginary part of the $\Phi^{\ell}_{1}$ and
$\Phi^{\ell}_{2}$ is 0 and this immediately implies that the imaginary part of
$\Phi_{5}$ must vanish at this order:

$\boxed{\sum_{abc}\alpha^{(5)}_{abc}(-1)^{a+b+c}a=0}~{}.$ (383)

In addition, it implies that $|1+2\phi^{\ell}_{3}|\leq 1$.505050More
precisely, it implies it only for even $\ell$, but we also have the same
condition $|1+2\phi^{\ell}_{3}|\leq 1$ separately for odd $\ell$ from (137).
Since $\int dz\,d^{\ell}_{22}(z)>0$ for even $\ell$ and $\int
dz\,d^{\ell}_{22}(z)<0$ for odd $\ell$, we get

$\boxed{\sum_{abc}\alpha^{(1)}_{cba}(-1)^{a+b+c}a=0}~{}.$ (384)

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# On the origin and the amplitude of T-square resistivity in Fermi liquids

Kamran Behnia  LPEM (CNRS-Sorbonne University), ESPCI Paris, PSL University,
75005 Paris, France

###### Abstract

In 1937, Baber, Landau and Pomeranchuk postulated that collisions between
electrons generates a contribution to the electric resistivity of metals with
a distinct T2 temperature dependence. The amplitude of this term is small in
common metals, but dominant in metals hosting either heavy carriers or a low
concentration of them. The link between the temperature dependence and the
size of the scattering phase space is straightforward, but not the microscopic
source of dissipation. To explain how electron-electron collisions lead to
momentum leak, Umklapp events or multiple electron reservoirs have been
invoked. This interpretation is challenged by a number of experimental
observations: the persistence of T-square resistivity in dilute metals (in
which the two mechanisms are irrelevant), the successful extension of
Kadowaki-Woods scaling to dilute metals, and the observation of a size-
dependent T-square thermal resistivity ($T/\kappa$) and its Wiedemann-Franz
(WF) correlation with T-square electrical resistivity. This paper argues that
much insight is provided by the case of normal liquid 3He where the T-square
temperature dependence of energy and momentum diffusivity is driven by
fermion-fermion collisions. The amplitude of T-square resistivity in 3He and
in metals share a common scaling. Thus, the ubiquitous T-square electrical
resistivity ultimately stems from the Fermi-liquid temperature dependence of
momentum diffusivity.

The contribution of electron-electron scattering to the temperature dependence
of electrical conductivity in conducting solids has a long history, which
begins in the late 1930s with its theoretical identification by Baber [1] and
(contemporaneously and independently) by Landau and Pomeranchuk [1, 2]. The
concept began to attract significant interest several decades later [3, 4]
with the accumulation of experimental data. In common metals, this effect is
not easy to observe and remains small compared with the contribution due to
electron-phonon scattering. The discovery of metals hosting strongly-
correlated electrons changed the experimental landscape in 1980s. In heavy-
fermion systems [5] electron-electron scattering dominates the temperature
dependence of resistivity at cryogenic temperatures. Kadowaki and Woods [6]
(following an idea first put forward by Rice about transition metals [3])
observed that the prefactor of T-square resistivity scales with the square of
the electronic specific heat. This scaling, based on the argument that both
quantities are amplified by enhancement in the density of states proved
influential during decades of experimental research on heavy fermion metals
(for an early example see [7]). It was also used as a road-map to track the
emergence of what was dubbed ‘Non-Fermi-liquid behavior’ in strongly-
correlated-electron systems [8]. The microscopic origin of dissipation did not
attract much debate. It was assumed that Umklapp events allowed by the large
Fermi surface of the metallic solids of interest are what drive the T-square
resistivity [9].

Recently, a modified version of the Rice or Kadowaki-Woods scaling was
proposed to explain the amplitude of the T-square prefactor in dilute metals,
which are zero-temperature conductors hosting a very low concentration of
mobile electrons [10, 11, 12]. These are either stoichiometric semi-metals
(such as bismuth) or semiconductors sufficiently doped to be on the metallic
side of the metal-insulator transition (such as oxygen-reduced strontium
titanate). In their case, the relevant scaling is between the prefactor of
T-square resistivity and the inverse of the square of the Fermi energy (and
not the specific heat which is contaminated by the unusually low carrier
density).

Two new experimental developments have accompanied the renewal of interest for
this theme (see ref.[13] for a review of possible origins of T-square
resistivity prior to them). The first is the observation that T-square
resistivity persists even when the microscopic origin of the dissipation
mechanism cannot be identified [10, 12]. The second is the observation of
T-square thermal resistivity in semi-metals close to the ballistic regime [14,
15]. They raise a number of questions: Is the T-square resistivity in dilute
metallic strontium titanate [16, 17, 10, 18] due to electron-electron
scattering ? If yes, given the absence of Umklapp events and multiple pockets,
what causes it? If not, what kind of electron-phonon scattering mechanism can
generate a T-square resistivity [19]? A similar question was raised long ago
in the case of bismuth. The observed T-square resistivity to was attributed to
scattering between electrons residing in different pockets[20]. But this was
subsequently contested by other authors [21, 22]. It is raised again by the
observation of T-square resistivity in another dilute metallic solid, namely
Bi2O2Se [12], implying that SrTiO3 [10] is not a lonely anomaly. More
fundamentally and beyond these specific cases, What is the reason behind the
success of scaling à la Kadowaki-Woods across a wide variety of families of
materials (provided that one takes in to account the carrier density)? How
does this fit with the assumption that the microscopic mechanism generating
momentum decay from electron-electron collisions varies from one case to the
other?

The aim of the present paper is to argue in favor of a route from electron-
electron scattering to T-square resistivity in metals starting from our
understanding of thermal transport in liquid 3He [23, 24, 25, Sykes1970, 27].
It has been known for decades that, well below the degeneracy temperature of
$\sim 1.5K$, collisions between fermions in liquid 3He generate a thermal
conductivity which decreases with the inverse of temperature [28, 29, 30, 31].
This is strictly equivalent to the T-square thermal resistivity of electrons,
as defined by metal physicists. This picture has no need for Umklapp events,
or a bottleneck provided by multiple fermionic reservoirs. Its relevance to
electrons in metals was experimentally demonstrated in the case of semi-
metallic antimony [15]. Its potentially broader relevance would solve a number
of pending puzzles in other specific materials.

I will begin by a short historical account of experimental and theoretical
investigations of T-square electrical resistivity. Then, I will recall (much
less frequent) studies detecting the thermal counterpart of T-square
resistivity. The next section is devoted to normal liquid 3He. Greywell’s
extended study of the evolution of specific heat [32] and thermal conductivity
[32] with pressure is reviewed. It remains the most detailed experimental
verification of the theory of momentum and energy diffusivity in Fermi
liquids. By comparing electrons in metals (in presence of lattice and its
disorder) and fermions in 3He (in absence of both) a common scaling emerges.
This leaves little doubt regarding the main driver of T-square thermal
resistivity. The neglected role of viscosity of the electron liquid [33, 34]
in the emergence of T-square electrical resistivity is highlighted by this
analogy.

##  I Ubiquity and scalability of T-square resistivity in Fermi liquids

Fig.1 shows the low-temperature resistivity of four different metals. Bismuth
and graphite are elemental semi-metals. Sr2RuO4 and UPt3 are correlated metals
each with an unconventional superconducting ground state. Low-temperature
resistivity of these four solids quadratically increases with temperature,
following :

$\rho=\rho_{0}+AT^{2}$ (1)

Figure 1: Examples of T-square resistivity: Resistivity as function of $T^{2}$
in bismuth [20](a), in graphite [21] (b), in UPt3 [35] (c) and in Sr2RuO4 [36]
(d).

The first term on the right side is residual resistivity, $\rho_{0}$,
attributed to impurity scattering. The second term, $A$ (in ohm.m.K-2), is
attributed to electron-electron scattering. It will be the focus of this
paper. The low-temperature resistivity of these four solids lacks any trace of
$T^{5}$ resistivity caused by electron-phonon scattering in the Bloch-
Grüneisen picture [37]. This is because $A$ is large enough to impede its
observation. In contrast, $A$ is orders of magnitude smaller in noble or
alkali metals and temperature-dependent resistivity is dominantly $T^{5}$.

Figure 2: $A$ vs. $\gamma^{2}$ scaling: Left:The amplitude of $A$ in several
transition metals it scales with $\gamma^{2}$. Rice’s 1968 figure [3]
[reproduced with permission granted by APS]. Right: Kadowaki and Woods [6]
found a similar scaling in the case of heavy-fermion metals [reproduced by
with permission granted by Elsevier]. Dilute metals are absent in both plots.

What sets the amplitude of $A$ in a given metal? Does it correlate with
$\rho_{0}$? Numerous experiments point to a negative answer to the latter
question and indicate that $A$ is an intrinsic property of a Fermi liquid. In
many heavy-fermion metals, the quality of crystals have improved over the
years, pulling down $\rho_{0}$ by more than one order of magnitude, yet $A$ is
similar in clean and dirty crystals. In URu2Si2, for example,in 1986, Palstra
el al. [38] found $\rho_{0}=33\mu\Omega.cm$ and $A=0.1\mu\Omega.cmK^{-2}$. A
quarter of century later, working on much cleaner crystals, Matsuda et al.
[39] reported $\rho_{0}=1.05\mu\Omega.cm$ and $A=0.099\mu\Omega.cmK^{-2}$. The
same is true for UPt3, where the magnitude of $A$ remains the same in crystals
in which $\rho_{0}$ differe by one order of magnitude [40]. One can go also
along the opposite direction by introducing disorder through electronic
irradiation, as in the case of metallic strontium titanate, where a twofold
enhancement in $\rho_{0}$ leaves $A$ unchanged [41].

The intrinsic nature of $A$ is backed by the success of scaling approaches,
which was initiated by Rice in 1968 [3]. He demonstrated that $A$ scales with
the square of the electronic T-linear specific heat $\gamma$ (whose unit is
J.K-1.mo$l^{-2}$) in seven elemental transition metals. 18 years later,
Kadowaki and Woods (KW) [6] applied this scaling to the newly-discovered
Heavy-fermion metals and found that a similar scaling works there too. They
also noticed a difference in the amplitude of the ratio. In heavy fermions, it
is ($\frac{A}{\gamma^{2}}=10^{-7}\Omega$.m($\frac{mol.K}{J})^{2}$). This is
more than one order of magnitude larger than what Rice had found in transition
metals ($\frac{A}{\gamma^{2}}=4\times
10^{-9}\Omega$.m($\frac{mol.K}{J})^{2}$). Theses historical plots are
reproduced in Fig. 2.

This difference between strongly-correlated and weakly-correlated systems
became fuzzier as the data became more abundant. In 2003, Tsuiji et al. [42]
showed that most Yb-based heavy fermions are closer to the Rice scaling line
than to the KW scaling line. Nevertheless, this scaling approach is
impressively effective over eight to nine orders of magnitude. It implied that
one could predict the rough amplitude of one measurable quantity (namely $A$),
thanks to knowing the amplitude of another measurable quantity (namely
$\gamma$).

Figure 3: The proper scaling: a) $A$ vs. $\gamma$ in several metallic elements
including low-density semi-metals. The blue solid line represents
$A\propto\gamma^{2}$. b) $A$ of the same metallic elements vs. the Fermi
energy, $E_{F}$, in kelvin. The red solid line represents $A\propto
E_{F}^{-2}$.

The Pauli exclusion principle is the ultimate reason behind both the
temperature dependence of electronic specific heat and electron-electron
contribution to the resistivity. Electrons giving rise to both properties are
those confined to a thermal window near the Fermi energy. This leads to the
following expression for the electron specific heat (per volume):

$\frac{C_{e}}{T}=\gamma=\frac{\pi^{2}}{2}\frac{nk_{B}^{2}}{E_{F}}$ (2)

Here $n$ is the carrier density. Now, the phase space for collisions between
two electrons is proportional to $(\frac{k_{B}T}{E_{F}})^{2}$. Since $A$ is
expressed in $\Omega$m K-2, it can be written as [10]:

$A=\frac{\hbar}{e^{2}}(\frac{k_{B}}{E_{F}})^{2}\ell_{quad}$ (3)

Here, $\hbar$ and $e$ are the Planck constant (divided by $2\pi$) and the
fundamental charge. The material-dependent parameter $\ell_{quad}$ has the
dimension of a length and condenses all the microscopic details of scattering.

According to these equations, $\gamma\propto$ E${}_{F}^{-1}$ and A $\propto$
E${}_{F}^{-2}$. As a consequence, the A$\propto\gamma^{2}$, provided that
$\ell_{quad}$ does not drastically vary from one metal to another. At the same
time, the scaling should collapse when the comparison is made between metals
with widely different carrier densities. As seen in Fig.3a, this is indeed the
case. As soon as one adds bismuth (Bi), graphite (C) and antimony (Sb), to
Rice’s plot, the scaling does not work. In these semi-metals, a single
electron is shared by thousands of atoms and the magnitude of $\gamma$ is
pulled down by the low carrier density. As a consequence, $A$ decreases with
$\gamma$.

In dilute metallic strontium titanate, where carrier density can be tuned by
many orders of magnitude, the simple KW scaling between $A$ and $\gamma$ does
not hold either [43]. The electronic specific heat with doping depends both on
the effective mass, which increases by a factor of $\approx$ 2, but much more
on the carrier density, which changes by several orders of magnitude. The
Sommerfeld coefficient, $\gamma$ is compatible with the experimental quantum
oscillations [44] and calculated band structure [17]. The magnitude of $A$
matches what was calculated by Hussey [45]. Nevertheless, $A$ decreases with
$\gamma$ as it does when comparing Bi, graphite and Sb in Fig. 3a.

To include dilute metals in this scaling scheme, it is necessary to replace
$\gamma^{2}$ by $E_{F}^{2}$ as the horizontal axis [10]. The Fermi energy is a
quantity accessible to the experimentalist. It can be quantified by measuring
the Fermi radius and the effective mass of electrons, or by a combined
knowledge of $\gamma$ and $n$. It is true that a multi-band metal has several
Fermi energies. In semi-metals, the Fermi energy of electrons and holes
differ, because of the mismatch in their masses. However, since we are dealing
with orders of magnitude, the average Fermi energy can do the job.

Fig.3b, is a plot of $A$ vs. $E_{F}^{2}$ for the same elements plotted in
Fig.3a. They are listed together with their relevant specific properties in
table 1. The scaling operates across more than five orders of magnitude of
$A$. This scaling was first put forward in ref.[10] and was extended in ref.
[11] and ref.[12].

The simple KW scaling (i.e $A/\gamma^{2}$) for correlated metals fails also
for dilute metals. A revealing case is CeNiSn, a low-density Kondo semi-metal
[46]. The prefactor of its T-square resistivity is $A=2.2\mu\Omega$cmK-2, two
orders of magnitude larger than what is expected according to its electronic
specific heat, which is $\gamma=40$mJ.mol-1K-2 [47] combined with the KW
$\frac{A}{\gamma^{2}}$ universal ratio. However, this discrepancy evaporates
if one extracts the Fermi energy from the slope of the Seebeck coefficient
[48], as noticed by Kuitra et al. [49]. The latter is indeed adequately large
in CeNiSn [50].

Hussey [45] was the first to put his finger on the inadequacy of the simple KW
scaling. He was partly motivated by the case of Na0.7CoO2, in which an
unexpectedly large $A/\gamma^{2}$ was resolved [51]. He proposed a modified
version of the KW scaling in which $A$ scales with the electronic specific
heat per volume and not per mole. However, finding a general expression for
$A$ remains a challenge, since it is transport coefficient and cannot be
derived out without specific and contestable assumptions about scattering
details. An alternative amendment to the KW scaling [52] consisted in
abandoning the idea of scaling $A$ with a quantity directly accessible to the
experiment.

Empirically, however, Eq. 3 gives a reasonable account of $A$ in numerous
metals over eight orders of magnitude [12]. The list includes numerous newly
discovered semi-metals, such as WTe2 [53], which have attracted attention for
other reasons. The material-dependent $\ell_{quad}$ resides in the range of
1-40 nm[12]. Each family of metal clusters along specific values of
$\ell_{quad}$. In elemental metals of Fig.3, $\ell_{quad}\approx 2$nm. In
layered metals, such as Sr2RuO4 [36] or La1.7Sr0.3Cu04[54], the out-of-plane
resistivity is orders of magnitude larger the in-plane resistivity. As a
consequence, $A$ and $\ell_{quad}$ are also anisotropic. In these cases, our
focus will be the in-plane transport.

Element | Carrier density (m3) |  $\gamma$ (mJ.mol-1.K${}^{-2})$ |  $E_{F}$ (K) |  $A$ (n$\Omega$.cmK-2) | ref.
---|---|---|---|---|---
Bi | 6$\times 10^{23}$ | 0.0085 | 220 | 12 | [20, 55, 56, 57]
C(graphite) | 6$\times 10^{24}$ | 0.0138 | 315 | 5 | [58, 21, 59]
Sb | 1.1$\times 10^{26}$ | 0.119 | 1030 | 0.6 | [56, 55, 15, 57]
Mo | 2.5$\times 10^{28}$ | 1.9 | $2.5\times 10^{4}$ | $1.5\times 10^{-3}$ | [60, 61]
W | 1.4$\times 10^{28}$ | 0.84 | $4.5\times 10^{4}$ | $7\times 10^{-4}$ | [62, 63, 61]
Pd | 5$\times 10^{28}$ | 9.43 | $4\times 10^{3}$ | 0.034 | [3, 64]
Al | 6$\times 10^{28}$ | 1.37 | $4.5\times 10^{4}$ | $5.3\times 10^{4}$ | [65, 66, 67]
Table 1: The experimentally resolved $A$ in a number of elements together with
a number of relevant physical properties. In semi-metals (Bi, graphite, Sb, Mo
and W), the carrier density is the sum of the density of electrons and holes
and the Fermi energy is the average of the Fermi energy of the holes and
electrons.

This brings us to the physical significance of this phenomenological length
scale. Mott [68] argued that $A$ is proportional to the collision cross
section of the two electrons, $\sigma_{cs}$. Therefore:

$\ell_{quad}\propto k_{F}\sigma_{cs}$ (4)

The Fermi wavevector, $k_{F}$, and the collision cross-section $\sigma_{cs}$
are expected to evolve in opposite directions with the evolution of carrier
density. In a dilute metal, the Fermi wave-vector is short and the collision
cross section (in real space) is large. The opposite is true in a dense metal.
We note that $\ell_{quad}$ in Bi and Al are roughly similar. We will come back
to this question in section 5.

A fundamental question is the following: Why should electron-electron
collision decay the charge flow ? Answers to this question fall in to two
categories.

First: If the wave-vector of one of the two electrons after the collision is
larger than the width of the Brillouin zone, then a finite amount of momentum
(equal to $\hbar G$, where $G$ is a unit vector of the reciprocal lattice)
will be lost. This Umklapp option was what Landau and Pomeranchuk had it in
mind in postulating T-square resistivity [2]. Umklapp events require a
sufficiently large Fermi surface, which is the case of all metals, put under
scrutiny by Rice [3] and by Kadowaki and Woods[6]. In the case of heavy-
fermion metals the role of Umklapp events in generating $T^{2}$-ressitivity
was explicitly recognized by Maebashi and Fukuyama [9].

Second: If there are two distinct electron reservoirs, then momentum exchange
between these two reservoirs may constitute a bottleneck for momentum loss.
This was the scenario explicitly put forward by Baber [1] and was invoked by
Hartman [20] in the case of bismuth in which the Fermi pockets are too small
to allow Umklapp events.

However, none of these two options work for dilute metallic strontium titanate
[10], which motivated a search for alternative mechanisms of T-square
resistivity [19] generated by electron-phonon and not electron-electron
scattering. It is true that strontium titanate is a non-trivial solid with
peculiar phonon modes [69]. However, it is not the only case of T-square
resistivity in absence of Umklapp or multiple pockets. Bi2O2Se[12] is yet
another. Moreover, specifically tailored scenarios fail to explain the success
of the scaling approach across many orders of magnitude irrespective of the
presence or absence of Umklapp events.

In search of an alternative solution, let us turn our attention to heat
transport.

##  II The Wiedemann-Franz law and the T-square thermal resistivity

In the zero temperature limit, thermal conductivity, like specific heat and in
contrast to electrical conductivity, vanishes. Disorder, or finite size lead
to a finite residual electrical resistivity, $\rho_{0}$ and a residual
T-linear thermal conductivity, $\frac{\kappa_{0}}{T}$. Their ratio obeys the
Wiedeamnn-Franz law :

$\rho_{0}\frac{\kappa_{0}}{T}=\frac{\pi^{2}}{3}\frac{k_{B}^{2}}{e^{2}}$ (5)

$L_{0}=\frac{\pi^{2}}{3}\frac{k_{B}^{2}}{e^{2}}$ is known as the Sommerfeld
value. The validity of this equality has been experimentally verified in all
bulk metals hitherto investigated.

The fact that the phase space for electron-electron scattering grows
quadratically with temperature has a signature in thermal transport. It leads
to a thermal conductivity to the inverse of the temperature. In order to make
a parallel with electrical resistivity, it is convenient to define thermal
resistivity as $WT=\frac{T}{\kappa}$ and then, one has:

$WT=WT_{0}+BT^{2}$ (6)

The T-square prefactor, $B$, is the thermal counterpart of $A$ in Eq. 1.
Several experiments have quantified $B$ in a number of metals [70, 71, 67, 72,
73, 74, 75, 76, 77, 14, 15]. These studies remain much less numerous than
those devoted to charge transport. A selection of their findings is shown in
Fig. 4.

Figure 4: T-square thermal resistivity in several metals: Low-temperature
electrical resistivity, $\rho$ and thermal resistivity normalized by the
Sommerfeld value, $L_{0}WT$ (in order to be expressed in the same units), for
Ni [70], Al [67], W [71], Sb [14], CeRhIn5 [75] and WP2 [15]. In all cases,
the two data sets join as the temperature tends to zero. In all cases, the
slope of the thermal resistivity is higher.

These experiments find that $L_{0}WT_{0}=\rho_{0}$, but $BL_{0}>A$. In other
words, The WF law is valid in the zero -temperature limit, but not at finite
temperature.

Historically, this finite-temperature deviation from the WF correlation has
been understood as a specific manifestation of the different responses of
momentum and energy flow to inelastic collisions [37]. The starting point is a
distinction between ‘vertical’ and ’horizontal’ collision events, A
‘horizontal event’, in which the loss of energy is concomitant with a loss of
momentum, would affect both electric and thermal conductivity. However, one
can conceive ‘vertical’ events, where a ‘hot’ electron loses the energy it had
to occupy a state slightly above the Fermi energy, without significant
modification of its momentum vector. Such events, which require finite
temperature, have little effect on electric resistivity, but drastically
reduce the thermal resistivity [37].

The distinction between ‘vertical’ and ’horizontal’ is also relevant to
electron-phonon scattering. The resistivity of noble metals is dominated by
electron-phonon scattering and follows a $T^{5}$ temperature dependence. Their
Lorenz number becomes significantly lower than the Sommerfeld value at
intermediate temperature range [78, 79]. The WF law is eventually recovered
above the Debye temperature, when scattering becomes elastic.

There is an important difference between being scattered by phonons and being
scattered by other electrons, however. In the case of e-ph scattering, the
temperature dependence of the phase space differs for energy and momentum
diffusion. Thermal resistivity follows $T^{3}$, the temperature dependence of
the phonon population. Electrical resistivity, on the other hand, follows
$T^{5}$. This is because the fraction of the phonons capable of inducing
’horizontal’ events shrinks by an additional factor of $T^{2}$. The $T^{5}$
e-ph electrical resistivity is the low-temperature asymptotic behavior
expected in the Bloch-Grünesisen model [37], successfully applied to the
experimental data in numerous elemental metals. The $T^{3}$ e-ph thermal
resistivity ($WT\propto T^{3}$, i.e. $W\propto T^{2}$) have been
experimentally resolved in several elemental metals [80, 81] as well as in the
semi-metallic WP2 [14].

In contrast, the phase space for e-e scattering follows $T^{2}$ for both
thermal and electric resistivity. The differentiation between ’vertical’ and
’horizontal’ scattering would only lead to a smaller prefactor for electrical
resistivity. Herring argued that this difference is bounded to a value less
than 2, that is $BL_{0}/A<2$. Li and Maslov, on the other hand, argued that in
a compensated semi-metal $BL_{0}/A$ can become arbitrarily small, if the
screening length is long enough compared to the Fermi wavelength [82]. These
theoretical proposals have not been confirmed by the experimental data. The
bound proposed by Herring is violated in several cases, such as W
($BL_{0}/A\approx 6$) [71] and WP2 ($BL_{0}/A\approx 5$) [14]. Moreover, the
largest reported deviation from the WF law occurs in Al ($BL_{0}/A\approx 10$)
[67], which is not a compensated semi-metal.

T-square thermal resistivity, in contrast with T-square electric resistivity,
does not require Umklapp events or multiple reservoirs. Normal e-e scattering
inside a Fermi pocket can generate T-square resistivity. This is another
dichotomy, distinct from horizontal and vertical events and capable of playing
a role in the observed downward deviation from the Wiedemann-Franz law. Let us
consider the case of thermal transport in a liquid of neutral fermions without
the annoying presence of a lattice.

##  III Energy and momentum diffusivity in 3He

Soon after the conception of Landau’s Fermi liquid theory [83], Abrisokov and
Khalatnikov wrote a paper on the kinetic properties of a Fermi liquid with the
case of 3He in mind [23]. In this paper, they derived expressions for the
temperature dependence of viscosity and thermal conductivity.

Kinematic viscosity is the diffusion constant for momentum and thermal
diffusivity is the diffusion constant for energy. In the simple kinetic theory
for classical gases, there is a simple expression for diffusivity :

$D=\frac{1}{3}\tau v^{2}_{m}$ (7)

Here, $v_{m}$ is the mean molecular velocity and $\tau$, the mean scattering
time. In a degenerate fermionic fluid, $v_{m}$ becomes the Fermi velocity and
$\tau$ is the inverse of the fermion-fermion scattering rate, which follows
$T^{-2}$. Thus, both the viscosity and the thermal diffusivity are expected to
decrease quadratically with temperature. Now, thermal conductivity is
proportional to the product of (energy) diffusivity and the specific heat and
the latter is T-linear. Therefore, one expects a thermal conductivity
following $T^{-1}$.

The temperature dependence obtained by such simplistic arguments are confirmed
by the outcome of the elaborate calculations performed by Abriokosov and
Khatalnikov [23]. They quantified the expected prefactors for the $T^{-2}$
viscosity and the $T^{-1}$ thermal conductivity. Brooker and Sykes [25,
Sykes1970] revisited the subject a decade later and found that the previous
result for theoretical thermal conductivity was too large by a factor of 2.
Moreover, they performed the first comparison between theory and experiment
and found [25] their theoretical $\kappa T$ to be still 1.5 times larger than
the experimentally-measured $\kappa T$ reported by Wheatley and his
collaborators [29, 28]. A year after, Dy and Pethick [85] carried out a new
set of calculations (assuming negligible Landau parameters when $\ell>2$) and
found a closer agreement with the experimental data.

An extensive study of thermal conductivity in 3He was published by Greywall in
1984 [30], a year after his study of specific heat [32]. They document in
detail how entropy transport evolves with the strength of correlations in this
emblematic Fermi liquid.

Figure 5: The original Fermi liquid: a) Phase diagram of 3He [86]. Black and
red rectangles designate the temperature range of the two sets of data shown
in the right panel. b) Thermal resistivity at two different pressures as a
function of temperature according to the data reported by Greywall [30].

The phase diagram of 3He is shown in Fig. 5a. The superfluid transition occurs
below 3 mK. The normal liquid is a degenerate liquid of fermions below 1 K.
Under a pressure of $\approx$3 MPa (30 atmospheres), interaction between
fermions becomes strong enough in order to solidify the liquid. Greywall
measured the thermal conductivity $\kappa$ of the liquid down to $\sim 7$ mK
at different pressures, ranging from vacuum up to the threshold of
solidification. He found that, in the first approximation, $\kappa$ is
proportional to the inverse of temperature, as expected and quantified the
product of thermal conductivity and temperature, $\kappa T$, at different
pressures.

Now, since $WT\equiv\frac{T}{\kappa}$, the expression $\kappa\propto T^{-1}$
is strictly equivalent to $WT\propto T^{2}$. Fig. 5b shows $WT$ extracted from
Greywall’s $\kappa(T)$ data at two different pressures (or molar volumes) as a
function of $T^{2}$. One can see that at low temperature, thermal resistivity
displays a T-square temperature dependence. At high pressure, when
interactions are stronger, the slope (i.e. $B$ of Eq.6) is larger. Moreover,
when the prefactor is larger, the temperature below which the T-square
behavior is visible shifts to lower temperatures. These features do not
surprise those familiar with the Kadowaki-Woods ratio in heavy-electron
metals. For example, near the quantum critical point of CeCoIn5 [87, 88] and
YRh2Si2 [89, 90], the electronic specific heat and the T-square prefactor
increase and the domain of validity of the T-square resistivity shrinks.

Figure 6: Tuning correlation with pressure- a) T-linear Specific heat as a
function of pressure [32]. b) $(\kappa T)^{-1}$ plotted against pressure,
according to experiment (black solid circles) [30] and theory (red solid
circles) [85]. c) The same quantity plotted as a function of the Fermi energy
at each pressure. d) The dimensionless parameter $A^{\prime}$ (see text) as a
function of the Fermi energy. In these panels, $(\kappa T)^{-1}$ and T-linear
specific heat are values reported by Greywall who extrapolating the data to
extract zero-temperature limit.

With increasing pressure, interaction between 3He atoms intensify and above a
threshold pressure, the liquid solidifies. This striking version of pressure-
tuned correlation inspired P. W. Anderson to state that “the Mott insulator is
a form of quantum solid, and the melting transition in 3He is our best example
of a Mott transition.” [91].

Fig. 6 shows how the specific heat and the fermion-fermion scattering (which
is proportional to $(\kappa T)^{-1}$) evolve with pressure. One can see that
both smoothly enhance as the solidification approaches. Red circles in Fig. 6
are values expected by theory [85] which are close to the experimentally-
measured data.

Calkoen and van Weert [31] observed that the magnitude of $(\kappa T)^{-1}$
[30] allows to extract a Landau parameter, which is in excellent agreement
with the one extracted from the specific heat[32]. They argued that at zero
temperature limit, one has:

$\kappa T|_{0}=\frac{5}{18\pi^{3}}\frac{p_{F}^{3}v_{F}^{2}}{A^{2}}$ (8)

Here, $p_{F}$ and $v_{F}$ are Fermi momentum and Fermi velocity. The parameter
$A$ (not to be mistaken with the prefactor of T-square resistivity) has the
dimensions of the Planck constant and was identified as a combination of
Landau Fermi-liquid parameters [31]. Extracting the Fermi temperature from the
specific heat per volume, one can monitor the evolution of $(\kappa T)^{-1}$
and $A^{\prime}=\frac{A}{\hbar}$ with the Fermi temperature. As seen in
Fig.6c-d, $(\kappa T)^{-1}$ steadily increases with the Fermi energy. This is
concomitant with the evolution of dimensionless $A^{\prime}$, intimately
linked with the Landau parameter $F^{a}_{0}$ [31].

Thus the amplitude and the pressure dependence of $\kappa T$ in 3He are well
understood. The prefactor of T-square thermal resistivity, $B$, is the inverse
of $\kappa T$ and therefore, Eq.8 can be rewritten as:

$B\equiv(\kappa T)^{-1}=\frac{9\pi^{3}}{10}\frac{\hbar A^{\prime
2}}{E_{F}^{2}k_{F}}$ (9)

Here, $A$ is replaced by dimensionless $A^{\prime}$ and the presence of the
Fermi energy $E_{F}=\frac{1}{2}p_{F}v_{F}$ is explicit.

Reference |  $\tau^{0}_{\eta}T^{2}$ (ps.K2) |  $\tau^{0}_{\kappa}T^{2}$ (ps.K2)
---|---|---
Greywall (1984) (p=0) [30] | – | 0.391
Wheatley (1975) (p=0)[92] | 1.24 | 0.51
Alvesalo et al. (1975) [93] | 0.65 | 0.23
Bertinatet al. (1974) [94] | 1.29 | 0.524
Black et al. (1971) [95] | 1.58 | –
Table 2: The rate of fermion-fermion scattering extracted from measurements of
viscosity and from measurements of thermal conductivity as reported by several
authors. The time scale associated with momentum diffusion is 2-3 times longer
than the scattering time associated with energy diffusion. Compare this with
what is seen in metals (Fig. 4) and the dichotomy between vertical and
horizontal events.

Several studies have been devoted to measuring the temperature dependence of
viscosity$\eta$. They have confirmed that the expected T-square decrease in
the amplitude of viscosity ($\eta\propto T^{-2}$). These studies allow
quantifying the fermion-fermion scattering rate giving rise to the
temperature-dependent viscosity ($\tau^{0}_{\eta}T^{2}$). Several authors have
compared it then with the one extracted from thermal
conductivity($\tau^{0}_{\kappa}T^{2}$). A summary of these numbers reported by
various authors is given in table 2. One can see that the scattering time for
momentum diffusivity is 2-3 times longer than the scattering time for energy
diffusivity. Like in the case of metals, and by a comparable factor, thermal
transport is punished more than momentum transport by fermion fermion
scattering.

Normal liquid 3He demonstrates not only the persistence of T-square thermal
resistivity in a single fermionic reservoir without Umklapp scattering, but
also the scaling between its amplitude and the square of the Fermi energy. Let
us now check the relevance of these facts to metals.

##  IV A common thread

Fig. 7a is a plot à la Kadowaki-Woods (or Rice), exclusively focused on
thermal transport. Each data point represents a metallic solid in which the
thermal resistivity prefactor $B$ has been quantified. It includes two weakly
correlated (W [71] and WP2 [14]) and three strongly correlated (UPt3 [73],
CeRhIn5 [75] and CeCoIn5 [76]) metals. Note that CeCoIn5 is not a Fermi liquid
at zero magnetic field. Its resistivity displays a T-square resistivity only
in presence of a finite magnetic field restoring the Fermi-liquid behavior
[88, 87]. The plot compares the amplitude of $B$ in these metals and
$B\equiv(\kappa T)^{-1}$ in 3He, discussed in the previous section, as a
function of their $\gamma$. The data points scatter close to the dashed line
which represents a $\gamma^{2.15}$ slope in the log-log scale. Note that 3He
data points evolves slightly faster, but reasonably close to the dashed line.

Figure 7: Continuity between 3He and metals: a) The magnitude of T-square
thermal resistivity, $B$, in 3He and in metals where the T-square thermal
resistivity has been measured plotted against the T-linear specific heat,
$\gamma$. The dashed line, which is a guide to eye has a slope of 2.15. b) The
T-square thermal resistivity prefactor multiplied by the Sommerfeld value
($L_{0}B$) in 3He and the T-square electrical resistivity prefactor ($A$) in
lightly-doped single-band strontium titanate as a function of the Fermi
temperature.

Up to now, the amplitude of T-square thermal resistivity in metals has been
analyzed starting from T-square electrical resistivity. It has been taken for
granted that the latter cannot arise by normal (i.e not Umklapp), intra-pocket
electron-electron scattering. Fig.7a which focuses on thermal transport,
provides an alternative and a short-cut root. Normal momentum-conserving
between fermionic quasi-particles generates T-square thermal resistivity in
metals as it does in 3He.

The idea that normal electron-electron scattering plays a role in generating
$B$ has found support by the results of recent experiments on thermal
transport in bulk antimony (Sb). In this semi-metal, electrons are quasi-
ballistic with a mean-free-path approaching (and scaling) with the sample
size. Jaoui et al.[15] quantified the prefactors of the thermal ($B$) and the
electric ($A$) T-square resistivities and found that the $BL_{0}/A$ ratio
evolves with the change in residual resistivity (or sample size). The cleaner
the sample, the larger was the $B/A$ ratio and the deviation from the WF
law[15]. This behavior finds a natural explanation in the hydrodynamic
scenario of heat transport as the driver of the deviation from the WF law at
the onset of ballistic regime. Indeed, according to a theoretical work by
Principi and Vignale [96] the WF ratio between the thermal and the electric
conductivity is reduced by a $1+\tau_{0}/\tau_{ee}$ factor, where $\tau_{0}$
is the momentum relaxing time and $\tau_{ee}$ is the electron-electron
scattering time. In Sb, at the onset of ballistic regime ($\sim 10K$), these
two time scales are comparable in amplitude. In cleaner samples, $\tau_{0}$ is
larger and the deviation from the WF law is more pronounced. Thus attributing
a role to normal electron-electron collisions in generating $B$ provides a
satisfactory explanation to the evolution of the ratio of the two prefactors
with sample size. In contrast, the traditional picture invoking horizontal and
vertical events fails to explain why the deviation between thermal and
electrical resistivity grows with sample size..

According to Principi and Vignale [96], the WF law is recovered when
$\tau_{0}\ll\tau_{ee}$. In most metals, this condition is easily satisfied in
a reasonable temperature range thanks to unavoidable disorder. In all metals,
the condition will be satisfied at sufficiently low temperature, because as
$T\rightarrow 0$, $\tau_{ee}^{-1}$ will diverge and $\tau_{0}^{-1}$ will stay
finite. Therefore, the T-square thermal resistivity should have an electrical
counterpart of the amplitude set of $\sim L_{0}B$ and T-square electrical
resistivity is expected even without Umklapp (or other momentum-relaxing)
collision events between electrons. The combination of Normal electron-
electron scattering and disorder will suffice for this.

In this picture, the (relative) success of the $A$ vs. $E_{F}^{-2}$ scaling
across different families of metals is a consequence of the approximate
validity of the WF law, which implies
$A\approx\frac{\pi^{2}}{3}(\frac{k_{B}}{e})^{2}B$. This, combined with Eq. 3
and Eq.9, yields the following expression for the phenomenological
$\ell_{quad}$ :

$\ell_{quad}\approx\frac{3\pi^{5}}{10}\frac{A^{\prime 2}}{k_{F}}$ (10)

In the weakly-interacting metals considered by Rice [3] $\ell_{quad}\approx
1.6$ nm and in the strongly-correlated metals scrutinized by Kadowaki and
Woods $\ell_{quad}\approx$ 40 nm [10, 12]. Many other metals fall somewhere in
between. According to Eq. 10, the magnitude of $\ell_{quad}$ in 3He Mpa.

Fig. 7b) plots $L_{0}B$ in ${}^{3}He$ and $A$ in lightly-doped strontium
titanate as a function of the Fermi energy. The two curves follow a similar
trend.

##  V Concluding remarks

Thus, when the thermal transport becomes the departing point, our puzzles find
solutions. The link between the magnitude of $A$ and the Fermi energy becomes
understandable. The persistence of T-square in dilute metals becomes
unsurprising.

The temperature dependence of resistivity in SrTiO3-δ does not reduce to this
issue. Well below the degeneracy temperature, the exponent of resistivity in
this metal is close to two, but it smoothly evolves with warming [97, 98]. In
order to explain resistivity in a wide temperature range, one needs to invoke
coupling between electrons and transverse optical phonons as recently shown by
two independent theoretical studies [19, 99] Nazaryan and Feigel’man [99], in
contrast to Kumar et al.[19] , incorporated in their calculations the
evolution of the effective mass with temperature found by experiment [98] and
gave an account of both resistivity and thermopower in a wide temperature
range, starting from the minimum energy of phonons. Coupling with phonons
cannot explain the persistence of T-square resistivity below this temperature
range, which unavoidably leads to electron-electron scattering. The present
approach not only provides an explanation for this persistence, but also for
its amplitude.

Note that electronic thermal conductivity is orders of magnitude lower than
the total thermal conductivity in SrTiO3-δ [100]. To check the validity of the
WF law one needs a way to separate the two. This has been only done in the
zero-temperature limit [101]. Quantifying the electronic thermal resistivity
and quantifying $B$ in this system remains an experimental challenge.

The case of graphene is beyond the scope of the present paper. In graphite,
T-square resistivity emerges below 6 K[21]. An investigation of resistivity in
graphene, focused on the relevance of Bloch-Grüneisen resistivity [102] did
not explore such low temperatures. Theory expects the phase space of e-e
scattering to follow a $T^{2}$ln$T$ temperature dependence [103, 4] in two
dimensions. Recently, $T^{2}$ resistivity was observed in a Moiré superlattice
of graphene on top of boron-nitride [104] and in twisted bilayer graphene
[105]. Comparing the amplitude of T-square in these studies with what has been
observed in three-dimensional solids emerges as a future task.

The distinction between e-ph and e-e scattering is not impermeable. There are
cases where one suspects that scattering between electrons is accompanied by
exchange of phonons [106, 107, 108]. It has been suggested that phonon
exchange amplifies the magnitude of $A$ in Al [106] or in A15 compounds [4].
It remains a task for theory to sort out the link between phonon exchange and
the electron-electron cross-section.

The interplay between disorder and electronic interaction is yet to be sorted
out. The recovery of the WF law in the picture put forward by Principi and
Vignale [96] requires a hierarchy between the two scattering time:
$\tau_{0}<\tau_{ee}$. Upon warming, $\tau_{ee}$ becomes steadily shorter and
eventually this inequality is no more satisfied. As seen in Fig. 1, the
T-square behavior seen in several metals extends to temperatures above
$AT^{2}\sim\rho_{0}$. This implies a crucial missing component.

The analogy between the thermal resistivity of metals and 3He is beyond
dispute. But how relevant is momentum diffusivity to electrical resistivity of
metals? The usual answer to this question is negative. The thermal current is
accompanied by a gradient of thermal energy, but the drift velocity is
identical along the charge current. In presence of ohmic response, the
electron viscosity [109] of a metallic Fermi liquid has seldom been considered
relevant to diffusive electric transport. However, Landauer [110, 111]
contested the assumption that current flows homogeneously in a macroscopic
sample hosting defects. He postulated that, within a screening distance of
each scattering center, electric field and charge current are both
inhomogeneous.

Let us end by noting a number of studies, which may prove relevant to the
subject matter of the present paper. Einzel and Parpia have already examined
the link between Drude conductivity in metals and the Poiseuille flow in
normal liquid 3He in presence of aerogel [112]. Nazraov et al. have shown that
dynamical electron-electron interactions, known to govern the conductivity of
nanoscale junctions [113], play a role in setting the zero-temperature
resistivity of a bulk metal [114]. Evidence for inhomogeneous charge flow is
provided by experiments imaging the flow of current in a two-dimensional
electron gas (2DEG) [115], which have found that it follows branching strands
driven by ripples in the energy landscape.

Thus, the viscosity of the electron liquid may be more accessible than
previously thought, and the ‘hydraulic analogy’ [116] deeper than a mere
metaphor.

Acknowledgements- I am grateful to Benoît Fauqué, Mikhail Fei’gelman,
Alexandre Jaoui, Xiao Lin, Dmitrii Maslov, Vladimir Mineev, and Giovanni
Vignale for their valuable comments. This work was supported by the Agence
Nationale de la Recherche (ANR-18-CE92-0020-01; ANR-19-CE30-0014-04).

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