# Breakdown of the Wiedemann-Franz law at the Lifshitz point of strained
Sr2RuO4

Veronika C. Stangier  Institute for Theory of Condensed Matter, Karlsruhe
Institute of Technology, Karlsruhe 76131, Germany  Erez Berg  Department of
Condensed Matter Physics, Weizmann Institute of Science, Rehovot, 76100,
Israel  Jörg Schmalian  Institute for Theory of Condensed Matter, Karlsruhe
Institute of Technology, Karlsruhe 76131, Germany  Institute for Quantum
Materials and Technologies, Karlsruhe Institute of Technology, Karlsruhe
76021, Germany

###### Abstract

Strain tuning Sr2RuO4 through the Lifshitz point, where the Van Hove
singularity of the electronic spectrum crosses the Fermi energy, is expected
to cause a change in the temperature dependence of the electrical resistivity
from its Fermi liquid behavior $\rho\sim T^{2}$ to $\rho\sim T^{2}{\rm
log}\left(1/T\right)$, a behavior consistent with experiments by Barber et al.
[Phys. Rev. Lett. 120, 076601 (2018)]. This expectation originates from the
same multi-band scattering processes with large momentum transfer that were
recently shown to account for the linear in $T$ resistivity of the strange
metal Sr3Ru2O7. In contrast, the thermal resistivity $\rho_{Q}\equiv
T/\kappa$, where $\kappa$ is the thermal conductivity, is governed by
qualitatively distinct processes that involve a broad continuum of compressive
modes, i.e. long wavelength density excitations in Van Hove systems. While
these compressive modes do not affect the charge current, they couple to
thermal transport and yield $\rho_{Q}\propto T^{3/2}$. As a result, we predict
that the Wiedemann-Franz law in strained Sr2RuO4 should be violated with a
Lorenz ratio $L\propto T^{1/2}{\rm log}\left(1/T\right)$. We expect this
effect to be observable in the temperature and strain regime where the
anomalous charge transport was established.

##  I Introduction

Sr2RuO4 is a fascinating material that combines electronic correlations and
unconventional superconductivity whose mechanism has yet to be
understoodMackenzie2003 ; Mackenzie2017 . Major progress in our understanding
of this material was achieved through the application of uniaxial stress. This
leads to a more than two-fold increase in the superconducting transition
temperatureHicks2014  to $T_{\rm c}^{\rm max}\approx 3.5\,{\rm K}$, a rich
phase diagram where the superconducting and time reversal breaking transitions
are splitGrinenko2021 , and puzzling behavior of thermodynamic
propertiesLi2021 . The maximum in the superconducting transition temperature
occurs for strain values $\varepsilon_{xx}^{{\rm*}}\approx 0.45$. This is at
or very near the Lifshitz transitionLifshitz1960  where a Van Hove singularity
of the electronic spectrum crosses the Fermi energy.

Evidence that the normal state of the system is equally affected by this
Lifshitz point was given by Barber _et. al_Barber2018  in measurements of the
electrical resistivity as function of strain $\varepsilon_{xx}$. The main
finding of these measurements are as follows: For strain values below and
above $\varepsilon_{xx}^{{\rm*}}$, the resistivity shows Fermi liquid behavior
with $\rho\approx\rho_{0}+AT^{2}$, while for
$\varepsilon_{xx}\approx\varepsilon_{xx}^{*}$ the resistivity is more
singular. The data between $T_{c}$ and about $40\>{\rm K}$ are consistent with
$\rho=\rho_{0}+AT^{2}{\rm log}\left(T_{0}/T\right)$. Over most of this
temperature regime, the residual resistivity $\rho_{0}$ is a small fraction of
the total resistivity. Hence, the system can be safely analyzed in the clean
limit. Similar results were obtained by tuning the system to the Van Hove
singularity by La3+ substituted ${\rm Sr}_{2-y}{\rm La}_{y}{\rm
RuO}_{4}$Kikugawa2004 ; Shen2007  or epitaxial strainBurganov2016 , however
with somewhat larger values for the residual resistivity $\rho_{0}$.

In this paper we determine the electrical and thermal transport behavior due
to electron-electron scattering of clean Sr2RuO4 near the strain-induced
Lifshitz point. Distinct scattering processes, both impacted by the presence
of a Van Hove singularity at the Fermi energy, affect charge and heat
transport differently, leading to a violation of the Wiedemann Franz
lawWiedemann1853 . Charge transport is determined by large momentum transfer
scattering that couples non-Van Hove to Van Hove states. Heat transport is
dominated by long-wavelength scattering due to compressive modes that build a
broad continuum due to the saddle point in the energy dispersion. For
perfectly clean systems and ignoring the onset of superconductivity or other
ordered states this behavior should continue down to lowest temperatures. The
Wiedemann Franz law should only be recovered once impurity scattering becomes
dominant. As the two scattering rates that govern electrical and thermal
transport are both caused by electron-electron interactions and the presence
of the Van Hove singularity, it seems natural that the violation of the
Wiedemann Franz law occurs in the same temperature regime $3.5\>{\rm K}\cdots
20-40\>{\rm K}$ where the deviation from the $T^{2}$ behavior in the
resistivity was observedBarber2018 . Our results call for thermal conductivity
measurements under strain to verify the importance of electron-electron
interactions of the quasi two-dimensional sheet of the Fermi surface.

Figure 1: Temperature dependence of the resistivity $\rho(T)\sim
T^{2}\log(D/T)$ (left) and thermal resistivity $\rho_{Q}(T)\sim T^{3/2}$
(right) divided by the Fermi liquid behavior $T^{2}$ at the Van Hove point
(red curves). The more singular behavior of the thermal resistivity is clearly
visible. $D$ is the effective bandwidth of the problem. As one moves away from
the Van Hove point, one expects a temperature scale $T^{\ast}$ below which
both transport coefficients recover ordinary Fermi liquid behavior following
Eqs(4) and (5). We use $T^{\ast}/D=0.01$ (green curves) and $T^{\ast}/D=0.03$
(blue curves). The inset shows the corresponding behavior for the Lorentz
ratio $L(T)$. The right index shows the characteristic temperature scale
$T^{*}$ for the crossover to Van Hove dominated scattering as the strain is
varied. We used the parameters given in Appendix B. In this Inset we also
indicate the regimes where the Lorenz number $L$ tends to a finite value or
vanishes as $T$ decreases.

The existence of logarithmic corrections to the $T^{2}$ behavior near van Hove
singularities was discussed in the past with $\rho\sim T^{2}{\rm
log^{2}}\left(1/T\right)$ in Ref.Hlubina1995  and $\rho\sim T^{2}{\rm
log}\left(1/T\right)$ in Ref.Hlubina1996 . Including impurity scattering then
changes the behavior to $\rho=\rho_{0}+BT^{3/2}$, where the $T$-dependent term
is, however, a small correction to the residual resistivity. A careful
analysis of the transport processes in systems with impurity scattering was
recently performed in Ref.Herman2019 . Interestingly, in this work a drop in
the Lorentz ratio near the Van Hove singularity was found. These results are
somewhat surprising as usually a single, _hot_ point on the Fermi surface does
not influence the transport properties of a system. A prime example is the
transport behavior near a density-wave instability where hot spots are Fermi
surface points connected by the ordering vector of the density waveHlubina1995
; Stojkovic1996 . While the scattering rate at these isolated points on the
Fermi surface is singular, the transport is dominated by generic, _cold_
regions of the Fermi surface that short circuit the contribution from the hot
spots and lead to Fermi liquid behavior with $\rho\sim T^{2}$Hlubina1995 .
Only the inclusion of impurity scattering changes the behavior to
$\rho=\rho_{0}+AT^{d/2}$Rosch1999 ; Syzranov2012 . However, in this case the
$T$-dependent term is a small correction to the dominant residual resistivity
$\rho_{0}$, in distinction to the experimental result of Ref.Barber2018 .

An interesting exception to the rule that hot spots are irrelevant for
transport was recently presented in Ref.Mousatov2020  to explain the linear in
$T$ resistivity of the strange metal Sr3Ru2O7. It was shown that scattering
processes $cc\leftrightarrow ch$, in which a cold electron ($c$) becomes hot
($h$) after colliding with another cold electron, exists everywhere on the
Fermi surface, i.e. it cannot be short circuited. For Sr3Ru2O7 the hot
electrons are made up of an exceptionally sharp peak in the density of states
that crosses the Fermi surface. The relevance of this scenario for Sr2RuO4 was
already mentioned in Ref.Mousatov2020 . Below we show that for the specific
Fermi surface geometry of Sr2RuO4 the $cc\leftrightarrow ch$ processes do
indeed yield the $\rho\sim T^{2}{\rm log}\left(1/T\right)$ of Ref.Hlubina1996
. One has to be careful however to include all bands that cross the Fermi
surface, as mere umklapp processes of a single band do not provide the
necessary phase space for $cc\leftrightarrow ch$ processes down to lowest
temperatures. Hence, our analysis shows that the electrical resistivity of
strained Sr2RuO4 can be understood in terms of the $cc\leftrightarrow ch$
approach of Ref.Mousatov2020 .

The modifications of the electrical resistivity at the strain-tuned Lifshitz
point are argued to be due to the divergent density of states at the Van Hove
singularity with large momentum transfer in the involved scattering processes.
Sr2RuO4 is of course a three-dimensional material. Hence, the logarithmic
divergence in the density of states is cut off at some energy scale
$t_{\perp}$ set by the inter-layer hopping. However, this energy scale was
shown in experiments to be only a few KelvinMackenzie2003 . This is consistent
with the three-dimensional electronic structure where the dispersion in the
c-direction is particularly weak for in-plane momenta near the van Hove
pointHaverkort2008 ; Roising2019 . With $t_{\perp}$ comparable to $T_{c}$ we
will ignore these effects in what follows. In addition to these density of
states effects, electrons near a Van Hove singularity are also expected to
yield singular negative corrections to the compressibility or other elastic
constants:

$\delta C\left(T\right)\sim-\frac{D}{v_{0}}\log\frac{D}{k_{B}T},$ (1)

where $v_{0}$ is the unit cell volume and $D$ the band width. Unless preempted
by other states of order, such as superconductivity, this should eventually
give rise to a lattice instability at some low temperature. These compressive
modes are related to a broad continuum in the long wave-length density
fluctuation spectrum of systems with Van Hove singularityGopalan1992 . Such
fluctuations are known to give rise to singular single particle scattering
rates, with frequency and temperature dependencies that depend on the details
of the band dispersionGopalan1992 ; Pattnaik1992 . The corresponding
contribution to the resistivity is small, however, since scattering of
electrons from these fluctuations involves a small momentum transfer.

The presence of distinct scattering processes that do and do not contribute to
the resistivity suggests to analyze different transport properties. After all,
the Wiedemann-Franz law, according to which the Lorenz ratio

$L=\frac{\kappa}{\sigma T}=\frac{\rho}{\rho_{Q}}$ (2)

of the thermal conductivity $\kappa=T/\rho_{Q}$ and the electronic
conductivity $\sigma=1/\rho$ approaches
$L_{0}=\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}$ , is caused by the
same scattering processes contributing to thermal and charge
transportZiman1960 . This is certainly the case for disordered
electronsCastellani1987 ; Michaeli2009 . In clean Fermi liquids, charge
current relaxation requires umklapp scattering, while heat current relaxation
transport does not. Hence, there is no reason to expect that $L\rightarrow
L_{0}$. However, given that both scattering rates are proportional to $T^{2}$
with $\tau_{J,Q}^{-1}=A_{J,Q}T^{2}$ for charge ($J$) and heat ($Q$) transport
processes, one still expects a constant Lorenz ratio $L\left(T\rightarrow
0\right)\rightarrow L_{0}\frac{A_{J}}{A_{Q}}$.

We show that the thermal transport at the Lifshitz point is governed by a
scattering rate $\tau_{Q}^{-1}\propto T^{3/2}$ caused by the continuum of
density fluctuations, while the resistivity follows the discussed
$\tau_{J}^{-1}\propto T^{2}\log\frac{D}{T}$. As a result, we obtain for the
Lorenz ratio

$L\propto T^{1/2}\log\frac{D}{T},$ (3)

which vanishes as $T\rightarrow 0$. Hence, we expect a strong breakdown of the
Wiedemann-Franz law at the Lifshitz point of strained Sr2RuO4. These results
are valid right at the Lifshitz transition. As the chemical potential moves
away from the Van Hove point one expects to recover the usual Fermi liquid
behavior. This is sketched in Fig. 1, where we use

$\rho\left(T\right)\approx A_{J}T^{2}\log\frac{D}{\sqrt{T^{2}+T^{*2}}}$ (4)

and

$\rho_{Q}\left(T\right)\approx
A_{Q}T^{2}\left(\frac{D^{2}}{T^{2}+T^{*2}}\right)^{1/4}.$ (5)

The energy scale $k_{\rm B}T^{*}$ is essentially the distance of the Van Hove
point to the Fermi energy and is shown in the right inset of Fig. 1 using
realistic parameters for the electronic structure of Sr2RuO4; see Appendix B
for details. The left inset shows the Lorentz number as function of
temperature. Our prediction for the violation of the Wiedemann-Franz law is
consistent with the numerical solution of the Boltzmann equation of
Ref.Herman2019 , where a suppression in the Lorentz number for weakly
disordered systems was seen near the Van Hove point.

It is of interest to contrast the behavior found here with the one of two-
dimensional Fermi liquids without Van Hove singularity. Then the single-
particle scattering rate is enhanced by a logarithmic term, compared to the
usual $T^{2}$ behavior $\tau^{-1}_{\rm qp}\sim T^{2}\log\left(D/T\right)$. As
this enhancement is due to small momentum transfer processes, it will not
affect the resistivityPal2012 , i.e. $\rho(T)\sim T^{2}$. However, the thermal
conductivity does couple to forward scattering processes and acquires an
additional logarithmic contribution $\rho_{Q}(T)\sim
T^{2}\log\left(D/T\right)$Lyakhov2003 . Hence one also finds a violation of
the Wiedemann-Franz law $L(T)\sim 1/\log\left(D/T\right)$. This violation,
however, is much weaker than the one we predict at a Van Hove point.

In what follows we briefly discuss the electronic structure of Sr2RuO4 within
a three-band model. We then summarize the behavior of long wavelength density
fluctuations near a Van Hove point. Finally we present our results for the
charge and heat transport. In the appendices we comment briefly on the
relation to Matthiessen’s rule, summarize the tight-binding parametrization of
the band structure, determine the single particle scattering rate, and present
our results for the current relaxation rate $\tau_{J}^{-1}\left(\omega\right)$
in the regime $\tau_{J}^{-1}(\omega=0)\ll\omega\ll D$.

##  II Model and Density response

As we include multi-band effects in our analysis we start from the following
three-band model with kinetic energy

$H_{0}=\sum_{\boldsymbol{k}\sigma}\psi_{\boldsymbol{k}\sigma}^{\dagger}{\cal
H}\left(\boldsymbol{k}\right)\psi_{\boldsymbol{k}\sigma}$ (6)

where
$\psi_{\boldsymbol{k}\sigma}=\left(d_{\boldsymbol{k},xy,\sigma},d_{\boldsymbol{k},xz,\sigma},d_{\boldsymbol{k},yz,\sigma}\right)^{T}$
with annihilation operators for electrons in the Ru $4d_{xy}$ as well as
$4d_{xz}$ and $4d_{yz}$ orbitals, respectively. For our analysis we use the
single particle Hamiltonian

${\cal
H}\left(\boldsymbol{k}\right)=\left(\begin{array}[]{ccc}\varepsilon_{\boldsymbol{k}xy}&0&0\\\
0&\varepsilon_{\boldsymbol{k}xz}&V_{\boldsymbol{k}}\\\
0&V_{\boldsymbol{k}}&\varepsilon_{\boldsymbol{k}yz}\end{array}\right),$ (7)

where we employ the dispersion relations obtained in Ref.Burganov2016  from
angular-resolved photoemission data for the unstrained system and follow
Ref.Barber2018  to account for the changes in the dispersion at finite applied
strain. Uniaxial strain $\epsilon_{xx}$ lifts the degeneracy between states at
momenta $(\pi,0)$ and $(0,\pi)$ and splits the Van Hove singularity into two
peaks. For $\epsilon_{xx}^{\ast}=0.45\%$ the Van Hove singularity at

$\boldsymbol{k}_{\rm VH}=(0,\pi)$ (8)

crosses the Fermi energy. The details of this analysis are summarized in
Appendix B. In Fig.2 we show our results for the strain dependence of the
Fermi surface and the density of states.

Figure 2: Density of states of the $\gamma$ band at zero strain (blue) and at
the strain value $\epsilon^{\ast}$ that corresponds to the Lifshitz point
where one of the strain-split Van Hove singularities crosses the Fermi energy
(red). The inset shows the Fermi energies at zero strain (left) and at the
Lifshitz point (right), where the hot parts of the Fermi surface are indicated
in red.

The details of this electronic structure will be important when we analyze the
kinematics of umklapp scattering events that are crucial for the electrical
resistivity. For the thermal transport long-wavelength density excitations of
a system with Van Hove singularity will become important. The density
excitation spectrum follows from

$\displaystyle{\rm Im}\Pi\left(\boldsymbol{q},\omega\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}p}{4\pi}\left(f\left(\varepsilon_{\boldsymbol{p}+\boldsymbol{q}}\right)-f\left(\varepsilon_{\boldsymbol{p}}\right)\right)$
(9) $\displaystyle\times$
$\displaystyle\delta\left(\omega-\varepsilon_{\boldsymbol{p}+\boldsymbol{q}}+\varepsilon_{\boldsymbol{p}}\right).$

As the low momentum regime is dominated by states near the saddle point of the
dispersion, we approximate the so called $\gamma$ band with dispersion
$\varepsilon_{\boldsymbol{k}xy}$ by

$\varepsilon_{\boldsymbol{k}_{\rm
VH}+\boldsymbol{p},xy}\approx\varepsilon_{{\rm
VH},{\boldsymbol{p}}}=\frac{p_{x}^{2}-p_{y}^{2}}{2m}.$ (10)

At $T=0$ the momentum integration can be performed easily, yieldingGopalan1992

${\rm
Im}\Pi\left(\boldsymbol{q},\omega\right)=-\frac{m}{2\pi}\left\\{\begin{array}[]{ccc}\frac{\omega}{\left|\varepsilon_{{\rm
VH},\boldsymbol{q}}\right|}&{\rm
if}&\left|\omega\right|<\left|\varepsilon_{{\rm
VH},{\boldsymbol{q}}}\right|\\\ {\rm sign}\left(\omega\right)&{\rm
if}&\left|\omega\right|>\left|\varepsilon_{{\rm
VH},{\boldsymbol{q}}}\right|\end{array}\right..$ (11)

In comparison to the usual electronic spectrum, where the density response
vanishes for $\left|\omega\right|>v_{F}q$, a flat continuum extends up to an
energy scale of the order of the electronic bandwidth $D$. At finite
temperatures follows

${\rm
Im}\Pi\left(\boldsymbol{q},\omega\right)=P\left(\frac{\omega}{\left|\varepsilon_{{\rm
VH},\boldsymbol{q}}\right|},\frac{T}{\left|\varepsilon_{{\rm
VH},\boldsymbol{q}}\right|}\right),$ (12)

with scaling function

$\displaystyle P\left(x,y\right)$ $\displaystyle=$
$\displaystyle\sqrt{y}\frac{m}{4\sqrt{\pi}}\left(Li_{1/2}\left(-e^{\frac{1}{4y}\left(2x-1\right)^{2}+\frac{2x}{y}}\right)\right.$
(13) $\displaystyle-$
$\displaystyle\left.Li_{1/2}\left(-e^{\frac{1}{4y}\left(2x-1\right)^{2}}\right)\right)$

and polylogarithmic function $Li_{s}\left(z\right)$. As we show in Fig. 3, the
finite temperature density response can be expressed to a good approximation
in the form Eq.(11) but with

$\left|\varepsilon_{{\rm
VH},\boldsymbol{q}}\right|\rightarrow\Omega_{\boldsymbol{q}}\left(T\right)=\sqrt{\varepsilon_{{\rm
VH},\boldsymbol{q}}^{2}+\left(k_{B}T\right)^{2}}.$ (14)

The corresponding real part of the density response yields
$\Pi\left(\boldsymbol{q},0\right)=-\frac{m}{\pi^{2}}\log\frac{D}{\Omega_{\boldsymbol{q}}\left(T\right)}$
which leads to Eq. (1) for the compressibility.

Figure 3: Scaling function $P(x,y)$ of Eq. (13) (solid lines) determining the
frequency and temperature dependence of the charge excitation spectrum near a
Van Hove point as function of $x=\omega/\left|\varepsilon_{{\rm
VH},\boldsymbol{q}}\right|$, in comparison with the simplified version of Eq.
(11) together with the substitution Eq. (14) (dashed lines) to include thermal
effects. We used $m=1$ and show results for $y=0.1$ (red) $y=5$ (green), and
$y=10$ (blue), where $y=T/\left|\varepsilon_{{\rm VH},\boldsymbol{q}}\right|$.

An important result that follows from this continuum of density excitations is
an anomalous single particle scattering rateGopalan1992 ; Pattnaik1992 . To
see this we analyze the imaginary part of the single-particle self energy
coupled to the above density fluctuations:

$\displaystyle{\rm Im}\Sigma\left(\boldsymbol{k},\omega\right)$
$\displaystyle=$ $\displaystyle
2\frac{U^{2}}{N}\sum_{\boldsymbol{k}^{\prime}}\left(f_{0}\left(\varepsilon\left(\boldsymbol{k}^{\prime}\right)\right)+n_{0}\left(\varepsilon\left(\boldsymbol{k}^{\prime}\right)-\omega\right)\right)$
(15) $\displaystyle\times$ $\displaystyle{\rm
Im}\Pi\left(\boldsymbol{k}-\boldsymbol{k}^{\prime},\omega-\varepsilon\left(\boldsymbol{k}^{\prime}\right)\right),$

where $U$ is the electron-electron interaction and $f_{0}$ and $n_{0}$ are the
Fermi and Bose distribution functions, respectively. While the momentum
transfer $\boldsymbol{k}-\boldsymbol{k}^{\prime}$ is small, the individual
Fermi momenta $\boldsymbol{k}$ and $\boldsymbol{k}^{\prime}$ do not
necessarily have to be located in the vicinity of the Van Hove point. Indeed,
the result ${\rm
Im}\Sigma\left(\boldsymbol{k},\omega\right)\propto\left|\omega\right|^{\gamma}$
for the single particle self energy depends sensitively whether
$\boldsymbol{k}$ and $\boldsymbol{k}^{\prime}$ are near or away from the
saddle point of the dispersion. In the former case the dispersion
$\varepsilon\left(\boldsymbol{k}\right)$ is given by Eq.(10). In this case
follows $\gamma=1$ which is the result obtained in Ref.Pattnaik1992 .
Alternatively we can analyze the self energy for generic momenta. Now it is
sufficient to assume a parabolic spectrum for
$\varepsilon\left(\boldsymbol{k}\right)$, which yields $\gamma=3/2$Gopalan1992
. A third option for momenta with parabolic dispersion and Fermi velocity
parallel to the directions of zeros of Eq.(10) yields a somewhat more singular
behavior with $\gamma=4/3$. We will show that the behavior with $\gamma=3/2$
is the one that determines the thermal conductivity. Further details of the
analysis of the single particle self energy are summarized in Appendix C.

##  III Electrical Resistivity

To calculate the resistivity we use the standard Boltzmann ansatz

$\frac{\partial f_{\boldsymbol{k}}}{\partial
t}+e\boldsymbol{E}\cdot\frac{\partial
f_{\boldsymbol{k}}}{\partial\boldsymbol{k}}=-{\cal
C}_{\boldsymbol{k}}\left[f\right]$ (16)

with scattering operator ${\cal C}_{\boldsymbol{k}}\left[f\right]$ and
determine the charge current

$\boldsymbol{j}=-\frac{e}{N}\sum_{\boldsymbol{k}\sigma}\boldsymbol{v}_{\boldsymbol{k}}f_{\boldsymbol{k}}$
(17)

at given electrical field. Next we expand $f_{\boldsymbol{k}}$ for small
deviations from equilibrium, parametrized by a function $\psi_{\bm{k}}$
(proportional to the electric field $\bm{E}$):

$f_{\boldsymbol{k}}=f_{0}\left(\varepsilon_{\boldsymbol{k}}\right)-T\frac{\partial
f_{0}\left(\varepsilon_{\boldsymbol{k}}\right)}{\partial\varepsilon_{\boldsymbol{k}}}\psi_{\boldsymbol{k}}.$
(18)

The linearized collision operator due to electron-electron scattering up to
second order in $U$ takes the usual form

$\displaystyle{\cal C}_{\boldsymbol{k}_{1}}\left[\psi\right]$ $\displaystyle=$
$\displaystyle\frac{2\pi}{\hbar}U^{2}\sum_{\boldsymbol{k}_{2}\boldsymbol{k}_{3}\boldsymbol{k}_{4}}\delta\left(\varepsilon_{\boldsymbol{k}_{1}}+\varepsilon_{\boldsymbol{k}_{2}}-\varepsilon_{\boldsymbol{k}_{3}}-\varepsilon_{\boldsymbol{k}_{4}}\right)$
(19) $\displaystyle\times$ $\displaystyle
L_{\varepsilon_{\boldsymbol{k}_{1}},\varepsilon_{\boldsymbol{k}_{2}},\varepsilon_{\boldsymbol{k}_{3}},\varepsilon_{\boldsymbol{k}_{4}}}\sum_{\boldsymbol{G}}\delta_{\boldsymbol{k}_{1}+\boldsymbol{k}_{2}-\boldsymbol{k}_{3}-\boldsymbol{k}_{4}-\boldsymbol{G}}$
$\displaystyle\times$
$\displaystyle\left(\psi_{\boldsymbol{k}_{1}}+\psi_{\boldsymbol{k}_{2}}-\psi_{\boldsymbol{k}_{3}}-\psi_{\boldsymbol{k}_{4}}\right).$

The sum over $\boldsymbol{G}$ goes over the entire reciprocal lattice. In
practice only a few terms contribute since four vectors of the first Brilouin
zone only reach a finite number of reciprocal lattice vectors. For notational
simplicity we suppressed the band index in the collision integral and assumed
the same interaction matrix element $U$ for intra- and inter-band scattering.
Below, we will discuss in detail which bands are included in the analysis. The
phase space restrictions of degenerate fermions are included through

$L_{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}}=f_{0}\left(-\varepsilon_{1}\right)f_{0}\left(-\varepsilon_{2}\right)f_{0}\left(\varepsilon_{3}\right)f_{0}\left(\varepsilon_{4}\right).$
(20)

Obviously $\psi_{\boldsymbol{k}}\propto\rm{const.}$ is a zero mode of the
collision operator due to charge conservation. Similarly,
$\psi_{\boldsymbol{k}}\propto\varepsilon_{\boldsymbol{k}}$ is a zero mode due
to energy conservation. However, $\psi_{\boldsymbol{k}}\propto k_{\alpha}$ is
not a zero mode as the momenta do not have to add up to zero. Umklapp
scattering processes spoil momentum conservation and yield a finite
resistivity.

When analyzing the resistivity we make the simplifying assumption that

$\psi_{\boldsymbol{k}}\approx\psi_{0}e\boldsymbol{E}\cdot\boldsymbol{v}_{\boldsymbol{k}}$
(21)

is determined by the velocity
$\boldsymbol{v}_{\boldsymbol{k}}=\frac{\partial\varepsilon_{\boldsymbol{k}}}{\partial\boldsymbol{k}}$.
This assumption is only justified if a given scattering process is
kinematically allowed everywhere on the Fermi surface. Otherwise the
coefficient $\psi_{0}$ will depend strongly on the position on the Fermi
surface, effects that occur as corrections of the current vertex in a
diagrammatic treatment of transport. As a result the longest scattering rate
that is allowed on the entire Fermi surface dominates the transport behavior;
it short circuits stronger scattering events that are kinematically only
allowed on a subset of the Fermi surface. While this hot spot reasoning is
well establishedHlubina1995 ; Hlubina1996 ; Stojkovic1996 ; Rosch1999 ;
Syzranov2012 ; Mousatov2020 , it appears at odds with the expectation based on
Matthiessen’s rule Matthiessen1864  where one has to add scattering rates, not
times; see Appendix A. With the ansatz Eq. (21) we finally arrive at the
following result for the low-frequency ( $\omega<\tau^{-1}_{J}$) conductivity

$\sigma\left(\omega\right)=\frac{n_{D}}{-i\omega+\tau_{J}^{-1}},$ (22)

with Drude weight
$n_{D}=\frac{e^{2}}{N}\sum_{\boldsymbol{k}\sigma}\frac{\partial^{2}\varepsilon_{\boldsymbol{k}}}{\partial
k_{\alpha}\partial k_{\beta}}f_{0}\left(\varepsilon_{\boldsymbol{k}}\right)$
and scattering rate for charge transport

$\displaystyle\tau_{J}^{-1}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{\hbar}\frac{U^{2}e^{2}}{4n_{D}k_{{\rm
B}}T}\sum_{\boldsymbol{k}_{1}\cdots\boldsymbol{k}_{4}}\delta\left(\varepsilon_{\boldsymbol{k}_{1}}+\varepsilon_{\boldsymbol{k}_{2}}-\varepsilon_{\boldsymbol{k}_{3}}-\varepsilon_{\boldsymbol{k}_{4}}\right)$
(23) $\displaystyle\times$ $\displaystyle
L_{\varepsilon_{\boldsymbol{k}_{1}},\varepsilon_{\boldsymbol{k}_{2}},\varepsilon_{\boldsymbol{k}_{3}},\varepsilon_{\boldsymbol{k}_{4}}}\sum_{\boldsymbol{G}}\delta_{\boldsymbol{k}_{1}+\boldsymbol{k}_{2}-\boldsymbol{k}_{3}-\boldsymbol{k}_{4}-\boldsymbol{G}}$
$\displaystyle\times$
$\displaystyle\left(v_{\boldsymbol{k}_{1}}+v_{\boldsymbol{k}_{2}}-v_{\boldsymbol{k}_{3}}-v_{\boldsymbol{k}_{4}}\right)^{2}.$

Due to the velocity term, momentum conserving scattering processes vanish in
the single-band case and, as expected, do not contribute to the resistivity.
For multiple bands momentum-conserving terms can, of course, change the
current.

We perform the momentum summation according to

$\frac{1}{N}\sum_{\boldsymbol{k}}F\left(\boldsymbol{k}\right)\approx\sum_{S}\int
d\epsilon\rho_{S}\left(\epsilon\right)\int_{S}\frac{d\varphi}{2\pi}F\left(\epsilon,\varphi\right).$
(24)

In the last step we sum over segments of the Fermi surface with essentially
constant density of state $\rho_{S}\left(\epsilon\right)\approx\rho_{F}$ or
with logarithmic density of state
$\rho_{S}\left(\epsilon\right)\approx\rho_{F}\log\left(D/|\epsilon|\right)$.
Finally we use

$\displaystyle I$ $\displaystyle=$ $\displaystyle\int d\epsilon_{1}\cdots
d\epsilon_{4}\delta\left(\epsilon_{1}+\epsilon_{2}-\epsilon_{3}-\epsilon_{4}\right)L_{\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4}}$
(25) $\displaystyle\times$
$\displaystyle\prod_{i=1}^{4}\rho_{S_{i}}\left(\epsilon_{i}\right)\approx\frac{2\pi^{3}}{3}\left(k_{B}T\right)^{3}\rho_{F}^{4}\log^{n}\frac{D}{T},$

where $n=0\cdots 4$ is the number of electrons near a Van Hove point that are
involved in a given scattering process. For example, for a $cc\rightarrow ch$
process, $n=1$.

Depending on the number of such hot electrons involved, distinct scattering
processes contribute differently to the resistivity. At first glance, the
contribution of a scattering process with $n$ hot electrons seems to be
$\rho^{(n)}\propto T^{2}\log^{n}\left(1/T\right)$, which would be dominated by
$\rho^{(4)}$ at low $T$, i.e. by the largest value of $n$. However, as
discussed above, we must i) ensure the existence of umklapp scattering for
given $n$, ii) check that those processes exist on the entire Fermi surface,
and iii) ensure that some of the states involved have a finite velocity,
needed to carry the current. We will see that this yields $n=0$ for a single
band problem, while $n=1$ if one includes additional bands that cross the
Fermi energy. Hence, including multi-band scattering events, the final result
for the electrical resistivity will be

$\rho(T)=A_{J}T^{2}\log\left(D/T\right),$ (26)

with $A_{J}\sim U^{2}\rho_{F}^{3}/n_{D}$. Eq.(22) also includes the low
frequency Drude response of the system where $\tau_{J}$ corresponds to the
zero frequency limit of the charge transport rate. In Appendix D we also
discuss $\tau_{J}^{-1}(\omega)$ in the opposite limit
$\tau_{J}^{-1}(0)\ll\omega\ll D$ within a memory function approach and obtain

$\tau_{J}^{-1}(\omega)\sim\omega^{2}\log\left(D/|\omega|\right),$ (27)

which determines the optical conductivity at these intermediate frequencies

$\sigma\left(\omega\right)=\frac{n_{D}}{-i\omega\left(1+\lambda\left(\omega\right)\right)+\tau_{J}^{-1}\left(\omega\right)}.$
(28)

Here, $\lambda\left(\omega\right)=\lambda_{0}\left(\log
2-\frac{\omega}{D}\arctan\left(\frac{D}{\omega}\right)\right)$ is the optical
mass enhancement that follows from Kramers-Kronig transformation. This
frequency dependency parallels the temperature dependency of the optical
scattering rate at low frequencies.

For a problem with two Van Hove points at $\boldsymbol{k}^{(1)}_{\rm
VH}=(\pi,0)$ and $\boldsymbol{k}^{(2)}_{\rm VH}=(0,\pi)$, i.e. without the
problems that occur in the single-band model relevant to Sr2CuO4 Eq.(26) was
already obtained in Ref.Hlubina1996 . In what follows we discuss the
kinematics of the single-band and multi-band problems separately.

Figure 4: A $\mathrm{cc\rightarrow cc}$ process, where two electrons scatter
from two cold initial states (purple) into two cold final states (orange) with
umklapp scattering, leads to a resistivity of $\rho\propto T^{2}$. The blue
arrows indicate the momentum transfer for each electron. The
$\mathrm{cc\rightarrow ch}$ process contributes by $\rho\propto
T^{2}\log\left(1/T\right)$ and dominates, if every point on the Fermi surface
participates in this scattering. The $\mathrm{ch\rightarrow ch}$ processes
always conserve momentum, therefore they do not contribute to the resistivity.

###  III.1 Resistivity for the $\gamma$-band only

As we have seen, electron-electron scatterings yield different contributions
to the resistivity, depending on the number of hot electrons involved. Which
of these scattering processes are possible depends on the geometry of the
Fermi surface. For simplicity, we start from a simplified model that includes
only the $\gamma$ band. In principle, the Fermi surface allows scattering
processes with $n=0,1,2$ and $4$ hot electrons. Of these, only
$cc\leftrightarrow cc$ (i.e. $n=0$) and $cc\leftrightarrow ch$ ($n=1$)
contribute to the resistivity as they allow umklapp processes. Fig. 4 shows
the possible scatterings schematically for zero, one and two hot electrons.

The $cc\leftrightarrow ch$ scattering dominates the resistivity if every point
on the cold Fermi surface can participate in such a scattering process. These
processes have to be umklapp processes in order to contribute to the
resistivity. It turns out that this is not possible for the entire Fermi
surface. For an electron close to the Van Hove point the momentum transfer is
not sufficiently large to surpass the gap between two neighboring Fermi
surfaces. For the tight-binding parameters relevant for Sr2RuO4, we obtain
that more than $10\%$ of the Fermi surface cannot participate in
$cc\leftrightarrow ch$ scattering processes. This implies that our assumption
of Eq. (21) for the distribution function is not justified.

States on the Fermi surface relatively close to the Van Hove point cannot
participate in $cc\rightarrow ch$ umklapp scattering and should at
sufficiently low temperatures lead to a resistivity that behaves as
$\rho_{cc\rightarrow cc}\propto T^{2}$. One can qualitatively understand the
crossover at higher temperature by considering two competing contributions to
the resistivity with $\rho_{cc\rightarrow cc}$ and $\rho_{cc\rightarrow
ch}\propto T^{2}\log\left(1/T\right)$ that add up to the total resistivity

$\rho_{ee}^{-1}\sim\rho_{cc\rightarrow cc}^{-1}+\rho_{cc\rightarrow ch}^{-1}.$
(29)

At lowest $T$ the resistivity exhibits the usual relation $\rho\propto T^{2}$,
while at higher temperatures the $cc\rightarrow ch$ process dominates and the
resistivity obtains the logarithmic correction $\rho\propto
T^{2}\log\left(1/T\right)$. The value of the crossover temperature can then be
estimates as $T^{\ast}\sim D\exp\left(-c\frac{1-x}{x}\right)$ where $x$ is the
fraction of the Fermi surface where $\mathrm{cc\leftrightarrow ch}$ scattering
processes are kinematically forbidden and $c$ is a numerical coefficient of
order unity.

###  III.2 Resistivity with inter-band scattering

So far we have restricted our analysis to the $\gamma$ band. However, to
explain the transport properties of $\mathrm{Sr_{2}RuO_{4}}$, one has to take
the whole Fermi surface into account. Since current is not conserved in inter-
band scattering, no matter if it is umklapp or not, these scattering events
contribute to the resistivity. Since the $\beta$ band is convex and only
includes cold regions (called $\mathrm{c_{\beta}}$), we can always find a
$\mathrm{c_{\beta}\rightarrow c_{\beta}}$ process, as long as the momentum
transfer is not too large. These processes fill the kinematic gap of the
single-band case such that now the entire $\gamma$-band can participate in
$\mathrm{c_{\beta}c_{\gamma}\rightarrow c_{\beta}h_{\gamma}}$ scattering,
either by umklapp or inter band scattering, as shown in Fig. 5. A similar
argumentation can be made for the $\alpha$ band. Hence, additional cold states
on the Fermi surface that can couple via inter-band collisions open up the
phase space for singular scattering processes. This yields the somewhat
surprising result that more cold states suppress the ability of these states
to short circuit the singular transport processes that involve Van Hove
points. Since no new forbidden areas in the $\alpha$ and $\beta$ bands emerge,
the resistivity follows Eq.(26) down to lowest temperatures.

Figure 5: When all bands are considered, every point on the cold Fermi surface
can participate in $\mathrm{cc\rightarrow ch}$ scattering, either by umklapp
or inter band scattering.

##  IV Thermal transport

Next, we analyze the thermal transport for a system at the Lifshitz point. For
the thermal transport we start again from the Boltzmann equation

$\frac{\partial f_{\boldsymbol{k}}}{\partial
t}+e\boldsymbol{v}_{\boldsymbol{k}}\cdot\frac{\partial
f_{\boldsymbol{k}}}{\partial\boldsymbol{r}}=-{\cal
C}_{\boldsymbol{k}}\left[f\right]$ (30)

with the same collision operator given in Eq.(19) for the electrical
resistivity. The heat current

$\boldsymbol{j}_{Q}=\frac{1}{N}\sum_{\boldsymbol{k}\sigma}\left(\varepsilon_{\boldsymbol{k}}-\mu\right)\boldsymbol{v}_{\boldsymbol{k}}f_{\boldsymbol{k}}$
(31)

is then determined as function of the temperature gradient, which enters the
Boltzmann equation through

$\frac{\partial
f_{\boldsymbol{k}}}{\partial\boldsymbol{r}}\approx-\frac{\partial
f_{0}\left(\varepsilon_{\boldsymbol{k}}\right)}{\partial\varepsilon_{\boldsymbol{k}}}\,\frac{(\varepsilon_{\boldsymbol{k}}-\mu)}{T}\frac{\partial
T}{\partial\boldsymbol{r}}\,.$ (32)

The crucial difference is of course that the momentum and the thermal current
do not couple, i.e. there is no need to analyze whether umklapp scattering
processes existLawrence1973 ; Schulz1995 ; Hartnoll2013 ; Principi2015 ;
Hartnoll2016 ; comment_thermal . Hence, small momentum transfer collisions,
where
$\boldsymbol{q}=\boldsymbol{k}_{3}-\boldsymbol{k}_{1}=\boldsymbol{k}_{2}-\boldsymbol{k}_{4}$
is small, contribute to the thermal resistivity. Of course, the two incoming
momenta $\boldsymbol{k}_{1}$ and $\boldsymbol{k}_{2}$ can be far from each
other, an issue that will be crucial when we determine the correct temperature
dependence of the thermal conductivity.

We hence ignore umklapp processes in the thermal transport and perform one
momentum sum using momentum conservation for the reciprocal lattice vector
$\boldsymbol{G}=\boldsymbol{0}$. This yields for time-independent thermal
gradients

$\displaystyle\boldsymbol{v}_{\boldsymbol{k}_{1}}\cdot\frac{\partial
f_{\boldsymbol{k}_{1}}}{\partial\boldsymbol{r}}$ $\displaystyle=$
$\displaystyle\frac{2\pi
U^{2}}{\hbar}\sum_{\boldsymbol{k}_{2}\boldsymbol{q}}L_{\varepsilon_{\boldsymbol{k}_{1}},\varepsilon_{\boldsymbol{k}_{2}},\varepsilon_{\boldsymbol{k}_{1}+\boldsymbol{q}},\varepsilon_{\boldsymbol{k}_{2}-\boldsymbol{q}}}$
(33) $\displaystyle\times$
$\displaystyle\delta\left(\varepsilon_{\boldsymbol{k}_{1}}+\varepsilon_{\boldsymbol{k}_{2}}-\varepsilon_{\boldsymbol{k}_{1}+\boldsymbol{q}}-\varepsilon_{\boldsymbol{k}_{2}-\boldsymbol{q}}\right)$
$\displaystyle\times$
$\displaystyle\left(\psi_{\boldsymbol{k}_{1}}+\psi_{\boldsymbol{k}_{2}}-\psi_{\boldsymbol{k}_{1}+\boldsymbol{q}}-\psi_{\boldsymbol{k}_{2}-\boldsymbol{q}}\right)\,.$

We consider a generic momentum $\boldsymbol{k}_{1}$ on the Fermi surface. For
small momentum transfer $\boldsymbol{k}_{1}+\boldsymbol{q}$ will then be near
$\boldsymbol{k}_{1}$. For the other incoming momentum $\boldsymbol{k}_{2}$ we
can, however, assume that it is located in the vicinity of the Van Hove point,
which implies that $\boldsymbol{k}_{2}-\boldsymbol{q}$ is also near that
point. Following our analysis for the charge transport, one might be tempted
to conclude that the resistivity has $n=2$ states near the Van Hove point,
which would imply a scattering rate that behaves as
$T^{2}\log^{2}\left(1/T\right)$. However, as we saw in the analysis of the
density fluctuation spectrum in Eqs.(11) - (13), small momentum transfer
processes have a much increased phase space which gives rise to a more
singular behavior for the thermal transport.

To proceed we first make the ansatz for the distribution function

$\psi_{\boldsymbol{k}}=\psi_{0}\frac{\varepsilon_{k}-\mu}{T}\boldsymbol{v}_{\boldsymbol{k}}\cdot\nabla
T$ (34)

which is analog to Eq.(21) for thermal transport. As before, we have to
demonstrate that a given scattering process is present everywhere on the Fermi
surface in order to justify the assumption that the coefficient $\psi_{0}$ is
weakly dependent on momentum. With Eq. (34) and given the vanishing velocity
at the Van Hove point, we can safely neglect $\psi_{\boldsymbol{k}_{2,4}}$ in
Eq. (33). If we use the identity
$f_{0}\left(\varepsilon\right)f_{0}\left(-\varepsilon^{\prime}\right)=\left(f_{0}\left(\varepsilon\right)-f_{0}\left(\varepsilon^{\prime}\right)\right)n_{0}\left(\omega\right)$
for $\varepsilon^{\prime}=\varepsilon+\omega$ we can express the sum over
$\boldsymbol{k}_{2}$, which runs over states near the Van Hove point, in terms
of the density spectrum of Eq. (9)

$\displaystyle\boldsymbol{v}_{\boldsymbol{k}_{1}}\cdot\frac{\partial
f_{\boldsymbol{k}_{1}}}{\partial\boldsymbol{r}}$ $\displaystyle=$
$\displaystyle-\frac{2\pi U^{2}T}{\hbar N}\frac{\partial
f_{0}\left(\varepsilon_{\boldsymbol{k}_{1}}\right)}{\partial\varepsilon_{\boldsymbol{k}_{1}}}\sum_{\boldsymbol{q}}\left(\psi_{\boldsymbol{k}_{1}}-\psi_{\boldsymbol{k}_{1}-\boldsymbol{q}}\right)$
(35) $\displaystyle\times$
$\displaystyle\left(n_{0}\left(\varepsilon_{\boldsymbol{k}_{1}-\boldsymbol{q}}-\varepsilon_{\boldsymbol{k}_{1}}\right)+f_{0}\left(\varepsilon_{\boldsymbol{k}_{1}-\boldsymbol{q}}\right)\right)$
$\displaystyle\times$ $\displaystyle{\rm
Im}\Pi\left(\boldsymbol{q},\varepsilon_{\boldsymbol{k}_{1}}-\varepsilon_{\boldsymbol{k}_{1}-\boldsymbol{q}}\right).$

Hence, we explicitly see how the coupling to compressive fluctuations affects
the thermal conductivity. One now finds that, up to numerical factors of order
unity, the collision operator is merely determined by the single-particle self
energy of Eq. (15).

$\boldsymbol{v}_{\boldsymbol{k}_{1}}\cdot\frac{\partial
f_{\boldsymbol{k}_{1}}}{\partial\boldsymbol{r}}=\tau_{\boldsymbol{k}_{1}Q}^{-1}T\frac{\partial
f_{0}\left(\varepsilon_{\boldsymbol{k}_{1}}\right)}{\partial\varepsilon_{\boldsymbol{k}_{1}}}\psi_{\boldsymbol{k}_{1}}$
(36)

with the heat transport scattering rate

Figure 6: ch$\rightarrow$ch process dominating the thermal transport with two
hot momenta near the Van Hove point (red) and two cold electrons with generic
dispersion (blue). For small momentum transfer, this process is allowed
everywhere on the Fermi surface. Figure 7: Self energy diagram determining the
thermal resistivity with hot momenta $\mathbf{k}_{2}$ and
$\mathbf{k}_{2}-\mathbf{q}$ with hyperbolic dispersion (red) and cold momenta
$\mathbf{k}_{1}$ and $\mathbf{k}_{1}+\mathbf{q}$ with parabolic dispersion
(blue).

$\tau_{\boldsymbol{k}_{1}Q}^{-1}\approx-2\mathrm{Im}\Sigma\left(\boldsymbol{k}_{1},\varepsilon_{{}_{\boldsymbol{k}_{1}}}\right).$
(37)

This process is kinematically allowed everywhere on the Fermi surface, and
amounts to an analysis of the self energy diagram shown in Fig. 7. In Appendix
C we summarize the analysis of the self energy, see also Refs.Gopalan1992 ;
Pattnaik1992 . While there exists more singular scattering in some parts of
the Fermi surface, a generic point obeys

$\tau_{\boldsymbol{k},Q}^{-1}=\frac{16U^{2}\rho_{F}^{2}}{3\sqrt{D}}T^{3/2}.$
(38)

This yields for the thermal conductivity

$\kappa=\frac{1}{TN}\sum_{\boldsymbol{k}\sigma}v_{\boldsymbol{k}}^{2}\tau_{\boldsymbol{k},Q}\left(-\frac{\partial
f_{0}\left(\varepsilon_{\boldsymbol{k}}\right)}{\partial\varepsilon_{\boldsymbol{k}}}\right)\left(\varepsilon_{\boldsymbol{k}}-\mu\right)^{2}.$
(39)

Performing the integrals in the usual manner yields for the thermal
resistivity introduced in Eq. (2) the result

$\rho_{Q}=A_{Q}D^{1/2}T^{3/2},$ (40)

where the coefficient $A_{Q}$ is of the same order of magnitude as $A_{J}$
that determines the charge transport in Eq. (26). This finally yields the
$T$-dependence of the Lorentz number given in Eq. (3) and the concomitant
violation of the Wiedemann Franz law.

##  V Conclusions

In conclusion, we analyzed the electrical and thermal transport of clean
Sr2RuO4 under strain near the Lifshitz point where a Van Hove singularity of
the $\gamma$-sheet of the Fermi surface crosses the Fermi energy. Based on the
observation of well defined quasiparticles in this material we perform a
quasi-classical Boltzmann transport theory. For larger frequencies, we also
present results for the optical conductivity within a memory function
approach. We find that both, the electrical and the thermal transport are
affected by the vicinity to the Lifshitz point. The known result for the
electrical resistivity, discussed already in Ref.Hlubina1996 , is physically
interpreted in terms of scattering processes where an electron near the Van
Hove point collides with a _cold_ electron away from the Van Hove point and
scatters into two other cold states. This gives rise to a logarithmic
enhancement of the charge transport rate $\tau_{J}^{-1}\sim
T^{2}\log\left(1/T\right)$ and hence of the electrical resistivity, consistent
with observations of Refs.Kikugawa2004 ; Shen2007 ; Burganov2016 ; Barber2018
. In particular, the observation by Barber et al.Barber2018 , with a very
small residual resistivity $\rho_{0}$, demonstrate that an understanding of
these results has to be achieved without resorting to impurity scattering
processes that usually increase the phase space for singular electron-electron
scatteringRosch1999 . For the specific electronic structure of Sr2RuO4 we show
that the logarithmic enhancement is present down to lowest temperatures if one
includes inter-band scattering processes. The reason is that intra-band
scattering requires umklapp processes which have more stringent phase space
requirements. Hence, the anomalous transport behavior seen in Sr2RuO4 is
particularly robust as there are several bands that cross the Fermi surface.
The situation is drastically different if one analyses thermal transport.
Systems with Van Hove singularities are characterized by a much enhanced phase
space of long-wavelength compressive modes. These compressive modes do not
couple to charge transport. However, they are able to relax the thermal
current and give rise to a thermal transport rate $\tau_{Q}^{-1}\propto
T^{3/2}$. As a consequence the Wiedemann-Franz law is violated with a Lorentz
number that vanishes with a powerlaw plus logarithmic corrections. The
experimental confirmation of this prediction would in particular demonstrate
the importance of the mentioned compressive modes for the low-energy
excitations of Sr2RuO4. This would also be of interest as it would be curious
to study whether these long-wavelength modes might enhance an existing or even
cause an instability towards a superconducting state. A related issue is
clearly the role of higher-order processes not included in our analysis. The
fact that the compressibility is guaranteed to vanish at a finite temperature
is clear evidence that the system will eventually undergo some kind of
instability. Our transport theory is however expected to be valid in the
regime that leads up to this instability.

Our prediction for a violation of the Wiedemann Franz law is only valid as
long as one is in the clean regime. At some low temperature, impurity
scattering effects should become important. Then we expect to recover at
lowest temperatures the Wiedemann-Franz law where $L$ approaches
$L_{0}=\frac{\pi^{2}}{3}\left(k_{B}/e\right)^{2}$. For the leading low-
temperature corrections, still dominated by impurity scattering events, one
expects $\tau_{Q,J}^{-1}=\tau_{{\rm imp}}^{-1}+C_{Q,J}T^{3/2}$ i.e. both
transport rates should follow the same temperature dependence. This behavior
could be relevant for the observations of La-substituted systemsShen2007 ;
Burganov2016 . Again, it’s confirmation would give strong evidence for the
importance of compressive modes for the low-energy electronic degrees of
freedom.

###### Acknowledgements.

We are grateful to M. Garst, S. A. Hartnoll, C. Hicks, and A. P. MacKenzie for
useful discussions. V. C. S. and J. S. were supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - TRR 288-422213477
Elasto-Q-Mat (project B01). E. B. was supported by the European Research
Council (ERC) under grant HQMAT (Grant Agreement No. 817799), the Minerva
foundation, and a Research grant from Irving and Cherna Moskowitz.

##  Appendix A Matthiesen’s rule and hot versus cold carriers

According to Matthiesen’s rule, the total resistivity is the sum of
resistivities of different scattering mechanisms, $\rho_{{\rm tot}}=\rho_{{\rm
imp}}+\rho_{{\rm e-ph}}+\rho_{{\rm e-e}}$, meaning that the dominant
contribution to the resistivity comes from the shortest life time. This seems
in sharp contrast to what was discussed in Eq.(29) and to our strategy to
ensure that a given scattering mode is not short circuited by scattering
processes on the Fermi surface with smaller rate. To clarify this issue we
consider the linearized Boltzmann equation in the operator form

$\hat{C}\left|\psi\right\rangle=\left|S\right\rangle,$ (41)

where $\hat{C}$ is the collision operator,
$\psi_{\boldsymbol{k}}=\left\langle\boldsymbol{k}\mid\psi\right\rangle$ the
correction to the distribution function and
$S_{\boldsymbol{k}}=\left\langle\boldsymbol{k}\mid S\right\rangle$ some
external source term. The source term due to an external electric field is
$S_{\boldsymbol{k}}\equiv-\frac{e}{T}\boldsymbol{E}\left(\boldsymbol{q},\omega\right)\cdot\boldsymbol{v}_{\boldsymbol{k}}$.
With the collision integral ${\cal C}_{\boldsymbol{k}}$ that we used in the
main text, the operator is defined as

$\hat{C}\psi_{\boldsymbol{k}}=\frac{1}{-T\frac{\partial
f^{\left(0\right)}\left(\varepsilon_{\boldsymbol{k}}\right)}{\partial\varepsilon_{\boldsymbol{k}}}}\int_{\boldsymbol{k}^{\prime}}\frac{\delta{\cal
C}_{\boldsymbol{k}}}{\delta\psi_{\boldsymbol{k}^{\prime}}}\psi_{\boldsymbol{k}^{\prime}}.$
(42)

It is a Hermitian operator with respect to the inner product

$\left\langle\phi\mid\psi\right\rangle=\int_{\boldsymbol{k}}w_{\boldsymbol{k}}\phi_{\boldsymbol{k}}^{*}\psi_{\boldsymbol{k}}$
(43)

with weight function $w$${}_{\boldsymbol{k}}=-T\frac{\partial
f^{\left(0\right)}\left(\varepsilon_{\boldsymbol{k}}\right)}{\partial\varepsilon_{\boldsymbol{k}}}>0$.
The entropy production can then we written as $\partial S/\partial
t=\left\langle\psi\left|\hat{C}\right|\psi\right\rangle$, i.e. the operator is
positive definite. The eigenvalues of the collision operator are the
scattering rates

$\hat{C}=\sum_{\lambda}\left|\lambda\right\rangle\tau_{\lambda}^{-1}\left\langle\lambda\right|,$
(44)

where $\lambda$ labels distinct modes of the scattering process. For a
rotation-invariant problem they could for example describe different angular
momentum modes.

If the system is governed by several, distinct scattering mechanism, such as
impurity, electron-phonon, or electron-electron scattering, the individual
operators add up

$\hat{C}=\hat{C}_{{\rm imp}}+\hat{C}_{{\rm el-ph}}+\hat{C}_{{\rm el-el}}.$

We write this as $\tau_{\lambda}^{-1}=\sum_{s}\tau_{\lambda,s}^{-1}$ where the
index $s$ indicates the scattering mechanisms. In order to determine the
distribution function we now have to invert the collision operator in the
subspace orthogonal to zero modes that indicate conserved quantities such as
particle number or energy. This is possible as long as these zero modes are
orthogonal to the source term $\left|S\right\rangle$. It follows

$\displaystyle\left|\psi\right\rangle$ $\displaystyle=$
$\displaystyle\hat{C}^{-1}\left|S\right\rangle$ (45) $\displaystyle=$
$\displaystyle\sum_{\lambda}^{\prime}\left|\lambda\right\rangle\frac{1}{\sum_{s}\tau_{\lambda,s}^{-1}}\left\langle\lambda\mid
S\right\rangle,$

where the prime in the sum indicates that the inversion has been performed in
the subspace orthogonal to the mentioned zero modes. In many instances, the
scattering rates for non-zero-modes depend weakly on the eigen-mode index
$\lambda$. Then one obtains the total scattering rate

$\tau_{{\rm tot}}^{-1}=\sum_{s}\tau_{s}^{-1}.$ (46)

This leads to Matthiesen’s rule. However, if one considers a system with one
dominant scattering mechanism, e.g. a clean system with electron-electron
scattering where phonons are frozen out at low temperatures, as we assume in
this paper, then

$\left|\psi\right\rangle=\sum_{\lambda}\left|\lambda\right\rangle\tau_{\lambda,{\rm
el-el}}\left\langle\lambda\mid S\right\rangle.$ (47)

For systems where distinct eigenmodes of the scattering process under
consideration vary strongly, as happens for strongly momentum dependent
scattering rates, the largest scattering time dominates and we obtain a
relation like given in Eq.(29).

To summarize, an electron that is exposed to different scattering mechanisms
must cope with all of them and the rates add according to Matthiesen’s rule;
different scattering mechanisms act like resistors in series. On the other
hand, if there are different modes of one dominant mechanism, the longest
scattering times dominate the conductivity; different modes act like resistors
that are in parallel where the weakest scattering short circuits the
transport.

##  Appendix B Tight binding parameters and strain dependence of the
electronic structure

We consider the electronic structure of Sr2RuO4 under arbitrary in-plane
strain. The Van Hove singularity near the Fermi energy is due to the so called
$\gamma$ band, made of $4d_{xy}$ states is:

$\displaystyle\varepsilon_{\boldsymbol{k},xy}$ $\displaystyle=$
$\displaystyle-2t_{1,x}\cos k_{x}-2t_{1,y}\cos k_{y}-\mu.$ $\displaystyle-$
$\displaystyle
2\left(t_{4,+}\cos\left(k_{x}+k_{y}\right)+t{}_{4,-}\cos\left(k_{x}-k_{y}\right)\right).$

It is natural to assume that for strained systems $t_{1,x}$ only depends on
the change of the nearest neighbor distance $a_{x}$, $t_{1,y}$ only on
$a_{y}$, while $t_{4,+}$ and $t_{4,-}$ are functions of the diagonal distance
$d_{+}$ and counter-diagonal distance $d_{-}$, respectively. For simplicity we
only consider in-plane strain. The nearest neighbor distances of the strained
system are $a_{\alpha}\approx a_{0}\left(1+\epsilon_{\alpha\alpha}\right)$ and
$d_{\pm}\approx\sqrt{2}a_{0}\left(1+\frac{1}{2}\left(\epsilon_{xx}+\epsilon_{yy}\pm
2\epsilon_{xy}\right)\right).$ Following Ref.Barber2018  it is reasonable to
use $t_{1,\alpha}=t_{1}\left(1-\alpha\epsilon_{\alpha\alpha}\right)$ and
$t_{4,\pm}/t_{4}=1-\frac{\alpha}{2}\left(\epsilon_{xx}+\epsilon_{yy}\pm
2\epsilon_{xy}\right).$ Assuming $\sigma_{xx}$ along the $x$ direction follows
$\epsilon_{xy}=0$ and $\epsilon_{yy}=-\nu_{xy}\epsilon_{xx}$ with Poisson
ratio $\nu_{xy}$ such that

$\displaystyle t_{1,x}$ $\displaystyle=$ $\displaystyle
t_{1}\left(1-\alpha\epsilon_{xx}\right),$ $\displaystyle t_{2,y}$
$\displaystyle=$ $\displaystyle
t_{2}\left(1+\alpha\nu_{xy}\epsilon_{xx}\right),$ $\displaystyle t_{4,\pm}$
$\displaystyle=$ $\displaystyle
t_{4}\left(1-\frac{\alpha}{2}\left(1-\nu_{xy}\right)\epsilon_{xx}\right).$
(49)

Using the elastic constants of Ref.Paglione2002  yields $\nu_{xy}=0.39$. From
Ref.Burganov2016  we take $t_{1}=0.119\,{\rm eV}$ and $t_{4}=0.41t_{1}$ as
well as $\mu=1.48t_{0}$.

The $4d_{xz}$ and $4d_{yz}$ orbitals form the so-called $\alpha$ and $\beta$
sheets of the Fermi surface. They are characterized by a dispersion

$h=\left(\begin{array}[]{cc}\varepsilon_{xz}&V\\\
V&\varepsilon_{yz}\end{array}\right)$ (50)

with

$\displaystyle\varepsilon_{\boldsymbol{k},xz}$ $\displaystyle=$
$\displaystyle\varepsilon_{x}^{\left(0\right)}-2t_{2,x}\cos k_{x}-2t_{3,y}\cos
k_{y}-\mu,$ (51) $\displaystyle\varepsilon_{\boldsymbol{k},yz}$
$\displaystyle=$ $\displaystyle\varepsilon_{y}^{\left(0\right)}-2t_{2,y}\cos
k_{y}-2t_{3,x}\cos k_{x}-\mu,$ $\displaystyle V_{\boldsymbol{k}}$
$\displaystyle=$
$\displaystyle-2t_{5,+}\cos\left(k_{x}+k_{y}\right)+2t_{5,-}\cos\left(k_{x}-k_{y}\right)$

with
$\text{$t_{2,3,\alpha}$}=t_{2,3}\left(1-\beta\epsilon_{\alpha\alpha}\right)$
and $t_{5,\pm}/t_{5}=1-\frac{\beta}{2}\left(\epsilon_{xx}+\epsilon_{yy}\pm
2\epsilon_{xy}\right)$. For the hopping elements of the unstrained system we
use the values given in Ref.Burganov2016 : $t_{2}=0.165\>{\rm eV},$
$t_{3}=0.08t_{2}$, and $t_{5}=0.13t_{2}$. Finally, for the unstrained system
holds
$\varepsilon_{x}^{\left(0\right)}-\mu=\varepsilon_{y}^{\left(0\right)}-\mu=-0.18t_{2}$.

##  Appendix C Single particle self energy

In this appendix we analyze the single particle self energy of Eq. (37) that
describes the scattering of electrons with momentum $\boldsymbol{k}_{1}$ with
density fluctuations described by Eqs. (11)-(13). We consider the imaginary
part of the self energy as given in Eq. (15).

We want to analyse a generic point on the Fermi surface, away from the Van
Hove singularity. Given the small momentum transfer we assume that
$\boldsymbol{k}$ and $\boldsymbol{k}^{\prime}$ are points nearby on the Fermi
surface of similar length $k_{F}$. Thus, we can safely assume a constant
density of states $\rho_{F}$ for these states. As follows from Eq.(12), the
momentum dependence of the density fluctuation spectrum is determined solely
by $\varepsilon_{{\rm VH},\boldsymbol{k}-\boldsymbol{k}^{\prime}}$ with saddle
point dispersion $\varepsilon_{{\rm VH},\boldsymbol{k}}$ of Eq.(10). If we use
$\boldsymbol{k}=k_{F}\left(\cos\theta,\sin\theta\right)$ and
$\boldsymbol{k}^{\prime}=k_{F}\left(\cos\phi,\sin\phi\right)$, we obtain for
small $\varphi=\theta-\phi$ that

$\varepsilon_{{\rm
VH},\boldsymbol{k}-\boldsymbol{k}^{\prime}}\approx-\varepsilon_{{\rm
VH},\boldsymbol{k}}\varphi^{2}+\frac{k_{x}k_{y}}{4m}\varphi^{3}.$ (52)

We first analyze the self energy at $T=0$ as function of frequency.
Considering without restriction $\omega>0$ the condition coming from the Fermi
and Bose functions is that $0<\epsilon_{c}\left(\boldsymbol{p}\right)<\omega.$
This yields

${\rm
Im}\Sigma\left(\boldsymbol{k},\omega\right)=2U^{2}\rho_{F}\int_{0}^{\omega}d\varepsilon_{c}\int_{0}^{2\pi}d\varphi{\rm
Im}\Pi\left(\boldsymbol{k}-\boldsymbol{p},\varepsilon_{c}\right).$ (53)

Let us first consider momenta $\boldsymbol{k}$ with $\varepsilon_{{\rm
VH},\boldsymbol{k}}\neq 0$, i.e. $\theta\neq\left(2l+1\right)\frac{\pi}{4}$.
With $s\equiv\left|\varepsilon_{{\rm
VH},\boldsymbol{k}}\right|\varphi^{2}\approx D\varphi^{2}$ follows

$\displaystyle{\rm Im}\Sigma\left(\boldsymbol{k},\omega\right)$
$\displaystyle=$
$\displaystyle-\frac{U^{2}\rho_{F}}{\sqrt{D}}\int_{0}^{\omega}d\epsilon\int_{0}^{\infty}\frac{ds}{\sqrt{s}}{\rm
Im}\Pi\left(\boldsymbol{k}-\boldsymbol{p},\epsilon\right)$ (54)
$\displaystyle=$
$\displaystyle-\frac{U^{2}\rho_{F}}{\sqrt{D}}\int_{0}^{\omega}d\epsilon\left(\int_{\epsilon}^{\infty}\frac{ds\epsilon}{s^{3/2}}+\int_{0}^{\epsilon}\frac{ds}{s^{1/2}}\right)$
$\displaystyle=$
$\displaystyle-\frac{8U^{2}\rho_{F}^{2}}{3\sqrt{D}}\left|\omega\right|^{3/2}.$

For the special momenta with $\theta=\left(2l+1\right)\frac{\pi}{4}$, where
$\varepsilon_{{\rm VH},\boldsymbol{k}}=0$ one should use $s\propto\varphi^{3}$
and it follows ${\rm
Im}\Sigma\left(\boldsymbol{k},\omega\right)\propto\left|\omega\right|^{4/3}$.
Those results were earlier obtained in Ref.Gopalan1992 . If, instead, the
dispersion $\varepsilon_{c}\left(\boldsymbol{k}\right)$ is assumed to be equal
to $\mathbf{\varepsilon_{{\rm VH},\boldsymbol{k}}}$ one finds ${\rm
Im}\Sigma\left(\boldsymbol{k},\omega\right)\propto\left|\omega\right|$. This
result was obtained in Ref.Pattnaik1992 .

At finite temperatures the analysis proceeds in an analogous manner and one
obtains

${\rm
Im}\Sigma\left(\boldsymbol{k},\varepsilon_{\boldsymbol{k}}\right)\approx-\frac{8U^{2}\rho_{F}^{2}}{3\sqrt{D}}T^{3/2}$
(55)

for generic momenta and corresponding results for special directions. This
result is crucial for the thermal transport behavior.

##  Appendix D Charge transport rate within the memory function formalism

The optical conductivity
$\mathbf{j}\left(\omega\right)=\sigma\left(\omega\right)\mathbf{E}\left(\omega\right)$
obtained by the Kubo formula is

$\sigma_{\alpha\beta}\left(\omega\right)=\frac{i}{\omega}\left(\left<T_{\alpha\beta}\right>+\chi_{j_{\alpha}j_{\beta}}\left(\omega\right)\right)$
(56)

with inverse mass tensor
$T_{\alpha\beta}=\frac{e^{2}}{N}\sum_{\mathbf{k}\sigma}\frac{\partial^{2}\varepsilon\left(\mathbf{k}\right)}{\partial
k_{\alpha}\partial k_{\beta}}c_{k\sigma}^{\dagger}c_{k\sigma}$ and current-
current susceptibility $\chi_{j_{\alpha}j_{\beta}}$. If the current is
conserved this reduces to
$\sigma_{\alpha\beta}\left(\omega\right)=\frac{i\left<T_{\alpha\beta}\right>}{\omega}$.
Then the real part of the conductivity diverges as $\omega$ approaches zero,
making it impossible to include a current-relaxing interaction by perturbation
theory in the low frequency limit. This problem can be tackled by introducing
the memory matrixGoetze72  as a correction to the $\omega^{-1}$-behavior and
regain the low frequency expansion of the conductivity

$\sigma_{\alpha\beta}\left(\omega\right)=\frac{i\left<T_{\alpha\beta}\right>}{\omega+M_{\alpha\beta}\left(\omega\right)}.$
(57)

The imaginary part of the memory function determines the scattering rate
$\tau_{J}^{-1}\left(\omega\right)$. The memory function contains the current-
relaxing processes and can more easily be treated within perturbation theory.
By comparing with Eq. (56) we find that the memory function up to first order
in the interaction is

$M_{\alpha\beta}\left(\omega\right)\approx\frac{\omega\chi_{j_{\alpha}j_{\beta}}\left(\omega\right)}{\left<T_{\alpha\beta}\right>}.$
(58)

This expansion is valid for $\omega\gg\tau_{J}^{-1}(0)$. We can rewrite this
as

$M_{\alpha\beta}\left(\omega\right)\approx\frac{\left<\left[F_{\alpha}F_{\beta}\right]\right>-\chi_{F_{\alpha}F_{\beta}}\left(\omega\right)}{\omega\left<T_{\alpha\beta}\right>},$
(59)

where $F_{\alpha}=\left[j_{\alpha},H\right]$ follows from the equation of
motion for $\chi_{j_{\alpha}j_{\beta}}\left(\omega\right)$. The scattering
rate is therefore given by

$\tau_{J}^{-1}\left(\omega\right)=\frac{-\mathrm{Im}\chi_{F_{\alpha}F_{\beta}}\left(\omega\right)}{\omega\left<T_{\alpha\beta}\right>}$
(60)

with

$\displaystyle\chi_{F_{\alpha}F_{\beta}}\left(\omega\right)$ $\displaystyle=$
$\displaystyle
e^{2}U^{2}\sum_{\mathbf{k}_{1}...\mathbf{k}_{4}}\left(v_{k_{1}}+v_{k_{2}}-v_{k_{3}}-v_{k_{4}}\right)^{2}$
(61) $\displaystyle\times$ $\displaystyle
L_{\varepsilon_{\mathbf{k}_{1}}\varepsilon_{\mathbf{k}_{2}}\varepsilon_{\mathbf{k}_{3}}\varepsilon_{\mathbf{k}_{4}}}\sum_{\mathbf{G}}\delta_{\mathbf{k}_{1}+\mathbf{k}_{2}-\mathbf{k}_{3}-\mathbf{k}_{4}-\mathbf{G}}$
$\displaystyle\times$
$\displaystyle\left[\frac{1}{\omega+\varepsilon_{\mathbf{k}_{1}}+\varepsilon_{\mathbf{k}_{2}}-\varepsilon_{\mathbf{k}_{3}}-\varepsilon_{\mathbf{k}_{4}}}\right.$
$\displaystyle-\left.\frac{1}{\omega-\varepsilon_{\mathbf{k}_{1}}-\varepsilon_{\mathbf{k}_{2}}+\varepsilon_{\mathbf{k}_{3}}+\varepsilon_{\mathbf{k}_{4}}}\right].$

The imaginary part of the susceptibility can be evaluated in analogy to
$\tau_{J}\left(0\right)$, where the external frequency replaces temperature.
Taking into account the proper phase space for umklapp scattering, this yields
a frequency depending scattering rate

$\tau_{J}^{-1}\left(\omega\right)=\lambda_{0}\frac{\omega^{2}}{D}\log\left(\frac{D}{\left|\omega\right|}\right).$
(62)

This result, valid in the regime $\tau_{J}^{-1}(\omega=0)\ll\omega\ll D$,
gives rise to the optical conductivity given in Eq.(28). The rate is therefore
determined by the same processes that give rise to the temperature dependence
of the resistivity. As was discussed in Ref.Rosch2000 ; Rosch2002  for higher
frequencies, shorter rates that do not rely on the condition of momentum
conservation may also affect the frequency dependence of the optical
conductivity. Here, the shorter single-particle rate $\sim\omega^{3/2}$ could
become relevant. For our problem those would only become relevant once
$\omega$ becomes comparable to the band width $D$. Thus is unclear whether
such an intermediate regime will be observable in experiment. At lowest
frequencies the result Eq.(62) is however the dominant one.

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