\section{Background}
The low-temperature behaviour of the heavy-fermion metal YbRh$_2$Si$_2$\, was
studied by means of thermal and electrical transport across its
field-induced quantum critical point (QCP). The thermal
conductivity of metals is typically measured down to temperatures
in the Kelvin range.~\cite{2Smith08,2Wakeham11} Because of the low
energy scales, measurements in the heavy-fermion metals have been
extended down to temperatures as low as 0.06\,K
(Ref.~\onlinecite{2Tanatar07}) or even 0.04\,K
(Ref.~\onlinecite{2Tomokuni11}). In our case, special efforts have
been made to measure the thermal conductivity at $B = 0$ and
0.02\,T down to 0.025\,K due to the very low N\'eel temperature
$T_{\rm N}$ = 0.07\,K (at $B = 0$). Furthermore, we have performed
substantially more temperature scans for $B=0$ in order to reduce
the statistical error of the data. At $B = 0.06$\,T, which is
close to the critical field $B_{\rm c}$, we were able to measure
the thermal conductivity down to 0.04\,K. At higher fields, the
measurements of the thermal conductivity were performed down to
0.06\,K.
\section{Contacts}\label{sec:contact}
Previous thermal transport studies pointed out the strong
influence of thermal contact resistances and demonstrated the
necessity of a careful contact
preparation.\cite{2Tanatar07,2Seyfarth05} Subsequent to polishing
the YbRh$_2$Si$_2$\, sample (sample 1) for optimization of their geometry for
our measurements, the sample surface was cleaned in an ultrasonic
bath and rinsed in ethanol. The contact pads were prepared by
evaporating a gold film (thickness of $\approx 50$\,nm) and
applying a lift-off technique which allowed for optimum pad width
and separation. Gold evaporation was conducted by using electron
beam evaporators in ultra-high vacuum which turned out to provide
superior contacts compared to thermal evaporation. Gold wires
(50\,$\mu$m) were then attached to the contact pads by silver
paint to make use of the pad area. The silver paint covered an
area of width of about 130\,$\mu$m, which is very small compared
to the distance between the contacts (2.9\,mm). This contributes
to a systematic standard error of about 4.5\% in the estimation of
the geometry factor. It is the same for all experiments. We did
not observe any indication of diminished thermal conductivity due
to contact resistances down to 0.025\,K. The same contacts were
used for thermal and electrical conductivity measurements.
\section{Systematic errors}\label{sec:error}
Although most measurements were performed on a sample cut
specifically for thermal transport, sample 1 ($4.2 \times 0.5
\times 0.1$ mm$^{3}$), additional electrical resistivity
measurements were done on a second sample (sample 2) with average
size $1.7 \times 0.41 \times 0.06$ mm$^{3}$ from the same piece of
crystal. This was necessary to systematically study the dependency
of the low-$T$ electrical resistivity on the current, {\it cf.}
Section IV. Because of the inhomogeneous thickness of the samples,
the geometry factor could not be determined with very high
precision.
\begin{figure}[b]
\centering
\includegraphics[width=0.5\textwidth]{figS1.pdf}
\caption{Temperature dependency of the electrical resistivities of
two YbRh$_2$Si$_2$\,\,single crystals from the same batch.
The data of sample 2 have been multiplied by a
factor $1.25 \pm 0.03$ and corrected by a difference
in residual resistivity of 0.22\,$\mu\Omega$cm.}
\label{figS1}
\end{figure}
However, the measured resistivities could perfectly be rescaled by
a factor $1.25 \pm 0.03$ and corrected by a difference in residual
resistivity of 0.22\,$\mu\Omega$cm. The former number allows to
estimate the uncertainty in the determination of the sample
dimensions, {\it i.e.}, the systematic standard error for the
geometry factor. It is about 2.5\%, and it is the same for all
measurements. The thermal and electrical transport coefficients
were measured on the same sample (sample 1) with the same contact
configuration, and the Lorenz ratio $L(T)/L_0 =\rho(T)/w(T)$ is
affected by an additional systematic error of about 4.5\% due to
the finite width of the contacts, see Section~\ref{sec:contact}.
Thus, a total systematic error of 7\% has to be considered, {\it
i.e.}, it will shift systematically all curves of Fig.~3 of the
main text. This explains why in the region of the field-induced
Fermi-liquid in the phase diagram, {\it e.g.}, at $B \ge 0.6$\,T
and below $T$ = 0.15\,K, the difference $w(T) - \rho(T)$ is not
exactly zero, and the Lorenz ratio does not reach exactly one.
\section{Electrical resistivity}
The electrical resistivity $\rho(T)$ was determined by a
four-point ac-technique in a $^3$He-$^4$He dilution refrigerator
down to $T\approx$ 0.02\,K. Figure~S\ref{figS1} shows the
resistivity data of samples 1 and 2 at $B = 0$ and 0.1\,T,
respectively. The data for the second sample, which has a lower
residual resistivity, have been scaled as described above. The
data agree nicely, and we conclude that both samples show the same
overall behaviour. The slightly larger resistivity in the
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{figS2.pdf}
\caption{Temperature dependency of the electrical resistivity of
sample 2. The kink at about 0.07\,K indicates the N\'eel
temperature $T_{\rm N}$\,. Below $T_{\rm N}$\, a $T^2$ dependency of the resistivity
is observed as expected~\cite{2Gegenwart02} ({\it cf.} inset where
the individual curves are shifted vertically).} 
\label{figS2}
\end{figure}
zero-field curve below 0.05\,K for sample 1 is due to heating
effects. Therefore, the second sample was measured at different
currents to investigate the influence of the current on the
curvature of $\rho(T)$. The results below 0.1\,K are displayed in
the inset of Fig.~S\ref{figS2} plotted as a function of $T^{2}$, a
temperature dependence expected below $T_{\rm N}$\, = 0.07\,K for $B = 0$
(Ref.~\onlinecite{2Gegenwart02}). The heating effect can be
neglected at a current of $5\,\mu$A. We used the resistivity data
with this current to evaluate the Lorenz ratio. As will be shown
in a forthcoming paper,~\cite{2Lausberg} the heating effect is
particularly strong in the vicinity of $B_c$. In addition, it is
increasing with increasing residual resistivity $\rho_0$. For a
single crystal of very high perfection with $\rho_0 \approx 0.5
~\mu\Omega$\,cm, no heating effect could be observed down to $T
\approx 0.02$\,K using a current of 50\,$\mu$A
(Ref.~\onlinecite{2Westerkamp08}). In this case, $\Delta \rho \sim
T$ was observed below $T= 0.1$\,K down to the lowest accessible
temperature of 0.02\,K.

In all samples the magnetoresistivity was measured and compared to
that of the samples investigated in
Refs.~\onlinecite{2Friedemann10,2Friedemann11}, to assure consistency
of the properties observed in our samples with those of the best
single crystals grown so far. As an example, the
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{figS3_new.pdf}
\caption{Magnetoresistivity of sample 2 measured with a current of
$5\: \mu$A. The upper inset magnifies the low field, low
temperature region of the main panel to emphasize that the
magnetoresistivity is positive inside the AF phase, but exhibits a
rapid drop across a crossover field $B^{*}$. This is further
illustrated in the lower inset for our lowest temperature $T =$
0.03 K. The rapid crossover across $B^*$ was systematically
analysed \cite{2Friedemann10,2Friedemann11}; in the zero-temperature
limit, it corresponds to a sharp jump because the crossover width
extrapolates to zero. The low-temperature data are also consistent
with the existence of a peak in the residual resistivity as a
secondary feature superimposed on the sharp jump, although its
interpretation is not clearcut as it may also be associated with
the N\'eel transition. } 
\label{figS3}
\end{figure}
magnetoresistivity of sample 2 is shown in Fig.~S\ref{figS3}: At
$T = 0.03$\,K the change in resistivity
associated with the crossover is about 10\% (see insets of the
same figure), as also found in the samples studied in
Ref.~\onlinecite{2Friedemann10} ({\it cf.} Fig.~S5 of the
Supporting Information of Ref.~\onlinecite{2Friedemann10}). The
dominant feature is a rapid crossover across $B^*$, corresponding
to a jump in the zero-temperature limit
\cite{2Friedemann10,2Friedemann11}. The isothermal resistivity vs.
the magnetic field at low temperatures are also consistent with
the existence of a peak near $B^*$ as a secondary feature
superimposed on the sharp jump, although it could also be
associated with the (classical) N\'eel transition.

\section{Thermal conductivity}
\subsection{Experimental details}
The thermal conductivity $w(T)$ was measured using a steady-state
two-thermometer, one-heater technique in a $^3$He-$^4$He dilution
refrigerator down to $T\approx$ 0.025~K. The heat loss along the
wires and suspensions for thermometers and heater is estimated to
be a factor of 1000 smaller than the heat flow through the sample.
$\kappa$ was calculated from the electrical power $P$ released by
a resistive heater, the temperature difference $\Delta T$ between
two contacts on the sample and the geometry factor $A$ as $\kappa
= A \cdot P/ \Delta T$. $\Delta T$ was measured by two RuO$_{2}$
resistance thermometers, that were calibrated against the primary
(``cold finger``) sensor during each temperature run, using the
resistances measured at zero heating power. A stability of better
than 0.1\,\% of both sample temperature read-outs was achieved in
the entire measurement range. For each stabilized bath temperature
a set of four different heat currents was applied which results in
temperature gradients $\Delta T/T$ = 1\% - 7\% along the sample.
The proportionality between the applied heater power and the
achieved temperature gradients at a constant bath temperature
proves that the system is in the regime of linear response. The
resulting raw data of $\kappa$ were then averaged. Our measurement
procedure implies the following uncertainties: i) The uncertainty
in $A$ represents a systematic error, that shifts all $\kappa (T)$
curves by a constant factor, as described in
Section~\ref{sec:error}; ii) the uncertainty in $P$ is negligible
because current and voltage at the heater can be measured with
high accuracy; iii) the uncertainty in $\Delta T$ results from the
measurement of the thermometer resistances and from errors in the
calibration. Concerning the last point, two or more temperature
runs have been performed at many fields, each with its own
calibration. For a given field, the calculated $\kappa(T)$ values
fall on top of each other within the scattering of the data.
Therefore, the error in the calibration is negligible compared to
the data scattering. In fact, the largest uncertainty arises from
the measurement of the thermometer resistances due to the limited
excitation current, especially at low $T$. It leads to scattering
of the raw data and is significantly reduced in our final data by
averaging over several points. As error bars we took the standard
deviation of the raw data as displayed in Figs.~2 and 3 of the
main text as well as Figs.~S\ref{figS4} and S\ref{figS5}. The
systematic error due to the sample and contact geometry is not
included, because it results only in a shift of all curves as
explained in Section III. We attribute the observation of $w_0$
being about 1\% larger than $\rho_0$ in the Fermi liquid regime at
high fields (Figs.~2{\sffamily\bfseries d} and
S4{\sffamily\bfseries f}) to this systematic error.

Finally, the noise level in the data of the thermal resistivity
increases substantially with increasing magnetic field, as can be
directly seen in Fig.~S\ref{figS4}. It is presumably due to the
vibrations of the set-up wires in
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth]{supp_resistivity.pdf}
\caption{Thermal and electrical resistivity $w = L_0 T/\kappa_{\rm
el}\approx L_0 T/\kappa$ and $\rho$ plotted as a function of
temperature, at $T \le 0.5$\,K for $B = 0.02$\,T
{\sffamily\bfseries a}, 0.08\,T {\sffamily\bfseries b}, 0.1\,T
{\sffamily\bfseries c}, 0.2\,T {\sffamily\bfseries d}, 0.4\,T
{\sffamily\bfseries e} and 0.6\,T {\sffamily\bfseries f} $(B \perp
c)$. The arrows in {\sffamily\bfseries b} through
{\sffamily\bfseries f} indicate the crossover to Fermi-liquid
behaviour ({\it cf.} Fig.~1{\sffamily\bfseries a} of the main
text). A significant drop is seen in the residual ($T\rightarrow
0$) thermal and electrical resistivities when going from $B =
0.02$\,T {\sffamily\bfseries a} to 0.08\,T {\sffamily\bfseries b}
and 0.1\,T {\sffamily\bfseries c}, which parallels the drop in the
isothermal magnetoresistivity across $B_{\rm c}$ ({\it cf.}
Fig.~S\ref{figS3}) as previously observed and attributed to an
abrupt increase in the charge carrier
concentration.~\cite{2Friedemann10,2Friedemann11} Also, the slight
rise of the residual thermal and electrical resistivities upon
increasing field ({\it cf.} {\sffamily\bfseries d} through
{\sffamily\bfseries f}) confirms the trend observed in
Ref.~\onlinecite{2Friedemann11} and ascribed there to the
magnetoresistivity in the paramagnetic phase of YbRh$_2$Si$_2$.}
\label{figS4}
\end{figure}
magnetic fields. This is the main reason why we could not perform
reliable measurements below 0.06\,K at fields larger than 0.06\,T.
\subsection{Phonon contribution}
The phonon contribution, $\kappa_{\rm ph}(T)$\,, to the measured thermal
conductivity $\kappa(T)$ can be separated from the electronic
part, $\kappa_{\rm el}(T)$\,, by avoiding the low-temperature range where a
significant inelastic scattering of the charge carriers has been
evidenced ({\it cf.} Fig.~1{\sffamily\bfseries b} of the main text
and Ref.~\onlinecite{2Tomokuni11}).

After having subtracted $\kappa_{\rm el}(T)$\, = $\kappa_{WF}(T) = L_0 T/\rho(T)$
from $\kappa(T)$ for 6\,K $< T < 12$\,K, {\it i.e.}, assuming the
validity of the Wiedemann-Franz (WF) law to hold in this
temperature range for the electronic heat transport, the phonon
contribution $\kappa_{\rm ph}(T)$\, is found to follow a $T^{\epsilon}$ dependence
with $\epsilon = 2 \pm 0.2$. As shown in Fig.~1{\sffamily\bfseries
b}, main text, an extrapolation of this power law to lower
temperatures indicates a negligible $\kappa_{\rm ph}(T)$\, below 1\,K, {\it i.e.},
within the temperature range of interest in the present work. The
uncertainty in the exponent has only little effect on the
estimated $\kappa_{\rm ph}(T)$\, below 5\,K. The insignificance of $\kappa_{\rm ph}(T)$\, below 1\,K
is further corroborated by the almost constant Lorenz ratio $L /
L_0 \approx 0.86$ within
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth]{supp_analysis.pdf}
\caption{Difference $(w - \rho)$ vs $T$ at $B = 0.02$\,T, 0.08\,T
and 0.1\,T {\sffamily\bfseries a} as well as 0.2\,T, 0.4\,T and
0.6\,T {\sffamily\bfseries b}. {\sffamily\bfseries c},
{\sffamily\bfseries d} Lorenz ratio $L/L_0 = \rho/w$ vs $T$ for
the same fields as in {\sffamily\bfseries a}, {\sffamily\bfseries
b}. The data for $B = 0.02$\,T are similar to those for $B = 0$
(\textit{cf.} Figs.~3{\sffamily\bfseries a} and
{\sffamily\bfseries c} of the main text). They provide convincing
evidence for an extra heat channel which adds to the one of the
electronic quasiparticles and ist most likely due to AF magnons
(see text). For $B = 0.6$\,T a Fermi-liquid phase forms below $T
\approx 0.15$\,K. This can also be anticipated for $B = 0.4$\,T
and 0.2\,T and even for $B = 0.1$\,T and 0.08\,T at
correspondingly lower crossover temperatures.} \label{figS5}
\end{figure}
the range $0.5$\,K $< T  < 1$\,K, presumably associated with a
measured thermal conductivity of purely electronic origin.
Assuming this Lorenz ratio to extend to above 2\,K, $\kappa_{\rm el}(T)$\, = $L\,
T/\rho(T)$ can easily be estimated. Subtracting this from the
measured $\kappa(T)$, the phonon part is obtained once more. Below
about 5\,K, $\kappa_{\rm ph}(T)$\, obtained by the latter procedure is found to be
in very good agreement with that from the former one.

We, therefore, conclude that  below about 10\,K, $\kappa_{\rm ph}(T)$\, of YbRh$_2$Si$_2$\,
follows a $T^{2}$ dependence, as commonly expected for a metal in
this temperature range due to dominating phonon scattering from
the conduction electrons.~\cite{2Ziman60} A nearly
$T^{2}$-dependence of $\kappa_{\rm ph}(T)$\, has indeed been observed for several
heavy-fermion compounds, e.g., CeB$_{6}$,~\cite{2Peysson85}
CeNiSn~\cite{2Kitagawa02} and CeAuAl$_{3}$~\cite{2Aoki00} in a
similar temperature range. In these systems the dominant phonon
wavelength, $\lambda_{\rm ph}$, is assumed to be shorter than the
mean free path of charge carriers, $l_{\rm el}$,
Ref.~\onlinecite{2Zimmerman59}. This assumption, thus, appears to
hold also for the YbRh$_2$Si$_2$\, single crystal studied here. On the other
hand, for CeCu$_{2}$Si$_{2}$,~\cite{2Franz78}
CeAl$_{3}$~\cite{2Ott84} and CeCu$_{6}$~\cite{2Peysson86} $\kappa_{\rm ph}(T)$\,
depends almost linearly on $T$, which hints at the opposite
relation between $\lambda_{\rm ph}$ and $l_{\rm el}$, {\it i.e.},
$\lambda_{\rm ph} > l_{\rm el}$, Ref.~\onlinecite{2Zimmerman59}.
\subsection{Electronic contribution}
The thermal conductivity obtained by subtracting $\kappa_{\rm ph}(T)$\, from the
measured $\kappa(T)$ is $\kappa_{\rm el}(T)$. In Fig.~S\ref{figS6}
we show that the temperature coefficient, $\kappa_{\rm el}/T$,
increases logarithmically upon cooling from $T = 2$\,K to about
\begin{figure}[t]
\centering
\includegraphics[width=8.4cm]{supp_kappa-vs-logT.pdf}
\caption{Temperature coefficient of the $B = 0$ electronic thermal
conductivity, $\kappa_{\rm el}/T$, of YbRh$_2$Si$_2$\, plotted as a function
of $T$ on a logarithmic scale to emphasize that it follows a
logarithmic behaviour on cooling from 2\,K to about 0.3\,K.}
\label{figS6}
\end{figure}
0.3\,K. A corresponding logarithmic divergence was observed also
for the Sommerfeld coefficient of the electronic specific heat,
$\gamma = C_{\rm el}/T$,~\cite{2Trovarelli00,2Custers03} and the
thermopower coefficient $S/T$.~\cite{2Hartmann10}

In the following, we are interested in the corresponding thermal
resistivity $w(T) = L_0 T/\kappa_{\rm el} (T)$, which is displayed
together with $\rho(T)$, in Fig.~2 of the main text for four
different fields. The results of $w(T)$ and $\rho(T)$ for other
magnetic fields are shown in Fig.~S\ref{figS4}. The data taken at
$B = 0.08$\,T through $0.6$\,T are complementary to those at $B =
0.3$\,T and 1 T presented in Figs.~2{\sffamily\bfseries c} and
2{\sffamily\bfseries d} of the main text. The data at $B =
0.02$\,T are likewise complementary to those at zero field
presented in Fig.~2{\sffamily\bfseries a} of the main text. Like
in the zero-field case, they show a low-temperature downturn in
the AF phase which contains the magnon contribution to the heat
conduction. Figure S\ref{figS5} displays the difference $w(T) -
\rho(T)$ {\sffamily\bfseries a},{\sffamily\bfseries b} and the
ratio $\rho(T)/w(T) = L/L_0$ {\sffamily\bfseries
c},{\sffamily\bfseries d} for the data shown in Fig.~S\ref{figS4}.
The existence of a magnon contribution to the thermal conductivity
at $B = 0.02$\,T is indicated by $(w - \rho)$ being negative and
$L/L_0$ being larger than one below $T \approx 0.03$\,K, very
similar to the results at $B = 0$ (\textit{cf.}
Figs.~3{\sffamily\bfseries a} and 3{\sffamily\bfseries c} of the
main text). Within the experimental uncertainties, the data at $B
= 0.6$\,T indicate Fermi-liquid behaviour and the validity of the
WF law below $T \approx 0.15$\,K. Fermi-liquid behaviour below a
crossover temperature which continuously decreases with decreasing
magnetic field is inferred for both $B = 0.4$\,T and 0.2\,T
{\sffamily\bfseries b},{\sffamily\bfseries d}. This implies
$L/$L$_{0} \rightarrow 1$ as $T \rightarrow 0$, with which also
the results obtained for $B = 0.1$\,T and even 0.08\,T are
compatible {\sffamily\bfseries a},{\sffamily\bfseries c}.

The data for $w$ and $\rho$ measured as a function of temperature
for fixed magnetic fields, shown in Fig.~2 of the main text and
Fig.~S\ref{figS4}, have been used to determine the isothermal
Lorenz ratio as a function of field displayed in
Fig.~3{\sffamily\bfseries e} of the main text. The isothermal
field dependence exhibits a minimum near $B^*$. This striking
behaviour is also seen in preliminary results obtained from direct
isothermal measurements of $w$ and $\rho$ as a function of field
in a sample with lower residual resistivity~\cite{2isothermal_new}.

\subsection{Magnon contribution}
For the measurements performed at $B = 0$
(Fig.~2{\sffamily\bfseries a}, main text) as well as 0.02\,T
(Fig.~S\ref{figS4}{\sffamily\bfseries a}) a downturn in the
thermal resistivity is found below $T \approx 0.1$\,K and $T
\approx 0.08$\,K, respectively, while at $B = 0.06$\,T a similar
but smaller feature occurs below $T \approx 0.07$\,K
(Fig.~2{\sffamily\bfseries b}, main text). One may ascribe this
drop in $w(T)$ to the freezing out of inelastic scatterings
provided by spin fluctuations in the electronic heat transport.
However, while ferromagnetic spin fluctuations remain unchanged,
antiferromagnetic ones grow at sufficiently low temperatures and
magnetic fields, as inferred from NMR Knight shift and
spin-lattice relaxation rate measurements, respectively.
\cite{2Ishida.02} Therefore, the only natural interpretation of
this drop involves heat carriers which add to the electronic ones.
The existence of a corresponding excess contribution to the
thermal conductivity, $\Delta \kappa (T)$, is proven by our
observation that for $B=0$ and $B=0.02$ T the thermal resistivity
becomes smaller than the electrical resistivity, {\it cf.}
Figs.~2{\sffamily\bfseries a} and S4{\sffamily\bfseries a}.
$\Delta \kappa (T)$ adds to $\kappa_{\rm el} (T)$ which we assume
to be given by $L_0 T/\rho$ at temperatures well below $T_{\rm N}$\,, {\it
i.e.}, we assume the Wiedemann-Franz law to describe the
electronic transport in the Fermi-liquid phase well below $T_{\rm
N}$. The observed $\Delta\kappa(T)$ falls into the range $2 -
5\cdot10^{-3}$\,W/Km between 0.025 and 0.03\,K. Because this value
of $\Delta\kappa$ is close to the experimental uncertainty, its
temperature dependence is hard to be experimentally determined.
Nevertheless, its existence has been confirmed by repeated
measurements. Lattice vibrations are unapt to account for this
extra thermal conductivity, as the largest expected $\kappa_{\rm ph}$\,, which is
limited by the sample dimension ($\approx 100\, \mu$m) and,
employing the lattice specific heat,~\cite{2Custers03} is estimated
to be less than $1\cdot 10^{-5}$\,W/Km in this range.

Another potential thermal heat channel is that of AF magnons. The
signature of these spin-wave excitations was clearly observed
\begin{figure}[t]
\centering
\includegraphics[width=0.52\textwidth]{figS7.pdf}
\caption{Specific heat of YbRh$_2$Si$_2$\,, shown as $\Delta C /T$ versus
$T^{2}$. $\Delta C(T) = C(T) - C_{\rm ph}(T) - C_{\rm Q} (T)$,
where $C_{\rm ph}$/$C_{\rm Q}$ denotes the phonon/nuclear
quadrupole contribution. The red line indicates a $T^{3}$
contribution to $\Delta C$ below 0.05\,K.~\cite{2Custers03}}
\label{figS7}
\end{figure}
\cite{2Custers03} in the zero-field specific heat examined down to
$T = 0.018$\,K for an YbRh$_2$Si$_2$\, single crystal being of similar quality
as samples 1 and 2. As seen in Fig.~S\ref{figS7}, below $T =
0.05$\,K the specific heat can be described by $\Delta C = C -
C_{\rm ph} - C_{\rm Q} = \gamma T + \beta T^{3}$, with $C_{\rm
ph}$ and $C_{\rm Q}$ being, respectively, the phonon and the
nuclear quadrupole contributions, $\gamma = 1.64$\,J/(K$^{2}$mol)
and $\beta = 132.2$\,J/(K$^{4}$mol). The huge electronic
contribution $C_{\rm el} = \gamma T$ denotes a heavy Landau FL
phase.~\cite{2Custers03} The Debye-like term $C_m \sim T^{3}$ is
characteristic of the contribution of long-wavelength AF acoustic
magnons. In the framework of the Debye theory and using the
measured $\beta$ value, we can estimate the ``magnetic Debye
temperature", $\Theta_m$, and the group velocity $v_{m}$ of the
magnons to be 4.2\,K and 36\,m/s, respectively. This Debye
temperature corresponds to the AF exchange interaction suitably
averaged over the three main spatial directions. The extracted
group velocity is substantially smaller than that for typical
Ce-based heavy-fermion AF metals, {\it e.g.}, about one-tenth of
the corresponding value for CeAl$_{2}$,~\cite{2Bredl78} reflecting
both the averaging over the spatial directions as well as the weak
N\'eel order of YbRh$_2$Si$_2$\,. The classical kinetic relation, $\kappa_{m}
= (1/3) C_{m} v_{m} l_{m}$ ($l_{m}$: magnon mean free path),
allows to estimate the magnon contribution to the thermal
conductivity, $\kappa_{m}$. In order to obtain a $\kappa_{m}$ of
$2 - 5\cdot10^{-3}$\,W/Km in the range $0.025 - 0.03$\,K as
observed in Fig.~2{\sffamily\bfseries a} of the main text and
Fig.~S\ref{figS4}{\sffamily\bfseries a}, the corresponding magnon
mean free path $l_{m}$ has to be in the range of $2 - 13$\,$\mu$m.
Since in the low-temperature limit $l_{m}$ is expected to be equal
in size to that of the AF domains, the latter are estimated to be
of the order of a few $\mu$m below about 0.03\,K which, indeed, is
a reasonable order of magnitude.~\cite{2Note}

The less pronounced downturn in $w(T)$ observed below $T =
0.07$\,K at $B = 0.06$\,T (Fig.~2{\sffamily\bfseries b}, main
text) is ascribed to overdamped AF magnons, which were shown, via
inelastic neutron scattering
experiments,\cite{2Wiltshire83,2Wiltshire85} to exist for
antiferromagnetically ordered materials substantially above the
N\'eel temperature. Heat transport by short-lived magnon
excitations has been reported, e.g., for the parent compound of
the 214 high-$T_{c}$ cuprates, La$_{2}$CuO$_{4}$, an $S = 1/2$, 2D
antiferromagnet with a N\'eel temperature $T_{\rm N} \approx
310$\,K.\cite{2Hess03} In particular, $S = 1$ chain systems, like
Y$_{2}$BaNiO$_{5}$ (Ref.~\onlinecite{2Kordonis05}) and
Ni$($C$_{2}$H$_{8}$N$_{2})_{2}$NO$_{2}($ClO$_{4})$ (NENP)
(Ref.~\onlinecite{2Sologubenko08}) have served as model systems in
this context.

We would like to note that the magnon contribution cannot be
avoided by performing thermal Hall measurements. In contrast to
the case of phonons, \cite{2Wakeham11} magnons may generate a
transverse thermal gradient. \cite{2Onose10}


\section{Extrapolation of the Lorenz ratio to zero temperature in
quantum critical systems}

We now discuss the various isofield and isothermal scans that are
used for the extrapolation of the Lorenz ratio in the vicinity o a
QCP. We will discuss the special case that is pertinent to YbRh$_2$Si$_2$,
namely a field-induced QCP separating an AF ordered phase and a
paramagnetic Fermi-liquid phase.

Fig.~S\ref{figS8}{\sffamily\bfseries a} illustrates different
temperature scans in different parts of the phase diagram. Scan A
starts from the quantum critical regime, but runs into the ordered
phase; by construction, it passes through the phase boundary. The
zero-temperature limit of $w(T)$, $\rho(T)$ and $L(T)/{L_0}$ so
extrapolated does not reflect the quantum critical behavior, but
instead only captures that of the ordered phase. Scan B also
starts from the quantum critical regime, but runs into the
low-temperature paramagnetic Fermi liquid phase; it passes through
a crossover temperature. Again, the zero-temperature limit so
extrapolated does not reflect the quantum critical behavior, but
instead only captures that of the paramagnetic Fermi liquid phase.
Scan C starts from the quantum critical regime, and goes all the
way to the QCP as the temperature is lowered. The zero-temperature
limit so extrapolated captures the properties of the QCP. In
practice, however, the same purpose can be achieved by carrying
out scans C$_1$ or C$_2$ (provided there is enough dynamical range
in temperature). These start in the quantum critical regime but
stop before running into the N{\' e}el temperature or the
crossover temperature to the paramagnetic Fermi liquid region. The
zero-temperature limit so extrapolated also manifests the behavior
of the QCP.

The different temperature scans will also be manifested in the
isothermal properties of $L/{L_0}$ as a function of the control
parameter, as illustrated in Fig.~S\ref{figS8}{\sffamily\bfseries
b}. Scan a starts from the paramagnetic Fermi-liquid regime, and
runs into the quantum critical regime. As it passes through the
crossover scale between the two regimes, we expect to see a
crossover in $L/{L_0}$ in a way that compliments what is seen in
the temperature scan B. Scan B starts from the paramagnetic
Fermi-liquid regime, passing through the quantum critical regime,
and runs into the ordered phase. Because it passes through both
the crossover temperature and the N\'eel temperature lines, this
isothermal scan will
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{figS8_final}
\caption{ Isofield {\sffamily\bfseries a} and isothermal
{\sffamily\bfseries b} scans for the extrapolation of the Lorenz
ratio in the vicinity of a QCP, as described in detail in the
text. } \label{figS8}
\end{figure}
reflect the features of both temperature scan B and temperature
scan A.

In this work, our primary focus is on temperature scans C$_1$
(Figs.\ 2{\sffamily\bfseries a}, 3{\sffamily\bfseries a},
3{\sffamily\bfseries c}), scans close to C (Fig.
2{\sffamily\bfseries b}), and scans C$_2$ ({\it e.g.} Figs.
S4{\sffamily\bfseries b} and {\sffamily\bfseries c}) through B
(Figs. 2{\sffamily\bfseries c}, 2{\sffamily\bfseries d},
3{\sffamily\bfseries b}, 3{\sffamily\bfseries d}). In addition,
isothermal scans a (Fig. 3{\sffamily\bfseries e}) are conducted.
Because of the masking effect of magnons on the electronic heat
transport, we do not attempt to analyze any scan b. Temperature
scans A cannot be performed in YbRh$_2$Si$_2$ because of the small
value of the critical field $B_c$.

\section{Theoretical aspects}
The validity of the WF law in the vicinity of a heavy-fermion
spin-density-wave QCP was briefly discussed in the main text.
Across such a QCP, the heavy quasiparticles remain intact in the
main part of the Fermi surface. In these ``cold'' regions, the
quasiparticles do not experience scattering by the AF spin
fluctuations, and their spectral weight---the quasiparticle
residue $Z$---stays finite as the control parameter moves through
the QCP (Fig.~S9). $L/{L_0}$ must be equal to one in this case.
Even the contributions from the ``hot'' regions, which experience
scatterings by the bosonic collective fluctuations, cannot violate
the WF law, as was illustrated by the case of ZrZn$_2$
(Ref.~\onlinecite{2Smith08}) where ferromagnetic fluctuations
influence the entire Fermi surface.

This is in contrast to the Kondo-destroying local quantum critical
description which is illustrated in Fig.~4 of the main text. In
the paramagnetic Fermi-liquid state, the conduction electron
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{figS9_final.pdf}
\caption{The evolution of the quasiparticle weights across a
heavy-fermion spin-density-wave quantum critical points. $Z_L$ of
the large Fermi surface stays nonzero as the system is tuned
through the QCP.} \label{figS9}
\end{figure}
self-energy, $\Sigma ({\boldsymbol k},{\omega},T)$ contains a pole
at the Kondo-resonance energy. It converts the $f$-moments into a
part of the Landau quasiparticles, thereby creating a large Fermi
surface. The quasiparticle residue at the large Fermi momenta,
$\pmb{k}^{\rm L}_{\rm F}$, is nonzero, $Z_L(\pmb{k}^{\rm L}_{\rm
F},\omega=0,T=0) \ne 0$. In the AF Fermi-liquid state, the Kondo
resonance is destroyed. The Fermi surface is given by that of the
conduction electrons only in the presence of a staggered magnetic
field. Away from the ``hot spots", there are Landau quasiparticles
associated with a small Fermi surface. The quasiparticle residue
at such generic small Fermi momenta, $\pmb{k}^{\rm S}_{\rm F}$, is
nonzero, $Z_S(\pmb{k}^{\rm S}_{\rm F},\omega=0,T=0) \ne 0$.

In these Fermi-liquid regimes, both the electrical and electronic
heat currents are predominantly carried by Landau quasiparticles.
At nonzero temperatures, the electronic heat carriers experience
inelastic scatterings. Because the non-Umklapp processes are
considerably more efficient in relaxing electronic heat current
than electrical current, even in anisotropic systems at low
temperatures, the electron-electron scatterings lead to an
electronic Lorenz ratio $L_{\rm el}/{L_0} < 1$ at nonzero
temperatures.~\cite{2Herring67} The effect is similar to the usual
case of electron-phonon scattering.~\cite{2Ziman60} The Umklapp
scatterings are expected to contribute equally efficiently to the
thermal and electrical resistivities. In the zero-temperature
limit, well-defined quasiparticles remain, but the inelastic
scatterers are frozen out; only the elastic scattering processes
remain, and $L_{\rm el}/{L_0}$ is equal to $1$. This is
illustrated in Fig.~S\ref{figS10}, where $L_{\rm el}/{L_0}$
reaches $1$ as the system moves away from the QCP (at $B_c$) into
the Fermi-liquid regimes on both sides.

In the quantum critical regime, low-energy electronic excitations
occur at both the small and large Fermi surfaces (Fig.~4, main
text). The single-electron self-energy at both $\pmb{k}^{\rm
S}_{\rm F}$ and $\pmb{k}^{\rm L}_{\rm F}$ vanishes at the
$\omega=0$ and $T=0$ limit, but has the scaling form as a function
of $\omega$ and $T$ as given in Eq.~1 (main text). This reflects
the fluctuations of the Fermi surfaces, which
\begin{figure}[b]
\centering
\includegraphics[width=0.45\textwidth]{figS10_final.pdf}
\caption{Expected isothermal behaviour of the Lorenz ratio across
a Kondo-destroying QCP. $B^*(T) = B(T^*)$ is the Kondo breakdown
scale at a nonzero but small $T$. At zero temperature an abrupt
dip occurs at $B^* = B_c$} \label{figS10}
\end{figure}
characterize the quantum fluctuations in the {\it entire} quantum
critical regime. We will consider the electrical and heat currents
carried by the electronic excitations at both $\pmb{k}^{\rm
S}_{\rm F}$ and $\pmb{k}^{\rm L}_{\rm F}$, all of which are
non-Fermi liquid in nature as specified by Eq.~1. These electronic
current carriers are subject to inelastic scatterings that are
associated with the quantum criticality. While such scattering
processes are many-body in nature, they can still be divided into
Umklapp and non-Umklapp processes. The non-Umklapp processes will
contribute considerably more to the thermal resistivity than to
the electrical resistivity. As a result, $L_{\rm el}/{L_0}$ will
be less than~1.

The end result is a dip of $L_{\rm el}/{L_0}$ near $B_c$.
Fig.~S\ref{figS10} illustrates the corresponding isothermal
behaviour of $L_{\rm el}/{L_0}$ for a given nonzero but low $T$,
where the dip is expected to be centered at $B^*(T) = B(T^*)$,
{\it cf.} Fig~1{\sffamily\bfseries a} of the main text. As $T
\rightarrow 0$, the dip becomes abrupt at $B_c$.


