text:
\section{Introduction}
\label{Section-Introduction}

Rapid progress in the ability to produce clean, stable, 2D layered van der Walls heterostructures made up of graphene and/or transition metal dichalcogenides (TMDs) has opened a new subfield of condensed matter physics \cite{Geim_2013, Novoselov_2006, Bistritzer_2011, Morell_2010, Li_2019, Kim_2017, Cao_2018a, Cao_2018b, Cao2020PRL, Cao2021, Yankowitz_2019,Kerelsky_2019, Lu_2019, Stepanov2020untying, Sharpe_2019, Chen_2020, Rozen2021entropic, AndreaYoungBernal, AndreaYoungRhombo, AndreaYoungRhombo2, Serlin_2020, Wu_2018, Wu_2019_TIPRL, KinFaiMak_TopologyTMD, Polshyn_2020, TBLGStrangeMetalExperiment1, TBLGStrangeMetalExperiment2, TBLGStrangeMetalExperiment3, SankarFengchengStrangeMetal, CalTechBernal, CalTechSymmetryBreakingTBLG, CalTechTwisted, Xie_2020, MacDonaldTLBGReview, Li_2021, Ghiotto_2021, Pan_2020, Pan_2021, Morales_Dur_n_2021, Ahn_2022, Kerelsky2021moireless, Khalaf2019}. The sensitivity of the band structures of these systems to external control parameters, especially twist angle and displacement field, gives an unprecedented experimental ability to engineer flat bands and control the location of geometric band features (e.g. Van Hove singularities and Lifshits transitions), and thus to tune the relative strength of interaction-driven physics.

This family of systems has already shown various correlated insulating states \cite{Cao_2018a, Lu_2019}, ferromagnetism \cite{Sharpe_2019, Chen_2020}, correlation-driven valley and iso-spin polarization \cite{AndreaYoungBernal, AndreaYoungRhombo}, anomalous quantum hall physics \cite{Serlin_2020}, topological insulator physics \cite{Wu_2019_TIPRL, KinFaiMak_TopologyTMD, Polshyn_2020}, metal-insulator transitions \cite{Li_2021, Ghiotto_2021, Pan_2020, Pan_2021, Morales_Dur_n_2021, Ahn_2022}, possible ``strange metal" resistance scaling at very low temperature \cite{TBLGStrangeMetalExperiment1, TBLGStrangeMetalExperiment2, TBLGStrangeMetalExperiment3, SankarFengchengStrangeMetal}, and most conspicuously, possibly-exotic superconductivity \cite{Cao_2018b, Wu_2018, AndreaYoungRhombo2, AndreaYoungRhombo, AndreaYoungBernal, CalTechBernal, CalTechSymmetryBreakingTBLG, CalTechTwisted}, including phases with verified non-spin-singlet pairing \cite{AndreaYoungBernal, AndreaYoungRhombo2}. The rich phase diagrams and high experimental control that characterize these systems has quickly made them into one of the most studied platforms in condensed matter physics. The above-listed discoveries demonstrate that geometric band features can have a profound influence on the effects of interactions on transport properties. In turn, this highlights the need for a refinement of the basic theories of phonon-limited resistivity as applied to these materials, accurately taking complex band geometry into account.

\begin{figure}[t!]
\includegraphics[angle=0,width=.45\textwidth]{ResultsSummary-BernalAndRhomboAtomicLattice.PNG}
\caption{We depict the lattice structure of Bernal bilayer graphene (top) and rhombohedral trilayer graphene (bottom). The left side of the image shows top-down views of the xy-plane, labelling atoms with their layer number ($\{1,2\}$) and sublattice index ($\{A,B\}$). The right side of the figure shows the stacking from a cross-section view.}
\label{GrapheneStackGeometries}
\end{figure}

\begin{figure*}[t]
\includegraphics*[angle=0,width=.9\textwidth]{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram.PNG}
\caption{
Overview of phonon scattering in Bernal bilayer and rhombohedral trilayer layered graphene systems. The top row (a,e) depicts qualitatively distinct kinematically-allowed scattering manifolds for different Bloch states at the same energy. In these figures, the black curve depicts a Fermi surfaced (near the hole-doped VHS), the pink dot denotes a reference Bloch state, and the colored points denote the points in $\vex{k}$-space that the reference state can scatter too while conserving conservation of energy and momentum. These are the ``scattering manifolds" [Sec.~\ref{Subsection-BasicTheory}], which depend on the geometry of the system. The coloring of the scattering manifolds encodes the transitions rates from the reference state. These plots demonstrate the nontrivial kinematics and geometry at play in scattering in these systems.
The middle layers (b,c,f,g) shows the density of states of the two systems, with labels showing how the Fermi surface geometry changes as the sample is doped.
The bottom layer (d,h) gives the central results of this work, the scaling of the resistivity with temperature in various regions of $n-T$ space. This is captured by a heat map of $d\log[\rho(n,T)]/d\log T.$ We emphasize the clear connection between the scaling behavior and the geometric features in the density of states. The doted line gives the naive $T^*_{BG}$ calculated with Eq.~(\ref{BGTransitionTemperature}) for an isotropic system. We see that at sufficiently large dopings, this approximation becomes accurate, but that our system exhibits a surprising suppression of $T^*_{BT}$ as doping is decreased and the Fermi surface qualitatively changes. These results are calculated with an inter-layer potential of $\Delta = 0.07\ eV$, and should be compared with Fig.~\ref{ResultsSummary-GaplessBernalAndRhomboPhaseDiagram}, which treats the zero-field case. The analogous results for a simple Dirac cone (gapped and ungapped) are given in Fig.~\ref{ResultsSummary-GappedAndGaplessDiracConePhaseDiagram} for further comparison.
}
\label{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram}
\end{figure*}

\begin{figure*}[t!]
\includegraphics[angle=0,width=.9\textwidth]{ResultsSummary-GaplessBernalAndRhomboPhaseDiagram.PNG}
\caption{
Resistivity scaling with $T$ due to phonon scattering in ungapped BBG and RTG, as in the absence of the inter-layer potential, to be compared with Fig.~\ref{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram}. As in Fig.~\ref{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram}, the top layers (a,b,d,e) plot the density of states of the system for various doping levels, labelled with Fermi surface geometries. The bottom row (c,f) provides a heat map of $d \log[\rho]/d \log T$ over $n-T$ space, mapping out the various regimes of resistivity scaling. As in the case of the gapped systems (Fig.~\ref{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram}), we see that the BG transition is strongly affected by the band geometry. Though this is more subtle without the applied field, we still see the effects clearly in the RTG case, which still exhibits an annular Fermi surface over a small doping window. As before, the dotted line gives the expected BG crossover for an isotropic system; for the ungapped systems plotted here, this estimate is quite accurate as long as the sample is sufficiently doped.
}
\label{ResultsSummary-GaplessBernalAndRhomboPhaseDiagram}
\end{figure*}

\begin{figure}[b!]
\includegraphics[angle=0,width=.45\textwidth]{ResultsSummary-GappedAndGaplessDiracConePhaseDiagram.PNG}
\caption{
Resistivity scaling with $T$ due to phonon scattering in simple gapped (right) and ungapped (left) Dirac cones, to be compared with Figs.~\ref{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram} and \ref{ResultsSummary-GaplessBernalAndRhomboPhaseDiagram}. As in those figures, the bottom row provides a heat map of $d \log[\rho]/d \log T$ over $n-T$ space. The top row plots the density of states of the system for various doping levels. The dotted line gives $T^*_{BG}$ [Eq.~(\ref{BGTransitionTemperature})].
}
\label{ResultsSummary-GappedAndGaplessDiracConePhaseDiagram}
\end{figure}

In particular, recent experiments in ABC-stacked rhombohedral trilayer graphene (RTG) and AB-stacked Bernal bilayer graphene (BBG) (Fig.~\ref{GrapheneStackGeometries}) have discovered superconductivity (SC) proximate to several correlated, iso-spin polarized phases \cite{AndreaYoungBernal, AndreaYoungRhombo, AndreaYoungRhombo2, CalTechBernal} in the vicinity of Van Hove singularities and Lifshitz transitions in the band structures. Additionally, there is evidence that some of the superconducting phases host unconventional, non-spin-singlet pairing.
Theories of SC in RTG and BBG based on Cooper pairing mediated by interaction with acoustic phonons have been put forth that propose likely explanations for the SC, explaining the presence of both spin-triplet and spin-singlet phases and providing roughly accurate transition temperatures \cite{YZBernal, YZRhombo, YZLongPaper}. Additionally, the proximity of SC phases to various interaction-driven phases has spurred comparison to strong correlation physics, and several other explanations centering $e-e$ interactions have been proposed \cite{EEAshvin,EEBerg,EEBitan,EEGuinea,EELeonid,EEZalatel,Szabo2021BBG, Qin2022, Dai2022}.

A time-tested method for ascertaining the relative importance of various scattering mechanisms in a material is to look for clues in the temperature dependence of the resistivity. This is because different mechanisms generally produce various characteristic contributions and finite-$T$ crossovers between these. For example, this debate is currently unfolding for twisted bilayer graphene, where it is still unclear whether observed linear-in-$T$ resistivity dependence is caused by phonons or an interaction-driven strange metal state, analogous to that famously seen in several highly-correlated systems \cite{SankarFengchengStrangeMetal, TBLGStrangeMetalExperiment1, TBLGStrangeMetalExperiment2, TBLGStrangeMetalExperiment3}.

The recent experiments in BBG and RTG show that moir\'{e}-induced correlation effects are not a necessary ingredient for SC in layered graphene systems, leaving phonon-induced pairing as the de-facto leading candidate for a universal SC mechanism in these systems. Especially since acoustic phonons give a consistent theory of SC in both BBG and RTG, it is important to understand and isolate the contribution to the resistivity that should be expected due to acoustic phonons in the absence of $e-e$ effects. In conventional superconductors, electron-phonon couplings extracted from SC tend to agree well with those extracted from transport measurements. Thus, an extensive quantitative comparision of the SC data and the transport data is an important step in elucidating the nature of the SC pairing. Further, since these systems demonstrate that superconductivity in 2D layered systems can be intertwined with the nontrivial Fermi surface geometry, they offer an arena to understand the extent to which these geometric features effect transport generally.

The general paradigm of acoustic-phonon-limited resistivity in isotropic (semi)metals is as follows \cite{SankarGrapheneKT1, SankarGrapheneKT2, SankarGrapheneKT3, SankarGrapheneKT4, SankarGrapheneKT5, Ziman, AshcroftAndMermin}. In the low-T regime, where the quantum statistics of the phonon are important, we expect $\rho \approx T^{d+2}$, where $d$ is the dimension of the sample. This characterizes the ``Bloch-Gr\"{u}neisen" (BG) regime, which corresponds to
\begin{align}
\label{BGTransitionTemperature}
    k_BT \ll k_BT^*_{BG} = \mathcal{C}_{BG}\cdot(2\hbar v_p k_F),
\end{align}
where $v_p$ is the phonon velocity, $k_F$ is the Fermi momen-
\FloatBarrier
\noindent
tum, and $\mathcal{C}_{BG} \approx \mathcal{O}(1)$ is a material-specific constant. (Further, $k_B T_{BG} \equiv 2\hbar v_p k_F$ is usually defined.) In the high-$T$ ``equipartition" (EP) regime, $T >> T^*_{BG}$, we instead expect linear-in-T resistivity. We note that single-layer graphene displays these properties elegantly, with $\mathcal{C}_{BG} \approx 1/6$ \cite{SankarGrapheneKT1, Efetov_2010}.

The goal of this paper is to give a precise theoretical calculation of the resistance due to acoustic phonon scattering in BBG and RTG systems in the presence of an inter-layer potential ($\Delta$). We give concrete predictions for the doping ($n$) and temperature ($T$) dependence of the resistivity of these systems in the limit of phonon-dominated transport. The inter-layer potential (produced by a displacement field) is required to induce SC in BBG, and tuning this potential can significantly alter the band structure and control the location of the Van Hove singularities, affecting both the SC and the interaction-driven phases \cite{AndreaYoungBernal, AndreaYoungRhombo}.

\begin{figure}[t!]
\includegraphics[angle=0,width=.47\textwidth]{ResultsSummary-LogRhoLowT.PNG}
\caption{
We plot a heat map of $\log[\rho]$ over doping density and temperature for the BBG (left), RTG (middle), and Dirac (right) systems for both the gapless (top) and gapped (bottom) cases. For BBG and RTG, the gapped systems correspond to a displacement field $\Delta = 0.07\ eV$, and the gapped Dirac system has $M = 0.05\ eV$. Several features are prominent. The dark blue in the lower corners shows the universal features of the BG-transition. All systems but the ungapped Dirac cone display density dependence throughout the high-$T$ regime, with a significant increase in resistivity near charge neutrality. This is especially prominent for the gapped RTG and BBG systems, where the resistivity spikes to $\rho\approx 3000 \Omega$.
}
\label{ResultsSummary-LogRhoLowT}
\end{figure}
\begin{figure}[t!]
\includegraphics[angle=0,width=.46\textwidth]{ResultsSummary-LogRhoHighT.PNG}
\caption{
Accurate approximate calculations of resistivity in the equipartition regime can be made efficiently with the protocol discussed in Sec.~\ref{Subsection-BGAndEPRegimes}. Here we plot these approximate results over a large range of $T$, extending the scope of Fig.~\ref{ResultsSummary-LogRhoLowT}. We see that the large spike in resistivity near charge neutrality is a relatively low-$T$ behavior and that the resistivity decreases at higher temperatures, as $T$ get high enough to excite carriers in the conduction band. We emphasize that even at high-$T$, the resistivity does \textit{not} return to simple linear scaling, but instead asymptotes to a constant value. [See also Figs.~\ref{ResultsSummary-Slices} and \ref{AdditionalData-ParticleDopedResistivitySlices}.]
}
\label{ResultsSummary-LogRhoHighT}
\end{figure}

We carry out our calculation in the framework of Boltzmann kinetic theory, treating the acoustic phonons via the Debye approximation but retaining the full electronic band structure obtained by the diagonalization of $\vex{k}\cdot\vex{p}$ Hamiltonians \cite{MacDonaldBandstructureBernal, MacDonaldBandstructureRhombo}. We are able to numerically solve the linearized Boltzmann equation in the anisotropic band geometry and give quantitative predictions for the resistance and thus for the BG crossover temperature, $T^*_{BG}.$ We emphasize that accurately treating the non-isotropic band structure is a significant technical complication, beyond the techniques of prominent earlier treatments of resistivity in 2D layered graphene structures \cite{SankarGrapheneKT1, SankarGrapheneKT2, SankarGrapheneKT3, SankarGrapheneKT4, SankarGrapheneKT5}. Further, these earlier treatments of multi-layer graphene do not include the effects of the inter-layer potential.

We find that the electronic structure of the layered systems significantly distorts the BG paradigm explained above. In particular, while the high-$T$ behavior of the scattering rate of an individual Bloch state is linear, $1/\tau_{\vex{k}} \propto k_B T$, band curvature effects can lead to a complicated non-linear T-dependence of the resistivity curves. In the cases of gapped systems, (the displacement field generating $\Delta > 0$ opens a gap), there is a large spike in resistivity near charge neutrality.  These band-curvature effects interfere with the BG crossover, and we find that the approximate power law for resistivity scaling

\begin{figure*}[p!]
\includegraphics*[angle=0,width=.9\textwidth]{ResultsSummary-BernalResistivitySlices2.PNG}
\includegraphics*[angle=0,width=.9\textwidth]{ResultsSummary-RhomboResistivitySlices.PNG}
\caption{
Resistivity data for hole-doped Bernal bilayer (top) and rhombohedral trilayer (bottom) graphene stacks, evaluated with the displacement field at $\Delta = 0.07\ eV$. Resistivity is given in ohms on a linear scale. The leftmost two columns give the results of our full numerical calculation for the resistivity of the two systems at various doping levels up to $30K$ and $120K$, respectively. The third column gives the same resistivity curves extended to $800K$, making use of the equipartition assumption discussed in Sec.~\ref{Subsection-NumericalComputation}. The far right columns indicate the doping levels of the curves shown in each row. From the full low-$T$ results (left) we may extract the effective BG crossover. We see that for large dopings, $\rho(T)$ exhibits a BG-EP crossover temperature $T^*_{BG}$ as high as $40-60 K$; however, there is a sharp drop in $T^*_{BG}$ to around $20 K$ upon the Lifshitz transition to the annular Fermi surface. The $T^*_{BG}$ continues to drop as we approach charge neutrality, dropping as low as $5-10 K$. We note that there is no sharp or discontinuous behavior at the Van Hove singularities. The high-$T$ equipartition results (center) show how band curvature effects lead to nonlinearity in $\rho(n,T)$ and how the effects can complicate the BG crossover. We see also that for dopings close to charge neutrality, we should expect a large spike in resistivity at moderate temperatures ($100-300 K$).
}
\label{ResultsSummary-Slices}
\end{figure*}
\FloatBarrier

\noindent
is strongly influenced by the band structure geometry. This is demonstrated in Figs.~\ref{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram} and \ref{ResultsSummary-GaplessBernalAndRhomboPhaseDiagram} (and most others in this paper). Further, we note that the anisotropy (i.e., trigonal warping in graphene systems) in the band structure alters the low-$T$ BG relaxation rate $T^4$ power law to a non-universal, $\vex{k}$-dependent $T$-dependence. While this nonlinear-in-$T$ equipartition-regime phonon-limited resistivity is unexpected in the context of Boltzmann theory, we note that it has been detected experimentally in both bilayer and trilayer twisted graphene systems \cite{Polshyn2019, Siriviboon_2021}.

Our paper is organized as follows. In Sec.~\ref{Section-SummaryOfMainResults}, we present an overview of the main results of the work, emphasizing the most important quantitative aspects for comparison with experiment and qualitative results that run counter to common expectations. We then provide a concise review of acoustic phonon scattering in kinetic theory and present an overview of the calculation of relaxation times in the BBG and RTG systems in Sec.~\ref{Section-ResistivityViaBoltzmannKineticTheory}. We emphasize the roles of anisotropy and band curvature, which requires more care than the case of an isotropic band. The non-linear $T$-dependence we report in the equipartition regime is unexpected ; Section ~\ref{Section-NonlinearHighTResistivity} provides more intuition for these effects. Our concluding discussion is presented in Sec.~\ref{Section-DiscussionAndConclusions}.

Some supporting details are relegated to appendices. Appendix~\ref{Appendix-HamiltoniansForStackedGrapheneSystems} presents the $\vex{k}\cdot\vex{p}$ Hamiltonians used to calculate the band structure of BBG and RTG. Appendix~\ref{Appendix-NumericalImplementationOfResistivityCalculation} discusses the numerical solution for the relaxation rates in the solution of the linearized Boltzmann equation. Appendix~\ref{Appendix-RelaxationTimeApproximationInNonIsotropicSystems} discusses the role of the relaxation time approximation in resistivity calculations. Finally, Appendix~\ref{Appendix-AdditonalData} presents additional $\rho(n,T)$ data for the systems of interest, supplementing the results presented in Sec.~\ref{Section-SummaryOfMainResults}.

\section{Summary of main results}
\label{Section-SummaryOfMainResults}

Our central results are the calculations of the doping ($n$) and temperature ($T$) dependence of the resistivity [$\rho(n,T)$] for Bernal bilayer and rhombohedral trilayer graphene in the presence of a displacement field, under the assumption that scattering is limited to acoustic phonons (which we treat in the Debye approximation.) In particular, we give quantitative predictions for the crossover from the Bloch-Gr\"{u}neisen regime to the equipartition regime.

We plot $log[\rho(n,T)]$ for low $T$ ($0-120K$) in Fig.~\ref{ResultsSummary-LogRhoLowT} and an approximate extension of these results to higher-$T$ ($0-800K$) in Fig.~\ref{ResultsSummary-LogRhoHighT}. Individual curves of $\rho(n,T)$ for fixed $n$ are given in Figs.~\ref{ResultsSummary-Slices}. The most obvious feature in this data is a strong spike in resistivity ($\rho \approx 3000 \Omega$) near to charge neutrality at low $T$. From Figs.~\ref{ResultsSummary-LogRhoHighT} and \ref{ResultsSummary-Slices}, we see that this is a low-$T$ phenomenon and that resistivity drops and levels out at higher $T$. However, we note that the high-$T$ resistivity is definitely \textit{not} given by a simple $T$-linear power law above the BG regime. In Figs.~\ref{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram}, \ref{ResultsSummary-GaplessBernalAndRhomboPhaseDiagram}, and \ref{ResultsSummary-GappedAndGaplessDiracConePhaseDiagram}, we plot $d\log[\rho(n,T)]/d\log(T)$ as an approximate scaling exponent for the resistivity. These plots act as a sort of ``phase diagram" for the various regimes of $T$-dependence in the resistivity profile. In particular, we find there is a region where the resistivity curve flattens out to be essentially constant with $T$, sometimes after a downturn. While this is counter to high-$T$ phonon expectations, this behavior has been measured in twisted bilayer \cite{Polshyn2019} and trilayer \cite{Siriviboon_2021} graphene systems. We stress that this is an effect entirely due to band curvature, which we discuss further in Sec.~\ref{Section-NonlinearHighTResistivity}.

Figures \ref{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram}, \ref{ResultsSummary-Slices}, \ref{ResultsSummary-LogRhoLowT},  \ref{ResultsSummary-GaplessBernalAndRhomboPhaseDiagram}, and \ref{ResultsSummary-GappedAndGaplessDiracConePhaseDiagram} all demonstrate the BG crossover mentioned in the introduction. At high dopings, where the Fermi surface is roughly circular, we find a $T^*_{BG} \approx 40-60K$, in line with expectations for a circular Fermi surface \cite{SankarGrapheneKT1, SankarGrapheneKT2, SankarGrapheneKT3, SankarGrapheneKT4, SankarGrapheneKT5}. However, we see a sharp drop to around $20 K$ at the Lifshitz transition to an annular Fermi surface, and $T^*_{BG}$ continues to drop as we approach charge neutrality. From Figs.~\ref{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram}, \ref{ResultsSummary-GaplessBernalAndRhomboPhaseDiagram}, and \ref{ResultsSummary-GappedAndGaplessDiracConePhaseDiagram}, it is clear that the band geometry created by applying a displacement field ($\Delta$) to the graphene layers causes significant alterations to the standard BG transition profile. Additionally, the curve-flattening discussed in the last paragraph can come into effect at $T$ comparable to the crossover temperature $T^*_{BG}$, making the $T^4 \rightarrow T$ transition difficult to observe.

Nevertheless, we predict that the phonon contribution to resistivity should become important at temperatures that vary between $10 K$ and $60 K$, depending on the doping, as shown in Fig.~\ref{ResultsSummary-Slices}. This should be compared with what is currently known from experiment: linear-in-$T$ resistivity dependence has not been observed under 20$K$ in RTG or under 1.5$K$ in BBG. It is important to note that the zero-$T$ contribution to resistivity from disorder ranges from about 30$\Omega$ to 70$Omega$ in these systems \cite{AndreaYoungBernal, AndreaYoungRhombo, AndreaYoungRhombo2, CalTechBernal}.

We also report results for phonon scattering in BBG and RTG in the absence of the displacement field. The data is all given in Fig.~\ref{ResultsSummary-LogRhoHighT}, and the effective resistivity power law is extracted in Fig.~\ref{ResultsSummary-GaplessBernalAndRhomboPhaseDiagram}, which should be compared with Fig.~\ref{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram}. We note that in the absence of the applied field, the high-resistivity spike near charge neutrality is significantly diminished. However, we still find high-$T$ nonlinearity in the resistivity curves. Resistivity curves for the ungapped cases analogous to Fig.~\ref{ResultsSummary-Slices} can be found in Appendix~\ref{Appendix-AdditonalData}.

\section{Resistivity via Boltzmann kinetic theory}
\label{Section-ResistivityViaBoltzmannKineticTheory}

The use of Boltzmann kinetic theory to calculate linear response resistivities due to phonon collisions with Bloch state electrons is well-established \cite{AshcroftAndMermin, Ziman, SankarGrapheneKT1, SankarGrapheneKT2}. In this section, we outline the structure of the theory and explain our calculation, appealing to the Dirac cone of single-layer graphene to display concepts and highlight departures of our theory from previous work. We first introduce the model in Sec.~\ref{Subsection-Model}, then we state the main results of the kinetic theory in Sec.~\ref{Subsection-BasicTheory} and use these results to give intuition into the Bloch-Gr\"{uneisen} crossover in Sec.~\ref{Subsection-BGAndEPRegimes}. Finally, Sec.~\ref{Subsection-NumericalComputation}
discusses the actual computation of the resistivity.

\subsection{Model}
\label{Subsection-Model}
We use the electronic single-particle Hamiltonian
\begin{align}
\label{SingleParticleHamiltonian}
    H^e &=
    \frac{1}{L^2}
    \sum_{\vex{k}}

    c^\dagger_{\vex{k}}

    H^e_{\vex{k}}

    c_{\vex{k}},
\end{align}
where $c^\dagger \equiv c^\dagger_{s,\xi,\sigma,l,\vex{k}}$ creates an electron with crystal momentum $\vex{k}$ (relative to Dirac point), spin $s$, valley $\xi$, sublattice $\sigma$, and layer $l$. In our models, $H^e_{\vex{k}} \equiv \delta_{s,s'}\delta_{\xi,\xi'}H^e_{\sigma,l,\sigma',l',\vex{k}}$ is a $\vex{k}$-dependent matrix coupling together layer and sublattice degrees of freedom, which are given in Appendix~\ref{Appendix-HamiltoniansForStackedGrapheneSystems}. The four degenerate spin-valley flavors remain decoupled in our calculation and contribute equally to the conductivity (inverse resistivity).

We are interested in the effects of the electron bands, so we restrict our model to in-plane longitudinal acoustic phonons and adopt a simple Debye description. This is further justified by the fact that the electron-optical-phonon couplings are weak in graphene multilayers due to sublattice polarization \cite{Wu_2018, Choi2019}. We thus take the phonon Hamiltonian to be
\begin{align}
\label{PhononHamiltonian}
H^p = \sum_{l,\vex{q}}\hbar\omega_{\vex{q}}a_{l,\vex{q}}^\dagger a_{l,\vex{q}},
\end{align}
where $\omega_{\vex{q}}$ is the phonon dispersion and we use the Debye approximation $\omega_{\vex{q}} \approx v_{p}|\vex{q}|$, where $v_p$ is the phonon velocity.

We couple the electrons to phonons via the well-known deformation potential coupling Hamiltonian \cite{Ziman, SankarGrapheneKT1, Coleman2015introduction}:
\begin{align}
\label{CouplingHamiltonian}
    H^{epc} &=
    \sqrt{\frac{D^2 \hbar}{2\rho_M L^2}}
    \sum_{l,\vex{q}}
    \frac{\hat{n}_{\vex{q},l}}{\sqrt{\omega_{\vex{q}}}}(-i\vex{q}\cdot\hat{e}_{\vex{q}})(a_{\vex{q},l} + a_{-\vex{q},l}^\dagger).
\end{align}
Above, $D$ is the deformation potential, $\rho_M$ is the mass density of monolayer graphene, and $\hat{e}_{\vex{q}}$ is the desplacement unit vector of the phonon. Throughout this work, we set $D = 25$ eV, $\rho_M = 7.6 \cdot 10^{-8} g/cm^2$, and $v_p = 2.6 \cdot 10^6 cm/s$ \cite{SankarGrapheneKT1, SankarGrapheneKT2, SankarGrapheneKT3, Efetov_2010}. Finally, the electron density operator is
\begin{align}
\label{DensityOperator}
    \hat{n}_{\vex{q},l} \equiv \sum_{\vex{k}}c^{\dagger}_{(\vex{k}+\vex{q}),l}c_{\vex{k},l}.
\end{align}
In Eqs.~(\ref{SingleParticleHamiltonian}) and (\ref{DensityOperator}), sums over unwritten $s,\sigma,\xi,l$ indices are implicit.

\begin{figure}[t!]
\includegraphics[angle=0,width=.48\textwidth]{KineticTheory-DiracScatteringManifoldEDIT.PNG}
\caption{
We plot the kinematically-allowed scattering manifolds for phonon-scattering on a Dirac cone, given by Eq.~(\ref{EnergyConservation}). The Fermi surface at $\mu = -0.25 eV$ is plotted in black and a reference point on the Fermi surface is identified with a pink dot. The colored points mark the set of k-space points that the reference point can scatter too while conserving energy and momentum. The color coding on the scattering manifold are proportional to the scattering rate between the two points. The figure is shown for descending temperatures: $300K$, $100K$, $30K$, $10K$. The Bloch-Gr\"{u}neisen transition is demonstrated by the fact that the $300K$ and $100K$ figures (top) only differ quantitatively by the scale of the color bar, while they are qualitatively distinct from the lower-temperature versions (bottom).
}
\label{KineticTheory-DiracScatteringManifold}
\end{figure}

\subsection{Kinetic theory}
\label{Subsection-BasicTheory}

In the so-called ``relaxation time approximation" \cite{AshcroftAndMermin} [also see Appendix~\ref{Appendix-RelaxationTimeApproximationInNonIsotropicSystems}] to Boltzmann kinetic theory, the resistivity tensor ($\rho$) is given by
\begin{align}
    \label{ResistivityDefinition}
    [\rho^{ij}(n,T)]^{-1}
    &=
    \frac{4e^2}{TL^2}
    \sum_{\vex{k} \in BZ}
    \tau_{\vex{k}}
    v^i_{\vex{k}}v^j_{\vex{k}}
    f(\varepsilon_{\vex{k}})[1-f(\varepsilon_{\vex{k}})],
\end{align}
where $T$ is temperature, $L$ is system length, $e$ is the electron charge, $v_{\vex{k}}^j$ are components of the velocity of the Bloch state $\vex{k}$, $f(\varepsilon)$ is the Fermi distribution function, and the $\tau_{\vex{k}}$ are the \textit{relaxation times} of the various Bloch states. If the band structure and Bloch states are known, the main challenge in the computation of the resistivity is the computation of the relaxation times. The leading factor of $4$ follows from the spin and valley degeneracies of the problem.

In Eq.~(\ref{ResistivityDefinition}), we have suppressed the band index ($b$) and taken the sum over $\vex{k}$ to mean a sum over all Bloch states: $\vex{k} \rightarrow (\vex{k},b)$. We will continue to use this notation and will explicitly mention when interband excitations or transitions are important.

Enforcing self-consistency of the relaxation time approximation on the Boltzmann equation [Appendix~(\ref{Appendix-RelaxationTimeApproximationInNonIsotropicSystems})], we find that
\begin{align}
    \label{RelaxationLengthSelfConsistency}
    \frac{1}{|\vex{v}_{\vex{k}}| L^2}
    \sum_{\vex{k'}\in BZ}
    \mathcal{W}_{\vex{k}\rightarrow\vex{k'}}
    \frac{1-f(\varepsilon_{\vex{k'}})}{1-f(\varepsilon_{\vex{k}})}
    \left[
    l_{\vex{k}} - l_{\vex{k'}}\cos\theta_{\vex{v}}
    \right]
    &= 1,
\end{align}
where $l_{\vex{k}} \equiv |\vex{v}_{\vex{k}}|\tau_{\vex{k}}$ are the ``relaxation lengths" (mean free paths), $\theta_{\vex{v}}$ is the angle between the Bloch velocities $\vex{v}_{\vex{k}}$ and $\vex{v}_{\vex{k'}}$, and $\mathcal{W}_{\vex{k}\rightarrow\vex{k'}}$ is the transition rate from state $\vex{k}$ to $\vex{k'}$. In the thermodynamic limit, Eq.~(\ref{RelaxationLengthSelfConsistency}) becomes an integral equation. For a finite-size system, it is a matrix equation that can be inverted to find the relaxation lengths. Again, band indices have been suppressed, but $\vex{k}$ and $\vex{k'}$ should be taken to stand for the Bloch states $(\vex{k},b)$ and $(\vex{k'},b')$.

In the case of the standard deformation potential phonon coupling Hamiltonian, a standard Fermi's golden rule calculation gives the transition rates
\begin{align}
    \nonumber
    \mathcal{W}_{\vex{k}\rightarrow\vex{k'}}
    &=
    \frac{\hbar D^2}{\rho_M v_p}
    |\vex{k'}-\vex{k}|
    \Delta(\varepsilon_{\vex{k}},\varepsilon_{\vex{k'}})
    \sum_{l}
    \bigg|\langle \psi_{\vex{k'},l}|\psi_{\vex{k},l}\rangle\bigg|^2
    \\
    \label{TransitionRates}
    &\equiv
    \hbar v_p|\vex{q}|
    \Delta(\varepsilon_{\vex{k}},\varepsilon_{\vex{k'}})
    \mathcal{C}_{\vex{k},\vex{k'}}.
\end{align}
with
\begin{align}
    \vex{q} &\equiv \vex{k'}-\vex{k},
    \\
\label{EnergyConservation}
    \Delta(\varepsilon_{\vex{k}},\varepsilon_{\vex{k'}})
    &\equiv

    \begin{aligned}
    N_{\vex{q}} &\delta(\varepsilon'-\varepsilon - \hbar v_p |\vex{q}|)
    \\
    +
    (N_{\vex{q}} + 1 ) &\delta(\varepsilon'-\varepsilon + \hbar v_p |\vex{q}|)
    \end{aligned}

    ,
    \\
\label{OccupationNumber}
    N_{\vex{q}} &\equiv \frac{1}{\exp(\hbar v_p |\vex{q}|/k_BT) - 1}.
\end{align}

The Dirac $\delta$-functions in Eq.~(\ref{EnergyConservation}) enforce conservation of energy and momentum and $N_{\vex{q}}$ gives the occupation numbers of phonons available for scattering. The first line in Eq.~(\ref{EnergyConservation}) refers to phonon absorption processes while the second refers to phonon emission.

For a given energy band geometry, the conservation laws in Eq.~(\ref{EnergyConservation}) determine a set of \textit{scattering manifolds} for each Bloch state, corresponding to absorption and emission of phonons. The summand in Eq.~(\ref{RelaxationLengthSelfConsistency}) then determines the rate of transition to each point on the scattering manifold. Written as a sum over the scattering manifold  ($SM$), Eq.~(\ref{RelaxationLengthSelfConsistency}) takes the form
\begin{align}
    \label{RelaxationLengthSelfConsistency2}
    \frac{\hbar v_p}{|\vex{v}_{\vex{k}}| L^2}
    \sum_{\vex{k'}\in SM}
    |\vex{q}|
    \mathcal{C}_{\vex{k},\vex{k'}}
    \mathcal{F}_{\vex{k},\vex{k'}}^{\mu,T}
    \left[
    l_{\vex{k}} - l_{\vex{k'}}\cos\theta_{\vex{v}}
    \right]
    &= 1,
\end{align}
with
\begin{align}
    \mathcal{F}_{\vex{k},\vex{k'}}^{\mu,T}
    \equiv
    \frac{1-f(\varepsilon_{\vex{k'}})}{1-f(\varepsilon_{\vex{k}})}
    \times
    \left\{
    \begin{aligned}
    N_{\vex{q}} \ \ \ \ \varepsilon_{\vex{k'}} > \varepsilon_{\vex{k}}
    \\
    N_{\vex{q}} + 1 \ \ \varepsilon_{\vex{k'}} < \varepsilon_{\vex{k}}
    \end{aligned}
    \right\},
\end{align}
where $\vex{k} \in SM$ indicates a summation over the scattering manifold of states picked out by the delta functions in Eq.~(\ref{EnergyConservation}). We emphasize that all implicit dependence of the relaxation lengths on the temperature or chemical potential are due to $\mathcal{F}_{\vex{k},\vex{k'}}^{\mu,T}$.

From Eqs.~(\ref{ResistivityDefinition}-\ref{TransitionRates}) we see that resistance scales linearly with $D^2/\rho_M$, so our results are easy to adjust for different values of these parameters. The dependence on $v_p$ is more involved, since it also affects the geometry of the scattering manifolds.

Fig.~\ref{KineticTheory-DiracScatteringManifold} shows how scattering rates can vary across the scattering manifold, using a Dirac cone as a simple example. It also visually demonstrates the transition between the BG and EP regimes, which we discuss next.

\begin{figure}[b]
\includegraphics[angle=0,width=.48\textwidth]{KineticTheory-DiracScatteringLengths.PNG}
\caption{
We plot the log of the relaxation lengths $\{l_{\vex{k}}\}$ for a Dirac cone band structure for various temperatures. We fix the chemical potential at $\mu = -0.25\ eV$ and perform the calculation at $T = 10\ K$ (top), $T = 30\ K$ (middle), and $T = 100\ K$ (bottom). The far left column shows the relaxation lengths plotted over energy, while the central and rightmost columns give a heat map of the relaxation lengths in momentum space for the bottom (center) and top (right) bands, respectively. We emphasize that at low T, states near the Fermi level become long-lived. Further, states near the Dirac point are always long-lived due to a vanishing scattering manifold.
}
\label{KineticTheory-DiracScatteringLengths}
\end{figure}

\subsection{Bloch-Gr\"{u}neisen and Equipartition regimes}
\label{Subsection-BGAndEPRegimes}

The low-$T$ BG regime is best understood in the case of an isotropic ($l_{\vex{k}} \rightarrow l_{\varepsilon_{\vex{k}}}$ and $\vex{v}_{\vex{k}} \parallel \vex{k}$) and quasi-elastic ($\varepsilon' \approx \varepsilon$) system, such as graphene \cite{SankarGrapheneKT1}. In this case, we can replace the velocity angle with the momentum angle ($\theta_{\vex{v}} = \theta_{\vex{k}}$) and
Eq.~(\ref{RelaxationLengthSelfConsistency}) simplifies to a direct formula for the relaxation time:
\begin{align}
    \label{IsotropicRelaxationTime}
    \frac{1}{\tau_{\vex{k}}}
    &=
    \frac{\hbar v_p}{L^2}
    \sum_{\vex{k'}\in FS}
    |\vex{q}|
    \mathcal{C}_{\vex{k},\vex{k'}}
    \mathcal{F}_{\vex{k},\vex{k'}}^{\mu,T}
    \left[
    1 - \cos\theta_{\vex{k}}
    \right],
\end{align}
where $\vex{k} \in FS$ indicates a summation over the Fermi surface, which is taken to be indistinguishable from the scattering manifold in the quasi-elastic approximation.

For small $\vex{q}$, $1-\cos\theta_{\vex{k}} \approx |\vex{q}|^2$ and $\mathcal{C}_{\vex{k},\vex{k'}} \approx 1$, and the summand of Eq.~(\ref{IsotropicRelaxationTime}) scales with $\vex{q}$ roughly as $|\vex{q}|^3$. For low $T$, the Fermi functions $1-f(\varepsilon_{\vex{k'}})$ and the phonon occupation function $N_{\vex{q}}$ effectively restrict the sum in Eq.~(\ref{IsotropicRelaxationTime}) to $\vex{k'}$ with $\hbar v_p |\vex{q}| \leq k_B T$. Summing $|\vex{q}|^3$ over the portion of the [$(d-1)$-dimensional] scattering manifold within a radius proportional to $T$ gives the famous power-law defining the BG regime:
\begin{align}
    \frac{1}{\tau_{\vex{k}}} \propto T^{d+2}.
\end{align}
However, if we do \textit{not} assume isotropy, then we must restore
\begin{align}
\label{Anisotropy}
    1 - \cos\theta_{\vex{k}} \rightarrow 1 - \frac{l_{\vex{k'}}}{l_{\vex{k}}}\cos\theta_{\vex{v}}
\end{align}
in Eq.~(\ref{IsotropicRelaxationTime}). The small-$|\vex{q}|$ limit of the right hand side of Eq.~(\ref{Anisotropy}) is not necessarily proportional to $|\vex{q}|^2$, since it depends on the way $l_{\vex{k'}} \rightarrow l_{\vex{k}}$ and $\vex{v}_{\vex{k'}} \rightarrow \vex{v}_{\vex{k}}$ as $\vex{k'} \rightarrow \vex{k}.$ We therefore expect anisotropy to introduce non-universal, $\vex{k}$-dependent modifications of the BG power law in the $T$-dependence of each relaxation time $\tau_{\vex{k}}.$

In the high-$T$ limit, expanding in small $\Delta\varepsilon/T$, we find
\begin{align}
\label{FExpansion}
    \mathcal{F}_{\vex{k},\vex{k'}}^{\mu,T} = \frac{k_B T}{\hbar v_p |\vex{q}|} + \mathcal{O}(\Delta\varepsilon/T),
\end{align}
and inserting into Eq.~(\ref{RelaxationLengthSelfConsistency}) gives
\begin{align}
    \label{RelaxationLengthHighT}
    \frac{k_B T}{|\vex{v}_{\vex{k}}| L^2}
    \sum_{\vex{k'} \in SM}
    \mathcal{C}_{\vex{k},\vex{k'}}
    \left[
    l_{\vex{k}} - l_{\vex{k'}}\cos\theta_{\vex{v}}
    \right]
    &=
    1 + \mathcal{O}(\Delta\varepsilon/T)^2.
\end{align}
Solving Eq.~(\ref{RelaxationLengthHighT}) order-by-order in $1/T$, we see that the high-$T$ form of the relaxation length is
\begin{align}
\label{EquipartitionLengths}
    l_{\vex{k}} &= \frac{c_{\vex{k}}}{k_B T} + \mathcal{O}(\Delta\varepsilon/T)^3.
\end{align}
We note that the $\mathcal{O}(1)$ term in the $\Delta\varepsilon/T$ expansion of $\mathcal{F}_{\vex{k},\vex{k'}}^{\mu,T}$ in Eq.~(\ref{FExpansion}) rather remarkably vanishes, preventing a  $\mathcal{O}(\Delta\varepsilon/T)^2$ term in Eq.~(\ref{EquipartitionLengths}). This implies that the high-$T$ scattering rate (due to phonons) of a given Bloch state should be purely linear, going to zero in the $T \rightarrow 0$ extrapolation.

The equipartition regime is the range of temperature for which Eq.~(\ref{EquipartitionLengths}) holds for all Block states $\vex{k}$. Unlike the case in the BG regime, the linear-in-$T$ power law for the relaxation rate of the EP regime is not affected by anisotropy - all band structure information is encoded in the ``length constants" $c_{\vex{k}}$.

\subsection{Resistivity computation}
\label{Subsection-NumericalComputation}

Equations (\ref{ResistivityDefinition}-\ref{OccupationNumber}) combined with knowledge of the Bloch states give all the tools necessary to make a resistivity prediction. We solve Eqs.~(\ref{RelaxationLengthSelfConsistency}) for scattering lengths for each Bloch state [see Appendix~\ref{Appendix-NumericalImplementationOfResistivityCalculation} for discussion.] We emphasize that in general, the relaxation lengths $\{l_{\vex{k}}\}$ implicitly depend on temperature and chemical potential through the Fermi functions and phonon occupation number ($N_{\vex{q}}$) in Eq.~(\ref{RelaxationLengthSelfConsistency}). Once the $\{l_{\vex{k}}\}$ are known for a given pair $(n,T)$, the resistivity can be computed through Eq.~(\ref{ResistivityDefinition}). We plot the relaxation lengths for a Dirac cone band structure in Fig.~\ref{KineticTheory-DiracScatteringLengths}, keeping $\mu$ fixed as we vary $T$. These results illustrate that states near the Fermi surface become long-lived at low $T$.

In Secs.~\ref{Subsection-BasicTheory}-\ref{Subsection-BGAndEPRegimes}, we have suppressed the band index in summations over Bloch states. The Bernal bilayer and rhombohedral trilayer $\vex{k}\cdot\vex{p}$ Hamiltonians have four and six bands, respectively, while the Dirac cone model has two. In each case, we have two ``low energy" bands near charge neutrality: a ``valence" (hole) band and a ``conduction" (particle) band. The higher energy bands, when present, are over $3.5eV$ from charge neutrality. We note that the Fermi distributions in Eq.~(\ref{ResistivityDefinition}) suppress excitations in these higher energy bands for the temperatures and dopings we are interested in. However, it is important to keep both the conduction and valence bands as charge carriers may be excited in both bands, especially in the gapless systems. In all the models we study, interband transitions between the conduction and valence bands are forbidden by kinematics (i.e. the phonon velocity is too low). Interband transitions into higher energy bands are kinematically allowed, but thermally irrelevant.

It is important to note that as we scan $T$ for fixed $n$, $\mu(n,T)$ can change, and this can be quite drastic near a gap. We must therefore calculate $\mu(n,T)$ self-consistently via
\begin{align}
    \label{SelfConsistentChemicalPotential}
    n = \frac{4}{L^2}\sum_{\vex{k} \in BZ}f(\varepsilon_{\vex{k}}).
\end{align}
The prefactor 4 above follows from the spin and valley degeneracies. We stress that accurately computing the $T$-dependence of $\mu(n,T)$ near the band edge requires keeping both the valence and conduction bands, even if $T$ is far too low to excite carriers across the gap.

\begin{figure}[t]
\includegraphics[angle=0,width=.48\textwidth]{KineticTheory-BernalScatteringLengths.PNG}
\caption{
We plot heat maps of the relaxation length in k-space for the Bernal bilayer band structure. We set $\mu = -0.066\ eV$ (left) and $\mu = -0.058\ eV$ (right). For each $\mu$, we do the calculation for $T = 10\ K$ (top), $T = 30\ K$ (middle), and $T = 100\ K$ (bottom). and We see that at low $T$, the specific geometry of the Fermi surface is very important to relaxation, but that this information tends to get washed out at higher $T$.
}
\label{KineticTheory-BernalScatteringLengths}
\end{figure}

\begin{figure}[t]
\includegraphics[angle=0,width=.48\textwidth]{HighTBandFlattening-EPScatteringLengths.PNG}
\caption{
We plot the equipartition ``length constants" over a wide range of energy for the various systems under study. The top row gives Bernal bilayer graphene, the middle row gives rhombohedral trilayer graphene, and the bottom row gives standard Dirac cone graphene. Gapless systems are on the left and gapped systems are on the right. With these values stored we can efficiently compute the resistivity of these systems up to very high temperatures using Eq.~\ref{EquipartitionLengths}, though our results will miss the low-$T$ BG physics, as discussed in Sec.~\ref{Subsection-BGAndEPRegimes}. This is how we generate the high-T results in Figs.~\ref{ResultsSummary-Slices}, \ref{ResultsSummary-LogRhoHighT}. We emphasize that the ungapped Dirac cone graphene, which has diverging $c_{\vex{k}}$ near charge neutrality due to a vanishing scattering manifold, is the outlier here. All other systems we consider have band curvature effects near charge neutrality that more than compensate for the vanishing scattering manifolds and suppress the divergence of $c_{\vex{k}}.$
}
\label{KineticTheory-EPLengthConstantPlots}
\end{figure}

\begin{figure}[t]
\includegraphics[angle=0,width=.46\textwidth]{HighTBandFlattening-GFunctionPlots.PNG}
\caption{
We plot the ``$g$-function", [$g(\varepsilon)$], defined in Eqs.~(\ref{GFunctionDefinition}-\ref{ConductivityFromG}) for Bernal bilayer (top), rhombohedral trilayer (center), and Dirac cone graphene (bottom). We plot $g(\varepsilon)$ for both ungapped (left) and gapped (right) cases. These graphs demonstrate clearly why single ungapped Dirac cone graphene has such a robust high-$T$ linear resistivity and why all the other systems display nonlinear resistivity effects at high-$T$.
}
\label{KineticTheory-GFunctionPlots}
\end{figure}

\begin{figure}[t]
\includegraphics[angle=0,width=.48\textwidth]{HighTBandFlattening.PNG}
\caption{
We plot heat maps of $d\log[\rho(n,T)]/d\log T$ for the various systems using the resistivities calculated in the high-$T$ equipartition calculation (the data from Fig.~\ref{ResultsSummary-LogRhoHighT}). This should be compared with Figs.~\ref{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram},\ref{ResultsSummary-GaplessBernalAndRhomboPhaseDiagram}, and \ref{ResultsSummary-GappedAndGaplessDiracConePhaseDiagram}. Since this is based on resistivity data from the equipartition regime, it does not contain any BG physics, and any deviations from linear scaling are due to band curvature effects. We emphasize that these figures show linear-in-$T$ scaling for dopings away from charge neutrality, but then demonstrate a flattening of the $\rho(T)$. We further emphasize that the flattening seen near the van-Hove singularities is present here as well, indicating it is an effect of band geometry in the thermal averaging and not a transition in the nature of the scattering.
}
\label{KineticTheory-EquipartitionPhaseDiagram}
\end{figure}

The main result of this work is the application of the above analysis to Bernal bilayer and rhombohedral trilayer graphene stacks. These results are presented and discussed in Sec.~\ref{Section-SummaryOfMainResults}. We use $\vex{k}\cdot\vex{p}$ Hamiltonians for these systems \cite{MacDonaldBandstructureBernal,MacDonaldBandstructureRhombo}, which we provide in Appendix~\ref{Appendix-HamiltoniansForStackedGrapheneSystems}. The band structure further gives the density of states and Fermi surface geometries depicted in Figs.~\ref{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram},\ref{ResultsSummary-GaplessBernalAndRhomboPhaseDiagram}.

The nontrivial band geometry of these systems gives scattering manifolds that depend qualitatively on not only the Fermi level, but also the specific Bloch state in question, as depicted in Fig.~\ref{IntroductionSuperfigure-GappedBernalAndRhomboPhaseDiagram}. Since the bands are not isotropic and the phonon scattering cannot be considered ``quasi-elastic" \cite{SankarGrapheneKT1}, we need to find the full solution of Eq.~(\ref{RelaxationLengthSelfConsistency}). Solving Eq.~(\ref{RelaxationLengthSelfConsistency}) for the $\{l_k\}$ repeatedly for many values of $n$ and $T$, we calculate the resistivity data given in Figs.~\ref{ResultsSummary-LogRhoLowT},\ref{ResultsSummary-Slices}. Data showing how scattering lengths vary throughout the band structure are given in Fig.~\ref{KineticTheory-BernalScatteringLengths}.

The equipartition regime scaling coefficients, $c_{\vex{k}}$, are given for all the models of interest in Fig.~\ref{KineticTheory-EPLengthConstantPlots}. In the case of Dirac cone graphene, we see that there is a divergence of $c_{\vex{k}}$ at the Dirac point, arising from a vanishing set of scattering states. However, in all the other models under consideration, band curvature effects near charge neutrality more than compensate for the vanishing scattering manifolds and suppress $c_{\vex{k}}$.

\section{Nonlinear $T$-dependence of resistivity}
\label{Section-NonlinearHighTResistivity}

The ``common knowledge" of high-$T$ phonon scattering is that the resistivity scales linearly with $T$ above the BG crossover regime \cite{AshcroftAndMermin, Ziman, SankarGrapheneKT1, SankarGrapheneKT5}. While it is true that each individual relaxation length has the high-$T$ scaling of Eq.~(\ref{EquipartitionLengths}), the $T$-dependence of the resistivity itself can be quite nonlinear. Indeed, our calculations for BBG and RTG predict a nonlinear $T$-dependence of the resistivity, especially in the vicinity of the gap. [See Figs.~\ref{ResultsSummary-Slices},\ref{ResultsSummary-LogRhoHighT}.]

Our calculations predict that the phonon scattering will crossover from the BG regime to the EP regime at an effective BG crossover temperature that varies from as high as $40\ K$ at high doping to as low as $<10\ K$ near charge neutrality. However, in the EP regime, we start to see sharp reductions in slope of the resistivity at temperatures as low as $40\ K$ [See Fig.~\ref{ResultsSummary-Slices}]. For dopings closer to charge neutrality, we see the resistivity peak and drop precipitously at $ T \approx 300\ K$. This behavior has already been observed in twisted bilayer graphene, \cite{Polshyn2019}, at temperatures and resistivity values qualitatively consistent with our results here.

We note that the non-linear $T$-dependence resembles the same sort of resistivity profiles that have been characterized as ``resistivity saturation" \cite{Poniatowski_2021, Sarkar_2018, Hwang_2019, Kivelson_1995} and are sometimes associated with a breakdown of kinetic theory at the Mott-Ioffe-Regel limit \cite{Gurvitch_1981, Millis_1999, Kivelson_1995, Hussey__2004}. However, we stress that our results are fully in the Boltzmann framework. The possibility that the apparent resistivity saturation type effect could arise purely from the electron-phonon coupling effects was pointed out in the literature before \cite{Millis_1999}, but the physics of this apparent saturation in the current work is qualitatively different, arising not from non-Boltzmann strong coupling physics, but from subtle band structure effects as discussed in our paper.

In the rest of this section, we provide some intuition for the non-linear $T$-dependence of the resistivity. As discussed above, the high-$T$ relaxation lengths are given in terms of the $n,T$-independent constants $\{c_{\vex{k}}\}.$ We can gain an understanding of the non-linearity of $\rho(T)$ by considering the function
\begin{align}
\label{GFunctionDefinition}
    \delta^{ij}g(\varepsilon)
    &\equiv
    \frac{1}{L^2}
    \sum_{\vex{k}}
    \frac{
    v^i_{\vex{k}}v^j_{\vex{k}}
    }
    {|v_{\vex{k}}|}
    c_{\vex{k}}
    \delta[\varepsilon-\varepsilon_{\vex{k}}].
\end{align}
In the EP regime, we can write the high-$T$ conductivity in terms of $g(\varepsilon)$:
\begin{align}
\label{ConductivityFromG}
    \sigma = \frac{1}{(k_B T)^2} \int d\varepsilon g(\varepsilon) f(\varepsilon)[1-f(\varepsilon)]
\end{align}
We see that even in the equipartition regime, we only have linear scaling of the resistivity if the integral over $\varepsilon$ in Eq.~(\ref{ConductivityFromG}) scales linearly with $T$, which will be true as long as $g(\varepsilon)$ has a good linear approximation in a window of width $k_B T$ about $\mu(n,T).$ The functions $g(\varepsilon)$ are plotted in Fig.~\ref{KineticTheory-GFunctionPlots}. We see that gapless Dirac cone graphene has a perfectly flat $g(\varepsilon),$ (though our figure shows finite-size effects near the Dirac point), giving the familiar, perfectly linear resistivity in the equipartition regime. On the other hand, we see that gapless Dirac cone graphene is the exception - all of the other systems studied exhibit band curvature that manifests nonlinearity in $g(\varepsilon).$ The gapless bilayer and trilayer systems exhibit $g(\varepsilon)$ that can be roughly approximated as linear over small $\varepsilon$-windows when sufficiently doped. However, we expect a qualitative change when $T \approx |\mu(n,T)|,$ and the integral in Eq.~(\ref{ConductivityFromG}) crosses the zero-energy point, where we expect the scaling of the integral in Eq.~(\ref{ConductivityFromG}) to crossover from linear-in-$T$ to quadratic-in-$T$. This would result in a crossover to a roughly $T$-independent resistivity when $T \approx \mu(n,T),$ which is indeed what we see in Fig.~(\ref{ResultsSummary-GaplessBernalAndRhomboPhaseDiagram}). All three gapped systems exhibit more curvature in $g(\varepsilon),$ even when far from charge neutrality, but may still be linearly approximated in a small $T$-window. However, sharp qualitative changes in $g(\varepsilon)$ occur at a band edge, so we expect sharp qualitative changes in the resistivity scaling when $T \approx |\mu(n,T)|-E_{gap}/2$ and again when $T \approx |\mu(n,T)|+E_{gap}/2$. For a system without a band gap, we would only expect a single kink. Figures~\ref{ResultsSummary-Slices} and \ref{AdditionalData-ParticleDopedResistivitySlices} - \ref{AdditionalData-NoFieldRhomboSlices} demonstrate this intuition. Two distinct kinks are visible in many resistivity curves in Figs.~\ref{ResultsSummary-Slices} and \ref{AdditionalData-ParticleDopedResistivitySlices}, which plot the data for gapped systems, while curves in Figs.~\ref{AdditionalData-NoFieldBernalSlices} and \ref{AdditionalData-NoFieldRhomboSlices} tend to have a single kink.

We emphasize that the nonlinear $T$-dependence of the resistivity is in general due to the curvature of the bands and not necessarily related to interband excitations \cite{Polshyn2019}. For instance, in the hole-doped systems in Fig.~\ref{ResultsSummary-Slices}, with the potential difference at $\Delta = 0.07\ eV$, the gap is approximately $0.1\ eV$ wide. However, nonlinear $T$-dependence is seen at temperatures as low as $40\ K$, which is far to cold to excite appreciable states in the conduction band.

\section{Discussion and conclusions}
\label{Section-DiscussionAndConclusions}

We have calculated the electrical DC resistivity of Bernal bilayer and rhombohedral trilayer graphene systems, due to scattering off of acoustic phonons. We extend previous study by using a detailed $\vex{k}\cdot\vex{p}$ band structure and focusing our attention on the roles of geometric features of the band structure of these systems, including those affected by a displacement field.

We develop a thoroughly nontrivial transport theory for carrier resistivity due to electron-acoustic phonon interaction in experimentally relevant RTG and BBG multilayer graphene systems. The theory, while using the standard graphene acoustic phonons and the conventional electron-phonon deformation potential coupling, includes the full effects of RTG and BBG band structures (even including an applied electric displacement field) non-perturbatively by employing a full $\vex{k}\cdot\vex{p}$ description. The qualitative importance of the van Hove singularities and the anisotropies in the graphene band structures are exactly incorporated in the theory by iteratively solving the integral Boltzmann transport equation. This leads to several qualitatively new features in the resistivity (e.g. inapplicability of the simple Bloch-Gr\"{u}neisen criteria for linear versus non-linear resistivity in temperature, apparent resistivity saturation behavior at higher temperatures, and other features as discussed in this paper), which have not been discussed in the transport literature of electronic materials before in any context. We provide concrete predictions for the doping and temperature dependence of resistivity in RTG and BBG multilayers, finding that simple considerations for a Bloch-Gr\"{u}neisen temperature separating the linear-in-$T$ high-temperature resistivity from the non-linear low-temperature resistivity does not apply because of the band geometry introducing strong modifications of the resistivity behavior.

Our results are important in two contexts. First, the experiments in BBG and RTG have shown that the exotic superconductivity and the various interaction-driven correlated states are closely related to the nontrivial geometric features of the band structures, including Fermi-surface reshapes and Van Hove singularities. This spotlights the enhanced effects of band geometry on scattering processes in complex 2D systems. As 2D layered heterostructures are currently ascendant in condensed matter physics, it is important to study the relationship between band geometry and transport directly and to modify intuitions gained in three dimensions. Second, it is crucial in the investigation of the origin of the superconductivity in moir\'{e}less layered graphene systems to understand the relative importance of various scattering mechanisms. Our work provides a clear and concrete picture of how the resistivity should behave in a phonon-dominated system. If strong deviations from these results are seen in experiment, that could serve as evidence that the scattering mechanisms other than phonons play dominant roles in transport. This would point to directions for non-phonon pairing in the observed superconductivity.

The doping and temperature dependence of the resistivity of these systems behave similarly and with many interesting features. We find that the BG crossover in the qualitative $T$-dependence of the scattering rates varies as a function of doping from as low as $5K$ to as high as $60K$. However, we note that this crossover temperature depends strongly on the geometric features of the band structure, and is sharply reduced by the emergence of the annular Fermi surface, which is related to the observed SC. Further, we find that band curvature effects also give rise to a non-linear $T$-dependence of the resistivity at temperatures in the intermediate range of $60K-300K$. While our results show an interesting sensitivity to changes in Fermi surface geometry, they are remarkably smooth at the Van Hove singularity.

Our results are qualitatively compatible with what is currently known in experiment \cite{AndreaYoungBernal, AndreaYoungRhombo, AndreaYoungRhombo2}. We have not yet seen evidence of the high-$T$ nonlinear equipartition resistivity in BBG or RTG, but very similar effects have been observed in twisted bilayer \cite{Polshyn2019} and trilayer \cite{Siriviboon_2021} graphene systems. While the BG crossover to linear scattering has not yet been observed in these systems at low temperatures, our results show that current experiment cannot rule out the possibility that these systems are dominated by phonon scattering. In particular, no linear-in-$T$ region has been observed below $20 K$ in RTG and the zero temperature resistivity varies from $20-70 \Omega$ (c.f. Fig.S6 in \cite{AndreaYoungRhombo}). Our predictions are compatible with these experimental results. However, our results do make it clear that extensive experimental resistivity data over wider ranges of doping and temperature (from $0$ to $300K$) should be sufficient to tell if there are strong deviations from the phonon-dominated picture. Comparison of our results with future additional experimental resistivity data could be a crucial step in discovering the origin of SC in these systems. Further, the low Fermi velocities and high density of states at the Van Hove singularity should enhance the effects of electron-electron interactions. Since our calculations do not predict sharp features to emerge at the Van Hove singularities in a purely phonon picture, observations of such features in the resistivity could serve as evidence for strong-coupling physics that could underlie the systems' superconductivity.

\acknowledgments

We thank Christopher David White, Jiabin Yu, Jay D. Sau, and Matthew S. Foster for helpful discussion. This work is supported by the Laboratory for Physical Sciences (S.M.D, Y.-Z.C., and S.D.S). It is also partially funded by JQI-NSF-PFC (Y.-Z.C.). F.W. is supported by startup funding of Wuhan University.

redpajama_set_name: RedPajamaArXiv


By teasing out signatures of particles just tenths of a millimeter from the center of a trillion-degree fireball that mimics the early universe, nuclear physicists from the Department of Energy's Lawrence Berkeley National Laboratory (Berkeley Lab) and from other sites are gaining new insights about fundamental physics. They achieved this by smashing together gold nuclei at the Relativistic Heavy Ion Collider (RHIC).
Particle collisions at RHIC, located at DOE's Brookhaven National Laboratory, regularly recreate tiny specs of quark-gluon plasma (QGP), the so-called subatomic "soup" that contains the building blocks of visible matter.
The collisions free the quarks and gluons from their confinement within ordinary particles—the protons and neutrons that make up the nucleus of atoms—so nuclear physicists can study their interactions and the force that holds them together in the universe today. The quarks and gluons in this plasma last existed as free particles in the very early universe, some 14 billion years ago.
The new measurements, detailed in a study published in Physical Review Letters, are the first to come from a precision upgrade to RHIC's STAR detector known as the "Heavy Flavor Tracker" (HFT). The paper gives details about the first direct measurement at RHIC of how a type of heavy particle containing a "charm" quark gets caught up in the flow of the expanding fireball. This measurement—a testament to the capabilities of the HFT—gives scientists a new window into understanding the interactions of the particles that make up the subatomic soup.
Particles containing heavy quarks are considered ideal probes for understanding quark-gluon plasma because they may interact differently with the plasma than light quarks do, offering up subtle clues about the plasma's properties. But the QGP spits out particles containing heavy quarks only rarely, amid thousands of other particles made of the lighter varieties of quarks.
The few heavy particles that do emerge decay into other particles almost instantly—mere fractions of a millimeter from the QGP fireball in which they were created. This rarity and rapid decay make heavy particles difficult to detect.
STAR's HFT, a state-of-the-art tracking device now sitting at the center of the house-sized detector, tracks these elusive but important heavy particles.
Designed and developed by a team of nuclear physicists led by senior scientist Howard Wieman at Lawrence Berkeley National Laboratory, now retired, the HFT is the first silicon detector at a collider that uses a technology found in digital cameras called monolithic active pixel sensor technology.
The ultrathin sensors in the HFT—unlike many of the particle detection components of STAR—sit very close to the central beampipe in which the collisions take place. While not quite close enough to detect the heavy charm quark itself, this location and the detector's high resolution (composed of 360 million pixels each measuring 20-by-20 millionths of a meter) allow it to pick up signs of the heavy particles' decay.
For this latest study, STAR physicists were tracking particles called kaons and pions that emerge when charm-quark-containing particles known as a D-zeros (D0s) decay.
"We use the HFT to look for kaons and pions that are very close to one another— within fractions of a millimeter of one another—whose paths from the collision emerge from a single point that's away from the collision vertex, but not very far— about 100-500 microns," said Flemming Videbaek, a Brookhaven Lab scientist and the contract project manager for the HFT.
That's the distance D0s travel before they decay, he explained. If the kaon and pion have just the right mass and trajectories emerging from such a point, the scientists can conclude that they originated from a D0 at that spot—and use these measurements to track the emergence of D0s from all around the QGP.
The Berkeley Lab and a Kent State University group, together with other groups—including the software and computing team at Brookhaven Lab—developed the software that allowed scientists to connect the particle traces from the HFT detector with other components of the house-sized STAR detector.
The findings, based on an analysis of tens of thousands of sought-after particle events in 1.1 billion collisions, were somewhat surprising, researchers noted.
When two spherical gold ions collide off-center, the pattern of the collision is an oblong overlap—something like a football standing on end. STAR physicists found more D0s emerging from the fat part of the "football" than from its pointy ends, implying that heavy quarks are thermalized in the quark-gluon soup produced in the high-energy collisions at RHIC. D0s are created in the earliest part of the collision, when quarks and gluons are free.
This pattern of so-called "elliptic flow" was familiar from measurements of lighter particles emerging from the QGP. The new observations suggest that heavy-quark interactions inside the quark-gluon plasma are so strong that the heavy quarks themselves become part of the quark-gluon "soup," said Xin Dong.
Note: This news release was adapted from a Brookhaven National Laboratory press release. View the Brookhaven Lab release.


text:
Bernstein: The Age of Anxiety & Serenade after Plato's "Symposium" ((Remastered))
アルバム 2017/03/03 (11曲)
シャッフル
Disc1(11曲)
The Age of Anxiety, Symphony No. 2 for Piano and Orchestra (after W. H. Auden): a. The Prologue: Lento moderato
The Age of Anxiety, Symphony No. 2 for Piano and Orchestra (after W. H. Auden): b. The Seven Ages: Variations 1 - 7
The Age of Anxiety, Symphony No. 2 for Piano and Orchestra (after W. H. Auden): c. The Seven Stages: Variations 8-14
The Age of Anxiety, Symphony No. 2 for Piano and Orchestra (after W. H. Auden): a. The Dirge: Largo
The Age of Anxiety, Symphony No. 2 for Piano and Orchestra (after W. H. Auden): b. The Masque: Extremely fast
The Age of Anxiety, Symphony No. 2 for Piano and Orchestra (after W. H. Auden): c. The Epilogue: Adagio - Andante - Con Moto
Serenade after Plato's "Symposium" for Solo Violin, Strings, Harp and Percussion: I. Phaedrus: Pausanias (Lento - Allegro marcato)
Serenade after Plato's "Symposium" for Solo Violin, Strings, Harp and Percussion: II. Aristophanes (Allegretto)
Serenade after Plato's "Symposium" for Solo Violin, Strings, Harp and Percussion: III. Erixymathus (Presto)
Serenade after Plato's "Symposium" for Solo Violin, Strings, Harp and Percussion: IV. Agathon (Adagio)
Serenade after Plato's "Symposium" for Solo Violin, Strings, Harp and Percussion: V. Socrates: Alcibiades (Molto tenuto - Allegro molto vivace)
アルバムアーティスト

redpajama_set_name: RedPajamaCommonCrawl


Rutabaga gnocchi is a new twist on rutabaga dishes to make at home. Their sweet, fresh and crisp flavour is great raw, and will make your house smell wonderful as you knead it into this gnocchi to simmer on the stove. Lemon and thyme make this dish bright and colourful to complement the sweetness of rutabaga in this savoury dish!
Use a sharp knife to slice off the peel of the rutabaga.
Chop the rutabaga into 1-inch (2.5 cm) cubes. Put in a medium saucepan, cover with water and bring to a boil. Boil for about 25 - 30 minutes until rutabaga is very soft and falls apart when pierced with a fork.
Drain, then return rutabaga to the pot. Use and immersion blender or potato masher to blend until completely smooth and lump free. Put in the fridge to cool completely.
Once cool, add 2 cups (475 mL:) of the puréed rutabaga to a mixing bowl, 2 cups (475 mL) of the flour, and the salt. Mix and add in more flour as needed until the dough stops sticking to the sides of the bowl. Flour a clean work surface, remove the dough from the bowl and lightly kneed until dough comes together. Add more flour as needed so that dough is soft but not sticky.
Divide the dough into 6 and roll each section into a rope. Use a knife or pastry cutter to slice the ropes into gnocchi. Set gnocchi aside on lightly floured baking tray until you are ready to use.
Bring a large pot of water to a rolling boil. Drop gnocchi into the pot one at a time, cooking in batches of about 20 so that they don't stick together. Cook for 4 minutes until gnocchi is floating. Use a slotted spoon to scoop out the cooked gnocchi and set aside in a bowl. Continue cooking the remaining batches.
In the meantime, finely mince the garlic, zest and juice the lemon, and remove thyme leaves from the stems. The easy way to remove thyme leaves, is to hold the top of the stem, and run your fingers down with the other hand, which easily pops off the leaves.
Heat olive oil over medium heat, and when hot, add the garlic and thyme leaves. Cook about 1-2 minutes until the garlic just starts to get a bit of colour and then remove from heat. It will continue to cook once removed.
Toss in the hot gnocchi, lemon zest, lemon juice, maple syrup, salt and pepper and toss. Serve immediately.
Make ahead tip: The gnocchi dough can be prepared and cut ahead of time. Store on a covered, lightly floured baking sheet in the fridge overnight. Or you can freeze, by arranging the gnocchi in a single layer on a lightly floured baking sheet. Put the sheet into the freezer so the gnocchi can freeze individually, about 3 hours. Gather the now frozen gnocchi, and store in an air tight container or freezer bag. They should last about 1 month. Pop frozen gnocchi directly into the boiling water, and cook until floating (about 4 - 6 minutes).
This recipe was developed by Sam from It Doesn't Taste Like Chicken. Sam is vegan and an artist, but she's not starving! Writing her vegan-focused blog from Toronto, Sam's goal is to do for vegan cooking what Julia Child did for French cooking. Make it common, on every table, in every house, and every mouth! Sam's recipes are designed to please your mind, body, soul, and taste buds.
Hi Tima,We love that you're trying rutabaga in your favourite recipes to accommodate your diet! We don't have any specific recommendations for those types of recipes, but we've seen many people see a lot of success with replacing traditional wheat flour recipes with GF blends using almond meal and/or coconut flour. We did a quick search for rutabaga pizza crust, and maybe this will do the trick? Best of luck!


Twelve-year-old Artemis Fowl is a millionaire, a genuis, and, above all, a criminal mastermind. But even Artemis doesn't know what he's taken on when he kidnaps a fairy, Captain Holly Short of the LEPrecon Unit. These aren't the fairies of bedtime stories, they're dangerous! Full of unexpected twists and turns,Artemis Fowl is a riveting, magical adventure.


Almost every GPS provider has a vehicle tracking system that offers detailed reporting. But what if you need quick answers? What if you want to make a management decision based on historical driver data? You can cull the detailed reports for all the information you need, or you can use a vehicle tracking system that offers analytics reports.
Graphic data display – It's easy to get overwhelmed with numbers when your vehicle tracking system logs literally every event. Although this level of detail can be very useful, sometimes you just need a snapshot of the big picture. Analytics allow you to view trends over any time period with easy-to-read graphics so you can quickly see whether a metric is increasing, decreasing, or remaining consistent.
Driver metrics – Run an analytics report for a single driver to evaluate their behavior over time. You can use this objective data at review time, or simply to keep an eye on specific employees.
Group metrics – Monitor a particular group or use analytics reports to motivate drivers to operate more efficiently.
Fleet metrics – Run a report on your entire fleet so you can get a true view of the big picture. If you notice that a particular metric is changing in an undesirable way, you can drill down to identify the source of the problem.
Fuel use – Are fuel costs going up or down? Do you need to trim the budget or keep a closer watch on engine idling? You won't know unless you have a report at your fingertips.
Equipment hours – Find out how much your equipment is being used and how that has changed over time. Perhaps a spike in activity will prompt a maintenance review.
Speeding – Which drivers are speeding and have they changed their behavior over time? Keeping unsafe behavior in check is one of the best ways to prevent accidents or costly damage to vehicles.
The more you know about your fleet, the better you can manage it. You can't make decisions to improve efficiency or reprimand a driver unless you have real data to support those conclusions. Your vehicle tracking system has all the information you need to make these important decisions, but unless you can quickly find it, chances are it gets lost in the shuffle. Analytics are essential for quick decision making and effective fleet management. If your vehicle tracking system doesn't offer this feature, you're missing out.
GPSTrackIt offers analytics reports in addition to dozens of other useful features that will help you better manage your business. View the live demos today to learn more.
What analytics reports would you use most frequently?


Synthetic Skin – an elastic collision of acoustic playing and tape music. The quartet of Gerald Preinfalk (saxophone), Yaron Deutsch (electric guitar), Andreas Lindenbaum (violoncello) and Uli Fussenegger (double bass) encounters tape pieces by Jérôme Noetinger and Electric Indigo, while the piece flexibly moves between three levels and playing styles: pure tape music, composed and improvised instrumental passages, which in turn reference the tape music and, being inspired by it, evolve.
The form reflects the hermetic premise of the concept: The commissioned audiotape pieces played in full in the course of Synthetic Skin are created autonomously, without specific stipulations. They initially encounter the quartet constellation as a foreign body, which the quartet approaches over the course of the piece and makes it something familiar, without changing it. In this field that opens up between extrapolation and integration, between monolithic building block and improvisation, a flexible network of highly heterogeneous tonal qualities takes shape. The material composed by Uli Fussenegger thus acts as a kind of connective tissue between two extremes, namely the rigidity of the finished tape music and the ephemeral quality of improvisation. And although the collective material was authored by various people, the question of authorship increasingly loses significance in this process of integration.


text:
Q: Pandas Sqlite query using variable With sqlite3 in Python if I want to make a db query using a variable instead of a fixed command I can do something like this :
name = 'MSFT'

c.execute('INSERT INTO Symbol VALUES (?) ', (name,))

And when I try to access the SQL db using pandas data frame
I can do this:
df = pd.read_sql_query('SELECT open FROM NYSEXOM', conn)

However I am unsure how to load data from SQL to a pandas data frame while referencing a variable. I have tried the following:
conn = sqlite3.connect('stocks.db')
dates= [20100102,20100103,20100104]
for date in dates:

    f = pd.read_sql_query('SELECT open FROM NYSEMSFT WHERE date = (?)', conn, (date,))

When I run this I get an error saying "Incorrect number of bindings supplied, The current statement uses 1, and there are 0 supplied"
How can I properly format a command to load SQL data into a pandas data frame using a variable reference?

A: You need to use the params keyword argument:
f = pd.read_sql_query('SELECT open FROM NYSEMSFT WHERE date = (?)', conn, params=(date,))

A: As @alecxe and @Ted Petrou have already said, use explicit parameter names, especially for the params parameter as it's a fourth parameter in the pd.read_sql_query() function and you used it as a third one (which is coerce_float)
But beside that you can improve your code by getting rid of the for date in dates: loop using the following trick:
import sqlite3

dates=['2001-01-01','2002-02-02']
qry = 'select * from aaa where open in ({})'

conn = sqlite3.connect(r'D:\temp\.data\a.sqlite')

df = pd.read_sql(qry.format(','.join(list('?' * len(dates)))), conn, params=dates)

Demo:
Source SQLite table:
sqlite> .mode column
sqlite> .header on
sqlite> select * from aaa;
open
----------
2016-12-25
2001-01-01
2002-02-02

Test run:
In [40]: %paste
dates=['2001-01-01','2002-02-02']
qry = 'select * from aaa where open in ({})'
conn = sqlite3.connect(r'D:\temp\.data\a.sqlite')

df = pd.read_sql(qry.format(','.join(list('?' * len(dates)))), conn, params=dates)
## -- End pasted text --

In [41]: df
Out[41]:
         open
0  2001-01-01
1  2002-02-02

Explanation:
In [35]: qry = 'select * from aaa where open in ({})'

In [36]: ','.join(list('?' * len(dates)))
Out[36]: '?,?'

In [37]: qry.format(','.join(list('?' * len(dates))))
Out[37]: 'select * from aaa where open in (?,?)'

In [38]: dates.append('2003-03-03')   # <-- let's add a third parameter

In [39]: qry.format(','.join(list('?' * len(dates))))
Out[39]: 'select * from aaa where open in (?,?,?)'

A: Another option is to use pythons string formatting explicitly and forget about the params argument. That is to say you do something like this
conn = sqlite3.connect("database.sqlite")

df = pd.read_sql_query("                                           \
           SELECT * FROM table WHERE name= %s AND                  \
           number=%f" %(some_string_variable,some_float_variable)  \
           ,conn))

redpajama_set_name: RedPajamaStackExchange


This unique, transition photo shows the change in Toomer's Corner from 1899 to 2012. This panoramic photograph captures the changes to this Auburn landmark over the past century. Add this to your vintage Auburn collection in your home, office or study.
Dimensions: Unframed Matted Photo measures 11x20. Framed and matted photo with or without coins measures approximately 15x28 and is available in classic walnut or dark mahogany.


Students should make sure to repeat their main point and tie everything up with a strong final sentence. Spending a day with an alien. Eating disorders and the problems emerge with it. Ethical implications of cloning. For example, they might be teaching the children about the Olympic Games in Ancient Greece.
Have you ever wondered what exactly is up with informative speech topics? Students listening to MP3 players with headphones at school. They are all in luck because another big advantage is that our entire team of writers has academic degrees and background.
Students committing cyber bullying should be expelled from school. Essential Modes of a Persuasive Speech In order for a persuasive speech to be more effective and become able to grab and change the minds of the listeners, it has to utilize the vital means of persuasion.
We understand the value of your time and money, which is why every question and answer on Exam-Labs has been verified by Cisco experts.
This informative report can give you an insight into everything you've ever wanted to know about informative speech topics. Tuesday, December 8, Persuasive Writing KS2 - 5 Things to Remember If you're looking for some quick fire tips for success at persuasive writing ks2 level, then look no further.
It's a writer's final chance to leave a lasting impression on their reader, and is the thing people marking the work will remember most. Government should introduce additional tax over the manufacturers of junk food items and snacks full of fats.
Should the products containing tobacco be banned? Is financial education important in present era?
Or maybe you have a different one that has to be written in the Harvard style? The reason is that we are more likely to care or respond when we actively know who or what is involved firsthand. We have carefully complied realistic exam questions and answers, which are updated frequently, and reviewed by industry experts.
They would look at how the layout of these texts might be persuasive, for example:A comprehensive, coeducational Catholic High school Diocese of Wollongong - Albion Park Act Justly, love tenderly and walk humbly with your God Micah Persuasive Writing KS2 - This great resource includes a super helpful powerpoint that guides you through the process of writing a great piece of persuasive writing, as well as a stuctured worksheet to help your children record their own piece of persuasive writing.
Dec 08, · Persuasive writing ks2 level requires some stylistic elements to achieve higher grades. By this age, students need to be able to attract their reader, hold their attention and demonstrate a sound knowledge of persuasive agronumericus.com: Fuda Shiro.
So, what is love? First let's get. matlab assignment help save figure making a thesis statement for a research paper schizophrenia online homework service kvs dissertation case study vs phenomenology. e-magazines. The library has several books that are part of the Accelerated Reading Program.
Many, but not all are marked with e A/R Label.


text:
Home Education Disparities in advanced math and science skills begin by kindergarten
Disparities in advanced math and science skills begin by kindergarten
Kids Listening to Music Using Headphones while Sitting on a Wooden Flooring Photo: Pexels
By Paul L. Morgan, The Conversation
Racial and ethnic disparities in advanced math and science skills occur far earlier in the U.S. than previously known. Our new study finds that 13% of white students and 16% of Asian students display advanced math skills by kindergarten. The contrasting percentage for both Black and Hispanic students is 4%.
These disparities then continue to occur throughout elementary school. By fifth grade, 13% of white students and 22% of Asian students display advanced math skills. About 2% of Black students and 3% of Hispanic students do so. Similar disparities occur in advanced science skills.
What explains these disparities? Factors that consistently explain these disparities include the family's socioeconomic status – such as parental education and household income – and the student's own understanding of math, science and reading during kindergarten.
We observed these findings in analyses of a nationally representative sample of about 11,000 U.S. elementary school students. The students were followed from the start of kindergarten until the end of fifth grade.
Fewer than 10% of U.S. scientists and engineers are Black or Hispanic.
Racial and ethnic disparities in advanced math and science skills are constraining the country's scientific innovation and economic competitiveness. Students who display advanced math skills early are more likely to later obtain doctoral degrees in science, technology, engineering and math fields – collectively called STEM – and to become scientists or inventors.
Yet little has been known about how early racial and ethnic disparities in advanced math and science skills emerge. This information could help inform efforts to support students of color at a key time of child development.
Currently, most efforts by researchers and policymakers to address Black and Hispanic underrepresentation in STEM begin in high school or college. Yet minority students' interest in STEM careers begins to decline by middle school, with many students viewing scientists as stereotypically white.
Recent work suggests that racial and ethnic disparities in advanced math skills are increasing in size in the U.S. by the upper elementary grades.
What still isn't known
We were able to identify the factors that mostly explained disparities in advanced math or science skills between Hispanic and white students during elementary school. These factors included the family's socioeconomic status, the student's emerging bilingualism, and the student's early knowledge about math, science and reading. However, these same factors explained only some of the disparities between Black and white students.
Other factors we did not study could be involved, including the greater likelihood of Black students to attend lower-quality schools. The emerging bilingualism of many Hispanic students may help facilitate advanced STEM skills through greater mathematical reasoning, procedural learning and problem-solving.
To increase STEM representation in high school, college and the workforce, efforts by educators and policymakers to support talented students of color may need to begin by the elementary grades.
black students
Black teachers
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here is some rough work for this month, I must say I always seem to like my rough work more that the final ,maybe this time will be different?
Back in summer 2016 I got invloved with a short film called Protector, the early costume ideas didn't make it into the final product but it's still cool to see my bird logo.
Ideas for the bird symbol were based on the eagle, the really detailed patturns were ideas for engravings to go on belt buckles armour and other clothing.
My train reader: loosly based on a gesture I skeched on the train, the guy was on his phone and not as engaged.
hmm I still kind of like the rough, but this isn't too bad.
I sometimes like to do small write ups to build a story on what I draw.


For a very long time, Pearl Jam was my favorite band. I don't listen to them nearly as much now, but I can't really think of any band I like better. I listened a little to each of the original albums in the car, but went straight for the live disc when I got home.
"Immortality" is in an early and unfinished state. It sounds like it doesn't have an ending.
Still, when I got to "Spin The Black Circle" in the car, I found myself at full volume, howling the words. Its nice to have something so dear to my younger self brought back to life. These CDs have never been far, but a remaster gives me another reason to listen to them. Its not a bold remix like the one included with TEN, but I guess it sounds better or something. With the speakers in clip, its hard to tell.


text:
The joy of pipa
Karl Gehrke
St. Paul, Minn.
April 18, 2008 9:45 p.m.
ListenPipa demonstration
Gao Hong is celebrating her 35th anniversary as a professional pipa player.
Photo courtesy of Gao Hong
When Gao Hong was an eight year old girl growing up in China, the last thing she wanted to do was play the pipa. She'd even stand in the rain without clothes in hopes of catching a cold and getting sick enough to avoid practicing for at least a few hours.
"When I was young, I wished that I never had to play the pipa again," Gao Hong laughs. "My mom forced me and I just hated it. It was too hard."
Gao Hong demonstrates the pipa in the Minnesota Public Radio studios.
MPR Photo by Karl Gehrke
Gao Hong's mother was insistent that she learn the pipa so she could become a professional musician and avoid the chaos of China's Cultural Revolution. After a few years of intense practicing, she became accomplished enough on the difficult instrument to join a song and dance troupe that performed in north central China. Gao Hong was 12 years old and 400 miles from home.
"I didn't even know how to comb my hair by myself," she remembers. "I had never slept by myself. I was crying every day. I was really, really very sad and very lonely."
The pipa is an instrument with a history going back over 2,000 years. It's shaped like a pear with four strings that are plucked with the fingers. The pipa is held upright with the body of the instrument resting in the player's lap.
Gao Hong describes the pipa as sort of a mix of guitar, banjo and mandolin. But unlike those instruments, she says, the pipa can imitate a wide array of sounds, creating musical pictures. She can make the pipa sound like flowing water, blowing wind, wild geese or even Chinese people talking.
Gao Hong arrived in the U.S. in 1994, settling in Northfield, Minnesota where she teaches at Carleton College. While she remains dedicated to the traditional Chinese music of the pipa, she's been collaborating over the past 14 years with a wide array of musicians from other cultures and creating unique musical blends.
Shubhendra Rao says he and Gao Hong had some difficult moments learning how the pipa and sitar could fit together musically.
Her Saturday performance at the Ted Mann Concert Hall and debut at Carnegie Hall later this month will feature an unlikely trio of pipa, and two instruments from India, sitar and tabla.
New Delhi Sitar player Shubhendra Rao has performed with Gao Hong since first hearing her play the pipa with the Ragamala Music and Dance Theater some ten years ago. He knew right away that she was a brilliant musician.
"I could also recognize that the pipa and the sitar could go well together," Rao says. "How to make that possible was a difficult procedure. We went through some frustrating moments and some good moments making it work. Now when we are able to play together it's a joy."
Twin Cities percussionist Marc Anderson describes Gao Hong as a force of nature. He plays with her in the multi-ethnic quartet, Speaking in Tongues, featuring musicians from Mexico and Ghana.
Percussionist Marc Anderson plays with Gao Hong in the band, Speaking in Tongues. He describes her as a force of nature.
"She has a very joyful spirit and a lot of passion," he says "She's one of those special players who has a combination of skill, technique and an adventurous spirit. You can really hear her heart when she plays."
Gao Hong has played her pipa throughout Europe, Asia, Australia and the U.S., but perhaps her most memorable concert took place about six years ago at Carleton College. Her mother flew in from China and for the first time heard Gao Hong play a full, solo concert.
"The Chinese always say, 'You're not good enough,'" Gao Hong explains. "My mom never said I was good, but after that concert she said, 'You're pretty good.'"
"That was her comment after 35 years of playing the pipa," Gao Hong laughs.
Gao Hong recently returned from a trip to China where she was honored with the title of guest professor at the Central Conservatory in Beijing, China's premiere school of music where she studied the pipa.
Proudly, she says, she's joining a prestigious group of guest professors that includes Yo-Yo Ma, Seiji Ozawa, Itzahk Perlman, Luciano Pavarotti and other great artists.
Fullscreen Slide
Biplab Bhattacharya is playing the tabla during Gao Hong's 35th anniversary concerts this month in Minneapolis and Carnegie Hall.
An Orpheus for soprano, clarinet and dancers
VocalEssence announces 40th season
Online releases reshape the music industry
Triple Espresso - strong till the end
Tapes 'n Tapes roll again


Watch out for certain perennials that are on sale now, however. Many you see for sale, like the ones I saw at Plant World yesterday, have been greenhouse forced, are in full bloom, and really need to be slowly hardened off before they go in the garden. The cheerfully blooming greenhouse-forced plants in the pictures below fall into that category.
Hardening off is a practice of letting plants sit outside in the shade for a few hours each day before they can go out full time in the sun. [Edit: A few days will do it. You could also simply leave them in a sheltered spot close to the house, which tends to provide a bit of a microclimate. Generally, any field-grown perennials you buy don't need to go through this process.] It gives the plants a chance to toughen up to real-life outdoor conditions: full sun, wind, cold.
So be aware when buying. Some are not quite ready for prime-time. Remember to harden off. In any case, beautiful as these babies are, perennials always transplant better when they are not in bloom.
Better to enjoy these drumstick primroses in the house, harden off, then plant them later when the bloom is over.


text:
1. Staff
Please use this identifier to cite or link to this item: https://doi.org/10.1039/c3nj01380d
Title: Synthesis and nonlinear optical switching of Bi2S3 nanorods and enhancement in the NLO response of Bi2S3@Au nanorod-composites
Authors: Chen, J.L.T.
Nalla, V.
Kannaiyan, G.
Mamidala, V.
Ji, W.
Vittal, J.J.
Citation: Chen, J.L.T., Nalla, V., Kannaiyan, G., Mamidala, V., Ji, W., Vittal, J.J. (2014-03). Synthesis and nonlinear optical switching of Bi2S3 nanorods and enhancement in the NLO response of Bi2S3@Au nanorod-composites. New Journal of Chemistry 38 (3) : 985-992. ScholarBank@NUS Repository. https://doi.org/10.1039/c3nj01380d
Abstract: Organic solvent dispersible bismuth(iii) sulfide (Bi2S 3) nanorods were prepared by the chemical decomposition of tris(monothiobenzoato) bismuth(iii), [Bi(SCOPh)3] as a single molecular precursor in the presence of trioctylphosphine oxide (TOPO) and 1-dodecanethiol (DdT). The rod surfaces of the as-prepared Bi2S 3 nanorods were attached to Au nanoparticles (NPs) via direct reduction of KAuCl4·2H2O in the presence of octylamine (OTA) and oleic acid (OA). The attachment of the Au NPs onto the Bi2S3 rod surfaces was found to be relatively stable. The prepared samples (both Bi2S3 and Bi2S 3@Au) were characterized by powder X-ray diffraction (PXRD), high resolution transmission electron microscopy (HRTEM), energy dispersive X-ray (EDX) analysis and UV-vis spectroscopy. Their nonlinear-absorption (NLA) responses were studied using laser pulses at 532, 780 and 1064 nm, and the observed enhancement in the NLO response of Bi2S3@Au was discussed. Optical switching behaviour was observed at all the wavelengths. © 2014 The Royal Society of Chemistry and the Centre National de la Recherche Scientifique.
Source Title: New Journal of Chemistry
URI: http://scholarbank.nus.edu.sg/handle/10635/95075
DOI: 10.1039/c3nj01380d
Appears in Collections: Staff Publications
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Hasn't anyone on HSN seen the recent news on Hoverboard? Why are they still selling them?
My question is "Why is anyone buying them"?
I would not want to take a chance , even though the ones HSN is selling , that they might be the next ones recalled.
I Purchased An "Air Ride Hoverboard" IVR In December 2015 – Item # 468900611 – Recently Saw A News Report From Tne "CPSC Chairman" Regarding Recalls On Hoverboards – Is My Hoverboard On This List For Recalling?
I noticed hoverboards are out of stock. Will you be getting more in stock soon ?
Hello , I certainly understand your concerns with the news reports. We have checked with our vendor for the AIR RIDE hoverboards (469-879) we are currently selling, and they have had no complaints of any fires with their products nor have we been made aware of any such issues from customers. We have confirmed that the lithium batteries in the AIR RIDE hoverboards have passed applicable safety testing. As with all of the products HSN sells, we will continue to work with our vendor to monitor any customer feedback and take appropriate action if needed.