Scattering mechanisms and electrical transport near an Ising nematic quantum critical point

Electrical transport properties near an electronic Ising-nematic quantum critical point in two dimensions are of both theoretical and experimental interest. In this work, we derive a kinetic equation valid in a broad regime near the quantum critical point using the memory matrix approach. The formalism is applied to study the effect of the critical fluctuations on the dc resistivity through different scattering mechanisms, including umklapp, impurity scattering, and electron-hole scattering in a compensated metal. We show that electrical transport in the quantum critical regime exhibits a rich behavior that depends sensitively on the scattering mechanism and the band structure. In the case of a single large Fermi surface, the resistivity due to umklapp scattering crosses over from $\rho \sim T^2$ at low temperature to sublinear at high temperature. The crossover temperature scales as $q_0^3$, where $q_0$ is the minimal wave vector for umklapp scattering. Impurity scattering leads to $\rho -{\rho }_0\sim T^{\alpha }$ (${\rho }_0$ being the residual resistivity), where $\alpha$ is either larger than 2 if there is only a single Fermi sheet present, or $4/3$ in the case of multiple Fermi sheets. Finally, in a perfectly compensated metal with an equal density of electrons and holes, the low-temperature behavior depends strongly on the structure of “cold spots” on the Fermi surface, where the coupling between the quasiparticles and order parameter fluctuations vanishes by symmetry. In particular, for a system where cold spots are present on some (but not all) Fermi sheets, $\rho \sim T^{5/3}$. At higher temperatures, there is a broad crossover regime where $\rho$ either saturates or $\rho \sim T$, depending on microscopic details. We discuss these results in the context of recent quantum Monte Carlo simulations of a metallic Ising nematic critical point, and experiments in certain iron-based superconductors.

Figure 1

(a) Diagrammatic representation of ${\dot{n}}_{\alpha \mathbf{k}}\left(\mathrm{\Omega }\right)$. The open/closed circles denote inflow/outflow of an external frequency $\mathrm{\Omega }$. The vertex function changes sign depending on if the $\alpha \mathbf{k}$ state is an initial (final) state of the scattering. [(b)–(d)] Two class of diagrams for the memory matrix $M_{\alpha \mathbf{k},\beta {\mathbf{k}}^{\prime }}\left(\mathrm{\Omega }\right)$. The red colored labels correspond to external momenta $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$. For convenience we omitted the flavor index on the fermionic Green's functions, and the summation over internal frequencies is implicit.

Figure 2

Three types of scattering processes when the relevant momentum states are in the vicinity of the Fermi surface. (a) Class I diagram describes direct scattering between $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$ mediated by exchanging an nematic boson ${\varphi }_{\mathbf{q}}$ with $\mathbf{q}={\mathbf{k}}^{\prime }-\mathbf{k}$. (b) and (c) are two-electron scatterings mediated by two nematic bosons, described by class I diagram. (b) describes the momentum exchange scattering, where two electrons with momentum $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$ exchange their momentum. (c) describes the head-on scattering, where two electrons located at $\mathbf{k}$ and $-\mathbf{k}$ are scattered into ${\mathbf{k}}^{\prime }$ and $-{\mathbf{k}}^{\prime }$.

Figure 3

(a) Resistivity due to impurity scattering for a system with a single Fermi surface (blue) and multiple Fermi sheets (red). In (b), we plot $dln\left(\rho -{\rho }_0\right)/dlnT$ for multisheet case to extract the power-law dependence on temperature. The dashed line marks the value $4/3$. The dispersion used for one-band case is ${\varepsilon }_{\mathbf{k}}=-2t\left(cos{k}_x+cos{k}_y\right)-4t^{\prime }cos{k}_{x}cos{k}_y-\mu$, with $t=1$, $t^{\prime }=-0.3$, and $\mu =-1$. In the multisheet case, we used three circularly shaped Fermi pockets of equal size, with two electronlike and one holelike dispersions. Impurity scattering strength is taken to be $g_{\text{imp}}=0.02{\varepsilon }_F$. The nematic-electronic coupling strength is ${\lambda }^2\approx 0.83{\varepsilon }_F$. The Fermi energy is defined as ${\varepsilon }_F={\langle v_F{k}_F\rangle }_{\text{FS}}$.

Figure 4

Two-particle collision involving a umklapp process. One of the final states is a different Brillouin zone. Therefore the total momentum changes by the crystal momentum $\mathbf{G}$. ${\mathbf{q}}_0$ is the minimal momentum transfer in an umklapp scattering event.

Figure 5

(a) Resistivity from umklapp scattering at and away near an Ising-nematic QCP, as a function of $r-r_c$ and $T$. Here, we use the same band dispersion studied for the single-band case for impurity scattering (see caption of Fig. 3). The coupling strength ${\lambda }^2=1.88{\varepsilon }_F$. (b) $dln\rho /dlnT$ as a function of $r-r_c$ and $T$. (c) Temperature cuts at the QCP (green) and in the disordered state ($r>r_c$, red). (d) Crossover temperature to Fermi liquid $T^2$ behavior as a function of the umklapp threshold wave vector $|{\mathbf{q}}_0{|}^3$. ${\mathbf{q}}_0$ was varied by changing the chemical potential $\mu$. We define $T_0$ as the temperature where $dln\rho /dlnT=1.8$.

Figure 6

$\rho \left(T\right)$ and its logarithm-derivative plot at the QCP in a perfectly compensated metal with different arrangements of cold spots on the Fermi pockets. The model consists of two circular pockets with equal radii, one electronlike and one holelike. The nematic form factor is either taken to be $cos2{\theta }_{\mathbf{k}}$ when the corresponding pocket has cold spots, or unity for the case with no cold spots. The coupling constant is ${\lambda }^2\approx 0.84{\varepsilon }_F$. Red curve: both pockets have cold spots along the diagonals. Blue curve: cold spots on the hole pocket, but not the electron pocket. Green curve: neither pocket has cold spots. Magenta curve: both pockets have cold spots, but also subject to impurity scattering of strength $g_{\text{imp}}\approx 0.1{\varepsilon }_F$. The black dashed lines in (b) correspond to exponents of $4/3$, $5/3$, and 2.

Figure 7

The function ${\mathrm{\Phi }}_{i,{\theta }_{\mathbf{k}}}\propto \delta n_{i,{\theta }_{\mathbf{k}}}$ on Fermi pocket $i=1,2$ in the presence of a current in the $x$ direction, for the same model as in Fig. 6. Here, ${\theta }_{\mathbf{k}}$ is the angle measured with respect to the $x$ direction. The solid (dashed) curve is the distribution function on the electron (hole) pocket. The colors correspond to the same cases shown in Fig. 6. All four distribution functions are calculated at the same temperatures $T\approx 0.002{\varepsilon }_F$.

Figure 8

Electron scattering on the Fermi surface due to small wave-vector nematic fluctuations. The intersection between the Fermi surface and the dashed lines are where the nematic form factor vanishes—the nematic “cold spots.” Small wave-vector scattering across the cold spots are strongly suppressed by the nematic form factor.

Figure 9

$\rho \left(T\right)$ in the high-temperature regime $T>{\mathrm{\Omega }}_L$, calculated for a compensated metal without cold spots. $r\left(T\right)\equiv {\xi }^{-2}\left(T\right)$ is the thermal mass for nematic fluctuations. The calculation assumes strong coupling ${\lambda }^2\gg {\varepsilon }_F$ and large fermion flavor $N$, so that ${\mathrm{\Omega }}_L\ll {\varepsilon }_F$, and our memory matrix approach remains valid.

Figure 10

Patches of momentum states. The blue circle is the Fermi surface. The width of the patch is determined by the typical scattering momentum due to critical nematica fluctuations. Near the QCP, the total quasiparticle density within each patch can be treated as a slow variable.

Figure 11

Interaction contributions to optical conductivity, to leading order in $1/N$. [(a) and (b)] DOS diagrams, (c) MT, and [(d) and (e)] AL diagrams. The current vertices are labeled by circles. Open/solid circles represent the inflow/outflow of the external frequency $\mathrm{\Omega }$.

Figure 12

Sketch of a two-electron scattering process for compensated metal. Blue Fermi pockets are electronlike and red Fermi pocket is holelike. The purple (green) arrows label the Fermi velocities in the initial (final) states. Such a process flips the direction of electrical current from upward to downward.

Figure 13

Eigenspectrum of the memory matrix $M_{\theta ,{\theta }^{\prime }}$ for a single Fermi surface when only quantum critical scattering is considered (black), when umklapp scattering is included (red), and when quenched random impurity is included (blue). In this calculation, we used the dispersion ${\varepsilon }_{\mathbf{k}}=-2t\left(cos{k}_x+cos{k}_y\right)-4t^{\prime }cos{k}_{x}cos{k}_y-\mu$, with $t=1$, $t^{\prime }=-0.3$, and $\mu =-1$. The nematic coupling strength is ${\lambda }^2\approx 2.62{\varepsilon }_F$, and impurity strength is $g_{\text{imp}}\approx 0.02{\varepsilon }_F$. The spectrum is obtained at temperature $T\approx 0.04{\varepsilon }_F$. We used 200 points to uniformly discretize the Fermi surface.