Low temperature $T$-linear resistivity due to umklapp scattering from a critical mode.

In this paper we consider a model which assumes a two dimensional Fermi sea where the fermions with dispersion $\epsilon_{\mathbf{k}}$ are being scattered by a critical bosonic mode with the propagator $P(q,\omega)=1/(Cq^{2}-i\omega/q)$ so that the spectral function $D(q,\omega)=Im[P(q,\omega)]$ is given by

To simplify notations, throughout this paper we measure energy, temperature or frequency in units of the nearest neighbor fermion hopping parameter $t$ and we set the lattice constant to be unity. The fermions are scattered by this mode with a coupling constant $g$. Equation 1 has arisen in several contexts. For example, this equation describes the critical fluctuations at a quantum critical point that separates a Fermi surface with four-fold symmetry to one with two-fold symmetry. This is sometimes called the electronic nematic phase transition or the Pomeranchuk instability. This model is considered to be one of the leading candidates for a quantum critical point in the Cuprates. Nematicity is also known to play a prominent role in the iron based superconductors.Hussey et al. (2018) It is also worth noting that nematicity, $T$-linear resistivity and van Hove singularity are ingredients that are showing up in twisted bilater graphene as well.Cao et al. (2020) Importantly, the singularity is for small momentum $q$. Oganesyan et al. (2001); Maslov et al. (2011); Hartnoll et al. (2014) This is very different from a charge or spin density wave or an antiferromagnetic instability, where the singularity occurs at a finite $q$.Hlubina and Rice (1995); Rosch (2000) Another example is the scattering of fermionic spinons by U(1) gauge fluctuations that arise in spin-charge separated models in certain quantum spin liquids. Lee and Nagaosa (1992) This model has also been discussed in the composite fermion description of the half-filled Landau level. Halperin et al. (1993) The $\omega/q$ term in the propagator $P(q,\omega)$ comes from Landau damping of the mode by the Fermi sea. The mode is overdamped with $\omega$ that scales as $q^{3}$. This leads to a fermion self energy $\Sigma(\omega)$ that goes in leading order as $\omega^{2/3}$, thereby violating the Landau criterion for a well defined quasi-particle. Lee and Nagaosa (1992) As a result, this model is considered to be an example of a non-Fermi liquid.

We have earlier suggested that this model gives rise to a transport scattering rate that goes as $T^{4/3}$. Lee and Nagaosa (1992) This was based on the assumption that the bosonic mode equilibrates to the lattice, but this assumption has been rightly criticized on the grounds that the total momentum is conserved in the scattering process and therefore there is no mechanism for relaxing the current. Maslov et al. (2011); Hartnoll et al. (2014) Some form of momentum relaxation such as disorder or umklapp scattering needs to be introduced. Ref. Maslov et al. (2011) considered impurity scattering and concluded that the leading T dependent correction for a singly connected convex 2D Fermi surface has a power greater than 2 and saturate for higher temperatures. In ref. Hartnoll et al. (2014),the problem was worked out for several versions of disorder including a linear coupling to the nematic order parameter. In this paper we focus on umklapp scattering as the mechanism of momentum relaxation. The motivation is the following. In the Cuprates, the Fermi level is close to a van Hove singularity so that the minimum umklapp vector $\Delta_{q}$ connecting neighboring Fermi surfaces is small near the antinode, as seen in fig 1. Due to the $q^{3}$ scaling in Eq. 1, the temperature scale when the umklapp scattering kicks in can be very low, being proportional to $\Delta_{q}^{3}$. It will be interesting to find out what the resistivity is above this low temperature scale. A more general remark is that because the mode is overdamped, the umklapp effect is not exponentially small at low $T$ as in the case of electron-phonon interaction, but will in general be power-law down to the lowest temperature. As we shall show, what we find is a $T^{2}$ behavior that crosses over to $T$ at a very low temperature scale. To see how this come about, we make a quick estimate of the scattering rate $\tilde{\gamma}$ of a fermion at the umklapp point ( $\theta=0$ ) with energy $T$ by emitting a boson mode with momentum $q$ to reach a final state at $\phi$ on the umklapp Fermi surface, a seen in the inset of Fig. 1

In the second step we replaced the Bose factor $n_{0}(\omega)+1$ by $T/\omega$ and the integration limit by $T$. If $T>C\Delta_{q}^{3}$, we can first perform the $\omega$ integration by extending the upper limit to $\infty$. The integrand is now $1/q^{2}$. We approximate $q^{2}\approx\Delta_{q}^{2}+(k_{F}\phi)^{2}$ , and the $\phi$ integration gives a constant. This shows that the scattering rate is proportional to $T/\Delta_{q}$. In the opposite case of $T<C\Delta_{q}^{3}$ it is easy to see that the upper limit of the $\omega$ integral gives an additional factor of $T$, leading to a $T^{2}$ behavior. As mentioned earlier, unlike the electron-phonon problem, the scattering rate is never exponentially activated. A more accurate calculation shown later gives the same result. It is worth remarking than since umklapp scattering involves a final state that carries current in the opposite direction, there is no significant distinction between a transport lifetime which is sensitive to the momentum relaxation of each collision and the collision time, in contrast to the case of forward scattering. Lee and Nagaosa (1992) On the other hand, since the scattering is concentrated at certain points on the Fermi surface, in order to calculate transport properties it is necessary to determine how the momentum distribution relaxes in the vicinity of these points. We need to solve the Boltzmann equation using the scattering processes as input. This is done in the following sections.

We will calculate the resisitivity, magneto-resisitivty, Hall conductivity and the Hall constant $R_{H}$. The key physics is that the important scattering process is concentrated near the antinodal point. As such, there is a lot of overlap in the mathematical framework with a different problem involving scattering by antiferromagnetic fluctuations at momentum $(\pi,\pi)$ where strong scattering are concentrated at hot spots. Hlubina and Rice (1995) This problem was thoroughly investigated by Rosch Rosch (2000) and his work was very helpful to us. In particular he found a linear T resistivity and linear B magnetoresistivity in certain parameter range. Just as in his case, we found that while some impurity scattering is needed to restrict the modification of the distribution function to the vicinity of a special point on the Fermi surface, the coefficient of the linear $T$ term is independent of disorder. In this paper we focus on the relatively low temperature regime where the impurity scattering rate is comparable to the inelastic scattering rate, ie. when the temperature dependent part of the resistivity is comparable to the zero temperature limit. We defer discussion of the high temperature limit, or the clean limit, to a later paper.

where $\Sigma^{\prime}$ and $\Sigma^{\prime\prime}$ are the real and imaginary part of the self energy. For $\Sigma^{\prime\prime}(\omega)\sim$ $\omega^{2/3}$ the spectral function is indeed very broad as a function of $\omega$, but at a given small $\omega$, it is actually a sharp function of $\epsilon_{k}-\mu$. This distinction is a familiar one in the modern ARPES literature, where the energy dispersion curve (EDC) is broad and the momentum dispersion curve (MDC) is sharp. The conventional derivation of the Boltzmann equation starting from the non-equilibrium Green’s function involves integrating over $\omega$. What Prange and Kadanoff proposed to do is to pick a patch of the Fermi surface specified by $k(\theta)$ which defines a contour along the Fermi surface parametrized by the direction $\theta$ of the Fermi momentum vector and integrate the perpendicular momentum and hence over $\epsilon_{k}$ instead. As a result, instead of the usual distribution function $f(\mathbf{k})$ they introduced the function $f(k(\theta),\omega)$. Note that in the usual formulation, the vector $\mathbf{k}$ is decomposed into $k(\theta)$ and the momentum in the perpendicular direction, which is then represented by $\epsilon_{k}-\mu$. Thus we can simply write down the standard Boltzmann equation with the understanding that in the momentum $\mathbf{k}$, the component perpendicular to the Fermi surface is replaced by $(\epsilon_{k}-\mu)/v_{F}$ and $(\epsilon_{k}-\mu)$ is in turn is replaced by $\omega$. We next introduce in the standard way the linear perturbation from the equilibrium by writing $f(\mathbf{k})$=$f_{0}(\epsilon_{\mathbf{k}})-(df_{0}(\epsilon_{\mathbf{k}})/d\epsilon_{\mathbf{k}})\tilde{\Phi}(\mathbf{k})$= $f^{0}_{\mathbf{k}}+f^{0}_{\mathbf{k}}(1-f^{0}_{\mathbf{k}})\tilde{\Phi}(\mathbf{k})/T$ where $f_{0}(\epsilon_{\mathbf{k}})=f^{0}_{\mathbf{k}}$ is the Fermi distribution and $\tilde{\Phi}(\mathbf{k})$ has the interpretation of a local shift of the chemical potential at patch $\theta$. This leads to the linearized Boltzmann equation Maslov et al. (2011):

where the electron-electron collision term is given by

where $\mathbf{G}$ denotes a set of reciprocal lattice vectors and the electron-impurity collision term is $I_{ei}=-\gamma_{0}\tilde{\Phi}(\mathbf{k})(df_{0}(\epsilon_{\mathbf{k}})/d\epsilon_{\mathbf{k}})$ where $\gamma_{0}$ is the impurity scattering rate. Note that we have described the scattering process as a electron-electron scattering via the exchange of the boson propagator $D(q,\omega)$, rather than the scattering between electrons via the emission and absorption of bosons. The reason is that the conservation of total momentum for the normal scattering ( $\mathbf{G}=0$) is clearly seen in Eq. II. The distribution will rapidly equilibrate to a Fermi function with a shifted overall momentum $\tilde{\Phi}(\mathbf{k})\propto\mathbf{k.E}$ so that its contribution to the collision integral is exactly zero. Had we considered the emission and absorption of bosons, we would have to keep track of the shifted boson distribution function as well, which introduces more complications. Next, we will show that the vanishing of the normal collision term is more general. As long as the scattering is predominantly forward, then in order to conserve momentum and energy, the scattering partner $\mathbf{p}$ must be approximately equal to either $\mathbf{k}$ or $-\mathbf{k}$. In the first case the last two terms in [ ] in Eq II cancel the first two terms to first order in $q$. In the second case, since $\tilde{\Phi}(\mathbf{-k})=-\tilde{\Phi}(\mathbf{k})$, the first two terms in [ ] in Eq II approximately cancel. A similarly argument applies for the next two terms corresponding to the outgoing pair $\mathbf{p+q}$ and $\mathbf{k-q}$. In the presence of impurity scattering which ensures that $\tilde{\Phi}(\mathbf{k})$ is smoothly varying, these arguments show that the contribution from normal scattering to the collision integral is negligible. This argument works also in the presence of a magnetic field. The Fermi surface is shifted along an axis that is rotate from the direction of the applied field $\mathbf{E}$ but the relation $\tilde{\Phi}(\mathbf{-k})=-\tilde{\Phi}(\mathbf{k})$ continues to hold. So from this point on we will keep only the umklapp term ( $\mathbf{G}\neq 0$ ) in Eq. II. On the other hand, in the clean limit, it is necessary to keep the normal scattering because the small residual terms left over from the approximate cancellation control the local equilibration and slow diffusion of the the disturbance of $\tilde{\Phi}(\mathbf{k})$ along the Fermi surface towards the umklapp points. This is beyond the treatment of the current paper and is the reason why the results of this paper are limited to relatively low temperatures.

Our next step is to integrate out $\mathbf{p}$ and the radial part of $\mathbf{k}$ taking advantage of the fact that $df_{0}(\omega)/d\omega)$ is a sharply peaked function. To be concrete we choose the origin in $\mathbf{k}$ space so that it is completely enclosed by the Fermi surface and we parametrize the Fermi surface by the angle $\theta$ as shown in Fig 1. Leaving details to the Appendix, we find

where $\nu(\theta)=k_{F}/(2\pi v_{F})$ is the density of states factor at angle $\theta$. In this equation, all variables are defined on the Fermi surface as parametrized by $\theta$ and $\Phi(\theta)$ is defined as $\tilde{\Phi}(k(\theta),\omega=0)$. The kernel of the integral equation $F(\theta,\theta^{\prime})$ depends on $\theta$ and $\theta^{\prime}$ only via the magnitude of the momentum transfer $q(\theta,\theta^{\prime})$ which connects the points $\theta$ and $\theta^{\prime}$ on the umklapp Fermi surface as shown in Fig. 1 . Here we focus on $\theta$ near zero, and the discussion should be repeated for $\theta$ close to the other three nearest neighbor umklapp Fermi surfaces. The kernel is given by

where $n_{0}(\omega)$ is the Bose function. Note that after integration out the scattering partner, the Boltzmann equation now takes the form of umklapp scattering by emission and absorption of the mode defined by $D(q,\omega).$ The kernel can be written in the form $F(\theta,\theta^{\prime})=2qf(a)$ where $a=Cq^{3}/T$ and the scaling function $f(a)$ is given by

where $f(a)=\pi/2a$ for small $a$ and $\pi^{2}/3a^{2}$ for large $a$. Hlubina and Rice (1995) For our numerical work we find it useful to approximate $f(a)$ by the form suggested by Rosch Rosch (2000) which matches the large and small $a$ limits and is accurate to 2 % everywhere: $f(a)=\pi^{2}(2\pi+a)/[a(4\pi^{2}+6\pi a+3a^{2})]$.

The coefficient of $\Phi(\theta$) on the right hand side of Eq. 6 can be interpreted as the scattering out rate $\gamma(\theta)$ for an electron at $\theta$.

We now estimate $\gamma(\theta)$ for small $\theta$. In Fig 1 the line connecting $\theta$ to the center of the next Brillouin zone defines $q_{0}(\theta)$ which is the shortest distance from $\theta$ to a point on the umklapp Fermi surface. Then $q(\theta,\theta^{\prime})$ can be approximated by $\sqrt{q_{0}^{2}(\theta)+(k_{F}\phi)^{2}}$ The $\theta^{\prime}$ integral can be replaced by an integral over $\phi$. For $T\ll Cq_{0}^{3}(\theta)$, we use the form for large $a$ given after Eq. 8. The integral over $\phi$ is rapidly convergent and we find $\gamma(\theta)\propto T^{2}/(v_{F}C^{2}q_{0}^{4}(\theta)).$ Note that it is a rapidly decaying function of $q_{0}(\theta)$.

Next we consider the case $T>Cq_{0}^{3}(\theta)$ and use the small $a$ limit of $f(a)$. The $\phi$ integration is easily done to give

where $\alpha=g^{2}/(2v_{F}C)$. The linear $T$ dependence of the scattering out rate near the umklapp point which onsets at very low temperature is one of the main results of this paper.

Acknowledgment: I thank Andrey Chubukov and Dmitrii Maslov for helpful discussions. This work has been supported by DOE office of Basic Sciences grant number DE-FG02-03ER46076.