Absence of a quantum limit to charge diffusion in bad metals

Good metals are characterized by diffusive transport of coherent quasiparticle states and the resistivity is much less than the Mott-Ioffe-Regel (MIR) limit, $\frac{ha}{e^2}$, where $a$ is the lattice constant. In bad metals, such as many strongly correlated electron materials, the resistivity exceeds the Mott-Ioffe-Regel limit and the transport is incoherent in nature. Hartnoll, loosely motivated by holographic duality (AdS/CFT correspondence) in string theory, recently proposed a lower bound to the charge-diffusion constant, $D\gtrsim \hslash v_F^2/\left(k_{B}T\right)$, in the incoherent regime of transport, where $v_F$ is the Fermi velocity and $T$ the temperature. Using dynamical mean-field theory (DMFT) we calculate the charge-diffusion constant in a single band Hubbard model at half filling. We show that in the strongly correlated regime the Hartnoll's bound is violated in the crossover region between the coherent Fermi-liquid region and the incoherent (bad metal) local moment region. The violation occurs even when the bare Fermi velocity $v_F$ is replaced by its low-temperature renormalized value $v_F^{*}$. The bound is satisfied at all temperatures in the weakly and moderately correlated systems as well as in strongly correlated systems in the high-temperature region where the resistivity is close to linear in temperature. Our calculated charge-diffusion constant, in the incoherent regime of transport, also strongly violates a proposed quantum limit of spin diffusion, $D_s\sim 1.3\hslash /m$, where $m$ is the fermion mass, experimentally observed and theoretically calculated in a cold degenerate Fermi gas in the unitary limit of scattering.

Figure 1

Energy dependent spectral function $A_d\left(\omega \right)$ for various values of the interaction strength $U$ and temperature $T$. Panel (a): weakly correlated regime. (Green) dashed line is the density of states for the noninteracting case. Panel (b): moderately correlated regime. Panels (c) and (d): strongly correlated regime. With increasing $U$, the integrated spectral weight under central peak (Kondo resonance) gets transferred to high-energy Hubbard bands which corresponds to destruction of quasiparticle states. In these cases, there is a temperature-dependent crossover between strong-coupling regime (Fermi liquid) and local-moment regime (bad metals). All energies and temperatures are measured in units of $W$, the half bandwidth.

Figure 2

The zero-temperature quasiparticle renormalization factor $Z$ and charge compressibility ${\kappa }_e$ as a function of $U$. The system becomes increasingly incompressible with increasing $U$ and the quasiparticle weight $Z$ smoothly decreases, as the transition to the Mott insulator is approached at $U_c\simeq 3.4$ [26]. $U$ is measured in units of $W$.

Figure 3

Temperature dependence of the charge compressibility ${\kappa }_e$ for various correlation strengths $U$. The dashed line is the noninteracting $(U=0)$ case calculated using Eq. (39) and dotted lines are quadratic fits to the Fermi-liquid form Eq. (43). The apparent kinklike behavior for $U=2.5$ is due to the sharp crossover between the Fermi liquid and the bad metal (local moments fixed point). Both $T$ and $U$ are measured in units of $W$.

Figure 4

Resistivity shows violation of the Mott-Ioffe-Regel limit $({\rho }_{\mathrm{MIR}}=ha/e^2)$ with increasing $U$. This is consistent with the picture that transport becomes increasingly incoherent with increasing correlation effects. Both $T$ and $U$ are measured in units of $W$.

Figure 5

Scaled diffusivity shows violation of Hartnoll's bound in the strongly correlated incoherent regime of transport in the low-temperature region. However, in strongly correlated systems at large temperatures $(T\gg T_K)$ and for weakly correlated systems at all temperatures the bound is respected. Kinklike behavior for $U=2.5$ is due to sharp crossover between the Fermi-liquid and the local-moment regimes. Both $T$ and $U$ are measured in units of $W$.

Figure 6

Scaled diffusion constant for charge transport also violates quantum limit for spin-diffusion constant in the incoherent regime of transport. Inset: Detailed plot near the origin shows small yet nonvanishing scaled diffusivity for $U=2.5$. $D_H/D_A$ traces out the temperature dependence, $1/3\alpha k_{B}T$, of the Hartnoll bound showing how for $T<0.3$ it is larger than the proposed bound on the spin-diffusion constant for cold atoms. Both $T$ and $U$ are measured in units of $W$.