Critical strange metal from fluctuating gauge fields in a solvable random model

Building upon techniques employed in the construction of the Sachdev-Ye-Kitaev model, which is a solvable (0 + 1)-dimensional model of a non-Fermi liquid, we develop a solvable infinite-ranged random-hopping model of fermions coupled to fluctuating $U\left(1\right)$ gauge fields. In a specific large-$N$ limit, our model realizes a gapless non-Fermi-liquid phase, which combines the effects of hopping and interaction terms. We derive the thermodynamic properties of the non-Fermi-liquid phase realized by this model and the charge transport properties of an infinite-dimensional version with spatial structure.

Figure 1

A cartoon of our model. It consists of $N$ clusters indexed by $i,j,...$, each of which contains $M$ sites indexed by $\alpha ,\beta ,...$. Random hopping occurs between all possible pairs of intracluster and intercluster sites, but only intercluster hops are coupled to dynamic $U\left(1\right)$ gauge-fields $A_{ij}$. The model is solved in the $M,N\to \infty$ limit with $M/N$ fixed.

Figure 2

Diagrammatic representation of the fermion ($\mathrm{\Sigma }$) and regularized gauge-field $[\tilde{\mathrm{\Pi }}=\mathrm{\Pi }\left(i{\mathrm{\Omega }}_m\right)-\mathrm{\Pi }(i{\mathrm{\Omega }}_m=0)]$ self-energies for the Dyson equation (10). The black lines are fermion propagators, the red lines are gauge-field propagators, and the dashed blue lines are contractions of the Gaussian random variables $t_{ij}^{\alpha \beta }$ coming from the disorder average. These are the only diagrams that contribute in the large-$M,N$ limit.

Figure 3

Plot of the exponent $x$ giving the frequency scaling of the IR fermion self-energy vs $2M/N$ at half-filling.

Figure 4

(a) Plot of the filling $q_0$ vs the asymmetry parameter $\mathcal{E}$ for $2M=N$. (b) Plot of the function $-\kappa \left(x\right)$ defined in (29) vs the self-energy exponent $x$.

Figure 5

(a) Plot of the scaling form of the fermion self-energy $C^{-1}\left(0\right)t^{1-x}{F}_{\mathrm{\Sigma }}({\omega }_n/T,0)\equiv -\mathrm{Im}\left[\mathrm{\Sigma }\left(i{\omega }_n\right)\right]/T^x$ vs ${\omega }_n/\left(\pi T\right)$ obtained by numerical solution of the imaginary-time Dyson equations for different values of $T$. For all curves, $t=g^2=1,2M=N$, and $\mu =0$ [corresponding to $x=0.230651$ from (14)]. The curves collapse onto one another at low frequencies, confirming a universal low-energy scaling form. The deviations from universality at higher frequencies and temperatures occur because of the finite size of the critical window $\mathrm{\Lambda }$ over which the low-energy solution is valid. (b) Plot of the quantity $\nu \left(T\right)/T^{2x}$ vs $T$ for the same values of parameters. This clearly shows $\nu \left(T\right)\propto T^{2x}$ over a large range of temperatures.

Figure 6

(a) Plot of $r$ as a function of $\left|H\right|$ in the Higgsed phase, obtained from numerical solution of (A5) in the $T\to 0$ limit. The orange line fits the numerical data with $r=r_c+h_2{\left|H\right|}^2+h_3{\left|H\right|}^3$, so $r-r_c\sim {\left|H\right|}^2$ as $\left|H\right|\to 0$. The values of parameters used are $t=t_H=g^2=g_H=1,2M=N$, and $\mu =0$. (b) Plot of the free energies per fermionic degree of freedom of the $\left|H\right|\ne 0$ solution (orange) and $H=0$ solution (blue) of (A5). The weak first-order behavior at very small $\left|H\right|$ is due to a small finite $T={10}^{-5}$ in the numerics and disappears as $T\to 0$. The values of other parameters used are the same as in (a).