Mean-field theory for the three-dimensional Coulomb glass

We study the low temperature phase of the three-dimensional Coulomb glass within a mean-field approach which reduces the full problem to an effective single site model with a nontrivial replica structure. We predict a finite glass transition temperature $T_c$, and a glassy low temperature phase characterized by permanent criticality. The latter is shown to assure the saturation of the Efros-Shklovskii Coulomb gap in the density of states. We find this pseudogap to be universal due to a fixed point in Parisi’s flow equations. The latter is given a physical interpretation in terms of a dynamical self-similarity of the system in the long time limit, shedding light on the concept of effective temperature. From the low temperature solution we infer properties of the hierarchical energy landscape, which we use to make predictions about the master function governing the aging in relaxation experiments.

Figure 1

Evolution of the correlation hole (plasma dip) in the distribution of the instantaneous fields $P\left(h\right)$ for moderate disorder $W=2$. The curves are plotted for the inverse temperatures $\beta =0.25,0.5,1,2,4,7$ (from top to bottom at $h=0$) above the glass transition $({\beta }_c=7.149)$. The plasma dip starts opening as a perturbation of order $O(T_c∕T)$ at high temperature and develops into a significant depletion already for $T>T_c$.

Figure 2

Universal scaling of the plasma dip in the distribution of instantaneous fields in large disorder at $T=T_c$.

Figure 3

Typical ladder diagrams contributing to the variance of the spin-spin correlation function. The single lines represent disorder averaged propagators, the vertices correspond to the variance of the irreducible polarizability.

Figure 4

(Color online) Evolution of the pseudogap in the distribution of the thermodynamic fields $P\left(y\right)$ (top panel) and the instantaneous fields $P\left(h\right)$ (bottom panel) for moderate disorder $W=2$. The curves are plotted for the inverse temperatures $\beta =7.2,7.6,8,10,14,20,40,60,80,100$ (from top to bottom) below the glass transition $({\beta }_c=7.149)$. Note the rapid opening of the pseudogap in $P\left(y\right)$ just below the transition. Most of the suppression of the instantaneous density of states is already formed as a plasma dip at the transition.

Figure 5

(Color online) Evolution of the gap in the distribution of the thermodynamic fields $P\left(y\right)$ (top panel) and the instantaneous fields $P\left(h\right)$ (bottom panel) in strong disorder $W=10$. The curves correspond to inverse temperatures $\beta =8.2,8.4,8.7,9,10,14,20,40,60,80,100$ below the glass transition $({\beta }_c\approx 8.134)$. The salient features are similar as for moderate disorder, cf. Fig. 4.

Figure 6

(Color online) The distribution of instantaneous fields $h$ (solid lines) and thermodynamic fields $y$ (dashed lines) for temperatures well above ($\beta =2$, red), close to ($\beta =7.2$, green) and well below ($\beta =100$, blue) the glass transition at moderate disorder $W=2$. Notice the presence of a pronounced plasma dip in $P\left(h\right)$ even at high temperature.

Figure 7

(Color online) Main panel: Zero-field cooled compressibility ${\kappa }_C$ (ZFC, blue line), equilibrium field cooled compressibility (FC, green line) and the replica symmetric result (RS, red line) for $W=2$. The extrapolation of the RS prediction below $T_c$ is unphysical, but closely follows ${\kappa }_C^{\mathrm{FC}}$, as one may expect. Inset: Tunneling conductance $G$ as predicted by Eq. (81) in the glass phase (RSB). The unphysical replica symmetric theory (RS) would predict a strong suppression of $G$ due to a constant Onsager term and an ensuing hard gap in $P_{\mathrm{RS}}\left(h\right)$.

Figure 8

(Color online) The coupling function $\lambda \left(b\right)$ for $\beta =20,40,60,80,100$, and disorder $W=2$. The result is nearly independent of $\beta$ as long as ${\beta }_c⪡b\lesssim 0.7\beta$.

Figure 9

(Color online) The coupling function $\lambda \left(b\right)$ at low temperature $T=0.01$, for different disorders, $W=2,10$. In the regime $b⪢{\beta }_c$, the coupling function is universal, i.e., essentially independent of the bare disorder.

Figure 10

(Color online) Illustration of the temporal flow of $m(x,z)$ for increasing Sompolinsky times $x=b∕\beta =1,0.9,0.8,\dots ,0.2$ for $\beta =100$ and $W=2$. An intermediate fixed point is clearly approached near $x=0.5$. The asymptotic prediction for the fixed point is plotted as a dashed line (black), for which we used an artificial power-law fitting ${\psi }_{3\mathrm{D}}$ beyond its universal-low energy regime.

Figure 11

(Color online) Illustration of the temporal flow of $p(x,z)$ for increasing Sompolinsky times $x=b∕\beta =1.0,0.9,\dots ,0.2$ for $\beta =100$ and $W=2$. The apparent fixed point occurring around $x=0.5$ comes close to the fixed point predicted with an artificial power law which fits ${\psi }_{3\mathrm{D}}$ beyond its universal low energy regime (dashed black line). The remaining small discrepancy is due to the approximate nature of the power-law fit.

Figure 12

Universal, asymptotic low temperature scaling function ${\Psi }_{3\mathrm{D}}\left(z\right)=P(1,y=zT)∕T^2=p(1,z)$ (full line) and the fixed point function $p^{*}\left(z\right)$ (dashed line). Both have the same curvature at large $z$.

Figure 13

(Color online) Scaled distribution of thermodynamic fields as a function of temperature. The curves correspond to $W=2$ and $\beta =20,40,60,80,100$ (from top to bottom). Top panel: Scaling with the Efros-Shklovskii exponent $\alpha =2$, $p(1,z)=P(1,y=zT)∕T^2$. The inset demonstrates that the differences $\Delta P=\mid P_{W=2}-P_{W=10}\mid$ due to the strength of disorder tend to zero in the low $T$ limit, as expected from the universality induced by the fixed point. Bottom panel: Improved scaling with an artificial power $\tilde{\alpha }=2.34$, $p(1,z)=P(1,y=zT)∕T^{\tilde{\alpha }}$.

Figure 14

$T=0$ solution of the coupling function $\lambda \left(x\right)={\left(\beta x\right)}^3\dot{\Lambda }\left(x\right)∕2\beta$ for the 3D electron glass. The fixed point controls the value $\lambda (x\to 0)={\lambda }^{*}$. The drop to 0 around $x_P\approx 0.78$ indicates a plateau in the coupling and overlap function.

Figure 15

(Color online) Equilibrium distribution of the rescaled overlap distance, $\widehat{d}=(1-Q)∕T^3$, in the 3D electron glass. There is a peak of strength $1-x_P$ at ${\widehat{d}}_{\mathrm{EA}}=0.8609$ corresponding to the intrastate (Edwards-Anderson) overlap, and an asymptotic power-law tail $P\left(\widehat{d}\right)\sim {\widehat{d}}^{-(2+\alpha )∕(1+\alpha )}={\widehat{d}}^{-4∕3}$.

Figure 16

Schematic representation of the energy landscape assumed in the trap model. The depth of the traps is exponentially distributed, deeper traps being rarer than shallow ones. To make contact with the replica solution we must assume that the states in a cluster have a common “ceiling” or threshold level beyond which the system easily escapes and becomes trapped in any other state within the cluster.

Figure 17

(Color online) Symbol lines: 3D data taken from Ref. 68, corresponding to uniformly distributed disorder with variance $\tilde{W}=⟨{(ϵ∕2)}^2⟩=0.29,0.58,1.15,2.30$ in units of $e^2∕4\kappa \ell$ (from bottom to top at low energies). The full lines (blue and red) are our mean-field predictions for $P\left(h\right)$ at $T=0.01$, for Gaussian disorder with $W=2$ and $W=10$, respectively. Dashed line: Prediction from the ES stability argument (Ref. 76), $\rho \left(E\right)=(3∕\pi )E^2$. In the plot, we use the standard units $e^2∕\kappa \ell$ and $\ell$ for energy and length, respectively.

Figure 18

(Color online) Universality of the SK model in random fields. Top: Scaling of the local field distribution $P\left(y\right)$ for various temperatures $10⩽\beta ⩽60$ and disorders $0⩽W⩽1.5$. Bottom: Unscaled distribution $P\left(y\right)$ at $\beta =20$ for various disorder strengths, $W=0,0.3,0.6,0.9,1.2$.