Quantum memory effects in disordered systems and their relation to $1/f$ noise

We propose that memory effects in the conductivity of metallic systems can be produced by the same two-level systems that are responsible for the $1/f$ noise. Memory effects are extremely long-lived responses of the conductivity to changes in external parameters such as density or magnetic field. Using quantum transport theory, we derive a universal relationship between the memory effect and the $1/f$ noise. Finally, we propose a magnetic memory effect, where the magnetoresistance is sensitive to the history of the applied magnetic field.

Figure 1

Figure showing the response of the conductivity to a change in the density $n_e$. The behavior is qualitatively similar for a change in magnetic field. The density is changed by $\delta n_e$ at $t=0$ and returned to its original value at $t=t_h$. The graph plots conductivity vs time for several different choices of $t_h$, but the same $\delta n_e$. There is a jump in the conductivity $\delta {\sigma }_1$ when the chemical potential is first changed and a second jump $\delta {\sigma }_2$ at $t=t_h$. The time scale is in arbitrary units. Figures offset slightly for clarity. The scale ${\sigma }_{*}$ is defined in Eq. (2.26). A positive ${\sigma }_2$ only appears when $t_h>\sqrt{t_i{t}_f}$ when 50% of the TLS are relaxed.

Figure 2

An illustration of semiclassical paths in the “interference” contribution to the probability to propagate from point $A$ to point $B$. The crossed circles represent static impurities and the reversed arrow indicates the complex conjugate of the amplitude.

Figure 3

Examples of an interference contribution to the variance of the conductivity. The crossed circles represent static impurities. (a) The pair of paths 1 and 2 contribute to the classical probability probability to propagate. Because the two paths are different they have a random phase, which means the sum over all paths is self-canceling. But, combined with the paths 3 and 4, the diagram makes a nonvanishing contribution to the variance of the conductivity. (b) A cooperon contribution, where the path 3 is the time reverse of path 1 and likewise for 4 and 2.

Figure 4

Figure showing the change in the geometric length of a path because of a shift in mobile impurity from position ${\vec{r}}_1$ to ${\vec{r}}_2$. The crossed circles represent static impurities and the solid dot shows two possible positions of a TLS.

Figure 5

Semiclassical paths demonstrating the memory effect. The impurity in the TLS is represented by a solid dot and the crossed circles represent static impurities. (a) A multiple scattering contribution to the scattering rate of the TLS with a random phase. (b) A contribution to the energy in the semiclassical picture. (c) An interference contribution to the covariance of the scattering rate and the energy.

Figure 6

Graph of the zero-bias anomaly in the conductivity. The conductivity and chemical potential are measured from the resting values. The curves are obtained by numerical integration of Eq. (3.35).

Figure 7

Graph showing the relaxation in a thin film of the conductivity singularity from the old Fermi level ${\mu }_i$ to the new Fermi level ${\mu }_f$. The curves are labeled by the fraction $K$ of TLS that have relaxed to the new equilibrium.

Figure 8

Plot of the magnetic memory effect. The curves plot the difference between $\sigma \left(0\right)$, the conductivity of a sample equilibrated in zero field, and $\sigma \left(\infty \right)$, the conductivity after the sample has equilibrated in a transverse field with magnetic length $L_{B_0}$. The curves are shown for different choices of the resting magnetic length $L_{B_0}$ and plotted in terms of the $L_B$, the magnetic field length when the conductivity is measured. They are obtained by numerical evaluation of Eq. (3.32).

Figure 9

The definition of the diagrammatic elements: (a) bare electron Green's function, (b) static impurity, (c) dressed electron Green's function, (d) and (e) the resummation for the cooperon and diffuson pole. The external fermion lines are amputated and the functions $\mathbf{D}$ and $\mathbf{C}$ are defined in Eqs. (3.11b) and (3.11a), respectively.

Figure 10

The definition of the diagrammatic elements: (a) current operator, (b) Keldysh Green's function, (c) electron distribution function, (d) expectation of the current operator. The factor of 2 comes from the spin summation. The Fermi function $f_F\left(\epsilon \right)\equiv {[1+exp(\epsilon /T)]}^{-1}$. Note the factor of $-i$ in the definition (c) of the average current.

Figure 11

The Hikami box subdiagrams. The external lines are amputated.

Figure 12

The diagrams contributing to the universal conductance fluctuations. They must be multiplied by the factor ${(-i)}^2$. Compare with Fig. 10.

Figure 13

The diagrams contributing to the noise. The TLS enters through subdiagram (b).

Figure 14

The diagrams contributing to the memory effect. The TLS enters through subdiagram (a). Note there is an overall factor of $i$ from the definition of $j$ in Fig. 10.

Figure 15

A normal electron-electron interaction, indicated by the double wavy line in figure (a), is effectively a delta function in time on the scales of interest. Therefore, diagrams of the form (b) do not contribute to the memory effect and there is no Fermi-liquid-type resummation.

Figure 16

Sketch of the electrochemical potential $\varphi \left(x\right)$ as a function of position $x$ in the transverse direction of the sample.