Disorder-enhanced superconductivity in a quasi-one-dimensional strongly correlated system

We perform an analytical and numerical study of a superconducting instability in quasi-one-dimensional (quasi-1D) disordered systems. Modeling them as an array of Luttinger liquids with Josephson-type interchain coupling, we employed renormalization-group analysis with an extensive search for parameters that support superconductivity enhancement. We have found that this phenomenon is possible in the parameters range that support a latent disorder-driven phase transition between charge- and spin-density-wave phases. Our results may explain the experimental observation of disorder-enhanced superconductivity.

Figure 1

The phase portrait of a system without disorder. The purple lines indicate the separatrix of the RG equations for $D=0$. The blue line indicates the SDW phase at $y=1$, whereas the black line indicates the CDW phase at $y=-1$.

Figure 2

$K_{\sigma }^{\left(0\right)}=0.6,K_{\rho }^{\left(0\right)}=2.2,u^{\left(0\right)}=8$ - a typical phase diagram for $K_{\sigma }<1$ displaying the SDW/CDW-boundary moved upwards by disorder and the Luttinger liquid phase destroyed.

Figure 3

$K_{\sigma }^{\left(0\right)}=1.1,K_{\rho }^{\left(0\right)}=1.5,u^{\left(0\right)}=1$ - a typical phase diagram for $K_{\sigma }>1$ with no SDW/CDW-boundary and a destroyed Luttinger liquid phase.

Figure 4

The variation in $y\left(l\right)$ and $K_{\sigma }\left(l\right)$ as a function of $l$ for a disordered system. For the weak disorder, the RG trajectories flow into a SDW phase, however as disorder is increased above a critical value the system enters into a CDW phase.

Figure 5

The temperatures of the second-order phase transitions between the TLL and spin gapped phases as a function of the disorder for the initial conditions given by $K_{\sigma }^{\left(0\right)}=0.8$, $y^{\left(0\right)}=0.25$, $K_{\rho }^{\left(0\right)}=1.95$, $u^{\left(0\right)}=8$, with $D^{\left(0\right)}=0.01,0.015,0.02,0.05,0.07,0.16,0.18,0.19,0.20,0.21,0.22,\text{and}0.25$. The red dashed line is a plot of Eq. (12), where $\nu =0.33$, and $\alpha =1.27$, to fit the data. For the weak (strong) disorder, blue (black) circles represent the temperature at which system goes into SDW (CDW) state. The critical disorder for the quantum phase transition is $D_{\text{cr}}^{\left(0\right)}\approx 0.12$. (Inset) Phase diagram in the $y^{\left(0\right)}-D^{\left(0\right)}$ plane for $K_{\sigma }^{\left(0\right)}=0.8$, $K_{\rho }^{\left(0\right)}=1.95$, and $u^{\left(0\right)}=8$. This is valid for low temperatures where the TLL phase is not relevant.

Figure 6

High-$T$ resistivity for different disorders. The graphs are cut at the minimum of the resistivity where the spin gap opens at lower temperatures because the system goes into a SDW phase. The resisitivity would continue upwards for the spin gapped set of RG equations (8) and (9).

Figure 7

This figure shows the minimum in resistivity occurs before any spin gapped or superconducting phase is reached. When the lines on the figure after the upturn stop, this indicates that the system has entered either a superconducting phase or a CDW phase. The only disorder which enters a CDW phase is the strongest disorder ($D^{\left(0\right)}=0.22$) as superconductivity emerges before the spin gapped phase for the lower disorders. For the CDW phase, the spin gapped RG equations must be solved, and the value at which superconductivity occurs must be found. The onset of superconductivity is discussed in more detail in the next section.

Figure 8

Resistivity minimum temperature as function of disorder. Three left symbols correspond to $y=+1$, small brown circles are scaled experimental results, and five right symbols correspond to a “real” resistivity minimum.

Figure 9

$T_{\text{peak}}$ as function of disorder, including the spin gapped mechanisms. This figure shows a plateau from the third weakest disorder to the next strongest disorder. The three weakest disorders occurs via a SDW mechanism. There is an almost linear increase in $T_{\text{peak}}$ as disorder increases from the gapless phases through to the CDW phase. The strongest disorder occurs via a CDW mechanism, which has a larger relative increase. This does suggest a CDW mechanism is favourable for disorder-induced superconductivity.