Superconductor-insulator transition and energy localization

We develop an analytical theory for generic disorder-driven quantum phase transitions. We apply this formalism to the superconductor-insulator transition and we briefly discuss the applications to the order-disorder transition in quantum magnets. The effective spin-$\frac{1}{2}$ models for these transitions are solved in the cavity approximation which becomes exact on a Bethe lattice with large branching number $K⪢1$ and weak dimensionless coupling $g⪡1$. The characteristic feature of the low-temperature phase is a large self-formed inhomogeneity of the order-parameter distribution near the critical point $K\ge K_c\left(g\right)$, where the critical temperature $T_c$ of the ordering transition vanishes. We find that the local probability distribution $P\left(B\right)$ of the order parameter $B$ has a long power-law tail in the region where $B$ is much larger than its typical value $B_0$. Near the quantum-critical point, at $K\to K_c\left(g\right)$, the typical value of the order parameter vanishes exponentially, $B_0\propto e^{-C/\left[K-K_c\left(g\right)\right]}$ while the spatial scale $N_{inh}$ of the order parameter inhomogeneities diverges as ${\left[K-K_c\left(g\right)\right]}^{-2}$. In the disordered regime, realized at $K<K_c\left(g\right)$ we find actually two distinct phases characterized by different behavior of relaxation rates. The first phase exists in an intermediate range of $K^{\ast }\left(g\right)<K<K_c\left(g\right)$. It has two regimes of energies: at low excitation energies, $\omega <{\omega }_d\left(K,g\right)$, the many-body spectrum of the model is discrete, with zero-level widths, while at $\omega >{\omega }_d$ the level acquire a nonzero width which is self-generated by the many-body interactions. In this phase the spin model provides by itself an intrinsic thermal bath. Another phase is obtained at smaller $K<K^{\ast }\left(g\right)$, where all the eigenstates are discrete, corresponding to full many-body localization. These results provide an explanation for the activated behavior of the resistivity in amorphous materials on the insulating side near the superconductor-insulator transition and a semiquantitative description of the scanning tunneling data on its superconductive side.

Figure 1

(Color online) Iteration of the cavity equations three times (here with $K=2$). The field $B_0$ on the root is proportional to the infinitesimal field $B$ added on the boundary. The corresponding susceptibility, given in Eq. (9), is equal to the partition function of the DP problem, obtained by summing over all the paths from the root to the boundary (such as the one shown by the dashed line). In this sum, each path has a weight equal to the product of terms associated with each edge that it visits.

Figure 2

(Color online) Main panel: phase diagram in plane $\left(g,T\right)$ for $K=4$. The full lines show the critical temperature as function of $g$. The low-temperature phase is superconducting and the high-temperature phase is insulating. The top curve is the naive mean-field prediction which gives the correct result above $T_{\text{RSB}}=0.0207$. The bottom curve is the result of the analysis on the Bethe lattice described in the text that includes the RSB effects in the DP problem, which occur at temperatures $T<T_{\text{RSB}}$. The insert shows the critical temperature as function of $K$ for $g=0.129$, a value which roughly corresponds to the experimental situation in disordered InO films (see Sec. 6). For this value of $g$, the replica symmetric solution gives a $K$-independent transition temperature $T_c=0.001$. This prediction of the replica symmetric theory is correct for $K>K^{\text{RSB}}\simeq 6$. For smaller $K$ the transition temperature starts to drop, the quantum-critical point corresponds to $K_c\simeq 2.2$. Notice that in a rather wide regime the replica symmetry is broken but the effect of the breaking on the transition temperature is small.

Figure 3

(Color online) The typical order parameter vanishes with an essential singularity at the superconductor-insulator transition. Upper curves show the logarithm of the typical fields (defined by $\text{ln}{B}_{typ}=\overline{\text{ln}B}$ obtained by population dynamics with small external fields: $h={10}^{-8},{10}^{-9},{10}^{-10}$ (from top to bottom). The lower curve shows the fit to $1/\left[{\left(g/g_c\right)}^m-1\right]$ dependence expected from Eq. (45).

Figure 4

(Color online) Distribution function, $B^{1+m}P\left(B\right)$ for $K=4$ and different values of $\left(g-g_c\right)/g_c$ shown next to each curve. As expected, $B^{1+m}P\left(B\right)$ is field-independent in a range of fields which is wide in logarithmic scale and becomes wider as the transition is approached.

Figure 5

(Color online) Comparison between the typical fields (measured by the harmonic mean) computed with the population dynamics of the full cavity Hamiltonian (5) and the simplified one in Eq. (6). The line corresponds to the results of the simplified mapping, the dots to the mapping given by the full cavity Hamiltonian.

Figure 6

(Color online) Laplace transform of the distribution function produced by a small external field at $g=g_c$ for $K=4$. Main panel shows the regime of small $s$, the inset shows the behavior at large $s$. The dashed lines show the asymptotic behavior at small and large $s$ discussed in the text.

Figure 7

(Color online) Average and typical fields induced by an external field at $g<g_c$ and $K=4$. The lower set of curves shows the typical response $B_{typ}/h=\text{exp}\left(\overline{\text{log}B}\right)/h$ for external fields $h={10}^{-6},{10}^{-7},{10}^{-8},{10}^{-9}$ (bottom to up). The upper set of curves shows the average response $B_{av}/h=\overline{B}/h$ for the same values of the external field. The curves show no singularity at the critical value of $g$ indicated by arrow. As discussed in the text, the average response to the external field is controlled by the far tail of the distribution function and is nonlinear so the ratio $B_{av}/h$ grows at $h\to 0$.

Figure 8

(Color online) Critical frequency, ${\omega }_d$, that separates zero-width levels from the levels with nonzero width, plotted as a function of $K$ for $g=0.129$. This value of $g$ was chosen because it corresponds to the experimentally relevant transition temperature at large $K:T_{\text{BCS}}=0.001$ (see Sec. 6). At $K<K^{\ast }$ all states have zero width. In the intermediate regime $K^{\ast }<K<K_c$ states to the left of ${\omega }_d\left(K\right)$ line have zero width (infinite decay time). The insert shows a blow-up region around $K_c$.

Figure 9

(Color online) Coupling of similar pairs of spins (dipoles). The spin at site 0 is strongly coupled to the spin at site $j$ at distance $R=3$ from it so that the excited states of these two spins are entangled. The spin at site $k$ is similarly strongly coupled with spin at site $l$. Each of these pairs of spins (shown by magenta arrows) forms two close energy levels. If the interaction between these pairs (shown by dotted magenta arrow) is significant, it might result in the propagation of these excitations.

Figure 10

(Color online) Blue line: dependence of the critical temperature $T_c\left(K\right)$ on $K$ of the fully frustrated system for $g=0.1$. Red dashed line: result of the simple mean field approximation for the problem without frustration. The maximum of the blue line corresponds to the case where the order parameter distribution decays with an exponent $x=1$ and $K=K^{\text{RSB}}$, where $K^{\text{RSB}}$ is determined for the unfrustrated model by Eq. (21). At $K<K^{\text{RSB}}$ the critical temperatures of the regular and the frustrated model coincide.

Figure 11

(Color online) Distribution function of the local order parameter for the strongly frustrated model with $K=4$ at three different values of the coupling constant. The number near each curve indicates the corresponding value of $\left(g/g_c-1\right)$. The plateau in $P\left(B\right)B^{m+1}$ demonstrates the existence of the scaling regime in Eq. (31).

Figure 12

(Color online) Typical amplitude of the order parameter as function of the proximity to the quantum-critical point.