Quantum Phases of a Strongly Disordered Two-Legged Josephson Ladder

TL;DR

The paper addresses how strong spatial disorder modifies the superconductor–insulator transition in a two-leg Josephson ladder with $E_J\sim E_C$ while preserving a $\mathbb{Z}_2$ leg-symmetry. It develops a strong-disorder real-space RG framework, tracking distributions of charging, Josephson, and phase-coupling parameters and introducing regular, bowtie, and doublet site types, to derive phase-favoring flow and effective Hamiltonians. The analysis yields a phase diagram with three disorder-dominated phases: a disordered superconductor, an intermediate Bose glass, and a spin-glass insulator, connected by KT-like and spin-chain–driven transitions; the intermediate BG phase illustrates a dimensional crossover from a ladder toward 2D-like behavior. The study highlights the crucial role of the ladder’s symmetry in enabling richer phases than in a simple chain and suggests experimental platforms (engineered Josephson ladders or cold-atom analogs) to observe the Bose glass and spin-glass–like insulator phenomena, with transport signatures distinguishing the phases.

Abstract

Disordered superconductors in low dimensions provide an exemplary manifestation for the role of quantum fluctuations in a many-body system. Specifically in Josephson arrays with comparable Josephson and charging energies ($E_J\sim E_C$), disorder tends to change the nature of the paradigmatic Superconductor-Insulator Transition (SIT) and potentially leads to formation of multiple distinct phases. We address this problem in a model of a two-legged Josephson ladder subjected to a wide spatial distribution of its parameters along the legs. In contrast, we assume the system to have a perfect $\mathbb{Z}_2$ symmetry to interchange between the legs, and investigate the effects of spatial randomness which preserves this symmetry in the strong-disorder limit. To this end, we apply a strong randomness real-space renormalization group technique and explore the resulting phase diagram. We identify three disorder-dominated phases, including an intermediate phase between a disordered superconductor and a disordered insulator. The latter insulating phase can be mapped to a XY spin-chain in a spin glass phase, while the intermediate phase turns out to be a Bose glass.

Figures (11)

• Figure 1: The basic system described in equations (\ref{['eqn:Hsum']})-(\ref{['eqn:Hrung']}). Different colors correspond to different Hamiltonian terms, and different hues stand for different energies (either Josephson for the bonds or Charging for the sites). Note that while every bond and every site have different energies, the up-down symmetry is preserved.
• Figure 2: The five unit cells involving R sites and B sites, and a section from a ladder. Top row (left to right): $R_{r=}, R_{r\times}, R_b$. Middle row (left to right): $B_r, B_b$. Bottom row: a connected chain of the four unit cells $R_{r=}$, $R_{r\times}$, $R_b$, $B_b$. A vertical line denotes $K \cos\left(\phi_u-\phi_d-\varphi_K\right)$; a horizontal or diagonal line are $J \cos\left(\phi_1-\phi_2\right)$; a bold dot is a superconducting grain.
• Figure 3: Unit cells involving a doublet (D) site. Top row (left to right): $D_r$, $D_b$, $D_{r\times}$ where the dashed arcs denote $\tilde{J}$-type bonds. Middle row (left to right): $D_d$, $D_{d\times}$; the former (with three lines) generated from $R_{r=}$ sites and the latter from $R_{r\times}$. Bottom row: a bond decimation of an $R_d-D_r$ configuration with D on site $i$. The red bond is decimated, and the resulting $R_{r\times}$ site has interactions dominated by the next-nearest-neighbor blue and cyan terms, not by the black nearest-neighbor term. The resulting $J_{i-1, i+1}$ is equal to $\tilde{J}^{NNN}_{i-1, i+1}$ up to a non-universal factor of order unity, and similarly for $\tilde{J}^{NNN}_{i-1, i+1, \times}$ and $J_{i-1, i+1,\times}$.
• Figure 4: Four RG steps performed one by one on a small ladder. The large energy scale is bold in red. The steps, by order: a bond-decimation step on an $R_b$ site; a site charge-locking step on a $B_b$ site; a site charge-locking step on an $R_b$ site, which results with a D site; a site charge-locking step on an $R_d$ site, which results with full decimation.
• Figure 5: An illustration of the distribution of $\kappa$. (a) The blue area marked with $\gamma_h$ is the region $\Omega e^{-d\Gamma}<K<\Omega$, or $1-d\Gamma<\left|\kappa\right|<1$, soon to undergo a site phase-locking step as the flow goes on. Two pairs of points, in black and in blue, show the addition process. The blue pair will be immediately site phase-locking decimated, as they landed outside of the circle; the black remains in the circle and will be rescaled to the adjacent gray point after the blue ring will be decimated. Another point, drawn in red, will be site phase decimated soon, as it is in the blue area. (b) A numerical approximation of the stable distribution $h(\kappa)$ in the case of almost no rescaling at all ($\gamma_g\gg 1$); the dashed reference line is the uniform case $h(\kappa)=\frac{1}{\pi}$ ($\gamma_g \ll 1$).