Exactly Solvable RD Model: RG Cycles Meet Fractality

Abstract

We consider the Bethe ansatz integrable Russian Doll (RD) model of superconductivity with time-reversal symmetry breaking, which exhibits a cyclic renormalization group. By obtaining an exact solution for the renormalization group flows, we investigate the phase structure in the one-pair sector, which includes localized, fractal, and delocalized phases. We show that the quantum number Q, arising from the Bethe ansatz equations, counts the number of cycles and parametrizes the towers of states. Using the action of the renormalization group on the eigenstates, we demonstrate that Q serves as an order parameter, providing a new mechanism for the formation of the fractal phase in the deterministic systems and an example of the interplay between fractality and cyclic RG.

Figures (7)

• Figure 1: Distribution of quantum number $Q$ over energies for $N = 1000$, $\theta = \pi/4$, $\delta = 1$ in (a) localized phase with $Q = 0$. (b) fractal phase with two branches of solutions, levels in the bulk of the spectrum are degenerate with respect to $Q$ (c) delocalized phase with all solution on outer branch. $Q_{int}$ corresponds to Eq.(\ref{['BAeq']}) where the sum is replaced by an integral, $Q_{exact}$ corresponds to $Q$ obtained for energies from numerical Hamiltonian diagonalization, they are quantized.
• Figure 2: Numerical RG flows of $\gamma$ (a) in the localized phase (b) in the delocalized phase (c) in the fractal phase with flow of $\gamma^{*}$ for initial $\theta_0 = \pi/4$.
• Figure 3: (a) IPR for $E = \varepsilon_1$ on $(\gamma, \theta)$ plane for $N=1000$, $\delta = 1$. (b) $\abs{Q_{min}-1}$ on $(\gamma, \theta)$ plane for $N = 1000$. The subtraction of 1 is for convenience of taking $\ln$ in the localized phase.
• Figure 4: Sketch of RG action on the eigenstate with $E \simeq \varepsilon_1$. $M$ denotes characteristic number of sites, where $\abs{\psi}^2$ is sufficient. RG step corresponds to the elimination of the site with the largest number. Blue arrows represent cycles in the fractal phase with logarithmic RG time, green arrows represent cycles in the delocalized with linear RG time.
• Figure 5: Phase structure of the RD model with flows of $\gamma^*$