Theta-term in Russian Doll Model: phase structure, quantum metric and BPS multifractality

TL;DR

The paper analyzes the Russian Doll Model, a TRS-breaking generalization of the reduced BCS Hamiltonian, through Bethe Ansatz integrability to map its phase structure in the two-parameter plane $(\theta,\gamma)$ with hopping scaled as $r=N^{-\gamma}$. It uncovers extended non-ergodic multifractal phases and reentrant transitions between localized, ergodic, and multifractal regimes, with a global charge $Q(\theta,\gamma)$ derived from BA equations guiding these transitions. By computing the quantum metric and Berry curvature across both deterministic and disordered versions, the work links geometric singularities to level interactions and resonance structures, and provides analytic and perturbative insights in the $N=2$ and large-$N$ limits. The second part connects the BA equations of RDM to the worldvolume theory of vortex strings in ${\cal N}=2$ SQCD at strong coupling, proposing a BPS multifractality regime in a specific 2d–4d subsector and drawing parallels to black hole microstate physics. Together, these results illuminate how integrable, topological, and holographic aspects intertwine to produce rich fractal phases and nonperturbative scaling, with potential implications for dense QCD and holography.

Abstract

We investigate the phase structure of the deterministic and disordered versions of the Russian Doll Model (RDM), which is a generalization of Richardson model of superconductivity in a finite system with time-reversal symmetry breaking parameter $θ$. It is one of the simplest examples of the cyclic RG where $\log N$ plays the role of the RG time. The deterministic model is integrable and shares the same Bethe Ansatz (BA) equations with the inhomogeneous twisted XXX spin chain. We analyze the quantum metric, the Berry curvature, and the fractal dimension in the sector with a single Cooper pair. A rich phase structure in the $(θ,γ)$ parameter plane is found, where $γ\log N$ quantifies the hopping term. For the deterministic RDM we clearly identify the extended domain of non-ergodic multifractal phase on the $(θ,γ)$ parameter plane supporting the reentrance transitions between the localized, ergodic, and multifractal phases. We find the pattern of phase transitions in the global charge $Q(θ,γ)$, which arises from the BA equation. In particular, in the multifractal phase in the deterministic model $Q(γ)$ exhibits the analogue of "charge concentration" and fortuity phenomena discussed in the context of black hole microstates at finite $N$. The BA equations in RDM exactly coincide with the equations defining the ground states in the theory on the worldvolume of the vortex strings in $N_F=2N_C$ ${\cal N}=2$ SQCD at a strong coupling point $\frac{1}{g_{YM}^2}=0$ with identification $θ_{RDM}= θ_{4D}-π$. We conjecture that the Hamiltonian of the RDM model describes the mixing in particular 2d-4d BPS sector of the Hilbert space. Our findings provide an example of the BPS multifractality regime for the probe operator in the sector of Hilbert space, and we comment on the possible application to dense QCD with $θ$ term.

Figures (17)

• Figure 1: $D_q$ on a ($\gamma, \theta$) plane for $q = 1.75$ a) in the bulk of spectrum b) on the edge
• Figure 2: $q$-dependence in $D_q(\gamma)$ for RDM without disorder, averaged for $N = 500 - 1000$ and $\ln{Q}$ staircase structure for level $n = 125$, with $\theta = \pi/4$.
• Figure 3: $\ln{(\abs{Q}+1)}$ for a fixed level on a ($\gamma, \theta$) plane
• Figure 4: Plot of r.h.s. and l.h.s. of \ref{['BAE']} vs $E$ for $N = 20$, $\delta = 0.4$, discontinuities correspond to $E = \varepsilon_l + x$ a) localized phase with $\varepsilon_{i+1}-\varepsilon_i = \delta \gg y$ b) example of level interaction $\varepsilon_{i+1}-\varepsilon_i = \delta < y$, where levels with $Q > 1$ arise
• Figure 5: Numerics for $D_q(\gamma)$ RDM with strong disorder for level $E = 1/3 + x$