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arXiv:2510.20758 · Journal of High Energy Physics

θ-term in Russian Doll Model: phase structure, quantum metric and BPS fractality

A. Gorsky, Ilya Liubimov

Published May 4, 2026• 3 citations
arXivHTMLPDF

Abstract

A bstract 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(TRS) breaking parameter θ . It is one of the simplest examples of the cyclic RG. The deterministic model is integrable and shares the same Bethe Ansatz (BA) equations with the inhomogeneous twisted XXX spin chain. Using the BA equation in the single Cooper pair sector, we analyze the quantum metric, the Berry curvature, and the fractal dimension of RDM eigenstates. A rich structure is found in the parameter plane ( θ , γ ), where γ log N quantifies the hopping term. For the deterministic RDM we identify the extended domain of the non-ergodic fractal phase on the ( θ , γ ) parameter plane where the quantum number Q ( θ , γ ), which arises from the BA equation, exhibits the staircase behavior. 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 = 2 N C $$ \mathcal{N} $$ N = 2 SQCD at a strong coupling point $$ \frac{1}{g_{\textrm{YM}}^2} $$ 1 g YM 2 = 0 with identification θ RDM = θ 4 D – π . The Hamiltonian of the RDM model is identified with one of the commuting families of non-local Hamiltonians of the twisted inhomogeneous XXX spin chain describing the vortex string; hence the exact fractality of the eigenmode of H RDM implies the fractality in the peculiar 2d-4d BPS sector of the SQCD Hilbert space. Our findings provide an example of the BPS fractality regime for the probe operator in the sector of Hilbert space with some amount of SUSY which is determined by the N -scaling of the effective Planck constant.

DOI
10.1007/jhep05(2026)002

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Motivated by the interplay of Bethe-Ansatz integrability and localization in the Richardson model of superconductivity, we consider a time-reversal symmetry breaking deformation of this model, known as the Russian Doll Model (RDM), and implement diagonal on-site disorder. The localization and ergodicity-breaking properties of the single-particle spectrum are analyzed using a large-energy renormalization group (RG) over the momentum-space spectrum. Based on the above RG, we derive an effective Hamiltonian of the model, discover a fractal phase of non-ergodic delocalized states – with the fractal dimension different from the paradigmatic Rosenzweig-Porter model – and explain it in terms of the developed RG equations and the matrix-inversion trick.
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.
Focusing on Bethe-Ansatz integrable models, robust to both time-reversal symmetry breaking and disorder, we consider the Russian Doll Model (RDM) for finite system sizes and energy levels. Suggested as a time-reversal-symmetry breaking deformation of Richardson’s model, the well-known and simplest model of superconductivity, RDM revealed an unusual cyclic renormalization group (RG) over the system size N N , where the energy levels repeat themselves, shifted by one after a finite period in \ln N ln N , supplemented by a hierarchy of superconducting condensates, with the superconducting gaps following the so-called Efimov (exponential) scaling. The equidistant single-particle spectrum of RDM made the above Efimov scaling and cyclic RG to be asymptotically exact in the wideband limit of the diagonal potential. Here, we generalize this observation in various respects. We find that, beyond the wideband limit, when the entire spectrum is considered, the periodicity of the spectrum is not constant, but appears to be energy-dependent. Moreover, we resolve the apparent paradox of shift in the spectrum by a single level after the RG period, despite the disappearance of a finite fraction of energy levels. We also analyze the effects of disorder in the diagonal potential on the above periodicity and show that it survives only for high energies beyond the energy interval of the disorder amplitude. Our analytic analysis is supported with exact diagonalization.
The recently proposed Russian doll BCS model provides a simple example of a many-body system whose renormalization group analysis reveals the existence of limit cycles in the running coupling constants of the model. The model was first studied using RG, mean field and numerical methods showing the Russian doll scaling of the spectrum, E n ∼ E 0 e −λ n , where λ is the RG period. In this paper we use the recently discovered exact solution of this model to study the low energy spectrum. We find that, in addition to the standard quasiparticles, the electrons can bind into Cooper pairs that are different from those forming the condensate and with higher energy. These excited Cooper pairs can be described by a quantum number Q which appears in the Bethe ansatz equation and has an RG interpretation.
The recently proposed Russian doll BCS model provides a simple example of a many-body system whose renormalization group analysis reveals the existence of limit cycles in the running coupling constants of the model. The model was first studied using RG, mean field and numerical methods showing the Russian doll scaling of the spectrum, En ∼ E0 e−λn, where λ is the RG period. In this paper we use the recently discovered exact solution of this model to study the low energy spectrum. We find that, in addition to the standard quasiparticles, the electrons can bind into Cooper pairs that are different from those forming the condensate and with higher energy. These excited Cooper pairs can be described by a quantum number Q which appears in the Bethe ansatz equation and has an RG interpretation.
We study a string-net ladder in the presence of a string tension. Focusing on the simplest non-Abelian anyon theory with a quantum dimension larger than two, we determine the phase diagram and find a Russian doll spectrum featuring size-independent energy levels as well as highly degenerate zero-energy eigenstates. At the self-dual points, we compute the gap exactly by using a mapping onto the Temperley-Lieb chain. These results are in stark contrast with the ones obtained for Fibonacci or Ising theories.