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 NF = 2NC <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">$ \mathcal{N} $</EquationSource> </InlineEquation> = 2 SQCD at a strong coupling point <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>1</mn> <msubsup> <mi>g</mi> <mi>YM</mi> <mn>2</mn> </msubsup> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$ \frac{1}{g_{\textrm{YM}}^2} $</EquationSource> </InlineEquation> = 0 with identification θRDM = θ4D – 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 HRDM 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.

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

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2026年5月4日

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Alexander Gorsky,
Ilya Liubimov

摘要

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 = 2N_C $N$ $ \mathcal{N} $ = 2 SQCD at a strong coupling point $\frac{1}{g_{YM}^{2}}$ $ \frac{1}{g_{\textrm{YM}}^2} $ = 0 with identification θ_RDM = θ_4D – 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.