We address a generalized Richardson model (Russian doll BCS model), which is characterized by the breaking of time-reversal symmetry. This model is known to be exactly solvable and integrable. We point out that the Russian doll BCS model, on the level of Hamiltonian, is also particle-hole symmetric. This implies that the same state can be expressed both in the particle and hole representations with two different sets of Bethe roots. We then derive exact relations between Bethe roots in the two representations, which can hardly be obtained staying on the level of Bethe equations. In a quasi-classical limit, similar identities for usual Richardson model, known from literature, are recovered from our results. We also show that these relations for Richardson roots take a remarkably simple form at half-filling and for a symmetric with respect to the middle of the interaction band distribution of one-body energy levels, since, in this special case, the rapidities in the particle and hole representations up to the translation satisfy the same system of equations.We address a generalized Richardson model (Russian doll BCS model), which is characterized by the breaking of time-reversal symmetry. This model is known to be exactly solvable and integrable. We point out that the Russian doll BCS model, on the level of Hamiltonian, is also particle-hole symmetric. This implies that the same state can be expressed both in the particle and hole representations with two different sets of Bethe roots. We then derive exact relations between Bethe roots in the two representations, which can hardly be obtained staying on the level of Bethe equations. In a quasi-classical limit, similar identities for usual Richardson model, known from literature, are recovered from our results. We also show that these relations for Richardson roots take a remarkably simple form at half-filling and for a symmetric with respect to the middle of the interaction band distribution of one-body energy levels, since, in this special case, the rapidities in the particle and hole representations up to the translation satisfy the same system of equations.

Dec 2, 20140 citations

The elementary excitations of the exactly solvable russian doll BCS model of superconductivity

Alberto Anfossi, Andr Le Clair, Gemrman Sierra

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.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.

Jan 1, 20059 citations

The elementary excitations of the exactly solvable Russian doll BCS model of superconductivity

Alberto Anfossi, André LeClair, Germán Sierra

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, 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.

arXiv:cond-mat/0503014Journal of Statistical Mechanics Theory and ExperimentMay 27, 200512 citations

Exactly solvable Russian-doll model: Renormalization-group cycles meet fractality

Ilya Liubimov, Alexander Gorsky

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.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.

cond-mat.stat-mechCross-listedarXiv:2603.17674Physical review. B./Physical review. BMar 19, 20260 citations

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

A. Gorsky, Ilya Liubimov

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.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.

arXiv:2510.20758Journal of High Energy PhysicsMay 4, 20263 citations

Algebraic Bethe ansatz for a discrete-state BCS pairing model

Jan von Delft, Rubik Poghossian

We show in detail how Richardson's exact solution of a discrete-state BCS (DBCS) model can be recovered as a special case of an algebraic Bethe-ansatz solution of the inhomogeneous XXX vertex model with twisted boundary conditions: by implementing the twist using Sklyanin's $\mathbf{K}$ -matrix construction and taking the quasiclassical limit, one obtains a complete set of conserved quantities ${\mathbf{H}}_{i}$ from which the DBCS Hamiltonian can be constructed as a second order polynomial. The eigenvalues and eigenstates of the ${\mathbf{H}}_{i}$ (which reduce to the Gaudin Hamiltonians in the limit of infinitely strong coupling) are exactly known in terms of a set of parameters determined by a set of on-shell Bethe ansatz equations, which reproduce Richardson's equations for these parameters. We thus clarify that the integrability of the DBCS model is a special case of the integrability of the twisted inhomogeneous XXX vertex model. Furthermore, by considering the twisted inhomogeneous XXZ model and/or choosing a generic polynomial of the ${\mathbf{H}}_{i}$ 's as Hamiltonian, more general exactly solvable models can be constructed. To make the paper accessible to readers that are not Bethe-ansatz experts, the introductory sections include a self-contained review of those of its feature which are needed here.We show in detail how Richardson's exact solution of a discrete-state BCS (DBCS) model can be recovered as a special case of an algebraic Bethe-ansatz solution of the inhomogeneous XXX vertex model with twisted boundary conditions: by implementing the twist using Sklyanin's $\mathbf{K}$-matrix construction and taking the quasiclassical limit, one obtains a complete set of conserved quantities ${\mathbf{H}}_{i}$ from which the DBCS Hamiltonian can be constructed as a second order polynomial. The eigenvalues and eigenstates of the ${\mathbf{H}}_{i}$ (which reduce to the Gaudin Hamiltonians in the limit of infinitely strong coupling) are exactly known in terms of a set of parameters determined by a set of on-shell Bethe ansatz equations, which reproduce Richardson's equations for these parameters. We thus clarify that the integrability of the DBCS model is a special case of the integrability of the twisted inhomogeneous XXX vertex model. Furthermore, by considering the twisted inhomogeneous XXZ model and/or choosing a generic polynomial of the ${\mathbf{H}}_{i}$'s as Hamiltonian, more general exactly solvable models can be constructed. To make the paper accessible to readers that are not Bethe-ansatz experts, the introductory sections include a self-contained review of those of its feature which are needed here.

arXiv:cond-mat/0106405Physical review. B, Condensed matterOct 4, 200262 citations

\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.

\mathbf{K}

-matrix construction and taking the quasiclassical limit, one obtains a complete set of conserved quantities

{\mathbf{H}}_{i}

from which the DBCS Hamiltonian can be constructed as a second order polynomial. The eigenvalues and eigenstates of the

{\mathbf{H}}_{i}

(which reduce to the Gaudin Hamiltonians in the limit of infinitely strong coupling) are exactly known in terms of a set of parameters determined by a set of on-shell Bethe ansatz equations, which reproduce Richardson's equations for these parameters. We thus clarify that the integrability of the DBCS model is a special case of the integrability of the twisted inhomogeneous XXX vertex model. Furthermore, by considering the twisted inhomogeneous XXZ model and/or choosing a generic polynomial of the

{\mathbf{H}}_{i}

's as Hamiltonian, more general exactly solvable models can be constructed. To make the paper accessible to readers that are not Bethe-ansatz experts, the introductory sections include a self-contained review of those of its feature which are needed here.