Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1709.08576 · Physical review. B./Physical review. B

Solvable two-dimensional superconductors with l -wave pairing

Luca Lepori, M. Roncaglia

Published Oct 2, 2018• 4 citations
arXivHTMLPDF

Abstract

We analyze a family of two-dimensional BCS Hamiltonians with general $l$-wave pairing interactions, classifying the models in this family that are Bethe-ansatz solvable in the finite-size regime. We show that these solutions are characterized by nontrivial winding numbers, associated with topological phases, in some part of the corresponding phase diagrams. By means of a comparative study, we demonstrate benefits and limitations of the mean-field approximation, which is the standard approach in the limit of a large number of particles. The mean-field analysis also allows us to extend part of the results beyond integrability, clarifying the peculiarities associable with the integrability itself.

DOI
10.1103/physrevb.98.144504

Related papers

Exactly solvable pairing model for superconductors with px+ipy -wave symmetry

Miguel Ibáñez-Berganza, Jon Links, Germán Sierra +1 more

We present the exact Bethe ansatz solution for the two-dimensional BCS pairing Hamiltonian with ${p}_{x}+i{p}_{y}$ symmetry. Using both mean-field theory and the exact solution we obtain the ground-state phase diagram parametrized by the filling fraction and the coupling constant. It consists of three phases that are denoted weak-coupling BCS, weak pairing, and strong pairing. The first two phases are separated by a topologically protected line where the exact ground state is given by the Moore-Read pfaffian state. In the thermodynamic limit the ground-state energy is discontinuous on this line. The other two phases are separated by the critical line, also topologically protected, previously found by Read and Green. We establish a duality relation between the weak and strong pairing phases, whereby ground states of the weak phase are ``dressed'' versions of the ground states of the strong phase by zero energy (Moore-Read) pairs and characterized by a topological order parameter.

arXiv:0810.0340•Physical Review B•May 1, 2009•80 citations

Exactly solvable pairing model with p+ip wave symmetry

Miguel Ibáñez-Berganza, Jon Links, Germán Sierra +1 more

We present the exact Bethe ansatz solution for the two-dimensional BCS pairing Hamiltonian with p_x + i p_y symmetry. Using both mean-field theory and the exact solution we obtain the ground-state phase diagram parameterized by the filling fraction and the coupling constant. It consists of three phases denoted weak coupling BCS, weak pairing, and strong pairing. The first two phases are separated by a topologically protected line where the exact ground state is given by the Moore-Read pfaffian state. In the thermodynamic limit the ground-state energy is discontinuous on this line. The other two phases are separated by the critical line, also topologically protected, previously found by Read and Green. We establish a duality relation between the weak and strong pairing phases, whereby ground states of the weak phase are dressed versions of the ground states of the strong phase by zero energy (Moore-Read) pairs and characterized by a topological order parameter.

•Oct 2, 2008•0 citations

Generalized Heine–Stieltjes and Van Vleck polynomials associated with two-level, integrable BCS models

Ian Marquette, Jon Links

We study the Bethe ansatz/ordinary differential equation (BA/ODE) correspondence for Bethe ansatz equations that belong to a certain class of coupled, nonlinear, algebraic equations. Through this approach we numerically obtain the generalized Heine–Stieltjes and Van Vleck polynomials in the degenerate, two-level limit for four cases of integrable Bardeen–Cooper–Schrieffer (BCS) pairing models. These are the s-wave pairing model, the p + ip-wave pairing model, the p + ip pairing model coupled to a bosonic molecular pair degree of freedom, and a newly introduced extended d + id-wave pairing model with additional interactions. The zeros of the generalized Heine–Stieltjes polynomials provide solutions of the corresponding Bethe ansatz equations. We compare the roots of the ground states with curves obtained from the solution of a singular integral equation approximation, which allows for a characterization of ground-state phases in these systems. Our techniques also permit the computation of the roots of the excited states. These results illustrate how the BA/ODE correspondence can be used to provide new numerical methods to study a variety of integrable systems.

arXiv:1206.2988•Journal of Statistical Mechanics Theory and Experiment•Aug 31, 2012•36 citations

Consequences of integrability breaking in quench dynamics of pairing Hamiltonians

Jasen A. Scaramazza, Pietro Smacchia, Emil A. Yuzbashyan

We study the collisionless dynamics of two classes of nonintegrable pairing models. One is a Bardeen-Cooper-Schrieffer model with separable energy-dependent interactions, and the other is a two-dimensional topological superconductor with spin-orbit coupling and a band-splitting external field. The long-time quantum quench dynamics at integrable points of these models are well understood. Namely the squared magnitude of the time-dependent order parameter $\mathrm{\ensuremath{\Delta}}(t)$ can vanish (Phase I), reach a nonzero constant (Phase II), or periodically oscillate as an elliptic function (Phase III). We demonstrate that nonintegrable models, too, exhibit some or all of these nonequilibrium phases. Remarkably, elliptic periodic oscillations persist, even though both their amplitude and functional form change drastically with integrability breaking. Striking new phenomena accompany loss of integrability. First, an extremely long timescale emerges in the relaxation to Phase III, such that short-time numerical simulations risk erroneously classifying the asymptotic state. This timescale diverges near integrable points. Second, an entirely new Phase IV of quasiperiodic oscillations of $|\mathrm{\ensuremath{\Delta}}|$ emerges in the quantum quench phase diagrams of nonintegrable pairing models. As integrability techniques do not apply for the models we study, we develop the concept of asymptotic self-consistency and a linear stability analysis of the asymptotic phases. With the help of these new tools, we determine the phase boundaries, characterize the asymptotic state, and clarify the physical meaning of the quantum quench phase diagrams of BCS superconductors. We also propose an explanation of these diagrams in terms of bifurcation theory.

arXiv:1812.04410•Physical review. B./Physical review. B•Feb 28, 2019•15 citations

Bethe Ansatz analysis of the open and closed p + ip-pairing BCS models

Yibing Shen

The p+ip-wave pairing BCS model has attracted much attention due to its potential applications inexperimental physics, and the developing theory of topological superconductors. In this thesis, weinvestigate the p + ip BCS model as an integrable model with exact Bethe Ansatz solutions. Theground-state energies of this model will be studied with various methods.  First a brief introduction on quantum integrability is given, followed by an introduction to theQuantum Inverse Scattering Method, algebraic Bethe Ansatz and BCS theory of superconductivity. Wethen introduce a two-dimensional BCS pairing Hamiltonian with px +ipy symmetry. In the frameworkof the Quantum Inverse Scattering Method, the p+ip Hamiltonian can be constructed from a specificsolution of the non-standard classical Yang-Baxter equation. This solution produces a family ofconserved operators satisfying quadratic relations. Remarks on topological superconductivity for thep+ip-wave pairing model are provided.  We move on to investigate methods for calculating the ground-state energy for the p+ip Hamil-tonian. We first consider the Hamiltonian isolated from its environment (closed model) through twoforms of Bethe Ansatz solution, which generally have complex-valued Bethe roots. A continuum limitapproximation, leading to an integral equation, is applied to compute the ground-state energy. Theevolution of the root distribution curve with respect to a range of parameters is presented, followed bya discussion of the limitations of this method. Despite the limitations, the ground-state energy andthe ‘gap’ equation derived from the integral approximation are consistent with mean-field results fromprevious study.  We then consider the Hamiltonian interacting with its environment (open model) and begin witha mean-field analysis. We discuss the relation between the Bethe roots and the conserved operatoreigenvalues. The Bethe Ansatz solution of the open model is studied numerically, revealing complicatedbehaviour of the Bethe roots without a clear indication for an integral approximation.  Next we consider an alternative approach based on the quadratic relations among the real-valuedconserved operator eigenvalues. An integral equation is first established for the closed model. Thisequation is shown to admit an exact solution associated with the ground state.  Finally we extend the aforementioned alternative approach to the open model. Combined withresults from mean-field analysis, we are able to establish an integral equation with an exact solutionthat corresponds to the ground state for the open case.

•Sep 27, 2019•0 citations

Pairing and superconductivity in quasi-one-dimensional flat-band systems: Creutz and sawtooth lattices

Si Min Chan, Benoît Grémaud, G. G. Batrouni

We study the pairing and superconducting properties of the attractive Hubbard model in two quasi-one-dimensional topological lattices---the Creutz and sawtooth lattices---which share two peculiar properties: each of their band structures exhibits a flat band with a nontrivial winding number. The difference, however, is that only the Creutz lattice is genuinely topological, due to a chiral (sublattice) symmetry, resulting in a quantized winding number and zero energy edge modes for open boundary conditions. We use a multiband mean field and exact density matrix renormalization group in our work. Our three main results are as follows: (a) For both lattice systems, the superconducting weight, ${D}_{s}$, is linear in the coupling strength, $U$, for low values of $U$. (b) For small $U, {D}_{s}$ is proportional to the quantum metric for the Creutz system but not for the sawtooth system because its sublattices are not equivalent. We have therefore extended the approach to this more complex situation and found an excellent agreement with the numerical results. (c) At moderate and large $U$, the conventional BCS mean field is no longer appropriate for such systems with inequivalent sublattices. We show that, for a wide range of densities and coupling strengths, these systems are very well described by a full multiband mean-field method where the pairing parameters and the local particle densities on the inequivalent sublattices are variational mean-field parameters.

arXiv:2105.12761•Physical review. B./Physical review. B•Jan 5, 2022•38 citations
We analyze a family of two-dimensional BCS Hamiltonians with general lll-wave pairing interactions, classifying the models in this family that are Bethe-ansatz solvable in the finite-size regime. We show that these solutions are characterized by nontrivial winding numbers, associated with topological phases, in some part of the corresponding phase diagrams. By means of a comparative study, we demonstrate benefits and limitations of the mean-field approximation, which is the standard approach in the limit of a large number of particles. The mean-field analysis also allows us to extend part of the results beyond integrability, clarifying the peculiarities associable with the integrability itself.
We present the exact Bethe ansatz solution for the two-dimensional BCS pairing Hamiltonian with px+ipy{p}_{x}+i{p}_{y}px​+ipy​ symmetry. Using both mean-field theory and the exact solution we obtain the ground-state phase diagram parametrized by the filling fraction and the coupling constant. It consists of three phases that are denoted weak-coupling BCS, weak pairing, and strong pairing. The first two phases are separated by a topologically protected line where the exact ground state is given by the Moore-Read pfaffian state. In the thermodynamic limit the ground-state energy is discontinuous on this line. The other two phases are separated by the critical line, also topologically protected, previously found by Read and Green. We establish a duality relation between the weak and strong pairing phases, whereby ground states of the weak phase are ``dressed'' versions of the ground states of the strong phase by zero energy (Moore-Read) pairs and characterized by a topological order parameter.
We present the exact Bethe ansatz solution for the two-dimensional BCS pairing Hamiltonian with p_x + i p_y symmetry. Using both mean-field theory and the exact solution we obtain the ground-state phase diagram parameterized by the filling fraction and the coupling constant. It consists of three phases denoted weak coupling BCS, weak pairing, and strong pairing. The first two phases are separated by a topologically protected line where the exact ground state is given by the Moore-Read pfaffian state. In the thermodynamic limit the ground-state energy is discontinuous on this line. The other two phases are separated by the critical line, also topologically protected, previously found by Read and Green. We establish a duality relation between the weak and strong pairing phases, whereby ground states of the weak phase are dressed versions of the ground states of the strong phase by zero energy (Moore-Read) pairs and characterized by a topological order parameter.
We study the Bethe ansatz/ordinary differential equation (BA/ODE) correspondence for Bethe ansatz equations that belong to a certain class of coupled, nonlinear, algebraic equations. Through this approach we numerically obtain the generalized Heine–Stieltjes and Van Vleck polynomials in the degenerate, two-level limit for four cases of integrable Bardeen–Cooper–Schrieffer (BCS) pairing models. These are the s-wave pairing model, the p + ip-wave pairing model, the p + ip pairing model coupled to a bosonic molecular pair degree of freedom, and a newly introduced extended d + id-wave pairing model with additional interactions. The zeros of the generalized Heine–Stieltjes polynomials provide solutions of the corresponding Bethe ansatz equations. We compare the roots of the ground states with curves obtained from the solution of a singular integral equation approximation, which allows for a characterization of ground-state phases in these systems. Our techniques also permit the computation of the roots of the excited states. These results illustrate how the BA/ODE correspondence can be used to provide new numerical methods to study a variety of integrable systems.
We study the collisionless dynamics of two classes of nonintegrable pairing models. One is a Bardeen-Cooper-Schrieffer model with separable energy-dependent interactions, and the other is a two-dimensional topological superconductor with spin-orbit coupling and a band-splitting external field. The long-time quantum quench dynamics at integrable points of these models are well understood. Namely the squared magnitude of the time-dependent order parameter $\mathrm{\ensuremath{\Delta}}(t)$ can vanish (Phase I), reach a nonzero constant (Phase II), or periodically oscillate as an elliptic function (Phase III). We demonstrate that nonintegrable models, too, exhibit some or all of these nonequilibrium phases. Remarkably, elliptic periodic oscillations persist, even though both their amplitude and functional form change drastically with integrability breaking. Striking new phenomena accompany loss of integrability. First, an extremely long timescale emerges in the relaxation to Phase III, such that short-time numerical simulations risk erroneously classifying the asymptotic state. This timescale diverges near integrable points. Second, an entirely new Phase IV of quasiperiodic oscillations of $|\mathrm{\ensuremath{\Delta}}|$ emerges in the quantum quench phase diagrams of nonintegrable pairing models. As integrability techniques do not apply for the models we study, we develop the concept of asymptotic self-consistency and a linear stability analysis of the asymptotic phases. With the help of these new tools, we determine the phase boundaries, characterize the asymptotic state, and clarify the physical meaning of the quantum quench phase diagrams of BCS superconductors. We also propose an explanation of these diagrams in terms of bifurcation theory.
The p+ip-wave pairing BCS model has attracted much attention due to its potential applications inexperimental physics, and the developing theory of topological superconductors. In this thesis, weinvestigate the p + ip BCS model as an integrable model with exact Bethe Ansatz solutions. Theground-state energies of this model will be studied with various methods.  First a brief introduction on quantum integrability is given, followed by an introduction to theQuantum Inverse Scattering Method, algebraic Bethe Ansatz and BCS theory of superconductivity. Wethen introduce a two-dimensional BCS pairing Hamiltonian with px +ipy symmetry. In the frameworkof the Quantum Inverse Scattering Method, the p+ip Hamiltonian can be constructed from a specificsolution of the non-standard classical Yang-Baxter equation. This solution produces a family ofconserved operators satisfying quadratic relations. Remarks on topological superconductivity for thep+ip-wave pairing model are provided.  We move on to investigate methods for calculating the ground-state energy for the p+ip Hamil-tonian. We first consider the Hamiltonian isolated from its environment (closed model) through twoforms of Bethe Ansatz solution, which generally have complex-valued Bethe roots. A continuum limitapproximation, leading to an integral equation, is applied to compute the ground-state energy. Theevolution of the root distribution curve with respect to a range of parameters is presented, followed bya discussion of the limitations of this method. Despite the limitations, the ground-state energy andthe ‘gap’ equation derived from the integral approximation are consistent with mean-field results fromprevious study.  We then consider the Hamiltonian interacting with its environment (open model) and begin witha mean-field analysis. We discuss the relation between the Bethe roots and the conserved operatoreigenvalues. The Bethe Ansatz solution of the open model is studied numerically, revealing complicatedbehaviour of the Bethe roots without a clear indication for an integral approximation.  Next we consider an alternative approach based on the quadratic relations among the real-valuedconserved operator eigenvalues. An integral equation is first established for the closed model. Thisequation is shown to admit an exact solution associated with the ground state.  Finally we extend the aforementioned alternative approach to the open model. Combined withresults from mean-field analysis, we are able to establish an integral equation with an exact solutionthat corresponds to the ground state for the open case.
We study the pairing and superconducting properties of the attractive Hubbard model in two quasi-one-dimensional topological lattices---the Creutz and sawtooth lattices---which share two peculiar properties: each of their band structures exhibits a flat band with a nontrivial winding number. The difference, however, is that only the Creutz lattice is genuinely topological, due to a chiral (sublattice) symmetry, resulting in a quantized winding number and zero energy edge modes for open boundary conditions. We use a multiband mean field and exact density matrix renormalization group in our work. Our three main results are as follows: (a) For both lattice systems, the superconducting weight, Ds{D}_{s}Ds​, is linear in the coupling strength, UUU, for low values of UUU. (b) For small U,DsU, {D}_{s}U,Ds​ is proportional to the quantum metric for the Creutz system but not for the sawtooth system because its sublattices are not equivalent. We have therefore extended the approach to this more complex situation and found an excellent agreement with the numerical results. (c) At moderate and large UUU, the conventional BCS mean field is no longer appropriate for such systems with inequivalent sublattices. We show that, for a wide range of densities and coupling strengths, these systems are very well described by a full multiband mean-field method where the pairing parameters and the local particle densities on the inequivalent sublattices are variational mean-field parameters.