Integrability of an extended d+id-wave pairing Hamiltonian

Abstract: We introduce an integrable Hamiltonian which is an extended d+id-wave pairing model. The integrability is deduced from a duality relation with the Richardson-Gaudin (s-wave) pairing model, and associated to this there exists an exact Bethe ansatz solution. We study this system using the continuum limit approach and solve the corresponding singular integral equation obtained from the Bethe ansatz solution. We also conduct a mean-field analysis and show that results from these two approaches coincide for the ground state in the continuum limit. We identify instances of the integrable system where the excitation spectrum is gapless, and discuss connections to non-integrable models with d+id-wave pairing interactions through the mean-field analysis.

• The paper demonstrates an exact Bethe ansatz solution for an extended d+id-wave pairing model via a duality transformation of Richardson-Gaudin conserved charges.
• It employs both discrete and continuum analyses to link exact eigenvalue equations with mean-field predictions, pinpointing the critical coupling for gapless excitations.
• The work establishes a universal ground state structure across integrable and non-integrable regimes, advancing the study of higher-angular-momentum superconductivity.

Integrability and Exact Solution of an Extended $d+id$-Wave Pairing Hamiltonian

Introduction

This paper presents the construction and analysis of an integrable extension of the $d+id$-wave pairing Hamiltonian, motivated by the search for exactly solvable models exhibiting higher angular momentum pairing symmetries analogous to $s$-wave (Richardson-Gaudin) and $p+ip$-wave models. The authors derive the integrable structure via a duality transformation of the conserved charges in the Richardson-Gaudin model and demonstrate the existence of an exact Bethe ansatz solution. The work further establishes a correspondence between exact results in the continuum limit and those obtained from a mean-field analysis, characterizing regimes of gapless excitations and drawing connections to non-integrable $d+id$-wave systems.

Model Construction and Integrability

The Hamiltonian studied extends the conventional $d+id$-wave pairing interaction by the inclusion of quadratic number-operator terms, maintaining integrability:

$$H = \sum_{j=1}^{L}\varepsilon_{j}N_{j} - G\sum_{j,k=1}^{L} \varepsilon_{j}\varepsilon_{k}\left( b_{j}^{\dagger}b_{k} + b_{j}b_{k}^{\dagger} + 2 N_{j}N_{k} \right),$$

where $b_j$, $b_j^\dagger$ are hard-core pair operators. By leveraging the integrability of the $s$-wave Richardson-Gaudin Hamiltonian through a reciprocal transformation on the single-particle energies and pair energies in the Bethe ansatz framework, and adjusting the coupling constants appropriately, the authors demonstrate that the above Hamiltonian possesses a complete set of commuting conserved operators and admits an exact solution. The eigenvalues and Bethe ansatz equations are likewise mapped from those of the $s$-wave model to this extended $d+id$-wave model.

The energy spectrum is expressed in terms of the roots $\{y_l\}$ of the Bethe ansatz equations:

$$\frac{1}{2G y_l} + \sum_{i=1}^L \frac{\varepsilon_i^2}{y_l (y_l - \varepsilon_i)} = 2\sum_{j \neq l}^M \frac{y_j}{y_l - y_j},\quad l = 1,\ldots, M$$

with the total energy given by

$$E = \sum_{l=1}^M y_l - 2G \sum_{l < j}^M y_l y_j - G \sum_{j=1}^L \varepsilon_j^2.$$

Continuum Limit and Solution of Singular Integral Equation

To characterize the bulk properties, the continuum limit of the Bethe ansatz equations is developed, leading to a singular integral equation for the density of roots $r(y)$ supported on an arc $\Gamma$ in the complex plane:

$$\frac{c}{y} + \frac{d}{y^2} - \int_{\Omega} \frac{\rho(\varepsilon)}{\varepsilon - y}\, d\varepsilon + \mathcal{P} \int_{\Gamma} \frac{2r(y')}{y' - y}\, |dy'| = 0,$$

where $\rho(\varepsilon)$ is the density of single-particle energies. For uniform $\varepsilon$ density, corresponding to free fermions in two dimensions, closed-form expressions for the arc and filling fraction are derived, and the phase structure is elucidated by analyzing the endpoints of the root distribution.

Crucially, the analysis reveals a critical coupling $g_c = 1$, at which the arc supporting Bethe roots collapses to the origin, and the ground state is characterized as a fully "dressed" vacuum. This scenario lacks an electrostatic analogy and contrasts sharply with the $p+ip$-wave model, where the arc remains finite at the Moore-Read line.

Mean-Field Analysis and Comparison

A comprehensive mean-field analysis is performed on a family of Hamiltonians interpolating between non-integrable $d+id$-wave models and the integrable extended model, parameterized by two couplings $G$ and $F$. In the integrable case ($F=G$), the mean-field gap and chemical potential equations reduce, in the continuum limit, to those found from the exact Bethe ansatz solution. The elementary excitation spectrum is found to become gapless at the point where the chemical potential vanishes ($\mu = 0$), identifying the critical point derived above.

For generic $F \neq G$ (non-integrable case), the mean-field results reveal that the critical line of gapless excitations exists only for $x<1/2$ in the pure $d+id$-wave case, while for the integrable extension gapless excitations occur for all filling fractions.

The projected mean-field ground state at the gapless point,

$$\left|\Psi_M\right\rangle = C\, \left( \sum_{j=1}^L b_j^\dagger \right)^M |0 \rangle,$$

precisely coincides with the Bethe ansatz ground state in the degenerate root limit, indicating universality of the ground state structure and critical behavior.

Theoretical and Practical Implications

The explicit construction of an integrable $d+id$-wave pairing Hamiltonian extends the paradigm of exactly solvable pairing models beyond the $s$-wave and $p+ip$ regimes, opening avenues for systematic study of higher angular momentum superconductivity in the presence of additional interactions. The direct confirmation of mean-field results by exact solution, in the thermodynamic limit, validates mean-field methods for the ground state in integrable settings and informs the expected breakdown of integrability for realistic models.

The identification of universal ground states at criticality has potential implications for the topological characterization of paired fermion systems as well as for connection to trial wavefunctions such as the Haldane-Rezayi and Moore-Read states. These results may inform the design of engineered quantum systems (e.g., cold atom or mesoscopic setups) with tunable interactions emulating higher-angular-momentum superconductivity.

Conclusion

The paper rigorously proves the integrability of a nontrivial extension of the $d+id$-wave pairing Hamiltonian and provides a detailed exact solution that is in full agreement with mean-field theory in the ground state. The work clarifies the structure of excitations, critical points for gapless modes, and the universality of ground-state properties across integrable and non-integrable regimes. This establishes an important theoretical reference point for future studies of pairing phenomena in strongly correlated, higher-angular-momentum fermionic systems (1206.2684).