Colloquium: Exactly solvable Richardson-Gaudin models for many-body quantum systems

The use of exactly solvable Richardson-Gaudin models to describe the physics of systems with strong pair correlations is reviewed. The article begins with a brief discussion of Richardson’s early work, which demonstrated the exact solvability of the pure pairing model, and then shows how that work has evolved recently into a much richer class of exactly solvable models. The Richardson solution leads naturally to an exact analogy between these quantum models and classical electrostatic problems in two dimensions. This analogy is then used to demonstrate formally how BCS theory emerges as the large-$N$ limit of the pure pairing Hamiltonian. Several applications to problems of relevance to condensed-matter physics, nuclear physics, and the physics of confined systems are considered. Some of the interesting effects that are discussed in the context of these exactly solvable models include (i) the crossover from superconductivity to a fluctuation-dominated regime in small metallic grains; (ii) the role of the nucleon Pauli principle in suppressing the effects of high-spin bosons in interacting boson models of nuclei, and (iii) the possibility of fragmentation in confined boson systems. Interesting insight is also provided into the origin of the superconducting phase transition both in two-dimensional electronic systems and in atomic nuclei, based on the electrostatic image of the corresponding exactly solvable quantum pairing models.

Figure 1

Two-dimensional representation of the pairon positions in ${}^{114}\mathrm{Sn}$ for a pure pairing-model Hamiltonian with single-particle energies as given in Table and with $g=-0.02\mathrm{MeV}$.

Figure 2

Evolution of the real and imaginary parts of $E_{\mu }\left(g\right)$, in units of $d=\omega ∕L$, for the equally spaced model with $M=L∕2=8$, as a function of the coupling constant $g$. For convenience, the energy levels are chosen in this figure as ${ϵ}_j=2j$. From 60.

Figure 3

Plot of the roots $E_{\mu }$ for the equally spaced pairing model in the complex $\xi$ plane. The discrete symbols denote the numerical values for $M=100$. All energies are in units of $\omega$. From 60.

Figure 4

Real part of $E_{\mu }$ for the equally spaced pairing model with $M=L∕2=20$ pairs and $N_G=0$, 1, 2 excitations. From 61.

Figure 5

Lowest 44 excitation energies $E_{\mathit{exc}}=E-E_{GS}$ for $M=20$ pairs at half filling. From 61.

Figure 6

An illustration of the algorithm for determining the number of roots $N_G$, for a case with $N_G=3$: (a) Top, initial state at $g=0$, where each solid circle (●) denotes a level occupied by a pair, each empty circle (엯) denotes an empty level, and the vertical dashed line with an $F$ near it denotes the position of the Fermi surface; bottom, the associated Young diagram; (b) real part of $E_{\mu }$. From 61.

Figure 7

Calculated results for ultrasmall superconducting grains as a function of the grain size. The upper panel gives the condensation energies $E_b^C$ for even $(b=0)$ and odd $(b=1)$ grains calculated using the number-projected BCS and exact wave functions. The lower panel gives the Matveev-Larkin gap, ${\mathrm{\Delta }}_{ML}$, obtained in number-projected BCS calculations (14) and in exact calculations.

Figure 8

Exact even and odd condensation energies $E_b^C$ for uniform equally spaced levels (dashed line), and the ensemble-averaged energies $⟨E_b^C⟩$ for randomly-spaced levels (solid line). The height of the fluctuation bars gives the variances $\delta E_b^C$. From 68.

Figure 9

Two-dimensional representation of the positions of the orbitons and pairons corresponding to a $6\times 6$ lattice at half filling, for three selected values of $g$. The orbitons are represented by open circles and the pairons by solid circles, with radii proportional to their charges.

Figure 10

Evolution of the pairon positions as a function of the scaled strength parameter $x$ for a model with ten $s$ and $d$ bosons subject to a Hamiltonian with linear single-boson energies and a repulsive boson pairing interaction. The circles at $x=0$ denote the positions of the $s$ and $d$ orbitons.

Figure 11

Occupation numbers for 1000 bosons confined in 50 harmonic-oscillator shells and interacting via a pure pairing interaction with strength $g=-0.0025$. The occupation of the $l=0$ state is off scale and thus not shown.

Figure 12

Occupation numbers for 1000 bosons confined in 50 harmonic-oscillator shells and interacting via a renormalized pairing interaction [see Eq. (62)] with $g=-1.0$.

Figure 13

Occupation numbers $n_0$ and $n_1$ for 1000 bosons confined to 50 harmonic-oscillator shells and interacting via a repulsive renormalized pairing interaction as a function of the scaled strength parameter $x$.