Bardeen-Cooper-Schrieffer interaction as an infinite-range Penson-Kolb pairing mechanism

We demonstrate that the well-known $(k\uparrow ,-k\downarrow )$ Bardeen-Cooper-Schrieffer interaction, when considered in real space, is equivalent to an infinite-range Penson-Kolb pairing mechanism coexisting with an attractive Hubbard term. Driven by this discovery and aiming at exploring the conduction properties, we investigate the dynamics of fermionic particles confined in a ring-shaped lattice. We assume that fermions are simultaneously influenced by the pairing interaction and by an Aharonov-Bohm electromagnetic phase, which is incorporated into the model in a highly nontrivial manner. Remarkably, the aforementioned model shows Richardson integrability for both integer and half-integer values of the applied magnetic flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$, thus permitting the exact solution of a genuine many-body problem. We discuss the ground-state properties of both two-particle and many-particle systems, drawing comparisons with results from the attractive Hubbard model. Our approach combines exact diagonalization, density matrix renormalization group techniques, and numerical solution of the Richardson equations. This comprehensive analysis allows us to study various key metrics, including the system's conductivity as a function of the interaction strength. In this way, the BCS-BEC transition is investigated in a continuous manner, thus permitting to shed light on fundamental aspects of superconducting pairing. Our findings can be experimentally tested in a condensed matter context or, with greater level of control, using atomtronics platforms.

Figure 1

Ground-state energy $E_G$ (measured in units of the hopping parameter $t$) of a free fermions system as a function of the normalized applied flux $f=\mathrm{\Phi }/{\mathrm{\Phi }}_0$. The ring contains $N=30$ lattice sites, while the particles number has been fixed to $M=18$ (a) or $M=16$ (b). Flux-one periodicity of the $E_G$ vs $f$ curves is evident for both the panels, so that $E_G(f+1)=E_G\left(f\right)$. Moreover, depending on the odd or even parity of $M/2$, the $E_G$ vs $f$ curves show a local minimum or a local maximum, respectively, in close vicinity of $f=0$.

Figure 2

(a) $Z_o\left(f\right)$ curve in the domain $f\in [0,1]$. (b) Absolute value of the Fourier coefficients $c_n^{\left(o\right)}$, with $n>0$, of the curve shown in (a).

Figure 3

(a) Density of states (DOS) of the problem of two fermions subject to attractive Hubbard interaction obtained by setting the interaction strength to $g=10$, the number of lattice sites to $N=16$, while fixing $f=0$. The energy spectrum is organized in two distinct parts: (i) The bound states region formed by fermions paired by the interaction; (ii) the unpaired states region, describing free fermions. The energy required to produce unpaired fermions starting from a fermions pair is controlled by the interaction strength $g$. Looking at the flux dependence of the energy levels, one observes that bound states present a ${\mathrm{\Phi }}_0/2$ periodicity, which is related with the formation of a fermions pair. Unpaired states, on the other hand, are characterized by ${\mathrm{\Phi }}_0$ periodicity of the energy-flux curves, reminiscent of the free fermions physics. The energy-flux curves for the ground state, the first excited state, and the bottom of the band formed by the unpaired states are presented in (b), (c), and (d), respectively. Panels (e), (f), (g) show the energy-flux curves for the ground state of the two-fermions problem as a function of the interaction strength $g$, while setting $N=16$. It is shown that the flux periodicity of the energy-flux curves evolves from ${\mathrm{\Phi }}_0$ towards ${\mathrm{\Phi }}_0/2$ as the interaction strength increases. A similar behavior is observed for fixed interaction strength $g$ and variable system size $N$.

Figure 4

(a) Density of states (DOS) of the problem of two fermions subject to a real-space BCS interaction obtained by setting the interaction strength to $g=10$, the number of lattice sites to $N=16$, while fixing $f=0$. In contrast with the Hubbard case, the energy spectrum presents a single bound state accompanied by a DOS region describing free fermions (unpaired states). The energy required to produce unpaired fermions starting from a fermions pair is controlled by the interaction strength $g$. Looking at the flux dependence of the energy levels, one observes that the bound state presents a ${\mathrm{\Phi }}_0/2$ periodicity, which is related with the formation of a fermions pair. Unpaired states, on the other hand, are characterized by ${\mathrm{\Phi }}_0$ periodicity of the energy-flux curves, reminiscent of the free fermions physics. The energy-flux curves for the ground state and the bottom of the band formed by the unpaired states are presented in (b) and (c), respectively. Panels (d), (e), (f) show the energy-flux curves for the ground state of the two-fermions problem as a function of the interaction strength $g$, while setting $N=16$. It is shown that the flux periodicity of the energy-flux curves evolves from ${\mathrm{\Phi }}_0$ towards ${\mathrm{\Phi }}_0/2$ as the interaction strength increases. A similar behavior is observed for fixed interaction strength $g$ and variable system size $N$.

Figure 5

$X_{1,2}$ defined in Eqs. (16) and (17) as a function of the pairing strength $g$. The curves have been obtained by solving the two-particle problem with $N=16$, while considering the attractive Hubbard interaction (a) or the real-space BCS pairing (b). For both the panels, it is observed that $X_1\propto X_2$ for $g\gtrsim 2$. In particular, taking $g\gtrsim 2$, one obtains $X_1\approx 0.39X_2$ for the Hubbard interaction (a) or $X_1\approx 0.17X_2$ for the real-space BCS pairing (b).

Figure 6

(a) Dependence on the system size $N$ of the absolute values of the Fourier coefficients $A_{1,2}$ involved in the ground-state energy expansion given in Eq. (18). The numerical analysis has been performed by considering two particles subject to attractive Hubbard interaction with pairing strength given by $g=1.2$. Full lines have been obtained by considering the interpolation curves $|A_1|\propto exp(-N/\lambda )$ and $|A_2|\propto N^{\gamma }$, being the interpolation parameters $\lambda \approx 3.34$ and $\gamma \approx -1.48$. (b) The same analysis reported in panel (a) is here specialized to the problem of two particles interacting via the real-space BCS interaction with pairing strength given by $g=1.2$. Full lines represent the interpolation curves $|A_1|\propto exp(-N/\lambda )$ and $|A_2|\propto N^{\gamma }$, with $\lambda \approx 3.24$ and $\gamma \approx -1.00$. (c) Absolute value of the Fourier coefficients $A_n$ with $n>0$ obtained by setting attractive Hubbard interaction with $g=1.2$ and system size $N=20$. (d) Absolute value of the Fourier coefficients $A_n$ with $n>0$ obtained by setting real-space BCS interaction with $g=1.2$ and system size $N=20$.

Figure 7

(a)-(c) The ground-state energy $E_G$ of a fermionic system subject to the real-space BCS pairing plotted as a function of the applied magnetic flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$. Blue circular points represent results obtained by using the density matrix renormalization group technique, while red triangles represent the numerical solution of the Richardson equations. Various curves correspond to systems with $M=14$ particles distributed in a ring with $N=16$ lattice sites and different interaction strengths $g\in \{1,2,4\}$ (refer to the legends for details). (d)–(f) $E_G$ vs $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ curves obtained by setting $M=16$, while keeping the other parameters unchanged with respect to the first row. (g) $X_2$ vs $g$ curve obtained by considering $M=14$ particles and $N=16$ lattice sites. (h) Absolute values of the Fourier coefficients $A_1, A_2$, and $A_4$ [see Eq. (18)] shown as a function of the interaction strength $g$, while considering $M=14$ particles and $N=16$ lattice sites.

Figure 8

Panels (a)–(d) illustrate the convergence of the ground-state energy relative to the bond dimension $m$ in the matrix product state, as employed by the DMRG algorithm. To monitor the numerical accuracy, we define the convergence metric $\eta =|(E_G^{\left(m\right)}-E_G^{\left(400\right)})/E_G^{\left(400\right)}|$, where $E_G^{\left(m\right)}$ represents the ground-state energy calculated with bond dimension $m$. All simulations are performed with DMRG sweeps set to 5. The legends within the figures detail the simulation parameters. The analysis reveals that for moderate interaction strengths [i.e., $g=1.0$, see panels (a) and (c)], the computational cost of simulating real-space BCS interaction parallels that of the Hubbard interaction. However, in systems with strong interactions [i.e., $g=6.0$, see panels (b) and (d)], real-space BCS interaction is simulated more efficiently than the attractive Hubbard model, a trend observed across various system sizes.

Figure 9

(a) $X_2$ vs $g$ curves obtained for systems of different sizes $N$ close to half-filling $\left[M/\left(2N\right)\approx 0.45\right]$. Specifically, square symbols ($\square$) represent a system size of $N=16$ with a particle number of $M=14$; triangle symbols ($▵$) indicate $N=20$ and $M=18$; circle symbols ($◯$) represent $N=24$ and $M=22$. The inset displays the $X_2$ vs $N$ curves for interaction values noted within the figure. (b) $N·X_2$ vs $g$ curves obtained for the same $N$ and $M$ values used in panel (a). For small to moderate interaction strengths (i.e., $g<1$), the curves are observed to collapse onto each other. The real-space BCS interaction has been considered for both the panels.