Basics of pairing correlations

In quantum mechanics of finite many-fermion systems, pairing correlations are best described in terms of number operators $\hat{N}_\mu=a^+_\mu a_\mu$, where $\mu$ represents any suitable set of single-particle (s.p.) quantum numbers. Thus we may have, e.g., $\mu\equiv\bm{k}\sigma$ for plane waves of spin-$\frac{1}{2}$ particles (electrons); $\mu\equiv\bm{r}\sigma\tau$ for spin-$\frac{1}{2}$ and isospin-$\frac{1}{2}$ nucleons localized in space at position $\bm{r}$; and $\mu\equiv n,\ell,j,m$ for fermions moving in a spherical potential well. Since $\hat{N}^2_\mu=\hat{N}_\mu$, the number operators are projective; hence, one can - at least in principle - devise an experiment that would project any quantum many-fermion state $\vert\Psi\rangle$ into its component with exactly one fermion occupying state $\mu$. As the rules of quantum mechanics stipulate, any such individual measurement can only give 0 or 1 (these are the eigenvalues of $\hat{N}_\mu$), whereas performing such measurements many times, one could experimentally determine the occupation probabilities $v^2_\mu=\langle\Psi\vert\hat{N}_\mu\vert\Psi\rangle$.

Along such lines, we can devise an experiment that would determine the simultaneous presence of two fermions in different orthogonal s.p. states $\mu$ and $\nu$. Since the corresponding number operators $\hat{N}_\mu$ and $\hat{N}_\nu$ commute, one can legitimately ask quantum mechanical questions about one-particle occupation probabilities $v^2_\mu$ and $v^2_\nu$, as well as about the two-particle occupation. The latter one reflects the simultaneous presence of two fermions in state $\vert\Psi\rangle$: $v^2_{\mu\nu}=\langle\Psi\vert\hat{N}_\mu\hat{N}_\nu\vert\Psi\rangle$. In this way, one can experimentally determine the pairing correlation between states $\mu$ and $\nu$ as the excess probability

$$P_{\mu\nu} = v^2_{\mu\nu} - v^2_{\mu}v^2_{\nu} ,$$ (1)

of finding two fermions simultaneously over that of finding them in independent, or sequential, measurements. Such a definition of pairing is independent of its coherence, collectivity, nature of quasiparticles, symmetry breaking, thermodynamic limit, or many other notions that are often associated with the phenomenon of pairing. In terms of occupations, pairing can be viewed as a measurable property of any quantum many-fermion state.

Obviously, no pairing correlations are present in a quantum state that is an eigenstate of $\hat{N}_\mu$ or $\hat{N}_\nu$, such as the Slater determinant. The beauty of the BCS ansatz is in providing us with a model $N$-fermion state, in which pairing correlations are explicitly incorporated:

$$\vert\Phi_N\rangle = {\cal{}N}_N\left(\sum_{\mu>0}s_\mu z_\mu a^+_{\tilde\mu} a^+_{\mu} \right)^{N/2}\vert\rangle ,$$ (2)

where the summation $\mu>0$ runs over the representatives of pairs ($\tilde\mu,\mu$) of s.p. states (that is, any one state of the pair is included in the sum, but not both), $z_{\tilde\mu} = z_\mu$ are real positive numbers, $s_{\tilde\mu} = - s_\mu$ are arbitrary complex phase factors, and ${\cal{}N}_N$ is the overall normalization factor.

It now becomes a matter of technical convenience to employ a particle-number mixed state,

$$\vert\Phi\rangle = {\cal{}N}\sum_{N=0,2,4,\ldots}^\infty\fra... ...u>0}s_\mu z_\mu a^+_{\tilde\mu} a^+_{\mu} \right)\vert\rangle,$$ (3)

in which the pairing correlations (1) are:

$$P_{\mu\nu} = v^2_{\mu}u^2_{\nu}\delta_{{\tilde\mu}\nu} \quad... ..._{\mu}} \quad\mbox{and}\quad u^2_{\nu}=\frac{1 }{1+z^2_{\nu}}.$$ (4)

In terms of the s.p. occupations, the state $\vert\Phi\rangle$ assumes the standard BCS form:

$$\vert\Phi\rangle= \prod_{\mu>0}\left(u_\mu + s_\mu v_\mu a^+_{\tilde\mu} a^+_{\mu}\right)\vert\rangle.$$ (5)

In this many-fermion state, the s.p. states $\tilde\mu$ and $\mu$ are paired, that is, $\vert\Phi\rangle$ can be viewed as a pair-condensate. For $z_{\mu}$=1, the pairing correlation $P_{{\tilde\mu}\mu}$ (1) equals 1/4; in fact, in this state, it is twice more likely to find a pair of fermions ( $v^2_{{\tilde\mu}\mu}$=1/2) than to find these two fermions independently ( $v^2_{{\tilde\mu}}v^2_{\mu}$=1/4). For particle-number conserving states (2), the occupation numbers can be calculated numerically; qualitatively the results are fairly similar, especially for large numbers of particles.

At this point, we note that the most general pair-condensate state (3) has the form of the Thouless state,

$$\vert\Phi\rangle = {\cal{}N}\exp\left(\frac{1}{2}\sum_{\nu\mu}Z^*_{\nu\mu} a^+_{\nu} a^+_{\mu} \right)\vert\rangle ,$$ (6)

in which pairs ($\tilde\mu,\mu$) do not appear explicitly. However, there always exists a unitary transformation $U_0$ of the antisymmetric matrix $Z$ that brings it to the canonical form $(U_0^+Z^*U_0^*)_{\nu\mu}= s_{\mu}z_{\mu}\delta_{{\tilde\mu}\nu}$ (the Bloch-Messiah-Zumino theorem[9,10]). Therefore, pairs are present in any arbitrary Thouless state (the so-called canonical pairs), and they can be made explicitly visible by a simple basis transformation.

The canonical pairs exist independently of any symmetry of the Thouless state. In the particular case of a time-reversal-symmetric state, $\hat{T}\vert\Phi\rangle=\vert\Phi\rangle$, they can be associated with the time-reversed s.p. states, $\tilde{\mu}\equiv\bar{\mu}$. The ground-states of even-even nuclei can be described in this manner. However, the appearance of pairing phase does not hinge on this particular symmetry - states in rotating nuclei (in which time-reversal symmetry is manifestly broken) can also be paired. In this latter case the canonical states are less useful, because they cannot be directly associated with the eigenstates of the HFB Hamiltonian.