Diagrammatic approach to orbital quantum impurities interacting with a many-particle environment

Recently it was shown that an impurity exchanging orbital angular momentum with a surrounding bath can be described in terms of the angulon quasiparticle [Phys. Rev. Lett. 118, 095301 (2017)]. The angulon consists of a quantum rotor dressed by a many-particle field of boson excitations and can be formed out of, for example, a molecule or a nonspherical atom in superfluid helium or out of an electron coupled to lattice phonons or a Bose condensate. Here we develop an approach to the angulon based on the path-integral formalism, which sets the ground for a systematic, perturbative treatment of the angulon problem. The resulting perturbation series can be interpreted in terms of Feynman diagrams, from which, in turn, one can derive a set of diagrammatic rules. These rules extend the machinery of the graphical theory of angular momentum—well known from theoretical atomic spectroscopy—to the case where an environment with an infinite number of degrees of freedom is present. In particular, we show that each diagram can be interpreted as a ‘skeleton’, which enforces angular momentum conservation, dressed by an additional many-body contribution. This connection between the angulon theory and the graphical theory of angular momentum is particularly important as it allows us to systematically and substantially simplify the analytical representation of each diagram. In order to exemplify the technique, we calculate the one- and two-loop contributions to the angulon self-energy, the spectral function, and the quasiparticle weight. The diagrammatic theory we develop paves the way to investigate next-to-leading order quantities in a more compact way compared to the variational approaches.

Figure 1

The only diagram contributing to the first-order self-energy in the diagrammatic expansion. The labels $\lambda$ and $\omega$ on each solid (dashed) line denote the angular momentum and the energy of the angulon (phonon), respectively.

Figure 2

Diagrams appearing at the second order of the diagrammatic expansion. The first and second diagrams are 1-particle irreducible and form the second order contribution to the self-energy ${\mathrm{\Sigma }}^{\left(2\right)}$, whereas the third diagram is not 1-particle irreducible and is accounted for in the Dyson sum for ${\mathrm{\Sigma }}^{\left(1\right)}$.

Figure 3

The angulon spectral function ${\tilde{\mathcal{A}}}_L\equiv {\mathcal{A}}_{L}B$ for $L=0,1,2$ as a function of the dimensionless density, $\tilde{n}=n{\left(mB\right)}^{-3/2}$, and of the dimensionless energy, $\tilde{\omega }=\omega /B$, parameters defined in the text. The left panel shows the first-order spectral function, Eq. (45), obtained from the Dyson equation of Eq. (32) and one-loop diagrams, Eq. (34). The right panel, on the other hand, includes two-loop contributions, Eqs. (35) and (36). The white dashed lines show the energy of the first-order quasiparticle states, derived from Eq. (48), showing the negative shift in the quasiparticle energy due to the inclusion of two-phonon processes in the high-density region. The dashed red region corresponds to an unphysical region with negative spectral weight, as described in the text. The notation for the state labels is also introduced in the text.

Figure 4

The angulon spectral function, ${\tilde{\mathcal{A}}}_L\equiv {\mathcal{A}}_{L}B$, as a function of the dimensionless energy, $\tilde{\omega }=\omega /B$, in the high-density regime $log\left(\tilde{n}\right)=5$, for different values of the boson-boson scattering length, [in units of ${\left(mB\right)}^{-1/2}]$. The lowest peaks for $L=0,1$ are shown, in the case of $a_{\text{bb}}=2.0{\left(mB\right)}^{-1/2}$ also a third peak $L=2$ is visible in the plotted range of energies. Smaller values of the boson-boson scattering length $a_{\text{bb}}$ correspond to a stronger renormalization of the angulon energy, as well as to an increased splitting between the peaks calculated using the first order theory (solid line) and the second order theory (dashed lines).

Figure 5

The angulon spectral weight for the $L=0, L=1$, and $L=2$ states, from left to right, calculated at one-loop level (black solid line) and at two-loop level (red solid line), as a function of the dimensionless density $\tilde{n}=n{\left(mB\right)}^{-3/2}$ and the dimensionless boson-boson scattering length ${\tilde{a}}_{\text{bb}}=a_{\text{bb}}{\left(mB\right)}^{1/2}$.

Figure 6

The diagrams in Fig. 2 and the corresponding analytic expressions for the geometric part can be readily mapped onto diagrams of the graphical theory of angular momentum [87]. These diagrams encode the geometric aspect of the self-energies and reflect the conservation of angular momentum. These diagrams still contain the summations over ${\mu }_i$ which can be eliminated using graphical techniques: the upper diagram is simplified by changing the sign of two nodes, thus reconstructing the $6j$ symbol, whereas the lower diagram is simplified making use of the separation technique for subdiagrams connected by two lines. As a result we arrive at the noticeably simpler results of Eq. (D3) and Eq. (D4).