Variational approaches to quantum impurities: from the Fröhlich polaron to the angulon

Figures & data

Figure 1. (a) The polaron energy as a function of the Fröhlich coupling constant, α, for the Chevy ansatz, Equation (4(4) $\mid {\psi }_{\mathit{p}}⟩=\sqrt{Z_{\mathit{p}}}\mid \mathit{p}⟩\mid 0⟩+\sum _{\mathit{k}}{\beta }_{\mathit{p}}\left(\mathit{k}\right)\mid \mathit{p}-\mathit{k}⟩{\hat{b}}_{\mathit{k}}^{†}\mid 0⟩,$(4) ) (red solid line), coherent state on top of single phonon excitation, Equations (12(12) $\mid {\mathrm{\Phi }}_{\mathit{p}}⟩=\mid \varphi ⟩\otimes \mid \mathit{p}⟩,$(12) ) and (16(16) $\mid \varphi ⟩=g\mid 0⟩+\sum _{\mathit{k}}\alpha \left(\mathit{k}\right){\hat{b}}_{\mathit{k}}^{†}\mid 0⟩.$(16) ) (black dotted line), and the Feynman variational method [27] (orange dash-dotted line). (b) Renormalization of the polaron mass as a function of the Fröhlich coupling constant, α, for the Chevy ansatz (red solid line), coherent state on top of single phonon excitation (black dot line), and the weak coupling theory [24] (purple circles). See the text (Colour online).

Figure 2. The polaron energy as a function of the Fröhlich coupling constant, α, for the Pekar ansatz, Equation (24(24) $\mid {\mathrm{\Psi }}_P⟩=\mid \varphi ⟩\otimes \mid {\xi }_B⟩,$(24) ) (blue dash line), and the Pekar diagonalisation technique, Equations (29(29) $\mid {\mathrm{\Psi }}_n⟩=\mid {\varphi }_n⟩\exp \left(-{\hat{X}}_{nn}\right)\mid 0⟩,$(29) ) and (33(33) ${\varphi }_n\left(\mathit{x}\right)=N_n{\mathrm{e}}^{-\beta r}(1+a_{1}r+\cdots a_n{r}^n),$(33) ) (green triangles). See the text (Colour online).

Figure 3. The angulon ground state energy as a function of the angulon coupling constant, α, for the Chevy ansatz [17,18] (red solid line), the Pekar ansatz [35] (blue dashed line), and the Pekar diagonalisation method of Equation (35(35) $|{\mathrm{\Psi }}_{jm}⟩=|jm⟩\exp [-{\hat{X}}_{jm}]|0⟩,$(35) ) (green triangles). The basis consists of the vectors with j=0,1,2. See the text (Colour online).

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