Linear response approximation in effective field theory for the calculation of elastically mediated interactions in one dimension

The linear response approximation, used within effective field theory to calculate mediated interactions between inclusions, is studied for an exactly solvable one-dimensional model. We show that it works poorly in the case of inclusions imposing absolute deformations to the field, while it works well for massless theories in the case of inclusions imposing relative deformations to the field.

Figure 1

Average elastic deformation produced by two inclusions imposing the value of the field (type A). The parameters are ${\varphi }_1=2,{\varphi }_2=3,a=1,R=5$, and $\xi =5$. The inclusions are drawn in red.

Figure 2

Average elastic deformation produced by two inclusions imposing a jump of the field (type B). The parameters are ${\alpha }_1=1,{\alpha }_2=1.5,a=1,R=5$, and $\xi =5$. The inclusions are drawn in red.

Figure 3

In a background (thin black line) set by the boundary values ${\varphi }^{\pm }$ separated by a distance $2L$, either a real inclusion of type A or its effective counterpart of type ${\text{A}}^{\prime }$ are placed at the origin. The Wilson coefficients of the effective inclusion are such that the deformation produced by the type A inclusion (thick gray line) and that produced by the type ${\text{A}}^{\prime }$ inclusion (dashed black line) match whatever the background. The discontinuity of the dashed line at the origin can be much larger for other parameter values. The parameters are ${\varphi }^{-}=2,{\varphi }^{+}=3,L=12,{\varphi }_0=3.5,a=2$, and $\xi =8$. The inclusions are drawn in red.

Figure 4

In a background (thin black line) set by the boundary values ${\varphi }^{\pm }$ separated by a distance $2L$, either a real inclusion of type B or its effective counterpart of type ${\text{B}}^{\prime }$ are placed at the origin. The Wilson coefficients of the effective inclusion are such that the deformation of the type-B inclusion (thick gray line) and the deformation of the ${\text{type-B}}^{\prime }$ inclusion (dashed black line) match whatever the background. Here, the parameters are ${\varphi }^{-}=2,{\varphi }^{+}=3,L=12,\alpha =1,a=2$, and $\xi =8$. The inclusions are drawn in red.

Figure 5

Response of an inclusion of type ${\text{A}}^{\prime \prime }$ (LRA) to a background, and comparison with the response of a real type-A inclusion. (a) The parameters are ${\varphi }^{-}=2,{\varphi }^{+}=3,L=12,{\varphi }_0=3.5,a=2$, and $\xi =8$, exactly as in Fig. 3. (b) Same parameters except that $\xi =25$ instead of $\xi =8$. In both cases the discrepancy between the real field profile (thick gray line) and the effective one (dashed line) is large.

Figure 6

Response of an inclusion of type ${\text{B}}^{\prime \prime }$ (LRA) to a background, and comparison with the response of a real type-B inclusion. (a) The parameters are ${\varphi }^{-}=2,{\varphi }^{+}=3,L=12,\alpha =1,a=2$, and $\xi =8$, exactly as in Fig. 4. (b) Same parameters except that $\xi =25$ instead of $\xi =8$. In the latter case the response of the ${\text{type-B}}^{\prime \prime }$ inclusion is very good (agreement between the thick gray line and the dashed line).