Topology of contact points in Lieb–kagomé model

Abstract

We analyse Lieb–kagomé model, a three-band model with contact points showing particular examples of the merging of Dirac contact points. We prove that eigenstates can be parametrized in a classification surface, which is a hypersurface of a 4-dimension space. This classification surface is a powerful device giving topological properties of the energy band structure; the analysis of its fundamental group proves that all singularities of the band structure can be characterized by four independent winding (integer) numbers. Lieb case separates: its classification surface differs and there is only one winding number.

Graphic abstract

Appendices

A: Hamiltonian in basis I

For completeness [2, 23, 39], let us recall that, in basis I, the Bloch Hamiltonian reads

$$\begin{aligned} H_I({\mathbf {k}})= \left( \begin{array}{ccc}0&{} 1+\mathop {\,\mathrm e}\nolimits ^{\mathbbm {i}k_x}&{}t'(1+\mathop {\,\mathrm e}\nolimits ^{\mathbbm {i}(k_x+k_y)})\\ 1+\mathop {\,\mathrm e}\nolimits ^{-\mathbbm {i}k_x}&{}0&{}1+\mathop {\,\mathrm e}\nolimits ^{\mathbbm {i}k_y}\\ t'(1+\mathop {\,\mathrm e}\nolimits ^{-\mathbbm {i}(k_x+k_y)})&{}1+\mathop {\,\mathrm e}\nolimits ^{-\mathbbm {i}k_y}&{}0 \end{array}\right) . \end{aligned}$$

The relation between both basis is $H({\mathbf {k}})=\mathop {\,\mathrm e}\nolimits ^{-\mathbbm {i}{\mathbf {k}}\cdot {\mathbf {r}}}H_S \mathop {\,\mathrm e}\nolimits ^{\mathbbm {i}{\mathbf {k}}\cdot {\mathbf {r}}}$ and $H_I({\mathbf {k}})= \mathop {\,\mathrm e}\nolimits ^{-\mathbbm {i}{\mathbf {k}} \cdot {\mathbf {R}}}H_S \mathop {\,\mathrm e}\nolimits ^{\mathbbm {i}{\mathbf {k}} \cdot {\mathbf {R}}}$, where $H_S$ is Schrödinger Hamiltonian of the crystal and the complete position operator ${\mathbf {r}}$ is the sum of the Bravais lattice position ${\mathbf {R}}$ and the intra-cell position ${\varvec{\delta }}$. Therefore,

$$\begin{aligned} H({\mathbf {k}})=\mathop {\,\mathrm e}\nolimits ^{-\mathbbm {i}{\mathbf {k}}\cdot \varvec{\delta }}H_I({\mathbf {k}}) \mathop {\,\mathrm e}\nolimits ^{\mathbbm {i}{\mathbf {k}}\cdot \varvec{\delta }}, \end{aligned}$$

with the intra-cell position operator

$$\begin{aligned} \varvec{\delta } = \left( \!\!\begin{array}{ccc}-\frac{{\mathbf {a}}_1}{2}&{}0&{}0 \\ 0&{}0&{}0\\ 0&{}0&{}\frac{{\mathbf {a}}_2}{2} \end{array}\!\!\right) \hbox { such that } \mathop {\,\mathrm e}\nolimits ^{\mathbbm {i}{\mathbf {k}}\cdot \varvec{\delta }}= \left( \!\!\begin{array}{ccc}\mathop {\,\mathrm e}\nolimits ^{-\mathbbm {i}k_x/2} &{}0&{}0\\ 0&{}1&{}0\\ 0&{}0&{}\mathop {\,\mathrm e}\nolimits ^{\mathbbm {i}k_y/2} \end{array}\!\!\right) . \end{aligned}$$

Working with $H_I$ would have the advantage of preserving Brillouin zone, so that any object be $2\pi$-periodic in $k_x$ and $k_y$ directions. However, $H_I$ is complex, so projectors representing eigenvectors would have eight components, instead of five with H.

B: Determination of the singularity at the origin of ${{\mathcal {S}}}$

First of all, we have defined a supplementary coordinate s, to get a bijection from $(a^4_{-1},a^6_{-1},a^4_0,a^6_0)$ to (X, Y, Z, s).

Definition of coordinate s We define $s(a^4,a^6,b^4,b^6)=(a^4)^2+(b^4)^2-(a^6)^2-(b^6)^2$, i.e. $s=V-W$. In Hopf coordinate, one gets

$$\begin{aligned} X= & {} \frac{r^2}{2}(\cos ^2t\sin (2u)+\sin ^2t\sin (2v))\;;\\ Y= & {} \frac{r^2}{2}\sin (2t)\sin (v-u)\;;\\ Z= & {} r^2\;;\\ s= & {} r^2\cos (2t). \end{aligned}$$

Let us explain why we have needed to introduce this coordinate.

Paths ${\mathop {{{\widetilde{\gamma }}}}}$ are never close from the singularity With a posteriori look at paths ${\mathop {{{\widetilde{\gamma }}}}}$, one observes that the Hopf coordinate $r=\sqrt{Z}$ comes close to zero for a very short part of the trajectory and remains close to 1 for the most part of it. We have measured rigorously the length of ${\mathop {{{\widetilde{\gamma }}}}}$ in ${{\mathcal {S}}}$ and found that, indeed, it tends to a non zero value when the radius of $\gamma$ tends to zero.

Artificial path merging towards the singularity In particular, path $s({\mathop {{{\widetilde{\gamma }}}}})$ remains far from O, the singularity in ${{{\mathcal {S}}}}_2$. With the conviction that approaching the (formerly unsettled) singularity in ${{\mathcal {S}}}$ implies approaching O in ${{{\mathcal {S}}}}_2$ and understanding that this would never occur with paths $s({\mathop {{{\widetilde{\gamma }}}}})$, we have designed a path $\gamma _2$ in ${{{\mathcal {S}}}}_2$ which is not a projection $s({\mathop {{{\widetilde{\gamma }}}}})$.

We have chosen $X=\nu \cos \theta$, $Y=\nu \sin \theta$, Z deduced from (15) and $s=0$, with $\nu >0$ a parameter. Thanks to bijection $(X,Y,Z,s)\leftrightarrow (a^4_{-1},a^6_{-1},a^4_0,a^6_0)$, we could map these coordinates as path $\gamma _2$ in ${{{\mathcal {S}}}}_2$ and path ${\mathop {{{\widetilde{\gamma }}}}}_2$ in ${{\mathcal {S}}}$. When $\nu \rightarrow 0$, the former makes a loop merging towards O, while ${\mathop {{{\widetilde{\gamma }}}}}_2$ merges towards (0, 0, 0, 0).

This indicates (0, 0, 0, 0) as a possible singularity. We definitively confirmed this by studying the intersection of ${{\mathcal {S}}}$ and r-$S_3$ spheres, as explained in the text.

C: Detailed description of ${{\mathcal {S}}}$

We first give the folding rules to which Hopf coordinates (t, u, v) obey (r is fixed to some arbitrary value and implicit).

Folding rules Hopf coordinates are defined with the following folding rules

$$\begin{aligned} (t,u,v)&\leftrightarrow&(-t,u,v\pm \pi ), \end{aligned}$$

(20)

$$\begin{aligned} (t,u,v)&\leftrightarrow&(\pi -t,u\pm \pi ,v), \end{aligned}$$

(21)

$$\begin{aligned} (t,u,v)&\leftrightarrow&(t,u\pm \pi ,v\pm \pi ), \end{aligned}$$

(22)

where ± depends on u and v and is chosen so that they lie in the prescribed ranges.

Within ranges $t\in [0,\frac{\pi }{2}[$, $u\in ]-\pi ,\pi ]$ and $v\in ]-\pi ,\pi ]$, these rules almost never apply. They only apply in the following cases, where we write intervals following order (t, u, v):

(20) sends $\{\frac{\pi }{2}\}\times ]{-}\pi ,0]\times ]{-}\pi ,\pi ]$ on $\{\frac{\pi }{2}\}\times ]0,\pi ]\times ]{-}\pi ,\pi ]$ and reciprocally;

(21) sends $\{0\}\times ]{-}\pi ,\pi ]\times ]{-}\pi ,0]$ on $\{0\}\times ]{-}\pi ,\pi ]\times ]0,\pi ]$ and reciprocally;

(22) sends $\{0\}\times ]{-}\pi ,0]\times ]{-}\pi ,0]$ on $\{0\}\times ]0,\pi ]\times ]0,\pi ]$ and reciprocally;

(22) sends $\{0\}\times ]0,\pi ]\times ]{-}\pi ,0]$ on $\{0\}\times ]{-}\pi ,0]\times ]0,\pi ]$ and reciprocally;

(22) sends $\{\frac{\pi }{2}\}\times ]{-}\pi ,0]\times ]{-}\pi ,0]$ on $\{\frac{\pi }{2}\}\times ]0,\pi ]\times ]0,\pi ]$ and reciprocally;

(22) sends $\{\frac{\pi }{2}\}\times ]0,\pi ]\times ]{-}\pi ,0]$ on $\{\frac{\pi }{2}\}\times ]{-}\pi ,0]\times ]0,\pi ]$ and reciprocally.

The (t, u, v) representation of the intersection of ${{\mathcal {S}}}$ with r-$S_3$ (with, for instance, $r=1$) is not only a tridimensional torus with periodic conditions, it is twisted by these complicated rules. In particular, the triviality of path ${\mathop {{{\widetilde{\gamma }}}}}_0$, defined in Sect. 4.8.6, is left unsolved.

Detailed descriptions of holes in ${{\mathcal {S}}}$ Unfolding t and looking at the intersection of ${{\mathcal {S}}}$ with r-$S_3$ sphere allows one to better distinguish holes. We show in Fig. 19 the view in the u direction, with $t\in [-\pi ,\pi ]$, showing two vertical range of six holes. Taking into accounts folding rules reduces this number to three.

Fig. 19

View of ${{\mathcal {S}}}$ in the u direction, with t unfolded twice

The same view is obtained when u and v are inverted and t is translated by $\frac{\pi }{2}$. This results in a view in the v direction, with $t\in [-\frac{\pi }{2},\frac{3\pi }{2}[$.

Altogether, counting the twelve holes in the t direction, one finds exactly 18 holes in ${{\mathcal {S}}}$.

D: Definitions of ${{\mathcal {R}}}$ and $\widetilde{{{\mathcal {R}}}}$

A simple way to construct ${{\mathcal {R}}}$ is to adjoin the surface shown in Fig. 20a to ${{{\mathcal {R}}}}'$ by matching the two circular edges to each elliptic hole boundary: one must match the hole boundary to one of the circular edge of Fig. 20a and the other hole boundary to the other circular edge, by deforming them.

A simple way to construct $\widetilde{{{\mathcal {R}}}}$ is to adjoin the surface shown in Fig. 20b to ${{{\mathcal {R}}}}'$ by matching the two circular edges to each elliptic hole boundary: one must match the hole boundary to one of the circular edge of Fig. 20b and the other hole boundary to the other circular edge, by deforming them.

Fig. 20

Two surfaces related to Klein bottle that allow the construction of ${{\mathcal {R}}}$ and $\widetilde{{{\mathcal {R}}}}$ from ${{{\mathcal {R}}}}'$. The plane represents surface ${{{\mathcal {R}}}}'$, where both elliptic holes are deformed to match the holes of the surface—embedded in a tridimensional space—which lies at the back of ${{{\mathcal {R}}}}'$

Instead, one can construct ${{\mathcal {R}}}$ and $\widetilde{{{\mathcal {R}}}}$ as quotient spaces, using equivalence relations that identify in ${{{\mathcal {R}}}}'$ paths $r({\mathop {{{\widetilde{\gamma }}}}})$ around left and right holes in, respectively, the same or the reverse direction.

E: Other winding numbers

Many classification surfaces can be built. Note that all combinations of winding integers can be obtained. For instance, mapping $z\mapsto z^2$ can be applied to $\tau _d({\mathop {{{\widetilde{\gamma }}}}})$, giving $z_{1d}^2= (a^6_0)^2-(a^4_0)^2+2\,\mathbbm {i}\,a^6_0a^4_0$, and leads to a surface, which is characterized by the same winding $\omega _4$.

In some of them, only diagonal contact points give non zero winding numbers, or reversely, only antidiagonal ones, in some of them all contact points are concerned. Some have two disconnected compounds, while others are connected. They are all projections from ${{\mathcal {S}}}$, thus corresponding winding numbers are product combinations of $\omega _i$, with $i=2..5$.

Alternative bidimensional surface ${{{\mathcal {S}}}}'_2$ We introduce ${\tilde{Z}}=\frac{3}{4}-Z-2X$. Then (15) is equivalent to

$$\begin{aligned} 3X^2{\tilde{Z}}=(Y^2+\frac{3}{2}X)^2\ , \end{aligned}$$

(23)

where the dependency in $(a^4,a^6,b^4,b^6)$ is hidden.

Fig. 21

Representation of surface ${{{\mathcal {S}}}}'_2$ defined by $3X^2\tilde{Z}=(Y^2+\frac{3}{2}X)^2$, showing a singularity at $P_0$. Images of the same loops as those defined in Fig. 11 are shown here. Contact lines are exactly parallel to ${\tilde{Z}}$-axis and contains singularity $P_0$

From $X\in [{-}\frac{3}{4},0]$, one get ${\tilde{Z}}\in [0,\frac{3}{4}]$. Then, taking advantage of these conditions, (23) is equivalent to

$$\begin{aligned} \sqrt{3}({-}X)(\frac{\sqrt{3}}{2}\pm \sqrt{{\tilde{Z}}})=Y^2. \end{aligned}$$

(24)

We note $P_0=(0,0,\frac{3}{4})$ the singular point. (23) or (24) define new surface ${{{\mathcal {S}}}}'_2$ (see Fig. 21), which is topologically identical to ${{{\mathcal {S}}}}_2$. ${{{\mathcal {S}}}}'_2$ is embedded in a tridimensional space and reveals singular point $P_0$, as shown in Fig. 21. All sections orthogonal to $YP_0{\tilde{Z}}$ axis or to $XP_0Y$ axis are delimited by parabolas. The mapping of paths $\gamma$ onto ${{{\mathcal {S}}}}'_2$ is characterized by winding number $\omega _2$.

Alternative surface $\widetilde{{{\mathcal {C}}}}$ We define $\widetilde{{{\mathcal {C}}}}$ the bidimensional surface, built as the disjunctive union of the two ellipses of equations $\forall (a^1,a^3)\in {\mathbb {R}}^2$,

$$\begin{aligned} \begin{array}{rcl} 3\Big (a^1-\frac{1}{2\sqrt{3}}\Big )^2+4(a^3)^2&{}\le &{}1\;;\\ 3\Big (a^1+\frac{1}{2\sqrt{3}}\Big )^2+4(a^3)^2&{}\le &{}1. \end{array} \end{aligned}$$

(25)

Note that $\widetilde{{{\mathcal {C}}}}$ is included in the disk centered at $(a^1,a^3)=(0,0)$ and with radius $\frac{\sqrt{3}}{2}$, which one deduces from Eqs. (7), (8), (9) (and actually from Eq. (3), which they follow).

All Bloch components $(a^1_m,a^3_m)$ follow (25) for all $t\in [0,1]$ and $m={-}1$ or $m=0$, but not for $m=1$. We construct two disconnected surfaces as projections of ${{\mathcal {S}}}$. The mappings write $(a^4_{-1},a^6_{-1},a^4_0,a^6_0) \mapsto (a^1_m,a^3_m)$. With $m={-}1$, mapping ${\mathop {{{\widetilde{\gamma }}}}}\rightarrow (a^1_{-1},a^3_{-1})$ defines surface ${\widetilde{C}}_a$; With $m=0$, mapping ${\mathop {{{\widetilde{\gamma }}}}}\rightarrow (a^1_0,a^3_0)$ defines surface ${\widetilde{C}}_d$. Paths on $\widetilde{{{\mathcal {C}}}}_a$ are non-trivial only if $\gamma$ circles a contact point between lower and middle bands. Paths on $\widetilde{{{\mathcal {C}}}}_d$ are non-trivial only if $\gamma$ circles a contact point between upper and middle bands.

Coefficients in (25) are deduced from a numerical determination of $\widetilde{{{\mathcal {C}}}}$. The intersection of the edges of the two ellipses, computed with (25), are found to be $S_\pm =(0,\pm \frac{\sqrt{3}}{4})$, which matches exactly limit $t'=0$, where paths are circular, with radius $\frac{\sqrt{3}}{4}$. In addition, the simplicity of these equations and their coefficients make them very plausible.

Although we have not proven it yet, confidence in the two-ellipse shape and in the properties of $\widetilde{{{\mathcal {C}}}}$ is complete, because our numerical determination is actually exact. $\widetilde{{{\mathcal {C}}}}$ is embedded in a bidimensional space and reveals two singular points, which are the intersection $S_\pm$, as shown in Fig. 22.

Fig. 22

Representation of surface $\widetilde{{{\mathcal {C}}}}_d$, in the bidimensional space spanned by $(a^1_{-1},a^3_{-1})$, it is exactly the union of two ellipses less their intersection. The edges of the two ellipses intersect at points $S_+$ and $S_-$, defined in the text, which are the singularities of the surface. We show a path for $t'=\frac{1}{2}$, corresponding to a circle around $M_{00}$, of radius $\frac{5}{2}$ (in dashed line), and another, for $t'=1$, corresponding to a circle around $\Gamma _{00}$, of radius $\frac{5}{2}$ (in solid line). Colored points are defined the same way as in Fig. 10, both red points are recovered by yellow ones, which confirms that the image of the path around $\Gamma _{00}$ is described twice, while that of the loop, which contains all contact points of a M-aggregate, is described four times (as would be that of a path around $M_{00}$ at $t'=0$)

When mingling both surfaces, paths are characterized by winding number $-\omega _4$. On the contrary, encoding by 1 paths in $\widetilde{{{\mathcal {C}}}}_a$ and by ${-}1$ those in $\widetilde{{{\mathcal {C}}}}_d$ exactly corresponds to $\omega _5$. Altogether, $\widetilde{{{\mathcal {C}}}}$ is globally characterized by $(-\omega _4,\omega _5)$. Eventually, one finds that the path is non-trivial in $\widetilde{{{\mathcal {C}}}}_a$ if $-\omega _4\omega _5>0$ and non-trivial in $\widetilde{{{\mathcal {C}}}}_d$ if $-\omega _4\omega _5<0$.

Relation between $\widetilde{{{\mathcal {C}}}}_a\times \widetilde{{{\mathcal {C}}}}_d$ and $\widetilde{{{\mathcal {S}}}}_1$ Ignoring scaling factors $\frac{-Y^2}{2\sqrt{3}(X+V)}$ or $\frac{-Y^2}{2\sqrt{3}(X+W)}$, one observes that mapping ${{{\mathcal {Q}}}}_i\rightarrow \widetilde{{{\mathcal {C}}}}_i$ corresponds to the conformal mapping $(x,y)\mapsto (x^2-y^2,-2xy)$, where x stands for $a^6_m$, y for $a^4_m$, $x^2-y^2$ for $a^3_m$ and $-2xy$ for $a^1_m$ (both last components must be divided by the scaling factor), with $m={-}1$ when $i=a$ and $m=0$ when $i=d$. Using complex notation $z_{1i}0$, introduced in 4.2.1, these mappings write $z_{0i}\mapsto \overline{(z_{0i})}^2$, for $i=a,d$.

Alternative surface ${{{\mathcal {C}}}}'$ We construct surface ${{{\mathcal {C}}}}'$ as aprojection of $\widetilde{{{\mathcal {S}}}}_1$ by the mapping $(a^4_{-1},a^6_{-1},a^4_0,a^6_0)\mapsto (a^6_{-1}a^4_0+a^6_0a^4_{-1}, a^6_{-1}a^6_0-a^4_{-1}a^4_0$. Using the complex notation $z_{1i}$, with $i=a,d$, it writes $(z_{0a},z_{0d})\mapsto z_{0a}z_{0d}$.

This surface is a circle with center $O=(0,0)$ and radius $\frac{3}{4}$, less two disconnected elliptic holes, shown in Fig. 23; the two ellipses are centered at $(\pm \frac{1}{3},0)$, their semi-minor axis is along Ox with length $\frac{1}{\sqrt{3}}$, their semi-major axis is along Oy with length $\frac{1}{3}$.

The parameters of ${{{\mathcal {C}}}}'$ are very likely to be exact, in particular, its circular edge can be proven. The confidence in the general shape is complete, since this surface has been found by exact numerical calculations.

Fig. 23

Representation of surface ${{{\mathcal {C}}}}'$, its outer edge is the circle centered at O, with radius $\frac{3}{4}$, its inner boundary is made of the two ellipses, described in the text. Images of the same non-trivial loops as in Fig. 13 are shown. Colored points are defined the same way as in Fig. 10, both red points are recovered by yellow ones, but here the images of both loops are described twice

One can build two winding numbers the same way we have done for ${{{\mathcal {R}}}}'$. Mingling the two holes, one gets $\omega _4$. Doing as for $\widetilde{{{\mathcal {R}}}}$, one gets $-\omega _5$. Altogether, ${{{\mathcal {C}}}}'$ is globally characterized by $(\omega _4,-\omega _5)$, as $\widetilde{{{\mathcal {C}}}}$. Eventually, one finds that the hole, around which the path turns, is given by $-\omega _4\omega _5$ with the same convention as for ${{{\mathcal {R}}}}'$.

Bidimensional surface $\widetilde{{{\mathcal {T}}}}$ We write $\widetilde{{{\mathcal {T}}}}$ the bidimensional surface, defined as the disjunctive union of the two ellipses of equations

$$\begin{aligned} \begin{array}{l} \frac{9}{4}\big (\cos \frac{\pi }{5}(a^4-\frac{1}{2\sqrt{3}})-\sin \frac{\pi }{5}\;a^8\big )^2 \\ \qquad + \frac{64}{9}\big (\sin \frac{\pi }{5}(a^4-\frac{1}{2\sqrt{3}})+\cos \frac{\pi }{5}\;a^8\big )^2 \le 1\;; \\ \frac{9}{4}\big (\cos \frac{\pi }{5}(a^4+\frac{1}{2\sqrt{3}})+\sin \frac{\pi }{5}\;a^8\big )^2 \\ \qquad + \frac{64}{9}\big (-\sin \frac{\pi }{5}(a^4+\frac{1}{2\sqrt{3}})+\cos \frac{\pi }{5}\;a^8\big )^2 \le 1\;; \end{array} \nonumber \\ \end{aligned}$$

(26)

these ellipses have symmetric axis, turned by $\pm \frac{\pi }{5}$ from the $a^4$ axis.

All Bloch components $(a^4_m,a^8_m)$ follow (26) for all $t\in [0,1]$ and $m={-}1$ or $m=0$, but not for $m=1$. We construct two disconnected surfaces as projections of ${{\mathcal {S}}}$. The mappings write $(a^4_{-1},a^6_{-1},a^4_0,a^6_0) \mapsto (a^4_m,a^8_m)$. With $m={-}1$, mapping ${\mathop {{{\widetilde{\gamma }}}}}\rightarrow (a^4_{-1},a^8_{-1})$ defines surface $\widetilde{{{\mathcal {T}}}}_a$; with $m=0$, mapping ${\mathop {{{\widetilde{\gamma }}}}}\rightarrow (a^4_0,a^8_0)$ defines surface $\widetilde{{{\mathcal {T}}}}_d$. $\widetilde{{{\mathcal {T}}}}_a$ are non-trivial only if $\gamma$ circles a contact point between lower and middle bands. Paths on $\widetilde{{{\mathcal {T}}}}_d$ are non-trivial only if $\gamma$ circles a contact point between upper and middle bands.

$\widetilde{{{\mathcal {T}}}}$ is embedded in a bidimensional space and reveals singular points, which are the intersections of the two ellipses, shown in Fig. 24. Confidence in the two-ellipse shape and in the properties of $\widetilde{{{\mathcal {T}}}}$ (26) is complete, because our numerical determination is actually exact. However, the coefficients in (26) must be improved because they lead to a wrong determination of the lower intersection, which is exactly found from the $t'=0$ limit to be $(0,{-}\frac{1}{4})$. Moreover, these coefficients give four intersections (among which three are close to the upper boundary), where two seems more plausible. Prescribing only two intersections would indeed settle new coefficients but we have not investigated this possibility because we could not prove any rigorous founds. Nevertheless, in case this prescription would give $(0,{-}\frac{1}{4})$ as the lower limit, it is very plausible that it be correct.

Apropos, one observes that $\widetilde{{{\mathcal {T}}}}_a$ (as well as $\widetilde{{{\mathcal {T}}}}_d$) is a negative view of ${{{\mathcal {R}}}}'$. Since coefficients in (26) lack of exactness, we are inclined to think that those in (18) too.

Fig. 24

Representation of surface $\widetilde{{{\mathcal {T}}}}_d$, in the bidimensional space spanned by $(a^4_{-1},a^8_{-1})$, it is exactly the union of two ellipses less their intersection. Four intersections of the ellipses, determined by coefficients in (26), are found but they are not reliable. We show images of the same paths than in Fig. 22 but the second one is trivial here. Colored points are defined the same way as in Fig. 10, here the conclusions written in Fig. 22 only apply to the non-trivial path

When mingling both surfaces, paths are characterized by winding number $\omega _3$. On the contrary, encoding by 1 paths in $\widetilde{{{\mathcal {T}}}}_a$ and by ${-}1$ those in $\widetilde{{{\mathcal {T}}}}_d$ exactly corresponds to $-\omega _5$. Altogether, $\widetilde{{{\mathcal {T}}}}$ is globally characterized by $(\omega _3,-\omega _5)$. Eventually, one finds that the path is non-trivial in $\widetilde{{{\mathcal {T}}}}_a$ if $-\omega _3\omega _5>0$ and non-trivial in $\widetilde{{{\mathcal {T}}}}_d$ if $-\omega _3\omega _5<0$.