bistable domains then first emerge along the $x-$axis at the points
$\tilde{\bf R}_{m}=(\pm\sqrt{2}\xi,0)$ and $\bar{\bf R}_{m}=(\pm
2\sqrt{2}\xi,0)$ when

$\kappa_{m}=\frac{e_{p}}{4\bar{C}\xi^{2}}=1.$ (96)

Upon increasing the pinning strength $\kappa_{m}$, e.g., via softening of the
vortex lattice as described by a decrease in $\bar{C}$, the unstable and
bistable domains $\mathcal{U}_{\tilde{\bf R}}$ and $\mathcal{B}_{\bar{\bf R}}$
expand away from these points, and eventually merge along the $y-$axis at
$\tilde{\bf R}_{s}=(0,\pm\sqrt{2}\xi(1+\epsilon))$, $\bar{\bf R}_{s}=(0,\pm
2\sqrt{2}\xi(1+\epsilon))$ when

$\kappa_{s}=\frac{e_{p}}{4\bar{C}\xi^{2}(1+\epsilon)^{2}}=\frac{\kappa_{m}}{(1+\epsilon)^{2}}=1,$
(97)

i.e., for $\kappa_{m}=(1+\epsilon)^{2}$. The evolution of the strong pinning
landscape from onset to merging takes place in the interval
$\kappa_{m}\in[1,(1+\epsilon)^{2}]$; pushing $\kappa_{m}$ beyond this
interval, we will analyze the change in topology and appearance of non-simply
connected unstable and bistable domains after the merging.

The quantity determining the shape of the unstable domain
$\mathcal{U}_{\tilde{\bf R}}$ is the Hessian determinant $D(\tilde{\bf R})$ of
the total vortex energy $e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$, see
Eqs. (36) and (1), respectively. At onset, the minimum of $D(\tilde{\bf R})$
touches zero for the first time; with increasing $\kappa_{m}$, this minimum
drops below zero and the condition $D(\tilde{\bf R})=0$ determines the
unstable ellipse that expands in $\tilde{\bf R}$-space. Viewing the function
$D(\tilde{\bf R})$ as a height function of a landscape in the $\tilde{\bf R}$
plane, this corresponds to filling this landscape, e.g., with water, up to the
height level $D=0$ with the resulting lake representing the unstable domain.
In the present uniaxially symmetric case, a pair of unstable ellipses grow
simultaneously, bend around the equipotential line near the radius
$\sim\sqrt{2}\xi$ and finally touch upon merging on the $y$-axis. In our
geometric interpretation, this corresponds to the merging of the two (water-
filled) valleys that happens in a saddle-point of the function $D(\tilde{\bf
R})$ at the height $D=0$. Hence, the merger point $\tilde{\bf R}_{s}$
correspond to saddles in $D(\tilde{\bf R})$ with

$D(\tilde{\bf R}_{s})=0,\quad\mathbf{\nabla}_{\tilde{\bf
R}}\,D(\mathbf{R})\big{|}_{\tilde{\bf R}_{s}}=0,$ (98)

and

$\mathrm{det}\bigl{[}\mathrm{Hess}\bigl{[}D(\tilde{\bf
R})\bigr{]}\bigr{]}\big{|}_{\tilde{\bf R}_{s}}<0,$ (99)

cf. Eq. (44).

In our calculation of $D(\tilde{\bf R})$, we exploit that the Hessian in (36)
does not depend on the asymptotic position $\bar{\bf R}$ and we can set it to
zero,

$\displaystyle D(\tilde{\bf R})$
$\displaystyle=\det\\{\mathrm{Hess}[\bar{C}\tilde{R}^{2}/2+e_{p}^{\scriptscriptstyle(i)}(\tilde{R})+\delta
e_{p}(\tilde{\bf R})]\bigr{\\}},$ (100)

where we have split off the anisotropic correction $\delta e_{p}(\tilde{\bf
R})=e_{p}(\tilde{\bf R})-e_{p}^{\scriptscriptstyle(i)}(\tilde{R})$ away from
the isotropic potential $e_{p}^{\scriptscriptstyle(i)}(\tilde{R})$ with
$\epsilon=0$. In the following, we perform a perturbative analysis around the
isotropic limit valid in the limit of weak anisotropy $\epsilon\ll 1$; this
motivates our use of polar (tip) coordinates $\tilde{R}$ and $\tilde{\phi}$.

The isotropic contribution $\mathrm{H}^{\scriptscriptstyle(i)}$ to the Hessian
matrix $\mathrm{H}$ is diagonal with components

$\displaystyle\mathrm{H}^{\scriptscriptstyle(i)}_{\tilde{R}\tilde{R}}(\tilde{R})$
$\displaystyle\equiv\partial_{\tilde{R}}^{2}[\bar{C}\tilde{R}^{2}/2+e_{p}^{\scriptscriptstyle(i)}(\tilde{R})]$
$\displaystyle=\bar{C}+\partial_{\tilde{R}}^{2}e_{p}^{\scriptscriptstyle(i)}(\tilde{R})$
(101)

and

$\displaystyle\mathrm{H}^{\scriptscriptstyle(i)}_{\tilde{\phi}\tilde{\phi}}(\tilde{R})$
$\displaystyle\equiv(\tilde{R}^{-2}\partial^{2}_{\tilde{\phi}\tilde{\phi}}+\tilde{R}^{-1}\partial_{\tilde{R}})[\bar{C}\tilde{R}^{2}/2+e_{p}^{\scriptscriptstyle(i)}(\tilde{R})]$
$\displaystyle=\bar{C}-f_{p}^{\scriptscriptstyle(i)}(\tilde{R})/\tilde{R}.$
(102)

The radial component
$\mathrm{H}^{\scriptscriptstyle(i)}_{\tilde{R}\tilde{R}}\propto(\kappa_{m}-1)$
vanishes at onset, while
$\mathrm{H}^{\scriptscriptstyle(i)}_{\tilde{\phi}\tilde{\phi}}$ remains
finite, positive, and approximately constant.

Figure 10: Unstable and bistable domains close to the onset of strong pinning
for a uniaxial defect (94) centered at the origin, with $\epsilon=0.1$ and
$\kappa_{m}-1=0.01$. The pinning potential is steepest at angles
$\tilde{\phi}=0,\,\pi$ and least steep at $\tilde{\phi}=\pm\pi/2$, hence
strong pinning is realized first in a small interval around
$\tilde{\phi}=0,\,\pi$ (solid black dots) where $\kappa_{m}(\tilde{\phi})\geq
1$. (a) The unstable domain $\mathcal{U}_{\tilde{\bf R}}$ in tip space is
bounded by red/blue solid lines (jump lines $\mathcal{J}_{\mathrm{\tilde{\bf
R}}}$, see Eq. (108)); dashed lines mark the associated landing lines
$\mathcal{L}_{\mathrm{\tilde{\bf R}}}$, see (114). (b) Focus on the unstable
domain near $\tilde{\phi}=0$ in polar coordinates $\tilde{R}$ and
$\tilde{\phi}$. The jumping (solid) and landing (dashed) lines have the
approximate shape of ellipses, see Eq. (111), in agreement with our analysis
of Sec. III.2. (c) The bistable domain $\mathcal{B}_{\bar{\bf R}}$ in
asymptotic space involves symmetric crescents centered at $\bar{\phi}=0,\,\pi$
and a narrow width $\propto(\kappa_{m}(\bar{\phi})-1)^{3/2}$, see Eq. (112),
in agreement with the analysis of Sec. III.3. (d) Focus on the bistable domain
at $\bar{\phi}=0$ in polar coordinates $\bar{R}$ and $\bar{\phi}$. Red/blue
colors indicate different vortex configurations as quantified through the
order parameter $\tilde{R}-\tilde{R}_{m}(\tilde{\phi})$.

The anisotropic component $\delta e_{p}(\tilde{\bf R})$ introduces corrections
$\propto\epsilon$; these significantly modify the radial entry of the full
Hessian while leaving its azimutal component
$\mathrm{H}_{\tilde{\phi}\tilde{\phi}}$ approximately unchanged; the off-
diagonal entries of the full Hessian scale as $\epsilon$ and hence contribute
in second order of $\epsilon$ to $D(\tilde{\bf R})$. As a result, the sign
change in the determinant

$D(\tilde{\bf R})\approx\mathrm{H}_{\tilde{R}\tilde{R}}(\tilde{\bf
R})\mathrm{H}_{\tilde{\phi}\tilde{\phi}}(\tilde{R})+\mathcal{O}\left(\epsilon^{2}\right),$
(103)

is determined by

$\mathrm{H}_{\tilde{R}\tilde{R}}(\tilde{\bf
R})=\mathrm{H}^{\scriptscriptstyle(i)}_{\tilde{R}\tilde{R}}(\tilde{R})+\partial^{2}_{\tilde{R}}\delta
e_{p}(\tilde{\bf R})$ (104)

for radii close to $\tilde{R}_{m}$ with
$\delta\tilde{R}=\tilde{R}-\tilde{R}_{m}\approx\mathcal{O}(\sqrt{\kappa_{m}-1})$.
We expand the potential (94) around the isotropic part
$e_{p}^{\scriptscriptstyle(i)}(\tilde{R})$,

$\delta e_{p}(\tilde{\bf
R})\approx-\epsilon\,[\partial_{\tilde{R}}e_{p}^{\scriptscriptstyle(i)}(\tilde{R})]\tilde{R}\sin^{2}\tilde{\phi},$
(105)

and additionally expand both $e_{p}^{\scriptscriptstyle(i)}(\tilde{R})$ and
$\delta e_{p}(\tilde{\bf R})$ around $\tilde{R}_{m}$, keeping terms
$\propto\epsilon\,\sqrt{(\kappa_{m}-1)}$. The radial entry of the anisotropic
Hessian matrix then assumes the form

$\mathrm{H}_{\tilde{R}\tilde{R}}(\tilde{\bf
R})\approx\bar{C}\,[1-\kappa_{m}(\tilde{\phi})]\\\
+\gamma\,[\delta\tilde{R}^{2}/2-\epsilon\,\sin^{2}{\tilde{\phi}}\,\tilde{R}_{m}\delta\tilde{R}]$
(106)

with
$\gamma=\partial^{4}_{\tilde{R}}e_{p}^{\scriptscriptstyle(i)}(\tilde{R})|_{\tilde{R}_{m}}$
and the angle-dependent Labusch parameter

$\kappa_{m}(\tilde{\phi})\equiv\frac{\max_{\tilde{R}}[-\partial_{\tilde{R}}^{2}e_{p}(\tilde{R},\tilde{\phi})|_{\tilde{\phi}}]}{\bar{C}}=\kappa_{m}-2\epsilon\sin^{2}\tilde{\phi}.$
(107)

The edges of the unstable region $\mathcal{U}_{\tilde{\bf R}}$ then can be
obtained by imposing the condition $\mathrm{H}_{\tilde{R}\tilde{R}}(\tilde{\bf
R})=0$ and the solution to the corresponding quadratic equation define the
jump positions $\tilde{R}_{\mathrm{jp}}(\tilde{\phi})$ (or boundaries
$\partial\mathcal{U}_{\tilde{\bf R}}$)

$\displaystyle\tilde{R}_{\mathrm{jp}}(\tilde{\phi})\approx\tilde{R}_{m}(\tilde{\phi})\pm\delta\tilde{R}(\tilde{\phi}).$
(108)

These are centered around the (‘large’) ellipse defined by

$\tilde{R}_{m}(\tilde{\phi})=\tilde{R}_{m}(1+\epsilon\sin^{2}\tilde{\phi})$
(109)

and separated by (cf. Eq. (20))

$\displaystyle
2\,\delta\tilde{R}(\tilde{\phi})=\sqrt{\frac{8\bar{C}}{\gamma}(\kappa_{m}(\tilde{\phi})-1)}$
(110)

along the radius. Making use of the form (107) of $\kappa_{m}(\tilde{\phi})$
and assuming a small value of $\kappa_{m}>1$ near onset, we obtain the jump
line in the form of a (‘small’) ellipse centered at $[\pm\tilde{R}_{m},0]$,

$\gamma\,\delta\tilde{R}^{2}+\epsilon\bar{C}\,\tilde{\phi}^{2}=\bar{C}(\kappa_{m}-1).$
(111)

Figure 11: Unstable and bistable domains before merging for a uniaxial defect
(94) centered at the origin, with $\epsilon=0.1$ and $1-\kappa_{s}\approx
0.01$. Strong pinning is realized everywhere but in a small interval around
$\tilde{\phi}=\pm\pi/2$ where $\kappa_{m}(\tilde{\phi})<1$. (a) The unstable
domain $\mathcal{U}_{\tilde{\bf R}}$ in the tip plane is bounded by the solid
red/blue jump lines $\mathcal{J}_{\mathrm{\tilde{\bf R}}}$, see Eq. (108) and
involves two strongly bent ellipses originating from angles
$\tilde{\phi}=0,\,\pi$ (black dots) and approaching one another close to
$\tilde{\phi}=\pm\pi/2$ (black crosses); red/blue dashed lines are landing
points as given by Eqs. (114). (b) Focus (in polar coordinates
$\tilde{R},\,\tilde{\phi}$) on the tips of the unstable domain near
$\tilde{\phi}=\pi/2$. (c) The bistable domain $\mathcal{B}_{\bar{\bf R}}$ in
the asymptotic space consists of thin symmetric crescents (colored in magenta)
originating from $\bar{\phi}=0,\,\pi$, with the delimiting black solid lines
given by Eq. (112). (d) Focus on the cusps of the bistable domain close to
$\bar{\phi}=\pi/2$ in polar coordinates $\bar{R},\,\bar{\phi}$. Red/blue
colors indicate different vortex configurations as quantified through the
order parameter $\tilde{R}-\tilde{R}_{m}(\bar{\phi})$.

Hence, we find that the anisotropic results are obtained from the isotropic
ones by replacing the circle $\tilde{R}_{m}$ by the ellipse $\tilde{\bf
R}_{m}(\tilde{\phi})$ and substituting $\kappa\to\kappa_{m}(\tilde{\phi})$ in
the width (20), see Figs. 10(a) and (b) evaluated for small values
$\kappa_{m}-1=0.01$ and $\epsilon=0.1$.

Analogously, the boundaries of the bistable domain $\mathcal{B}_{\bar{\bf R}}$
can be found by applying the same substitutions to the result (25), see Figs.
10(c) and (d),

$\displaystyle\bar{R}(\bar{\phi})\approx\bar{R}_{m}(\bar{\phi})\pm\delta\bar{R}(\bar{\phi})$
(112)

with $\bar{R}_{m}(\bar{\phi})=\bar{R}_{m}(1+\epsilon\sin^{2}\bar{\phi})$ and
the width

$\displaystyle
2\,\delta\bar{R}(\bar{\phi})=\frac{2}{3}\sqrt{\frac{8\bar{C}}{\gamma}}(\kappa_{m}(\tilde{\phi})-1)^{3/2}.$
(113)

The landing line $\mathcal{L}_{\tilde{\bf R}}$ is given by (see Eq. (23) and
note that the jump point is shifted by $\tilde{u}_{\mathrm{jp}}$ away from
$\tilde{x}_{m}$, see Eq. (19))

$\displaystyle\tilde{R}_{\mathrm{lp}}(\tilde{\phi})\approx\tilde{R}_{m}(\tilde{\phi})\mp
2\,\delta\tilde{R}(\tilde{\phi}).$ (114)

An additional complication is the finite angular extension of the unstable and
bistable domains $\mathcal{U}_{\tilde{\bf R}}$ and $\mathcal{B}_{\bar{\bf
R}}$; these are limited by the condition $\kappa_{m}(\phi_{\max})=1$,
providing us with the constraint

$\tilde{\phi}_{\max}=\bar{\phi}_{\max}\approx\pm\sqrt{\frac{\kappa_{m}-1}{2\epsilon}}$
(115)

near the strong pinning onset with $(\kappa_{m}-1)\ll\epsilon$. The resulting
domains $\mathcal{U}_{\tilde{\bf R}}$ have characteristic extensions of scale
$\propto\sqrt{\kappa_{m}-1}$, see Fig. 10.

Figure 12: Unstable and bistable domains for a uniaxial defect (94) after
merging, with $\epsilon=0.1$ and $\kappa_{s}-1\approx 0.01$. (a) The unstable
domain $\mathcal{U}_{\tilde{\bf R}}$ in tip plane is enclosed between the jump
lines $\mathcal{J}_{\mathrm{\tilde{\bf R}}}$ (solid red/blue, see Eq. (108))
and takes the shape of a deformed ring with a wider (narrower) width at
strongest (weakest) pinning near the solid dots (crosses). Red/blue dashed
lines mark the landing positions $\mathcal{L}_{\mathrm{\tilde{\bf R}}}$ of the
vortex tips and are given by Eq. (114). (b) Focus on the narrowing in the
unstable domain close to the merger points (crosses) at $\tilde{\phi}=\pi/2$
in the polar coordinates $\tilde{R},\,\tilde{\phi}$. (c) The bistable domain
$\mathcal{B}_{\bar{\bf R}}$ in asymptotic space is a narrow ring (colored in
magenta) thicker (thinner) at points of strongest (weakest) pinning near
$\bar{\phi}=0,\ \pi$ ($\bar{\phi}=\pm\pi/2$); black lines correspond to Eq.
(112). (d) Focus on the constriction in the bistable domain close to
$\bar{\phi}=\pi/2$ in polar coordinates $\bar{R},\,\bar{\phi}$. Red/blue
colors indicate different vortex configurations as quantified through the
order parameter $\tilde{R}-\tilde{R}_{m}(\bar{\phi})$.

Close to merging (marked by crosses in the figure) at $\phi=\pm\pi/2$, we
define the deviation $\delta\phi=\pi/2-\phi$ with $\delta\phi\ll 1$, and
imposing the condition $\kappa_{m}(\phi_{\max})=1$, we find

$\delta\tilde{\phi}_{\max}=\delta\bar{\phi}_{\max}\approx\sqrt{1-\frac{\kappa_{m}-1}{2\epsilon}}\approx\sqrt{\frac{1-\kappa_{s}}{2\epsilon}}.$
(116)

The corresponding geometries of $\mathcal{U}_{\tilde{\bf R}}$ and
$\mathcal{B}_{\bar{\bf R}}$ are shown in Fig. 11 for $1-\kappa_{s}\approx
0.01$ and $\epsilon=0.1$. Finally, $\delta\tilde{\phi}_{\max}$ vanishes at
merging for $\kappa_{s}=1$ (or $\kappa_{m}-1\approx 2\epsilon$), in agreement,
to order $\epsilon$, with the exact result (97).

Pushing the Labusch parameter beyond the merger with $\kappa_{s}>1$ or
$\kappa_{m}>(1+\epsilon)^{2}\approx 1+2\epsilon$, the unstable and bistable
regimes $\mathcal{U}_{\tilde{\bf R}}$ and $\mathcal{B}_{\bar{\bf R}}$ change
their topology: they develop a (non-simply connected) ring-like geometry with
separated inner and outer edges that are a finite distance apart in the radial
direction at all angles $\tilde{\phi}$ and $\bar{\phi}$. The situation after
the merger is shown in Fig. 12 for $\kappa_{s}-1\approx 0.01$ and
$\epsilon=0.1$, with the merging points $\tilde{\bf R}_{s}$ and $\bar{\bf
R}_{s}$ marked by crosses.

The merging of the unstable domains at the saddle point $\tilde{\bf R}_{s}$ is
a general feature of irregular pinning potentials. In the next section, we
will analyze the behavior of the unstable domains close to a saddle point
$\tilde{\bf R}_{s}$ of the Hessian determinant $D(\tilde{\bf R})$ and obtain a
universal description of their geometry close to this point. We will see that
the geometry associated with this merger is of a hyperbolic type described by
$\gamma\tilde{u}^{2}+\delta\tilde{v}^{2}=2\bar{C}(\kappa_{s}-1)$, $\gamma>0$
and $\delta<0$ (assuming no skew). The change in topology then is driven by
the sign change in $\kappa_{s}-1$: before merging, $\kappa_{s}<1$, the
hyperbola is open along the unstable (radial) direction $\tilde{u}$, thus
separating the two unstable regions, while after merging, $\kappa_{s}>1$, the
hyperbola is open along the transverse direction $\tilde{v}$, with the ensuing
passage defining the single, non-simply connected, ring-like unstable region.

##  V Merger points

The merging of unstable and bistable domains is a general feature of irregular
pinning potentials that is relevant beyond the simple example of a weakly
anisotropic uniaxial defect discussed above. Indeed, while the exact
geometries of $\mathcal{U}_{\tilde{\bf R}}$ and $\mathcal{B}_{\bar{\bf R}}$
depend on the precise shape of the pinning potential, their behavior close to
merging is universal. Below, we will study this universal behavior by
generalizing the expansions of Sec. III to saddle points $\tilde{\bf R}_{s}$
of the determinant $D(\tilde{\bf R})$. As with the onset of strong pinning,
the merger of two domains induces a change in topology in the unstable and
bistable domains; we will discuss these topological aspects of onsets and
mergers in Secs. V.4 and VI below.

###  V.1 Expansion near merger

Following the strategy of Sec. III, we expand the energy functional around a
saddle point $\tilde{\bf R}_{s}$ of the determinant $D(\tilde{\bf R})$ in
order to obtain closed expressions for the unstable and bistable domains at
merging. In doing so, we again define local coordinate systems
$(\tilde{u},\tilde{v})$ and $(\bar{u},\bar{v})$ in tip- and asymptotic space
centered at $\tilde{\bf R}_{s}$ and $\bar{\bf R}_{s}$, where the latter is
associated with $\tilde{\bf R}_{s}$ through the force balance equation (38) in
the original laboratory system. Furthermore, we fix our axes such that
$D(\tilde{\bf R}_{s})$ is a local maximum along the (unstable) $u$\- and a
local minimum along the (stable) $v$-direction of the saddle; the mixed term
$\propto\tilde{u}\tilde{v}$ is absent from the expansion (as the Hessian
matrix is symmetric). Furthermore, the vanishing slopes at the saddle point,
see (98), imply the absence of terms $\propto\tilde{u}^{3}$ and
$\propto\tilde{u}^{2}\tilde{v}$ in the expansion and dropping higher-order
terms (corresponding to double-primed terms in (40)), we arrive to the
expression

$\displaystyle e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf
R})=\frac{\bar{C}}{2}(1-\kappa_{s})\,\tilde{u}^{2}+\frac{\bar{C}+\lambda_{+,s}}{2}\,\tilde{v}^{2}+\frac{a_{s}}{2}\,\tilde{u}\tilde{v}^{2}$
$\displaystyle\quad+\frac{\alpha_{s}}{4}\,\tilde{u}^{2}\tilde{v}^{2}+\frac{\beta_{s}}{6}\,\tilde{u}^{3}\tilde{v}+\frac{\gamma_{s}}{24}\,\tilde{u}^{4}-\bar{C}\bar{u}\tilde{u}-\bar{C}\bar{v}\tilde{v},$
(117)

with $\kappa_{s}\equiv-\lambda_{-}(\tilde{\bf R}_{s})/\bar{C},\
\lambda_{+,s}\equiv\lambda_{+}(\tilde{\bf R}_{s})$ and the remaining
coefficients defined in analogy to Eq. (III.2.1).

The most important term in the expansion (V.1) is the curvature term
$\bar{C}(1-\kappa_{s})\,\tilde{u}^{2}/2$ along the unstable direction $u$. As
before in Sec. III.2, see Eq. (III.2.1), the coefficient $(1-\kappa_{s})$
changes sign at some value of the pinning strength and will serve as the small
parameter in our considerations. The higher-order terms in the expansion (V.1)
are constrained by the saddle condition (99), implying that (cf. (48) and
(50))

$\gamma_{s}\delta_{s}-\beta_{s}^{2}<0$ (118)

with

$\delta_{s}\equiv\alpha_{s}-\frac{2a_{s}^{2}}{\bar{C}+\lambda_{+,s}}$ (119)

(for the saddle point there is no condition on the trace of the Hessian). The
mapping of the two-dimensional pinning energy (V.1) to an effective one-
dimensional Landau theory (178) of the van der Waals kind is discussed in
Appendix A.2, both before and after merging.

###  V.2 Unstable domain $\mathcal{U}_{\tilde{\bf R}}$

Figure 13: Jump lines $\mathcal{J}_{\mathrm{\tilde{\bf R}}}$ (solid red/blue)
and landing lines $\mathcal{L}_{\mathrm{\tilde{\bf R}}}$ (dashed red/blue) in
tip space $\tilde{\bf R}$ (in units of $\xi$), with the hyperbola
$\mathcal{J}_{\mathrm{\tilde{\bf R}}}$ defining the edge
$\partial\mathcal{U}_{\tilde{\bf R}}$ of the unstable domain
$\mathcal{U}_{\tilde{\bf R}}$, before (a) and after (b) merging, for
$1-\kappa_{s}=\pm 0.01$. Parameters are
$\lambda_{-,s}=-0.25\,e_{p}/\xi^{2},\lambda_{+,s}=0$, and $a_{s}\approx
0.035\,e_{p}/\xi^{3}$, $\alpha_{s}=-0.025\,e_{p}/\xi^{4}$, $\beta_{s}=0$,
$\gamma_{s}\approx 0.68\,e_{p}/\xi^{4}$. A finite skew parameter
$\beta_{s}=0.025e_{p}/\xi^{4}$ tilts the hyperbola away from the axes (dotted
curves). Crosses correspond to the vertices (125) and (129) of the hyperbola
before and after merging. Pairs of solid and open circles connected via long
arrows are examples of pairs of jumping- and landing tip positions. After
merging, see (b), the unstable domain $\mathcal{U}_{\tilde{\bf R}}$ is
connected along the $\tilde{v}$-axis, dividing the tip coordinate plane into
two separate regions. The jumping and landing hyperbolas coincide at their
vertices before merging, see (a), but not thereafter, see (b), where the
jumping and landing hyperbolas are separated (vertices on
$\mathcal{L}_{\mathrm{\tilde{\bf R}}}$ are marked with open red/blue stars)
and no contact point is present. Note the rotation by 90 degrees of the
unstable direction with respect to Figs. 11(b) and 12(b).

####  V.2.1 Jump line $\mathcal{J}_{\mathrm{\tilde{\bf R}}}$

The boundary of the unstable domain $\mathcal{U}_{\tilde{\bf R}}$ is
determined by the jump condition $D(\tilde{\bf R}_{s,\mathrm{jp}})=0$. Making
use of the expansion (V.1) and keeping only terms quadratic in
$\tilde{u},\tilde{v}$, the edges $\delta\tilde{\bf
R}_{s,\mathrm{jp}}=(\tilde{u}_{s,\mathrm{jp}},\tilde{v}_{s,\mathrm{jp}})$ of
$\mathcal{U}_{\tilde{\bf R}}$ (measured relative to $\tilde{\bf R}_{s}$) are
given by the solutions of the quadratic form (cf. (53))

$[\gamma_{s}\,\tilde{u}^{2}+2\beta_{s}\,\tilde{u}\tilde{v}+\delta_{s}\,\tilde{v}^{2}]_{\tilde{\bf
R}_{s,\mathrm{jp}}}=2\bar{C}(\kappa_{s}-1).$ (120)

Equation (120) describes a hyperbola (centered at $\tilde{\bf R}_{s}$) as its
associated determinant is negative, see Eq. (118). Again, (120) can be cast in
the form of a matrix equation

$\delta\tilde{\bf R}_{s,\mathrm{jp}}^{T}M_{s,\mathrm{jp}}\delta\tilde{\bf
R}_{s,\mathrm{jp}}=\bar{C}(\kappa_{s}-1),$ (121)

with $M_{s,\mathrm{jp}}$ given by

$\displaystyle M_{s,\mathrm{jp}}$
$\displaystyle=\begin{bmatrix}\gamma_{s}/2&~{}~{}~{}\beta_{s}/2\\\
\beta_{s}/2&~{}~{}~{}\delta_{s}/2\end{bmatrix}$ (122)

with $\det M_{s,\mathrm{jp}}=(\gamma_{s}\delta_{s}-\beta_{s}^{2})/4<0$. As
shown in Fig. 13, the geometry of the unstable domain $\mathcal{U}_{\tilde{\bf
R}}$ changes drastically when $1-\kappa_{s}$ changes sign. Before merging,
i.e., for $1-\kappa_{s}>0$, the unstable domain (top and bottom regions in
Fig. 13(a)) is disconnected along the stable $v$-direction and the two
red/blue branches of the hyperbola (120) describe the tips of
$\mathcal{U}_{\tilde{\bf R}}$. When $\kappa_{s}$ goes to unity, the tips of
the unstable domain merge at the saddle point $\tilde{\bf R}_{s}$. After
merging, the unstable domain extends continuously from the top to the bottom
in Fig. 13(b) with a finite width along the unstable $u$-direction, similarly
to the isotropic case shown in Fig. 5(c). Correspondingly, the two (red and
blue) branches of the hyperbola (120) now describe the edges of
$\mathcal{U}_{\tilde{\bf R}}$.

Solving the quadratic equation (120) before merging, i.e., $1-\kappa_{s}>0$,
we find solutions $\tilde{u}_{s,\mathrm{jp}}(\tilde{v})$ away from a gap along
the stable $v$-direction,

$\tilde{u}_{s,\mathrm{jp}}(|\tilde{v}|\geq\tilde{v}_{s,c})=-\frac{1}{\gamma_{s}}\Bigl{[}\beta_{s}\tilde{v}\\\
\pm\sqrt{2\gamma_{s}\bar{C}(\kappa_{s}-1)-(\gamma_{s}\delta_{s}-\beta_{s}^{2})\tilde{v}^{2}}\Bigr{]},$
(123)

i.e., Eq. (123) has real solutions in the (unbounded) interval
$|\tilde{v}|\geq\tilde{v}_{s,c}$, with

$\tilde{v}_{s,c}=\sqrt{2\gamma_{s}\bar{C}(1-\kappa_{s})/|\gamma_{s}\delta_{s}-\beta_{s}^{2}|}.$
(124)

For the uniaxial defect (94) before merging, this gap corresponds to a
splitting of $\mathcal{U}_{\tilde{\bf R}}$ along the stable angular direction,
producing two separated domains as shown in Fig. 11(a). The coordinates
$\left(\tilde{u}_{s,\mathrm{jp}}(\pm\tilde{v}_{s,c}),\pm\tilde{v}_{s,c}\right)$
give the positions of the vertices $\delta\tilde{\bf R}^{<}_{s,c,\pm}$
(relative to $\tilde{\bf R}_{s}$) of the hyperbola before merging,

$\displaystyle\delta\tilde{\bf R}^{<}_{s,c,\pm}$
$\displaystyle=\pm\left(-\beta_{s}/\gamma_{s},1\right)\,\tilde{v}_{s,c}.$
(125)

These are marked as black crosses in Fig. 13(a) (note the rotation in the
geometry as compared with Fig. 11(a)). We denote the distance between these
vertices by $\delta v^{<}$, defining a gap of width
$\propto\sqrt{1-\kappa_{s}}$ given by

$\delta v^{<}=2|\delta\tilde{\bf
R}^{<}_{s,c,\pm}|=2\sqrt{\left(\gamma_{s}+\frac{\beta_{s}^{2}}{\gamma_{s}}\right)\frac{\bar{C}(1-\kappa_{s})}{|\gamma_{s}\delta_{s}-\beta_{s}^{2}|}}.$
(126)

After merging, i.e., for $\kappa_{s}-1>0$, the (local) topology of
$\mathcal{U}_{\tilde{\bf R}}$ has changed as the gap along $v$ closes and
reopens along the unstable $u$-direction; as a result, the two separated
domains of $\mathcal{U}_{\tilde{\bf R}}$ have merged. The two branches of the
hyperbola derived from (120) are now parametrized as

$\tilde{v}_{s,\mathrm{jp}}(|\tilde{u}|\geq\tilde{u}_{s,e})=-\frac{1}{\delta_{s}}\Bigl{[}\beta_{s}\tilde{u}\\\
\pm\sqrt{2\delta_{s}\bar{C}(\kappa_{s}-1)-(\gamma_{s}\delta_{s}-\beta_{s}^{2})\tilde{u}^{2}}\Bigr{]},$
(127)

with

$\tilde{u}_{s,e}=\sqrt{2\delta_{s}\bar{C}(\kappa_{s}-1)/|\gamma_{s}\delta_{s}-\beta_{s}^{2}|}.$
(128)

The corresponding unstable domain is shown in Fig. 13(b). For the uniaxial
defect (94) after merging, this gap now corresponds to the finite width of
$\mathcal{U}_{\tilde{\bf R}}$ along the radial direction, as shown in Fig.
12(a). The coordinates
$\left(\pm\tilde{u}_{s,e},\tilde{v}_{s,\mathrm{jp}}(\pm\tilde{u}_{s,e})\right)$
for the vertices $\tilde{\bf R}^{>}_{s,e,\pm}$ read

$\displaystyle\delta\tilde{\bf R}^{>}_{s,e,\pm}$
$\displaystyle=\pm\left(1,-\frac{\beta_{s}}{\delta_{s}}\right)\,\tilde{u}_{s,e}$
(129)

and correspond to the points of closest approach in the branches of the
hyperbola (120); these are again marked as black crosses in Fig. 13(b) but are
no longer associated with critical points (we index these extremal points by
‘e’). Their distance $\delta u^{>}$ is given by

$\delta u^{>}=2|\delta\tilde{\bf
R}^{>}_{s,e,\pm}|=2\sqrt{\left(\delta_{s}+\frac{\beta_{s}^{2}}{\delta_{s}}\right)\frac{\bar{C}(\kappa_{s}-1)}{|\gamma_{s}\delta_{s}-\beta_{s}^{2}|}},$
(130)

i.e., the smallest width in $\mathcal{U}_{\tilde{\bf R}}$ grows as
$\propto\sqrt{\kappa_{s}-1}$.

As discussed above and shown in Fig. 13, the solutions of the quadratic form
(120) before and after merging are unbounded for every value of
$\kappa_{s}-1$. As a consequence, neglecting the higher order terms in the
determinant $D(\tilde{\bf R})$ is valid only in a narrow neighborhood of the
saddle $\tilde{\bf R}_{s}$, where the boundaries of $\mathcal{U}_{\tilde{\bf
R}}$ have the shape of a hyperbola. Away from the saddle, these higher order
terms are relevant in determining the specific shape of the unstable and
bistable domain, e.g., the ring-like structures of $\mathcal{U}_{\tilde{\bf
R}}$ and $\mathcal{B}_{\bar{\bf R}}$ in Figs. 11 and 12.

####  V.2.2 Landing line $\mathcal{L}_{\mathrm{\tilde{\bf R}}}$

To find the second bistable vortex tip configuration $\tilde{\bf
R}_{s,\mathrm{lp}}$ associated to the edges of $\mathcal{B}_{\bar{\bf R}}$
before and after merging, we repeat the steps of Sec. III.2.2. For the jump
vector $\Delta\tilde{\bf R}_{s}=\tilde{\bf R}_{s,\mathrm{lp}}-\tilde{\bf
R}_{s,\mathrm{jp}}$, we find the result

$\displaystyle\Delta\tilde{u}_{s}(\tilde{v})=-3\left(\gamma_{s}\,\tilde{u}_{s,\mathrm{jp}}(\tilde{v})+\beta_{s}\,\tilde{v}\right)/\gamma_{s},$
(131)
$\displaystyle\Delta\tilde{v}_{s}(\tilde{v})=-\left[a_{s}/(\bar{C}+\lambda_{s,+})\right]\tilde{v}\,\Delta\tilde{u}_{s}(\tilde{v}),$
(132)

cf. Eqs. (65) and (66) above. Here, we make use of the parametrization for the
jump coordinate $\tilde{u}_{s,\mathrm{jp}}(\tilde{v})$ in (123) before
merging; after merging, the above result is still valid but should be
expressed in terms of the parametrization
$\tilde{v}_{s,\mathrm{jp}}(\tilde{u})$ in Eq. (127).

The landing positions $\tilde{\bf R}_{s,\mathrm{lp}}=\tilde{\bf
R}_{s,\mathrm{jp}}+\Delta\tilde{\bf R}_{s}$ arrange along the branches
$\mathcal{L}_{\mathrm{\tilde{\bf R}}}$ of a hyperbola in $\tilde{\bf R}$-space
that are described by the matrix equation

$\delta\tilde{\bf
R}_{s,\mathrm{lp}}^{\mathrm{T}}M_{s,\mathrm{lp}}\,\delta\tilde{\bf
R}_{s,\mathrm{lp}}=\bar{C}(\kappa_{s}-1),$ (133)

with the landing matrix now given by

$M_{\mathrm{s,lp}}=\frac{1}{4}M_{s,\mathrm{jp}}+\begin{bmatrix}0&0\\\
0&~{}~{}~{}\displaystyle{\frac{3}{4}\Bigl{(}\frac{\delta_{s}}{2}-\frac{\beta_{s}^{2}}{2\gamma_{s}}\Bigr{)}}\end{bmatrix}$
(134)

with $\det M_{\mathrm{s,lp}}=(\gamma_{s}\delta_{s}-\beta_{s}^{2})/16<0$.
Before merging, the vertices of the landing and jumping hyperbolas coincide
and the jump (131)–(132) vanishes at these points. Moreover, as for the
contact points (67) close to onset of strong pinning, the tangent to the
jumping and landing hyperbolas at the vertices is parallel to the
$u$-direction, as is visible in Fig. 13(a).

For $\kappa_{s}=1$, the tips of $\mathcal{U}_{\tilde{\bf R}}$ merge and both
the jumping and landing hyperbolas coincide at $\tilde{\bf R}_{s}$. After
merging, i.e., for $\kappa_{s}-1>0$, the condition
$\Delta\tilde{u}_{s}=\Delta\tilde{v}_{s}=0$ cannot be realized along the
hyperbola (120) and the jumping and landing lines separate completely; as a
result, both the jumping distance $\Delta\tilde{\bf R}_{s}$ as well as the
jump in energy $\Delta e_{\mathrm{pin}}$ are always finite (see also Appendix
A.2). Indeed, after merging the landing hyperbola (133) has vertices

$\delta\tilde{\bf
R}_{s,v,\pm}=\pm\left(1,-\frac{\gamma_{s}\beta_{s}}{(4\gamma_{s}\delta_{s}-3\beta_{s}^{2})}\right)\,\tilde{u}_{s,v},$
(135)

with

$\tilde{u}_{s,v}=\sqrt{\frac{2\bar{C}(\kappa_{s}-1)(4\gamma_{s}\delta_{s}-3\beta_{s}^{2})}{\gamma_{s}(\gamma_{s}\delta_{s}-\beta_{s}^{2})}}$
(136)

different from the jumping hyperbola in (129). At these points, the stable and
unstable hyperbolas are tangent to the $v$-direction, as is visible in Fig.
13(b).

In section Sec. V.4 below, we will take a step back from the local analysis of
the unstable domain $\mathcal{U}_{\tilde{\bf R}}$ close to a saddle point
$\tilde{\bf R}_{s}$ and consider the evolution of its geometry across the
merging transition from a global perspective using specific examples.
Elaborating on the analysis of Sec. III.5, we will provide a simple argument
explaining the absence of contact points between jump and landing lines after
merging. Furthermore, we discuss the two possible roles of mergers as changing
the number of components of $\mathcal{U}_{\tilde{\bf R}}$ or changing the
connectivity of $\mathcal{U}_{\tilde{\bf R}}$ between simply and non-simply
connected areas. Before doing so, we discuss the behavior of the bistable
region $\mathcal{B}_{\bar{\bf R}}$ close to merging.

###  V.3 Bistable domain $\mathcal{B}_{\bar{\bf R}}$

Figure 14: Bistable domain $\mathcal{B}_{\bar{\bf R}}$ in asymptotic space
$\bar{\bf R}$ before (a) and after (b) merging, for $1-\kappa_{s}=\pm 0.01$
and parameters as in Fig. 13. (a) Before merging, the bistable domain
$\mathcal{B}_{\bar{\bf R}}$ consists of two parts, corresponding to the two
unstable regions $\mathcal{U}_{\tilde{\bf R}}$ in Fig. 13(a). These terminate
in the cusps at $\bar{\bf R}_{s,c,\pm}^{<}$ that approach one another along
the dashed parabola (139) to merge at $\kappa_{s}=1$. Red/blue colors indicate
different vortex configurations as quantified through the order parameter
$\tilde{u}-\tilde{u}_{m}(\bar{v})$, while magenta is associated to the
bistable region $\mathcal{B}_{\bar{\bf R}}$. Colored dots mark the asymptotic
positions associated to the pairs of jump positions in Fig. 13(a). (b) After
merging, the bistable domain is continuously connected; the cusps/critical
points have vanished and the dashed parabola turns into the branch cutting
line. The black crosses now mark the positions of strongest pinching of
$\mathcal{B}_{\bar{\bf R}}$, the colored dots mark the asymptotic positions
associated to the pairs of tip positions in Fig. 13(b).

The set of asymptotic positions corresponding to $\mathcal{U}_{\tilde{\bf R}}$
before and after merging, i.e., the bistable domain $\mathcal{B}_{\bar{\bf
R}}$, can be found by systematically repeating the steps in Sec. III.3.
Applying the force balance equation
$\nabla_{\mathbf{R}}e_{\mathrm{pin}}(\mathbf{R};\bar{\bf
R})\Big{|}_{\tilde{\bf R}}=0$ to the energy expansion (V.1), we find the
counterpart of Eqs. (69),

$\displaystyle\bar{C}\bar{u}$
$\displaystyle=\bar{C}(1-\kappa_{s})\tilde{u}+\frac{a_{s}}{2}\tilde{v}^{2}+\frac{\gamma_{s}}{6}\tilde{u}^{3}+\frac{\beta_{s}}{2}\tilde{u}^{2}\tilde{v}+\frac{\alpha_{s}}{2}\tilde{u}\tilde{v}^{2},$
$\displaystyle\bar{C}\bar{v}$
$\displaystyle=(\bar{C}+\lambda_{s,+})\tilde{v}+a_{s}\,\tilde{u}\tilde{v}+\frac{\beta_{s}}{6}\tilde{u}^{3}+\frac{\alpha_{s}}{2}\tilde{u}^{2}\tilde{v},$
(137)

relating tip and asymptotic positions close to merging. As for the unstable
domain, the topology of $\mathcal{B}_{\bar{\bf R}}$ depends on the sign of
$1-\kappa_{s}$. The bistable domain $\mathcal{B}_{\bar{\bf R}}$ before merging
is shown in Fig. 14(a) for $1-\kappa_{s}=0.01$. It consists of two parts,
corresponding to the two pieces of $\mathcal{U}_{\tilde{\bf R}}$ for
$1-\kappa_{s}>0$, that terminate at the cusps $\bar{\bf R}_{s,c,\pm}^{<}$. The
latter are related to the vertices $\tilde{\bf R}_{s,c,\pm}^{<}$ of the
jumping hyperbola through the force balance equation (137),

$\delta\bar{\bf
R}_{s,c,\pm}^{<}\approx\left[\left(a_{s}/2\,\bar{C}\right)\,\tilde{v}^{2}_{s,c},\,\pm\left(1+\lambda_{s,+}/\bar{C}\right)\tilde{v}_{s,c}\right].$
(138)

For finite values of $(1-\kappa_{s})$, the cusps are separated by a distance
$2|\delta{\bar{\bf R}}_{s,c,\pm}^{<}|\approx
2\left(1+\lambda_{s,+}/\bar{C}\right)\tilde{v}_{s,c}\propto\sqrt{1-\kappa_{s}}$.
They approach one another along the parabola

$\bar{u}_{s,0}\approx\frac{a}{2\bar{C}}\frac{1}{(1+\lambda_{+}/\bar{C})^{2}}\bar{v}_{s,0}^{2},$
(139)

see the black dashed line in Fig. 14, with higher-order corrections appearing
at finite skew $\beta\neq 0$. After merging, this line lies within
$\mathcal{B}_{\bar{\bf R}}$ and defines the branch crossing line, cf. Eq.
(77).

After merging, when $\kappa_{s}-1>0$, the cusps have vanished and the edges
have rearranged to define a connected bistable region, see Fig. 14(b). The
extremal points of the two edges are found by evaluating the force balance
equation (137) at the vertices $\tilde{\bf R}_{s,e,\pm}^{>}$, Eq. (129), to
lowest order,

$\delta\bar{\bf
R}_{s,e,\pm}^{>}\approx\frac{\beta_{s}}{\delta_{s}}\left[\frac{a_{s}}{2\,\bar{C}}\frac{\beta_{s}}{\delta_{s}}\,\tilde{u}_{s,e}^{2},\,\mp\left(1+\frac{\lambda_{s,+}}{\bar{C}}\right)\,\tilde{u}_{s,e}\right].$
(140)

For finite values of $(\kappa_{s}-1)$, these points are separated by a
distance $2|\delta\bar{\bf R}_{s,e,\pm}^{>}|\approx
2\left(1+\lambda_{s,+}/\bar{C}\right)(\beta_{s}/\delta_{s})\tilde{u}_{s,e}\propto\sqrt{\kappa_{s}-1}$.
Note that the extremal points $\bar{\bf R}_{s,e,\pm}^{>}$ are no longer
associated to cusps or critical points as these have disappeared in the
merging process. When the skew parameter vanishes as in Fig. 14,
$\beta_{s}=0$, higher-order terms in $(\kappa_{s}-1)$ in the force-balance
equation (137) become relevant in determining the positions $\bar{\bf
R}_{s,e,\pm}^{>}$, separating them along the unstable $u$-direction. In this
case, we obtain a different scaling for their distance, i.e., $|\delta\bar{\bf
R}_{s,e,\pm}^{>}|\propto\left(1-\kappa_{s}\right)^{3/2}$.

###  V.4 Topological aspect of mergers

In order to discuss the topological aspect of a merger, it is convenient to
consider some specific examples. In Sec. IV, we have analyzed the case of a
uniaxial defect with a quadrupolar anisotropy $\delta
e_{p}\propto\epsilon\sin^{2}\tilde{\phi}$ in the pinning potential, see (105),
that produced a degenerate onset at symmetric points $[\pm\tilde{x}_{m},0]$.
Here, we choose again a weakly anisotropic defect centered in the origin but
with a dipolar deformation $\delta e_{p}\propto\epsilon\cos\tilde{\phi}$ that
results in an angle-dependent Labusch parameter

$\kappa_{m}(\tilde{\phi})=\kappa_{m}-\epsilon\cos\tilde{\phi},$ (141)

see Eq. (107). The strong pinning onset of such a defect then appears in an
isolated point on the negative $x$-axis, with the unstable ellipse
$\mathcal{U}_{\tilde{\bf R}}$ deforming with increasing $\kappa_{m}$ into a
horseshoe that is open on the positive $x$-axis—the closing of the horseshoe
to produce a ring, see Fig. 15, then corresponds to the local merger shown in
Fig. 13. With this example in mind, we can repeat the discussion in Sec.
III.5: The unstable eigenvector $\mathbf{v}_{-}(\mathbf{R}_{\mathrm{jp}})$
points radially outwards from the origin over the entire horseshoe, including
the merging region at positive $x$. On the other hand, the tangent to the
boundary $\partial\mathcal{U}_{\tilde{\bf R}}$ rotates forward and back along
the horseshoe as shown in Fig. 15 (we attribute a direction to
$\partial\mathcal{U}_{\tilde{\bf R}}$ with the convention of following the
boundary with the unstable region on the left); in fact, over most of the
boundary, the tangent is simply orthogonal to $\mathbf{v}_{-}$, with both
vectors rotating together when going along $\partial\mathcal{U}_{\tilde{\bf
R}}$. At the ends of the horseshoe, however, the tangent locally aligns
parallel (anti-parallel) to $\mathbf{v}_{-}$ and the two vectors rotate (anti-
clockwise) with respect to one another, with the total winding equal to
$2\pi$. After the merger, this winding has disappeared, with the resulting
ring exhibiting no winding in the tangent fields on the inner/outer boundary;
as a result, the contact points between the jump and landing lines have
disappeared.

Furthermore, the merger changes the topology of $\mathcal{U}_{\tilde{\bf R}}$
from the simply-connected horseshoe to the non-simply connected ring, while
the number of components in $\mathcal{U}_{\tilde{\bf R}}$ has not changed.
Note that the change in the relative winding is not due to crossing the
singularity of the vector field $\mathbf{v}_{-}$ as alluded to in Sec.
III.5—rather, it is the merger of the horseshoe tips that rearranges the
boundaries of $\mathcal{U}_{\tilde{\bf R}}$ and make them encircle the
singularity.

Figure 15: Left: Unstable region $\mathcal{U}_{\tilde{\bf R}}$ for a defect
with dipolar asymmetry. Upon the onset of strong pinning, an unstable ellipse
appears to the left of the defect center (black solid dot). With increasing
pinning strength (decreasing $\bar{C}$) the ellipse grows and deforms into a
horseshoe geometry. The unstable eigenvector field $\mathbf{v}_{-}$ (red
arrows) points radially outward away from the defect center. The tangent field
to the boundary $\partial\mathcal{U}_{\tilde{\bf R}}$ (black arrows) follows
the unstable direction at an angle of $\pi/2$ over most of
$\partial\mathcal{U}_{\tilde{\bf R}}$, with the exception of the two turning
points where the tangent rotates by $\pi$ with respect to $\mathbf{v}_{-}$,
producing a relative winding of $2\pi$. Right: After the merger of the turning
points the unstable region $\mathcal{U}_{\tilde{\bf R}}$ changes topology and
assumes the shape of a ring. The windings of the tangent field with respect to
the eigenvector-field $\mathbf{v}_{-}$ vanish separately for both boundaries
of $\mathcal{U}_{\tilde{\bf R}}$.

In the above example, we have discussed a merger that changes the
connectedness of $\mathcal{U}_{\tilde{\bf R}}$. On the other hand, as we are
going to show, a merger might leave the connectedness of
$\mathcal{U}_{\tilde{\bf R}}$ unchanged, while modifying the number of
components, i.e., the number of disconnected parts, in
$\mathcal{U}_{\tilde{\bf R}}$. Let us again consider a specific example in the
form of an anisotropic defect with a warped well shape, producing several (in
general subsequent) onsets and mergers; in Fig. 16, we consider a situation
with three onset points and subsequent individual mergers. After the onset,
the three ellipses define an unstable region $\mathcal{U}_{\tilde{\bf R}}$
with three disconnected parts that are simply-connected each. This
configuration is characterized by its number of components measuring $C=3$. As
two of the three ellipses merge, the number of components of
$\mathcal{U}_{\tilde{\bf R}}$ reduces to $C=2$, the next merger generates a
horseshoe that is still simply-connected with $C=1$. The final merger produces
a ring; while the number of components remains unchanged, $C=1$, the unstable
area assumes a non-simply connected shape with a ‘hole’; we associate the
index $H=1$ with the appearance of this hole within $\mathcal{U}_{\tilde{\bf
R}}$. In physics terms, the last merger producing a hole in
$\mathcal{U}_{\tilde{\bf R}}$ is associated with the appearance of a pinned
state; the unstable ring separates stable tip positions that are associated
with pinned and free vortex configurations residing at small and large radii,
respectively.

Figure 16: The unstable domain $\mathcal{U}_{\tilde{\bf R}}$ starting out with
$C=3$ components in (a) changes topology in three steps: after the first (b)
and second (c) mergers the number of components $C$ has changed from three in
(a) to two in (b) to one in (c), leading to a horseshoe shape of
$\mathcal{U}_{\tilde{\bf R}}$. The third merger closes the horseshoe to
produce the ring geometry in (d) characterized by the coefficients $C=1$ and
$H=1$ ($H$ denotes the number of ‘holes’ in $\mathcal{U}_{\tilde{\bf R}}$);
the Euler characteristic $\chi=C-H$ changes by unity in every merger.

Defining the (topological) characteristic $\chi\equiv C-H$, we see that $\chi$
changes by unity at every onset and merger, either through an increase (for an
onset) or decrease (for a merger) in the number of components $C\to C\pm 1$,
or through the appearance of a hole (in a merger) $H\to H+1$. Indeed, the
quantity $\chi$ is known as the Euler characteristic of a manifold and
describes its global topological properties; it generalizes the well known
Euler characteristic of a polyhedron to surfaces and manifoldsNakahara (2003),
see Sec. VI below. Finally, Morse theory Nash and Sen (2011) connects the
Euler characteristic with the local differential properties (minima, maxima,
saddles) of that manifold, hence establishing a connection between local
onsets and mergers (at minima and saddles of $D(\tilde{\bf R})$) and the
global properties of $\mathcal{U}_{\tilde{\bf R}}$ such as the appearance of
new pinned states. In Sec. VI below, we consider the general case of a random
pinning landscape in two dimensions and discuss the connection between local
differential and global topological properties of $\mathcal{U}_{\tilde{\bf
R}}$ in the light of Morse theory—the topology of bistable domains
$\mathcal{B}_{\bar{\bf R}}$ then follows trivially.

##  VI $\mathcal{U}_{\tilde{\bf R}}$ of a two-dimensional pinscape

We consider a two-dimensional pinning landscape $e_{p}(\mathbf{R})$, e.g., as
produced by a superposition of several (anisotropic Lorentzian) defects
residing in the $z=0$ plane. In the figures 17 and 18, we analyse two specific
cases with $n=3$ and $n=2$ defects as given in Eq. (94) with $\epsilon=0.1$
and positions listed in Tables 1 and 2; these produce unstable landscapes
$\mathcal{U}_{\tilde{\bf R}}$ of considerable complexity already, see Figs.
17(a) and 18(a). Our defects are compact with $e_{p}(\mathbf{R})\to 0$
vanishing at $R\to\infty$; as a result, $e_{\mathrm{pin}}$ becomes flat at
infinity. Note that a dense assembly of uniformly distributed individual
defects produces a random Gaussian pinning landscape, as has been shown in
Ref. Willa  _et al._ , 2022.

Here, we are interested in the evolution of the unstable and bistable domains
$\mathcal{U}_{\tilde{\bf R}}$ and $\mathcal{B}_{\bar{\bf R}}$ associated with
the 2D pinning landscape $e_{\mathrm{pin}}$; we focus on the unstable domain
$\mathcal{U}_{\tilde{\bf R}}$, with the properties of the bistable domain
$\mathcal{B}_{\bar{\bf R}}$ following straightforwardly from the solution of
the force balance equation (2). Unlike the analysis above that is centered on
special points of $\mathcal{U}_{\tilde{\bf R}}$, ellipses near onset and
hyperbolas near mergers, here, we are interested in the global properties of
the unstable region produced by a generic (though still two-dimensional)
pinscape.

As discussed in Sec. III above, the unstable region $\mathcal{U}_{\tilde{\bf
R}}$ associated with strong pinning is determined by the condition
$D(\tilde{\bf R})=0$ of vanishing Hessian determinant, more precisely, by the
competition between the lowest eigenvalue $\lambda_{-}(\tilde{\bf R})$ of the
Hessian matrix $\mathrm{H}_{ij}$ of the pinning potential $e_{p}(\mathbf{R})$
and the effective elasticity $\bar{C}$, see Eq. (37). In order to avoid the
interference with the second eigenvalue $\lambda_{+}(\tilde{\bf R})$ of the
Hessian matrix, we consider the shifted (by $\bar{C}$) curvature function

$\Lambda_{\bar{C}}(\tilde{\bf R})\equiv\bar{C}+\lambda_{-}(\tilde{\bf R}),$
(142)

i.e., the relevant factor of the determinant $D(\tilde{\bf
R})=[\bar{C}+\lambda_{-}(\tilde{\bf R})][\bar{C}+\lambda_{-}(\tilde{\bf R})]$.
The condition

$\Lambda_{\bar{C}}(\tilde{\bf R})=0$ (143)

then determines the boundaries of $\mathcal{U}_{\tilde{\bf R}}$.

Table 1:  Positions and relative weights of 3 uniaxially anisotropic Lorentzian defects in Fig. 17 as given by Eq. (94). | $~{}x/\xi$ |  $y/\xi$ |  weight
---|---|---|---
defect #1 | $1.14$ | $1.07$ | 0.65
defect #2 | $-0.98$ | $-0.19$ | 1
defect #3 | $0.20$ | $-0.67$ | 1
Table 2:  Positions and relative weights of 2 uniaxially anisotropic Lorentzian defects in Fig. 18 as given by Eq. (94). | $~{}x/\xi$ |  $y/\xi$ |  weight
---|---|---|---
defect #1 | $-1.32$ | $0.33$ | 1
defect #2 | $1.48$ | $-0.76$ | 1

The above problem can be mapped to the problem of cutting a surface, where
$\Lambda_{\bar{C}}(\tilde{\bf R})$ is interpreted as a height-function over
$\mathbb{R}^{2}$ that is cut at zero level; the elasticity $\bar{C}$ then
plays the role of a shift parameter that moves the function
$\lambda_{-}(\tilde{\bf R})$ downwards in height with decreasing $\bar{C}$
(that corresponds to increasing the relative pinning strength of the pinscape
in physical terms). As $\bar{C}$ is decreased to compensate the absolute
minimum of $\lambda_{-}(\tilde{\bf R})<0$, $\bar{C}+\lambda_{-}(\tilde{\bf
R})=0$, strong pinning sets in locally at $\tilde{\bf R}_{m}$ for the first
time in the form of an unstable ellipse $\mathcal{U}_{\tilde{\bf R}}$, see
Fig. 17(b) for our specific example with three defects; the Labusch parameter
$\kappa(\tilde{\bf R})$ evaluated at the point $\tilde{\bf R}_{m}$ defines
$\kappa_{m}$, the parameter tuned in Fig. 17. Decreasing $\bar{C}$ further,
this ellipse grows and deforms, while other local minima of
$\lambda_{-}(\tilde{\bf R})$ produce new disconnected parts of
$\mathcal{U}_{\tilde{\bf R}}$, a situation illustrated in Fig. 17(c) where
four ‘ellipses’ have appeared around (local) minima (blue filled dots). A
further increase in pinning strength (decrease in $\bar{C}$) continuous to
deform these ‘ellipses’ and adds three new ones. As the first saddle drops
below the zero level (red cross), two components merge and the number of
components decreases; in Fig. 17(d), we have three below-zero saddles and only
four components remain, $C=4$. In Fig. 17(e) four further mergers have reduced
$C$ to 1 as the corresponding saddles drop below zero level. This produces a
single non-simply connected component, i.e., $C=1$ and a hole, increasing the
number of holes $H$ from zero to one. The last merger leading to (f) finally
leaves $C=1$ but cuts the stable region inside the ring into two, increasing
the number of holes to $H=2$.

Figure 17: (a) Grayscale image of the pinning potential landscape
$e_{p}(\tilde{\bf R})$, with the three diamonds marking the positions of the
defects. (b)–(f) Shifted curvature function $\Lambda_{\bar{C}}(\tilde{\bf R})$
versus tip position $\tilde{\bf R}$ for increasing values of $\kappa_{m}$
(decreasing $\bar{C}$) as we proceed from (b) to (f). We make use of the
topographic interpretation with positive values of $\Lambda_{\bar{C}}$ marked
as landmass (greenish colors, with low/high elevation in dark/light green) and
negative values of $\Lambda_{\bar{C}}$ constituting $\mathcal{U}_{\tilde{\bf
R}}$ in flat light blue (height levels are shown by thin black lines). The
pinscape in (a) produces a curvature landscape with $7$ minima (solid dots),
$4$ maxima (open dots), and $10$ saddles (crosses). Several unstable regions
$\mathcal{U}_{\tilde{\bf R}}$ appear (solid dots turn blue) and merge (crosses
turn red) to change the topology of $\mathcal{U}_{\tilde{\bf R}}$. The Euler
characteristic $\chi(\mathcal{U}_{\tilde{\bf R}})=m-s+M=1-0+0=1$ in (b)
changes to $\chi(\mathcal{U}_{\tilde{\bf R}})=4$ in (c) and (d), drops to
$\chi(\mathcal{U}_{\tilde{\bf R}})=0$ in (e) and $\chi(\mathcal{U}_{\tilde{\bf
R}})=-1$ in (f); indeed, $\mathcal{U}_{\tilde{\bf R}}$ in (f) has one
component $C=1$ and two holes $H=2$, reproducing $\chi(\mathcal{U}_{\tilde{\bf
R}})=C-H=-1$.

This sequence of onsets and mergers is conveniently described in the
topographic language introduced in section IV that interprets stable tip
regions as land mass (green with bright regions indicating higher mountains in
Fig. 17) and unstable regions as lakes (flat blue with (below-water) height
levels indicated by thin black lines), with the height $\Lambda_{\bar{C}}=0$
defining the water level. The sequence (b) to (f) then shows the flooding of
the landscape as pinning increases ($\bar{C}$ decreasing), with white dot
minima turning blue at strong pinning onsets and white cross saddles turning
red at mergings; maxima in the landscape are shown as black open circles. Note
that we distinguish critical points (minima, saddles) residing below (blue and
red) and above (white) water level. Similarly, a (local) maximum above sea
level (black open dot) turns into a blue open dot as it drops belop sea level;
such an event is missing in Fig. 17 but can be produced with other
configurations of defects, see Fig. 18 where the curvature landscape for two
defects is shown.

The above discussion relates the local differential properties of the function
$\Lambda_{\bar{C}}(\tilde{\bf R})<0$, minima and saddles, to the global
topological properties of $\mathcal{U}_{\tilde{\bf R}}$, its number of
components $C(\mathcal{U}_{\tilde{\bf R}})$ and holes
$H(\mathcal{U}_{\tilde{\bf R}})$. This connection between local and global
properties is conveniently discussed within Morse theory Nash and Sen (2011).
Before presenting a general mathematical formulation, let us discuss a simple
heuristic argument producing the result relevant in the present context; in
doing so, we make use of the above topographic language.

Starting with the minima of the function $\Lambda_{\bar{C}}(\tilde{\bf R})$, a
new disconnected component appears in $\mathcal{U}_{\tilde{\bf R}}$ whenever
the minimum drops below sea level as $\bar{C}$ is decreased, that produces an
increase $C\to C+1$. With the further decrease of $\bar{C}$, these
disconnected regions expand and merge pairwise whenever a saddle point of
$\Lambda_{\bar{C}}(\tilde{\bf R})$ goes below sea level, thereby inducing a
change in the topology of $\mathcal{U}_{\tilde{\bf R}}$ by either reducing the
number of components $C\to C-1$ (keeping $H$ constant) or leaving it unchanged
(changing $H\to H+1$), see, e.g., the example with the horseshoe closing up on
itself in Sec. V.4. The below sea-level minima and saddles of
$\Lambda_{\bar{C}}(\tilde{\bf R})$ can naturally be identified with the
vertices and edges of a graph; the edges in the graph then define the
boundaries of the graph’s faces (the same way as the vertices are the
boundaries of the edges). For a connected graph, Euler’s formula then tells us
that the number $V$ of vertices, $E$ of edges, and $F$ of faces are
constrained via $V-E+F=1$ (not counting the outer face extending to infinity)
and a graph with $C$ components satisfies the relation $C=V-E+F$ as follows
from simple addition.

We have already identified minima and saddles of $\Lambda_{\bar{C}}(\tilde{\bf
R})<0$ with vertices and edges of a graph; denoting the number of below sea-
level minima and saddles by $m$ and $s$, we have $V=m$ and $E=s$. It remains
to express the number $F$ of faces in terms of critical points of the surface
$\Lambda_{\bar{C}}(\tilde{\bf R})<0$. Indeed, the faces of our graph are
associated with maxima of the function $\Lambda_{\bar{C}}(\tilde{\bf R})$:
following the boundaries of a face, we cross the corresponding saddles with
the function $\Lambda_{\bar{C}}(\tilde{\bf R})$ curving upwards away from the
edges, implying that the faces of our graph include maxima of
$\Lambda_{\bar{C}}(\tilde{\bf R})$. These maxima manifest in two possible
ways: either the face contains a single below sea-level maximum or a single
above sea-level landscape. The above sea-level landscape comprises at least
one maximum but possibly also includes other extremal points that we cannot
analyse with our knowledge of the below sea-level function
$\Lambda_{\bar{C}}(\tilde{\bf R})<0$ only; we therefore call the above sea-
level landscape a (single) hole. The appearance of a single maximum or hole is
owed to the fact that faces are not split by a below sea-level saddle as these
have already been accounted for in setting up the graph.

Let us denote the number of (below sea-level) maxima by $M$ and the number of
holes by $H$, then $F=H+M$. Combining this last expression with Euler’s
formula and regrouping topological coefficients $C(\mathcal{U}_{\tilde{\bf
R}})$ and $H(\mathcal{U}_{\tilde{\bf R}})$ on one side and extremal points
$m[\Lambda_{\bar{C}}(\tilde{\bf R})]$, $s[\Lambda_{\bar{C}}(\tilde{\bf R})]$,
and $M[\Lambda_{\bar{C}}(\tilde{\bf R})]$ on the other, we arrive at the Euler
characteristic $\chi\equiv C-H$ and its representation through local
differential properties,

$\chi(\mathcal{U}_{\tilde{\bf R}})\equiv[C-H]_{\mathcal{U}_{\tilde{\bf
R}}}=[m-s+M]_{\Lambda_{\bar{C}}(\tilde{\bf R})<0}.$ (144)

The result (144) follows rigorously from the Euler-Poincaré theoremNakahara
(2003); Nash and Sen (2011) in combination with Morse’s theorem Nash and Sen
(2011), with the former expressing the Euler characteristic
$\chi(\mathcal{U}_{\tilde{\bf R}})$ through the so-called Betti numbers
$b_{i}(\mathcal{U}_{\tilde{\bf R}})$,

$\chi(\mathcal{U}_{\tilde{\bf
R}})\equiv\sum_{i=0}^{2}(-1)^{i}b_{i}(\mathcal{U}_{\tilde{\bf R}}),$ (145)

where the $i$-th Betti number $b_{i}(\mathcal{U}_{\tilde{\bf
R}})=\mathrm{Dim}[H_{i}(\mathcal{U}_{\tilde{\bf R}})]$ is given by the
dimension or rank of the $i$-th (singular) homology group
$H_{i}(\mathcal{U}_{\tilde{\bf R}})$. In colloquial terms, the Betti numbers
$b_{i}$ count the number of ‘holes’ in the manifold with different dimensions
$i$: the zeroth Betti number gives the number of components $b_{0}=C$ of
$\mathcal{U}_{\tilde{\bf R}}$, the first Betti number $b_{1}=H$ counts the
holes, and the second Betti number refers to cavities, here $b_{2}=0$ for our
open manifold. Hence, we find that the Euler characteristic is given by the
number of components and holes in $\mathcal{U}_{\tilde{\bf R}}$,

$\chi(\mathcal{U}_{\tilde{\bf R}})=C(\mathcal{U}_{\tilde{\bf
R}})-H(\mathcal{U}_{\tilde{\bf R}}),$ (146)

in agreement with the discussion in Sec. V.4 and (144).

Figure 18: (a) Grayscale image of the pinning potential landscape
$e_{p}(\tilde{\bf R})$, with the two diamonds marking the positions of the
defects. (b)–(f) Shifted curvature function $\Lambda_{\bar{C}}(\tilde{\bf R})$
(in topographic coloring, see caption of Fig. 17) versus tip position
$\tilde{\bf R}$ for increasing values of $\kappa_{m}$ as we proceed from (b)
to (f). The pinscape in (a) produces a curvature landscape with $6$ minima
(solid dots), $4$ maxima (open dots), and $9$ saddles (crosses). Upon
increasing $\kappa_{m}$, several unstable regions $\mathcal{U}_{\tilde{\bf
R}}$ appear (solid dots turn blue) and merge (crosses turn red) to change the
topology of $\mathcal{U}_{\tilde{\bf R}}$. The Euler characteristic
$\chi(\mathcal{U}_{\tilde{\bf R}})=m-s+M=1=C$ in (b), remains
$\chi(\mathcal{U}_{\tilde{\bf R}})=1$ in (c), but with $C=2$ and $H=1$,
changes to $\chi(\mathcal{U}_{\tilde{\bf R}})=-1$ in (d), and
$\chi(\mathcal{U}_{\tilde{\bf R}})=-3$ with one component $C=1$ and four holes
$H=4$ in (e). In going from (e) to (f) two of the maxima (black open dots turn
blue) drop below zero, producing a characteristic
$\chi(\mathcal{U}_{\tilde{\bf R}})=6-9+2=-1$; indeed, $\mathcal{U}_{\tilde{\bf
R}}$ in (f) has one component $C=1$ and two holes $H=2$, reproducing
$\chi(\mathcal{U}_{\tilde{\bf R}})=C-H=-1$.

Morse theoryNash and Sen (2011) then provides a connection between the
topological properties of the manifold $\mathcal{U}_{\tilde{\bf R}}$ and the
local differential properties of the surface $\Lambda_{\bar{C}}(\tilde{\bf
R})<0$ defining it: with $C_{i}$ the number of critical points with index $i$
of the surface $\Lambda_{\bar{C}}(\tilde{\bf R})<0$ (the index $i$ counts the
number of negative eigenvalues of the Hessian matrix evaluated at the critical
point), the Euler characteristic $\chi(\mathcal{U}_{\tilde{\bf R}})$ relates
the manifold’s topology to the number and properties of critical points,

$\chi(\mathcal{U}_{\tilde{\bf
R}})=\sum_{i=0}^{2}(-1)^{i}C_{i}(\Lambda_{\bar{C}}<0).$ (147)

For our 2D manifold the coefficients $C_{i}$ count the minima $C_{0}=m$, the
number of saddles $C_{1}=s$, and $C_{2}=M$ refers to the number of maxima,
hence,

$\chi(\mathcal{U}_{\tilde{\bf R}})=[m-s+M]_{\Lambda_{\bar{C}}<0}$ (148)

and the combination with (146) produces the result (144) anticipated above.

Summarizing, knowing the number of critical points $m$, $M$, and $s$ of the
seascape, i.e., its local differential properties, we can determine the global
topological aspects of the pinning landscape via the evaluation of the Euler
characteristic $\chi(\mathcal{U}_{\tilde{\bf R}})$ with the help of Eq. (148).
The latter then informs us about the number $C$ of unstable domains in
$\mathcal{U}_{\tilde{\bf R}}$ where locally pinned states appear and the
number of holes $H$ in $\mathcal{U}_{\tilde{\bf R}}$ where globally distinct
pinned states show up. Furthermore, the outer boundaries of the lakes, of
which we have $C$ components, are to be associated with instabilities of the
free vortex state, while inner boundaries (or boundaries of holes, which count
$H$ elements) tell about instabilities of pinned states, hence the Betti
numbers $C$ and $H$ count different types of instabilities. It would then have
been nice to determine the separate topological coefficients $C$ and $H$
individually—unfortunately, $\chi(\mathcal{U}_{\tilde{\bf R}})$ as derived
from local differential properties provides us only with the difference $C-H$
between locally and globally pinned areas and not their individual values.
Nevertheless, using Morse theory, we could connect our discussion of local
differential properties of the pinning landscape in Secs. III.1 and V.1 with
the global pinning properties of the pinning energy landscape as expressed
through the topology of the unstable domain $\mathcal{U}_{\tilde{\bf R}}$.

Regarding our previous examples, the isotropic and uniaxial defects, we remark
that for the latter the two simultaneous mergers on the $y$-axis produce a
reduction in $C=2\to 1$ and an increase of $H=0\to 1$ and hence a jump from
$\chi=2$ to $\chi=0$ in one step, as expected for two simultaneous mergers.
The symmetry of the isotropic defect produces a (degenerate) critical line at
$\tilde{R}_{m}$ rather than a critical point; adding a small perturbation
$\propto x^{3}$ breaks this symmetry and produces the horseshoe geometry
discussed in Sec. V.4 above that is amenable to the standard analysis.

A last remark is in place about the topological properties in dual space,
i.e., of bistable regions $\mathcal{B}_{\bar{\bf R}}$. Here, the mergers
produce another interesting phenomenon as viewed from the perspective of its
thermodynamic analogue. Indeed, the merger of deformed ellipses in tip-space
corresponds to the merger of cusps in asymptotic space, what translates to the
vanishing of critical points and a smooth continuation of the first-order
critical and spinodal lines in the thermodynamic analogue, see also Sec. V.3.
We are not aware of a physical example in thermodynamics that produces such a
merger and disappearance of critical points.

##  VII Summary and outlook

Strong pinning theory is a quantitative theory describing vortex pinning in
the dilute defect limit where this complex many-body system can be reduced to
an effective single-pin–single-vortex problem. The accuracy offered by this
theory then allows for a realistic description of the shape of the pinning
potential $e_{p}(\mathbf{R})$ associated with the defects. While previous work
focused on the simplest case of isotropic defects, here, we have generalized
the strong pinning theory to the description of arbitrary anisotropic pinning
potentials. Surprisingly, going from an isotropic to an anisotropic defect has
quite astonishing consequences for the physics of strong pinning—this reminds
about other physical examples where the removal of symmetries or degeneracies
produces new effects.

While the strong pinning problem is quite a complex one requiring the use of
numerical tools in general, we have identified several generic features that
provide the essential physics of the problem and that are amenable to an
analytic treatment. Specifically, these are the points of strong pinning onset
and the merger points, around which the local expansions of the pinning
potential $e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$ in the tip coordinate
$\tilde{\bf R}$ allow us to find all the characteristics of strong pinning. In
particular, we identify the instability region $\mathcal{U}_{\tilde{\bf R}}$
in the vortex tip space (with coordinates $\tilde{\bf R}$) and the bistable
region $\mathcal{B}_{\bar{\bf R}}$ in the space of asymptotic vortex positions
$\bar{\bf R}$ as the main geometric objects that determine the critical
pinning force density $F_{\mathrm{pin}}$, from which the critical current
density $j_{c}$, the technologically most relevant quantity of the
superconductor, follows straightforwardly. While the relevance of the bistable
region $\mathcal{B}_{\bar{\bf R}}$ was recognized in the past Labusch (1969);
Larkin and Ovchinnikov (1979); Blatter  _et al._ (2004), the important role
played by the unstable region $\mathcal{U}_{\tilde{\bf R}}$ went unnoticed so
far.

When going from an isotropic defect to an anisotropic one, the strong pinning
onset changes dramatically: while the unstable region $\mathcal{U}_{\tilde{\bf
R}}$ grows out of a circle of radius $\sim\xi$ and assumes the shape of a ring
at $\kappa>1$ for the isotropic situation, for an anisotropic defect the onset
appears in a point $\tilde{\bf R}_{m}$ and grows in the shape of an ellipse
with increasing $\kappa_{m}>1$; the location where this onset appears is given
by the Hessian of $e_{\mathrm{pin}}$, specifically, the point $\tilde{\bf
R}_{m}$ where its determinant touches zero first,
$\det\\{\mathrm{Hess}[e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})|_{\bar{\bf
R}}]\\}_{\tilde{\bf R}_{m}}=0$. The boundary of this ellipse defines the jump
positions $\mathcal{J}_{\mathrm{\tilde{\bf R}}}$ associated with the strong
pinning instabilities; when combined with the landing ellipse
$\mathcal{L}_{\mathrm{\tilde{\bf R}}}$, these two ellipses determine the jump
distance $\delta\tilde{u}$ of the vortex tip, from which follows the jump in
the pinning energy $\Delta e_{\mathrm{pin}}\propto\delta\tilde{u}^{4}$, which
in turn determines $F_{\mathrm{pin}}$ and $j_{c}$.

The bistable region $\mathcal{B}_{\bar{\bf R}}$ in asymptotic vortex space
comes into play when calculating the average critical force density
$F_{\mathrm{pin}}$ opposing the vortex motion: while the vortex tip undergoes
a complex trajectory including jumps, the vortex motion in asymptotic space
$\bar{\bf R}$ is described by a straight line. As this trivial trajectory in
$\bar{\bf R}$-space traverses the bistable region $\mathcal{B}_{\bar{\bf R}}$,
the vortex tip jumps upon exiting $\mathcal{B}_{\bar{\bf R}}$, that produces
the jump $\Delta e_{\mathrm{pin}}$ and hence $F_{\mathrm{pin}}$. Again, the
shape of $\mathcal{B}_{\bar{\bf R}}$ changes when going from the isotropic to
the anisotropic defect, assuming a ring of finite width around a circle of
radius $\sim\xi$ in the former case, while growing in the form of a crescent
out of a point for the anisotropic defect.

The new geometries associated with $\mathcal{U}_{\tilde{\bf R}}$ and
$\mathcal{B}_{\bar{\bf R}}$ then produce a qualitative change in the scaling
behavior of the pinning force density
$F_{\mathrm{pin}}\propto(\kappa_{m}-1)^{\mu}$ near onset, with the exponent
$\mu$ changing from $\mu=2$ to $\mu=5/2$ when going from the isotropic to the
anisotropic defect. This change is due to the change in the scaling of the
geometric size of $\mathcal{B}_{\bar{\bf R}}$, with the replacement of the
fixed radius $\sim\xi$ of the ring by the growing size of the crescent
$\sim\xi(\kappa_{m}-1)^{1/2}$ [the exponent $\mu$ assumes a value $\mu=3$ for
trajectories cutting the crescent along its short dimension of size
$\xi(\kappa_{m}-1)$]. Furthermore, for directed defects, the pinning force
density $F_{\mathrm{pin}}(\theta)$ depends on the impact angle $\theta$
relative to the unstable direction $u$ and is aligned with $u$, except for a
small angular regime close to $\theta=\pi/2$. This results in a pronounced
anisotropy in the critical current density $j_{c}$ in the vicinity of the
strong pinning onset.

A fundamental difference between the strong pinning onsets in the isotropic
and in the anisotropic case are the geometries of the unstable
$\mathcal{U}_{\tilde{\bf R}}$ and bistable $\mathcal{B}_{\bar{\bf R}}$
regions: these are non-simply connected for the isotropic case (rings) but
simply connected for the anisotropic defect (ellipse and crescent). The
resolution of this fundamental difference is provided by the second type of
special points, the mergers. Indeed, for a general anisotropic defect, the
strong pinning onset appears in a multitude of points, with unstable and
bistable regions growing with $\kappa_{m}>1$ and finally merging into larger
areas. Two examples illustrate this behavior particularly well, the uniaxial
defects with a quadrupolar and a dipolar deformation, see Secs. IV and V.4. In
the first case, symmetric onset points on the $x$ axis produce two
ellipses/crescents that grow, approach one another, and finally merge in a
ring-shaped geometry that is non-simply connected. In the case of a dipolar
deformation, we have seen $\mathcal{U}_{\tilde{\bf R}}$ grow out of a single
point with its ellipse expanding and deforming around a circle, assuming a
horseshoe geometry, that finally undergoes a merging of the two tips to
produce again a ring; similar happens when multiple $\mathcal{U}_{\tilde{\bf
R}}$ domains grow and merge as in Figs. 16 (a warped defect) and 18(c) (a 2D
pinning landscape where four unstable domains have merged to enclose an
‘island’).

These merger points are once more amenable to an analytic study using a proper
expansion of $e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$ in $\tilde{\bf R}$
around the merger point $\tilde{\bf R}_{s}$, the latter again defined by the
local differential properties of the determinant
$\det\\{\mathrm{Hess}[e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})|_{\bar{\bf
R}}]\\}$, this time not a minimum but a saddle. Rather than elliptic as at
onset, at merger points the geometry is hyperbolic, with the sign change
associated with increasing $\kappa_{s}\equiv\kappa(\tilde{\bf R}_{s})$ across
unity producing a reconnection of the jump- and landing lines
$\mathcal{J}_{\mathrm{\tilde{\bf R}}}$ and $\mathcal{L}_{\mathrm{\tilde{\bf
R}}}$.

While the expansions of $e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$ are
describing the local pinning landscape near onset and merging (and thus
produce generic results), the study of the combined set of onset- and merger-
points describe the global topological properties of $\mathcal{U}_{\tilde{\bf
R}}$ as discussed in Sec. VI: every new (nondegenerate) onset increases the
number of components $C$ in $\mathcal{U}_{\tilde{\bf R}}$, while every merger
either decreases $C$ or increases $H$, the number of ‘holes’ or ‘islands’ (or
nontrivial loops in a non-simply connected region) in the pinning landscape.
It is the ‘last’ merging producing a non-simply connected domain that properly
defines a new pinned state; in our examples these are the closing of the two
deformed ellipses in the uniaxial defect with quadrupolar deformation and the
closing of the horseshoe in the defect with a dipolar deformation. Formally,
the relation between the local differential properties of the curvature
function $\Lambda_{\bar{C}}(\tilde{\bf R})=\bar{C}+\lambda_{-}(\tilde{\bf R})$
[with $\lambda_{-}(\tilde{\bf R})$ the lower eigenvalue of the Hessian of
$e_{p}(\tilde{\bf R})$], its minima, saddles, and maxima, are related to the
global topological properties of $\mathcal{U}_{\tilde{\bf R}}$ as described by
its Euler characteristic $\chi=C-H$ through Morse theory, see Eq. (144). Such
topological structures have recently attracted quite some interest, e.g., in
the context of Fermi surface topologies and topological Lifshitz transitions
Volovik (2017); Kane (2022).

The physics around the onset points as expressed through an expansion of
$e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$ resembles a Landau theory with
$\tilde{\bf R}$ playing the role of an order parameter and $\bar{\bf R}$ the
dual variable corresponding to a driving field—here, $\bar{\bf R}$ drives the
vortex lattice across the defect and $\tilde{\bf R}$ describes the deformation
of the pinned vortex. The endpoints of the crescent $\mathcal{B}_{\bar{\bf
R}}$ correspond to critical end points as they appear in the Landau theory of
a first-order transition line, e.g., the Ising model in an external field or
the van der Waals gas. The boundary lines of $\mathcal{B}_{\bar{\bf R}}$
correspond to spinodal lines where phases become unstable, e.g., the
termination of overheated/undercooled phases in the van der Waals gas. The
existence of critical end points tells that ‘phases’, here in the form of
different pinning branches, are smoothly connected when going around the
critical point, similar as in the gas–liquid transition of the van der Waals
gas. As the ‘last’ critical point vanishes in a merger, a well defined new
phase, here a new pinned branch, appears.

Perspectives for future theoretical work include the study of correlations
between anisotropic defects (see Ref. Buchacek  _et al._ , 2020 addressing
isotropic defects) or the inclusion of thermal fluctuations, i.e., creep (see
Refs. Buchacek  _et al._ , 2019a and Gaggioli  _et al._ , 2022). Furthermore,
our discussion of the extended pinscape in Sec. VI has been limited to a two-
dimensional pinning potential. In reality, defects are distributed in all
three dimensions that considerable complicates the corresponding analysis of a
full three-dimensional disordered pinning potential, with the prospect of
interesting new results.

On the experimental side, there are several possible applications for our
study of anisotropic defects. For a generic anisotropic defect, the inversion
symmetry may be broken. In this case, the pinning force along opposite
directions is different in magnitude, as different jumps are associated to the
boundaries of the bistable region $\mathcal{B}_{\bar{\bf R}}$ away from onset,
i.e., at sufficiently large values of $\kappa_{m}$. Reversing the current, the
different critical forces then result in a ratchet effect Villegas  _et al._
(2003); de Souza Silva  _et al._ (2006). This leads to a rectification of an
ac current and hence a superconducting diode effect. While for randomly
oriented defects the pinning force is averaged and the symmetry is
statistically restored, for specially oriented defects, the diode effect will
survive. Indeed, introducing nanoholes into the material, vortex pinning was
enhanced Wang  _et al._ (2013); Kwok  _et al._ (2016) and a diode effect has
been observed recently Lyu  _et al._ (2021). Generalizing strong pinning
theory to this type of defects then may help in the design of superconducting
metamaterials with interesting functionalities. Furthermore, vortex imaging
has always provided fascinating insights into vortex physics. Recently, the
SQUID-on-tip technique has been successful in mapping out a 2D pinning
landscape in a film Embon  _et al._ (2015) (including the observation of
vortex jumps) that has inspired a new characterization of the pinscape through
its Hessian analysis Willa  _et al._ (2022); the adaptation of this current-
driven purely 2D setup to the 3D situation described in the present paper is
an interesting challenge.

Finally, we recap the main benefits of this work in a nutshell: For one, we
have established a detailed connection of the strong pinning transition with a
the concept of first-order phase transitions in thermodynamics, with the main
practical result that the scaling of the pinning force density
$F_{\mathrm{pin}}\propto(\kappa_{m}-1)^{\mu}$ comes with an exponent $\mu=5/2$
when working with generic defects of arbitrary shapes. Second, we have found a
mechanism, the breaking of a defect’s inversion symmetry, that produces
rachets and a diode effect in superconducting material. Third, we have
uncovered the geometric structure and its topological features that is
underlying strong pinning theory, including a proper understanding of the
appearance of distinguished pinned states. While understanding these geometric
structures seems to be of rather fundamental/scholarly interest at present,
future work may establish further practical consequences that can be used in
the development of superconducting materials with specific functional
properties.

## Acknowledgments

We thank Tomáš Bzdušek, Gian Michele Graf, and Roland Willa for discussions
and acknowledge financial support of the Swiss National Science Foundation,
Division II.

##  Appendix A Effective $1$D Landau theory

The Landau-type pinning energies (18) and (V.1) for the vector order parameter
$(\tilde{u},\tilde{v})$ involves a soft variable $\tilde{u}$ with a vanishing
quadratic term $\propto(1-\kappa_{m})\>\tilde{u}^{2}$, as well as a stiff one,
$\tilde{v}$, characterized by a finite elasticity. By eliminating the stiff
direction $\tilde{v}$, we can arrive at a 1D Landau expansion for the order
parameter $\tilde{u}$ that provides us with the desired results for the
unstable and bistable domains $\mathcal{U}_{\tilde{\bf R}}$ and
$\mathcal{B}_{\bar{\bf R}}$ near onset and merging in a very efficient manner.

###  A.1 Close to onset

We start with the two-dimensional Landau-type energy functional (III.2.1)

$\displaystyle e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf
R})=\frac{\bar{C}\left(1-\kappa_{m}\right)}{2}\,\tilde{u}^{2}+\frac{\bar{C}+\lambda_{+}}{2}\,\tilde{v}^{2}+\frac{a}{2}\,\tilde{u}\tilde{v}^{2}$
$\displaystyle\quad+\frac{\alpha}{4}\,\tilde{u}^{2}\tilde{v}^{2}+\frac{\beta}{6}\,\tilde{u}^{3}\tilde{v}+\frac{\gamma}{24}\,\tilde{u}^{4}-\bar{C}\,\bar{u}\tilde{u}-\bar{C}\,\bar{v}\tilde{v}$
(149)

written in terms of the tip coordinates $\tilde{u},\tilde{v}$ measured
relative to $\tilde{\bf R}_{m}$, the position of the minimal determinant
$D(\tilde{\bf R})$ at strong pinning onset, and with $\tilde{u}$ and
$\tilde{v}$ aligned with the stable and unstable directions, respectively. The
expansion (149) is anisotropic: the quadratic (elastic) coefficient along the
unstable $\tilde{u}$-direction vanishes at the onset of strong pinning, while
the one along the stable $\tilde{v}$-direction stays positive and large,
allowing us to ‘integrate out’ the latter. The asymptotic coordinates
$\bar{u},\>\bar{v}$ assume the role of the driving (conjugate) fields for the
tip positions (or order parameters) $\tilde{u},\>\tilde{v}$; the latter then
are determined by the force equations $\partial_{\tilde{\bf
R}}e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})=0$,

$\displaystyle\bar{C}\bar{u}$
$\displaystyle=\bar{C}(1-\kappa)\tilde{u}+\frac{a}{2}\tilde{v}^{2}+\frac{\gamma}{6}\tilde{u}^{3}+\frac{\beta}{2}\tilde{u}^{2}\tilde{v}+\frac{\alpha}{2}\tilde{u}\tilde{v}^{2},$
(150) $\displaystyle\bar{C}\bar{v}$
$\displaystyle=(\bar{C}+\lambda_{+})\tilde{v}+a\,\tilde{u}\tilde{v}+\frac{\beta}{6}\tilde{u}^{3}+\frac{\alpha}{2}\tilde{u}^{2}\tilde{v},$
(151)

see Eq. (69), with $\delta\bar{\bf R}=(\bar{u},\bar{v})$ measured relative to
$\bar{\bf R}_{m}$. Inspection of Eqs. (150) and (151) shows that near the
strong pinning onset, the Ansatz
$\tilde{u},~{}\tilde{v},~{}\bar{v}\propto\sqrt{\kappa_{m}-1}$ and
$\bar{u}\propto(\kappa_{m}-1)$ produces a consistent solution. Solving the
second equation (151) for the stiff degree of freedom $\tilde{v}$, we then
find that

$\tilde{v}\approx\frac{\bar{C}\bar{v}}{\bar{C}+\lambda_{+}\\!+a\tilde{u}}\approx\frac{\bar{v}}{1\\!+\\!\lambda_{+}/\bar{C}}\Bigl{(}1-\frac{a/\bar{C}}{1\\!+\\!\lambda_{+}/\bar{C}}\>\tilde{u}\Bigr{)}$
(152)

which is precise to order $(\kappa_{m}-1)$. Inserting $\tilde{v}$ back into
the force-balance equation (150) for the unstable component $\tilde{u}$, we
find a cubic equation for $\tilde{u}$ (precise to order
$(\kappa_{m}-1)^{3/2}$) that is driven by a combination of $\bar{u}$ and
$\bar{v}^{2}$,

$\bar{C}\bar{u}-\frac{(a/2)\,\bar{v}^{2}}{(1+\lambda_{+}/\bar{C})^{2}}\approx\left[\bar{C}(1-\kappa_{m})+\frac{(\delta/2)\,\bar{v}^{2}}{(1+\lambda_{+}/\bar{C})^{2}}\right]\tilde{u}\\\
+\frac{(\beta/2)\,\bar{v}}{(1+\lambda_{+}/\bar{C})}\tilde{u}^{2}+\frac{\gamma}{6}\tilde{u}^{3}.$
(153)

Upon integration, we finally arrive at the effective one-dimensional Landau
expansion for the 1D order parameter $\tilde{u}$ that is precise to order
$(\kappa_{m}-1)^{2}$ (up to an irrelevant shift $\propto\bar{v}^{2}$),

$e_{\mathrm{pin}}^{\mathrm{eff}}(\tilde{u};\bar{u},\bar{v})=\frac{r(\bar{v})}{2}\tilde{u}^{2}+\frac{w(\bar{v})}{6}\tilde{u}^{3}+\frac{\gamma}{24}\tilde{u}^{4}-h(\bar{u},\bar{v})\tilde{u},$
(154)

with the coefficients $r,\>w$, and $h$ defined as

$\displaystyle\begin{split}r(\bar{v})&=\left[\bar{C}(1-\kappa_{m})+\frac{\delta}{2}\frac{\bar{v}^{2}}{(1+\lambda_{+}/\bar{C})^{2}}\right],\\\
w(\bar{v})&=\beta\frac{\bar{v}}{(1+\lambda_{+}/\bar{C})},\\\
h(\bar{u},\bar{v})&=\bar{C}\bar{u}-\frac{a}{2}\frac{\bar{v}^{2}}{(1+\lambda_{+}/\bar{C})^{2}}.\end{split}$
(155)

The Landau-type energy function (154) belongs to the van der Waals (gas-
liquid) universality class; its first-order transition line maps to the branch
crossing line in the strong pinning problem, its spinodals correspond to the
arcs of the crescent defining the bistable region $\mathcal{B}_{\bar{\bf R}}$,
and its critical points map to the two cusps of $\mathcal{B}_{\bar{\bf R}}$,
i.e., in the strong pinning problem, the spinodals end in two critical points.
The cubic term $w\tilde{u}^{3}/6$ is determined by the skew parameter $\beta$;
in the absence of such a skew, i.e., for a $\pm\tilde{v}$-symmetric unstable
ellipse $\mathcal{U}_{\tilde{\bf R}}$, we have $\beta=0$ and our problem
assumes an Ising-type $\mathbb{Z}_{2}$ symmetry.

Let us begin with the determination of the critical coefficients $r_{c}$,
$w_{c}$, and $h_{c}$. These are found by setting the first three derivatives
of $e_{\mathrm{pin}}^{\mathrm{eff}}(\tilde{u})$ to zero [two spinodals
(implying $\partial_{\tilde{u}}e_{\mathrm{pin}}^{\mathrm{eff}}=0$ and
$\partial^{2}_{\tilde{u}}e_{\mathrm{pin}}^{\mathrm{eff}}=0$) coalescing into a
single point
($\to\partial^{3}_{\tilde{u}}e_{\mathrm{pin}}^{\mathrm{eff}}=0$)]. Setting the
cubic derivative to zero, we find the order parameter

$\tilde{u}_{c}=-w_{c}/\gamma\approx-(\beta/\gamma)\tilde{v}_{c},$ (156)

where we have used Eq. (155) and the transformation
$\bar{v}\leftrightarrow\tilde{v}$ in (152) to leading order.

The vanishing of the second derivative relates the critical coefficients
$r_{c}$ and $w_{c}$,

$r_{c}=w_{c}^{2}/2\gamma,$ (157)

(where we have made use of $\tilde{u}_{c}$). Inserting the dependencies
$r(\bar{v})$ and $w(\bar{v})$, see Eq. (155), we find that

$\frac{\bar{v}_{c}^{2}}{(1+\lambda_{+}/\bar{C})^{2}}=\frac{\gamma\bar{C}(\kappa_{m}-1)}{2\det
M_{\mathrm{jp}}},$ (158)

with $\det M_{\mathrm{jp}}=(\gamma\delta-\beta^{2})/4$. Using again Eq. (152)
to leading order, we find that

$\tilde{v}_{c}\approx\sqrt{\frac{2\gamma\bar{C}(\kappa_{m}-1)}{\gamma\delta-\beta^{2}}},$
(159)

cf. Eq. (57). The critical endpoints of the 1D Landau theory then correspond
to the touching points (67) of the unstable domain $\mathcal{U}_{\tilde{\bf
R}}$

$\delta\tilde{\bf R}_{c,\pm}=\pm\left(-\beta/\gamma,1\right)\,\tilde{v}_{c},$
(160)

found before, see Eq. (67) with (57).

Finally, the vanishing of the first derivative defines the critical drive

$h_{c}=[r\tilde{u}+w\tilde{u}^{2}/2+\gamma\tilde{u}^{3}/6]_{c}=-\frac{w_{c}^{3}}{6\gamma^{2}}.$
(161)

Making use of the coefficients (155), this translates to the critical drive
$\bar{u}_{c}$

$\bar{u}_{c}=(a/2\bar{C})\tilde{v}_{c}^{2}-\frac{w_{c}^{3}}{6\bar{C}\gamma^{2}}$
(162)

and its combination with the result for $\bar{v}_{c}$ tells us that the
critical drives match up, to leading order, with the cusps (III.3) of the
bistable domain at $\bar{\bf R}_{c,\pm}$,

$\displaystyle\delta\bar{\bf R}_{c,\pm}$ $\displaystyle=$
$\displaystyle(\bar{u}_{c},\pm\bar{v}_{c})$ $\displaystyle\approx$
$\displaystyle\left[\left(a/2\bar{C}\right)\,\tilde{v}^{2}_{c},\,\pm(1+\lambda_{+}/\bar{C})\tilde{v}_{c}\right].$

Next, we find the entire boundary of the unstable region
$\mathcal{U}_{\tilde{\bf R}}$ that is defined as the points where local minima
and maxima of $e_{\mathrm{pin}}^{\mathrm{eff}}$ coalesce, i.e., where
$\partial^{2}_{u}e_{\mathrm{pin}}^{\mathrm{eff}}=0$,

$r+w\tilde{u}_{\mathrm{jp}}+\frac{\gamma}{2}\tilde{u}_{\mathrm{jp}}^{2}=0.$
(164)

Making use of the Landau coefficients (155) as well as the relation between
$\tilde{v}$ and $\bar{v}$ in (152), we recover the equation (53) for the
ellipse (we drop corrections $\propto\left(\kappa_{m}-1\right)^{3/2}$)

$\gamma\tilde{u}_{\mathrm{jp}}^{2}+2\beta\tilde{u}_{\mathrm{jp}}\tilde{v}_{\mathrm{jp}}+\delta\tilde{v}_{\mathrm{jp}}^{2}\approx
2\bar{C}(\kappa_{m}-1).$ (165)

In order to find the shape of the bistable region $\mathcal{B}_{\bar{\bf R}}$,
we exploit the fact that for fixed drives $\bar{u}$ and $\bar{v}$, the
bistable and the unstable vortex tip configurations are local extrema of
$e_{\mathrm{pin}}^{\mathrm{eff}}$, implying that
$\partial_{\tilde{u}}e_{\mathrm{pin}}^{\mathrm{eff}}=0$ and hence

$r\tilde{u}+\frac{w}{2}\tilde{u}^{2}+\frac{\gamma}{6}\tilde{u}^{3}=h,$ (166)

what corresponds to the force-balance equation (153) expressed in terms of the
coefficients (155). The cubic equation (166) with its left side
$\propto(\kappa_{m}-1)^{3/2}$ depends on $\bar{u}$ through the drive $h$.
According to (155), the two terms in the drive are of order $(\kappa_{m}-1)$
and hence have to cancel one another to lowest order. As a result, we find
that the bistable domain is centered around the parabola

$\bar{u}=\frac{a}{2\bar{C}}\frac{\bar{v}^{2}}{(1+\lambda_{+}/\bar{C})^{2}},$
(167)

that matches up with Eq. (70) found in Sec. III. Finding the precise form of
the bistable region $\mathcal{B}_{\bar{\bf R}}$, we have to solve Eq. (166) to
cubic order in $\sqrt{\kappa_{m}-1}$ with the help of an expansion around the
center parabola (167), what amounts to repeating the analysis leading to the
results (71) and (72) in Sec. III.3.

Finally, we find the landing line $\mathcal{L}_{\tilde{\bf R}}$ defined as the
second bistable tip position at fixed $\bar{u}$ and $\bar{v}$. We make use of
the cubic equation (166) and represent it in the factorized form (with the
inflection point at $\tilde{u}_{\mathrm{jp}}$ having multiplicity two)

$(\tilde{u}-\tilde{u}_{\mathrm{jp}})^{2}(\tilde{u}-\tilde{u}_{\mathrm{lp}})=0,$
(168)

and $\tilde{u}_{\mathrm{lp}}$ the landing position of the tip introduced in
Sec. III.2.2. A somewhat tedious but straightforward calculation shows that
the stable solution $\tilde{u}_{\mathrm{lp}}$ satisfies the quadratic equation

$r-\frac{3}{8}\frac{w^{2}}{\gamma}+\frac{w}{4}\tilde{u}_{\mathrm{lp}}+\frac{\gamma}{8}\tilde{u}_{\mathrm{lp}}^{2}=0$
(169)

and thus arranges along the ellipse

$\frac{\gamma}{8}\tilde{u}_{\mathrm{lp}}^{2}+\frac{\beta}{4}\tilde{u}_{\mathrm{lp}}\tilde{v}_{\mathrm{lp}}+\left(\frac{\delta}{2}-\frac{3}{8}\frac{\beta^{2}}{\gamma}\right)\tilde{v}_{\mathrm{lp}}^{2}=\bar{C}(\kappa_{m}-1)$
(170)

when expressed in the original two-dimensional tip space; this coincides with
the original result (63).

In a last step, we may go over to an Ising-type Landau expansion by measuring
the order parameter $\bar{u}$ with reference to the skewed line

$\tilde{u}_{m}(\bar{v})=\left(-\frac{\beta}{\gamma}\right)\frac{\bar{v}}{(1+\lambda_{+}/\bar{C})},$
(171)

i.e.,

$\tilde{u}^{\prime}=\tilde{u}-\tilde{u}_{m}(\bar{v}).$ (172)

The 1D effective Landau expansion now reads, with precision to order
$(\kappa_{m}-1)^{2}$,

$e_{\mathrm{pin}}^{\mathrm{eff}}(\tilde{u}^{\prime};\bar{u},\bar{v})=\frac{r^{\prime}}{2}\tilde{u}^{\prime
2}+\frac{\gamma}{24}\tilde{u}^{\prime 4}-h^{\prime}\tilde{u}^{\prime},$ (173)

with the new coefficients

$\displaystyle r^{\prime}=r-\frac{w^{2}}{2\gamma},\quad
h^{\prime}=h-\frac{w^{3}}{3\gamma^{2}}+\frac{rw}{\gamma}.$ (174)

The condition $h^{\prime}=0$ now defines the equilibrium state of the
thermodynamic problem that translates into the branch crossing line where the
bistable vortex tip positions have equal energy. Using the definitions (155)
and (174) for $h$ and $h^{\prime}$, we find that the branch crossing line
$\bar{u}_{0}(\bar{v}_{0})$ in the original two-dimensional asymptotic space
reads

$\bar{u}_{0}=\frac{a}{2\bar{C}}\frac{\bar{v}_{0}^{2}}{(1+\lambda_{+}/\bar{C})^{2}}-\frac{\beta}{\gamma}\biggl{[}(\kappa_{m}-1)\frac{\bar{v}_{0}}{1+\lambda_{+}/\bar{C}}\\\
+\left(\frac{\delta}{2}-\frac{\beta^{2}}{3\gamma}\right)\frac{1}{\bar{C}}\frac{\bar{v}_{0}^{3}}{(1+\lambda_{+}/\bar{C})^{3}}\biggr{]},$
(175)

extending the result (77) from Sec. III to finite values of $\beta$ with an
additional term $\propto\left(\kappa_{m}-1\right)^{3/2}$.

###  A.2 Close to merging

Let us study the strong pinning problem close to merging, as described by the
two-dimensional Landau-type energy functional (V.1),

$\displaystyle e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf
R})=\frac{\bar{C}(1-\kappa_{s})}{2}\,\tilde{u}^{2}+\frac{\bar{C}+\lambda_{+,s}}{2}\,\tilde{v}^{2}+\frac{a_{s}}{2}\,\tilde{u}\tilde{v}^{2}$
$\displaystyle\quad+\frac{\alpha_{s}}{4}\,\tilde{u}^{2}\tilde{v}^{2}+\frac{\beta_{s}}{6}\,\tilde{u}^{3}\tilde{v}+\frac{\gamma_{s}}{24}\,\tilde{u}^{4}-\bar{C}\bar{u}\tilde{u}-\bar{C}\bar{v}\tilde{v}.$
(176)

As found before for strong pinning close to onset, the energy functional (176)
is anisotropic with respect to vortex displacements in the stable and unstable
direction. Following the strategy of Sec. A.1, we can use the force-balance
equation (137) to relate the tip position along the $v$-axis to $\bar{v}$ and
$\tilde{u}$,

$\displaystyle\tilde{v}\approx\frac{\bar{v}}{1+\lambda_{+,s}/\bar{C}}\left(1-\frac{a_{s}/\bar{C}}{1+\lambda_{+,s}/\bar{C}}\>\tilde{u}\right).$
(177)

Inserting (177) into the force-balance equation for the unstable component
$\tilde{u}$ and integrating, we find that the resulting effective 1D Landau
theory is identical in form to the one close to onset,

$e_{\mathrm{pin}}^{\mathrm{eff}}(\tilde{u};\bar{u},\bar{v})=\frac{r_{s}}{2}\tilde{u}^{2}+\frac{w_{s}}{6}\tilde{u}^{3}+\frac{\gamma_{s}}{24}\tilde{u}^{4}-h_{s}\tilde{u},$
(178)

with a proper replacement of all coefficients involving the parameters
appropriate at merging,

$\displaystyle\begin{split}r_{s}&=\left[\bar{C}(1-\kappa_{s})-\frac{|\delta_{s}|}{2}\frac{\bar{v}^{2}}{(1+\lambda_{+,s}/\bar{C})^{2}}\right],\\\
w_{s}&=\beta_{s}\frac{\bar{v}}{(1+\lambda_{+,s}/\bar{C})},\\\
h_{s}&=\bar{C}\bar{u}-\frac{a_{s}}{2}\frac{\bar{v}^{2}}{(1+\lambda_{+,s}/\bar{C})^{2}}.\end{split}$
(179)

The difference to (155) is the sign change in the term
$\propto|\delta_{s}|\bar{v}^{2}$. This implies a modification of the main
equation determining the shape of $\mathcal{U}_{\tilde{\bf R}}$ (from which
$\mathcal{B}_{\bar{\bf R}}$ follows via the force balance equation (38)), with
the elliptic equation (165) transforming to the hyperbolic expression

$\gamma_{s}\tilde{u}_{\mathrm{jp}}^{2}+2\beta_{s}\tilde{u}_{\mathrm{jp}}\tilde{v}_{\mathrm{jp}}-|\delta_{s}|\tilde{v}_{\mathrm{jp}}^{2}\approx
2\bar{C}(\kappa_{s}-1).$ (180)

The results for the jumping and landing hyperbolas in $\tilde{\bf R}$-space
and for the edges of the bistable domain in $\bar{\bf R}$-space before and
after merging can be derived by following the strategy of Sec. A.1 above and
agree with the corresponding results from Sec. V.1.

We close with a final remark on the disappearance of critical points after
merging. The critical points are found in the standard manner by setting the
first three derivatives of
$e_{\mathrm{pin}}^{\mathrm{eff}}(\tilde{u};\bar{u},\bar{v})$ to zero. This
works fine before merging when $1-\kappa_{s}>0$ and we find that criticality
is realized for tip and asymptotic positions as given by Eqs. (125) and (138)
in Sec. V.1. However, after merging, the cubic derivative
$\partial^{3}_{\tilde{u}}e_{\mathrm{pin}}^{\mathrm{eff}}$ never vanishes,
signalling the absence of a critical point, in agreement with the discussion
in Secs. V.3 and V.2.2. The merger thus leads to the disappearance of the two
critical (end-)points in asymptotic space, with the attached first-order lines
(the branch crossing line) joining up into a single line that is framed by two
separated spinodals. We are not aware of such a disappearance of critical
points in a merging process within the standard discussion of thermodynamic
phase transitions.

## References

  * Campbell and Evetts (1972) A. Campbell and J. Evetts, Advances in Physics 21, 199 (1972).
  * Kassner (2015) M. E. Kassner,  _Fundamentals of Creep in Metals and Alloys_ (Elsevier Science & Technology Books, Amsterdam, 2015).
  * Gorchon  _et al._ (2014) J. Gorchon, S. Bustingorry, J. Ferré, V. Jeudy, A. Kolton, and T. Giamarchi, Phys. Rev. Lett. 113, 027205 (2014).
  * Abrikosov (1957) A. Abrikosov, Sov.Phys.JETP 5 (1957) 5, 1174 (1957).
  * Burgers (1940) J. Burgers, Proc. Phys. Soc. 52, 23 (1940).
  * Bloch (1932) F. Bloch, Z. Physik 74, 295 (1932).
  * Landau and Lifshitz (1935) L. D. Landau and E. Lifshitz, Phys. Z. Sowjet 8, 153 (1935).
  * Labusch (1969) R. Labusch, Crystal Lattice Defects 1, 1 (1969).
  * Larkin and Ovchinnikov (1979) A. I. Larkin and Y. N. Ovchinnikov, Journal of Low Temperature Physics 34, 409 (1979).
  * Blatter  _et al._ (2004) G. Blatter, V. B. Geshkenbein, and J. A. G. Koopmann, Phys. Rev. Lett. 92, 067009 (2004).
  * Thomann  _et al._ (2012) A. U. Thomann, V. B. Geshkenbein, and G. Blatter, Physical Review Letters 108, 217001 (2012).
  * Willa  _et al._ (2015) R. Willa, V. Geshkenbein, R. Prozorov, and G. Blatter, Physical Review Letters 115, 207001 (2015).
  * Buchacek  _et al._ (2019a) M. Buchacek, R. Willa, V. B. Geshkenbein, and G. Blatter, Physical Review B 100, 014501 (2019a).
  * Ertas and Nelson (1996) D. Ertas and D. Nelson, Physica C 272, 79 (1996).
  * Vinokur  _et al._ (1998) V. Vinokur, B. Khaykovich, E. Zeldov, M. Konczykowski, R. Doyle, and P. Kes, Physica C 295, 209 (1998).
  * Buchacek  _et al._ (2019b) M. Buchacek, Z. L. Xiao, S. Dutta, E. Y. Andrei, P. Raychaudhuri, V. B. Geshkenbein, and G. Blatter, Phys. Rev. B 100, 224502 (2019b).
  * Buchacek  _et al._ (2020) M. Buchacek, V. B. Geshkenbein, and G. Blatter, Physical Review Research 2, 043266 (2020).
  * Blatter  _et al._ (1994) G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, Rev. Mod. Phys. 66, 1125 (1994).
  * Thomann  _et al._ (2017) A. U. Thomann, V. B. Geshkenbein, and G. Blatter, Physical Review B 96, 144516 (2017).
  * Willa  _et al._ (2016) R. Willa, V. B. Geshkenbein, and G. Blatter, Phys. Rev. B 93, 064515 (2016).
  * Gaggioli  _et al._ (2022) F. Gaggioli, G. Blatter, and V. B. Geshkenbein, Phys. Rev. Research 4, 013143 (2022).
  * Buchacek  _et al._ (2018) M. Buchacek, R. Willa, V. B. Geshkenbein, and G. Blatter, Physical Review B 98, 094510 (2018).
  * Kwok  _et al._ (2016) W.-K. Kwok, U. Welp, A. Glatz, A. E. Koshelev, K. J. Kihlstrom, and G. W. Crabtree, Rep. Prog. Phys. 79, 116501 (2016).
  * Willa  _et al._ (2018a) R. Willa, A. Koshelev, I. Sadovskyy, and A. Glatz, Supercond. Sci. Technol. 31, 014001 (2018a).
  * Willa  _et al._ (2018b) R. Willa, A. E. Koshelev, I. A. Sadovskyy, and A. Glatz, Physical Review B 98, 054517 (2018b).
  * Willa  _et al._ (2022) R. Willa, V. B. Geshkenbein, and G. Blatter, Phys. Rev. B 105, 144504 (2022).
  * Buchacek (2020) M. Buchacek,  _The Strong Pinning Paradigm Applied to Vortices in Type-II Superconductors_, PhD Thesis (Institute for Theoretical Physics, ETH Zurich, Switzerland, 2020).
  * Bardeen and Stephen (1965) J. Bardeen and M. J. Stephen, Physical Review 140, A1197 (1965).
  * Nakahara (2003) M. Nakahara,  _Geometry, Topology and Physics (Graduate Student Series in Physics)_ (Taylor and Francis, 2003) pp. 88,118.
  * Nash and Sen (2011) C. Nash and S. Sen,  _Topology and Geometry for Physicists_ (Dover Publications Inc., 2011) pp. 105, 228.
  * Volovik (2017) G. Volovik, Low Temp. Phys. 43, 47 (2017).
  * Kane (2022) C. Kane, Phys. Rev. Lett. 128, 076801 (2022).
  * Villegas  _et al._ (2003) J. Villegas, S. Savel’ev, F. Nori, E. Gonzalez, J. Anguita, R. García, and J. Vicent, Science 302, 1188 (2003).
  * de Souza Silva  _et al._ (2006) C. de Souza Silva, J. Van de Vondel, M. Morelle, and V. Moshchalkov, Nature 440, 651 (2006).
  * Wang  _et al._ (2013) Y. Wang, M. Latimer, Z. Xiao, R. Divan, L. Ocola, G. Crabtree, and W. Kwok, Phys. Rev. B 87, 220501(R) (2013).
  * Lyu  _et al._ (2021) Y. Lyu, J. Jiang, Y. Wang, Z. Xiao, S. Dong, Q. Chen, M. Milošević, H. Wang, R. Divan, J. Pearson, P. Wu, F. Peeters, and W.-K. Kwok, Nature Communications 12, 2703 (2021).
  * Embon  _et al._ (2015) L. Embon, Y. Anahory, A. Suhov, D. Halbertal, J. Cuppens, A. Yakovenko, A. Uri, Y. Myasoedov, M. L. Rappaport, M. E. Huber, A. Gurevich, and E. Zeldov, Scientific Reports 5, 7598 (2015).