# Superconducting multi-vortices and a novel BPS bound
in chiral perturbation theory

Fabrizio Canfora<EMAIL_ADDRESS>Universidad San Sebastián, sede
Valdivia, General Lagos 1163, Valdivia 5110693, Chile  Centro de Estudios
Científicos (CECS), Casilla 1469, Valdivia, Chile  Marcela Lagos
<EMAIL_ADDRESS>Universidad San Sebastián, Avenida del Cóndor 720,
Santiago, Chile  Aldo Vera<EMAIL_ADDRESS>Vicerrectoría de
Investigación, Universidad Mayor, Manuel Montt 367, Santiago, Chile

###### Abstract

We derive a novel BPS bound from chiral perturbation theory minimally coupled
to electrodynamics at finite isospin chemical potential. At a critical value
of the isospin chemical potential, a system of three first-order differential
field equations (which implies the second-order field equations) for the gauge
field and the hadronic profile can be derived from the requirement to saturate
the bound. These BPS configurations represent magnetic multi-vortices with
quantized flux supported by a superconducting current. The corresponding
topological charge density is related to the magnetic flux density, but is
screened by the hadronic profile. Such a screening effect allows to derive the
maximal value of the magnetic field generated by these BPS magnetic vortices,
being $B_{\text{max}}=2,04\times 10^{14}\,\text{G}$. The solution for a single
BPS vortex is discussed in detail, and some physical consequences, together
with the comparison with the magnetic vortices in the Ginzburg-Landau theory
at critical coupling, are described.

##  I Introduction

The phase diagram of the low energy limit of quantum chromodynamics (QCD)
under non-trivial external conditions (such as finite density, low
temperatures and in presence of strong magnetic fields) is a very hard nut to
crack (see R0 -Pisarski1 , and references therein). Not only analytic
perturbative methods fail, but also lattice QCD may experience some problems
(see, for instance, Refs. sign1 -QGBook ).

An extremely important region of the phase diagram corresponds to low
temperatures, vanishing baryonic chemical potential ($\mu_{B}=0$) and finite
isospin chemical potential ($\mu_{I}\neq 0$) IsospinR1 -Brandt . There are
many reasons behind the relevance of this sector. First, when $\mu_{I}\neq 0$
but $\mu_{B}=0$, there is no sign problem, so that detailed results on the
phase diagram are available in this region. Second, in the seminal paper by
Son and Stephanov IsospinR1 , it has been shown that, at finite values of
$\mu_{I}$, a transition occurs from the hadronic phase to a superfluid phase
and, eventually, to a superconducting phase. It is expected that at low
isospin density the ground state is a pion condensate IsospinR1 , Massimo2 ,
Massimo3 , while a Fermi liquid with Cooper pairing should appear at higher
density. These results have been confirmed by lattice studies and still offers
many surprises, as we will see.

In particular, the analysis of the interplay between strong interactions and
electromagnetic fields still exhibit difficult unsolved problems at low
energies and temperatures. At these scales, the main role is played by the
topological solitons of QCD [4] -WeinbergBook . Topological solitons are
characterized by a non-vanishing topological charge, which prevents such
classical configurations from decaying into the trivial vacuum. Especially
taking into account the Cooper pairing appearing at finite $\mu_{I}$, it is
clear that the most natural topological solitons that should appear are
superconducting vortices with quantized magnetic flux. Indeed, vortex-like
solitons are expected both in superfluids and in superconductors. At finite
density, many vortex-like solutions have been constructed numerically, and
many of the following references have been important sources of inspiration as
far as the present work is concerned; Schmitt2 -Nitta4 .

Analytical results have also been obtained in Refs. US1 -US6 . However, these
analytical solutions (which describe inhomogeneous baryonic condensates with
the shapes of tubes and layers) have been obtained at finite baryon density
but not at finite isospin density111While, in the present case, we need
vanishing baryonic density but non-vanishing isospin density. and, moreover,
do not possess the required quantized magnetic flux characteristic of vortices
in superconductors. One of the most important open theoretical questions in
this area appears when we consider the situation in the Ginzburg-Landau (GL)
theory. In such a theory, depending on the value of the Higgs coupling, one
can be in a type I or in a type II superconductor and, exactly at the
transition, a remarkable phenomenon appears. One can saturate a BPS bound for
the free energy per unit of length of the vortices, where the magnetic flux
plays the role of the topological charge, and multi-vortices solutions (where
the vortices can be placed at arbitrary locations) appear Weinberg .

Since $\mu_{I}$ is responsible for the Cooper pairing IsospinR1  (and so
$\mu_{I}$ plays a similar role to the Higgs coupling), one would expect that
at a special value of $\mu_{I}$ it should be possible to saturate a suitable
BPS bound (where the magnetic flux is expected to be the corresponding
topological charge) in such a way that multi-vortices solutions should appear.
Such a special value of $\mu_{I}$ should be a signal of a change of behavior
from type I to type II (in which case flux tubes can penetrate the
superconductor). Despite all the efforts, such a BPS completion has not been
found. In this paper we will fill this gap.

We will work with chiral perturbation theory ($\chi$PT), which describes the
low energy limit of QCD; see CPT1 -Massimo1 , and references therein. As will
be clarified in the following sections, the present novel results are well
within the range of validity of $\chi$PT. Hence, our starting point is the
non-linear sigma model (NLSM) coupled to the Maxwell theory in the case of
$SU(2)$ isospin global symmetry, which is one of the most relevant effective
field theories [4] , BaMa . The analysis of soliton crystals in $\chi$PT in
presence of magnetic fields and/or at finite isospin chemical potential is a
very active topic (see Schmitt2 -Nitta4 , and references therein). The
conditions allowing the appearance of topologically stable non-homogeneous
condensates, besides its intrinsic interest, are also relevant for the
thermodynamics of dense nuclear/quark matter R0 -R2 . Moreover, when the
$SU(2)$ valued scalar field is assumed to be homogeneous, and the
corresponding spatial fluctuations are neglected, many important physical
effects are missed (see ex4d6 , ex4d4 , and references therein).

Until very recently, the more common techniques (which allow to describe
stable topological solitons such as instantons, monopoles and vortices in
Yang-Mills-Higgs and the abelian-Higgs models) did not work in the case of the
gauged $\chi$PT. The reason is that, at the level of the gauged NLSM, there is
no obvious BPS bound to saturate for static configurations. Indeed, the
obvious bounds (where one would put, on the right-hand side, the baryonic
charge density or the magnetic flux density) cannot be saturated. On the other
hand, the recent results in Ref. US7  have shown that when the obvious BPS
bound cannot be saturated, a novel less obvious BPS bound exist that can
actually be saturated and it has all the expected properties that a BPS bound
should have. In the present paper, we will see a quite remarkable
manifestation of this phenomenon; we will find a novel BPS bounds for a
special value of $\mu_{I}$ in the gauged $\chi$PT which can actually be
saturated by superconducting multi-vortices with quantized magnetic flux.

The reason why this bound has not been found before is very tricky. The
topological charge density, which is needed for the BPS completion, is not the
magnetic flux density (as it happens in the GL case), but rather a magnetic
flux density dressed by the hadronic profile itself. Such a dressing screens
the magnetic field $B_{z}$ providing with a maximal possible value for
$B_{z}$. A further by-product of the present results is that the Landau-
Peierls criteria is avoided. Also, it is worth emphasizing that magnetized
topological solitons are extremely relevant in many situations, such as heavy
ions collisions and neutron stars (see Refs. sign1 -QGBook , and strongB1
-strongB3 ).

The techniques introduced in Refs. US1 -US6 , allowed the construction of
analytical solutions describing multi-solitons at finite density for both in
the NLSM as well as in the Skyrme model minimally coupled to the Maxwell
theory (see, for details, Refs. US8 -US14 ). However, in such an approach, the
electromagnetic field (although self-consistently generated by the non-
homogeneous condensates) has necessarily the electric components of the same
order as the magnetic components. Therefore, in all the situations in which
the magnetic field dominates, this framework must be modified. This is the aim
of the present paper, in which the technique introduced in US7  will be used
to derive a novel BPS bound that can be saturated for magnetized
configurations.

This paper is organized as follows: In Section II the gauged $\chi$PT will be
introduced together with the Ansatz for the matter fields. In Section III we
derive a novel BPS bound from the gauged $\chi$PT. In Section IV we construct
magnetized vortex solutions from the BPS equations and we discuss its main
physical properties. In Section V we show in detail the case of a single
vortex. Section VI is devoted to the conclusions. In the Appendix we will
provide with mathematical arguments supporting the existence of such BPS
multi-vortices configurations.

##  II Preliminaries

The gauged $\chi$PT (up to order $\mathit{O}(p^{2})$) 4m , is described by the
action

$\displaystyle
I[U,A]=\frac{1}{4e^{2}}\int_{\mathcal{M}}d^{4}x\sqrt{-g}\left(K\text{Tr}\left[L^{\mu}L_{\mu}\right]-F_{\mu\nu}F^{\mu\nu}\right)\
,$ (1) $\displaystyle L_{\mu}=U^{-1}D_{\mu}U=L_{\mu}^{j}t_{j}\ ,\quad
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\ ,\quad
t_{j}=i\sigma_{j}\ .$

Here $U(x)\in SU(2)$ is the pionic field, $A_{\mu}$ is the Maxwell potential
and $\sigma_{i}$ are the Pauli matrices. The coupling $K$ is a positive
constant given by $K=(\frac{ef_{\pi}}{2})^{2}$, being $f_{\pi}$ the pions
decay constant and $e$ the electric charge. Here we will use
$f_{\pi}=93\,\text{MeV}$ and $e=\sqrt{4\pi\alpha}=0.303$. In our convention
$c=1$ and $\hslash=1$. In Eq. (1), $D_{\mu}$ denotes the covariant derivative,
defined as

$D_{\mu}U=\nabla_{\mu}U+A_{\mu}U\hat{O}\
,\quad\hat{O}=U^{-1}\left[t_{3},U\right]\ ,$ (2)

being $\nabla_{\mu}$ the partial derivative.222Note that in some references a
normalization factor $1/2$ is used in front of $A_{\mu}$ in Eq. (2). The
isospin chemical potential can be introduced to the model through the
covariant derivative in the following form

$D_{\mu}U\rightarrow\bar{D}_{\mu}U=D_{\mu}U+\mu_{I}[t_{3},U]g_{\mu t}\ ,$ (3)

where $\mu_{I}$ is the value of isospin chemical potential. It is important to
point out that critical values for the isospin chemical potential have been
derived from effective models (such as the Skyrme model or ladder-QCD) showing
different phase transitions (see, for instance, Refs. Barducci -Cao ). In the
next section, we will compute a critical value for $\mu_{I}$ in $\chi$PT that
allows the formation of a topologically stable inhomogeneous condensate made
of magnetized multi-vortices.333In this section we will not include the pions
mass since this chiral limit illustrate the emergence of this phenomenon in a
very neat way. The inclusion of the pions mass will be discussed in Section
IV.

The electromagnetic current obtained from Eq. (1) is given by

$J_{\mu}=-\frac{K}{2}\text{Tr}(\hat{O}L_{\mu})\ .$ (4)

The pionic field in the exponential representation is written as

$\displaystyle U=\cos(\alpha)\boldsymbol{1}+\sin(\alpha)n_{i}t^{i}\ ,$ (5)
$\displaystyle n_{i}=\\{\sin\Theta\cos\Phi,\sin\Theta\sin\Phi,\cos\Theta\\}\
,$

where $\alpha=\alpha(x^{\mu})$, $\Theta=\Theta(x^{\mu})$, $\Phi=\Phi(x^{\mu})$
are the three degrees of freedom of the $U$ field, and $\boldsymbol{1}$
denotes the $2\times 2$ identity matrix. In terms of this parametrization, the
covariant derivative $D_{\mu}U$ in Eq. (2) reads

$D_{\mu}\alpha=\partial_{\mu}\alpha\ ,\quad
D_{\mu}\Theta=\partial_{\mu}\Theta\ ,\quad\
D_{\mu}\Phi=\partial_{\mu}\Phi-2A_{\mu}\ .$ (6)

Hence, one can see that the scalar degree of freedom $\Phi(x^{\mu})$ plays the
role of the phase of the complex Higgs field in the GL theory (while $\alpha$
and $\Theta$ are not affected by the covariant derivative).

Since we are interested in multi-vortices (with quantized magnetic field along
the third spatial direction) in $\chi$PT minimally coupled to the Maxwell
theory, the natural Ansatz is

$\displaystyle\alpha=\alpha(x_{1},x_{2})\ ,\quad\Phi=\Phi(x_{1},x_{2})\
,\quad\Theta=\frac{\pi}{2}\ ,$ (7) $\displaystyle
A_{\mu}dx^{\mu}=A_{1}\,dx_{1}+A_{2}\,dx_{2}\ ,\qquad A_{i}=A_{i}(x_{1},x_{2})\
,$ (8)

which leads to a magnetic field in the $x_{3}$ direction. This choice is very
convenient since, as it will be clear from the following computations, the
combination $\sin(\alpha)n_{i}t^{i}$, plays the role of the complex Higgs
field $\phi$ in the GL free energy:

$\sin(\alpha)n_{i}t^{i}\backsim\phi\ ,$

while the choice of the gauge potential is forced by the requirement to have a
magnetic field along $x_{3}$.

##  III A novel BPS bound form $\chi$PT

In this section, for simplicity, we will assume a flat metric in Cartesian
coordinates, but our results can be obtained in the same way using arbitrary
coordinate systems, as we will see in the next section.

When one replaces in Eq. (1) the Ansatz defined in Eqs. (5), (7) and (8),
(taking into account the non-trivial isospin chemical potential in Eq. (3))
one obtains the lagrangian density $\mathcal{L}$, which will play the role of
free energy density $\digamma$, that is

$\digamma=-\mathcal{L}=\frac{1}{e^{2}}\biggl{[}\frac{K}{2}\biggl{\\{}\left(\overrightarrow{\nabla}\alpha\right)^{2}+\sin^{2}(\alpha)\left(\overrightarrow{D}\Phi\right)^{2}\biggl{\\}}+\frac{1}{2}(\vec{B}^{2})-2K\mu_{I}^{2}\sin^{2}(\alpha)\
\biggl{]}\ ,$ (9)

where we have denoted

$\left(\overrightarrow{\nabla}\alpha\right)^{2}=\sum_{i=1}^{2}\left(\partial_{i}\alpha\right)^{2}\
,\quad\left(\overrightarrow{D}\Phi\right)^{2}=\sum_{i=1}^{2}\left(D_{i}\Phi\right)^{2}\
.$

In order to derive a BPS bound from Eq. (9), it is convenient to rewrite the
free energy density as follows:

$e^{2}\digamma=\frac{K}{2}\biggl{\\{}\left(\overrightarrow{\nabla}\alpha\right)^{2}+\sin^{2}(\alpha)\left(\overrightarrow{D}\Phi\right)^{2}\biggl{\\}}+\frac{1}{2}(\vec{B}^{2})+2K\mu_{I}^{2}\cos^{2}(\alpha)-2K\mu_{I}^{2}\
.$ (10)

One can see that the above expression would be positive definite if one could
eliminate the last constant term. From the viewpoint of the differential
equations, the constant term is certainly not important (and one could simply
drop it). However, such a term is telling us something physical: in the
present case, the appropriate thermodynamical potential is not the free energy
$\boldsymbol{F}$, but rather the Gibbs free energy $\boldsymbol{G}$

$\boldsymbol{G=F}+PV\ ,\qquad\boldsymbol{F}=\int d^{3}x\,\digamma\
,\quad\boldsymbol{G}=\int d^{3}x\,G\ ,$ (11)

being $V$ the volume and $P$ the pressure fixed by the chemical potential,
namely $P=2K\mu_{I}^{2}$. Thus, the Gibbs free energy density $G$ reads

$e^{2}G=\frac{K}{2}\biggl{\\{}\left(\overrightarrow{\nabla}\alpha\right)^{2}+\sin^{2}(\alpha)\left(\overrightarrow{D}\Phi\right)^{2}\biggl{\\}}+\frac{1}{2}(\vec{B}^{2})+2K\mu_{I}^{2}\cos^{2}(\alpha)\
,$ (12)

so that

$\frac{\boldsymbol{G}}{L}\boldsymbol{=}\int dx_{1}dx_{2}G\ .$ (13)

Eq. (13) has to do with the fact that we are interested in superconducting
vortices whose Gibbs free energy does not depend on the third direction. From
now on, we will work with $G$ since it is a positive definite quantity, and it
simplifies the discussion of the boundary conditions.

###  III.1 Boundary Conditions and flux quantization

The requirement to have a finite Gibbs free energy per unit of length gives
rise to boundary conditions that are close analogs of the boundary conditions
used for vortices in the GL theory.

Let us introduce the usual coordinate system to discuss boundary conditions
for a single vortex at the origin of the cylindrical coordinates:

$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\theta^{2}+\left(dx_{3}\right)^{2}\ .$

In order for the integral in Eq. (13) to be convergent for large values of the
two-dimensional radius, we must require

$\alpha\underset{r\rightarrow\infty}{\rightarrow}\frac{\pi}{2}+c_{1}\pi\ ,\
c_{1}\in\mathbb{N}\
,\qquad\qquad\overrightarrow{D}\Phi\underset{r\rightarrow\infty}{\rightarrow}0\
\Leftrightarrow\overrightarrow{A}\
\underset{r\rightarrow\infty}{\rightarrow}\frac{1}{2}\overrightarrow{\partial}\Phi\
.$ (14)

On the other hand, close to the origin, regularity requires

$\alpha\underset{r\rightarrow 0}{\rightarrow}c_{2}\pi\ ,\ c_{2}\in\mathbb{N}\
,\qquad\qquad\overrightarrow{A}\ \underset{r\rightarrow 0}{\rightarrow}0\ .$
(15)

The condition in Eq. (15), from the viewpoint of the chiral field $U$ in Eqs.
(5) and (7), reads:

$U\underset{r\rightarrow 0}{\rightarrow}\pm\boldsymbol{1}\ .$ (16)

In fact, Eqs. (15) and (16) must be satisfied at the locations of all the
vortices as one can always choose a local coordinates system where the
position of the $j-$th vortex is the origin:

$U\underset{r\rightarrow r_{j}}{\rightarrow}\pm\boldsymbol{1}\ ,\quad\forall\
j=1,..,N\ ,$ (17)

where $N$ is the total number of vortices and $r_{j}$ is the position of the
$j-$th vortex.

On the other hand, one can derive the quantization of the magnetic flux in the
usual way. The chiral field $U$ in Eq. (5) must be single valued:

$U(r,\theta)=U(r,\theta+2\pi)\ \ ,$ (18)

and then, taking into account Eq. (7), one arrives at the conclusion that

$\Phi(r,\theta)=\Phi(r,\theta+2\pi)+2n\pi\ ,\ \ n\in\mathbb{N}\ ,$ (19)

which is natural taking into account that $\Phi$ plays the role of the phase
of the complex Higgs field in the GL theory (see Eq. (6)). This implies that
the following integral is an integer

$\frac{1}{2\pi}\oint_{\Gamma_{\infty}}\overrightarrow{\partial}\Phi\cdot
d\overrightarrow{l}=n\ ,$ (20)

where $\Gamma_{\infty}$ is the circle at spatial infinity. It is worth
emphasizing here that the above integral would be an integer for any closed
curve $\Gamma$ enclosing all the points $\left\\{r_{j}\right\\}_{j=1,..,N}$,
where $\sin(\alpha)$ vanishes. On the other hand, the finiteness of the Gibbs
free energy requires Eq. (14). Consequently, due to the Stokes theorem, as in
the case of the GL theory, one obtains that $n$ is exactly the magnetic flux,
and it is quantized.

###  III.2 Improper gauge transformations

In order to derive a BPS bound, we rewrite Eq. (12) as

$\displaystyle e^{2}G=$
$\displaystyle\frac{K}{2}\biggl{(}(\partial_{1}\alpha)^{2}+(\partial_{2}\alpha)^{2}+\sin^{2}(\alpha)[(D_{1}\Phi)^{2}+(D_{2}\Phi)^{2}])\biggl{)}+\frac{1}{2}B_{z}^{2}+2K\mu_{I}^{2}\cos^{2}(\alpha)$
$\displaystyle=$
$\displaystyle\frac{K}{2}\biggl{(}(\partial_{1}\alpha+\sin(\alpha)D_{2}\Phi)^{2}+(\partial_{2}\alpha-\sin(\alpha)D_{1}\Phi)^{2}\biggl{)}+\frac{1}{2}\biggl{(}B_{z}-2\mu_{I}\sqrt{K}\cos(\alpha)\biggl{)}^{2}$
$\displaystyle-K\sin(\alpha)(\partial_{1}\alpha D_{2}\Phi-\partial_{2}\alpha
D_{1}\Phi)+2\mu_{I}\sqrt{K}\cos(\alpha)B_{z}\ .$

Note that, locally, we can gauge away the phase $\Phi$ as follows

$\displaystyle
A_{\mu}\rightarrow\widehat{A}_{\mu}=A_{\mu}+\frac{1}{2}\partial_{\mu}\Phi\ ,$
(21) $\displaystyle\Phi\rightarrow\widehat{\Phi}=\Phi-\Phi=0\ .$ (22)

However, it is very important to emphasize that $\Phi$ itself is a multi-
valued function precisely due to the non-vanishing magnetic flux (see Eqs.
(19) and (20)). Therefore, Eqs. (21) and (22) do not define a proper gauge
transformation. In order to clarify which is the price to be paid (in
particular, which extra terms do appear) when one performs a gauge
transformation with a multi-valued gauge parameter, let us consider the case
of a single vortex (which will be analyzed in more detail in Section V). The
corresponding Ansatz is

$\alpha=\alpha(r)\ ,\quad A_{r}=A_{z}=0\ ,\quad A_{\theta}=A(r)\
,\quad\Phi=n\theta\ ,$ (23)

where the cylindrical coordinates are

$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\theta^{2}+dz^{2}\ .$ (24)

The spatial coordinates $\\{r,\theta,z\\}$, have the ranges

$0<r<\infty\ ,\quad 0\leq\theta<2\pi\ ,\quad-\infty<z<\infty\ .$

As one can see from Eq. (6), the interaction between the matter field and the
gauge field appears through the combination $\partial_{\mu}\Phi-2A_{\mu}$. It
is clear that with the Ansatz in Eq. (23), the conditions of regularity at the
origin in Eq. (15) reduce to

$A_{\theta}=A(r)\underset{r\rightarrow 0}{\rightarrow}0\
,\qquad\sin\alpha(r)\underset{r\rightarrow 0}{\rightarrow}0\ ,$

which means, of course, that the origin is the position of the vortex. If one
now performs the improper gauge transformation in Eqs. (21) and (22) one would
get

$\widehat{A}_{\theta}=A(r)+\frac{1}{2}n\underset{r\rightarrow
0}{\rightarrow}\frac{1}{2}n\ ,\qquad\sin\alpha(r)\underset{r\rightarrow
0}{\rightarrow}0\ .$

This is not an innocuous shift of the gauge potential since it introduces a
singularity. This is most easily seen in Cartesian coordinates, where

$\Phi=n\theta=n\arctan\left(\frac{y}{x}\right)\ .$

A gauge potential $A_{\mu}$ proportional to $\partial_{\mu}\Phi$ with the
$\Phi$ here above is nothing but the gauge potential of an Aharonov-Bohm gauge
field. As it is well known, such a gauge potential, despite being locally a
pure gauge, it corresponds to the magnetic field of a very thin solenoid along
the $z$ axis. Therefore, the magnetic field, $B_{z}^{AB}$, associated with the
Aharonov-Bohm gauge field is

$B_{z}^{AB}=n\delta(x,y)\ ,$ (25)

where $\delta(x,y)$ is the Dirac delta at the position of the thin solenoid.
This argument is true actually at the positions of all the vortices. The
reason is that, close to the position of any of the vortices present in the
configuration, one can choose a coordinate system with the origin at the
position of such vortex and repeat this argument. Consequently, the price to
pay if one decides to “gauge away” the function $\Phi$ (which is convenient
from the computational point of view) in a configuration with $N$ vortices is
that one has to include explicitly in the magnetic field the “Aharonov-Bohm
contributions” from all the vortices:

$B_{z}^{AB}=\sum_{j=1}^{N}n_{j}\delta\left(x-x_{j},y-y_{j}\right)\ ,$ (26)

(note that we have absorbed the factors of $4\pi$ in the $n_{j}$ in order to
simplify the notation) which corresponds to a $\Phi$ of the following form

$\Phi=\sum_{j=1}^{N}n_{j}\arg\left(z-z_{j}\right)\ ,$

where $z=x+iy$, and $z_{j}$ is the position of the $j-$th vortex in complex
coordinates (compare, for instance, with the expression below Eq. (3.2) of
Ref. YsonnYang ). Of course, this same result has been obtained in a more
rigorous way in the classic references Taubes1  and Taubes2  . In particular,
the above expression corresponds to the typical sum over the Dirac deltas,
which appear in the mathematical literature (see, for instance, the right hand
side of Eq. (3.3) in Ref. YsonnYang ). Although the present argument is less
rigorous than the one in the above mentioned references, it offers a very
intuitive physical picture: if one eliminates $\Phi$ with an improper gauge
transformation, then the “Aharonov-Bohm contributions” from all the vortices
in Eq. (26) must be include

###  III.3 BPS completion

The elimination of $\Phi$ simplifies the derivation of the BPS bound. One can
rewrite the free energy density as

$\displaystyle e^{2}G=$
$\displaystyle\frac{K}{2}\biggl{(}(\partial_{1}\alpha+\sin(\alpha)\widehat{A}_{2})^{2}+(\partial_{2}\alpha-\sin(\alpha)\widehat{A}_{1})^{2}\biggl{)}+\frac{1}{2}\biggl{(}\widehat{B}_{z}-2\mu_{I}\sqrt{K}\cos(\alpha)\biggl{)}^{2}$
(27)
$\displaystyle-K\sin\alpha(\partial_{1}\alpha\widehat{A}_{2}-\partial_{2}\alpha\widehat{A}_{1})+2\mu_{I}\sqrt{K}\cos(\alpha)\widehat{B}_{z}\
.$

In order to obtain the sought BPS completion, we need to write the last two
terms in the above equation as a total derivative. This is only possible at
the following special critical value for the isospin chemical potential

$\mu_{I}^{\text{c}}=\sqrt{K}=\frac{ef_{\pi}}{2}=14,1\,\text{MeV}\ .$ (28)

In fact, quite remarkably, this particular value is well within the range of
validity of $\chi$PT IsospinR7 , allowing that

$-K\int
dx^{1}dx^{2}\biggl{[}\sin(\alpha)(\partial_{1}\alpha\,\widehat{A}_{2}-\partial_{2}\alpha\,\widehat{A}_{1})-\cos(\alpha)\widehat{B}_{z}\biggl{]}=\int
d\omega\
,\qquad\omega=\cos(\alpha)(\widehat{A}_{1}dx^{1}+\widehat{A}_{2}dx^{2})\ ,$

where we have used that
$\widehat{B}_{z}=F_{12}=\partial_{1}\widehat{A}_{2}-\partial_{2}\widehat{A}_{1}$.
Thus, the free energy density is minimized when the following first order
equations are satisfied

$\displaystyle\partial_{1}\alpha+\sin(\alpha)\,\widehat{A}_{2}$
$\displaystyle=0\ ,$ (29)
$\displaystyle\partial_{2}\alpha-\sin(\alpha)\,\widehat{A}_{1}$
$\displaystyle=0\ ,$ (30) $\displaystyle\widehat{B}_{z}-2K\cos(\alpha)$
$\displaystyle=0\ ,$ (31)

saturating the BPS bound.

It is interesting to note that, for small $\alpha$, the above BPS system
reduces exactly to the BPS system of multi-vortices in critical
superconductors. In fact, considering

$\left|\alpha\right|\ll 1\ ,\ \ \sin(\alpha)\sim\alpha\ ,\ \cos(\alpha)\sim
1-\frac{\alpha^{2}}{2}\ ,$

we obtain the well-known system

$\displaystyle\partial_{1}\alpha+\alpha\,\widehat{A}_{2}$ $\displaystyle=0\ ,$
$\displaystyle\partial_{2}\alpha-\alpha\,\widehat{A}_{1}$ $\displaystyle=0\ ,$
$\displaystyle\widehat{B}_{z}-K(1-\frac{\alpha^{2}}{2})$ $\displaystyle=0\ .$

This fact explains why the boundary conditions at the positions of the
vortices (where $\alpha$ is small) are the same as in the GL case.

In order to solve the BPS system in Eqs. (29), (30) and (31), it is convenient
to define the function $H(\alpha)=\ln{(\tan{(\alpha/2)})}$. It follows that

$\widehat{A}_{1}=\frac{1}{\sin\alpha}\partial_{2}\alpha=\partial_{2}H(\alpha)\
,\quad\widehat{A}_{2}=-\frac{1}{\sin\alpha}\partial_{1}\alpha=-\partial_{1}H(\alpha)\
.$

Using the above expression for the Maxwell potential, the first order
equations in Eqs. (29), (30) and (31), can be written as second order
differential equation for $H(\alpha)$, namely,

$\nabla^{2}H=K\tanh(H)+\sum_{j=1}^{N}n_{j}\delta\left(x-x_{j},y-y_{j}\right)\
,$ (32)

where the second term on the right-hand side corresponds to the “Aharonov-Bohm
contributions” from all the vortices.444The comparison between the above non-
linear equation determining BPS superconducting vortices in $\chi$PT at finite
isospin chemical potential and the one characterizing superconducting vortices
in the GL theory at critical coupling (together with a sketch of a proof of
the existence of solutions of Eq. (32) for any distribution of vortices) will
be delayed in the Appendix. This equation (without the $\delta-$like terms)
can be obtained from the variation of the following effective action:

$\mathcal{L}_{\text{eff}}(H)=\frac{1}{2}(\vec{\nabla}H)^{2}+K\ln{(2\cosh H)}\
.$ (33)

##  IV Multi-vortices at critical values

The novel BPS bound presented in the previous section presents many peculiar
properties. In this section, we detail the most important ones.

###  IV.1 Strong magnetic field and finite isospin chemical potential

1. First, the topological charge density, which appears naturally in the BPS bound is not simply the magnetic flux density, as one would naively expect; instead, it is a “dressed magnetic flux” modulated by the hadronic profile (this possibility has been pointed out for the first time in Ref. US7 ). The presence of the pions generates a screening effect on the magnetic flux, tending to suppress the magnetic field of the vortices in the condensate.

2. The inhomogeneous condensate is stabilized by the topological charge as it happens for the usual GL vortices, and, as in the GL theory, the possibility to have multi-vortices configurations at arbitrary locations appears at a special value of the relevant parameter (which is the Higgs coupling in the GL case, while the isospin chemical potential in the present case). From the viewpoint of differential equations, this could be enough. However, from the physical viewpoint, the fact that these are topologically non-trivial minima of the Gibbs free energy (and not of the free energy) means that they can be realized in actual experiments keeping the pressure fixed at the value $P=2K(\mu_{I}^{\text{c}})^{2}=858.474\,(\text{MeV})^{4}$.

3. Unlike what happens in the GL theory, where the derivation of a bound on the maximal magnetic field requires quite some work (see, for instance, Refs. YsonnYang -Taubes2 ), the present novel BPS bound allows to find very easily the maximum $B_{\max}$ value for the magnetic field beyond which the condensate ceases to exist. Indeed, looking at Eq. (31), one gets

$B_{\text{max}}=2K=2\biggl{(}\frac{ef_{\pi}}{2}\biggl{)}^{2}=397,03\,(\text{MeV})^{2}=2,04\times
10^{14}\,\text{G}\ .$ (34)

Note that this maximum value for the magnetic field is of the order of what is
expected for magnetars ($\sim 10^{13}\,\text{G}$ to $\sim 10^{15}\,\text{G}$).

4. The behavior of the electromagnetic field, the current and the energy density can be easily described thanks to availability of the BPS equations. First, as we have point out above, the magnetic field only possesses component in the $z$ direction. This magnetic field is generated by a self-sustained current, given by

$J_{\mu}=2K\sin^{2}(\alpha)D_{\mu}\Phi\ .$ (35)

From Eq. (35) one can see that the current is not-null even when the
electromagnetic field is suppressed. In fact, there is a persistent current
generated by the coupling with pions, given by
$J_{\mu}^{(0)}=2K\sin^{2}(\alpha)\partial_{\mu}\Phi$.

5. The stationary point of the free energy density $\digamma$ can be determined through the variation of Eq. (10) with respect to the field $\alpha$, $\Phi$ and $A_{\mu}$, leading to the following set of second order field equations

$\displaystyle\triangle\alpha-\sin(\alpha)\cos(\alpha)\biggl{(}(\vec{D}\Phi)^{2}-4\mu_{I}^{2}\biggl{)}=0\
,$ $\displaystyle\triangle\Phi-2\
\vec{\nabla}\cdot\vec{A}+2\cot(\alpha)\vec{\nabla}\alpha\cdot\vec{D}\Phi=0\ ,$
(36) $\displaystyle\partial_{j}F^{ij}-2K\sin^{2}(\alpha)D^{i}\Phi=0\ ,$

where $\triangle$ is the Laplacian operator. As expected, the first order BPS
equations in Eqs. (29), (30) and (31) imply the second order system in Eqs.
(36). Moreover, one can check that the BPS equations also imply the field
equations obtained directly varying the action in Eq. (1) with respect to the
$U$ field and the spatial components of the Maxwell potential $\vec{A}$.

###  IV.2 Inclusion of the pions mass

The present formalism can be applied even when the gauged $\chi$PT includes a
pions mass term. As it has been discussed many times in the literature, it is
possible to introduce a mass term for the pions in many different ways; see,
for instance, mass1 -mass5 . The reason is that, if one write the chiral field
as

$U=\sigma\boldsymbol{1}+\pi_{i}t^{i}\
,\qquad\sigma^{2}+\sum_{i=1}^{3}\left(\pi_{i}\right)^{2}=1\ ,$

the mass term for the pions in the lagrangian can be seen as an interaction
potential for $\sigma$, namely,
$\mathcal{L}_{\text{mass}}=V\left(\sigma\right)$, and the mass of the pions
$m_{\pi}$ reads

$m_{\pi}^{2}=-\left.\frac{\partial^{2}V}{\partial\sigma^{2}}\right|_{\sigma=1}\
.$

It is worth emphasizing that it is impossible to distinguish between the
potentials by just knowing the pion mass (as it has been discussed in details
in the above references). Here we will consider a mass term proposed in Refs.
mass1  and mass2 ,

$\mathcal{L}_{\text{mass}}=-\frac{2Km_{\pi}^{2}}{e^{2}}\biggl{(}1-\frac{1}{4}\text{Tr}[U]^{2}\biggl{)}\
,$ (37)

which has several benefits. Such a mass term, at the same time, supports
domain walls solutions (hadronic layers) and (for small pions field) gives the
usual pions mass term. The appearance of hadronic layers at finite density is
confirmed both by numerical simulations and by astronomical observations (see
Dorso -Schmitt1 , and references therein). For the above reasons, there are
very strong experimental evidence in favor of the mass term in Eq. (37).

Using the parametrization in Eqs. (5) and (7), the mass term in Eq. (37) reads

$\mathcal{L}_{\text{mass}}=\frac{2Km_{\pi}^{2}}{e^{2}}\sin^{2}(\alpha)\ ,$
(38)

and the free energy density in Eq. (10) becomes

$\displaystyle e^{2}\digamma=$
$\displaystyle\frac{K}{2}\biggl{\\{}\left(\overrightarrow{\nabla}\alpha\right)^{2}+\sin^{2}(\alpha)\left(\overrightarrow{D}\Phi\right)^{2}\biggl{\\}}+\frac{1}{2}(\vec{B}^{2})+2K\mu_{I}^{2}\cos^{2}(\alpha)+2Km_{\pi}^{2}\sin^{2}(\alpha)-2K\mu_{I}^{2}$
$\displaystyle=$
$\displaystyle\frac{K}{2}\biggl{\\{}\left(\overrightarrow{\nabla}\alpha\right)^{2}+\sin^{2}(\alpha)\left(\overrightarrow{D}\Phi\right)^{2}\biggl{\\}}+\frac{1}{2}(\vec{B}^{2})+2K\left(\mu_{I}^{2}-m_{\pi}^{2}\right)\cos^{2}(\alpha)-2K\left(\mu_{I}^{2}-m_{\pi}^{2}\right)\
.$ (39)

Consequently, the inclusion of the mass term for the pions manifests itself in
a negative shift555It is worth noting that the isospin chemical potential and
the pions mass always appear in the combination $\mu_{I}^{2}-m_{\pi}^{2}$ (see
for instance IsospinR7  and references therein). This is a further
confirmation that the mass term in Eq. (38) is correct. of the $\mu_{I}^{2}$:

$\mu_{I}^{2}\rightarrow\mu_{I}^{2}-m_{\pi}^{2}\ .$

Hence, all the results in the previous sections still hold provided the shift
in the above equation is taken into account. The critical value for the
isospin chemical potential when the pions mass is taken into account reads

$\mu_{I}^{\text{c}}=\sqrt{K+m_{\pi}^{2}}<1.1m_{\pi}\ .$

It is important to emphasize that, even with the inclusion of the mass term,
our value for the critical isospin chemical potential $\mu_{I}^{\text{c}}$ are
within the range of validity of $\chi$PT (namely,
$0.1\,m_{\pi}\lesssim\mu_{I}^{\text{c}},<1.1m_{\pi}$). In fact, it is known
(see IsospinR7 , and references therein) that $\chi$PT needs improvement
starting from $\mu_{I}\sim 1.3m_{\pi}$.

##  V A single vortex

In this section, we will study the case of a single vortex. This is great
interest because it allows to explicitly show many of the most important
properties of our solution.

The starting point is the pionic field in Eq. (23) together with the metric in
Eq. (24). The Gibbs free energy density for a single vortex reduces to

$\displaystyle e^{2}G=$
$\displaystyle\frac{K}{2r^{2}}\\{r^{2}(\alpha^{\prime})^{2}+\sin^{2}(\alpha)(n-2A)^{2}\\}+\frac{1}{2r^{2}}(A^{\prime})^{2}+2K\mu_{I}^{2}\cos^{2}(\alpha)$
$\displaystyle=$
$\displaystyle\frac{K}{2r^{2}}\\{r\alpha^{\prime}+\sin(\alpha)(n-2A)\\}^{2}+\frac{1}{2r^{2}}\\{A^{\prime}+\bar{\mu}r\cos(\alpha)\\}^{2}-K\frac{1}{r}\sin(\alpha)(n-2A)\alpha^{\prime}-\bar{\mu}\frac{1}{r}\cos(\alpha)A^{\prime}\
,$

where we have defined $\bar{\mu}^{2}=4K\mu_{I}^{2}$. Now, we can follow the
same steps presented in the previous sections in a straightforward way.
Integrating the last expression, we obtain the Gibbs free energy

$\displaystyle\frac{\boldsymbol{G}}{L}=$ $\displaystyle
2\pi\int\frac{1}{2e^{2}r}\\{K(r\alpha^{\prime}+\sin(\alpha)(n-2A))^{2}+(A^{\prime}+\bar{\mu}r\cos(\alpha))^{2}\\}dr$
$\displaystyle-\frac{2\pi}{e^{2}}\int\left\\{K\sin(\alpha)(n-2A)\alpha^{\prime}+\bar{\mu}\cos(\alpha)A^{\prime}\right\\}dr\
.$ (40)

The second term in Eq. (40) is a total derivative for the particular value of
the isospin chemical potential presented in the previous section, that is
$\bar{\mu}^{\text{c}}=2K\Rightarrow\mu_{I}^{\text{c}}=\sqrt{K}$. In fact, for
this value

$-K\sin(\alpha)(n-2A)\alpha^{\prime}-\bar{\mu}\cos(\alpha)A^{\prime}=\\{K\cos(\alpha)(n-2A)\\}^{\prime}\
.$

It follows from Eq. (40) that the Gibbs free energy is minimized when the
following BPS equations are satisfied

$\displaystyle r\alpha^{\prime}+\sin(\alpha)(n-2A)=0\ ,$ (41) $\displaystyle
A^{\prime}+2Kr\cos(\alpha)=0\ ,$ (42)

where, also in this case, one can check that the above first order BPS system
implies the second-order field equations corresponding to the stationary
points of the Gibbs free energy in Eq. (40).

The topological charge of this configuration reads

$Q=\frac{2\pi}{e^{2}}\int\\{K\cos(\alpha)(n-2A)\\}^{\prime}dr=\frac{2\pi
K}{e^{2}}n\ .$ (43)

Note that, as we have pointed out in the previous section, the topological
charge density of the vortices here presented is different from the magnetic
flux density (due to the presence of the term $\cos(\alpha)$ in Eq. (43)).
However, once the boundary conditions are implemented, the topological charge
is in fact proportional to the magnetic flux (times a dimensional constant
that depends on the couplings of the theory), as can be seen from Eqs. (20)
and (43). Now, if the solitons are confined in a finite volume, the
topological charge and the magnetic flux may be different.666For instance, if
one analyzes this system on a finite sample, one does not necessarily need to
require that $\cos(\alpha)=0$ on the boundary. This more general case will be
studied in a forthcoming paper.

Fig. 1 shows the behavior of the pionic profile $\alpha$ as well as the
Maxwell potential characterized by the function $A$ for the single vortex with
$n=1$.

Figure 1: Profiles $\alpha$ and $A$ of the single vortex with $n=1$.

To integrate the system in Eqs. (41) and (42) we have used the boundary
conditions in Eqs. (14) and (15), with $c_{1}=0$ and $c_{2}=1$. To obtain
clearer plots, we have fixed the value of the coupling constants to
$K=\frac{1}{7}$ and $f_{\pi}=2$. Fig. 2 shows the Gibbs free energy density of
a single vortex for different values of $n$.

Figure 2: Gibbs free energy density for a single vortex with $n=1$, $n=2$ and
$n=3$.

Regarding the electromagnetic field self-sustained by the vortex, the magnetic
field is in the $z$ direction, given by

$B_{z}=-\frac{A^{\prime}}{r}\ ,$ (44)

while the current in the $\theta$ direction is

$J_{\theta}=2K\sin^{2}(\alpha)(n-2A)\ .$ (45)

Fig. 3 shows the magnitude of the magnetic field of the single vortex for
different values of $n$.

Figure 3: Magnetic field of the single vortex for for $n=1$, $n=2$ and $n=3$.

##  VI Final remarks

In the present manuscript, we have shown that, at a critical value of the
isospin chemical potential, superconducting multi-vortices configurations
appear in gauged $\chi$PT. This has been achieved through the derivation of a
novel BPS bound in $\chi$PT minimally coupled to electrodynamics, which is
saturated by these multi-vortices configurations. This bound has not been
discovered before because the topological charge density is not the magnetic
flux density but rather is a magnetic flux density “dressed” by the hadronic
profile. Such a dressing discloses the screening effects of the hadrons on the
magnetic field, allowing us to derive an upper bound on the intensity of the
magnetic field itself. These configurations are supported by a persistent
current, which does not vanishes when the Maxwell field is suppressed, showing
that the condensate has superconducting characteristics. These BPS vortices
exist for a particular value of the isospin chemical potential which is within
the range of validity of $\chi$PT. The present results open many interesting
questions on which we hope to come back in a future publication. First of all,
in the pioneering paper Weinberg , E. Weinberg, using in a very clever way the
index theorem, was able to determine the dimension of the moduli space of BPS
multi-vortices in the GL theory at critical coupling. Then, there are two
natural questions. The first is whether the dimension in the present case is
the same as in the GL theory (the arguments presented in the Appendix suggest
a positive answer). The second is whether the index appearing in the present
case is related to the magnetic flux density, or the dressed magnetic flux
density. Another closely related direction of research is the remarkable idea
proposed by N. Manton (see Manton1 -Samols , and references therein),
according to which one can describe the dynamics of BPS solitons in terms of
the geodesics of the geometry of the moduli space. Hence, it is an extremely
interesting issue to analyze the geometry of the moduli space in the present
case. On a more phenomenological side, the analysis of the coupling of up and
down quarks to these BPS multi-vortices is very relevant. We think that all
the above open questions deserve further investigation.

###### Acknowledgements.

F. C. has been funded by FONDECYT Grant No. 1240048. M. L. has been funded by
FONDECYT Iniciación No. 11241079.

##  Appendix A On the existence of multi-vortices configurations

In this Appendix, we outline two mathematical strategies (which can be
promoted to rigorous arguments, as it will be discussed in a future
publication) leading to the proof of the existence of multi-vortices
configurations in the $\chi$PT coupled to Maxwell at the critical value of the
isospin chemical potential defined in the previous sections. Besides the
intrinsic importance to prove rigorously the existence of multi-vortices
solutions, the first strategy presented in this Appendix shows in a very clear
way the difference between the present case, in which the hadronic profile
screens substantially the magnetic field, and the usual GL case.

###  A.1 First Strategy

As it has been explained in the previous sections, the BPS equations in Eqs.
(29), (30) and (31) are equivalent to the master equation in Eq. (32) which is
a semi-linear elliptic equation in two coordinates. Moreover, in the case of
multi-vortices in the GL theory at critical coupling, the BPS equations are
equivalent to a single master equation, which is also a semi-linear elliptic
equation in two coordinates (see Taubes1 , Taubes2 , and references therein).
The differences between the two master equations only lie in the non-linear
terms. Thus, let us write the two master equations in a unified fashion:

$\displaystyle\nabla^{2}H=\Psi_{I}\left(H\right)+\sum_{j=1}^{N}n_{j}\delta\left(x-x_{j},y-y_{j}\right)\
,\ \ I=1,2\ ,$ (46) $\displaystyle\Psi_{1}\left(H\right)=K\tanh(H)\
,\qquad\Psi_{2}\left(H\right)=-1+\exp H\ ,$

where $\Psi_{1}\left(H\right)$ corresponds to the BPS vortices in $\chi$PT
presented in this paper while $\Psi_{2}\left(H\right)$ corresponds to the BPS
vortices in GL theory at critical coupling. Now, we will describe a technique
(which is quite standard in the theory of non-linear elliptic equations; see
Ref. GilbargTrudinger ) based on the Banach-Caccioppoli theorem (see Ref.
BanachCaccioppoli1  and references therein). The idea to use a strategy based
on the Banach-Caccioppoli theorem is that it highlights a very important
difference between $\Psi_{1}\left(H\right)$ and $\Psi_{2}\left(H\right)$
which, physically, is related to the screening of the magnetic field from the
hadronic profile, which is stronger in $\chi$PT than in GL theory at critical
coupling. The mathematical consequence is that in the case of
$\Psi_{1}\left(H\right)$ one can apply directly the Banach-Caccioppoli
theorem, while in the case of $\Psi_{2}\left(H\right)$ the existence and
uniqueness parts require far more refined techniques (see Taubes1 , Taubes2 ,
and references therein).

Let us begin with the definition of an auxiliary linear problem:

$\displaystyle\nabla^{2}u=\Psi_{I}\left(V\right)+\sum_{j=1}^{N}n_{j}P_{j}\ ,\
n_{j}\in\mathbb{N}\ ,\ \ \left.u\right|_{\partial\Omega}=0\ ,\ \ V\in X\ ,\
N<\infty\ ,\ $ (47) $\displaystyle\Psi_{1}\left(V\right)=K\tanh V\
,\quad\Psi_{2}\left(V\right)=-1+\exp V\
,\quad\nabla^{2}=\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\ ,$ (48)

where the functions $P_{j}$ are defined as follows:

$\displaystyle P_{j}\left(x,y\right)=\Lambda\ ,\ \
\left(x,y\right)\in\Omega_{j},\ \ P_{j}\left(x,y\right)=0\ ,\ \
\left(x,y\right)\in\Omega\diagdown\Omega_{j}\ ,$ (49)
$\displaystyle\text{measure}\left(\Omega_{j}\right)=\frac{1}{\Lambda^{2}}\ ,\
\ \Omega_{j}\subset\Omega\ \forall j=1,..,N\ ,\ \ \Omega_{j}\cap\Omega_{k}=0\
for\ j\neq k\ ,$ (50) $\displaystyle P_{j}\ \in L^{2}\left(\Omega\right)\ \ \
\forall j=1,..,N\ ,$ (51)
$\displaystyle\Omega=\left(0,a\right)\times\left(0,a\right)\ .$ (52)

The conditions in Eqs. (49), (50) and (51) describe the functions $P_{j}$
(where $\Omega_{j}$ is a small circle of area $\frac{1}{\Lambda^{2}}$ around
the point of coordinates $(x_{j},y_{j})$). It is clear that when $\Lambda$ is
a large number, the $P_{j}$ tend to be very peaked $(x_{j},y_{j})$: the idea
to introduce the functions $P_{j}$ is to define a limit in which, as $\Lambda$
becomes bigger and bigger, $P_{j}$ becomes closer and closer to a Dirac delta.
There are actually many different ways to represent the Dirac delta as the
limit of bounded (even smooth) functions. This issue will be discussed in more
detail in a future publication.

In the linear elliptic equation in Eq. (47), $u=u(x,y)$ is the unknown, while
$V=V(x,y)$ is supposed to be known. For the moment, we will take as domain
$\Omega$ a square of side $a<K^{-1/2}$ in Eq. (52) (then we will explain how
to obtain similar results in bigger squares).

The first technical step is to choose the function $V$ (the “source term” of
the auxiliary problem) in a suitable functional space $X$ in such a way that
the solution $u(x,y)$ of the auxiliary problem belongs to the same functional
space of the function $V$.

The second technical step is to define a mapping $\Phi$ such that for any
$V_{k}\in X$ the mapping $\Phi$ sends $V_{k}$ into the solution $u_{k}$ of the
auxiliary problem in Eq. (47) with $V=V_{k}$:

$u_{k}=\Phi\left(V_{k}\right)|\qquad\nabla^{2}u_{k}=\Psi_{I}\left(V_{k}\right)+\sum_{j=1}^{N}n_{j}P_{j}\
,\ n_{j}\in\mathbb{N}\ ,\ \ \left.u_{k}\right|_{\partial\Omega}=0\ .$

The third technical step is to show that such a mapping is a contraction so
that, due to the Banach-Caccioppoli theorem, it possesses a unique fixed
point. Then, it is trivial to see that such a fixed point is a solution of the
original non-linear problem in Eq. (46).

####  A.1.1 First step

Here we need to put the basis to define such a mapping. The easiest choice is
to assume that the functional space $X$ (to which the source $V$ in Eq. (47)
belongs) is $L^{2}\left(\Omega\right)$:

$X=L^{2}\left(\Omega\right)\ ,$

where $L^{2}\left(\Omega\right)$ is the space of square integrable function on
$\Omega$. Moreover, one observes the following. If

$\nabla^{2}u=f\ ,\ \ \left.u\right|_{\partial\Omega}=0\ ,\ \ f\in
L^{2}\left(\Omega\right)\Rightarrow\ u\in L^{2}\left(\Omega\right)\ ,$ (53)

then (due to $\left.u\right|_{\partial\Omega}=0$) one gets

$\displaystyle\int_{\Omega}uf=\int_{\Omega}u\nabla^{2}u=-\left[\int_{\Omega}\left(\frac{\partial
u}{\partial x}\right)^{2}+\int_{\Omega}\left(\frac{\partial u}{\partial
y}\right)^{2}\right]$
$\displaystyle\Rightarrow\left[\int_{\Omega}\left(\frac{\partial u}{\partial
x}\right)^{2}+\int_{\Omega}\left(\frac{\partial u}{\partial
y}\right)^{2}\right]\leq\left\|f\right\|_{L^{2}\left(\Omega\right)}\left\|u\right\|_{L^{2}\left(\Omega\right)}\
,$ (54)

where we have used the Schwarz inequality (which will be used often in the
following computations). Using the obvious identity

$\int_{0}^{x}\frac{\partial
u}{\partial\sigma}\left(\sigma,y\right)d\sigma=u(x,y)\ ,$

it follows that for any $u(x,y)$ which solves the problem in Eq. (53) one gets

$\displaystyle\left(\left\|u\right\|_{L^{2}\left(\Omega\right)}\right)^{2}=\int_{\Omega}u^{2}dxdy=\int_{\Omega}\left[\int_{0}^{x}\frac{\partial
u}{\partial\sigma}\left(\sigma,y\right)d\sigma\right]^{2}dxdy$
$\displaystyle\Rightarrow\left(\left\|u\right\|_{L^{2}\left(\Omega\right)}\right)^{2}\leq\int_{\Omega}\left[\left(\int_{0}^{x}\left(\frac{\partial
u}{\partial\sigma}\left(\sigma,y\right)\right)^{2}d\sigma\right)^{1/2}\left(\int_{0}^{x}1^{2}d\sigma\right)^{1/2}\right]^{2}dxdy\overset{def}{=}\Upsilon\
.$ (55)

It is possible to further bound from above $\Upsilon$ as follows:

$\displaystyle\Upsilon$ $\displaystyle\leq$ $\displaystyle
a\int_{\Omega}\left(\int_{0}^{a}\left(\frac{\partial
u}{\partial\sigma}\left(\sigma,y\right)\right)^{2}d\sigma\right)dxdy=a\int_{0}^{a}dx\left[\int_{0}^{a}dy\left(\int_{0}^{a}\left(\frac{\partial
u}{\partial\sigma}\left(\sigma,y\right)\right)^{2}d\sigma\right)\right]$
$\displaystyle=$ $\displaystyle
a\int_{0}^{a}dx\left[\int_{\Omega}\left(\frac{\partial u}{\partial
x}\right)^{2}dxdy\right]dx=a^{2}\int_{\Omega}\left(\frac{\partial u}{\partial
x}\right)^{2}dxdy\ ,$
$\Rightarrow\left(\left\|u\right\|_{L^{2}\left(\Omega\right)}\right)^{2}\leq
a^{2}\int_{\Omega}\left(\frac{\partial u}{\partial x}\right)^{2}dxdy\ .\ $
(56)

Hence, from Eqs. (54) and (56) it follows that

$\displaystyle\int_{\Omega}\left(\frac{\partial u}{\partial x}\right)^{2}\leq
a\left(\int_{\Omega}\left(\frac{\partial u}{\partial
x}\right)^{2}dxdy\right)^{1/2}\left\|f\right\|_{L^{2}\left(\Omega\right)}$
$\displaystyle\Rightarrow\left(\int_{\Omega}\left(\frac{\partial u}{\partial
x}\right)^{2}dxdy\right)^{1/2}\leq
a\left\|f\right\|_{L^{2}\left(\Omega\right)}\ ,$ (57)

and this is true $\forall\ u$ which is a solution of the problem in Eq. (53).
Similarly, we get

$\left(\int_{\Omega}\left(\frac{\partial u}{\partial
y}\right)^{2}dxdy\right)^{1/2}\leq
a\left\|f\right\|_{L^{2}\left(\Omega\right)}\ .$ (58)

Moreover, from Eqs. (56) and (57) together with (58), we also get

$\left\|u\right\|_{L^{2}\left(\Omega\right)}\leq
a^{2}\left\|f\right\|_{L^{2}\left(\Omega\right)}\ .$ (59)

As it is well known (see, for instance, GilbargTrudinger ) the problem in Eq.
(53) possesses an existence and uniqueness theorem: $\forall f\in
L^{2}\left(\Omega\right)$ there is a unique $u$ which solves the problem in
Eq. (53). One can also say that, actually, the solution $u$ actually belongs
to a well-behaved subspace of $L^{2}\left(\Omega\right)$:

$u\in\mathbb{W\ },\qquad\mathbb{W}\overset{def}{=}\left\\{u\in
W^{2,2}\left(\Omega\right),\ \left.u\right|_{\partial\Omega}=0\right\\}\ ,$

where $W^{2,2}\left(\Omega\right)$ is the (2,2) Sobolev space on the domain
$\Omega$ (for a definition of Sobolev space, see GilbargTrudinger  and
references therein).

####  A.1.2 Second step

Here we define such a mapping. As a particular case of the problem in Eq.
(53), let us consider the problem in Eq. (47) with
$\Psi_{1}\left(V\right)=K\tanh V$, where

$V\in L^{2}\left(\Omega\right)\ .$

Thus, in this case,

$f=K\tanh V+\sum_{j=1}^{N}n_{j}P_{j}\in L^{2}\left(\Omega\right)\ .$

The existence and uniqueness result for the problem in Eq. (53) provides with
the following:

$\forall\,V\in L^{2}\left(\Omega\right)\ ,\ \left.\exists\right|\
u\in\mathbb{W\subset\ }L^{2}\left(\Omega\right)\ ,$

which solves the problem in Eq. (47) with $\Psi_{1}\left(V\right)=K\tanh V$.
We are in a position to define a mapping $\Phi$ such that for any $V\in
L^{2}\left(\Omega\right)$ the mapping $\Phi$ sends $V$ into the unique
solution $u$ of the auxiliary problem in Eq. (47):

$\displaystyle u=\Phi\left(V\right)|$ (60)
$\displaystyle\nabla^{2}u=\Psi_{1}\left(V\right)+\sum_{j=1}^{N}n_{j}P_{j}\ ,\
n_{j}\in\mathbb{N}\ ,\quad u|_{\partial\Omega}=0\
,\quad\Psi_{1}\left(V\right)=K\tanh V\ .$ (61)

Let us consider now two different “sources” $V_{1}$ and $V_{2}$ together with
the corresponding images under the mappings $u_{1}=\Phi\left(V_{1}\right)$ and
$u_{2}=\Phi\left(V_{2}\right)$. From the above definitions it follows that

$\nabla^{2}\left(u_{1}-u_{2}\right)=K\left(\tanh V_{1}-\tanh V_{2}\right)\in
L^{2}\left(\Omega\right)\ ,\ \
\left.\left(u_{1}-u_{2}\right)\right|_{\partial\Omega}=0\ ,\quad
u_{1}-u_{2}\in\mathbb{W\ }.$

Then, from Eq. (59), it follows that

$\displaystyle\left\|u_{1}-u_{2}\right\|_{L^{2}\left(\Omega\right)}\leq
a^{2}\left\|K\left(\tanh V_{1}-\tanh
V_{2}\right)\right\|_{L^{2}\left(\Omega\right)}$
$\displaystyle\Rightarrow\left\|u_{1}-u_{2}\right\|_{L^{2}\left(\Omega\right)}\leq
Ka^{2}\left\|\tanh V_{1}-\tanh V_{2}\right\|_{L^{2}\left(\Omega\right)}\
=Ka^{2}\left(\int_{\Omega}\left|\tanh V_{1}-\tanh
V_{2}\right|^{2}dxdy\right)^{1/2}\ .$ (62)

Then, using the elementary result from calculus (namely, the mean value
theorem) that for any continuous differentiable function $h(x)$ we have

$\displaystyle h(x)-h(y)$ $\displaystyle=$ $\displaystyle
h^{\prime}(z)\left(x-y\right)\ ,\ h^{\prime}(z)=\frac{dh}{dz}\ ,\ \
z\in\left(x,y\right)\ $ $\displaystyle\Rightarrow\left|\tanh V_{1}-\tanh
V_{2}\right|$ $\displaystyle=$
$\displaystyle\frac{1}{\cosh^{2}\left(z\right)}\ \left|V_{1}-V_{2}\right|\ ,\
\ \frac{1}{\cosh^{2}\left(z\right)}\leq 1\ \forall z\in\mathbb{R}\ ,$ (63)

for some $z$. Consequently, from Eq. (62) together with the mean value theorem
(and taking into account that the derivative of the hyperbolic tangent is
always less than or equal to 1 in absolute value) we get

$\left\|u_{1}-u_{2}\right\|_{L^{2}\left(\Omega\right)}\leq
Ka^{2}\left(\int_{\Omega}\left|V_{1}-V_{2}\right|^{2}dxdy\right)^{1/2}\ .$
(64)

####  A.1.3 Third step

Here we prove that the mapping defined in Eqs. (60) and (61) is a contraction
thanks to Eq. (64). Indeed, from the definition in Eqs. (60) and (61) together
with Eq. (64) we obtain that

$\displaystyle\forall\ V_{1}\ ,\ V_{2}\ $ $\displaystyle\in$ $\displaystyle
L^{2}\left(\Omega\right)\ ,$
$\displaystyle\left\|\Phi\left(V_{1}\right)-\Phi\left(V_{2}\right)\right\|_{L^{2}\left(\Omega\right)}$
$\displaystyle\leq$ $\displaystyle
Ka^{2}\left\|V_{1}-V_{2}\right\|_{L^{2}\left(\Omega\right)}\ .$ (65)

Consequently, if we take $Ka^{2}<1$, we see that the mapping defined here
above is a contraction so that, due to the Banach-Caccioppoli theorem, it
possesses a unique fixed point:

$\left.\exists\right|V^{\ast}\quad\text{such
that}\quad\Phi\left(V^{\ast}\right)=V^{\ast}\ .$ (66)

If we explicitly write the condition to be a fixed point of the mapping
defined in Eqs. (60) and (61), we see that $V^{\ast}$ satisfies the following
equation:

$\nabla^{2}V^{\ast}=\Psi_{1}\left(V^{\ast}\right)+\sum_{j=1}^{N}n_{j}P_{j}\ ,\
n_{j}\in\mathbb{N}\ ,\ \ \left.V^{\ast}\right|_{\partial\Omega}=0\ ,\ \
\Psi_{1}\left(V^{\ast}\right)=K\tanh V^{\ast}\ ,$

namely, the fixed point $V^{\ast}$ of the mapping defined here above is the
solution of the sought non-linear elliptic differential equation. Note that
such simple argument would not work in the case of the BPS vortices in the GL
theory. The reason is that, in the GL case, $\Psi_{2}\left(V\right)=-1+\exp
V$, and if we follow the same procedure, when we apply the mean value theorem
we would get

$\left|\exp V_{1}-\exp V_{2}\right|=\left(\exp z\right)\
\left|V_{1}-V_{2}\right|\ ,$

for some $z$. However, unlike what happens with the
$\Psi_{1}\left(V\right)=K\tanh V$, which characterizes the BPS vortices in
$\chi$PT introduced in the present work, $\left|\exp z\right|$ is not less
than or equal to one. This is the reason why the proof of existence and
uniqueness of BPS multi-vortices in the GL case requires very refined
techniques (see Refs. Taubes1  and Taubes2 ). This technical difference is
related to the fact that, in the case of $\chi$PT, the hadronic profile
screens considerably the magnetic field.

It is worth emphasizing that the above argument is not complete. First of all,
one should define properly the limit of $\Lambda\rightarrow\infty$ in order to
obtain $P_{j}\rightarrow\delta(x-x_{j},y-y_{j})$. This issue appears to be
within the reach of usual techniques in functional analysis (see Refs.
GilbargTrudinger  and BanachCaccioppoli1 ). Secondly, the smoothness of the
solution must be improved, as we would like $V^{\ast}$ to be smooth, far from
the positions of the vortices. The usual “bootstrap arguments” or “elliptic
regularity theory” in Sobolev spaces (which have been already used in the GL
case: see YsonnYang -Taubes2 ) appear to work in the present case as well
without much change. Thirdly, we would like to eliminate the constraint
$Ka^{2}<1$ on the size of the domain $\Omega$ allowing domains of generic
sizes. This can be certainly be achieved using more refined inequalities in
functional analysis (see BanachCaccioppoli2  and references therein).

It is important to remark here that the idea of the present paper is not to
give a fully rigorous proof of the existence and uniqueness (for this a
separate more mathematical paper is necessary) but to provide with sound
evidences that such BPS multi-vortices configurations do exist. In the present
work we have discussed (in this Appendix) not only strong mathematical
arguments in favor of the existence of such BPS configurations but also (in
the main text) we have constructed numerical solutions for the single vortex
case.

###  A.2 Second Strategy

The first strategy (which has been presented here above) is interesting since
it emphasizes how the physical properties, which has been discussed in the
main text (such as the hadronic screening of the magnetic field) manifest
themselves in the possibility of using the contraction theorem directly.
However, the easiest and fastest strategy to prove rigorously that such multi-
vortices exist is actually to follow exactly the same steps of Taubes1 . A
direct analysis reveals that many of the steps can be repeated without change
in the present case. In particular, we can start from our main equation

$\nabla^{2}H=K\tanh(H)+\sum_{j=1}^{N}n_{j}\delta\left(x-x_{j},y-y_{j}\right)\
.$ (67)

Then, the functions $u_{0}$ and $g_{0}$ in Eqs. (3.2) and (3.4) of Ref.
Taubes1  can be introduced in the present case as well in order to take care
of the Dirac deltas. More precisely, one can choose (the constant parameters
appearing in) $u_{0}$ in such a way that its Laplacian cancels exactly the sum
over the Dirac deltas, leaving the smooth function $g_{0}$ only. Then, (as in
Eq. (3.5) of Taubes1 ) the new unknown $v$ can be defined as follows:

$H=u_{0}+v\ .$

If the above expression is replaced into Eq. (67) one arrives at

$\nabla^{2}v=g_{0}+K\tanh(u_{0}+v)\ .$

The above equation for $v$ can be derived by the following action

$\mathcal{S}_{\text{eff}}(v)=\frac{1}{2}(\vec{\nabla}v)^{2}+g_{0}v+K\ln{(2\cosh(u_{0}+v))}\
,$ (68)

to be compared with Eq. (3.7) of Ref. Taubes1 . The most important step in the
analysis of Ref. Taubes1  is the proof that the action in his Eq. (3.7) is a
convex functional. In particular, in Eq. (4.28) of Taubes1  a fundamental role
is played by the strict convexity of the non-linear part of the action (the
exponential in that case). On the other hand, one can observe that the term
$K\ln{(2\cosh(u_{0}+v))}$, in Eq. (68), is a strict convex function of its
argument as well. Moreover, there are many technical steps of Ref. Taubes1
which actually do not depend on the explicit form of the non-linear term in
the action in his Eq. (3.7): such steps can be imported directly to the
present situations. At last, there are steps (such as the inequalities in Eqs.
(4.6) and (4.7) of Taubes1 ) which can be easily generalized to the present
case thanks both to Eq. (63) discussed in the first part of the Appendix and
to the fact that $\ln{(2\cosh X)}$ is well approximated by $\left|X\right|$.

The above sketchy arguments show that the strategy of Ref. Taubes1  can be
adapted to the present case, leading to a rigorous proof of the existence of
multi-vortex solutions. We will come back to this issue in a future
publication.

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License Attribution 4.0 International (CC BY 4.0).

GeoExT 2024: Second International Workshop on Geographic Information
Extraction from Texts at ECIR 2024, March 24, 2024, Glasgow, Scotland

[<EMAIL_ADDRESS>orcid=0009-0008-7082-7122 ] [1]

[<EMAIL_ADDRESS>orcid=0000-0002-3643-6493 ]

[<EMAIL_ADDRESS>orcid=0000-0003-3575-0365 ]

[<EMAIL_ADDRESS>orcid=0000-0003-3273-7499 ]

[<EMAIL_ADDRESS>orcid=0000-0001-9506-4266 ]

[1]Corresponding author.

# Do Sentence Transformers Learn Quasi-Geospatial Concepts from General Text?

Ilya Ilyankou  SpaceTimeLab, University College London, UK  Aldo Lipani
Stefano Cavazzi  Ordnance Survey, Southampton, UK  Xiaowei Gao  James Haworth

(2024)

###### Abstract

Sentence transformers [1] are language models designed to perform semantic
search. This study investigates the capacity of sentence transformers, fine-
tuned on general question-answering datasets for asymmetric semantic search,
to associate descriptions of human-generated routes across Great Britain with
queries often used to describe hiking experiences. We find that sentence
transformers have some zero-shot capabilities to understand quasi-geospatial
concepts, such as route types and difficulty, suggesting their potential
utility for routing recommendation systems.

###### keywords:

Semantic search sentence transformers language models hiking

##  1 Introduction

Semantic search is different from traditional, keyword search in that it tries
to capture the intent of the searcher [2]. For a good semantic search system,
the sentence ‘The fox sat on the mat’ should be similar to ‘An animal rested
upon the rug’, but different from ‘Fox News on Sat’.

SentenceTransformers [1] is a Python library that contains a collection of
predominantly BERT-family [3] transformer-based neural network models that are
fine-tuned for semantic search. This study investigates the extent to which
asymmetric semantic search models, designed for shorter queries and longer
documents [4], fine-tuned on general (non-geospatial) question-answering
datasets, can understand vague, subjective, and complex quasi-geospatial
concepts. For example, do such models consistently associate a query ‘a walk
with a variety of landscapes’ with documents describing longer walks going
through various terrains over the documents describing shorter, solely urban
or rural walks?

This topic is important because when searching for (hiking) activities, people
tend to describe their physical abilities or desired experiences [5, 6, 7]
over using precise geospatial terms.

##  2 Methodology

In our experiment, we take existing user-generated routes across Great
Britain, add geospatial context to generate their textual descriptions, use
Sentence Transformers to create vector embeddings for each description, and
compare these vectors with embeddings of user queries.

###  2.1 Data

We use user-generated hiking routes from the Ordnance Survey’s OS Maps
app—specifically, a subset of 501,294 routes classified by Ballatore et al.
[8]. We remove very short (under 1 km) and very long (over 50 km) routes to
focus on those that would typically be of interest to leisure hikers, and can
be completed in a day. After further removing routes with obvious GPS signal
issues, we have 496,723 routes whose average length is 11,289 metres. We then
assign a set of attributes to each route using geopandas [9] and several
Ordnance Survey datasets. A full list of attributes is shown in Table 1 in
Appendix.

###  2.2 Generating descriptions

We use simple template language to generate 3-4 sentence descriptions for each
route based on route attributes. Descriptions mention length, shape, start and
end points, total elevation gain and steepness. We explicitly state the walk
is ‘predominantly uphill’ or ‘predominantly downhill’ where the total
elevation gain exceeds (or is less than) the total elevation loss by at least
100 metres; this categorisation applies to approximately 8% of all routes. The
descriptions also include the types of areas the routes go near or through,
including coast, surface water, woodland, green space, urban areas, and
national parks. For each area type, we mention for how long in percentages,
swapping numbers (e.g., ‘60 percent’) for words (e.g., ‘sixty percent’ or
‘most’) roughly half the time. We do this because language models are known to
under-perform when required to work with numbers [10].

The resulting descriptions are between 112 and 589 characters long, with mean
and median of 299 and 296 characters respectively. The longest description
reads:

> ‘This is a twenty-five km walk that begins in Yelverton, West Devon, Devon
> and ends in Plymouth, City of Plymouth. Total elevation gain is seven
> hundred and thirty-nine metres, and elevation grade is 2.9. The walk is
> predominantly downhill. About 25 percent of the walk is within a national
> park, about thirty percent of the walk is in a wooded area, about thirty-
> three percent of the walk goes through an urban area, about 12 percent of
> the walk is within green space, about twenty percent of the walk is along
> the coast, about forty-three percent of the walk is alongside a body of
> water.’

Other examples of descriptions of various lengths are shown in Table 2 in
Appendix.

###  2.3 Matching queries with descriptions

We calculate vector representations, or embeddings, of all textual
descriptions and user queries using msmarco-{MiniLM-L6|distilbert}-cos-v5 and
multi-qa-{MiniLM-L6|distilbert|mpnet-base}-cos-v1 models. These are based on
MiniLM [11], DistilBERT [12], and MPNet [13] architectures, and fine-tuned on
MS MARCO [14] (about 500k records) and/or a compilation of question-answering
datasets which we refer to here as Multi-QA [15] (about 215M records). Neither
collection is specific to the geospatial domain. Models tuned on MS MARCO
support input sequences of up to 384 tokens (word pieces), and those tuned on
Multi-QA support up to 512 tokens; our inputs fit comfortably within both
limits, with the longest description of 589 characters represented by 129
tokens.

We use 20 queries (see Table 3 in Appendix) that resemble questions (e.g.,
‘what is a walk for an expert hiker’), and calculate cosine similarity between
all queries and route descriptions to rank the relevance of each description
for each query. The queries are inspired by various research papers that
studied hiking experiences [5, 7, 6, 16, 8, 17].

###  2.4 Visualising results

Our experiment is difficult to assess using standard information retrieval
quality metrics such as mean reciprocal rank (MRR) or mean average precision
(MAP) [18] given we cannot easily label an individual route description
(document) as relevant to the query or not. Instead, we are interested in the
overall patterns of how the documents are ranked.

We decided to plot cumulative means of relevant route attributes for ranked
documents, sorted from best to worst match (one can think of it as plotting
average@k of relevant route attribute means for all k between 1 and 496,723).
As such, the y-value of the left-most point of each line chart represents the
attribute value of the top-matching document, while the y-value of the right-
most point represents the mean attribute value for the whole dataset. An
increasing cumulative mean (i.e., a line going up) signifies higher-ranking
documents (on the left of the x-axis) typically having lower values than
lower-ranking documents (to the right), and vice versa. We utilise a
logarithmic scale for the x-axis to highlight the cumulative means of top-
ranking documents, while also presenting the overall trend for all documents.

##  3 Results

The results are mixed. Unsurprisingly, sentence transformers do better when
user queries have similar terminology to the documents. All five models are
able to associate a ‘seaside walk’ with routes described as having longer
stretches along the coast. Four models clearly associate ‘a walk for someone
who enjoys town walks’ with routes going through urban areas.

Figure 1 shows cumulative mean length, grade, and elevation gain for more
complex queries targeting easier walks produced by multi-qa-mpnet-base-cos-v1.
The queries mentioning a ‘beginner hiker’ and a ‘person with limited mobility’
are indeed associated with shorter and flatter routes; the results are less
clear for the query mentioning an ‘elderly person’.

Figure 1: Cumulative mean length, grade, and elevation gain for what should be
easier routes

Conversely, Figure 2 shows results produced by the same model for queries
aimed at more challenging hiking experiences. While the top-10 or so results
for an ‘expert hiker’ are indeed longer walks, both the slope and total
elevation gain patterns are not convincing. A ‘sporty person’ will receive
similarly disappointing suggestions. But ‘someone who likes climbing uphill’
will be pleasantly surprised, given both the grade and total elevation gain
are much higher for best matches.

Figure 2: Cumulative mean length, grade, and elevation gain for what should be
harder routes

Peculiarly, even when fine-tuned on the same Multi-QA dataset, MiniLM,
DistilBERT, and MPNet models are in total disagreement over the walks that can
be completed in under an hour (Figure 3). We generally expect these to be
routes of under 5 km [19]. While MiniLM shows a logical pattern of top-
matching routes being shorter, DistilBERT ranks the results in a near reverse
order; cumulative mean length of routes ranked by MPNet seems to hover around
the dataset mean.

Figure 3: Three models, even though fine-tuned on the same dataset, disagree
on which walks can be completed in under 1 hour

None of the models are good at associating ‘long’ and ‘very long’ walks with
higher kilometre values. Regrettably, most models associate ‘someone seeking
greater challenges’ with descriptions of shorter and flatter walks, and
queries for people ‘preferring wilderness to man-made’ barely relate to walks
going through national parks. Full results are available in the Appendix.

##  4 Conclusion

Sentence transformers, fine-tuned on a corpus of general question-answer pairs
for asymmetric semantic search, demonstrate some zero-shot ability to
associate short and subjective queries looking for particular hiking
experiences with synthetically composed route descriptions. One tested model
was able to relate ‘beginner hikers’ and those with ‘limited mobility’ to
shorter and flatter walks; another was able to associate ‘walks that can be
completed in under an hour’ with shorter walks. Models fine-tuned on the same
dataset sometimes showed very different results, signalling that architectures
and pre-training matter.

In future work, a more systematic approach to evaluate sentence transformers
and other language models for geospatial understanding should be introduced.
We suggest focusing on four aspects: model architecture, datasets for fine-
tuning, geospatial descriptions, and evaluation. Firstly, a wider array of
sentence transformers should be tested to identify which architectures, BERT-
based and beyond, achieve better results. Secondly, existing general question-
answering datasets used for fine-tuning should be evaluated both independently
and in combination to see how dataset size, theme, and quality affect
learning. Although the user queries we tested are not ‘traditionally’
geospatial, will using smaller datasets of primarily geographic questions make
sentence transformers better understand hiking and other active living
experiences? Thirdly, we recognise that using a generic template to describe
routes is just one of many ways to represent geospatial data as text. As such,
we suggest exploring more sophisticated approaches of generating descriptions
for routes and other geospatial objects (e.g., specific locations represented
as points, and general areas represented as polygons), given that various
sources (people, websites) describe such objects differently. And lastly, a
more formal way of evaluating ranked results should be explored, accounting
for the fact that user queries tested here are more subjective and incomplete
than those used in typical information retrieval tasks.

###### Acknowledgements.

This work was supported by the Ordnance Survey, and the Engineering and
Physical Sciences Research Council [grant no. EP/Y528651/1].

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##  Appendix A Tables

Table 1: Route attributes Attribute |  Description | Dataset
---|---|---
length_m |  Route length in metres | -
total_gain |  Total elevation gain (ascent) | OS Terrain 5
total_loss |  Total elevation loss (descent) | OS Terrain 5
grade |  Hiking grade, calculated as ($\texttt{total\\_gain}\div\texttt{length\\_m}\times\texttt{100})$ | -
is_circular |  True if route start and end points are within 500m of each other | -
is_out_and_back |  True if route is circular, and its 25m buffer has a high overlap with itself | -
start_place |  Name of the nearest place to the route’s start point, limited to 1km | OS Open Names
end_place |  Name of the nearest place to the route’s end point, limited to 1 km | OS Open Names
along_surfacewater |  Percent of route length that falls within 50m of a body of water | OS Zoomstack
along_coast |  Percent of route length that lies within 150m of GB boundary | OS Zoomstack
is_coastal |  True if at least 50% of the walk length is along the coast | -
in_national_parks |  Percent of route length that falls within a national park | OS Zoomstack
in_greenspace |  Percent of route length that falls within a green space boundary | OS Zoomstack
in_woodland |  Percent of route length that falls within a woodland boundary | OS Zoomstack
in_urban |  Percent of route length that falls within an urban area boundary | OS Zoomstack
Table 2: Examples of generated descriptions of various character lengths Description | Length
---|---
This is a circular, ten km walk that begins and ends in Priddy, Somerset. Total elevation gain is 130 metres, and elevation grade is 1.3. | 138
This is a circular, 12 km walk that begins and ends in Tarrant Gunville, Dorset. Total elevation gain is one hundred and ninety-one metres, and elevation grade is 1.6. | 168
This is a 2 km coastal walk that begins in Glenuig Bay, Highland and ends in Smirisary, Highland. Total elevation gain is 142 metres, and elevation grade is 4.9. About seven percent of the walk is in a wooded area, about 62 percent of the walk is along the coast. | 263
This is a circular, twenty-five km walk that begins and ends in Milton, Stirling. Total elevation gain is 1846 metres, and elevation grade is 7.4. The walk is entirely within a national park, about twenty percent of the walk is in a wooded area, about ten percent of the walk is alongside a body of water. | 305
This is a 6 km walk that begins in Pebbly Hill, Cotswold, Gloucestershire and ends in Stow-on-the-Wold, Cotswold, Gloucestershire. Total elevation gain is 215 metres, and elevation grade is 3.5. The walk is predominantly uphill. About seven percent of the walk is in a wooded area, about 8 percent of the walk goes through an urban area. | 337
This is a 22 km walk that begins in Rampart Head, Cumberland and ends in Little Caldew, Cumberland. Total elevation gain is 222 metres, and elevation grade is 1.0. About 6 percent of the walk is in a wooded area, about 9 percent of the walk goes through an urban area, about 7 percent of the walk is within green space, about 18 percent of the walk is along the coast, about twenty-seven percent of the walk is alongside a body of water. | 437
This is a nineteen km walk that begins in Millbrook, Caerffili - Caerphilly and ends in Ynysfro Reservoirs, Casnewydd - Newport. Total elevation gain is seven hundred and twenty-four metres, and elevation grade is 3.7. The walk is predominantly downhill. About seventeen percent of the walk is in a wooded area, about forty-five percent of the walk goes through an urban area, about eight percent of the walk is within green space, about 17 percent of the walk is alongside a body of water. | 490
This is a 7 km coastal walk that begins in Rhoose Cardiff International Airport, Rhoose / Y Rhws, Bro Morgannwg - the Vale of Glamorgan and ends in Storehouse Point, Bro Morgannwg - the Vale of Glamorgan. Total elevation gain is 194 metres, and elevation grade is 2.7. About thirteen percent of the walk is in a wooded area, about 19 percent of the walk goes through an urban area, about 17 percent of the walk is within green space, 77 percent of the walk is along the coast, about twelve percent of the walk is alongside a body of water. | 539
Table 3: User queries Query |  Relevant attributes
---|---
what is a short walk |  length_m
what is a very short walk |  length_m
what is a long walk |  length_m
what is a very long walk |  length_m
what is a walk by the seaside |  is_coastal
what is a walk through the woods |  in_woodland
what is an urban walk |  in_urban
what is a country walk |  in_urban, in_greenspace
what is a walk for a beginner hiker |  length_m, grade, total_gain
what is a walk for an expert hiker |  length_m, grade, total_gain
what is a walk for a sporty person |  length_m, grade, total_gain
what is a walk for a person with limited mobility |  length_m, grade, total_gain
what is a walk for an elderly person |  length_m, grade, total_gain
what is a walk that can be completed in an hour |  length_m
what is a walk for someone who likes climbing uphill |  grade, total_gain
what is a walk with a variety of landscapes |  in_urban, in_woodland, in_national_parks, along_surfacewater, along_coast
what is a walk for someone seeking greater challenges |  length_m, grade, total_gain
what is a walk for someone who enjoys town walks |  in_urban
what is a walk for someone who is interested in nature |  in_urban, in_woodland, in_national_parks, along_surfacewater, along_coast
what is a walk for someone who prefers wilderness to man-made |  in_urban, in_woodland, in_national_parks, along_surfacewater, along_coast

##  Appendix B Full results