# Skyrme crystals with massive pions

Derek<EMAIL_ADDRESS>Paul<EMAIL_ADDRESS>(corresponding author), and Martin<EMAIL_ADDRESS>
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK

(16th May 2023)

###### Abstract

The crystalline structure of nuclear matter is investigated in the standard
Skyrme model with massive pions. A semi-analytic method is developed to
determine local minima of the static energy functional with respect to
variations of both the field and the period lattice of the crystal. Four
distinct Skyrme crystals are found. Two of these were already known – the
cubic lattice of half-skyrmions and the $\alpha$-particle crystal – but two
are new. These new solutions have lower energy per baryon number and less
symmetry, being periodic with respect to trigonal but not cubic period
lattices. Minimal energy crystals are also constructed under the constraint of
constant baryon density, and its shown that the two new non-cubic crystals
tend to chain and multi-wall solutions at low densities.

##  1 Introduction

The Skyrme model [1] is a nonlinear field theory of $\pi$ mesons that
accommodates nucleons as topological solitons. It emerges as a low energy
effective theory of QCD in the regime where the number of colours (or
equivalently, the rank of the gauge group) grows large [2, 3]. The model has
only one field, taking values in the Lie group ${\mathrm{SU}}(2)$. Field
configurations are classified topologically by an integer-valued homotopy
invariant $B$ which is interpreted physically as the baryon number of the
configuration. There is a topological lower bound on static energy of the form
$E\geq E_{\textrm{top}}B$, where $E_{\textrm{top}}$ is some positive constant,
originally due to Faddeev [4] and subsequently improved by one of us [5].
(Improved in this context means that the constant $E_{\textrm{top}}$ is
increased.) Let $E(B)$ denote the minimum static energy among all fields of
baryon number $B$. The energy bound $E(B)=E_{\textrm{top}}B$ is never
attained, but numerical studies suggest that the ratio $E(B)/B$ decreases
monotonically, and hence converges to some limit $E_{*}$ as
$B\rightarrow\infty$. This suggests that, as $B$ grows large, minimal energy
Skyrme fields may tend to some regular, spatially periodic crystalline
structure, with baryon number $B$ and energy $E_{*}B$ per unit cell.

Such a crystal structure was first proposed by Klebanov [6]. He found a
crystal of $B=1$ skyrmions arranged in a simple cubic (SC) lattice so that
every unit skyrmion is internally oriented to be in the attractive channel
with respect to its nearest neighbours. Manton snd Goldhaber [7] later found
that at high densities this crystal undergoes a phase transition to a body
centred cubic (BCC) lattice of half-skyrmions with a lower energy per baryon
($E/B$) than Klebanov’s SC crystal. Then, independently, Kugler and Shtrikman
[8] and Castillejo et al. [9] determined a new solution with lower $E/B$,
wherein skyrmions are initially arranged in a face centred cubic (FCC) lattice
and relax to a SC lattice of half-skyrmions. In all of these studies, the
model has massless pions, $m_{\pi}=0$, the unit cell has baryon number $B=4$,
and the energy functional has been varied only over cubic period lattices,
that is, only the side length of the cube is varied.

The phase structure of the massless pion Skyrme model has been studied by
Jackson and Verbaarschot [10]. Perapechka and Shnir [11] investigated phase
transitions in the Skyrme model with $m_{\pi}>0$. (They also considered the
effect of incorporating a non-standard sextic term in the energy functional.
Such terms are of current interest because they arise in so-called “near BPS”
variants of the model [12], but will not be of central relevance to our
considerations.) Two candidates have been previously proposed as the minimal
$E/B$ crystal with massive pions: the cubic lattice of half-skyrmions [8, 9]
and the $\alpha$-particle lattice [13]. Once again, these studies impose
periodicity with respect to a cubic period lattice, and vary only the side
length of the cube.

In this paper, we study Skyrme crystals in the model with $m_{\pi}>0$,
minimizing the energy with respect to variations of both the Skyrme field
$\varphi:{\mathbb{R}}^{3}/\Lambda\rightarrow{\mathrm{SU}}(2)$ and its period
lattice $\Lambda$. To achieve this, we identify every 3-torus
${\mathbb{R}}^{3}/\Lambda$ with the fixed 3-torus
${\mathbb{T}}^{3}=S^{1}\times S^{1}\times
S^{1}={\mathbb{R}}^{3}/{\mathbb{Z}}^{3}$ by means of the obvious
diffeomorphism $f:{\mathbb{T}}^{3}\rightarrow{\mathbb{R}}^{3}/\Lambda$, and
equip ${\mathbb{T}}^{3}$ with the pullback of the Euclidean metric on
${\mathbb{R}}^{3}/\Lambda$, $g=f^{*}g_{\textrm{Euc}}$. Varying over all period
lattices $\Lambda$ is then equivalent to varying over all flat metrics $g$ on
the fixed torus ${\mathbb{T}}^{3}$. This approach was introduced in [14], in
which the interpretation of the gradient of the energy with respect to the
metric $g$ as the stress tensor of the field was repeatedly exploited. In the
current paper, we will find it convenient to think of the metric variational
problem more concretely, by identifying $g$ with the constant symmetric
positive definite matrix $(g_{ij})$ representing it with respect to the
canonical coordinate system on ${\mathbb{T}}^{3}$.

So, the numerical task we set ourselves is, for fixed topological degree $B$,
to minimize $E(\varphi,g)$ among all degree $B$ maps
$\varphi:{\mathbb{T}}^{3}\rightarrow{\mathrm{SU}}(2)$ and flat metrics $g$ on
${\mathbb{T}}^{3}$. It is known that, for fixed $g$, the function
$\varphi\mapsto E(\varphi,g)$ attains a minimum (in a function space of rather
low regularity) [15]. The complementary problem of minimising in $g$ for fixed
$\varphi$ was first studied in [14] and numerically implemented in [16, 17].
In [14] it was shown that in a two-dimensional toy model (the baby Skyrme
model), any critical metric $g$ is automatically a local minimimum of $E$. The
problem of extending this result to the Skyrme model was discussed, but
unfortunately the proof used in two dimensions did not generalise. Existence
of critical metrics was not addressed in [14].

In the present paper we obtain a much stronger result. We show in Corollary 4
that, for fixed $\varphi$ satisfying very mild assumptions, there is a unique
flat metric with respect to which $E(\varphi,g)$ is minimal, and hence a
unique period lattice $\Lambda$ (up to automorphism) with respect to which
$\varphi$ has minimal energy per unit cell. In the special case $m_{\pi}=0$,
we can even write down this metric explicitly. In the (more interesting)
massive case, we can resort to a gradient based numerical minimization scheme
to find $g$. Applying a similar scheme to minimize over
$\varphi:{\mathbb{T}}^{3}\rightarrow{\mathrm{SU}}(2)$ in tandem, we can find
the energetically optimal field and period lattice for a given $B$, without
ever imposing any symmetry assumptions on the lattice.

The results reveal that, for $m_{\pi}>0$, the energetically optimal lattice
(with $B=4$ per unit cell) does not have cubic symmetry. In fact there are two
crystal solutions with trigonal period lattices (orthorhombic with side
lengths $L_{1}=L_{2}\neq L_{3}$) which have lower energy than the lowest
strictly cubic lattice. This fact persists if, instead of minimizing over all
flat $g$, we minimize only over the subset of metrics with fixed total volume.
This is equivalent to minimizing under the constraint of fixed average baryon
density, a problem of phenomenological interest [18]. As might be expected,
the energy difference between the trigonal and cubic lattices becomes
negligible as baryon density grows very large, but is significant at lower
densities.

The rest of the paper is structured as follows. In section 2, we formulate the
model mathematically, considering in detail how its energy functional depends
on the metric on physical space. In section 3, we prove existence and
uniqueness of an energy minimizing metric $g$ for any given fixed field. In
section 4 we describe our numerical scheme in detail, while section 5 presents
the results of this scheme. In section 6 we determine minimal energy crystals
under the constraint of fixed baryon density. Finally, section 7 presents some
concluding remarks.

##  2 The Skyrme model

We wish to study the Skyrme model under the assumption that the Skyrme field
$\varphi:{\mathbb{R}}^{3}\rightarrow{\mathrm{SU}}(2)$ is periodic with respect
to some $3$-dimensional lattice

$\Lambda=\\{n_{1}\mbox{\boldmath{$X$}}_{1}+n_{2}\mbox{\boldmath{$X$}}_{2}+n_{3}\mbox{\boldmath{$X$}}_{3}:\mbox{\boldmath{$n$}}\in{\mathbb{Z}}^{3}\\},$
(1)

that is, $\varphi({\bf x}+\mbox{\boldmath{$X$}})=\varphi({\bf x})$ for all
${\bf x}\in{\mathbb{R}}^{3}$ and $\mbox{\boldmath{$X$}}\in\Lambda$, where
$\mbox{\boldmath{$X$}}_{1},\mbox{\boldmath{$X$}}_{2},\mbox{\boldmath{$X$}}_{3}$
is an oriented basis for ${\mathbb{R}}^{3}$,. In this case, we may equally
well interpret the field as a map
$\varphi:{\mathbb{R}}^{3}/\Lambda\rightarrow{\mathrm{SU}}(2)$, where the torus
${\mathbb{R}}^{3}/\Lambda$ inherits a metric $g_{\Lambda}$ from the Euclidean
metric on ${\mathbb{R}}^{3}$. It is convenient to identify
${\mathbb{R}}^{3}/\Lambda$ with the standard torus
${\mathbb{T}}^{3}={\mathbb{R}}^{3}/{\mathbb{Z}}^{3}$ via the diffeomorphism

$f_{\Lambda}:{\mathbb{T}}^{3}\rightarrow{\mathbb{R}}^{3}/\Lambda,\qquad(x_{1},x_{2},x_{3})\mapsto
x_{1}\mbox{\boldmath{$X$}}_{1}+x_{2}\mbox{\boldmath{$X$}}_{2}+x_{3}\mbox{\boldmath{$X$}}_{3},$
(2)

and denote by $g$ the pullback of the metric $g_{\Lambda}$ by $f_{\Lambda}$.
Explicitly,

$g=g_{ij}\mathrm{d}x_{i}\mathrm{d}x_{j},\qquad
g_{ij}=\mbox{\boldmath{$X$}}_{i}\cdot\mbox{\boldmath{$X$}}_{j}.$ (3)

Note that the matrix $(g_{ij})$ is a symmetric, positive definite real
$3\times 3$ matrix. We denote the set of such matrices ${\mathrm{SPD}}_{3}$
and note that every such matrix arises as the metric on ${\mathbb{T}}^{3}$
corresponding to some lattice $\Lambda$ and lattices producing the same matrix
are related by an oriented isometry of ${\mathbb{R}}^{3}$. Hence, instead of
considering the Skyrme model on ${\mathbb{R}}^{3}/\Lambda$ for all lattices
$\Lambda$, we may restrict to the standard torus ${\mathbb{T}}^{3}$ but
consider all flat metrics on ${\mathbb{T}}^{3}$,
$g=g_{ij}\mathrm{d}x_{i}\mathrm{d}x_{j}$, $(g_{ij})\in{\mathrm{SPD}}_{3}$. It
is convenient to abuse notation slightly and denote the matrix $(g_{ij})$ by
$g$.

The energy of a Skyrme field
$\varphi:({\mathbb{T}}^{3},g)\rightarrow{\mathrm{SU}}(2)$ is defined in stages
as follows. Denote by $\mu$ the left Maurer-Cartan form on ${\mathrm{SU}}(2)$,
that is, the ${\mathfrak{su}}(2)$-valued one-form that maps a tangent vector
$X\in T_{U}{\mathrm{SU}}(2)$ to the vector $\mu(X)\in
T_{{\mathbb{I}}_{2}}{\mathrm{SU}}(2)$ whose image under left translation by
$U$ is $X$. The pullback of $\mu$ by $\varphi$,

$\varphi^{*}\mu=\varphi^{-1}\mathrm{d}\varphi=:L_{i}\mathrm{d}x_{i},$ (4)

is usually called the left current. We equip ${\mathfrak{su}}(2)$ with the
usual $Ad$ invariant inner product
$(X,Y)_{{\mathfrak{su}}(2)}=-\frac{1}{2}\operatorname{Tr}(XY)$ and define the
Dirichlet energy of $\varphi$ to be

$E_{2}(\varphi,g)=\int_{{\mathbb{T}}^{3}}|\varphi^{*}\mu|_{g}^{2}{\rm
vol}_{g}=-\frac{1}{2}\int_{{\mathbb{T}}^{3}}g^{ij}\operatorname{Tr}(L_{i}L_{j})\sqrt{|g|}{\rm
vol}_{0}$ (5)

where $g^{ij}$ are the components of the inverse matrix $g^{-1}$, $|g|=\det g$
and ${\rm vol}_{0}=\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}$
is the canonical volume form on ${\mathbb{T}}^{3}$. We are following the
standard convention that a numerical subscript on an energy functional denotes
the degree of its integrand considered as a polynomial in spatial derivatives.
It is important to note that we regard $E_{2}$ as a function of both $\varphi$
and $g$.

To define the Skyrme term $E_{4}(\varphi,g)$, we introduce the
${\mathfrak{su}}(2)$-valued two-form $\Omega$ on ${\mathrm{SU}}(2)$ by
$\Omega(X,Y)=[\mu(X),\mu(Y)]$. Then

$E_{4}(\varphi,g)=\frac{1}{4}\int_{{\mathbb{T}}^{3}}|\varphi^{*}\Omega|_{g}^{2}{\rm
vol}_{g}=-\frac{1}{16}\int_{{\mathbb{T}}^{3}}g^{ik}g^{jl}\operatorname{Tr}([L_{i},L_{j}][L_{k},L_{l}])\sqrt{|g|}{\rm
vol}_{0}.$ (6)

We will also include a potential term

$E_{0}(\varphi,g)=\int_{{\mathbb{T}}^{3}}V(\varphi){\rm
vol}_{g}=\int_{{\mathbb{T}}^{3}}V(\varphi)\sqrt{|g|}{\rm vol}_{0}$ (7)

where $V:{\mathrm{SU}}(2)\rightarrow[0,\infty)$ is some smooth function. The
usual choice is

$V(U)=m_{\pi}^{2}\operatorname{Tr}({\mathbb{I}}_{2}-U),$ (8)

which has the effect of giving the pions of the theory (small amplitude waves
about the vacuum $\varphi={\mathbb{I}}_{2}$) mass $m_{\pi}$.

In summary, the Skyrme energy of a field $\varphi:{\mathbb{T}}^{3}\rightarrow
SU(2)$ and metric $g\in{\mathrm{SPD}}_{3}$ is

$E(\varphi,g)=E_{2}(\varphi,g)+E_{4}(\varphi,g)+E_{0}(\varphi,g).$ (9)

Since ${\mathrm{SU}}(2)$ is diffeomorphic to $S^{3}$ the homotopy class of the
field $\varphi$ is, by the Hopf degree theorem, labelled by its topological
degree $B\in{\mathbb{Z}}$,

$B=\frac{1}{2\pi^{2}}\int_{{\mathbb{T}}^{3}}\varphi^{*}{\rm
vol}_{{\mathrm{SU}}(2)},$ (10)

where ${\rm vol}_{{\mathrm{SU}}(2)}$ is the volume form on ${\mathrm{SU}}(2)$
defined by the bi-invariant metric $h$ which, at $U={\mathbb{I}}_{2}$
coincides with $(\cdot,\cdot)_{{\mathfrak{su}}(2)}$ (equivalently, the round
metric of radius $1$ on $S^{3}$). The mathematical problem that this paper
addresses is to minimize $E(\varphi,g)$, with respect to both $\varphi$ and
$g$, among all fields of fixed degree $B$ and all metrics
$g\in{\mathrm{SPD}}_{3}$.

##  3 Existence and uniqueness of minimizing metrics

For fixed $g\in{\mathrm{SPD}}_{3}$, $({\mathbb{T}}^{3},g)$ is a fixed, compact
oriented Riemannian $3$-manifold, and it follows from a direct application of
the calculus of variations that the functional $\varphi\mapsto E(\varphi,g)$
attains a minimum in each degree class in the space of finite energy maps in
the Sobolev space $W^{1,2}({\mathbb{T}}^{3},{\mathrm{SU}}(2))$ [15]. In this
section we address the complementary variational problem: we fix a map
$\varphi:{\mathbb{T}}^{3}\rightarrow{\mathrm{SU}}(2)$ and establish existence,
and uniqueness, of a minimizer of the function
${\mathrm{SPD}}_{3}\rightarrow{\mathbb{R}}$, $g\mapsto E(\varphi,g)$ which,
for brevity, we will denote $E(g)$.

It is convenient (and makes the result more comprehensive) to include the
possibility of a sextic term in the Skyrme energy,

$E_{6}(\varphi,g)=\int_{{\mathbb{T}}^{3}}|\varphi^{*}\Xi|^{2}{\rm vol}_{g}$
(11)

where $\Xi$ is some $3$-form on ${\mathrm{SU}}(2)$, for example, a constant
multiple of ${\rm vol}_{{\mathrm{SU}}(2)}$. Terms of this kind are of
phenomenological interest since they arise in so-called near BPS variants of
the Skyrme model [12]. To specialize to the model of primary interest, we
simply choose $\Xi=0$.

We begin by analyzing in more detail the $g$ dependence of the terms in $E$.
We first note that

$E_{2}(\varphi,g)=g^{ij}\sqrt{\det
g}\int_{{\mathbb{T}}^{3}}(L_{i},L_{j})_{{\mathfrak{su}}(2)}{\rm
vol}_{0}=\sqrt{\det g}\operatorname{Tr}(Hg^{-1})$ (12)

where

$H_{ij}(\varphi)=\int_{{\mathbb{T}}^{3}}(L_{i},L_{j})_{{\mathfrak{su}}(2)}{\rm
vol}_{0}$ (13)

is a symmetric positive semi-definite matrix depending on $\varphi$ but
independent of $g$. Let us assume that $\varphi$ is $C^{1}$ (so that this
matrix is well-defined) and is immersive somewhere, meaning that there is some
point $p\in{\mathbb{T}}^{3}$ at which $\mathrm{d}\varphi_{p}$ is invertible.
Note that this follows immediately for all maps with $B\neq 0$. By continuous
differentiability, it follows that $\varphi$ is immersive on some
neighbourhood of $p$. Then the matrix $H$ is actually positive definite, for
if not, there exists ${\bf v}\in{\mathbb{R}}^{3}$ with ${\bf v}\cdot H{\bf
v}=0$, whence

$\int_{{\mathbb{T}}^{3}}|{\bf
v}\cdot\mbox{\boldmath{$L$}}|_{{\mathfrak{su}}(2)}^{2}{\rm vol}_{0}=0$ (14)

and hence $\mathrm{d}\varphi(v_{i}\partial/\partial x_{i})=0$ almost
everywhere. This contradicts immersivity of $\varphi$ on a neighbourhood of
$p$. We conclude that $H\in{\mathrm{SPD}}_{3}$.

To understand $E_{4}(g)$ we appeal to an isomorphism peculiar to $3$
dimensions. Having chosen a $3$-form ${\rm
vol}_{0}=\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}$ on
${\mathbb{T}}^{3}$, there is an isomorphism
$(T_{p}{\mathbb{T}}^{3})\otimes{\mathfrak{su}}(2)\rightarrow(\Lambda^{2}T^{*}_{p}{\mathbb{T}}^{3})\otimes{\mathfrak{su}}(2)$
defined by $X\mapsto\iota_{X}{\rm vol}_{0}$. Denote by $X_{\varphi}$ the
section of $T{\mathbb{T}}^{3}\otimes{\mathfrak{su}}(2)$ whose image under this
isomorphism is $\varphi^{*}\Omega$,

$\varphi^{*}\Omega=\iota_{X_{\varphi}}{\rm vol}_{0},$ (15)

and note that $X_{\varphi}$ depends on $\varphi$, but is independent of $g$.
We may similarly define a $g$-dependent section $X_{\varphi}^{g}$ of
$T{\mathbb{T}}^{3}\otimes{\mathfrak{su}}(2)$ by using the isomorphism between
$2$-forms and tangent vectors defined by ${\rm vol}_{g}$ instead of ${\rm
vol}_{0}$:

$\varphi^{*}\Omega=\iota_{X_{\varphi}^{g}}{\rm
vol}_{g}=\iota_{\sqrt{|g|}X_{\varphi}^{g}}{\rm vol}_{0}.$ (16)

Clearly $X_{\varphi}^{g}=X_{\varphi}/\sqrt{|g|}$. An alternative
interpretation of $X_{\varphi}^{g}$ is that it is the vector field metrically
dual to the Hodge dual of $\varphi^{*}\Omega$ with respect to the metric $g$,
that is,

$*_{g}\varphi^{*}\Omega=\frac{1}{\sqrt{|g|}}\operatorname{\flat}_{g}X_{\varphi},$
(17)

where $*_{g}:\Lambda^{2}T^{*}{\mathbb{T}}^{3}\rightarrow
T^{*}{\mathbb{T}}^{3}$ denotes the Hodge isomorphism and
$\operatorname{\flat}_{g}:T{\mathbb{T}}^{3}\rightarrow T^{*}{\mathbb{T}}^{3}$
the metric isomorphism defined by $g$. Hence

$E_{4}(\varphi,g)=\frac{1}{4}\int_{{\mathbb{T}}^{3}}|X_{\varphi}/\sqrt{|g|}|_{g}^{2}{\rm
vol}_{g}=\frac{1}{4\sqrt{|g|}}\int_{{\mathbb{T}}^{3}}g(X_{\varphi},X_{\varphi}){\rm
vol}_{0}=\frac{1}{\sqrt{\det g}}\operatorname{Tr}(Fg)$ (18)

where

$F_{ij}(\varphi)=\frac{1}{4}\int_{{\mathbb{T}}^{3}}(X_{\varphi,i},X_{\varphi,j})_{{\mathfrak{su}}(2)}{\rm
vol}_{0}$ (19)

is another symmetric positive semi-definite matrix depending on $\varphi$ but
independent of $g$, and $X_{\varphi}=:X_{\varphi,i}\partial/\partial x_{i}$.
In terms of the left currents

$X_{\varphi,i}=\frac{1}{2}\epsilon_{ikl}[L_{k},L_{l}],$ (20)

and so the matrix $F$ takes the explicit form

$F_{ij}(\varphi)=-\frac{1}{32}\epsilon_{ikl}\epsilon_{jmn}\int_{{\mathbb{T}}^{3}}\operatorname{Tr}([L_{k},L_{l}][L_{m},L_{n}]){\rm
vol}_{0}.$ (21)

Once again, our non-degeneracy assumption on $\varphi$ (that it is $C^{1}$ and
somewhere immersive) implies that $F$ is positive definite. For if not, there
exists ${\bf v}\in{\mathbb{R}}^{3}$ such that ${\bf v}\cdot F{\bf v}=0$,
whence ${\bf v}\cdot X_{\varphi}=0$ and so
$*_{g}\varphi^{*}\Omega(v_{i}\partial/\partial x_{i})=0$. But then
$\varphi^{*}\Omega$ vanishes on every plane in $T{\mathbb{T}}^{3}$
$g$-orthogonal to ${\bf v}$, which contradicts nondegeneracy of $\Omega$ and
immersivity of $\varphi$.

The remaining terms of $E$ are more straightforward.

$E_{0}(\varphi,g)=\int_{{\mathbb{T}}^{3}}V(\varphi){\rm
vol}_{g}=C_{0}\sqrt{\det g},$ (22)

where

$C_{0}(\varphi)=\int_{{\mathbb{T}}^{3}}V(\varphi){\rm vol}_{0}\geq 0$ (23)

is a constant. Finally, we note that $\varphi^{*}\Xi=f_{\Xi}{\rm vol}_{0}$ for
some real function $f_{\Xi}:{\mathbb{T}}^{3}\rightarrow{\mathbb{R}}$
independent of $g$. Then

$f_{\Xi}{\rm vol}_{0}=(*_{g}\varphi^{*}\Xi){\rm
vol}_{g}=(*_{g}\varphi^{*}\Xi)\sqrt{\det g}{\rm vol}_{0},$ (24)

so $*_{g}\varphi^{*}\Xi=f_{\Xi}/\sqrt{\det g}$. Hence

$E_{6}(\varphi,g)=\int_{{\mathbb{T}}^{3}}\varphi^{*}\Xi\wedge*_{g}\varphi^{*}\Xi=\frac{C_{6}}{\sqrt{\det
g}}$ (25)

where

$C_{6}(\varphi)=\int_{{\mathbb{T}}^{3}}f_{\Xi}^{2}{\rm vol}_{0}\geq 0$ (26)

is a constant. Note that we allow the possibility that $C_{0}$ or $C_{6}$ is
$0$, to accommodate versions of the model with no potential or sextic term.

In summary, for a fixed $C^{1}$ map
$\varphi:{\mathbb{T}}^{3}\rightarrow{\mathrm{SU}}(2)$ which is immersive
somewhere, the total Skyrme energy as a function of the metric $g$ on
${\mathbb{T}}^{3}$ is

$E(g):=E(\left.\varphi\right|_{\textrm{fixed}},g)=\sqrt{\det
g}\operatorname{Tr}(Hg^{-1})+\frac{1}{\sqrt{\det
g}}\operatorname{Tr}(Fg)+C_{0}\sqrt{\det g}+\frac{C_{6}}{\sqrt{\det g}},$ (27)

where $H,F\in{\mathrm{SPD}}_{3}$ and $C_{0},C_{6}\in[0,\infty)$ are constants.
We wish to prove that the function
$E:{\mathrm{SPD}}_{3}\rightarrow{\mathbb{R}}$ attains a unique global minimum,
and has no other critical points. Before doing so, we note that
$E=\widetilde{E}\circ\sigma$ where

$\widetilde{E}:{\mathrm{SPD}}_{3}\rightarrow{\mathbb{R}},\qquad\widetilde{E}(\Sigma)=\operatorname{Tr}(H\Sigma^{-1})+\operatorname{Tr}(F\Sigma)+\frac{C_{0}}{\det\Sigma}+{C_{6}}{{\det\Sigma}}$
(28)

and $\sigma$ is the map

$\sigma:{\mathrm{SPD}}_{3}\rightarrow{\mathrm{SPD}}_{3},\qquad
g\mapsto\Sigma=\frac{g}{\sqrt{\det g}}.$ (29)

Since $\sigma$ is a diffeomorphism, we may equivalently prove that
$\widetilde{E}:{\mathrm{SPD}}_{3}\rightarrow{\mathbb{R}}$ attains a unique
global minimum and has no other critical points. We do this in two stages.

###### Proposition 1

The function $\widetilde{E}:{\mathrm{SPD}}_{3}\rightarrow{\mathbb{R}}$ of
equation (28) attains a global minimum.

###### Proof.

Clearly $\widetilde{E}$ is bounded below (by $0$). Let
$E_{*}=\inf\widetilde{E}$. We must show that there exists
$\Sigma_{*}\in{\mathrm{SPD}}_{3}$ with $\widetilde{E}(\Sigma_{*})=E_{*}$.

Consider the map

$f:(0,\infty)^{3}\times O(3)\rightarrow{\mathrm{SPD}}_{3},\qquad
f({\mbox{\boldmath{$\lambda$}}},\mathscr{O})=\mathscr{O}D_{\mbox{\boldmath{$\lambda$}}}\mathscr{O}^{T},\quad
D_{\mbox{\boldmath{$\lambda$}}}:=\operatorname{diag}(\lambda_{1},\lambda_{2},\lambda_{3}).$
(30)

This map is surjective: given any $\Sigma\in{\mathrm{SPD}}_{3}$ we may take
$\lambda_{i}$ to be its eigenvalues and $\mathscr{O}$ to be an orthogonal
matrix whose columns are its corresponding eigenvectors. Hence, it suffices to
prove that

$(\widetilde{E}\circ
f)({\mbox{\boldmath{$\lambda$}}},\mathscr{O})=\operatorname{Tr}(\mathscr{O}^{T}H\mathscr{O}D_{\mbox{\boldmath{$\lambda$}}}^{-1})+\operatorname{Tr}(\mathscr{O}^{T}F\mathscr{O}D_{\mbox{\boldmath{$\lambda$}}})+\frac{C_{0}}{\lambda_{1}\lambda_{2}\lambda_{3}}+C_{6}\lambda_{1}\lambda_{2}\lambda_{3}$
(31)

attains the value $E_{*}$.

Let $({\mbox{\boldmath{$\lambda$}}}_{n},\mathscr{O}_{n})$ be a sequence in
$(0,\infty)^{3}\times O(3)$ such that

$(\widetilde{E}\circ
f)({\mbox{\boldmath{$\lambda$}}}_{n},\mathscr{O}_{n})\rightarrow E_{*}.$ (32)

Such a sequence exists since $f$ is surjective. Consider the six functions

$(\mathscr{O}^{T}H\mathscr{O})_{ii},(\mathscr{O}^{T}F\mathscr{O})_{ii}:O(3)\rightarrow(0,\infty),\qquad
i=1,2,3,$ (33)

mapping $\mathscr{O}$ to the diagonal entries of
$\mathscr{O}^{T}H\mathscr{O},\mathscr{O}^{T}F\mathscr{O}\in{\mathrm{SPD}}_{3}$,
and note that these functions are strictly positive since $H,F$ are positive
definite. Since these functions are smooth and $O(3)$ is compact, there exists
$\alpha>0$ such that, for all $\mathscr{O}\in O(3)$,
$(\mathscr{O}^{T}H\mathscr{O})_{ii},(\mathscr{O}^{T}F\mathscr{O})_{ii}\geq\alpha$.
Hence, for all
$({\mbox{\boldmath{$\lambda$}}},\mathscr{O})\in(0,\infty)^{3}\times O(3)$,

$(\widetilde{E}\circ
f)({\mbox{\boldmath{$\lambda$}}},\mathscr{O})\geq\alpha\left(\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}+\frac{1}{\lambda_{3}}+\lambda_{1}+\lambda_{2}+\lambda_{3}\right).$
(34)

We may assume that $(\widetilde{E}\circ
f)({\mbox{\boldmath{$\lambda$}}}_{n},\mathscr{O}_{n})\leq E_{*}+1$ for all
$n$, so ${\mbox{\boldmath{$\lambda$}}}_{n}\in[K^{-1},K]^{3}$ for all $n$,
where $K=E_{*}/\alpha$. Hence, the sequence
$({\mbox{\boldmath{$\lambda$}}}_{n},\mathscr{O}_{n})$ takes values in a
compact subset of $(0,\infty)^{3}\times O(3)$, and so has a convergent
subsequence, converging to
$({\mbox{\boldmath{$\lambda$}}}_{*},\mathscr{O}_{*})$ say, which, without loss
of generality, we may assume is
$({\mbox{\boldmath{$\lambda$}}}_{n},\mathscr{O}_{n})$ itself. So
$(\widetilde{E}\circ
f)({\mbox{\boldmath{$\lambda$}}}_{n},\mathscr{O}_{n})\rightarrow E_{*}$ and
$({\mbox{\boldmath{$\lambda$}}}_{n},\mathscr{O}_{n})\rightarrow({\mbox{\boldmath{$\lambda$}}}_{*},\mathscr{O}_{*})$.
But $(\widetilde{E}\circ f)$ is continuous, so $(\widetilde{E}\circ
f)(\lambda_{*},\mathscr{O}_{*})=E_{*}$.

It follows that $\widetilde{E}(\Sigma_{*})=E_{*}$ where

$\Sigma_{*}=\mathscr{O}_{*}D_{{\mbox{\boldmath{$\lambda$}}}_{*}}\mathscr{O}^{T}_{*},$
(35)

which completes the proof. ∎

We note in passing that the minimizing metric whose existence follows from
Proposition 1 is

$g_{*}=\sigma^{-1}(\Sigma_{*})=\frac{\Sigma_{*}}{\det\Sigma_{*}}.$ (36)

It remains to prove that $\widetilde{E}$ has no other critical points. We
achieve this by proving that $\widetilde{E}$ is strictly convex, in the
following sense:

###### Definition 2

A function $f:M\rightarrow{\mathbb{R}}$ on a Riemannian manifold $M$ is convex
if, for all non-constant geodesics $\gamma(t)$ in $M$,
$(f\circ\gamma)^{\prime\prime}(t)\geq 0$, and strictly convex if, for all such
geodesics, $(f\circ\gamma)^{\prime\prime}(t)>0$.

To apply this definition to $\widetilde{E}$, we must equip
${\mathrm{SPD}}_{3}$ with a Riemannian metric, $G$. The correct choice for our
purposes is

$G_{\Sigma}:T_{\Sigma}{\mathrm{SPD}}_{3}\times
T_{\Sigma}{\mathrm{SPD}}_{3}\rightarrow{\mathbb{R}},\qquad
G_{\Sigma}(\xi_{1},\xi_{2})=\operatorname{Tr}(\Sigma^{-1}\xi_{1}\Sigma^{-1}\xi_{2}),$
(37)

where we have identified $T_{\Sigma}{\mathrm{SPD}}_{3}$ with ${\rm Sym}_{3}$,
the vector space of symmetric $3\times 3$ real matrices. We will exploit
several useful properties of the metric $G$, established in [19]. It is
invariant under the $GL(3,{\mathbb{R}})$ action

$GL(3,{\mathbb{R}})\times{\mathrm{SPD}}_{3}\rightarrow{\mathrm{SPD}}_{3},\qquad(A,\Sigma)\mapsto
A\Sigma A^{T}$ (38)

on ${\mathrm{SPD}}_{3}$. It is also inversion invariant, that is,

$\iota:{\mathrm{SPD}}_{3}\rightarrow{\mathrm{SPD}}_{3},\qquad\iota(\Sigma)=\Sigma^{-1}$
(39)

is an isometry. The general geodesic through ${\mathbb{I}}_{3}$ takes the form

$\gamma(t)=\exp(t\xi),\qquad\xi\in{\rm Sym}_{3},$ (40)

and hence a general nonconstant geodesic through $\Sigma$ is

$\gamma(t)=A\exp(t\xi)A^{T},$ (41)

where $A\in GL(3,{\mathbb{R}})$ satisfies $AA^{T}=\Sigma$, and $\xi\neq 0$.
Finally, it is complete and between any pair of distinct points $\Sigma_{1}$,
$\Sigma_{2}$, there is a geodesic, unique up to parametrization.

###### Proposition 3

The function $\widetilde{E}:{\mathrm{SPD}}_{3}\rightarrow{\mathbb{R}}$ of
equation (28) is strictly convex with respect to the metric $G$.

###### Proof.

Given a constant $M\in{\mathrm{SPD}}_{3}$, consider the function

$f_{M}:{\mathrm{SPD}}_{3}\rightarrow{\mathbb{R}},\qquad
f_{M}(\Sigma)=\operatorname{Tr}(M\Sigma).$ (42)

Let $\gamma$ be an arbitrary non-constant geodesic, as in (41). Then

$\displaystyle(f_{M}\circ\gamma)^{\prime\prime}(0)$ $\displaystyle=$
$\displaystyle\frac{d^{2}\>}{dt^{2}}\bigg{|}_{t=0}\operatorname{Tr}(MA\exp(t\xi)A^{T})=\operatorname{Tr}(MA\xi^{2}A^{T})$
(43) $\displaystyle=$
$\displaystyle\operatorname{Tr}((A\xi)^{T}M(A\xi))=\sum_{i=1}^{3}{\bf
v}_{i}\cdot M{\bf v}_{i}$

where ${\bf v}_{i}$ are the columns of $A\xi$. Since $M$ is positive definite,
it follows that $(f_{M}\circ\gamma)^{\prime\prime}(0)\geq 0$, and equals $0$
only if ${\bf v}_{1}={\bf v}_{2}={\bf v}_{3}=0$. But $\xi\neq 0$ (the geodesic
is nonconstant) so at least one ${\bf v}_{i}\neq 0$. Hence
$(f_{M}\circ\gamma)^{\prime\prime}(0)>0$ for all nonconstant geodesics
$\gamma$. It follows that $(f_{M}\circ\gamma)^{\prime\prime}(T)>0$ for all
nonconstant geodesics and all $T\in{\mathbb{R}}$, since for all geodesics
$\gamma$ and constants $T$, $\widetilde{\gamma}(t)=\gamma(t+T)$ is a geodesic.

Similarly, $\det:{\mathrm{SPD}}_{3}\rightarrow{\mathbb{R}}$ is convex since,
for all nonconstant geodesics

$\displaystyle(\det\circ\gamma)^{\prime\prime}(0)$ $\displaystyle=$
$\displaystyle\frac{d^{2}\>}{dt^{2}}\bigg{|}_{t=0}(\det
A)^{2}\det\exp(t\xi)=(\det
A)^{2}\frac{d^{2}\>}{dt^{2}}\bigg{|}_{t=0}\exp(t\operatorname{Tr}\xi)$ (44)
$\displaystyle=$ $\displaystyle(\det A)^{2}(\operatorname{Tr}\xi)^{2}\geq 0$

It follows that

$\widetilde{E}=f_{H}\circ\iota+f_{F}+C_{0}\det\circ\iota+C_{6}\det$ (45)

is strictly convex, since $H,F\in{\mathrm{SPD}}_{3}$, $\iota$ is an isometry,
and $C_{0},C_{6}\geq 0$. ∎

Propositions 1 and 3 quickly yield the desired result.

###### Corollary 4

Let $\varphi:{\mathbb{T}}^{3}\rightarrow{\mathrm{SU}}(2)$ be a fixed $C^{1}$
map that is immersive somewhere. Then the function
${\mathrm{SPD}}_{3}\rightarrow{\mathbb{R}}$ mapping a flat metric $g$ on
${\mathbb{T}}^{3}$ to the Skyrme energy $E(\varphi,g)$ attains a unique global
minimum, and has no other critical points.

###### Proof.

As previously established $E(\varphi,g)=\widetilde{E}(\sigma(g))$ where
$\widetilde{E}$ is the function defined in (28) and $\sigma$ is a
diffeomorphism of ${\mathrm{SPD}}_{3}$. By Proposition 1, $\widetilde{E}$
attains a minimum at some $\Sigma_{*}\in{\mathrm{SPD}}_{3}$, whence $E$
attains a global minimum at $g_{*}=\sigma^{-1}(\Sigma_{*})$. Assume, towards a
contradiction, that $E$ has a second critical point $g_{**}\neq g_{*}$. Then
$\widetilde{E}$ has a second critical point at
$\Sigma_{**}=\sigma(g_{**})\neq\Sigma_{*}$. Let
$\gamma:[0,1]\rightarrow{\mathrm{SPD}}_{3}$ be a geodesic (with respect to
$G$) with $\gamma(0)=\Sigma_{*}$ and $\gamma(1)=\Sigma_{**}$. Then, by Rolle’s
Theorem applied to
$(\widetilde{E}\circ\gamma)^{\prime}:[0,1]\rightarrow{\mathbb{R}}$, there
exists $t\in(0,1)$ at which $(\widetilde{E}\circ\gamma)^{\prime\prime}(t)=0$.
But this contradicts Proposition 3. ∎

In the case of the standard Skyrme model without a potential ($V=0$ and
$\Xi=0$), we can find the minimizing metric $g_{*}$ explicitly. We note that,
in this case

$\widetilde{E}(\Sigma)=\operatorname{Tr}(H\Sigma^{-1}+F\Sigma),$ (46)

whence

$\mathrm{d}\widetilde{E}_{\Sigma}(\xi)=\operatorname{Tr}(-H\Sigma^{-1}\xi\Sigma^{-1}+F\xi)=\operatorname{Tr}((F-\Sigma^{-1}H\Sigma^{-1})\xi).$
(47)

Hence, the unique critical point $\Sigma_{*}$ of $\widetilde{E}$ satisfies

$\displaystyle F$ $\displaystyle=$
$\displaystyle\Sigma_{*}^{-1}H\Sigma_{*}^{-1}$
$\displaystyle\Rightarrow\quad(F\Sigma_{*})^{2}$ $\displaystyle=$
$\displaystyle FH$ $\displaystyle\Rightarrow\quad\Sigma_{*}$ $\displaystyle=$
$\displaystyle(F^{1/2})^{-1}H^{1/2}$ (48)

where the matrix square root function
${\mathrm{SPD}}_{3}\rightarrow{\mathrm{SPD}}_{3}$, $M\mapsto M^{1/2}$, is
defined spectrally. Then

$g_{*}=\frac{\Sigma_{*}}{\det\Sigma_{*}}=\left(\frac{\det F}{\det
H}\right)^{1/2}F^{-1/2}H^{1/2}.$ (49)

We henceforth set $\Xi=0$. In the case $V\neq 0$, of primary interest, we have
not been able to solve for the minimum of $E(g)$ explicitly. Instead, we
resort to a numerical method described in the next section.

##  4 The numerical method

We now return to the problem of primary interest: to minimize $E(\varphi,g)$
among all smooth maps ${\mathbb{T}}^{3}\rightarrow{\mathrm{SU}}(2)$ of fixed
degree $B$, and all flat metrics $g\in{\mathrm{SPD}}_{3}$. Our numerical
scheme is similar to ones introduced in [16, 17] and based on the idea of
arrested Newton flow. For fixed $\varphi$, we interpret $E$ as a potential
energy on the manifold ${\mathrm{SPD}}_{3}$ and solve Newton’s law of motion

$\ddot{g}=-\operatorname{grad}_{g}E(\varphi,g)$ (50)

with initial data $g(0)=g_{0}$. This solution begins to run “downhill” in
${\mathrm{SPD}}_{3}$. We monitor $E(\varphi,g(t))$ and, at any time $t_{*}$
where $\frac{d}{dt}{E}(\varphi,g(t))>0$ we arrest the flow, that is, stop and
restart it at the current position but with velocity $0$. The flow converges
to the unique minimising metric $g_{\varphi}$. We minimise over $\varphi$ by a
similar technique, solving

$\ddot{\varphi}=-\operatorname{grad}_{\varphi}E(\varphi,g_{\varphi})$ (51)

with initial data $\varphi(0)=\varphi_{0}$ (here
$\operatorname{grad}_{\varphi}E$ is the derivative in the first argument,
$\varphi$, with $g$ treated as constant). Again, we arrest the flow if $E$
starts to increase.

In practice, we discretize space, placing $\varphi$ on a cubic grid of $N^{3}$
points with lattice spacing $h=1/N$ and periodic boundary conditions. We
replace the spatial derivatives of $\varphi$ occurring in $E$ by finite
difference approximations. This reduces the (51) to a system of ODEs in
${\mathbb{R}}^{4N^{3}}$, which we then solve using a 4th order Runge-Kutta
scheme with fixed time step $\delta t$. The arresting criterion is that
$E(t+\delta t)>E(t)$. After each iteration of the Runge-Kutta scheme, the
metric $g_{\varphi}$ is recalculated in a similar way by solving the ODE (50),
using the metric $g_{\varphi}$ from the previous iteration as initial datum.
Once the flow for $g$ has converged, the next iteration of $\varphi$ is
calculated, and so on. Each flow is deemed to have converged to a static
solution if the sup norm of $\operatorname{grad}E$ falls below some tolerance.
The numerical results presented hereafter were obtained with $N=201$. The time
steps $\delta$ used were 0.0017 for the flow in $\varphi$ and 0.1 for the flow
in $g$. The tolerances were $10^{-5}$ for the flow in $\varphi$ and $10^{-7}$
for the flow in $g$.

Implementing the method also entails choosing (formal) Riemannian metrics on
${\mathrm{SPD}}_{3}$ and $C^{\infty}({\mathbb{T}}^{3},SU(2))$. These are
required to make sense of both $\operatorname{grad}E$ and the connexions
$\nabla$ occurring implicitly in (50) and (51). On ${\mathrm{SPD}}_{3}$ we
choose the Euclidean metric induced by identifying ${\mathrm{SPD}}_{3}$ as a
subset of ${\mathbb{R}}^{9}$ in the obvious way, that is

$(\xi_{1},\xi_{2})_{{\mathrm{SPD}}_{3}}=\operatorname{Tr}(\xi_{1}^{T}\xi_{2}).$
(52)

Note that this differs from the metric $G$ used in section 3; it is simpler
for the current purpose. On $C^{\infty}({\mathbb{T}}^{3},SU(2))$ we choose the
$L^{2}$ metric defined by the volume form ${\rm vol}_{0}$,

$\langle\eta_{1},\eta_{2}\rangle_{L^{2}}=\int_{{\mathbb{T}}^{3}}h(\eta_{1},\eta_{2}){\rm
vol}_{0},$ (53)

which is independent of $g$.

For the purposes of numerics, it is convenient to use the sigma model
formulation of the model, that is, exploit the isometry between
$({\mathrm{SU}}(2),h)$ and $S^{3}$ with its round metric of unit radius. So,
we identify

${\mathrm{SU}}(2)\ni\left(\begin{array}[]{cc}\varphi_{0}+i\varphi_{3}&i\varphi_{1}+\varphi_{2}\\\
i\varphi_{1}-\varphi_{2}&\varphi_{0}-i\varphi_{3}\end{array}\right)\leftrightarrow(\varphi_{0},\varphi_{1},\varphi_{2},\varphi_{3})\in
S^{3}$ (54)

and interpret the Skyrme field as a unit length vector in Euclidean
${\mathbb{R}}^{4}$. The terms occurring in the Skyrme energy are easily
converted to this formulation,

$\displaystyle E_{2}(\varphi)$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{T}}^{3}}g^{ij}(\partial_{i}\varphi\cdot\partial_{j}\varphi)\sqrt{|g|}{\rm
vol}_{0}$ (55) $\displaystyle E_{4}(\varphi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{{\mathbb{T}}^{3}}(g^{ij}g^{kl}-g^{il}g^{jk})(\partial_{i}\varphi\cdot\partial_{j}\varphi)(\partial_{k}\varphi\cdot\partial_{l}\varphi)\sqrt{|g|}{\rm
vol}_{0}$ (56) $\displaystyle E_{0}(\varphi)$ $\displaystyle=$ $\displaystyle
2m_{\pi}^{2}\int_{{\mathbb{T}}^{3}}(1-\varphi_{0})\sqrt{|g|}{\rm vol}_{0},$
(57)

as are the matrices defined in section 3,

$\displaystyle H_{ij}$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{T}}^{3}}(\partial_{i}\varphi\cdot\partial_{j}\varphi){\rm
vol}_{0}$ (58) $\displaystyle F_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\varepsilon_{ikl}\varepsilon_{jmn}\int_{{\mathbb{T}}^{3}}\left\\{(\partial_{k}\varphi\cdot\partial_{m}\varphi)(\partial_{l}\varphi\cdot\partial_{n}\varphi)-(\partial_{k}\varphi\cdot\partial_{n}\varphi)(\partial_{l}\varphi\cdot\partial_{m}\varphi)\right\\}{\rm
vol}_{0}.$ (59)

Now, the gradient $E=E_{2}+E_{4}+E_{0}$, regarded as a function on
$C^{\infty}({\mathbb{T}}^{3},S^{3})$, is

$(\operatorname{grad}_{\varphi}E)(\varphi,g)_{C^{\infty}}=\sqrt{|g|}P_{\varphi}\left(-2g^{ij}\partial_{i}\partial_{j}\varphi-2(g^{li}g^{jk}-g^{ij}g^{kl})\partial_{i}(\partial_{j}\varphi\cdot\partial_{k}\varphi\,\partial_{l}\varphi)-2m_{\pi}^{2}{\bf
e}_{0}\right)$ (60)

where ${\bf e}_{0}=(1,0,0,0)$ and
$P_{\varphi}:{\mathbb{R}}^{4}\rightarrow{\mathbb{R}}^{4}$ is the projector
orthogonal to $\varphi$, that is

$P_{\varphi}({\bf v})={\bf v}-(\varphi\cdot{\bf v})\varphi.$ (61)

The gradient of $E$ regarded as a function on ${\mathrm{SPD}}_{3}$ is easily
deduced from (28) and the diffeomorphism
$\sigma:{\mathrm{SPD}}_{3}\rightarrow{\mathrm{SPD}}_{3}$. We note that

$\mathrm{d}\widetilde{E}_{\Sigma}(\xi)=\operatorname{Tr}((F-\Sigma^{-1}H\Sigma^{-1}-\frac{C_{0}\Sigma^{-1}}{\det\Sigma})\xi),$
(62)

and so

$(\operatorname{grad}\widetilde{E})(\Sigma)=F-\Sigma^{-1}H\Sigma^{-1}-\frac{C_{0}\Sigma^{-1}}{\det\Sigma}.$
(63)

Furthermore

$\mathrm{d}\sigma_{g}(\xi)=\frac{1}{\sqrt{\det
g}}\left(\xi-\frac{1}{2}\operatorname{Tr}(g^{-1}\xi)g\right),$ (64)

whence

$\displaystyle\mathrm{d}E_{g}(\xi)$ $\displaystyle=$
$\displaystyle\mathrm{d}\widetilde{E}_{\sigma(g)}\circ\mathrm{d}\sigma_{g}(\xi)$
(65) $\displaystyle=$
$\displaystyle\operatorname{Tr}\left\\{\operatorname{grad}\widetilde{E}(|g|^{-1/2}g)|g|^{-1/2}(\xi-\frac{1}{2}\operatorname{Tr}(g^{-1}\xi)g)\right\\}$
$\displaystyle=$
$\displaystyle\operatorname{Tr}\left\\{|g|^{-1/2}\left[\operatorname{grad}\widetilde{E}(|g|^{-1/2}g)-\frac{1}{2}\operatorname{Tr}(g\operatorname{grad}\widetilde{E}(|g|^{-1/2}g))g^{-1}\right]\xi\right\\},$

and so

$(\operatorname{grad}_{g}E)(\varphi,g)_{{\mathrm{SPD}}_{3}}=|g|^{-1/2}\left[\operatorname{grad}\widetilde{E}(|g|^{-1/2}g)-\frac{1}{2}\operatorname{Tr}(g\operatorname{grad}\widetilde{E}(|g|^{-1/2}g))g^{-1}\right].$
(66)

##  5 Skyrme crystal solutions

This section presents the results of the numerical scheme just described in
the charge $B=4$ sector, concentrating on the model with pion mass
$m_{\pi}=1$. Our approach is to treat the pion mass as a continuous variable
parameter $m_{\pi}=t\geq 0$: we minimize the energy

$E_{(t)}(\varphi,g)=E_{2}(\varphi,g)+E_{4}(\varphi,g)+t\int_{{\mathbb{T}}^{3}}(1-\varphi_{0}){\rm
vol}_{g}$ (67)

starting in the massless case $t=0$, and then increasing $t$ gradually to $1$.

We begin by reviewing the lowest energy solution known in the massless case,
$t=0$, found independently by Kugler and Shtrikman [8, 20] and Castillejo et
al. [9]. This can be found by starting with the initial field [9]

$\varphi_{0}=-c_{1}c_{2}c_{3},\quad\varphi_{1}=s_{1}\sqrt{1-\frac{s_{2}^{2}}{2}-\frac{s_{3}^{2}}{2}+\frac{s_{2}^{2}s_{3}^{2}}{3}},$
(68)

$\varphi_{2},\varphi_{3}$ obtained by cyclic permutation, where $s_{i}=\sin
2\pi x_{i}$ and $c_{i}=\cos 2\pi x_{i}$, and initial metric
$g={\mathbb{I}}_{3}$. This quickly converges under arrested Newton flow to a
solution with $\varphi$ close to (68) and $g=L^{2}{\mathbb{I}}_{3}$ with
$L=4.61$, corresponding to a cubic period lattice of side length $L$. This
solution is depicted in figure 1. It represents a simple cubic lattice of
half-skyrmions. That is, $\varphi$ maps each of the eight subcubes of side
length $L/2$ to either the upper ($\varphi_{0}\geq 0$) or lower
($\varphi_{0}\leq 0$) hemisphere of $S^{3}$, contributing charge $B=1/2$ to
the total topological charge of the unit cell. For this reason, we refer to
this solution as the $1/2$-crystal and denote it $(\varphi_{1/2},g_{1/2})$.

(a)

(b)

Figure 1: The $1/2$-crystal solution of the massless Skyrme model. The baryon
density is depicted in 1(a) and isosurface plots of the $\varphi_{0}$ field,
where
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\varphi_{0}=0.9}$
and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi_{0}=-0.9}$,
are shown in 1(b).

It is important to note that $E_{(0)}=E_{2}+E_{4}$ is invariant under the
natural action of ${\mathrm{SO}}(4)$ on the target three-sphere, that is, for
all $(\varphi,g)\in\mathscr{M}$ and all $R\in{\mathrm{SO}}(4)$,
$E_{(0)}(R\varphi,g)=E_{(0)}(\varphi,g)$. Hence the solution described above
is just one critical point of $E_{(0)}$ lying in a $6$-dimensional family of
critical points, its orbit under ${\mathrm{SO}}(4)$. If we now switch on the
pion mass, that is, consider $E_{(t)}$ for small $t>0$, we may ask which (if
any) of these critical points survive the perturbation. It is useful to switch
perspective slightly: rather than fixing the perturbation and considering what
happens to all points in the ${\mathrm{SO}}(4)$ orbit of the $1/2$-crystal, it
is convenient to fix the field and metric to be the $1/2$-crystal, and
consider what happens to this fixed configuration under the ${\mathrm{SO}}(4)$
orbit of the perturbation. That is, we ask for which $p\in S^{3}$, if any,
does the $1/2$-crystal $(\varphi_{1/2},g_{1/2})$ lie in a curve
$(\varphi(t),g(t))$ of critical points of the $t$-parametrized family of
functions

$E_{(t)}^{p}(\varphi,g)=E_{2}(\varphi,g)+E_{4}(\varphi,g)+t\int_{{\mathbb{T}}^{3}}(1-p\cdot\varphi){\rm
vol}_{g}.$ (69)

(We recover the original function $E_{(t)}$ by choosing $p=(1,0,0,0)$.) To
answer this question, we will need to understand the symmetries of the
$1/2$-crystal in some detail.

Denote by $O$ the subgroup of ${\mathrm{SO}}(3)$ consisting of matrices that
map the integer lattice ${\mathbb{Z}}^{3}$ to itself. This is a finite group
of order $24$, the rotational symmetries of the cube. The manifold
${\mathbb{T}}^{3}$ is itself an abelian Lie group whose group operation is
translation modulo ${\mathbb{Z}}^{3}$. The semidirect product
$\operatorname{Aut}({\mathbb{T}}^{3})=O\ltimes{\mathbb{T}}^{3}$

$(A_{1},{\bf v}_{1})\bullet(A_{2},{\bf v}_{2})=(A_{1}A_{2},A_{1}{\bf
v}_{2}+{\bf v}_{1})$ (70)

acts on ${\mathbb{T}}^{3}$ by $(A,{\bf v}):{\bf x}\mapsto A{\bf x}+{\bf v}$.
This induces a right action of $\operatorname{Aut}({\mathbb{T}}^{3})$ on
$\mathscr{M}=C^{\infty}({\mathbb{T}}^{3},S^{3})\times{\mathrm{SPD}}_{3}$ by
$S:(\varphi,g)\mapsto(\varphi\circ S,S^{*}g)$ or, more explicitly,

$(A,{\bf v})\cdot(\varphi({\bf x}),g)=(\varphi(A({\bf x}+{\bf v})),A^{T}gA).$
(71)

Having completed these preliminaries, we observe that the energy of the
massless Skyrme model $E_{(0)}:\mathscr{M}\rightarrow{\mathbb{R}}$ is
invariant under the left action of
$G={\mathrm{SO}}(4)\times\operatorname{Aut}({\mathbb{T}}^{3})$ on
$\mathscr{M}$,

$(R,S)\cdot(\varphi,g)=(R\circ\varphi\circ S^{-1},(S^{-1})^{*}g).$ (72)

The $1/2$-crystal $(\varphi_{1/2},g_{1/2})$ is a critical point (in fact a
minimum) of $E_{(0)}$. Its stabilizer $\Gamma$ (the subgroup of
${\mathrm{SO}}(4)\times\operatorname{Aut}({\mathbb{T}}^{3})$ that leaves it
fixed) is an order $192$ group generated by

$\begin{array}[]{ll}R(\varphi)=(\varphi_{0},\varphi_{2},\varphi_{3},\varphi_{1}),&S({\bf
x})=(x_{2},x_{3},x_{1}),\\\
R(\varphi)=(\varphi_{0},\varphi_{2},-\varphi_{1},\varphi_{3}),&S({\bf
x})=(x_{2},-x_{1},x_{3}),\\\
R(\varphi)=(-\varphi_{0},-\varphi_{1},\varphi_{2},\varphi_{3}),&S({\bf
x})=(x_{1}+\frac{1}{2},x_{2},x_{3}).\end{array}$ (73)

The image of the natural projection $\pi:\Gamma\to SO(4)$ is naturally
isomorphic to the octahedral group $O_{h}$, and the kernel is isomorphic to
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$.

Once we turn on the perturbation, the symmetry group of the energy function
$E_{(t)}^{p}$ is broken to
${\mathrm{SO}}(3)_{p}\times\operatorname{Aut}({\mathbb{T}}^{3})$ where

${\mathrm{SO}}(3)_{p}=\\{R\in{\mathrm{SO}}(4):Rp=p\\}.$ (74)

Let us define the reduced stabilizer of the $1/2$-crystal to be

$\Gamma_{p}=\Gamma\cap({\mathrm{SO}}(3)_{p}\times\operatorname{Aut}({\mathbb{T}}^{3})),$
(75)

and the set of fixed points of $\Gamma_{p}$ in $\mathscr{M}$ to be

$\mathscr{M}^{\Gamma_{p}}=\\{(\varphi,g)\in\mathscr{M}:\forall
q\in\Gamma_{p},\>q\cdot(\varphi,g)=(\varphi,g)\\}.$ (76)

Formally, this is a submanifold of $\mathscr{M}$, and it contains
$(\varphi_{1/2},g_{1/2})$ for all $p$, by construction. By the Principle of
Symmetric criticality, a point $(\varphi,g)\in\mathscr{M}^{\Gamma_{p}}$ is a
critical point of $E_{(t)}^{p}$ if (and only if) it is a critical point of its
restriction $E_{(t)}^{p}|:\mathscr{M}^{\Gamma_{p}}\rightarrow{\mathbb{R}}$.
For generic choices of $p\in S^{3}$ we expect $\Gamma_{p}$ to be trivial, so
that $\mathscr{M}^{\Gamma_{p}}=\mathscr{M}$, and this observation confers no
advantage. The interesting case is when the intersection of
$\mathscr{M}^{\Gamma_{p}}$ with the $G$ orbit of $(\varphi_{1/2},g_{1/2})$ is
(locally) just $(\varphi_{1/2},g_{1/2})$. Then $(\varphi_{1/2},g_{1/2})$ is an
isolated critical point of
$E_{(0)}|:\mathscr{M}^{\Gamma_{p}}\rightarrow{\mathbb{R}}$. If, as seems
likely, it is also a nondegenerate critical point of $E_{(0)}^{p}|$ (meaning
that the Hessian of $E_{(0)}^{p}|$ at $(\varphi_{1/2},g_{1/2})$ is
nondegenerate), then the persistence of a critical point for $t>0$
sufficiently small follows from the Inverse Function Theorem applied to
$\mathrm{d}E_{(t)}^{p}|$. That is, there exists $\varepsilon>0$ and a (unique)
smooth curve
$\gamma:(-\varepsilon,\varepsilon)\rightarrow\mathscr{M}^{\Gamma_{p}}$ such
that $\gamma(0)=(\varphi_{1/2},g_{1/2})$ and
$\mathrm{d}E_{(t)}^{p}|_{\gamma(t)}=0$ for all
$t\in(-\varepsilon,\varepsilon)$.

To summarize, we expect $(\varphi_{1/2},g_{1/2})$ to smoothly deform into a
critical point of $E_{(t)}^{p}$ (as $t$ increases from $0$) if $p$ is chosen
so that a neighbourhood of $(\varphi_{1/2},g_{1/2})$ in
$\mathscr{M}^{\Gamma_{p}}$ intersects the $G$ orbit of
$(\varphi_{1/2},g_{1/2})$ only at $(\varphi_{1/2},g_{1/2})$. Let us call this
condition the isolation condition. The next task is to understand this
condition on $p$ at an algebraic level.

Assume that $p$ fails the isolation condition. Then there exists a regular
curve $q:(-\varepsilon,\varepsilon)\rightarrow G$ with $q(0)=e$ such that, for
all $t$, $q(t)\cdot(\varphi_{1/2},g_{1/2})\in\mathscr{M}^{\Gamma_{p}}$ or,
more explicitly, for all $Q\in\Gamma_{p}$, and
$t\in(-\varepsilon,\varepsilon)$

$\displaystyle Q\cdot q(t)\cdot(\varphi_{1/2},g_{1/2})$ $\displaystyle=$
$\displaystyle q(t)\cdot(\varphi_{1/2},g_{1/2})$
$\displaystyle\Rightarrow\quad[q(t)^{-1}Qq(t)]\cdot(\varphi_{1/2},g_{1/2})$
$\displaystyle=$ $\displaystyle(\varphi_{1/2},g_{1/2}).$ (77)

Hence, for all $Q\in\Gamma_{p}$ and $t$, $q(t)^{-1}Qq(t)\in\Gamma$. But
$\Gamma$ is discrete (in fact, finite), so for all $t$ and $Q$,

$\displaystyle q(t)^{-1}Qq(t)$ $\displaystyle=$ $\displaystyle
q(0)^{-1}Qq(0)=Q$ (78) $\displaystyle\Rightarrow\quad Qq(t)Q^{-1}$
$\displaystyle=$ $\displaystyle q(t).$ (79)

The derivative of this equation at $t=0$ implies that there exists some
nonzero $\xi\in\mathfrak{g}$ (the Lie algebra of $G$), namely
$\xi=\dot{q}(0)$, such that $Ad_{Q}\xi=\xi$. Conversely, given a nonzero
$\xi\in\mathfrak{g}$ such that $Ad_{Q}\xi=\xi$ for all $Q\in\Gamma_{p}$, we
can construct a curve $\gamma(t)=\exp(t\xi)$ such that
$\gamma(t)\cdot(\varphi_{1/2},g_{1/2})$ remains in $\mathscr{M}^{\Gamma_{p}}$.
Hence, the isolation condition is that, for all
$\xi\in\mathfrak{g}\backslash\\{0\\}$, there exists some $Q\in\Gamma_{p}$ such
that $Ad_{Q}\xi\neq\xi$. More succinctly: $p$ satisfies the isolation
condition if and only if the adjoint representation of $\Gamma_{p}$ on
$\mathfrak{g}$ contains no copies of the trivial representation.

This reduces the problem to one in the representation theory of subgroups of
$O_{h}$. Given a subgroup $H$ of $O_{h}\subset{\mathrm{SO}}(4)$, we determine
whether its action on $\mathbb{R}^{4}$ contains copies of the trivial
representation. If not, it cannot arise as $\pi(\Gamma_{p})$ for any choice of
$p$. If it does, $\pi^{-1}(H)$ is a candidate for $\Gamma_{p}$ for any $p$ in
a one-dimensional invariant subspace of the action. This produces a short list
of candidate subgroups. For each of these we count copies of the trivial
representation in the adjoint representation of $\pi^{-1}(H)$ on
$\mathfrak{g}$. If there are none, this is a candidate for $\Gamma_{p}$ for
$p$ satisfying the isolation condition.

The results are summarized in table 1. We find 28 points $p$ for which
$(\varphi_{1/2},g_{1/2})$ is an isolated critical point of $E_{(0)}^{p}$ in
$\mathscr{M}^{\Gamma_{p}}$, falling into 4 distinct classes. One class is
$p\in\\{(1,0,0,0),(-1,0,0,0)\\}$. The other three classes all have $p_{0}=0$
and hence $(p_{1},p_{2},p_{3})\in S^{2}\subset{\mathbb{R}}^{3}$, pointing
along some symmetry line of the unit cube: towards the centre of a face (e.g.
$p=(0,0,0,1)$), the centre of an edge (e.g. $p=(0,1,1,0)/\sqrt{2}$) or a
vertex (e.g. $p=(0,1,1,1)/\sqrt{3}$).

$p$ | $|\Gamma_{p}|$ | $\pi(\Gamma_{p})$ | Description as subgroup of $O_{h}$ | $g$ | label
---|---|---|---|---|---
$(1,0,0,0)$ | 96 | $O$ | Orientation preserving | $\operatorname{diag}(a,a,a)$ |  $1/2$-crystal
$(0,0,0,1)$ | 32 | $C_{4v}$ | Maps a face to itself | $\operatorname{diag}(a,a,b)$ | sheet
$(0,0,1,1)/\sqrt{2}$ | 16 | $C_{2v}$ | Maps an edge to itself | $\operatorname{diag}(a,b,b)$ | chain
$(0,1,1,1)/\sqrt{3}$ | 24 | $C_{3v}$ | Maps a vertex to itself | $\operatorname{diag}(a,a,a)$ |  $\alpha$-crystal
Table 1: Points $p\in S^{3}$ for which the $1/2$-crystal is isolated in
$\mathscr{M}^{\Gamma_{p}}$, and hence is expected to continue to a critical
point of the massive Skyrme model. The leftmost column gives one
representative point in each class. Subsequent columns record the order of the
corresponding stabilizer $\Gamma_{p}\subset\Gamma$, the image of $\Gamma_{p}$
in $\pi(\Gamma)=O_{h}$, its description as a subgroup of the group of
symmetries of the cube, the most general metric consistent with the symmetry,
and a descriptive label of the corresponding crystal.

To do numerics, we switch back to the viewpoint of internally rotating the
field $\varphi_{1/2}$, rather than the energy functional, that is, we set
$p=(1,0,0,0)$ in $E_{(t)}^{p}$ and start with the configuration

$\varphi=Q\varphi_{1/2},\qquad g=g_{1/2},$ (80)

where $Q$ is any ${\mathrm{SO}}(4)$ matrix whose top row is $p$ (the inverse
of an ${\mathrm{SO}}(4)$ matrix mapping $(1,0,0,0)\mapsto p$). We then
minimize $E_{(t)}$ using arrested Newton flow for a sequence of pion masses
$m_{\pi}=t$ starting at $t=0$ and ending at $t=1$. As expected each of the 4
types of critical point smoothly continues. Somewhat unexpectedly, they are
all, as far as we can determine, local minima of $E_{(t)}$; none are saddle
points. We have checked this by perturbing the solutions with random
perturbations breaking all symmetries, finding that they always relax back to
the solutions presented.

(a) $1/2$-crystal

(b) $\alpha$-crystal

(c) sheet-crystal

(d) chain-crystal

Figure 2: Skyrme crystals in the model with pion mass $m_{\pi}=1$. The top row
is the isosurface plots of the baryon density. The bottom row is isosurface
plots of the $\varphi_{0}$ field, where
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\varphi_{0}=0.9}$
and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varphi_{0}=-0.9}$.

The solutions at pion mass $m_{\pi}=1$ are depicted in figure 2, labelled as
in the final column of table 1. Ordered by energy, we find sheet $<$ chain $<$
$\alpha$-crystal $<$ $1/2$-crystal, though the chain and $\alpha$-crystals are
so close in energy that their order is somewhat uncertain. The energies per
baryon per unit cell are

$\displaystyle\frac{E_{1/2}}{B}$ $\displaystyle=1.2417\times
12\pi^{2}=147.058,$ (81) $\displaystyle\frac{E_{\alpha}}{B}$
$\displaystyle=1.2368\times 12\pi^{2}=146.479$
$\displaystyle\frac{E_{\textrm{chain}}}{B}$ $\displaystyle=1.2368\times
12\pi^{2}=146.479$ $\displaystyle\frac{E_{\textrm{sheet}}}{B}$
$\displaystyle=1.2365\times 12\pi^{2}=146.451.$

Neither the sheet-crystal nor the chain-crystal has an isotropic metric,
meaning these crystals do not have a cubic period lattice. The
$\alpha$-crystal and the $1/2$-crystal do have cubic period lattices, as is
consistent with our symmetry analysis (see column 5 of table 1). The minimal
metrics are

$\displaystyle g_{1/2}$ $\displaystyle=L^{2}{\mathbb{I}}_{3},$ $\displaystyle
L$ $\displaystyle=3.202,$ (82) $\displaystyle g_{\alpha}$
$\displaystyle=L^{2}{\mathbb{I}}_{3},$ $\displaystyle L$
$\displaystyle=3.278,$ $\displaystyle g_{\textrm{chain}}$
$\displaystyle=\operatorname{diag}(L_{1}^{2},L_{2}^{2},L_{2}^{2}),$
$\displaystyle L_{1}$ $\displaystyle=3.221,\quad L_{2}=3.312,$ $\displaystyle
g_{\textrm{sheet}}$
$\displaystyle=\operatorname{diag}(L_{1}^{2},L_{1}^{2},L_{2}^{2}),$
$\displaystyle L_{1}$ $\displaystyle=3.222,\quad L_{2}=3.442$

from which we deduce that the unit cells for the sheet and chain crystals are
trigonal (cuboidal with one pair of periods equal), but with opposite types of
distortion: the sheet’s unit cell is a stretched cube, the chain’s a squashed
cube. Interestingly, the ordering of the volumes of the solutions’ unit cells
is the reverse of the ordering of their energies, with the sheet-crystal
occupying the greatest volume and the 1/2-crystal the least.

Restricting the kinetic energy functional of the model to the isospin orbit of
a given static solution we obtain a left invariant metric on
${\mathrm{SO}}(3)$ called the isospin inertia tensor, which is of some
significance in the method of rigid body quantization [21, 22]. The kinetic
energy associated with the potential energy $E_{2}+E_{4}+E_{0}$ given in
(55)-(57) is

$T(\varphi,\dot{\varphi})=\int_{{\mathbb{T}}^{3}}\big{[}\dot{\varphi}\cdot\dot{\varphi}+g^{ij}\left\\{(\dot{\varphi}\cdot\dot{\varphi})\,(\partial_{i}\varphi\cdot\partial_{j}\varphi)-(\dot{\varphi}\cdot\partial_{i}\varphi)\,(\dot{\varphi}\cdot\partial_{j}\varphi)\right\\}\big{]}\sqrt{|g|}{\rm
vol}_{0}.$ (83)

Writing $\dot{\varphi}=X^{i}J_{i}\varphi$, with $J_{i}$ being the basis for
$\mathfrak{so}(3)$ given by

$J_{1}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&-1\\\
0&0&1&0\end{pmatrix},\quad J_{2}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&1\\\
0&0&0&0\\\ 0&-1&0&0\end{pmatrix},\quad J_{3}=\begin{pmatrix}0&0&0&0\\\
0&0&-1&0\\\ 0&1&0&0\\\ 0&0&0&0\end{pmatrix},$ (84)

we find that $T(\varphi,\dot{\varphi})=\frac{1}{2}X^{i}U_{ij}X^{j}$, where $U$
is the symmetric matrix given by

$U_{ij}=2\int_{{\mathbb{T}}^{3}}\big{[}\delta_{ij}\varphi_{k}\varphi_{k}-\varphi_{i}\varphi_{j}+g^{kl}(\delta_{ij}-\varphi_{i}\varphi_{j})\partial_{k}\varphi_{0}\partial_{l}\varphi_{0}\\\
+g^{kl}(\varphi_{m}\varphi_{m}\partial_{k}\varphi_{i}\partial_{l}\varphi_{j}+\varphi_{0}\varphi_{j}\partial_{k}\varphi_{0}\partial_{l}\varphi_{i}+\varphi_{0}\varphi_{i}\partial_{l}\varphi_{0}\partial_{k}\varphi_{j})\big{]}\sqrt{|g|}{\rm
vol}_{0}$ (85)

and repeated indices are summed from 1 to 3. We find that, except for the
$1/2$-crystal, this matrix is not isotropic:

$\displaystyle U^{1/2}=\begin{pmatrix}165.2&0&0\\\ 0&165.2&0\\\
0&0&165.2\end{pmatrix},$ (86) $\displaystyle
U^{\alpha}=\begin{pmatrix}135.5&0&0\\\ 0&135.5&0\\\ 0&0&167.3\end{pmatrix}$
$\displaystyle U^{\textrm{chain}}=\begin{pmatrix}135.6&0&0\\\ 0&135.7&0\\\
0&0&167.2\end{pmatrix},$ $\displaystyle
U^{\textrm{sheet}}=\begin{pmatrix}135.8&0&0\\\ 0&135.8&0\\\
0&0&166.8\end{pmatrix}.$

As far as we are aware, in addition to the $1/2$-crystal, only the
$\alpha$-crystal has been previously determined in the massive Skyrme model
[13]. Neither of these is the minimal energy crystal.

It is interesting to track the energy as a function of pion mass, see figure
3. As $m_{\pi}$ increases, all of the crystals’ energies increase relative to
that of the one-skyrmion. This is an indication that classical binding
energies will be small (and hence close to experimental values) when $m_{\pi}$
is large. Amongst the various crystal solutions, we find that the sheet, chain
and $\alpha$-crystals remain close in energy, with stable order, but the gap
to the $1/2$-crystal (which always has highest energy) grows with $m_{\pi}$.

Figure 3: Comparison of the normalized energies per baryon per unit cell of
the four Skyrme crystals for increasing pion mass $m_{\pi}$. Energies are
presented in units of the energy of the $B=1$ skyrmion at the relevant pion
mass (which grows monotonically with $m_{\pi}$).

##  6 Skyrme crystals at prescribed average baryon density

If we are to use Skyrme crystals as a model of dense nuclear matter (for
example, in astrophysical contexts) it is important to understand the
properties of the lowest energy configuration among all those with a fixed
average baryon density, treating this density as a parameter of our system.
This problem was first approached by Hen and Karliner [23] in the context of
the baby Skyrme model. Therein they extremized the baby Skyrme energy
functional with respect to variations of the period lattice at a constant
skyrmion density. This method was carried out at various densities, producing
an energy-density curve. However, they did not address the nature of the
critical points they obtained, stating that they could in fact turn out to be
maxima or saddle points. Our method is similar but it is more general and
robust.

Let us fix $B$, the baryon number per unit cell. Then the average baryon
density of a configuration $(\varphi,g)$ is

$\rho_{B}=\frac{B}{\int_{{\mathbb{T}}^{3}}{\rm vol}_{g}}=\frac{B}{\sqrt{\det
g}}.$ (87)

Hence, finding the minimal crystal with fixed baryon density (and baryon
number $B$ per unit cell) amounts to minimizing
$E:\mathscr{M}\rightarrow{\mathbb{R}}$ over a level set of $\det g$. Once
again, we can address the partial minimization problem where we fix the field
$\varphi:{\mathbb{T}}^{3}\rightarrow SU(2)$ (assumed to be $C^{1}$ and
somewhere immersive) and a density $\rho_{B}=B/\nu$ then seek a minimum of
$E(\varphi,\cdot):\det^{-1}(\nu^{2})\rightarrow{\mathbb{R}}$. It turns out
that, like the unconstrained minimization problem studied in section 3, this
problem has a unique global minimum and no other critical points.

###### Proposition 5

Let $\varphi:{\mathbb{T}}^{3}\rightarrow{\mathrm{SU}}(2)$ be a fixed $C^{1}$
map that is immersive somewhere and $\nu>0$ be a constant. Then the function
${\mathrm{SPD}}_{3}\supset\det^{-1}(\nu^{2})\rightarrow{\mathbb{R}}$ mapping
each flat metric $g$ on ${\mathbb{T}}^{3}$ of volume $\nu$ to the Skyrme
energy $E(\varphi,g)$ attains a unique global minimum, and has no other
critical points.

###### Proof.

As before, it is equivalent to prove that the associated function

$\widetilde{E}:{\det}^{-1}(\nu^{-1})\rightarrow{\mathbb{R}},\quad\widetilde{E}=E\circ\sigma^{-1}$
(88)

attains a unique global minimum and has no other critical points, where
$\sigma:{\mathrm{SPD}}_{3}\rightarrow{\mathrm{SPD}}_{3}$ is the diffeomorphism
$\sigma(g)=g/\sqrt{\det g}$. Now

$\widetilde{E}(\Sigma)=\operatorname{Tr}(H\Sigma^{-1})+\operatorname{Tr}(F\Sigma)+C_{0}\nu+C_{6}\nu^{-1}$
(89)

where $H,F\in{\mathrm{SPD}}_{3}$ and $C_{0},C_{6}\in[0,\infty)$ are the
$\varphi$-dependent constants previously defined. Existence of a global
minimum of $\widetilde{E}$ follows mutatis mutandis from Proposition 1, since
the bound (34) still holds irrespective of the extra constraint
$\lambda_{1}\lambda_{2}\lambda_{3}=\nu^{-1}$ (equivalent to
$\det\Sigma=\nu^{-1}$). This confines the minimizing sequence to a compact
subset of the hypersurface $\lambda_{1}\lambda_{2}\lambda_{3}=\nu^{-1}$ in
$(0,\infty)^{3}\times O(3)$, whence a convergent subsequence can be extracted,
whose limit attains the infimum of $\widetilde{E}$ by continuity.

It remains to prove uniqueness. Assume towards a contradiction that
$\widetilde{E}:{\det}^{-1}(\nu^{-1})\rightarrow{\mathbb{R}}$ has two distinct
critical points $\Sigma_{*}$, $\Sigma_{**}$. Then there exists a geodesic

$\gamma(t)=A\exp(\xi t)A^{T}$ (90)

in $({\mathrm{SPD}}_{3},G)$ with $\gamma(0)=\Sigma_{*}$ and
$\gamma(1)=\Sigma_{**}$. Now

$\det(\gamma(t))=(\det A)^{2}e^{t\operatorname{Tr}\xi}$ (91)

and $\det(\gamma(0))=\det(\gamma(1))$, so $\operatorname{Tr}\xi=0$ and we
conclude that $\det(\gamma(t))$ is constant. Hence the geodesic $\gamma$
remains on the level set ${\det}^{-1}(\nu^{-1})$. Since $\Sigma_{*}$,
$\Sigma_{**}$ are critical points of
$\widetilde{E}:{\det}^{-1}(\nu^{-1})\rightarrow{\mathbb{R}}$, and $\gamma$ is
tangent to the level set for all $t$,
$(\widetilde{E}\circ\gamma)^{\prime}(0)=0=(\widetilde{E}\circ\gamma)^{\prime}(1)$,
so by Rolle’s Theorem $(\widetilde{E}\circ\gamma)^{\prime\prime}$ vanishes
somewhere on $(0,1)$, contradicting the convexity of $\widetilde{E}$
(Proposition 3). Hence no second critical point may exist. ∎

In the course of the proof above we established that all level sets of $\det$
are connected totally geodesic submanifolds of $({\mathrm{SPD}}_{3},G)$, and
hence the restriction of $\widetilde{E}$ to any such level set is strictly
convex. Note that, in general, the restriction of a convex function to a
submanifold may fail to be convex, so total geodesicity of the level sets is
crucial here.

We can again solve the minimization problem for
$E:{\det}^{-1}(\nu^{2})\rightarrow{\mathbb{R}}$ numerically by arrested Newton
flow, but now we must take care to project the gradient of $E$ tangent to the
level set. Given a curve $g(t)$ in $\det^{-1}(\nu^{2})$,

$\frac{d\>}{dt}\bigg{|}_{t=0}\det
g(t)=\nu^{2}\operatorname{Tr}(g(0)^{-1}\dot{g}(0))=0$ (92)

so $\dot{g}(0)$ is orthogonal to $g(0)^{-1}$ with respect to the Euclidean
metric $\langle X,Y\rangle=\operatorname{Tr}(X^{T}Y)$. Hence
$T_{g}\det^{-1}(\nu^{2})=\langle g^{-1}\rangle^{\perp}$. Now

$E(g(t))=\nu\operatorname{Tr}(Hg^{-1})+\frac{\operatorname{Tr}(Fg)}{\nu}+C_{0}\nu+\frac{C_{6}}{\nu},$
(93)

and hence

$\mathrm{d}E_{g}(v)=\frac{1}{\nu}\langle F-\nu^{2}g^{-1}Hg^{-1},v\rangle.$
(94)

It follows that, with respect to the metric on $\det^{-1}(\nu^{2})$ induced by
the Euclidean metric,

$(\operatorname{grad}E)(g)=\frac{1}{\nu}\left\\{F-\nu^{2}g^{-1}Hg^{-1}-\frac{\operatorname{Tr}((F-\nu^{2}g^{-1}Hg^{-1})g^{-1})}{\operatorname{Tr}(g^{-1}g^{-1})}g^{-1}\right\\}.$
(95)

We solve the Newton flow $\ddot{g}=-(\operatorname{grad}E)(g)$ numerically,
projecting $g$ back onto $\det^{-1}(\nu^{2})$ after each time step by radial
dilation ($g\mapsto(\nu^{2}/\det g)^{1/3}g$), arresting if $E(g(t+\delta
t))>E(g(t))$, and terminating if the sup norm of $\operatorname{grad}E$ falls
below a prescribed tolerance. As for the unconstrained problem, we apply this
algorithm after each iteration of the arrested Newton flow for the field
$\varphi:{\mathbb{T}}^{3}\rightarrow S^{3}$.

Figure 4: The energy per baryon per unit cell of the Skyrme crystals in the
model with pion mass $m_{\pi}=1$ as a function of cell volume.

Applying this approach at various densities to the four crystals found in
section 5 at pion mass $m_{\pi}=1$, we observe that the three lower energy
crystals tend to finite-energy solutions at low densities. The
$\alpha$-crystal tends to the $B=4$ $\alpha$-particle skyrmion on
$\mathbb{R}^{3}$ [24], see figure 4. This phase transition has already been
observed by Silva Lobo [25] in the massless model and by Adam et al. [18] in
the massive model with sextic term. The sheet-crystal tends to a double-
layered square sheet on $\mathbb{T}^{2}\times\mathbb{R}$, similar to the
$2$-wall massless solution found by Silva Lobo and Ward [26]. Finally, the
chain-crystal becomes a linear chain on $\mathbb{R}^{2}\times S^{1}$, which
appears to be a previously unknown solution.

At low densities the sheet solution is clearly energetically preferred over
other solutions. This qualitative result was predicted earlier in [27].
However, there are some important differences between between our result and
[27]. Our result comes from a minimisation over all Skyrme fields and lattice
geometries, whereas [27] used the more restrictive Atiyah-Manton approximation
for the Skyrme field and assumed symmetric lattice geometries. Second, our
minimal-energy Skyrme sheet has a square geometry, whereas those constructed
in [27] had a hexagonal geometry. Finally, our results are for the model with
$m_{\pi}=1$, whereas [27] considered $m_{\pi}=0$.

As one might expect, the four crystals become energetically indistinguishable
in the large density limit. As far as we can determine, the curves in figure 4
never cross, so the crystals maintain their energy ordering at all densities.

##  7 Conclusion

In this paper, we developed methods to obtain Skyrme crystals in a general
class of Skyrme models, and presented a detailed numerical study of crystals
in the standard Skyrme model with massive pions. To achieve this, we minimized
the model’s energy with respect to variations of both the field and its period
lattice in ${\mathbb{R}}^{3}$. A key idea is to reformulate the latter
variation as a variation over all flat metrics on the fixed unit torus
${\mathbb{T}}^{3}$. We obtained strong results on the partial minimization
problem in which the field is fixed and only the metric varied: under a mild
nondegeneracy assumption on the field, there exists a unique flat metric that
globally minimizes the Skyrme energy, and no other critical metrics. This
result holds also if we constrain the problem to vary only over metrics of
fixed volume, a variant relevant to constructing Skyrme crystals of prescribed
average baryon density. Our methods impose no symmetry on the period lattice a
priori, and hence go beyond previous studies which imposed a cubic unit cell.

We find that the minimal energy crystal (with baryon number $4$ per unit cell)
has trigonal but not cubic periodicity. At low densities it tends to a double
sheet solution. The next lowest energy crystal is also trigonal and not cubic,
tending to a chain solution at low densities. Both these crystals are new.
Above them in energy are two already known solutions, the $\alpha$-crystal and
the $1/2$-crystal. All these crystals, except the most energetic, the
$1/2$-crystal, have anisotropic isospin inertia tensors. The existence of four
distinct crystals can be understood semi-analytically by means of the
Principle of Symmetric Criticality and the Inverse Function Theorem.

The methods detailed in this paper could be applied to the study of isospin
asymmetric nuclear matter within the Skyrme model. The next step would be to
investigate neutron crystals by considering the quantum corrections to the
energy due to the quantization of the isospin degrees of freedom, and improve
on the work done on the massless model by Baskerville [21]. In particular, one
could determine “nuclear pasta” phases in neutron stars [28] by considering
the quantization of generalized Skyrme crystals in the low density regime. The
chain-crystal we have found could correspond to the so-called “spaghetti”
phase, and the sheet-crystal the “nuclear lasagne”.

## Acknowledgments

We would like to thank the solitons@work community for discussions following
the Solitons (non)Integrability Geometry X (SIG X) and Geometric Models of
Nuclear Matter (GMNM) conferences in June/July 2022, which motivated part of
this paper. P.N.L. is supported by a PhD studentship from UKRI, Grant No.
EP/V520081/1.

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