# Metal-insulator transition and local-moment collapse in negative charge-
transfer CaFeO3 under pressure

I. Leonov  M. N. Miheev Institute of Metal Physics, Russian Academy of
Sciencee s, 620108 Yekaterinburg, Russia  Ural Federal University, 620002
Yekaterinburg, Russia

###### Abstract

We compute the electronic structure, spin and charge state of Fe ions, and
structural phase stability of paramagnetic CaFeO3 under pressure using a fully
self-consistent in charge density DFT+dynamical mean-field theory method. We
show that at ambient pressure CaFeO3 is a negative charge-transfer insulator
characterized by strong localization of the Fe $3d$ electrons. It crystallizes
in the monoclinic $P2_{1}/n$ crystal structure with a cooperative breathing
mode distortion of the lattice. While the Fe $3d$ Wannier occupations and
local moments are consistent with robust charge disproportionation of Fe ions
in the insulating $P2_{1}/n$ phase, the physical charge density difference
around the structurally distinct Fe A and Fe B ions with the “contracted” and
“expanded” oxygen octahedra, respectively, is rather weak, $\sim$0.04. This
implies the importance of the Fe $3d$ and O $2p$ negative charge transfer and
supports the formation of a bond-disproportionated state characterized by the
Fe A $3d^{5-\delta}\underline{L}^{2-\delta}$ and Fe B $3d^{5}$ valence
configurations with $\delta\ll 1$, in agreement with strong hybridization
between the Fe $3d$ and O $2p$ states. Upon compression above $\sim$41 GPa
CaFeO3 undergoes the insulator-to-metal phase transition (IMT) which is
accompanied by a structural transformation into the orthorhombic $Pbnm$ phase.
The phase transition is accompanied by suppression of the cooperative
breathing mode distortion of the lattice and, hence, results in the melting of
bond disproportionation of the Fe ions. Our analysis suggests that the IMT
transition is associated with orbital-dependent delocalization of the Fe $3d$
electrons and leads to a remarkable collapse of the local magnetic moments.
Our results imply the crucial importance of the interplay of electronic
correlations and structural effects to explain the properties of CaFeO3.

##  I Introduction

The transition metal oxides with perovskite crystal structure have attracted
much attention due to their diverse electronic and magnetic properties,
allowing for a broad range of applications, e.g., in electronics and
spintronics, and energy storage Mott_1990 ; Imada_1998 ; Tokura_2000 ;
Dagotto_2001 ; Salamon_2001 ; Tokura_2014 . A particular interest is given to
the understanding of a Mott metal-insulator transition (MIT) driven by
correlation effects, the description of which has been a long-standing
challenge Mott_1990 ; Imada_1998 . In this context metal oxides with an
unusually high valence state of the transition metal, Ni3+ (nickelates) and
Fe4+ (ferrates) are notable because of a complex interplay of ligand holes and
electronic correlations in the $3d$ shell of Ni and Fe ions Catalan_2008 ;
AnnuRevMaterRes.46.305 ; RepProgPhys.81.46501 ; Takano_1991 ; Bocquet_1992 ;
Woodward_2000 ; Takeda_2000 , which leads to their complex physical behavior.

Such materials are often regarded as “negative charge transfer” compounds,
implying the importance for a charge transfer from oxygen to the transition
metal atom, leaving holes at oxygen ligands, to explain their electronic
properties Zaanen_1985 ; Mazin_2007 . In particular, the rare-earth nickelate
perovskites $R$NiO3 ($R$ = rare earth, $R^{3+}$) with a high oxidation state
of nickel, Ni3+ $3d^{7}$ exhibit a sharp metal-insulator transition (except
for LaNiO3) upon cooling and under pressure Catalan_2008 ;
AnnuRevMaterRes.46.305 ; RepProgPhys.81.46501 ; Torrance_1992 ;
Garcia_Munoz_1992 . The MIT transition is accompanied by a structural
transformation from the orthorhombic metallic ($Pbnm$, distorted GdFeO3-type)
to monoclinic insulating ($P2_{1}/n$) crystal structure, with a cooperative
breathing mode distortion of NiO6 octahedra in the insulating $P2_{1}/n$ phase
Torrance_1992 ; Garcia_Munoz_1992 . While analysis of the Ni-O bond lengths
and x-ray absorption spectroscopy suggests charge disproportionation of Ni
ions in the insulating $R$NiO3 phases, their electronic state is more
accurately described in terms of bond disproportionation, with alternating Ni
ions which (nearly) adopt a Ni2+ $3d^{8}$ (Ni2+ ions with local moments) and
$3d^{8}\underline{L}^{2}$ (nonmagnetic spin-singlet) electronic configuration
($\underline{L}$ denotes a hole in the O $2p$ band) Park_2012 ; Johnston_2014
; Park_2014 ; Subedi_2015 ; Bisogni_2016 ; Hampel_2017 ; Hampel_2019 ;
Georgescu_2019 ; Peil_2019 . We also note that the complex interplay between
(negative) charge transfer and correlation effects can result in unusual
charge or bond-disproportionation of the $A$-site ions of the perovskite ABO3
structure, e.g., in BiNiO3 Ishiwata_2002 ; Azuma_2007 ; Carlsson_2008 ;
Azuma_2011 ; Oka_2013 ; Saha_Dasgupta_2019 ; Leonov_2020b , PbFeO3
Tsuchiya_2006 ; Ye_2021 , and PbCoO3 Sakai_2017 ; Liu_2020 ; Hariki_2021 .

In the present work, we focus on a negative charge transfer material CaFeO3
with a nominal Fe $3d^{4}$ $(t_{2g}^{3}e_{g}^{1})$ electronic state that
exhibits the MIT upon cooling below $\sim$290 K (at ambient pressure)
Catalan_2008 ; AnnuRevMaterRes.46.305 ; RepProgPhys.81.46501 ; Woodward_2000 ;
Takeda_2000 . The MIT occurs upon compression of CaFeO3 above $\sim$30 GPa
Takano_1991 . It shows a complex screw antiferromagnetic structure below the
Néel temperature $\sim$115 K Woodward_2000 ; Takeda_2000 . Similarly to the
rare-earth nickelates, the MIT in CaFeO3 is accompanied by a structural
transformation from the orthorhombic metallic ($Pbnm$) to monoclinic
insulating ($P2_{1}/n$) crystal structure, with a cooperative breathing mode
distortion of FeO6 octahedra with the difference in Fe-O bond lengths
$\sim$0.1Å in the insulating $P2_{1}/n$ phase Woodward_2000 ; Takeda_2000 .

Given the electronic and structural complexity of CaFeO3 (e.g., doped with Ca
and La) and its similarity to the rare-earth nickelates, this system has been
intensively studied using both experimental and theoretical methods
Bocquet_1992 ; Woodward_2000 ; Takeda_2000 ; Takano_1991 ; Xu_2001 ;
Patrakeev_2003 ; Galakhov_2010 ; Fujioka_2012 ; Rogge_2018 ; Rogge_2019 ;
Jana_2019 ; Wang_2019 ; Onose_2020 ; Shein_2005 ; Saha-Dasgupta_2005 ;
Alexandrov_2008 ; Dalpian_2018 ; Cammarata_2012 ; Zhang_2018 ; Varignon_2019 ;
Varignon_2019b . Previous calculations were mostly employed band-structure
methods supplemented with the on-site Coulomb correlations for the Fe $3d$
states within density-functional theory (DFT)+$U$ method Anisimov_1991 ;
Liechtenstein_1995  with a major focus to the ground state properties of the
insulating $P2_{1}/n$ phase with a long-range magnetic ordering Shein_2005 ;
Saha-Dasgupta_2005 ; Alexandrov_2008 ; Cammarata_2012 ; Zhang_2018 . Using the
DFT+$U$ and more advanced strongly constrained appropriately normed exchange
and correlation functional (SCAN) methods Anisimov_1991 ; Liechtenstein_1995 ;
Sun_2015  the authors address the questions about the origin of a band gap,
charge and bond-disproportionation, ligand hole effects, Mott vs. charge
transfer insulator behaviors, etc. in CaFeO3 Dalpian_2018 ; Varignon_2019 ;
Varignon_2019b .

Recently, the electronic structure of CaFeO3 has been discussed Merkel_2021
in the context of a five-orbital tight-binding model using the DFT+dynamical
mean-field theory (DFT+DMFT) method Metzner_1989 ; Georges_1996 ; Kotliar_2006
; Anisimov_1997 ; Haule_2007 ; Pourovskii_2007 ; Amadon_2008 ; Aichhorn_2009 .
DFT+DMFT has been proven to be among the most advanced theoretical methods for
studying the electronic properties of strongly correlated materials, such as
correlated transition metal oxides, heavy-fermions, and Fe-based
superconductors, e.g., to study the Mott metal-insulator phase transition,
collapse of local moments, large orbital-dependent renormalizations, etc.
Haule_2007 ; Pourovskii_2007 ; Amadon_2008 ; Aichhorn_2009 ; Leonov_2015a ;
Greenberg_2018 ; Leonov_2019 ; Leonov_2015b ; Leonov_2015c ; Zhong_2015 ;
Leonov_2016 ; Backes_2016 ; Medici_2017 ; Seth_2017 ; Facio_2018 ;
Skornyakov_2017 ; Lechermann_2018 ; Arribi_2018 ; Skornyakov_2018 ;
Lechermann_2019 ; Jang_2019 ; Mandal_2019 ; Leonov_2020 ; Koemets_2021 ;
Leonov_2021 ; Leonov_2021b ; Lantz_2015 ; Bhandary_2021  The DFT+DMFT results
for the tight-binding model Merkel_2021  show the competition between high-
spin and low-spin homogeneous and an inhomogeneous charge-disproportionated
state, implying the importance of Hund’s coupling Georges_2013 ; Kunes_2011 ;
Isidori_2019 . Nonetheless, the electronic properties of CaFeO3 and, in
particular, the interplay of the electronic structure, magnetic states, and
phase stability of CaFeO3 near the MIT are still poorly understood.

We address this problem in our study by using a fully self-consistent in
charge density DFT+DMFT method Haule_2007 ; Pourovskii_2007 ; Amadon_2008 ;
Aichhorn_2009 ; Leonov_2015a ; Leonov_2015c ; Leonov_2020  implemented with
plane-wave pseudopotentials Giannozzi_2009 . By employing DFT+DMFT it becomes
possible to capture all generic aspects of the interplay between the
electronic correlations, magnetic states, and crystal structure of CaFeO3
under pressure. In particular, we explore the electronic structure, magnetic
properties, spin and charge state of Fe ions, and structural phase stability
of _paramagnetic_ CaFeO3 near the pressure-induced MIT. Our results show the
importance of the Fe $3d$ and O $2p$ negative charge transfer due to strong
covalency and supports the formation of the bond-disproportionated state in
the insulating phase of CaFeO3

##  II Method

We employ the state-of-the-art DFT+DMFT method Georges_1996 ; Kotliar_2006 ;
Anisimov_1997  with full self-consistency over the charge density Haule_2007 ;
Pourovskii_2007 ; Amadon_2008 ; Aichhorn_2009 ; Leonov_2015a ; Leonov_2015c ;
Leonov_2020  to explore the electronic structure, spin and charge state of Fe
ions, and lattice properties of paramagnetic CaFeO3 under pressure. We start
by constructing the effective low-energy Hamiltonian for the partially
occupied Fe $3d$ and O $2p$ states and consider the multiorbital Hubbard
Hamiltonian with the local electron-electron interaction part (in the density-
density approximation) for the Fe $3d$ orbitals

$\displaystyle\hat{H}_{\mathrm{int}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{mm^{\prime}\sigma}U_{mm^{\prime}}\hat{n}_{m\sigma}\hat{n}_{m^{\prime}\bar{\sigma}}$
(1) $\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{m\neq
m^{\prime}\sigma}(U_{mm^{\prime}}-J_{mm^{\prime}})\hat{n}_{m\sigma}\hat{n}_{m^{\prime}\sigma},$

where $\hat{n}_{m\sigma}$ is the occupation number operator with spin $\sigma$
and (diagonal) orbital indices $m$. $U_{mm^{\prime}}$ and $J_{mm^{\prime}}$
are the Coulomb repulsion and Hund’s exchange coupling matrix elements. In Eq.
1 we assume summation over all Fe sites. For the partially filled Fe $3d$ and
O $2p$ orbitals we construct a basis set of atomic-centered symmetry-
constrained Wannier functions Marzari_2012 ; Anisimov_2005 ; Trimarchi_2008
(the localized Fe $3d$ orbitals are constructed using the Fe $3d$ band set,
while the O $2p$ orbitals were defined over the full energy window spanned by
the O $2p$–Fe $3d$ band complex Greenberg_2018 ; Leonov_2019 ). This makes it
possible to explicitly account charge transfer effects between the partially
occupied Fe $3d$ and O $2p$ states and strong Coulomb correlations of the Fe
$3d$ electrons. We use the continuous-time hybridization expansion (segment)
quantum Monte Carlo algorithm in order to solve the realistic many-body
problem Gull_2011 . The DFT+DMFT calculations are performed in the
paramagnetic (PM) state at an electronic temperature $T=387$ K. The Coulomb
$U_{mm^{\prime}}$ and Hund’s exchange $J_{mm^{\prime}}$ matrix elements are
parametrized in terms of the Slater integrals $F_{0}$, $F_{2}$, and $F_{4}$
which are expressed in terms of the average Coulomb repulsion $U=F_{0}$ and
Hund’s exchange coupling $J_{H}=(F_{2}+F_{4})/14$, with a fixed ratio
$F_{4}/F_{2}=0.63$. We take the average Hubbard $U=6$ eV and Hund’s exchange
$J_{H}=0.86$ eV in accordance with previous DFT+DMFT studies Greenberg_2018 ;
Leonov_2019 . The fully localized double-counting correction evaluated from
the self-consistently determined local occupations was used,
$\hat{H}_{DC}=U(N-\frac{1}{2})-J(N_{\sigma}-\frac{1}{2})$, where $N_{\sigma}$
is the total Fe $3d$ occupation with spin $\sigma$ and
$N=N_{\uparrow}+N_{\downarrow}$. The Coulomb interaction is treated in the
density-density approximation, neglecting by spin-flip and pair-hopping terms
in the multiorbital Hubbard Hamiltonian. The spin-orbit coupling is neglected
in our calculations. The spectral functions were computed using the maximum
entropy method. In order to estimate the quasiparticle mass enhancement
$m^{*}/m$ of the Fe $3d$ states we perform analytic continuation of the self-
energy $\Sigma(\omega)$ determined self-consistently on the Matsubara contour
using Padé approximants. In order to analyse a degree of localization of the
Fe $3d$ electrons we compute the local spin susceptibility
$\chi(\tau)=\langle\hat{m}_{z}(\tau)\hat{m}_{z}(0)\rangle$ within DMFT, where
$\hat{m}_{z}(\tau)$ is the instantaneous magnetization on the Fe $3d$ site at
the imaginary time $\tau$ ($\tau$ denotes an imaginary-time evolution ranging
from 0 to $\beta=1/k_{B}T$ in the path integral formalism). In DFT we use the
generalised gradient Perdew-Burke-Ernzerhof approximation for the correlation
exchange functional Perdew_1996 .

##  III Results and discussion

We start by computing the electronic properties and structural phase stability
of PM CaFeO3 under pressure. In our study, we adopt the crystal structure
parameters determined experimentally for the monoclinic $P2_{1}/n$ phase (at
ambient pressure and temperature $\sim$15 K) and orthorhombic $Pbnm$
structures (taken above the insulator-to-metal and charge-disproportionation
transition, at 300 K) Woodward_2000 ; Takeda_2000 , and evaluate the DFT+DMFT
total energies as a function of the unit-cell volume Leonov_2015a ;
Leonov_2015c ; Leonov_2020 . We fit the total-energy results using the third-
order Birch-Murnaghan equation of states separately for the low- and high-
volume regions Murnaghan_1944 ; Birch_1947 .

Figure 1: Total energy of PM CaFeO3 calculated by DFT+DMFT at $T=387$ K as a
function of lattice volume. The lattice volume collapse associated with the
$P2_{1}/n$-$Pbnm$ structural phase transition is depicted by red shading.
Inset: Enthalpy difference (relative to the high-spin $P2_{1}/n$ solution)
obtained by DFT+DMFT as a function of pressure. The $P2_{1}/n$-$Pbnm$
structural transition at $\sim$41 GPa is shown by a vertical dotted line.

Our results for the total energy calculations and compressional behavior of
CaFeO3 are summarized in Fig. 1. The $P2_{1}/n$ phase is found to be
energetically favorable, i.e., thermodynamically stable at ambient pressure,
with a total-energy difference of $\sim$13 meV/atom with respect to the $Pbnm$
phase. This implies that at low pressure and temperature conditions CaFeO3
crystallizes in the $P2_{1}/n$ phase which is characterized by the cooperative
breathing-mode distortion of the crystal structure, in agreement with the
x-ray and neutron diffraction experiments Woodward_2000 ; Takeda_2000 . Given
the higher electronic entropy of the metallic $Pbnm$ phase this also suggests
a structural phase transition from $P2_{1}/n$ to the $Pbnm$ phase upon
heating, consistent with experiments Woodward_2000 ; Takeda_2000 . The
calculated equilibrium lattice volume for the $P2_{1}/n$ phase is
$V_{0}=375.4$ a.u.3 (by $\sim$4% larger than that in the experiment) and bulk
modulus $K_{0}=151$ GPa (with $K^{\prime}\equiv dK/dP$ fixed to
$K^{\prime}=4.0$). Our results for the instantaneous magnetic moments of the
structurally distinct Fe A and Fe B sites with the “contracted” and “expanded”
oxygen octahedra, respectively, are 3.18 and 4.65$\mu_{\mathrm{B}}$ (see Fig.
2). These values are close to the calculated fluctuating moments $\sim$2.98
and 4.58$\mu_{\mathrm{B}}$, evaluated as the imaginary-time average of local
spin susceptibility
$\chi(\tau)=\langle\hat{m}_{z}(\tau)\hat{m}_{z}(0)\rangle$,
$m_{\mathrm{loc}}=(k_{B}T\int\chi(\tau)d\tau)^{1/2}$. This suggests strong
localization of the Fe $3d$ electrons at low pressures, implying the crucial
importance of electronic correlations to explain the electronic properties of
CaFeO3. We also note that the calculated local moments are in good agreement
with the experimental estimates of 2.5-3.5$\mu_{\mathrm{B}}$ and
3.5-5.0$\mu_{\mathrm{B}}$ for the Fe A and B sites obtained from the fit of
neutron diffraction data using different spiral magnetic structures
Woodward_2000 ; Takeda_2000 .

Figure 2: The local magnetic moments of the $P2_{1}/n$ (AP) and $Pbnm$ (HP)
structural phases of PM CaFeO3 obtained by DFT+DMFT at $T=387$ K as a function
of the unit-cell volume. The unit-cell volume collapse associated with the MIT
and the $P2_{1}/n$-$Pbnm$ structural transition is depicted by red shading.

In agreement with experiment, we obtain a negative charge-transfer insulator
Zaanen_1985 ; Mazin_2007  with an energy gap of $\sim$0.6 eV (see the left
panel of Fig. 3). Moreover, our analysis of the Fe $3d$ Wannier occupations
give a remarkable charge disproportionation between the Fe A and Fe B sites
(due to a sufficiently different oxygen environment of these sites in the
insulating $P2_{1}/n$ phase, with the difference in Fe-O bond lengths
$\sim$0.1 ÅWoodward_2000 ; Takeda_2000 ). In fact, at $\sim$6.7 GPa the total
Fe $3d$ Wannier occupation for the Fe A site is only $\sim$4.3, while for the
Fe B sites it is 5.1, implying a rather large charge difference of $\sim$0.8
electron. We note that this charge difference is about 40% of the ideal ionic
Fe3+-to-Fe5+ charge disproportionation, roughly consistent with the bond
valence sum estimate of $\sim$1.1 Woodward_2000 ; Takeda_2000 , as well as is
in agreement with the significantly different local magnetic moments for the
Fe A and Fe B sites (see above). Interestingly that previous estimates of a
charge disproportionation in the low-temperature charge-ordered phases of the
mixed-valence oxides such as Fe3O4 and rare-earth nickelates $R$NiO3 give
$\sim$20-40% of the ideal ionic disproportionation Wright_2002 ; Leonov_2006 ;
Jeng_2006 ; Leonov_2005 ; Senn_2012 ; Perversi_2019 ; Torrance_1992 ;
Garcia_Munoz_1992 ; Catalan_2008 , consistent with our results.

In agreement with this, our analysis of the eigenvalues of the reduced Fe $3d$
density matrix, i.e., an estimate of a fluctuating valence state of Fe ions,
shows that the Fe B ions (with “expanded” oxygen octahedra) are $3+$. In fact,
for the Fe B ions $3d^{5}$ configuration has a predominant weight of about 74%
with a $\sim$7% admixture of the $3d^{4}$ and $\sim$18% of the $3d^{6}$
configurations (due to quantum mixing and temperature effects). On the other
hand, for the Fe A ions the $3d^{4}$ and $3d^{5}$ configurations are nearly
equally weighted, of $\sim$48% and 34%, respectively, in accordance with above
estimates. That is, the electronic state of the Fe A ions consists of heavily
mixed $d^{4}$ and $d^{5}\underline{L}$ states, reflecting the importance of Fe
$3d$ to O $2p$ negative charge transfer due to strong covalency in CaFeO3.

While our results for the Fe $3d$ Wannier occupations and local moments give a
robust charge disproportionation in the $P2_{1}/n$ insulating phase, a
difference of the total charge density $\rho({\bf r})$ around the structurally
distinct Fe A and Fe B ions is rather weak. In particular, our result for the
corresponding charge difference within the Fe-ion radius of 0.86Å give an
order of magnitude smaller value of $\sim$0.04. This implies the importance of
the Fe $3d$ and O $2p$ negative charge transfer and suggests the formation of
a bond-disproportionated state characterized by the Fe
$3d^{5-\delta}\underline{L}^{2-\delta}$ and $3d^{5}$ electronic configurations
with $\delta\ll 1$ for the “compressed” Fe A and “expanded” Fe B sites,
respectively, in the insulating phase of CaFeO3. This result is consistent
with a substantial Fe-O covalence and strong hybridization between the
unoccupied Fe $e_{g}$ and O $2p$ state Zaanen_1985 ; Mazin_2007 .

Figure 3: Orbitally-resolved spectral functions of PM CaFeO3 calculated by
DFT+DMFT for the $P2_{1}/n$ (left panel) and $Pbnm$ (right panel) phases for
different lattice volumes.

Our results for the orbitally-resolved spectral functions of PM CaFeO3 are
shown in Fig. 3. The energy gap lies between the occupied Fe B $e_{g}$ states
strongly mixed with the O $2p$ and the unoccupied Fe A $3d$ states. The latter
strongly hybridize with the empty O $2p$ states. The occupied O $2p$ band
appears at about -6 eV to near the Fermi level. The top of the valence band
has a strongly mixed Fe $3d$ and O $2p$ character, with a resonant peak in the
filled Fe $e_{g}$ states located at about -0.4 eV below the Fermi level. This
behavior can be ascribed to the formation of a Zhang-Rice bound state
Zhang_1988 .

Fe $t_{2g}$ and $e_{g}$ Wannier orbital occupancies are 0.55 and 0.25 per
spin-orbit for the Fe A and are 0.52 and 0.5 for the Fe B ions, respectively.
These findings clearly indicate that at ambient pressure the Fe B ions are in
the high-spin (HS) state. In fact, in an ionic picture, the Fe3+ ions have a
$3d^{5}$ configuration with three electrons in the $t_{2g}$ and two in the
$e_{g}$ orbitals (in the octahedral crystal field), forming the $S=5/2$ state.
This result is in agreement with our results for the decomposition of the
electronic state into atomic spin-state configurations within DFT+DMFT. In
particular, for the Fe B ions the HS state has a predominant weight of
$\sim$97% with a small admixture of $\sim$3% due to the intermediate-spin (IS)
state in the insulating $P2_{1}/n$ phase (at ambient pressure). On the other
hand, the Fe A ions show a strong mixture of the HS and IS state
configurations whose corresponding weights are $\sim$47% and 43%.

Figure 4: Orbitally-resolved local spin correlations functions
$\chi(\tau)=\langle\hat{m}_{z}(\tau)\hat{m}_{z}(0)\rangle$ of the $P2_{1}/n$
phase of CaFeO3 calculated by DFT+DMFT for different pressures.

Under pressure the energy gap of $P2_{1}/n$ CaFeO3 gradually decreases,
resulting in an insulator-to-metal phase transition, within the $P2_{1}/n$
structure upon compression to $\sim$0.74 $V_{0}$, above $\sim$85 GPa. The
phase transition is accompanied by a remarkable anomaly of our DFT+DMFT total-
energy and local-moment results. In fact, the local moments of the Fe A and B
ions are significantly different and are seen to retain their high-spin values
down to $\sim$0.74 $V_{0}$, while upon further compression collapse to a
nearly same moment value of $\sim$2.3$\mu_{\mathrm{B}}$, at 102 GPa. The phase
transition is accompanied by a large transfer of the Fe $3d$ spectral weight,
with a formation of the Fe $3d$ quasiparticle peak near the Fermi level. It is
associated with a substantial redistribution of charge within the Fe $3d$
shell. Fe $t_{2g}$ orbital occupations are found to gradually increase with
pressure ($t_{2g}$ occupation is about 0.66 per spin-orbit for the Fe A and
0.69 for the Fe B sites at 102 GPa). This comes along with a significant
depopulation of the Fe $e_{g}$ states. Fe A and B $e_{g}$ occupations are 0.17
and 0.19 per spin-orbit, respectively, while the Wannier Fe $3d$ total
occupancies change to 4.6 and 4.87 for the Fe A and Fe B ions. This implies a
significant reduction of charge disproportionation upon metallization of the
$P2_{1}/n$ structure, to about 0.26 at 102 GPa.

Our results for the local (dynamical) spin-spin correlation function
$\chi(\tau)=\langle\hat{m}_{z}(\tau)\hat{m}_{z}(0)\rangle$ reveal significant
delocalization of the Fe $3d$ electrons under pressure as seen in Fig. 4.
$\chi(\tau)$ is strongly suppressed, quickly decaying with the imaginary time
$\tau$ (it is most pronounced for the Fe B $e_{g}$ sates). On the other hand,
a relatively large value of $\chi(\tau)$ of $\sim$0.3–0.4 $\mu_{B}^{2}$ at
$\tau\simeq\beta/2$ (with $\beta=1/k_{B}T=30$ eV-1 at $T=387$ K) for the Fe A
and B $t_{2g}$ states suggests the proximity of Fe $3d$ magnetic moments to
orbital-selective localization.

However, we note that the calculated critical pressure of the MIT $p_{c}\sim
85$ GPa in the $P2_{1}/n$ phase of CaFeO3 is significantly, by almost three
times, higher than that found in the experiments Takano_1991 . It is know
experimentally that the MIT in CaFeO3 is accompanied by a structural change to
the orthorhombic $Pbnm$ phase Takano_1991 ; Bocquet_1992 ; Woodward_2000 ;
Takeda_2000 . In agreement with this, our DFT+DMFT total-energy calculations
show that upon compression to $\sim$0.83 $V_{0}$, the $P2_{1}/n$ lattice
becomes energetically unfavorable, with the orthorhombic $Pbnm$ phase being
thermodynamically stable. We therefore conclude that above $\sim$41 GPa CaFeO3
undergoes a phase transition from monoclinic $P2_{1}/n$ to the orthorhombic
$Pbnm$ crystal structure (see Fig. 1). In agreement with experiment, the phase
transition is accompanied by metallization of CaFeO3. Interestingly, the
calculated local magnetic moments for the $Pbnm$ phase are nearly equal to
that in the highly compressed metallic $P2_{1}/n$ phase of CaFeO3,
$\sim$2.22$\mu_{\mathrm{B}}$ at $\sim$0.7 $V_{0}$. The phase transition is
accompanied by a remarkable collapse of the unit-cell volume of $\frac{\Delta
V}{V}\sim 4$% and results in suppression of the cooperative breathing mode
distortion of the lattice. In fact, in the $Pbnm$ phase Fe sites have a
regular oxygen coordination with no evidence for charge disproportionation. We
also note that the charge disproportionated state is found to be unstable
within our DFT+DMFT calculations for the $Pbnm$ lattice. The calculated bulk
modulus for the high-pressure $Pbnm$ phase is about 197 GPa. This suggests
that the $P2_{1}/n$ to $Pbnm$ phase transition is accompanied by a remarkable
_increase_ of the bulk modulus upon the pressure-induced insulator-to-metal
phase transition, implying a crucial importance of the interplay of electronic
correlations and the lattice to explain the properties of CaFeO3.

Figure 5: Orbitally-resolved local spin correlations functions
$\chi(\tau)=\langle\hat{m}_{z}(\tau)\hat{m}_{z}(0)\rangle$ of the $Pbnm$ phase
of CaFeO3 calculated by DFT+DMFT for different pressures.

The $Pbnm$ phase of CaFeO3 is a strongly correlated metal, characterized by a
Fermi-liquid-like behavior with a large orbital-dependent damping of
$Im[\Sigma(\omega)]\sim 0.52$ and 0.09 eV for the Fe $t_{2g}$ and $e_{g}$
quasiparticle states at the Fermi energy, at about 93 GPa. The latter
indicates strong incoherence of the Fe $3d$ electronic states caused by
proximity of the $3d$ statse to orbital-selective localization. We evaluate
the quasiparticle mass enhancement $\frac{m^{*}}{m}=1-\partial
Im\Sigma(\omega)/\partial\omega|_{\omega=0}$ using extrapolation of the self-
energy $\Sigma(\omega)$ to $\omega=0$ eV, which gives a quantitative measure
of the correlation strength. At 93 GPa we obtain $m^{*}/m\sim 5.0$ and 1.7 for
the Fe $t_{2g}$ and $e_{g}$ bands, respectively. We also note that similarly
to the insulating $P2_{1}/n$ phase the $Pbnm$ phase of CaFeO3 exhibits a
remarkable collapse of the local magnetic moments under pressure at about 43
GPa (which is almost equal to the critical pressure $p_{c}$ of the $P2_{1}/n$
to $Pbnm$ structural transition). This implies that the $P2_{1}/n$ to $Pbnm$
phase transition is accompanied by a collapse of the local magnetic moments of
the Fe ions.

The MIT transition results in a significant redistribution of the spectral
weight near the Fermi level. It leads to the formation of a quasiparticle peak
originating from the Fe $t_{2g}$ states with the lower and upper Hubbard bands
located at -1.2 eV and 1.5 eV, respectively (see the right panel of Fig. 3).
It gives a strong depopulation of the Fe $e_{g}$ orbital occupations. Thus,
the Fe $e_{g}$ orbital occupations change from 0.38 (at ambient pressure in
the $Pbnm$ phase) to 0.17 per spin-orbit under pressure increase to 93 GPa. On
the other hand, it results in a gradual increase of the Fe $t_{2g}$
occupations, while the total Wannier Fe $3d$ occupation in the $Pbnm$ phase
remains essentially unchanged with pressure, $\sim$4.73. We note that this
value nearly equals to the average Wannier Fe $3d$ total occupancy of the Fe
ions in the metallic $P2_{1}/n$ phase.

In addition, our analysis of the atomic spin-state configurations and local
spin susceptibilities shows that the phase transition is associate with a
spin-state transition accompanied by a significant delocalization of the Fe
$3d$ states (see Figs. 4 and 5). In particular, in the metallic $Pbnm$ phase
at 93 GPa (i.e., above the MIT) the Fe ions adopt the IS state with a
predominant weight of $\sim$56%, with a strong admixture $\sim$34% of the LS
state. Our analysis of fluctuating valence state of the Fe ions yields a
predominant contribution of the $3d^{5}$ configuration, $\sim$47%, with a
considerable admixture of the $3d^{4}$ and $3d^{6}$ states of $\sim$34% and
16%, respectively. This implies that the electronic state of the Fe ions in
the $Pbnm$ phase consists of heavily mixed $d^{5}\underline{L}$ and $d^{4}$
states, highlighting negative charge transfer due to strong covalency in
CaFeO3. We therefore identify the metallic $Pbnm$ phase of CaFeO3 as a
negative charge-transfer material with a local Fe
$3d^{5-\delta}\underline{L}^{1-\delta}$ electron configuration (with
$\delta\ll 1$), implying the important role of Fe $3d$ to O $2p$ negative
charge-transfer.

Overall, our results for the electronic structure, spin and charge state of Fe
ions, and phase stability of CaFeO3 under pressure agree well with available
experimental studies. Our DFT+DMFT calculations give a microscopic
understanding of the pressure-induced evolution of the electronic structure
and lattice properties of CaFeO3. It reveals the complex interplay between the
electronic correlations and lattice effects in the vicinity of a pressure-
induced MIT in CaFeO3. We note that our results are compatible with the
scenario of bond disproportionation, which comprises the “compressed” Fe A
$3d^{5-\delta}\underline{L}^{2-\delta}$ and “expanded” Fe B $3d^{5}$ sites
with $\delta\ll 1$ alternating in the insulating $P2_{1}/n$ phase and a
homogeneous Fe $3d^{5-\delta}\underline{L}^{1-\delta}$ state in the metallic
$Pbnm$ phase.

##  IV Conclusion

In conclusion, we employed the DFT+DMFT computational approach to study the
effects of electronic correlations on the electronic structure, spin and
charge state of Fe ions, and structural phase stability of _paramagnetic_
CaFeO3 across the pressure-induced insulator-to-metal phase transition. In
agreement with experiments, the DFT+DMFT calculations show that at low
pressure and temperature CaFeO3 is a negative charge-transfer insulator
characterized by strong localization of the Fe $3d$ electrons. It crystallizes
in the monoclinic $P2_{1}/n$ crystal structure with a cooperative breathing
mode distortion of the lattice. While our analysis of the Fe $3d$ Wannier
occupations and local moments for the insulating $P2_{1}/n$ phase give a
robust charge disproportionation, a charge difference evaluated as an integral
of the charge density around the structurally distinct Fe A and Fe B ions is
rather weak, $\sim$0.04. This implies the importance of the Fe $3d$ and O $2p$
negative charge transfer and supports the formation of a bond-
disproportionated state characterized by the Fe
$3d^{5-\delta}\underline{L}^{2-\delta}$ and $3d^{5}$ valence configurations
with $\delta\ll 1$ for the “compressed” Fe A and “expanded” Fe B sites,
respectively, in the insulating phase of CaFeO3. This behavior is consistent
with a substantial Fe-O covalence and strong hybridization between the
unoccupied Fe $e_{g}$ and O $2p$ states.

In agreement with experiment, under critical pressure of $\sim$41 GPa CaFeO3
is found to make the insulator-to-metal phase transition which is accompanied
by a structural transformation into the orthorhombic $Pbnm$ phase. The phase
transition is accompanied by suppression of the cooperative breathing mode
distortion of the lattice and, hence, results in the melting of bond-
disproportionation of the Fe ions. Our analysis suggests that the MIT
transition is accompanied by orbital-dependent delocalization of the Fe $3d$
electrons and leads to a remarkable collapse of the local magnetic moments.
Our results imply the crucial importance of the interplay of electronic
correlations and structural effects to explain the electronic properties and
structural phase equilibrium of CaFeO3. We believe that this topic deserves
further detailed theoretical and experimental considerations.

###### Acknowledgements.

We acknowledge support by Russian Foundation for the Basic Research (Project
No. 20-42-660027). The theoretical analysis of the electronic structure and
DFT calculations were supported by the state assignment of Minobrnauki of
Russia (theme “Electron” No. AAAA-A18-118020190098-5).

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#  A hyperbolic analogue of the Atiyah-Hitchin manifold

Paul Sutcliffe
Department of Mathematical Sciences,
Durham University, Durham DH1 3LE, United Kingdom.
Email<EMAIL_ADDRESS>
(December 2021)

###### Abstract

The Atiyah-Hitchin manifold is the moduli space of parity inversion symmetric
charge two $SU(2)$ monopoles in Euclidean space. Here a hyperbolic analogue is
presented, by calculating the boundary metric on the moduli space of parity
inversion symmetric charge two $SU(2)$ monopoles in hyperbolic space. The
calculation of the metric is performed using a twistor description of the
moduli space and the result is presented in terms of standard elliptic
integrals.

##  1 Introduction

Non-abelian magnetic monopoles are topological soliton solutions of Yang-
Mills-Higgs gauge theories in three-dimensional space. In this paper the gauge
group is taken to be $SU(2)$, with the Higgs field in the adjoint
representation, and a vanishing Higgs potential. The number of monopoles is a
positive integer $N$, referred to as the charge, since the magnetic charge of
an $N$-monopole is equal to $N$ in suitable units. In the above system,
magnetic monopoles are BPS solitons, with a moduli space of static
$N$-monopole solutions that describes monopoles with arbitrary positions and
internal relative phases.

Some time ago, Manton proposed that the dynamics of $N$ slowly moving
monopoles could be described by geodesic motion in the $N$-monopole moduli
space, equipped with a metric induced from the field theory kinetic energy
[1]. This approach was later validated by rigorous mathematical analysis [2].
To study the scattering of a pair of monopoles within this scheme requires
knowledge of the metric on the 2-monopole moduli space. By fixing the centre
of mass, the problem reduces to finding the metric on the 4-dimensional
manifold of parity inversion symmetric 2-monopoles. This manifold is known as
the Atiyah-Hitchin manifold, following their ingenious calculation of the
metric by exploiting the fact that this is a hyperkähler manifold [3]. Their
study of the geodesics of this manifold revealed remarkable properties of
monopole dynamics, including the right angle scattering of monopoles in a
head-on collision: a phenomenon that is now ubiquitous in topological soliton
systems.

Atiyah observed that many of the properties of monopoles in Euclidean space
survive the transmutation to hyperbolic space [4]. Furthermore, if the
curvature of hyperbolic space is tuned to a specific value, determined by the
magnitude of the Higgs field at spatial infinity, then hyperbolic monopoles
correspond to circle-invariant Yang-Mills instantons in 4-dimensional
Euclidean space, with the number of monopoles equal to the number of
instantons. This simplifies a number of features in comparison to the case of
generic constant negative curvature and will therefore be assumed from now on.

An important distinction between Euclidean and hyperbolic monopoles concerns
their dynamics. The moduli space of static hyperbolic $N$-monopoles cannot be
naturally equipped with the metric induced from the field theory kinetic
energy, because the induced metric diverges. Braam and Austin defined an
alternative metric on the hyperbolic $N$-monopole moduli space [5], obtained
as a boundary metric using the abelian connection of the hyperbolic monopole
on the sphere at infinity, but its relation to monopole scattering was
unclear. Recently, this metric has been placed in a more general context of
boundary metrics on soliton moduli spaces and evidence presented to support
the interpretation of the geodesics of such metrics in terms of soliton
dynamics [6]. This motivates the present work, to obtain a hyperbolic analogue
of the Atiyah-Hitchin manifold by calculating the boundary metric on the
moduli space of parity inversion symmetric charge two hyperbolic monopoles.
The method exploits a description of the moduli space in terms of spectral
curves in mini-twistor space, and yields an explicit expression for the metric
in terms of standard elliptic integrals.

##  2 Spectral curves of hyperbolic monopoles

The mini-twistor space of three-dimensional hyperbolic space is the space of
its oriented geodesics. Each oriented geodesic may be labelled by a pair of
Riemann sphere coordinates, $(\eta,\zeta)$, where the geodesic starts at
$-1/\bar{\eta}$ and ends at $\zeta$, both regarded as points on the sphere at
infinity in the unit ball model of hyperbolic space. The mini-twistor space is
therefore $\mathbb{CP}^{1}\times\mathbb{CP}^{1}-\bar{\Delta},$ where
$\bar{\Delta}$ is the anti-diagonal, $\bar{\eta}\zeta=-1$, that must be
removed because the start and end point of a geodesic are different.

There is a bijective correspondence between hyperbolic monopoles and spectral
curves, which are algebraic curves in mini-twistor space satisfying certain
reality and non-singularity conditions [4]. Briefly, the spectral curve of an
$N$-monopole is a biholomorphic curve in mini-twistor space of bidegree
$(N,N)$, so it may be written in the form

$S(\eta,\zeta)=\sum_{i=0,j=0}^{N}c_{ij}\eta^{i}\zeta^{j}=0.$ (2.1)

The reality condition on the complex constants $c_{ij}$, that follows from
reversing the orientation of the geodesic, is

$\overline{c_{ij}}=(-1)^{N+i+j}c_{N-j,N-i}.$ (2.2)

The spectral curve describes all geodesics along which a certain linear
operator, constructed from the monopole fields, has a normalizable solution.
This is equivalent to imposing non-singularity conditions on the algebraic
curve that can be written in terms of relations between integrals of
holomorphic differentials around particular cycles. Roughly speaking, these
conditions identify the geodesics that pass through the locations of the
monopoles. As an example, the spectral curve of a 1-monopole is given by

$2\eta\zeta(X_{1}-iX_{2})+\zeta(1+|{\bf X}|^{2}-2X_{3})-\eta(1+|{\bf
X}|^{2}+2X_{3})-2(X_{1}+iX_{2})=0,$ (2.3)

where ${\bf X}=(X_{1},X_{2},X_{3})$ is the point inside the unit ball at which
the monopole is located (given by the vanishing of the Higgs field). This
spectral curve gives all the geodesics that pass through the point ${\bf X}.$

A monopole is parity inversion symmetric if it invariant under changing the
sign of all the Cartesian coordinates together with the sign of the Higgs
field. The spectral curve of a parity inversion symmetric $N$-monopole
satisfies

$S(\eta,\zeta)=\eta^{N}\zeta^{N}\overline{S(-\bar{\eta}^{-1},-\bar{\zeta}^{-1})}.$
(2.4)

For example, the spectral curve (2.3) of the 1-monopole is parity inversion
symmetric only if the monopole is located at the origin, ${\bf X}={\bf 0}$.

The reason for using the spectral curve description of monopoles in this paper
is that there is an integral formula for the boundary metric in terms of the
spectral curve [7]. Let the most general $N$-monopole spectral curve
$S(\eta,\zeta)$ be given in terms of real parameters
$\alpha_{1},\alpha_{2},...,\alpha_{4N-1}$, that are coordinates on the
$(4N-1)$-dimensional moduli space. A real function $h(z,\bar{z})$ is
constructed by evaluating the spectral curve on the anti-diagonal
$\bar{\Delta},$ via the definition

$h(z,\bar{z})=\bar{z}^{N}S(-\bar{z}^{-1},z).$ (2.5)

The formula for the boundary metric $ds^{2}=g_{ij}d\alpha_{i}d\alpha_{j}$ is
[7]

$g_{ij}=\frac{3}{4\pi}\int\Re\bigg{\\{}\bigg{(}\frac{\partial}{\partial\alpha_{i}}\frac{\partial}{\partial
z}\log
h\bigg{)}\bigg{(}\frac{\partial}{\partial\alpha_{j}}\frac{\partial}{\partial\bar{z}}\log
h\bigg{)}\bigg{\\}}\,idzd\bar{z},$ (2.6)

where the integration is over the whole plane and $\Re$ denotes the real part.
The normalization factor has been fixed by the requirement that the metric of
a single monopole restricted to a radial half-line from the origin is
$d\rho^{2}$, where $\rho$ is the geodesic distance from the origin [8]. To
date, the only calculation of a boundary metric for hyperbolic monopoles is
the simplest case of the three-dimensional 1-monopole moduli space, where the
result is 3-dimensional hyperbolic space [5]. Interpreting the geodesics of
this manifold in terms of monopole dynamics provides a description of the free
motion of a single monopole. In the next section the first example relevant
for hyperbolic monopole scattering is constructed, by calculating the boundary
metric on the hyperbolic analogue of the 4-dimensional Atiyah-Hitchin manifold
for charge two monopoles with parity inversion symmetry.

##  3 The metric

The identification of a hyperbolic $N$-monopole with a circle-invariant Yang-
Mills $N$-instanton allows the ADHM construction of instantons [9] to be
adapted to a construction for hyperbolic monopoles [10]. Within this
framework, the data for a hyperbolic $N$-monopole consists of an $N$-component
row vector $L$ of quaternions, together with a triplet $M_{1},M_{2},M_{3}$ of
real $N\times N$ symmetric matrices. These are assembled to form the
$(N+1)\times N$ matrix of quaternions

$\widehat{M}=\begin{pmatrix}L\\\ iM_{1}+jM_{2}+kM_{3}\end{pmatrix},$ (3.1)

that is required to satisfy the condition that
$\widehat{M}^{\dagger}\widehat{M}$ is the identity matrix, together with the
constraint that $L(iM_{1}+jM_{2}+kM_{3})=\mu L$, for some pure quaternion
$\mu.$ Here $\dagger$ denotes the quaternionic conjugate transpose.

The 1-parameter family of 2-monopole data of interest for the current
application is given by

$M_{1}=\begin{pmatrix}0&0\\\ 0&0\end{pmatrix},\
M_{2}=\frac{1}{2}(1-a)\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\
M_{3}=\frac{1}{2}(1+a)\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},\
L=\sqrt{\frac{1-a^{2}}{2}}(1,-i),$ (3.2)

with parameter $a\in[0,1)$. It is easy to verify that this data satisfies the
two conditions stated above, with $\mu=ak.$ Applying the adapted ADHM
construction to this data produces an explicit formula [10] for the Higgs
field of a pair of monopoles, one on the positive $x_{3}$-axis and one on the
negative $x_{3}$-axis, each with geodesic distance $\rho$ to the origin given
by

$\rho=2\tanh^{-1}\bigg{(}\sqrt{\frac{3-a^{2}-\sqrt{(1-a)^{2}(9-a^{2})}}{2a}}\bigg{)}.$
(3.3)

The above is true if $a>0$, so that the two distinct monopole positions can be
identified with the two distinct zeros of the Higgs field. If $a=0$ then
$\rho=0$ and the Higgs field of the monopole vanishes only at the origin. In
this case the 2-monopole is axially symmetric about the $x_{1}$-axis. Copying
the notation from Euclidean 2-monopoles [3], each parity inversion symmetric
2-monopole is characterized by three orthogonal, unoriented lines passing
through the origin, that may be viewed as the body-fixed axes
$e_{1},e_{2},e_{3}$. The $e_{3}$-axis is the line passing through both zeros
of the Higgs field, so the $x_{3}$-axis for the above data. The $e_{1}$-axis
is the symmetry axis when the two zeros of the Higgs field coalesce, being the
$x_{1}$-axis for this data.

The formula for the spectral curve in terms of adapted ADHM data is [11]

$S(\eta,\zeta)=\mbox{det}\big{(}\eta\zeta(M_{1}-iM_{2})+\zeta(1-M_{3})-\eta(1+M_{3})-(M_{1}+iM_{2})\big{)}.$
(3.4)

Inserting the data (3.2) gives

$\frac{1}{4}\left(1-a\right)^{2}\left({\eta}^{2}{\zeta}^{2}+1\right)+\frac{1}{4}(3+a)(1-a)\left({\eta}^{2}+{\zeta}^{2}\right)-2\left(1+a\right)\eta\zeta=0.$
(3.5)

It is easily checked that this spectral curve satisfies the condition (2.4) to
be parity inversion symmetric. If $a=0$ then the axial symmetry of the curve
around the $x_{1}$-axis is apparent from the invariance under rotations

$(\eta,\zeta)\mapsto\bigg{(}\frac{\cos(\gamma/2)\eta+i\sin(\gamma/2)}{i\sin(\gamma/2)\eta+\cos(\gamma/2)},\frac{\cos(\gamma/2)\zeta+i\sin(\gamma/2)}{i\sin(\gamma/2)\zeta+\cos(\gamma/2)},\bigg{)},$
(3.6)

for an arbitrary angle $\gamma.$ In the limit $a\to 1$ the curve degenerates
to a product $\eta\zeta=0,$ corresponding to a pair of single monopoles on the
positive and negative $x_{3}$-axis, at infinite geodesic distance from the
origin, as obtained by inserting ${\bf X}=(0,0,\pm 1)$ into the formula (2.3)
for the 1-monopole spectral curve.

The full 4-dimensional family of parity inversion symmetric 2-monopoles is
obtained by applying an arbitrary $SO(3)$ rotation to the data (3.2). In terms
of Euler angles $\theta,\phi,\psi$ define the $SO(3)$ matrix

$Q=\left[\begin{array}[]{ccc}\cos\psi\cos\theta\cos\phi-\sin\psi\sin\phi&\sin\psi\cos\theta\cos\phi+\cos\psi\sin\phi&\sin\theta\cos\phi\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr-\cos\psi\cos\theta\sin\phi-\sin\psi\cos\phi&-\sin\psi\cos\theta\sin\phi+\cos\psi\cos\phi&-\sin\theta\sin\phi\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr-\cos\psi\sin\theta&-\sin\psi\sin\theta&\cos\theta\end{array}\right]$
(3.7)

and obtain the required data by applying the rotation

$\displaystyle\begin{pmatrix}M_{1}\\\ M_{2}\\\ M_{3}\end{pmatrix}\mapsto
Q\begin{pmatrix}M_{1}\\\ M_{2}\\\ M_{3}\end{pmatrix},\quad
L\mapsto\sqrt{\frac{1-a^{2}}{2}}\left(1,k\cos\psi\sin\theta+(j\sin\psi-i\cos\psi\cos\theta)e^{k\phi})\right).$
(3.8)

Using the formula (3.4) gives the spectral curve of the 4-dimensional moduli
space of parity inversion symmetric 2-monopoles, with coefficients

$\displaystyle c_{00}=\overline{c_{22}}=-\frac{1}{4}{{\rm
e}^{-2i\phi}}\left(\left(1-a\right)^{2}\left(i\cos\theta\,\sin
2\psi-\cos^{2}\psi\cos^{2}\theta+\sin^{2}\psi\right)+4a\sin^{2}\theta\right)$
$\displaystyle
c_{01}=c_{10}=-\overline{c_{12}}=-\overline{c_{21}}=\frac{1}{4}{{\rm
e}^{-i\phi}}\sin\theta\left(\left(1-a\right)^{2}\left(i\sin
2\psi-2\cos\theta\cos^{2}\psi\right)-8a\cos\theta\right)$ $\displaystyle
c_{02}=c_{20}=\frac{1}{4}\left(1-a\right)^{2}\sin^{2}\theta\cos^{2}\psi+\frac{1}{4}\left(3-2a\cos
2\theta-{a}^{2}\right)$ $\displaystyle
c_{11}=\left(1-a\right)^{2}\sin^{2}\theta\cos^{2}\psi-2a\cos 2\theta-2.$ (3.9)

Note that for $N=2$ the symmetry of the coefficients, $c_{ij}=c_{ji}$, follows
directly from the combined application of parity inversion symmetry (2.4) and
the reality condition (2.2).

As with the Atiyah-Hitchin manifold, the $SO(3)$ and reflection symmetries
imply that the metric has the form

$ds^{2}=\Lambda_{0}^{2}\,da^{2}+\Lambda_{1}^{2}\,\sigma_{1}^{2}+\Lambda_{2}^{2}\,\sigma^{2}_{2}+\Lambda_{3}^{2}\,\sigma^{2}_{3},$
(3.10)

where $\Lambda_{i}^{2}(a)$ are functions of $a$ only, and $\sigma_{i}$ are the
standard 1-forms on $SO(3)$,

$\displaystyle\sigma_{1}$ $\displaystyle=$
$\displaystyle-\sin\psi\,d\theta+\cos\psi\sin\theta\,d\phi$ (3.11)
$\displaystyle\sigma_{2}$ $\displaystyle=$
$\displaystyle\cos\psi\,d\theta+\sin\psi\sin\theta\,d\phi$ (3.12)
$\displaystyle\sigma_{3}$ $\displaystyle=$ $\displaystyle
d\psi+\cos\theta\,d\phi\,.$ (3.13)

For $i=1,2,3$ the function $\Lambda_{i}^{2}(a)$ is the moment of inertia about
the body-fixed axis $e_{i}$. The spectral curve coefficients (3.9), together
with the integral formula (2.6), provide all the ingredients required for a
computation of the metric on the 4-dimensional moduli space of parity
inversion symmetric charge two hyperbolic monopoles.

Any convenient orientation may be chosen for the calculation of the metric
functions. In particular, once any differentiation with respect to Euler
angles required in (2.6) has been performed, then the Euler angles may be set
to values that will facilitate the calculation of the integral. A particularly
convenient choice is $\theta=\pi/2,\,\phi=\psi=0$, so given any function of
the Euler angles, say $f(\theta,\phi,\psi)$, it is helpful to introduce the
notation $f_{\bullet}=f(\pi/2,0,0).$

The calculation of the metric function $\Lambda_{0}^{2}(a)$ requires no
differentiation with respect to Euler angles, so they may be fixed immediately
to simply the spectral curve to

$S(\eta,\zeta)_{\bullet}=-a(\eta^{2}\zeta^{2}+1)+\eta^{2}+\zeta^{2}-(1-a^{2})\eta\zeta.$
(3.14)

This gives the real function

$h(z,\bar{z})_{\bullet}=1+|z|^{4}+(1-a^{2})|z|^{2}-a(z^{2}+\bar{z}^{2}),$
(3.15)

and hence the integral formula

$\displaystyle\Lambda_{0}^{2}(a)$ $\displaystyle=$
$\displaystyle\frac{3}{4\pi}\int\bigg{|}\frac{\partial}{\partial
z}\frac{\partial}{\partial a}\log h\bigg{|}^{2}_{\bullet}\,idzd\bar{z}.$
(3.16) $\displaystyle=$
$\displaystyle\frac{3}{4\pi}\int\frac{\big{|}a^{2}\bar{z}(\bar{z}^{2}-z^{2})+2a\bar{z}(|z|^{4}-1)+\bar{z}^{3}(2|z|^{2}+1)-z(|z|^{2}+2)\big{|}^{2}}{\big{(}1+|z|^{4}+(1-a^{2})|z|^{2}-a(z^{2}+\bar{z}^{2})\big{)}^{4}}idzd\bar{z}.\qquad$
(3.17)

By making the substitution $z=\sqrt{p}e^{i\chi},$ the integration over $\chi$
may be performed by using the results that the integrals

$I_{n}(b)=\int_{0}^{2\pi}\frac{\cos^{n}\chi}{(1-b\cos\chi)^{4}}\,d\chi,$
(3.18)

with parameter $b\in[0,1),$ are given by

$I_{0}(b)=\frac{\pi(2+3b^{2})}{(1-b^{2})^{7/2}},\quad I_{1}(b)=\frac{\pi
b(4+b^{2})}{(1-b^{2})^{7/2}},\quad
I_{2}(b)=\frac{\pi(1+4b^{2})}{(1-b^{2})^{7/2}}.\quad I_{3}(b)=\frac{\pi
b(3+2b^{2})}{(1-b^{2})^{7/2}}.\quad$ (3.19)

Applying these identities to (3.17) yields

$\Lambda_{0}^{2}(a)=\int_{0}^{\infty}\bigg{(}\left(2{a}^{2}+2\right)\left({p}^{10}+1\right)+\left(-6{a}^{4}+18{a}^{2}+8\right)\left({p}^{9}+p\right)+\left(6{a}^{6}-31{a}^{4}+42{a}^{2}+19\right)\left({p}^{8}+{p}^{2}\right)+\left(-2{a}^{8}+11{a}^{6}-33{a}^{4}+9{a}^{2}+31\right)\left({p}^{7}+{p}^{3}\right)+\left({a}^{8}+2{a}^{6}+9{a}^{4}-92{a}^{2}+40\right)\left({p}^{6}+{p}^{4}\right)+\left(-{a}^{10}+9{a}^{8}-8{a}^{6}+32{a}^{4}-147{a}^{2}+43\right){p}^{5}\bigg{)}\\\
\bigg{(}\left(1+{p}^{2}+\left(-{a}^{2}+2a+1\right)p\right)\left(1+{p}^{2}+\left(-{a}^{2}-2a+1\right)p\right)\bigg{)}^{-7/2}3p\,dp,$
(3.20)

which is an elliptic integral, as the integrand is a rational function of $p$
and the square root of a quartic in $p$. This elliptic integral representation
for $\Lambda_{0}^{2}(a)$ is perfectly acceptable for both numerical evaluation
and the calculation of a series expansion about $a=0$, to any desired order.
For example, the series to octic order is

$\Lambda_{0}^{2}(a)=3-{\frac{10\sqrt{3}\pi}{27}}+\left(7-{\frac{56\sqrt{3}\pi}{81}}\right){a}^{2}+\left(11-{\frac{758\sqrt{3}\pi}{729}}\right){a}^{4}+\left(15-{\frac{9176\sqrt{3}\pi}{6561}}\right){a}^{6}+\left(19-{\frac{34670\sqrt{3}\pi}{19683}}\right){a}^{8}+...$
(3.21)

Fig.1 displays a plot of $\Lambda_{0}^{2}(a).$ In the figure on the left the
octic approximation (3.21) is shown as the dashed line, and is found to be an
excellent approximation, even up to reasonably large monopole separations. The
asymptotic behaviour for large monopole separation ($a\sim 1$) is shown in the
figure on the right. The limit is

$\lim_{a\to 1}(1-a)^{2}\Lambda_{0}^{2}(a)=\frac{1}{2},$ (3.22)

so that for $a\sim 1$ the limiting behaviour, together with (3.3), gives
$\Lambda_{0}^{2}da^{2}\sim 2d\rho^{2},$ associated with a pair of monopoles,
each with geodesic distance $\rho$ to the origin.

Figure 1: The metric function $\Lambda_{0}^{2}(a)$ with octic approximation
(left), and asymptotics (right).

With some effort, the elliptic integral (3.20) can be written in terms of
standard elliptic integrals. The required change of variable is

$p=\frac{1+a+u\sqrt{(3+a)(1-a)}}{1+a-u\sqrt{(3+a)(1-a)}},$ (3.23)

and leads to the formula

$\displaystyle\Lambda_{0}^{2}(a)=2{a}^{-2}\left(a+3\right)^{-2}\left(1-a\right)^{-2}\left(3-a\right)^{-5/2}\left(1+a\right)^{-7/2}$
$\displaystyle\bigg{(}-24{F(\varphi,k)}a\left(a+3\right)\left(1-a\right)^{2}\left({a}^{4}-2{a}^{3}-4{a}^{2}+18a+3\right)$
$\displaystyle+3\Pi(1,\varphi,k)\left(1-a\right)^{3}\left(a+3\right)\left({a}^{6}-27{a}^{4}+99{a}^{2}-9\right)$
$\displaystyle+\sqrt{3-a}\left(1+a\right)^{3/2}\left({a}^{2}+3\right)\left({a}^{6}-9{a}^{4}+63{a}^{2}+9\right)\bigg{)}.$
(3.24)

Here $F(\varphi,k)$ and $\Pi(1,\varphi,k)$ are the standard elliptic integrals
of the first and third kind, defined by

$F(\varphi,k)=\int_{0}^{\sin\varphi}\frac{du}{\sqrt{(1-u^{2})(1-k^{2}u^{2})}},\quad\Pi(1,\varphi,k)=\int_{0}^{\sin\varphi}\frac{du}{\sqrt{(1-u^{2})^{3}(1-k^{2}u^{2})}},$
(3.25)

with amplitude $\varphi$ and elliptic modulus $k$ given by

$\varphi=\sin^{-1}((1+a)/2),\qquad\quad k=\sqrt{\frac{16a}{(1+a)^{3}(3-a)}}.$
(3.26)

A similar calculation for $\Lambda_{3}^{2}(a)$, using the integral formula

$\Lambda_{3}^{2}(a)=\frac{3}{4\pi}\int\bigg{|}\frac{\partial}{\partial
z}\frac{\partial}{\partial\psi}\log h\bigg{|}^{2}_{\bullet}\,idzd\bar{z},$
(3.27)

produces

$\Lambda_{3}^{2}(a)=\frac{3}{8}\int_{0}^{\infty}\bigg{(}{p}^{10}+1-2\left({a}^{2}+a+1\right)\left({p}^{9}+p\right)+\left(2{a}^{4}+12{a}^{2}+8a-1\right)\left({p}^{8}+{p}^{2}\right)+2\left(-{a}^{6}+{a}^{5}+{a}^{4}-10{a}^{3}-15{a}^{2}+5a+7\right)\left({p}^{7}+{p}^{3}\right)+\left({a}^{8}-8{a}^{6}+52{a}^{4}-100{a}^{2}-8a+41\right)\left({p}^{6}+{p}^{4}\right)+4\left(-5{a}^{6}-{a}^{5}+31{a}^{4}+10{a}^{3}-30{a}^{2}-4a+14\right){p}^{5}\bigg{)}\\\
\bigg{(}1+{p}^{2}+\left(-{a}^{2}+2a+1\right)p\bigg{)}^{-7/2}\bigg{(}1+{p}^{2}+\left(-{a}^{2}-2a+1\right)p\bigg{)}^{-5/2}(1-a)^{4}\,dp,$
(3.28)

with an octic approximation

$\Lambda_{3}^{2}(a)=\frac{2}{9}\sqrt{3}\pi-1+\left(\frac{3}{2}-\frac{4}{9}\sqrt{3}\pi\right)a+\left(-\frac{7}{2}+{\frac{26\sqrt{3}\pi}{27}}\right){a}^{2}+\left(6-{\frac{40\sqrt{3}\pi}{27}}\right){a}^{3}+\left(-{\frac{26}{3}}+{\frac{478\sqrt{3}\pi}{243}}\right){a}^{4}+\left({\frac{34}{3}}-{\frac{596\sqrt{3}\pi}{243}}\right){a}^{5}+\left(-14+{\frac{6434\sqrt{3}\pi}{2187}}\right){a}^{6}+\left({\frac{50}{3}}-{\frac{7504\sqrt{3}\pi}{2187}}\right){a}^{7}+\left({\frac{25726\sqrt{3}\pi}{6561}}-{\frac{58}{3}}\right){a}^{8}+...$
(3.29)

Again the transformation (3.23) allows the result to be written in terms of
standard elliptic integrals, yielding

$\Lambda_{3}^{2}(a)=\frac{(1-a)^{2}\left(8(1-a)F(\varphi,k)-(1-a)(3+a)(3+a^{2})\Pi(1,\varphi,k)+(1+a)^{5/2}\sqrt{3-a}\,\right)}{2(1+a)^{7/2}\sqrt{3-a}},$
(3.30)

with the same amplitude and elliptic modulus as earlier, given by (3.26).

In the limit of infinite monopole separation there is axial symmetry around
the $e_{3}$-axis and

$\lim_{a\to 1}\Lambda_{3}^{2}(a)=0.$ (3.31)

In the Euclidean case the analogous moment of inertia tends to a non-zero
constant in the limit of infinite monopole separation, as motion in the
relative phase turns the monopoles into dyons with opposite electric charge,
which contributes to the kinetic energy. In the hyperbolic case the boundary
metric detects this contribution for any finite monopole separation, but not
in the limit of infinite separation. This is because the boundary metric is
obtained by a renormalization of the divergent kinetic energy induced from the
field theory [6], and in the limit $a\to 1$ the associated tangent vector has
finite length and therefore does not contribute to the divergence of the
kinetic energy. This corresponds to the fact that a single hyperbolic monopole
can be promoted to a finite energy hyperbolic dyon.

Performing the calculation

$\Lambda_{2}^{2}(a)=\frac{3}{4\pi}\int\bigg{|}\frac{\partial}{\partial
z}\frac{\partial}{\partial\theta}\log h\bigg{|}^{2}_{\bullet}\,idzd\bar{z},$
(3.32)

reveals that $\Lambda_{2}^{2}(a)=\Lambda_{3}^{2}(-a)$. This is a reflection of
the fact that making the replacement $a\mapsto-a$ in the spectral curve swaps
the $e_{2}$ and $e_{3}$ axes. In particular, the axial symmetry at $a=0$
requires that $\Lambda_{2}^{2}(0)=\Lambda_{3}^{2}(0)$.

Using the relation $\Lambda_{2}^{2}(a)=\Lambda_{3}^{2}(-a)$ together with the
formula (3.30) gives that

$\lim_{a\to 1}(1-a)\Lambda_{2}^{2}(a)=2.$ (3.33)

Therefore in the limit of large monopole separations the asymptotic behaviour
is

$\Lambda_{2}^{2}(a)\sim 2/(1-a),\qquad\mbox{for \ }\ a\sim 1.$ (3.34)

The remaining component of the metric can be calculated using the formula

$\Lambda_{1}^{2}(a)=\frac{3}{4\pi}\int\bigg{|}\frac{\partial}{\partial
z}\frac{\partial}{\partial\phi}\log h\bigg{|}^{2}_{\bullet}\,idzd\bar{z},$
(3.35)

to give the elliptic integral representation

$\Lambda_{1}^{2}(a)=\int_{0}^{\infty}\bigg{(}2({p}^{7}+p)+\left(-4{a}^{2}+4\right)\left({p}^{6}+{p}^{2}\right)+\left(3{a}^{4}-10{a}^{2}+5\right)\left({p}^{5}+{p}^{3}\right)+\left(-{a}^{6}+7{a}^{4}-11{a}^{2}+5\right){p}^{4}\bigg{)}\\\
\bigg{(}1+{p}^{2}+\left(-{a}^{2}+2a+1\right)p\bigg{)}^{-5/2}\bigg{(}1+{p}^{2}+\left(-{a}^{2}-2a+1\right)p\bigg{)}^{-5/2}12a^{2}\,dp$
(3.36)

with octic expansion

$\Lambda_{1}^{2}(a)=\left(12-{\frac{40\sqrt{3}\pi}{27}}\right){a}^{2}+\left({\frac{68}{3}}-{\frac{832\sqrt{3}\pi}{243}}\right){a}^{4}+\left({\frac{100}{3}}-{\frac{3928\sqrt{3}\pi}{729}}\right){a}^{6}+\left(44-{\frac{48224\sqrt{3}\pi}{6561}}\right){a}^{8}+...$
(3.37)

Note that $\Lambda_{1}^{2}(0)=0,$ corresponding to the axial symmetry about
the $e_{1}$ axis when $a=0.$ In terms of standard elliptic integrals the
formula for $\Lambda_{1}^{2}(a)$ is

$\Lambda_{1}^{2}(a)=2\bigg{(}\left(3+a\right)\left(1-a\right)^{2}\left(8a\left(1-3a\right)F(\varphi,k)+\left(3+30{a}^{2}-a^{4}\right)\Pi(1,\varphi,k)\right)\\\
-\left(1+a\right)^{5/2}\sqrt{3-a}\left(3-20{a}^{2}+a^{4}\right)\bigg{)}\left(1+a\right)^{-7/2}\left(3-a\right)^{-3/2}\left(1-a\right)^{-1}\left(3+a\right)^{-1}.$
(3.38)

Using the result

$\lim_{a\to 1}(1-a)\Lambda_{1}^{2}(a)=2,$ (3.39)

confirms that in the limit of large monopole separations ($a\sim 1$) the
asymptotic behaviour is $\Lambda_{1}^{2}(a)\sim
2/(1-a)\sim\Lambda_{2}^{2}(a)$, reflecting the properties of point-like
monopoles.

Figure 2: The metric functions
$\Lambda_{1}^{2}(a),\Lambda_{2}^{2}(a),\Lambda_{3}^{2}(a)$ with octic
approximations (left), and asymptotics (right).

All three metric functions
$\Lambda_{1}^{2}(a),\Lambda_{2}^{2}(a),\Lambda_{3}^{2}(a)$ are displayed in
Fig.2. In the figure on the left the octic expansions are shown as the dashed
lines, and again provide an excellent approximation, with barely discernible
errors at this scale. In the figure on the right the dashed line shows the
common asymptotic behaviour of $\Lambda_{1}^{2}(a)$ and $\Lambda_{2}^{2}(a).$
These plots demonstrate a qualitative agreement with the corresponding metric
functions for the Atiyah-Hitchin manifold [3], modulo the asymptotic value of
$\Lambda_{3}^{2}$, as discussed earlier. This justifies the interpretation of
this 4-manifold as a hyperbolic analogue of the Atiyah-Hitchin manifold and
confirms the similarity of Euclidean and hyperbolic monopole scattering within
the geodesic approximation.

##  4 A rounded cone and a tapered trumpet

The Atiyah-Hitchin manifold contains totally geodesic 2-dimensional
submanifolds, known as the rounded cone and the trumpet, that are surfaces of
revolution. In this section their hyperbolic analogues are presented, together
with a description in terms of rational maps.

Fixing the principal axis $e_{1}$ to be the $x_{3}$-axis corresponds to
setting $\theta=\pi/2$ and $\psi=0$ and is a totally geodesic 2-dimensional
submanifold $\Sigma_{\rm cone}$. The associated spectral curve is

$a(e^{2i\phi}\eta^{2}\zeta^{2}+e^{-2i\phi})+\eta^{2}+\zeta^{2}-(1-a^{2})\eta\zeta=0,$
(4.1)

which shows that $\phi$ has period $\pi$ for this submanifold, rather than
period $2\pi$ in the full manifold. It is therefore convenient to introduce
the coordinate $\widetilde{\phi}=2\phi\,$ on $\Sigma_{\rm cone}$, with period
$2\pi.$ The metric induced from (3.10) is then

$ds^{2}=\Lambda_{0}^{2}\,da^{2}+\frac{1}{4}\Lambda_{1}^{2}\,d\widetilde{\phi}^{2},$
(4.2)

which is a rounded cone. Using the series expansions (3.21) and (3.37) gives
the leading order behaviour at the vertex $a=0$,

$ds^{2}=\frac{1}{27}(81-{10\sqrt{3}\pi})(da^{2}+a^{2}\,d\widetilde{\phi}^{2})+...$
(4.3)

confirming that the cone is smooth at the vertex. This vertex corresponds to
the axially symmetric 2-monopole, with the $x_{3}$-axis being the axis of
symmetry. The generating geodesics of the rounded cone pass through the vertex
and describe the right angle scattering of monopoles in a head-on collision,
with $\widetilde{\phi}$ constant during the evolution, except for a jump by
$\pi$ as the geodesic passes over the cone. This gives a jump of $\pi/2$ in
$\phi$ and hence the monopoles separate at right angles to the direction of
their approach.

The rational map of a hyperbolic $N$-monopole is a degree $N$ based rational
map between Riemann spheres, ${\cal R}(z)$, with the base point condition
${\cal R}(\infty)=0.$ The rational map may be viewed as scattering data in the
background of the hyperbolic $N$-monopole, along the geodesic associated with
the point $(\eta,\zeta)=(0,z)$ in mini-twistor space. The scattering data, and
hence the rational map, is defined only up to the equivalence relation of
multiplication by a phase. There is a bijective correspondence between these
equivalence classes of rational maps and the $(4N-1)$-dimensional moduli space
of hyperbolic $N$-monopoles [4, 12].

To obtain the rational map from the ADHM data of the hyperbolic monopole,
introduce the rank one Hermitian matrix

$H=(1-M_{3})^{-1/2}\bigg{(}1+M_{3}-(M_{1}-iM_{2})(1-M_{3})^{-1}(M_{1}+iM_{2})\bigg{)}(1-M_{3})^{-1/2},$
(4.4)

and let $v$ be its unit-length eigenvector with non-zero eigenvalue $\lambda.$
The formula for the rational map is [11]

${\cal R}(z)=\lambda
v^{\dagger}\bigg{(}z-(1-M_{3})^{-1/2}(M_{1}+iM_{2})(1-M_{3})^{-1/2}\bigg{)}^{-1}\bar{v}.$
(4.5)

Applying this to $\Sigma_{\rm cone}$ provides its rational map description

${\cal
R}(z)=\frac{1-a^{2}}{z^{2}-ae^{-i\widetilde{\phi}}}=\frac{1-|c|^{2}}{z^{2}-c},$
(4.6)

where the complex coordinate $c=ae^{-i\widetilde{\phi}}$, that lies inside the
unit disc, has been introduced. Restricting $c$ to the real interval $(-1,1)$
is a geodesic that describes right angle scattering, with an exchange of the
$e_{2}$ and $e_{3}$ axes as $c$ passes through zero.

To construct a second 2-dimensional geodesic submanifold, begin by setting
$\theta=\phi=0$ to fix the principal axis $e_{3}$ to be the $x_{3}$-axis. This
defines a 2-dimensional submanifold $\Sigma_{\rm trumpet}^{+}$ with spectral
curve

$\frac{1}{4}\left(1-a\right)^{2}\left(e^{i\widetilde{\psi}}{\eta}^{2}{\zeta}^{2}+e^{-i\widetilde{\psi}}\right)+\frac{1}{4}(3+a)(1-a)\left({\eta}^{2}+{\zeta}^{2}\right)-2\left(1+a\right)\eta\zeta=0,$
(4.7)

where $\widetilde{\psi}=2\psi$ has period $2\pi.$ The metric on $\Sigma_{\rm
trumpet}^{+}$ is

$ds^{2}=\Lambda_{0}^{2}\,da^{2}+\frac{1}{4}\Lambda_{3}^{2}\,d\widetilde{\psi}^{2}.$
(4.8)

This surface of revolution is a tapered cylinder with the boundary at $a=0$
given by a circle of radius $\frac{1}{6}\sqrt{2\sqrt{3}\pi-9}$. The radius of
the circular cross-section decreases as $a$ increases and tends to zero as
$a\to 1.$ The rational map associated with $\Sigma_{\rm trumpet}^{+}$ is

${\cal R}(z)=\frac{8(1+a)z}{(1-a)(3+a)z^{2}+(1-a)^{2}e^{-i\widetilde{\psi}}}.$
(4.9)

Setting $\theta=\psi=\pi/2\,$ fixes the principal axis $e_{2}$ to be the
$x_{3}$-axis and defines another 2-dimensional submanifold $\Sigma_{\rm
trumpet}^{-}$ with spectral curve

$\frac{1}{4}\left(1+a\right)^{2}\left(e^{i\widetilde{\psi}}{\eta}^{2}{\zeta}^{2}+e^{-i\widetilde{\psi}}\right)+\frac{1}{4}(3-a)(1+a)\left({\eta}^{2}+{\zeta}^{2}\right)-2\left(1-a\right)\eta\zeta=0,$
(4.10)

where now $\widetilde{\psi}=2\phi+\pi$, which again has period $2\pi.$ The
metric on $\Sigma_{\rm trumpet}^{-}$ is

$ds^{2}=\Lambda_{0}^{2}\,da^{2}+\frac{1}{4}\Lambda_{2}^{2}\,d\widetilde{\psi}^{2}.$
(4.11)

This surface of revolution also has a boundary at $a=0$ given by a circle of
radius $\frac{1}{6}\sqrt{2\sqrt{3}\pi-9}$. This time the radius of the
circular cross-section increases as $a$ increases and tends to infinity as
$a\to 1.$ The rational map description of $\Sigma_{\rm trumpet}^{-}$ is

${\cal R}(z)=\frac{8(1-a)z}{(1+a)(3-a)z^{2}+(1+a)^{2}e^{-i\widetilde{\psi}}}.$
(4.12)

Joining the two surfaces $\Sigma_{\rm trumpet}^{+}$ and $\Sigma_{\rm
trumpet}^{-}$ at their common boundary produces a totally geodesic
2-dimensional submanifold $\Sigma_{\rm trumpet}$, that is a tapered trumpet.
The spectral curves (4.7) and (4.10) are related by the transformation
$a\to-a$, as are the rational maps (4.9) and (4.12). The spectral curve and
rational map for $\Sigma_{\rm trumpet}$ are therefore given by (4.7) and (4.9)
with the range of $a$ extended to $(-1,1)$. The same is true for the metric
(4.8) because of the relation $\Lambda_{2}^{2}(a)=\Lambda_{3}^{2}(-a).$
Defining the complex coordinate $C=e^{-i\widetilde{\psi}}(1-a)/(3+a)$, that
lies inside the unit disc with the origin removed, allows the rational map for
$\Sigma_{\rm trumpet}$ to be written as

${\cal R}(z)=\frac{z(1-|C|^{2})/|C|}{z^{2}+C}.$ (4.13)

The narrow end of the trumpet corresponds to $C$ close to the origin, with the
monopoles on the $x_{3}$-axis, and the broad end of the trumpet is $C$ close
to the unit circle, with the monopoles in the plane $x_{3}=0$. On the circle
$|C|=\frac{1}{3}$ the 2-monopole is axially symmetric, with the axis of
symmetry lying in the plane $x_{3}=0$.

The generating curves of the tapered trumpet are all the geodesics where the
monopoles approach in the plane $x_{3}=0$, form an axially symmetric
2-monopole, and scatter at right angles to emerge along the $x_{3}$-axis.
Geodesics that pass through the trumpet, from the broad end to the narrow end,
generically correspond to dyons separating along the $x_{3}$-axis with motion
in the relative phase. Geodesics that start and end at the broad end describe
monopoles that scatter in the plane $x_{3}=0$, but this can be via
intermediate dyonic motion along the $x_{3}$-axis for an arbitrarily long
period of time.

##  5 Conclusion

The qualitative similarity between the Atiyah-Hitchin manifold and its
hyperbolic analogue indicates that hyperbolic monopole dynamics mirrors the
Euclidean picture, within a geodesic description. However, despite the
resemblance between these two manifolds, there is an important geometric
distinction. The Atiyah-Hitchin manifold is hyperkähler, unlike its hyperbolic
relative, and this difference is most clearly reflected in the methods used to
calculate the two metrics. The condition that a metric of the form (3.10) is
hyperkähler implies a set of first-order differential equations for the metric
functions [13]. This system of differential equations can be solved in terms
of complete elliptic integrals to determine the metric [3]. These differential
equations do not hold for the metric functions of the hyperbolic analogue,
removing this option for their construction and leaving brute-force
integration as the only alternative. Fortunately, by tuning the curvature of
hyperbolic space to a specific value (minus one in the units used here), the
twistor data is sufficiently amenable that the integration required to
calculate the boundary metric is tractable. The result is more complicated
than the elegant Atiyah-Hitchin metric, being written in terms of incomplete
rather than complete elliptic integrals, this being another reflection of its
geometric inferiority. An attempt could be made to try and calculate the
boundary metric for monopoles in hyperbolic space with an arbitrary constant
negative curvature, but even if tractable, the result will be an even more
unpleasant generalisation of the formulae presented here.

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