# Magnetism and superconductivity in doped triangular-lattice Mott insulators

Kun Woo Kim  Department of Physics, Chung-Ang University, 06974 Seoul,
Republic of Korea  T. Pereg-Barnea  Department of Physics, McGill University,
Montreal, Quebec H3A 2T8, Canada

(September 3, 2024)

###### Abstract

Inspired by recent advances in the fabrication of surface superlattices, and
in particular the triangular lattice made of tin (Sn) atoms on silicon, we
study an extended Hubbard mode on a triangular lattice. The observations of
magnetism in these systems justify the inclusion of a strong on-site repulsion
and the observation of superconductivity suggests including an effective,
nearest-neighbor attractive interaction. The attractive interaction mimics the
effect of strong on-site repulsion near half filling, which can be seen in
strong coupling vertex calculations such as the Eliashberg method. With this
extended Hubbard model on a triangular lattice with its geometrical
frustration, we find a rich phase diagram of various magnetic orders and
pairing functions, within the framework of self-consistent mean field theory.
We uncover the competition among magnetism and unconventional
superconductivity, and their coexistence for triplet pairings. We follow the
Fermi surface of the system as the system is doped away from half filling and
find nesting vectors and a Lifshitz transition which provide an intuitive
understanding of the phase transitions between the many orders we consider.

##  I Introduction

Recent theoretical studies of superconductivity in triangular lattice systems
have been motivated by a series of experimental investigation [1, 2, 3, 4]
where Sn adatoms placed on the surface of Si(111) form a two-dimensional
triangular lattice, showing some evidence for unconventional chiral d-wave
superconductivity. The magnetic ordering of the same superlattice system has
been intensively studied as well both theoretically and experimentally [5, 6,
7, 8]. These kind of systems combine the geometric frustration of the
triangular lattice with strong electronic correlations coexist and can, at
least in theory, bring about a variety of states of matter such as a spin
liquid, collinear antiferromagnet and spiral magnetic order at half filling as
well as several unconventional superconducting orders away from half filling.
Here, we explore the competition between magnetism and superconductivity in a
wide range of doping and interaction strengths.

The studies of magnetic orders in a triangular lattice [9, 10, 11, 12] and
others investigate the possibility of unconventional superconductivity [13,
14, 15, 16, 17] and point to the possibility of both singlet and triplet
superconductors with and without topological numbers. In this manuscript we
focus on the competition between many order parameters, both magnetic and
superconducting. We do so in self-consistent mean field theory where the on-
site Hubbard $U$ interaction favors magnetic order while pairing on bonds is
favored by an attraction term on nearest neighbor sites. This attraction is an
effective description of strong correlations; the result of a coulomb
repulsion and fermi surface nesting [14]. The tight binding model in the
triangular model comes with the long range hoping sharpening the saddle of
dispersion relation. We confirm not only the appearance of chiral d-wave
superconductivity and collinear antiferromagnetic ordering as reported in
experiments, but also find triplet pairing and the spiral magnetic ordering at
other fillings and interaction strength. Crucially, we provide direct and
intuitive understanding of magnetic phases from Fermi surface nesting and can
relate the favored superconducting state to a synergy between the Fermi
surface and the pairing function such that nodes are avoided and gaps are
maximized.

The organization of the manuscript is the following. In section II we
introduce a model with on-site repulsion and nearest neighbor attraction, and
then the mean field Hamiltonian with magnetism and superconductivity is
constructed. In section III we construct the grand potential and derive the
self consistency relations for multiple order parameters. In sections IV and V
we present our results and discuss them.

Figure 1:  (a) A triangular lattice with six nearest neighbors of the central
atom (in red) marked with different colors. (b) The dispersion relation
$E(k_{x},k_{y})$ and the filling
$\nu=\langle\hat{n}_{i\uparrow}+\hat{n}_{i\downarrow}\rangle\equiv\langle\hat{n}_{i}\rangle$.
Energy in unit of the nearest neighbor hopping amplitude,
$|t_{1}|=52.8(\text{meV})$. (c) The relation between filling and the energy,
and the DoS and energy.

##  II Extended Hubbard model on a triangular lattice

The kinetic part of our Hamiltonian is composed of tight binding hopping
parameters $t_{l}$ between the $l^{\text{th}}$ neighbors on the triangular
lattice, proposed to match ARPES data in Ref. [8]. The parameters where chosen
to fit the lowest energy band of the Sn/Si(111) surface:

$\displaystyle\hat{H}_{0}=\sum_{l=1}^{6}t_{l}\sum_{\langle
ij\rangle_{l}}\hat{c}^{\dagger}_{i}\hat{c}_{j},$ (1)

where $t_{1}=-52.7\text{meV}$, and the longer-range hopping amplitudes
$t_{2}/t_{1},...,t_{6}/t_{1}$ are -0.3881, 0.1444, -0.0228, 0 and -0.0318,
respectively. The indices $\langle i,j\rangle_{l}$ run over all pairs of $l$th
nearest neighbor sites. The Fermi surface is depicted as a function of filling
in Fig. 1(b) along with the density of states (DoS) as a function of energy.
We include the onsite and extended Hubbard interaction terms:

$\displaystyle\hat{H}_{\text{int}}=\sum_{i}U_{0}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}+\sum_{\langle
ij\rangle,\sigma\sigma^{\prime}}U_{1}\hat{n}_{i\sigma}\hat{n}_{j\sigma^{\prime}},$
(2)

such that $U_{0}>0$ is repulsive, and $U_{1}<0$ is attractive. The on-site
repulsive interaction favors magnetism. It is convenient to express the
Hubbard $U_{0}$ term as (Appendix A):

$\displaystyle\hat{H}^{(\text{on})}=\sum_{i}U_{0}\left[\frac{1}{4}\hat{n}_{i}\hat{n}_{i}-\hat{S}_{im}\hat{S}_{im}\right],$
(3)

where $\hat{n}_{i}=\hat{n}_{i\uparrow}+\hat{n}_{i\downarrow}$ is the occupancy
at site $i$, with $\langle\hat{n}_{i}\rangle\in[0,2]$. The spin operator at
site $i$ in direction $\hat{m}$ is defined as
$\hat{S}_{im}=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}\hat{c}^{\dagger}_{i\sigma}(\vec{\sigma}\cdot\hat{m})_{\sigma\sigma^{\prime}}\hat{c}_{i\sigma^{\prime}}$,
where $\vec{\sigma}$ is the vector of Pauli matrices. We can therefore define
a local spin order parameter as $\vec{m}=\langle\hat{S}_{im}\rangle$, where
the direction of $\vec{m}$ may give either ferromagnetic order or
antiferromagnetic (AF) order with collinear or spiral spin directions as
depicted in Fig.2(b). The magnitude $m=|\vec{m}|$ represents the strength of
the order. The mean field Hamiltonian is then given by:

$\displaystyle\hat{H}^{(\text{on})}_{\text{MF}}=$ $\displaystyle-
U_{0}N_{\text{lat}}\left[\frac{1}{4}\langle\hat{n}_{i}\rangle^{2}-|\vec{m}|^{2}\right]$
$\displaystyle+U_{0}\sum_{i}\begin{pmatrix}\hat{c}^{\dagger}_{i\uparrow}&\hat{c}^{\dagger}_{i\downarrow}\end{pmatrix}\left[\frac{1}{2}\langle\hat{n}_{i}\rangle\sigma_{0}-\vec{m}\cdot\vec{\sigma}\right]\begin{pmatrix}\hat{c}_{i,\uparrow}\\\
\hat{c}_{i,\downarrow}\end{pmatrix},$ (4)

where the first term describes the onsite potential energy cost of having the
magnetic order while the second term provides a possible energy benefit from a
spin splitting. Note that the band energy shifts by
$\frac{1}{2}\langle\hat{n}_{i}\rangle\sigma_{0}$ due to the mean repulsive
energy from the onsite interaction. To take into account the collinear and
spiral AF magnetism we extended the unit cell to include 2 or 3 atoms with
rotated spins in our calculations.

The attractive interaction $U_{1}<0$ can induce superconductivity or a charge
density wave. We focus on superconductivity using BCS-like the self-consistent
mean field:

$\displaystyle\hat{H}^{(\text{nn})}_{\text{MF}}=$ $\displaystyle-
U_{1}\sum_{\langle
ij\rangle,\sigma\sigma^{\prime}}|\Delta_{ji,\sigma^{\prime}\sigma}|^{2}$
$\displaystyle+U_{1}\sum_{\langle
ij\rangle,\sigma,\sigma^{\prime}}\left[c_{i,\sigma}^{\dagger}c_{j,\sigma^{\prime}}^{\dagger}\Delta_{ji,\sigma^{\prime}\sigma}+\Delta^{*}_{ij,\sigma\sigma^{\prime}}c_{j\sigma^{\prime}}c_{i\sigma}\right].$
(5)

The superconducting order parameter $\Delta_{ji,\sigma^{\prime}\sigma}=\langle
c_{j\sigma^{\prime}}c_{i\sigma}\rangle$ may describe either singlet or triplet
spin pairing with spatial symmetry of $s$-, $p$-,$d$\- or $f$-wave. We
consider superconductivity and magnetism together to determine which
combination of order parameters yields the lowest grand potential. It is worth
noting that in order to consider superconductivity and magnetism
simultaneously, the Hamiltonian in Eq.(4) needs to be written in the
Bogoliubov-de Gennes (BdG) form with particle-hole symmetry (see Appendix B).
The mean field hamiltonian is

$\displaystyle\hat{H}_{\text{MF}}=E_{0}(n_{i},\vec{m},\tilde{\Delta}^{(s/t)}_{ji,\sigma^{\prime}\sigma})$
$\displaystyle+\frac{1}{2}\sum_{k}^{BZ/2}\Psi^{\dagger}\begin{pmatrix}H_{kk}&&&\tilde{\Delta}_{k,-k}\\\
&H_{-k,-k}&\tilde{\Delta}_{-k,k}&\\\
&-\tilde{\Delta}^{*}_{k,-k}&-H_{kk}^{*}&\\\
-\tilde{\Delta}^{*}_{-k,k}&&&-H_{-k-k}^{*}\end{pmatrix}\Psi,$ (6)

where
$\tilde{\Delta}_{k^{\prime}k^{\prime\prime}}=U_{1}\Delta_{k^{\prime}k^{\prime\prime}}$
and $H_{k^{\prime}k^{\prime\prime}}$ are 2 by 2 matrices in spin space. The
BdG Hamiltonian is written in the basis
$\Psi^{\dagger}=\left(c^{\dagger}_{k\uparrow},c^{\dagger}_{k\downarrow},c^{\dagger}_{-k\uparrow},c^{\dagger}_{-k\downarrow},c_{k\uparrow},c_{k\downarrow},c_{-k\uparrow},c_{-k\downarrow}\right)$.
The constant energy
$E_{0}(n_{i},\vec{m},\tilde{\Delta}^{(s/t)}_{ji,\sigma^{\prime}\sigma})$
contains the usual BCS ground state energy as well as terms resulting from the
anti-commutation relation of the operators which compose the magnetic order
parameters:

$\displaystyle E_{0}$
$\displaystyle=-U_{0}N_{\text{lat}}\left[\frac{1}{4}\langle\hat{n}_{i}\rangle^{2}-|\vec{m}|^{2}\right]$
$\displaystyle+\frac{1}{2}U_{0}N_{\text{lat}}\langle\hat{n}_{i}\rangle-
U_{1}\sum_{\langle
ij\rangle,\sigma\sigma^{\prime}}|\Delta_{ji,\sigma^{\prime}\sigma}|^{2},$ (7)

The Hamiltonian written in this structure visibly satisfies the particle hole
symmetry which is represented by the operator $\mathcal{P}=K\tau_{x}$, where
$K$ is complex conjugation and the Pauli matrix $\tau_{x}$ acts exchanges
particles and holes, such that $\mathcal{P}H_{BdG}\mathcal{P}^{-1}=-H_{BdG}$
(see Appendix B for the discussion of the PHS in the basis $\Psi^{\dagger}$).
The block diagonal Hamiltonian in Eq. (6) is

$\displaystyle
H_{kk}=\left[\epsilon_{\vec{k}}+\frac{1}{2}U_{0}\langle\hat{n}_{i}\rangle\right]\sigma_{0}-\vec{m}\cdot\vec{\sigma},$
(8)

where $\epsilon_{\vec{k}}$ is the Fourier transform of Eq. (1) (see 111
$\epsilon_{\vec{k}}=-2t_{1}\left[\cos(k_{x})+2\cos(\frac{\sqrt{3}}{2}k_{y})\cos(\frac{k_{x}}{2})\right]\\\
-2t_{2}\left[\cos(\sqrt{3}k_{y})+2\cos(\frac{3}{2}k_{x})\cos(\frac{\sqrt{3}}{2}k_{y})\right]\\\
-2t_{3}\left[\cos(2k_{x})+2\cos(k_{x})\cos(\sqrt{3}k_{y})\right]\\\
-4t_{4}\left[\cos(\frac{5}{2}k_{x})\cos(\frac{\sqrt{3}}{2}k_{y})+\cos(2k_{x})\cos(\sqrt{3}k_{y})\right.\\\
\left.+\cos(\frac{k_{x}}{2})\cos(\frac{3\sqrt{3}}{2}k_{y})\right]-2t_{6}\left[\cos(2\sqrt{3}k_{y})+2\cos(3k_{x})\cos(\sqrt{3}k_{y})\right]$.
where the fifth nearest neighbor hopping strength $t_{5}=0$. for its explicit
expression). The summation over crystal momentum is on the half of the
Brillouin zone because our basis contains both $|k\sigma\rangle$ and
$|\text{-}k\sigma\rangle$ state. The BdG Hamiltonian (8 by 8) in the full
basis is block diagonal with two 4 by 4 blocks with opposite sign of
eigenvalues. Note that one could choose to either work with this $8\times 8$
block-diagonal Hamiltonian and sum over half of the Brillouin zone or work
with only one of the blocks and sum over the entire Brillouin zone. The
pairing potential $\Delta_{k,-k}$ is given by:

$\displaystyle(\Delta_{k,-k})_{\sigma\sigma^{\prime}}=\sum_{\vec{\delta}_{ji}}\Delta_{ji,\sigma^{\prime}\sigma}e^{-i\vec{k}\cdot\vec{\delta}_{ji}},$

where the sum is over the six nearest neighbor vectors,
$\vec{\delta}_{ji}=\vec{r}_{j}-\vec{r}_{i}\in\\{(1,0),(\frac{1}{2},\frac{\sqrt{3}}{2}),(-\frac{1}{2},\frac{\sqrt{3}}{2}),(-1,0),(-\frac{1}{2},-\frac{\sqrt{3}}{2}),(\frac{1}{2},-\frac{\sqrt{3}}{2})\\}$,
with angles
$\theta_{ji}\in\\{0,\frac{\pi}{3},\frac{2\pi}{3},\pi,\frac{4\pi}{3},\frac{5\pi}{3}\\}$
relative to the $x$-axis. The symmetry of the pairing function in real space
(odd or even with respect to exchanging the sites $i$ and $j$) determines
whether the spins of the Cooper pair is in the singlet or triplet
configuration such that $\Delta_{ji,\sigma\sigma^{\prime}}$ is antisymmetric
to the exchange of both position and spin.

$\displaystyle\Delta_{ji,\sigma\sigma^{\prime}}=\chi^{(\text{s/t})}_{\sigma\sigma^{\prime}}\otimes\phi^{(\text{even/odd})}_{ji}.$
(9)

and:

$\displaystyle\chi^{(s)}=\begin{pmatrix}0&+\Delta_{s}\\\
-\Delta_{s}&0\end{pmatrix},\,\,\,\chi^{(t)}=\begin{pmatrix}\Delta_{\uparrow\uparrow}&\Delta_{t}\\\
\Delta_{t}&\Delta_{\downarrow\downarrow}\end{pmatrix},$ (10)

We label the spatial pairing functions by their angular momentum; the phase
winding number around a central atom such that for s-wave $\phi^{(s)}_{ji}=1$
and for angular momentum $l$ a chiral pairing function winds $l$ times and is
given by $\phi^{(l)}_{jj^{\prime}}\propto e^{il\theta_{jj^{\prime}}}$.
However, since we do not want to impose chirality a priori, we minimize the
mean field energy with two pairing functions for each angular momentum:

$\displaystyle\phi^{(p_{x})}_{ji}=\cos\theta_{ji},\,\,\,$
$\displaystyle\phi^{(p_{y})}_{ji}=\sin\theta_{ji},$ (11)
$\displaystyle\phi^{(d_{x^{2}-y^{2}})}_{ji}=\cos 2\theta_{ji},\,\,\,$
$\displaystyle\phi^{(d_{xy})}_{ji}=\sin 2\theta_{ji},$ (12)
$\displaystyle\phi^{(f_{x(x^{2}-3y^{2})})}_{ji}=\cos 3\theta_{ji},\,\,\,$
$\displaystyle\phi^{(f_{y(3x^{2}-y^{2})})}_{ji}=\sin 3\theta_{ji},$ (13)

such that the chiral $p$-, $d$-, or $f$-wave is obtained if both pairing
functions are non-zero and there’s a $\pi/2$ phase difference between them.
Otherwise we end up with a non-chiral state. In the case of $f$-wave, the
$f_{y(3x^{2}-y^{2})}$ is zero for nearest neighbor links in the triangular
lattice and we therefore end up with a non-chiral, real order parameter of the
form of $f_{x(x^{2}-3y^{2})}$. A chiral $f$-wave order would require longer
range attraction.

##  III Self-consistency equations for order parameters

The grand potential is obtained from the grand canonical partition function
$Z=\text{Tr}\,e^{-\beta(\hat{H}_{\text{MF}}-\mu\hat{N})}$ in the diagonal
basis.

$\displaystyle\hat{H}_{\text{MF}}-\mu\hat{N}=E_{0}+\frac{1}{2}\sum_{k,\alpha}^{BZ/2}\left[\zeta_{k,\alpha}\hat{\gamma}^{\dagger}_{k\alpha}\hat{\gamma}_{k\alpha}-\zeta_{k,\alpha}\hat{\gamma}_{k\alpha}\hat{\gamma}^{\dagger}_{k\alpha}\right],$
(14)

where $\alpha$ labels the eigen values and $\zeta_{k,\alpha}$ is an eigenvalue
of one sub-block BdG Hamiltonian (6). Due to particle hole symmetry, the
eigenvalues in the two sub blocks of the Hamiltonian each have a pair of
identical eigenvalues with opposite sign. Summing over all (single particle)
eigen states we obtain the grand potential:

$\displaystyle\Omega=-\frac{1}{\beta}\ln
Z=E_{0}-\frac{1}{\beta}\sum^{\frac{1}{2}\text{BZ}}_{k,\alpha}\ln\left[2\cosh\frac{\beta\zeta_{k,\alpha}}{2}\right],$
(15)

where the temperature $\beta=k_{B}T=0.1|t_{1}|$ is taken and the sum above is
only over positive $\zeta_{k,\alpha}$ values. Note that this temperature of
about 61K is required for convergence at our momentum resolution. The minimum
of the grand potential is the self consistent solution for the order
parameters and we therefore set the grand potential derivatives with respect
to the order parameters to zero. We write the derivatives as (see Appendix C):

$\displaystyle\frac{\partial\Omega}{\partial m_{j}}$
$\displaystyle=\frac{\partial E_{0}}{\partial m_{j}}+\frac{\partial}{\partial
m_{j}}\left[\Omega-E_{0}\right]=0,$ (16)
$\displaystyle\frac{\partial\Omega}{\partial\Delta_{\nu}^{*}}$
$\displaystyle=\frac{\partial
E_{0}}{\partial\Delta_{\nu}^{*}}+\frac{\partial}{\partial\Delta_{\nu}^{*}}\left[\Omega-
E_{0}\right]=0,$ (17)

where $j\in\\{x,y,z\\}$ and
$\nu\in\\{s,t,\uparrow\uparrow,\downarrow\downarrow\\}$. We solve the above
conditions through iterations and update all order parameters at every step.
Using the fact that the energy $E_{0}$ contains $\sum_{j=x,y,z}m_{j}^{2}$ and
$\Delta_{\nu}\Delta_{\nu}^{*}$, we can update an order parameter through
iteration. That is,

$\displaystyle m_{j}^{\text{(new)}}$
$\displaystyle=m_{j}^{\text{(old)}}\left[1-\eta\frac{\partial\Omega}{\partial{m_{j}}}\left(\frac{\partial
E_{0}}{\partial m_{j}}\right)^{-1}\right]_{\text{old}},$ (18)
$\displaystyle\Delta_{\nu}^{\text{(new)}}$
$\displaystyle=\Delta_{\nu}^{\text{(old)}}\left[1-\eta\frac{\partial\Omega}{\partial\Delta^{*}_{\nu}}\left(\frac{\partial
E_{0}}{\partial\Delta^{*}_{\nu}}\right)^{-1}\right]_{\text{old}}.$ (19)

where the right hand side is computed using a set of order parameters to be
updated. $\eta\,(\simeq 0.2)$ controls the rate of approaching speed toward a
convergence over iterations. When converged,
$m_{j}^{\text{(new)}}=m_{j}^{\text{(old)}}$ and
$\left.\partial_{m_{j}}\Omega\right|_{\text{old}}=0$. The partial
differentiation is numerically obtained by computing the difference of grand
potentials, $\partial_{m_{j}}\Omega\simeq\lim_{\Delta m_{j}\rightarrow
0}\Delta\Omega/\Delta m_{j}$. The mean field Hamiltonian is also a function of
filling $\nu$, see Eq.(4). Along with other order parameters, the filling is
updated in each iteration by computing

$\displaystyle\nu=-\frac{1}{N_{\text{lat}}}\frac{\partial\Omega}{\partial\mu}=\langle\hat{n}_{i\uparrow}+\hat{n}_{i\downarrow}\rangle.$
(20)

Figure 2:  (a) The phase diagram at $U_{1}/U_{0}=-0.5$. (b) Three types of
magnetic orderings. (c) The collinear AF and spiral AF appear from the Fermi
surface nesting depicted. The ferromagnetism appears from the Stoner’s
criterion.

##  IV Results: topological superconductivity and magnetism

Magnetic orders–: The Stoner criterion gives a heuristic rule for when
magnetic order might develop; if the density of states at the Fermi level
exceeds a critical value set by the onsite repulsive interaction, magnetic
order could develop to reduce the interaction energy. This intuition is indeed
in line with our findings - Fig. 1 (c) shows that at filling near $\nu=0.77,2$
the DoS is peaked and this is where we find ferromagnetism. Another important
factor in determining whether or not magnetism will appear and the kind of
magnetism that will develop is the shape of the Fermi surface. A magnetic
order which reduces the periodicity is most effective if its ordering vector
connects many states near the Fermi level, i.e., opens a gap through nesting.

Around, $\nu\sim 1$, collinear antiferromagnetism appears. This order reduces
the periodicity by folding the Brillouin zone in one of three directions; one
such spin configuration is shown in Fig. 2(b). The periodicity of the
collinear antiferromagnet (CAF) in real space is $\sqrt{3}a$, which doubles
the size of the unit cell. Thus, the size of the nesting vector
$|\vec{Q}_{\text{CAF}}|=2\pi/\sqrt{3}a$ in momentum space is half of the
shortest reciprocal lattice vector. The possible nesting Fermi surface lines
are indicated in Fig. 2(c). However, since the CAF ordering appears only in
one direction, the Fermi surface is not fully gapped by this order, and the
system remains metallic.

Figure 3:  (a) The phase diagram at $U_{1}/U_{0}=-0.8$. Chiral d-wave and
p-wave superconductivity appear next to the anti-ferromagnetic orderings. At
lower and higher filling, f-wave and s-wave superconductivity appear,
respectively. (b) The magnitude of pairing
$|\Delta^{\text{(pairing)}}_{k,-k\sigma\sigma^{\prime}}|$ is plotted for the
orbital symmetries. The overlaid Fermi surface (blue lines) at filling
$\nu=1/3,1,5/3$ show that nodal lines of the pairing are avoided maximizing
the superconducting energy gap opening.

Right next to the CAF order, the spiral antiferromagnetic (SAF) is the most
energetically favorable phase. The filling is a bit higher, and the
corresponding Fermi surface is marked by green in Fig. 2(c). The size of the
nesting vector is ${4\pi\over 3a}$ which is smaller than that of the CAF case,
while the directions of the three vectors are rotated by $\pi/2$. The spiral
AF gaps the Fermi surface completely, lowering the grand potential even more
than the CAF.

The development of the three magnetic orderings in a small doping range can be
understood from the density of states (DoS) and the shape of the Fermi surface
of the long range hopping tight binding model in the triangular lattice model
for filling $\nu\in[0.8,1.5]$. The result of the self consistent mean field
with multiple order parameters is shown in Fig. 2 (a) for $U_{0}/|t_{1}|=6$
and $U_{1}/|t_{1}|=-3$. The bandwidth of the hopping model in Eq. (1) ($\sim
10|t_{1}|$) as shown in Fig.1 is the larger than employed interaction
strength. We note that previous authors [8] found that the CAF is preferred
over the SAF at half filling $\nu=1$ using a more sophisticated numerical
method. When the attractive nearest neighbor attraction $U_{1}$ increases
further, topological superconductivity appears next to the antiferromagnetic
orders.

Superconductivity–: The study of superconductivity in the one band triangular
lattice models was ignited by the experimental observation of
superconductivity in Sn/Si(111) [1, 2]. We therefore set out to find out what
kind of superconductivity can emerge from our effective nearest neighbor
attraction while on-site repulsion is present. This approach to interaction
based unconventional superconductivity provides an intuitive understanding on
how order parameters compete to balance decreasing the interaction energy
while increasing the kinetic energy through orderings. We therefore minimize
our grand potential by considering both magnetism and superconductivity
together.

Figure 4:  (a) The phase diagram as a function of filling $\nu$ and the
attractive interaction strength $U_{1}$ for $U_{0}=-6|t_{1}|$. (b,c) The
corresponding superconducting and magnetic order parameters are plotted in the
same domain. They show the portion of order parameters for magnetism
coexisting with triplet pairing superconductivity, p+ip and f-wave SC.

Figure 3(a) shows the phase diagram as a function of filling with
$U_{1}/U_{0}=-0.8$. The $f$-wave SC appears first and as can be seen in Fig.
3(b), it has three pairs of line nodes. For filling $\nu\in[0.2,0.7]$, the
Fermi surface is located where the nodal lines are avoided and therefore the
$f$-wave pairing fully gaps the system, reducing the interaction energy which
results from $U_{1}$.

As the filling is increased beyond 0.7, the Fermi surface undergoes a Lifshitz
transition from six pockets to a single snow-flake-shaped surface at the
center of the Brillouin zone. Once the transition has occurred, the $f$-wave
can not fully gap the Fermi surface and the preferred pairing changes. The
Fermi surface that favors the CAF order for reducing the on-site interaction
energy favors the chiral $d+id$ superconductor for reducing the $U_{1}$
interaction energy. The competition between the two phases is studied by
minimizing the grand potential with the two orders present and this yields the
result that $d+id$ superconductors appears before the collinear
antiferromagnet.

Around $\nu=3/4$ the spiral antiferromagnet is favored (despite the
significant $U_{1}$) but at higher filling of $\nu>1.5$ another topological
superconductor with $p+ip$ structure is developed. This order parameter has
nodal points which are avoided by the small Fermi surface around the zone
center as shown in Fig. 3(b)(bottom-left) such that the Fermi surface is fully
gapped.

Lastly, when every site is nearly fully filled, $\nu\sim 2$, the s-wave
pairing potential which has a circular line node away from $\vec{k}=0$ is
preferred. At the high filling, there is a transition from the ferromagnetic
ordering to the $s$-wave superconductivity with increasing attractive extended
interaction strength $|U_{1}|$, see Fig. 4(a).

We note that the order of superconducting phases (chiral p-wave and f-wave) we
find is inconsistent with that of Wolf et al. in Refs. [14, 16] but it is
consistent with that of Cheng et al. in Ref. [13]. While we rely on an
effective interaction for our mean field calculation, we believe that the
filling in which we find the various orders is very plausible since it is
compatible with the intuition provided by the Fermi surface shapes at the
various filling as discussed above.

From magnetic ordering to superconducting–: The phase diagram is drawn in Fig.
4(a) for $U_{1}/U_{0}\in[-0.2,1.2]$. At small $U_{1}$ only magnetic orders
appear. For $U_{1}/U_{1}<-0.4$ superconductivity begins to appear. While the
singlet pairing does not coexist with the magnetic orderings, the triplet
pairing may. We verify the coexistence and also plot the superconductor and
the magnetic order parameters separately in Fig. 4(b) and (c), respectively.
In particular $f$-wave superconductivity coexists with ferromagnetism near
$\nu=0.75$, and the CAF and SAF orders give way to $p+ip$ pairing when
$|U_{1}|$ is increased.

##  V Conclusions

In this work we explored the possibility of magnetic and superconducting
orders in the extended Hubbard model on a triangular lattice. Our model
includes a repulsive on-site Hubbard interaction $U_{0}$ and an effective
nearest neighbor attractive interaction $U_{1}$. We treat the model with
variational, self-consistent mean field theory which considers a large set of
magnetic and superconducting orders together. We map the phase diagram and
find a ferromagnetic phase and two antiferromagnetic phases when the
attractive interaction is weak. for higher values of the attractive
interaction we find superconducting states with $s$-, $d$-, $p$\- and $f$-wave
symmetry. The $p$-wave and $d$-wave superconducting order parameters are found
to be chiral/topological.

Near filling $\nu=1-1.5$ the collinear anti-ferromagnetism and the spiral
magnetism are consistent with previous studies [8], are found to coexist with
a $p+ip$ triplet topological superconductivity when the attractive interaction
is significant. Our finding of $f$-wave and $p+ip$-wave superconductivity at
low and high filling is also consistent with a previous study [13], and we
find a $d+id$ topological superconductor to emerge when long range hopping is
included in the kinetic energy.

Our study has been inspired by recent advances in surface manipulation and in
particular the creation of a triangular superlattice of tin atoms on a silicon
surface [9, 10, 11, 12]. However, we expect these results to hold for other
similar compounds. In particular, our finding of chiral topological
superconductivity could lead to the realization of Majorana zero modes at
vortex cores in these compounds. The proximity of these phases to other, non-
topological phases suggests that the existence of Majorana modes could be
controlled through gating and external fields.

Lastly we would like to add a note about temperature. Due to our limited
momentum resolution we could not perform calculations at a very low
temperature and used $\beta=k_{B}T=0.1|t_{1}|=5.28\text{(meV)}$ which is $\sim
61\text{(K)}$. This meant that in order to see gaps open we needed to work
with large interaction values. We therefore believe that at lower temperature
one could see superconductivity with even weaker interactions.

Acknowledgements - K.W.K. acknowledges that this research was supported by the
Chung-Ang University Research Grants in 2021. TPB acknowledges support from
the Natural Sciences and Engineering Research Council of Canada (NSERC).

##  Appendix A Mean-Field Decomposition of The Hubbard Interaction

The onsite Hubbard interaction can be written as the density and spin
operators using
$\hat{n}_{i\uparrow}=\sum_{\alpha\beta}\frac{1}{2}(1+\sigma_{z})_{\alpha\beta}\hat{c}^{\dagger}_{i\alpha}\hat{c}_{i\beta}$.
More in general, the interaction can be written on the basis of arbitrary spin
direction $\sigma_{l}=\hat{l}\cdot\vec{\sigma}$. Dropping the site index,

$\displaystyle\hat{n}_{\uparrow}\hat{n}_{\downarrow}$
$\displaystyle=\sum_{\alpha\beta\gamma\delta}\frac{1}{4}(1+\sigma_{l})_{\alpha\beta}(1-\sigma_{l})_{\gamma\delta}\hat{c}^{\dagger}_{\alpha}\hat{c}_{\beta}\hat{c}^{\dagger}_{\gamma}\hat{c}_{\delta},$
$\displaystyle=\sum_{\alpha\beta\gamma\delta}\frac{1}{4}\left(\delta_{\alpha\beta}\delta_{\gamma\delta}-(\sigma_{l})_{\alpha\beta}(\sigma_{l})_{\gamma\delta}\right)\hat{c}^{\dagger}_{\alpha}\hat{c}_{\beta}\hat{c}^{\dagger}_{\gamma}\hat{c}_{\delta},$

where from the first line to the second the following relation is used:
$\sum_{\alpha\beta\gamma\delta}(\sigma_{l})_{\gamma\delta}\delta_{\alpha\beta}\hat{c}^{\dagger}_{\alpha}\hat{c}_{\beta}\hat{c}^{\dagger}_{\gamma}\hat{c}_{\delta}=\sum_{\alpha\beta\gamma\delta}(\sigma_{l})_{\alpha\beta}\delta_{\gamma\delta}\hat{c}^{\dagger}_{\gamma}\hat{c}_{\delta}\hat{c}^{\dagger}_{\alpha}\hat{c}_{\beta}$.
As a result,

$\displaystyle\hat{H}^{\text{(on)}}$
$\displaystyle=U_{0}\sum_{i}\left[\frac{1}{4}\hat{n}_{i}\hat{n}_{i}-\hat{S}_{il}\hat{S}_{il}\right],$
(21)

where $\hat{n}_{i\sigma}=\hat{c}^{\dagger}_{i\sigma}\hat{c}_{i\sigma}$,
$\hat{S}_{il}=\frac{1}{2}\sum_{\alpha,\beta}\hat{c}^{\dagger}_{i\alpha}(\hat{l}\cdot\vec{\sigma})_{\alpha\beta}\hat{c}_{i\beta}$.
Neglecting the fluctuating part of the interaction, the mean field
approximation is

$\displaystyle\hat{n}_{i}^{2}$ $\displaystyle\simeq
2\hat{n}_{i}\langle\hat{n}_{i}\rangle-\langle\hat{n}_{i}\rangle^{2},$ (22)
$\displaystyle\hat{S}_{il}^{2}$ $\displaystyle\simeq
2\hat{S}_{il}\langle\hat{S}_{il}\rangle-\langle\hat{S}_{il}\rangle^{2}.$ (23)

The mean field Hamiltonian (4) is obtained.

##  Appendix B Particle-Hole Symmetry and Grand Potential

The BdG Hamiltonian (6) can be written as the sum of two sub blocks:

$\displaystyle\hat{H}-\mu\hat{N}$
$\displaystyle=\frac{1}{2}\sum_{k}^{\text{BZ/2}}\begin{pmatrix}\hat{C}^{\dagger}_{k}&\hat{C}_{-k}\end{pmatrix}\begin{pmatrix}\tilde{H}_{k,k}&\tilde{\Delta}_{k,-k}\\\
-\tilde{\Delta}_{-k,k}^{*}&-\tilde{H}_{-k,-k}^{*}\end{pmatrix}\begin{pmatrix}\hat{C}_{k}\\\
\hat{C}_{-k}^{\dagger}\end{pmatrix}$
$\displaystyle+\frac{1}{2}\sum_{k}^{\text{BZ/2}}\begin{pmatrix}\hat{C}_{k}&\hat{C}^{\dagger}_{-k}\end{pmatrix}\begin{pmatrix}-\tilde{H}_{k,k}&-\tilde{\Delta}_{k,-k}\\\
\tilde{\Delta}^{*}_{-k,k}&\tilde{H}_{-k,-k}^{*}\end{pmatrix}^{*}\begin{pmatrix}\hat{C}_{k}\\\
\hat{C}_{-k}^{\dagger}\end{pmatrix},$ (24)

where
$\hat{C}^{\dagger}_{k}=\begin{pmatrix}c^{\dagger}_{k\uparrow}&c^{\dagger}_{k_{\downarrow}}\end{pmatrix}$
and $\tilde{H}_{k,k}=H_{k,k}-\mu$. The hermiticity of the sub block
Hamiltonians are guaranteed because $\Delta^{T}=-\Delta$. By the
diagonalization,

$\displaystyle\begin{pmatrix}\tilde{H}_{k,k}&\tilde{\Delta}_{k,-k}\\\
-\tilde{\Delta}_{-k,k}^{*}&-\tilde{H}_{-k,-k}^{*}\end{pmatrix}$
$\displaystyle=U\begin{pmatrix}\zeta_{1}&0\\\
0&\zeta_{2}\end{pmatrix}U^{\dagger},$ (25)
$\displaystyle\begin{pmatrix}-\tilde{H}_{k,k}&-\tilde{\Delta}_{k,-k}\\\
\tilde{\Delta}_{-k,k}^{*}&\tilde{H}_{-k,-k}^{*}\end{pmatrix}^{*}$
$\displaystyle=U^{*}\begin{pmatrix}-\zeta_{1}&0\\\
0&-\zeta_{2}\end{pmatrix}U^{T},$ (26)

which verifies that the BdG Hamiltonian (6) contains pairs of eigenvalues with
the same magnitude and the opposite sign. Note that in general
$\zeta_{1}\neq-\zeta_{2}$ when $\tilde{H}_{k,k}\neq\tilde{H}_{-k,-k}$. Thus,
within each sub block Hamiltonian, there is no particle hole symmetry as it is
sometimes assumed in literature. The full spin and particle-hole basis must be
employed for a system without inversion symmetry. In the eigen state basis,
the Hamiltonian can be written as

$\displaystyle\hat{H}-\mu\hat{N}$
$\displaystyle=\frac{1}{2}\sum_{k}^{\text{BZ/2}}\begin{pmatrix}\hat{\gamma}_{k,1}^{\dagger}&\hat{\gamma}_{k,2}^{\dagger}\end{pmatrix}\begin{pmatrix}\zeta_{k,1}&0\\\
0&\zeta_{k,2}\end{pmatrix}\begin{pmatrix}\hat{\gamma}_{k,1}\\\
\hat{\gamma}_{k,2}\end{pmatrix}$
$\displaystyle+\frac{1}{2}\sum_{k}^{\text{BZ/2}}\begin{pmatrix}\hat{\gamma}_{k,1}&\hat{\gamma}_{k,2}\end{pmatrix}\begin{pmatrix}-\zeta_{k,1}&0\\\
0&-\zeta_{k,2}\end{pmatrix}\begin{pmatrix}\hat{\gamma}_{k,1}^{\dagger}\\\
\hat{\gamma}_{k,2}^{\dagger}\end{pmatrix},$ (27)

where
$\begin{pmatrix}\hat{\gamma}_{k,1}^{\dagger}&\hat{\gamma}_{k,2}^{\dagger}\end{pmatrix}=\begin{pmatrix}\hat{C}_{k}^{\dagger}&\hat{C}_{-k}\end{pmatrix}U$
and $U^{*}=U^{\dagger T}$ are used. This is the mean field Hamiltonian Eq.
(14) used for the construction of the grand canonical partition function.

Let us deduce the PHS relation that the block Hamiltonian matrix satisfies.
The second term in Eq. (B) can be written as

$\displaystyle\hat{H}-\mu\hat{N}=\frac{1}{2}\sum_{k}^{\text{BZ/2}}\begin{pmatrix}\hat{C}^{\dagger}_{k}&\hat{C}_{-k}\end{pmatrix}\begin{pmatrix}\tilde{H}_{k,k}&\tilde{\Delta}_{k,-k}\\\
-\tilde{\Delta}_{-k,k}^{*}&-\tilde{H}_{-k,-k}^{*}\end{pmatrix}\begin{pmatrix}\hat{C}_{k}\\\
\hat{C}_{-k}^{\dagger}\end{pmatrix}$
$\displaystyle+\frac{1}{2}\sum_{k}^{\text{BZ/2}}\begin{pmatrix}\hat{C}^{\dagger}_{-k}&\hat{C}_{k}\end{pmatrix}\tau_{x}\begin{pmatrix}-\tilde{H}_{k,k}&-\tilde{\Delta}_{k,-k}\\\
\tilde{\Delta}^{*}_{-k,k}&\tilde{H}_{-k,-k}^{*}\end{pmatrix}^{*}\tau_{x}\begin{pmatrix}\hat{C}_{-k}^{\dagger}\\\
\hat{C}_{k}\end{pmatrix}.$ (28)

The first term on the right hand side is the sum over momentum in one half of
the Brillouin zone, and the second term is over the other half. And, the
Hamiltonian in the second term is related to the first one by the following
relation:

$\displaystyle H_{\text{bld}}(-k)$
$\displaystyle=\tau_{x}\begin{pmatrix}-\tilde{H}_{k,k}&-\tilde{\Delta}_{k,-k}\\\
\tilde{\Delta}^{*}_{-k,k}&\tilde{H}_{-k,-k}^{*}\end{pmatrix}^{*}\tau_{x},$
(29) $\displaystyle=-\tau_{x}\mathcal{K}H_{\text{bld}}(k)\tau_{x}\mathcal{K},$
(30)

where subscript ${}_{\text{bld}}$ is to indicate the Hamiltonian matrices in
Eq. (B). $H_{\text{bdg}}(k)$ is the Hamltonian matrix in the first term on the
right hand side.

Next, let us deduce the PHS relation when the Hamiltonian matrix is
constructed in the extended basis (also see pedagogical note in Ref.[19]) as
in Eq. (6). There is an unitary transformation between creation operators in
real and momentum space:

$\displaystyle\Psi_{k}=\begin{pmatrix}\vec{C}_{k}\\\
\vec{C}^{\dagger}_{k}\end{pmatrix}=\begin{pmatrix}V&\\\
&V^{*}\end{pmatrix}\begin{pmatrix}\vec{C}_{r}\\\
\vec{C}_{r}^{\dagger}\end{pmatrix}.$ (31)

where the Fourier transformation
$(V)_{ij}=\frac{1}{\sqrt{N_{\text{lat}}}}e^{-i\vec{k}_{i}\cdot\vec{r}_{j}}$
and

$\displaystyle\vec{C}_{k}$ $\displaystyle=(c_{k_{1}},\cdots,c_{k_{N}})^{T},$
(32) $\displaystyle\vec{C}^{\dagger}_{k}$
$\displaystyle=(c^{\dagger}_{k_{1}},\cdots,c^{\dagger}_{k_{N}})^{T},$ (33)
$\displaystyle\vec{C}_{r}$ $\displaystyle=(c_{r_{1}},\cdots,c_{r_{N}})^{T},$
(34) $\displaystyle\vec{C}^{\dagger}_{r}$
$\displaystyle=(c^{\dagger}_{r_{1}},\cdots,c^{\dagger}_{r_{N}})^{T},$ (35)

where spin and sublattice degree of freedom can be added in the operator
vectors. Let $\Psi_{r}=\begin{pmatrix}\vec{C}_{r}\\\
\vec{C}_{r}^{\dagger}\end{pmatrix}$. Likewise,

$\displaystyle\Psi^{\dagger}_{k}=\Psi_{r}^{\dagger}\begin{pmatrix}V^{\dagger}&\\\
&V^{*\dagger}\end{pmatrix}.$ (36)

Note that $\Psi_{k}^{\dagger}$ and $\Psi_{k}$ take the same form with the one
used in the mean field Hamiltonian, Eq. (6). The Hamiltonian operator is

$\displaystyle\hat{H}$
$\displaystyle=\Psi^{\dagger}_{k}H_{k}\Psi_{k}=\Psi_{r}^{\dagger}\begin{pmatrix}V^{\dagger}&\\\
&V^{*\dagger}\end{pmatrix}H_{k}\begin{pmatrix}V&\\\
&V^{*}\end{pmatrix}\Psi_{r},$ (37)
$\displaystyle=-\Psi_{r}^{\dagger}\tau_{x}\mathcal{K}\begin{pmatrix}V^{\dagger}&\\\
&V^{*\dagger}\end{pmatrix}H_{k}\begin{pmatrix}V&\\\
&V^{*}\end{pmatrix}\tau_{x}\mathcal{K}\Psi_{r},$ (38)
$\displaystyle=-\Psi_{r}^{\dagger}\begin{pmatrix}V^{\dagger}&\\\
&V^{*\dagger}\end{pmatrix}\tau_{x}\mathcal{K}H_{k}\tau_{x}\mathcal{K}\begin{pmatrix}V&\\\
&V^{*}\end{pmatrix}\Psi_{r},$ (39)
$\displaystyle=\Psi^{\dagger}_{k}\tau_{x}(-H_{k}^{*})\tau_{x}\Psi_{k},$ (40)

where in the second line we used
$H_{r}=-\tau_{x}\mathcal{K}H_{r}\tau_{x}\mathcal{K}$. In the third line

$\displaystyle\tau_{x}\mathcal{K}\begin{pmatrix}V&\\\
&V^{*}\end{pmatrix}\tau_{x}\mathcal{K}=\begin{pmatrix}V&\\\
&V^{*}\end{pmatrix}$ (41)

is used. As shown in the last line, the PHS relation in the momentum space is
$H_{k}=-\tau_{x}\mathcal{K}H_{k}\mathcal{K}\tau_{x}$. This verifies that when
Hamiltonian is written in the basis containing the creation and annihilation
operators with the same set of indices as in $\Psi_{r}$ and $\Psi_{k}$, the
particle-hole symmetry relation is equally applied in the form of
$\mathcal{P}H_{r}\mathcal{P}=-H_{r}$ and $\mathcal{P}H_{k}\mathcal{P}=-H_{k}$.
Note the sign difference in momentum compared to Eq. (29), which applies to
block diagonal Hamiltonian matrices in $H_{k}$.

##  Appendix C Self-Consistency Relations

The mean field Hamiltonian includes order parameters whose values are
determined self consistently. We start by some initial value for each order
parameter and update the values at every iteration step in the following way.
With the ’old’ set of order parameter values we calculate the mean field
energies which are then used to calculate the grand potential. The ’new’ set
of order parameter values are then calculated from the grand potential and the
process is continued until conversions, when the values do not change between
iterations.

Calculating the order parameter from the grand potential can be demonstrated
for ferromagnetic order in the $z$ direction, $m_{z}$. Differentiating the
grand potential by $m_{z}$ gives,

$\displaystyle{d\Omega\over dm_{z}}=\frac{\partial(\Omega-E_{0})}{\partial
m_{z}}+{dE_{0}\over dm_{z}},$ (42)

where $\partial_{m_{z}}E_{0}=U_{0}N_{\text{lat}}\left(2m_{z}\right)$, and

$\displaystyle\frac{\partial(\Omega-E_{0})}{\partial m_{z}}$
$\displaystyle=-\beta^{-1}\frac{1}{Z}\frac{\partial Z}{\partial m_{z}}$
$\displaystyle=-\frac{U_{0}}{Z}\text{Tr}\left[\left(\sum_{i}\hat{c}^{\dagger}_{i\uparrow}\hat{c}_{i\uparrow}-\hat{c}^{\dagger}_{i\downarrow}\hat{c}_{i\downarrow}\right)e^{-\beta(\hat{H}-\mu\hat{N})}\right]$
$\displaystyle\equiv-
U_{0}\langle\sum_{i}^{N_{lat}}\hat{c}^{\dagger}_{i\uparrow}\hat{c}_{i\uparrow}-\hat{c}^{\dagger}_{i\downarrow}\hat{c}_{i\downarrow}\rangle,$
(43)

Therefore, when the minimum of the grand potential is found,
$\partial_{m_{z}}\Omega=0$,

$\displaystyle
m_{z}=\frac{1}{2}\langle\hat{c}^{\dagger}_{i\uparrow}\hat{c}_{i\uparrow}-\hat{c}^{\dagger}_{i\downarrow}\hat{c}_{i\downarrow}\rangle,$
(44)

which is consistent with the way $\hat{S}_{iz}$ operator is defined in Eq.(3).
For CAF and the SAF, we prepare the order parameter in an enlarged unit cell.
The relative spin angle within the unit cell is arranged in such a way that it
realizes the desired spin ordering and the strength of the magnetism is the
order parameter which is found self-consistently.

The self-consistent relation for the superconducting order parameter similarly
follows. Let us begin from the real space expression Eq.(5).

$\displaystyle\frac{\partial(\Omega-
E_{0})}{\partial\Delta_{ji,\sigma^{\prime}\sigma}}$
$\displaystyle=\frac{U_{1}}{Z}\text{Tr}\left[\left(\hat{c}^{\dagger}_{i\sigma}\hat{c}^{\dagger}_{j\sigma^{\prime}}\right)e^{-\beta(\hat{H}-\mu\hat{N})}\right],$
(45)
$\displaystyle=U_{1}\langle\hat{c}^{\dagger}_{i\sigma}\hat{c}^{\dagger}_{j\sigma^{\prime}}\rangle,$
(46)

where we pick a specific site index $i,j$ for the pairing order parameter,
hence there is no summation. Since
$\partial_{\Delta_{ji,\sigma^{\prime}\sigma}}E_{0}=-U_{1}\Delta_{ji,\sigma^{\prime}\sigma}^{*}$
we arrive at,

$\displaystyle\Delta_{ji,\sigma^{\prime}\sigma}^{*}=\langle\hat{c}^{\dagger}_{i\sigma}\hat{c}^{\dagger}_{j\sigma^{\prime}}\rangle.$
(47)

Note the positions of indices,
$(\Delta^{\dagger})_{ij,\sigma\sigma^{\prime}}=\Delta_{ji,\sigma^{\prime}\sigma}^{*}$.
For a general search of superconductivity with a certain pairing symmetry, we
transform the Hamiltonian to momentum space using
$\hat{c}_{i\sigma}^{\dagger}=\frac{1}{\sqrt{N_{\text{lat}}}}\sum_{k}e^{ikr_{i}}\hat{c}_{k\sigma}^{\dagger}$:

$\displaystyle
H_{\text{sc}}=U_{1}\sum^{\text{BZ}}_{k,\sigma\sigma^{\prime}\delta}\Delta_{\delta,\sigma^{\prime}\sigma}\hat{c}^{\dagger}_{k\sigma}c^{\dagger}_{-k\sigma^{\prime}}e^{-ik\delta}+h.c.$
(48)

where $\delta$ goes over the six nearest neighbor vectors. The summation in
momentum is then divided by half. This is to prepare the BdG Hamiltonian where
the basis $\Psi^{(\dagger)}$ includes both $k$ and $-k$.

$\displaystyle H_{\text{sc}}$
$\displaystyle=U_{1}\sum^{\text{BZ/2}}_{k,\sigma\sigma^{\prime}}\left(\sum_{\delta}\Delta_{\delta,\sigma^{\prime}\sigma}e^{-ik\delta}\right)\hat{c}^{\dagger}_{k\sigma}c^{\dagger}_{-k\sigma^{\prime}}$
$\displaystyle+U_{1}\sum^{\text{BZ/2}}_{k,\sigma\sigma^{\prime}}\left(\sum_{\delta}\Delta_{\delta,\sigma^{\prime}\sigma}e^{ik\delta}\right)\hat{c}^{\dagger}_{-k\sigma}c^{\dagger}_{k\sigma^{\prime}}+h.c.$
(49)

where the bracket in the first and the second term are
$\Delta_{k-k,\sigma\sigma^{\prime}}$ and $\Delta_{-kk,\sigma\sigma^{\prime}}$
in the BdG Hamiltonian Eq.(6). The order parameter
$\Delta_{\delta,\sigma^{\prime}\sigma}=\chi_{\sigma^{\prime}\sigma}\phi_{\delta}$.
Where the spin configuration is in $\chi_{\sigma,\sigma^{\prime}}$ and the
desired order parameter structure is $\phi_{\delta}$ which encodes a uniform
order parameter whose magnitude and phase may depend on the bond direction.
The magnitude of the order parameter is included in
$\chi_{\sigma^{\prime}\sigma}$ and we therefore find it iteratively by
performing the differentiation of the grand potential,

$\displaystyle\frac{\partial(\Omega-
E_{0})}{\partial\chi_{\sigma^{\prime}\sigma}}$
$\displaystyle=\frac{U_{1}}{Z}\text{Tr}\left[\left(\sum_{k\delta}^{\text{BZ}}\phi_{\delta}e^{-ik\delta}\hat{c}^{\dagger}_{k\sigma}\hat{c}^{\dagger}_{-k\sigma^{\prime}}\right)e^{-\beta(\hat{H}-\mu\hat{N})}\right],$
$\displaystyle=U_{1}\sum_{k\delta}^{\text{BZ}}\phi_{\delta}e^{-ik\delta}\langle\hat{c}^{\dagger}_{k\sigma}\hat{c}^{\dagger}_{-k\sigma^{\prime}}\rangle,$
(50)
$\displaystyle=U_{1}\sum_{k\delta}^{\text{BZ}}\phi_{\delta}e^{-ik\delta}\frac{1}{N_{\text{lat}}}\langle\sum_{i}e^{-ikr_{i}}\hat{c}^{\dagger}_{i\sigma}\sum_{j}e^{ikr_{j}}\hat{c}^{\dagger}_{j\sigma^{\prime}}\rangle,$
(51)

where in the last line we bring it back to real space expression using
$\hat{c}_{k\sigma}^{\dagger}=\frac{1}{\sqrt{N_{\text{lat}}}}\sum_{i}e^{-ikr_{i}}\hat{c}_{i\sigma}^{\dagger}$.
the summation over momentum yields the delta function
$\delta(r_{j}-r_{i}-\delta)$. As a result,

$\displaystyle\frac{\partial(\Omega-
E_{0})}{\partial\chi_{\sigma^{\prime}\sigma}}$
$\displaystyle=U_{1}\sum_{r_{i}\delta}\phi_{\delta}\langle\hat{c}^{\dagger}_{i\sigma}\hat{c}^{\dagger}_{i+\delta_{\sigma}^{\prime}}\rangle,$
$\displaystyle=U_{1}N_{\text{lat}}\sum_{\delta}\phi_{\delta}\langle\hat{\Delta}^{\dagger}_{\delta,\sigma\sigma^{\prime}}\rangle,$
$\displaystyle=U_{1}N_{\text{lat}}\langle\hat{\chi}^{\dagger}_{\sigma\sigma^{\prime}}\rangle\sum_{\delta}|\phi_{\delta}|^{2},$
(52)

where the summation over momentum is restored to the whole BZ by combining $k$
and $-k$ terms. Because the constant energy term provides

$\displaystyle\partial_{\chi_{\sigma^{\prime}\sigma}}E_{0}$
$\displaystyle=-U_{1}\sum_{ij}\partial_{\chi_{\sigma^{\prime}\sigma}}|\Delta_{\delta,\sigma^{\prime}\sigma}|^{2},$
$\displaystyle=-U_{1}\chi_{\sigma^{\prime}\sigma}^{*}N_{\text{lat}}\sum_{\delta}|\phi_{\delta}|^{2}.$
(53)

where
$\Delta_{\delta,\sigma^{\prime}\sigma}=\chi_{\sigma^{\prime}\sigma}\phi_{\delta}$
is used in the first line. Therefore, at the minimum of the grand potential,
when $\partial_{\chi_{\sigma^{\prime}\sigma}}\Omega=0$ we have,

$\displaystyle\chi^{*}_{\sigma^{\prime}\sigma}=\langle\hat{\chi}^{\dagger}_{\sigma\sigma^{\prime}}\rangle,$
(54)

where
$\langle\hat{\chi}^{\dagger}_{\sigma\sigma^{\prime}}\rangle=\langle\hat{\chi}^{*}_{\sigma^{\prime}\sigma}\rangle$.
Therefore,

$\displaystyle\Delta^{\dagger}_{\delta,\sigma\sigma^{\prime}}=\langle\hat{\Delta}^{\dagger}_{\delta,\sigma\sigma^{\prime}}\rangle,$
(55)

This verifies the superconductivity self consistency relation.

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  * Note [1] $\epsilon_{\vec{k}}=-2t_{1}\left[\mathop{cos}\nolimits(k_{x})+2\mathop{cos}\nolimits(\frac{\sqrt{3}}{2}k_{y})\mathop{cos}\nolimits(\frac{k_{x}}{2})\right]\\\ -2t_{2}\left[\mathop{cos}\nolimits(\sqrt{3}k_{y})+2\mathop{cos}\nolimits(\frac{3}{2}k_{x})\mathop{cos}\nolimits(\frac{\sqrt{3}}{2}k_{y})\right]\\\ -2t_{3}\left[\mathop{cos}\nolimits(2k_{x})+2\mathop{cos}\nolimits(k_{x})\mathop{cos}\nolimits(\sqrt{3}k_{y})\right]\\\ -4t_{4}\left[\mathop{cos}\nolimits(\frac{5}{2}k_{x})\mathop{cos}\nolimits(\frac{\sqrt{3}}{2}k_{y})+\mathop{cos}\nolimits(2k_{x})\mathop{cos}\nolimits(\sqrt{3}k_{y})\right.\\\ \left.+\mathop{cos}\nolimits(\frac{k_{x}}{2})\mathop{cos}\nolimits(\frac{3\sqrt{3}}{2}k_{y})\right]-2t_{6}\left[\mathop{cos}\nolimits(2\sqrt{3}k_{y})+2\mathop{cos}\nolimits(3k_{x})\mathop{cos}\nolimits(\sqrt{3}k_{y})\right]$. where the fifth nearest neighbor hopping strength $t_{5}=0$.
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# An Algorithm to Solve the Generalized Conjugacy Problem in Relatively
Hyperbolic Groups

Chris Karpinski 111McGill University, email:
<EMAIL_ADDRESS>
###### Abstract

In this note, we provide a (super-exponential time) algorithm to solve the
generalized conjugacy problem in relatively hyperbolic groups, given
solvability of the generalized conjugacy problem in each of the parabolic
subgroups.

††2010 Mathematics Subject Classification: 20F65, 20F67, 20F70, 05C25.††Key
words and phrases. Algorithmic problems, generalized conjugacy problem,
relatively hyperbolic groups.††This work was supported by NSERC grant
RGPIN-2016-06154

##  1 Introduction

The generalized conjugacy problem (GCP), also known as the simultaneous
conjugacy problem or the conjugacy problem for finite lists, asks whether
there exists an algorithm such that given as input a recursive presentation
for a recursively presented group $G$ and two finite lists
$\\{a_{1},...,a_{m}\\},\\{b_{1},...,b_{m}\\}$ of elements of $G$ decides
whether or not these two lists are conjugate, that is, if there exists an
element $x\in G$ such that $a_{i}^{x}=b_{i}$ for all $i=1,...,m$ (recall that
the notation $a^{x}$ means $x^{-1}ax$).

The GCP is known to be solvable in several classes of groups. Ladra and Silva
gave a solution to the GCP in finitely generated virtually free groups in [7].
Bezverkhnii showed in [1] that the GCP is solvable in Artin groups of large
type. The solvability of the GCP in braid groups and in general linear groups
was studied by Hahn, Lee and Park in [6]. An important class of groups for
which the GCP is solvable is the class of hyperbolic groups, as was first
shown by Bridson and Howie in [3]. However, Bridson and Howie were unable to
produce a sub-exponential time algorithm that dealt with lists of torsion
elements. In [4], a linear run-time algorithm was constructed by Buckley and
Holt that solves the GCP for lists of torsion elements as well, thereby
establishing a linear run-time algorithm to solve the GCP in hyperbolic
groups.

The solvability of the GCP in the more general relatively hyperbolic groups
(with sufficient algorithmic solvability conditions on the parabolic
subgroups, as usual) has remained an open problem. In this paper, we give an
algorithm to solve the GCP in relatively hyperbolic groups, with parabolic
subgroups $\\{H_{1},...,H_{N}\\}$ each having solvable GCP.

Notation and Setup

Throughout this paper, we fix a group $G$ generated by a finite set $X$
hyperbolic relative to finitely generated subgroups $\\{H_{1},...,H_{N}\\}$.
We assume that the GCP is solvable in each parabolic subgroup $H_{i}$. We
assume that $X$ satisfies the conditions of Lemma 3.1 from [8] (note that as
shown in [8], such finite generating set $X$ always exists) and that $X$ is
symmetrized, that is, $X=X^{-1}$. We will denote $\Gamma:=\Gamma(G;X)$ the
(non-relative) Cayley graph of $G$ with respect to $X$ and
$\hat{\Gamma}:=\Gamma(G;X\cup\mathcal{H})$ (where
$\mathcal{H}=\cup_{i=1}^{N}H_{i}$) the relative Cayley graph of $G$ with
respect to $X,\mathcal{H}$. Given an element $g\in G$, as usual
$|g|_{X},|g|_{X\cup\mathcal{H}}$ denote the word lengths of $g$ with respect
to the generating sets $X$ and $X\cup\mathcal{H}$, respectively. The notation
$B_{r}(x)$ denotes the closed ball of radius $r$ about the point $x$ in
$\hat{\Gamma}$ with its natural combinatorial metric and similarly
$B_{r}^{X}(x)$ denotes the closed ball of radius $r$ about the point $x$ in
$\Gamma$ with its natural combinatorial metric.

Main result

Our goal will be to prove the following theorem.

######  Theorem 1.

There exists a recursive function $f:\mathbb{N}\rightarrow\mathbb{N}$ such
that the following holds. Let
$\\{a_{1},...,a_{m}\\},\\{b_{1},...,b_{m}\\}$ be conjugate lists of elements
in $G$ and let $\mu=\max\\{|a_{i}|_{X},|b_{i}|_{X}:i=1,...,m\\}$. Then there
exists an element $x$ conjugating $\\{a_{1},...,a_{m}\\}$ to
$\\{b_{1},...,b_{m}\\}$ such that $|x|_{X}\leq f(\mu)$.

Theorem 1 immediately produces an algorithm to solve the GCP in $G$, namely,
for each $x\in B^{X}_{f(\mu)}(1)$ check if $a_{i}^{x}=b_{i}$ for all
$i=1,...,m$ using the solution to the word problem in $G$ (which exists since
the word problem is solvable in each $H_{i}$; see section 5.1 in [8]). If
these checks fail for all $x\in B^{X}_{f(\mu)}(1)$ then by Theorem 1 the lists
are not conjugate, otherwise the lists are conjugate. The function $f$ that we
will produce will be super-exponential in $\mu$. We shall not attempt to
produce algorithms with optimal efficiency in this paper, though we conjecture
that there exists a polynomial time algorithm to solve the GCP in relatively
hyperbolic groups provided the GCP can be solved in polynomial time in the
parabolic subgroups.

To prove Theorem 1, we will make use of the following results from Section 3.4
of [8] (see [8] for the definitions of symmetric pairs of geodesics,
$k$-similar geodesics, synchronous components/vertices, etc.):

######  Theorem (8, Theorem 3.37).

For any $k\geq 0$, there exists a constant $\chi=\chi(k)$ with the following
property. Let $(p,q)$ be a symmetric pair of $k$–similar geodesics in
$\hat{\Gamma}$, $u,v$ a pair of synchronous vertices on $p$ and $q$,
respectively. Then $d_{X}(u,v)\leq\chi$.

######  Lemma (8, Lemma 3.43).

For any $k\geq 0,\lambda\geq 1,c\geq 0$, there exists a constant
$\eta=\eta(\lambda,c,k)$ such that the following condition holds. Let $(p,q)$
be an arbitrary symmetric pair of $k$–similar $(\lambda,c)$–quasi–geodesics
without backtracking in $\hat{\Gamma}$ such that no synchronous components of
$p$ and $q$ are connected. Then for any $i=1,...,N$ and any $H_{i}$–component
$e$ of $p$, we have $d_{X}(e_{-},e_{+})\leq\eta\ell([p_{-},e_{+}])$ where
$[p_{-},e_{+}]$ is the segment of $p$.

Acknowledgements

The author is very grateful to Inna Bumagin for introducing the author to the
subject of this note, for continuous support and advice, for providing helpful
suggestions on earlier drafts of this note and for many useful discussions
which have aided the author tremendously in producing this work.

##  2 Proof of Theorem 1

###  2.1 Bounding the relative length of conjugating elements

We first show that for arbitrary conjugate lists
$\\{a_{1},...,a_{m}\\},\\{b_{1},...,b_{m}\\}$ we may bound above the relative
length of a shortest (in relative length) conjugating element.

######  Theorem 2.

Let $\\{a_{1},...,a_{m}\\},\\{b_{1},...,b_{m}\\}$ be two conjugate lists. Let
$\mu=\max\\{|a_{i}|_{X},|b_{i}|_{X}:i=1,...,m\\}$. Let $x$ be an element of
minimal relative length such that
$\\{a_{1},...,a_{m}\\}^{x}=\\{b_{1},...,b_{m}\\}$. Then
$|x|_{X\cup\mathcal{H}}\leq(|X|^{\chi(\mu)}+1)^{|X|^{\mu}+1}$, where
$\chi(\mu)$ is the constant from Theorem 3.37 in [8].

###### Proof.

We follow a similar idea to the proof of Theorem 3.3 in [3].

First, note that we may assume that the $a_{i}$ are all distinct, as if there
are any repetitions in the list, then we may simply remove the repeated
elements from $\\{a_{1},...,a_{m}\\}$ and $\\{b_{1},...,b_{m}\\}$ and this
will not affect the conjugacy of the original list (the list with repeated
elements).

We begin by fixing a geodesic word representing $x$:
$x=g_{1}g_{2}...g_{\ell}$, where $g_{i}\in X\cup\mathcal{H}$ for all
$i=1,...,\ell$. We shall also denote $x_{j}:=g_{1}...g_{j}$ for each
$j=1,...,\ell$. Suppose, for contradiction, that
$\ell>(|X|^{\chi(\mu)}+1)^{|X|^{\mu}+1}$.

Note that for each $i=1,...,m$, the equality $a_{i}^{x}=b_{i}$ produces a
geodesic quadrilateral $Q_{i}$ in the relative Cayley graph $\hat{\Gamma}$ as
shown in Figure 1, where the $a_{i},b_{i}$ are represented by relative
geodesics, which are labeled by fixed geodesic words over $X\cup\mathcal{H}$
that we have chosen for the $a_{i},b_{i}$. In each $Q_{i}$, the yellow paths
$\alpha_{i},\beta_{i}$ (which form the top and bottom edges, respectively, of
the geodesic quadrilateral $Q_{i}$) are $\mu$-similar symmetric geodesics in
$\hat{\Gamma}$ labeled by the geodesic word $g_{1}...g_{\ell}$ representing
$x$.

For any $j=1,...,\ell$, we examine the synchronous subpaths
$p_{ij}\subseteq\alpha_{i}$ and $q_{ij}\subseteq\beta_{i}$ which are each
labeled by the subword $x_{j}$ of $x$ (see Figure 1). Denoting
$u_{ij}:=(p_{ij})_{+},v_{ij}:=(q_{ij})_{+}$, we have that $u_{ij},v_{ij}$ are
synchronous vertices on $\alpha_{i},\beta_{i}$ so by Theorem 3.37 in [8] we
have $d_{X}(u_{ij},v_{ij})\leq\chi(\mu)$, where $\chi(\mu)$ is a constant from
Theorem 3.37 in [8] depending only on $\mu$. Note that
$v_{ij}^{-1}u_{ij}=a_{i}^{x_{j}}$, so we obtain
$|a_{i}^{x_{j}}|_{X}\leq\chi(\mu)$ for each $i=1,...,m$ and $j=1,...,\ell$.

Figure 1: The geodesic quadrilateral $Q_{i}$ representing the conjugacy
$a_{i}^{x}=b_{i}$ and the synchronous subpaths $p_{ij},q_{ij}$ representing
the subword $x_{j}$ of $x$

Thus, for each $j=1,...,\ell$, the tuple $(a_{1}^{x_{j}},...,a_{m}^{x_{j}})$
is an element of $(B^{X}_{\chi(\mu)}(1))^{m}$, the cardinality of which is
bounded above by $(|X|^{\chi(\mu)}+1)^{m}$. Let us now bound above $m$ in
terms of $\mu$.

As $|b_{i}|_{X}\leq\mu$ for all $i$ and since the $b_{i}$ are all distinct (as
the $a_{i}$ are all distinct and the lists are conjugate), we have
$m\leq|B^{X}_{\mu}(1)|\leq|X|^{\mu}+1$.

Therefore, as $j$ ranges through $\\{1,...,\ell\\}$, the number of distinct
tuples $(a_{1}^{x_{j}},...,a_{m}^{x_{j}})$ that may be produced is at most
$(|X|^{\chi(\mu)}+1)^{|X|^{\mu}+1}$. Since
$\ell>(|X|^{\chi(\mu)}+1)^{|X|^{\mu}+1}$, by the Pigeonhole Principle, there
exist $1\leq s<t\leq\ell$ such that
$(a_{1}^{x_{s}},...,a_{m}^{x_{s}})=(a_{1}^{x_{t}},...,a_{m}^{x_{t}})$, i.e.
$a_{i}^{x_{s}x_{t}^{-1}}=a_{i}$ for all $i=1,...,m$. This yields that
$x_{s}x_{t}^{-1}x$ conjugates $\\{a_{1},...,a_{m}\\}$ to
$\\{b_{1},...,b_{m}\\}$. However, note that
$|x_{s}x_{t}^{-1}x|_{X\cup\mathcal{H}}\leq|x_{s}|_{X\cup\mathcal{H}}+|x_{t}^{-1}x|_{X\cup\mathcal{H}}=s+\ell-t<\ell$,
contradicting the minimality of $\ell$.

We therefore conclude that $\ell\leq(|X|^{\chi(\mu)}+1)^{|X|^{\mu}+1}$.

∎

######  Remark 3.

Note that Theorem 2 gives a short proof of the solvability of the GCP in $G$
when $G$ is a hyperbolic group (where $\mathcal{H}=\\{1\\}$).

###  2.2 Bounding parabolic components of conjugating elements

We bound parabolic components of minimal relative length conjugating elements
to establish the following theorem, which yields Theorem 1.

######  Theorem 4.

There is a recursive function $g:\mathbb{N}\rightarrow\mathbb{N}$ such that
the following holds. Suppose that
$\\{a_{1},...,a_{m}\\},\\{b_{1},...,b_{m}\\}$ are conjugate lists of elements.
Let $\mu=\max\\{|a_{i}|_{X},|b_{i}|_{X}:i=1,...,m\\}$. There exists a
conjugating element $x$ between the two lists such that
$|x|_{X}\leq(|X|^{\chi(\mu)}+1)^{|X|^{\mu}+1}g(\mu)$.

###### Proof.

Let $x$ be a shortest (in relative length) conjugating element between the two
lists. By Theorem 2, we have
$|x|_{X\cup\mathcal{H}}\leq(|X|^{\chi(\mu)}+1)^{|X|^{\mu}+1}$. Fixing a
geodesic word $w$ representing $x$, as in the proof of Theorem 2, we have $m$
geodesic quadrilaterals representing simultaneous conjugacy diagrams as shown
in Figure 2, where each orange geodesic path has label the fixed geodesic word
that we have chosen for $x$ and each $a_{i},b_{i}$ are represented by fixed
geodesics. In each geodesic quadrilateral the geodesics whose label represents
$x$ are a symmetric pair of $\mu$-similar geodesics.

We consider two cases:

Case 1: There does not exist an $H_{i}$-syllable of $w$ such that the
$H_{i}$-component corresponding to this syllable is connected to its
synchronous component in each quadrilateral $Q_{j}$. Then for each syllable
$w$, we may choose some quadrilateral $Q_{i}$ such that the component labeled
$h$ corresponding to this syllable is not connected to its synchronous
component on the opposite side of $Q_{i}$. Looking at the symmetric pair of
geodesics labeled by this component $h$, these geodesics are
$\chi(\mu)$-similar by Theorem 3.37 in [8] because their endpoints are
synchronous vertices. By Lemma 3.43 in [8] we have
$|h|_{X}\leq\eta(1,0,\chi(\mu))$. Thus, we may bound above the $X$-length of
each syllable of $w$ by $\eta(1,0,\chi(\mu))$. We therefore have that
$|x|_{X}\leq|x|_{X\cup\mathcal{H}}\eta(1,0,\chi(\mu))\leq(|X|^{\chi(\mu)}+1)^{|X|^{\mu}+1}\eta(1,0,\chi(\mu))$.

Case 2: There exists a syllable of $w$ such that the component corresponding
to this syllable is connected to its synchronous component in each
quadrilateral $Q_{i}$. Let $h_{1},...,h_{n}$ be the labels all of the
components which are connected to their synchronous components in each of the
quadrilaterals $Q_{i}$ (see Figure 2) (note that the $h_{i}$ may belong to
different parabolic subgroups).

Figure 2: The case of connected synchronous components in each conjugacy
diagram $Q_{i}$ (Case 2)

Denote the connectors between the component labeled $h_{j}$ with its
synchronous component in $Q_{i}$ by $g_{ij}$ and $f_{ij}$ for each $i=1,...,m$
and $j=1,...,n$ (see Figure 2).

We have that for each $j=1,...,n$,
$\\{g_{1j},g_{2j},...,g_{mj}\\}^{h_{j}}=\\{f_{1j},f_{2j},...,f_{mj}\\}$.
Therefore, the lists
$\\{g_{1j},g_{2j},...,g_{mj}\\},\\{f_{1j},f_{2j},...,f_{mj}\\}$ are elements
in some parabolic subgroup $H$ which are conjugate by an element of $H$. Since
the GCP is solvable in $H$, there exists a conjugating element $s_{j}\in H$
such that
$\\{g_{1j},g_{2j},...,g_{mj}\\}^{s_{j}}=\\{f_{1j},f_{2j},...,f_{mj}\\}$ and
$|s_{j}|_{X}\leq\theta(\mu)$ for some recursive function $\theta$ (this
$\theta$ depends on the parabolic subgroup but we may just take the max of the
$\theta$s over each of the parabolic subgroups to remove this dependence on
the subgroups, and also the argument of $\theta$ should depend on
$\max\\{|g_{ij}|_{X},|f_{ij}|_{X}:i=1,...,m\\}$, but we have
$|g_{ij}|_{X},|f_{ij}|_{X}\leq\chi(\mu)$ for all $i,j$ so we may assume the
upper bound is a function of $\mu$). Let $x^{\prime}\in G$ be formed by
replacing each label $h_{j}$ by $s_{j}$. Note that
$\\{a_{1},...,a_{m}\\}^{x^{\prime}}=\\{b_{1},...,b_{m}\\}$ (because $s_{j}$
conjugates each $\\{g_{1j},g_{2j},...,g_{mj}\\}$ to
$\\{f_{1j},f_{2j},...,f_{mj}\\}$) and
$|x^{\prime}|_{X\cup\mathcal{H}}=|x|_{X\cup\mathcal{H}}$. In $x^{\prime}$ we
may bound the $X$-length of the $s_{j}$ by $\theta(\mu)$ and for the
components of the geodesics representing $x^{\prime}$ which are not connected
to their synchronous component in some $Q_{i}$, then as in Case 1 we can bound
the $X$-length of such components by $\eta(1,0,\chi(\mu))$. Thus, each
component of our word representing $x^{\prime}$ has $X$-length bounded above
by $\max\\{\theta(\mu),\eta(1,0,\chi(\mu))\\}$. Hence
$|x^{\prime}|_{X}\leq|x^{\prime}|_{X\cup\mathcal{H}}\max\\{\theta(\mu),\eta(1,0,\chi(\mu))\\}=|x|_{X\cup\mathcal{H}}\max\\{\theta(\mu),\eta(1,0,\chi(\mu))\\}\leq(|X|^{\chi(\mu)}+1)^{|X|^{\mu}+1}\max\\{\theta(\mu),\eta(1,0,\chi(\mu))\\}$.

Therefore, we set $g=\max\\{\theta,\eta(1,0,\chi(-))\\}$. Note that this is a
recursive function since $\theta$ is a recursive function (by the solvability
of the GCP in each of the parabolic subgroups), $\eta(1,0,-)$ is a recursive
function by the proof of Lemma 3.43 in [8] and $\chi$ is a recursive function
by the proof of Theorem 3.37 in [8].

∎

######  Remark 5.

Note that if the parabolic subgroups are abelian, then we do not have Case 2
in the proof of Theorem 4, as if we had Case 2, then we would have
$g_{ij}=f_{ij}$ for all $i,j$ because they are conjugate elements in an
abelian group. Therefore, we could obtain a new conjugating element between
$\\{a_{1},...,a_{m}\\},\\{b_{1},...,b_{m}\\}$ by deleting each of the
connected synchronous components shown in Case 2. This would be a shorter
conjugating element than $x$, a contradiction.

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