text:
The Legault government has loaded a large burden on young people's shoulders, using them to justify its dubious decisions while simultaneously blaming them for the spike in COVID cases.
By Sarah Guilbault
The way that youth are often depicted, as portrayed by news media, often reflects neither the diversity of their opinions nor the whole of their reality.
The Legault government—and, all-too-often, journalists—pigeonhole students as though all students shared the same viewpoint. Many feel alienated from decisions that will have a profound impact upon their future. They want a voice, and they want to be heard.
"Secondary 3, 4 and 5 students throughout Quebec will be in school daily and that, for me, is really good news, because that's where our children ought to be," Quebec Premier François Legault stated, March 23.
The Legault government claims to listen, understand and to know what they need. Many students believe the opposite. High-school students continue to confront constantly changing rules and class schedules. They find themselves in a stressful situation that it's impossible to adapt to.
"It's really hard to follow schedules that constantly change. It's exhausting to always have to change your routine."
• Amelys, 15, Red Zone
"It might work in theory, but honestly, the constant changes give me a headache. One day at home; the next at school, it makes it hard to cope."
• Pénélope, 16, Red Zone
For students to succeed at their studies, they need a safe, stable environment, no matter whether the pursue their studies at home or in class.
"Without enough support, we have to cope with many things at once, which makes it hard to learn."
• Chloé, 16, Red Zone
According to the Legault government, all students want to go back to school.
"It's going to have a huge impact on academic success, on mental health. We're working on it we're working swiftly to make it happen," stated Education Minister Jean-François Roberge, March 9.
Many high-school students still find the Legault government's decisions on their behalf risky and make some of them anxious about the health of their family.
"The lockdown affected my mental health [like many youth] but going back to school was worse. I felt like my mental health was in freefall due to the stress and worry about everything."
• Naïka, 15, Red Zone
"Opening schools right now is like playing with fire. It's like saying to someone a beach where big breakers constantly roll in: 'Oh, the wave is gone, or it has calmed down, so go build a sand castle on the shore."
• Mahdi, 16, Red Zone
Many students would prefer to continue their studies online from home. They find it disconcerting to have to manage two environments at once. Few youths wanted movie theatres to reopen. Many who participate in organized sports would prefer to wait before team sports resume.
"Each time we seen the number of cases fall, the government decides to reopen restaurants, movie theatres and so forth. That might let us go back to normal life, but they often prove bad decisions [in my view] and we never make any progress.
I feel really helpless, faced with this situation. We're forced to go to school, despite the many risks. I'm in Secondary 4, the most important year, I have to go, otherwise my school year goes down the drain."
• Bailin, 16, Red Zone
Don't forget that many students are also highly stressed and worried about the prospect of going back to class. The Legault government spotlights the students who want to return to school while ignoring the opinions and well-being of the others.
School is a microcosm of society. Just like adult society, you find some young people there who don't believe in the virus, some who are anti-mask and others who fear for their health.
No one would imagine forcing 1,000 employees to work at close together in cramped, dilapidated, poorly-ventilated quarters—while we force students who aren't allowed to leave the school building if they don't feel safe, without even the opportunity to complain about unsafe safety standards. It would be better to stop blaming youths whenever case counts start climbing, when they're simply trying to do their best to follow the rules.
"We're taking a degree of risk, with respect to the Red Zone, where Secondary 3, 4 and 5 students will attend school daily, but we figure that it's a risk worth taking, because it will have an impact on young people's learning, socialization and mental health," stated Dr. Horacio Arruda, March 23.
"There are too many people in the same place at the same time. Even if they implemented measures to reduce the number of cases, I think that sending everyone [back to school] will only make a bad situation worse."
• Max, 15, Red Zone
"Most of the rules meant to protect us are contradictory. We have to keep our distance in class, but once we're in the schoolyard or even on the stairs, we're all jammed together. It's even worse, given that all students are now back at school."
Students are forced follow rules that endanger them and puts protecting their family's health on their shoulders—a huge responsibility to bear. While Minister Roberge insisted, January 12, that there would be no compromise on student safety, we now learn that masks distributed in the schools are toxic and that much-vaunted ventilation studies were neither conducted nor validated by the public health service.
Since the outset of the pandemic, students have been flexible, have adapted to a host of changes and have faced as many, if not more, restrictions as adults with all of the efforts that they have undertaking, and all they ask is to have a choice as to whether they study online at home or at school.
Newer PostTribune libre : Les étudiants se sentent ignorés et traités comme des boucs émissaires
Older PostDélestage précipité dans les services sociaux


This year will be a competition like you've never seen before with an additional 25th Anniversary category, more prizes, and live regional events.
Your entry must be built with a Cooler Master Case.
All Cooler Master cases are acceptable regardless of size or year released.
A bonus 25th Anniversary Award is available for any entries that uses one of the pre-selected cases – See 25 Anniversary Award below.
Your entry must be built without the use of a pre-existing Case.
There is no limitation on what materials can be used to house components.
Final project must contain at least one Cooler Master product that is clearly identifiable (fans are acceptable).
If a pre-existing case is used in your mod process said case must not be recognizable in the final project.
If a pre-existing case is used in your mod process no manufacturer logos from said case can be displayed in the final project.
A bonus 25th Anniversary Award is available for any entries that resembles a replica of one of the pre-selected cases – see 25th Anniversary Award below.
Every completed entry will be entered for public voting to choose the winner. The winner of the People's Choice will be the entry that receives the most amount of likes during the voting period taking place on this site.
You must be a legal adult in your country of residence.
You must complete the registration process by March 31st 2017.
You must not have started your mod project before July 1st, 2016.
You must complete your mod before the final submission deadline, May 3rd, 2017.
You must fully complete your worklog before the final submission deadline.
Your worklog must show the entire process of the mod from start to finish with both pictures/photos and text.
You must complete the final submission process by May 3rd, 2017.
You must agree to all terms and conditions.
You must not have entered your submission in any other Cooler Master global competition.
Your final submission must be a working PC.
You can enter multiple submissions in multiple categories but you must register each project separately and fully complete a worklog and the final submission process for each one.
Cooler Master Case Mod World Series is open to residents of all countries worldwide.
All participants must be considered a legal adult in their country of residence.
Multiple entries by a single individual are acceptable but must be registered separately.
Winners will be announced online and contacted via email.
As a participant, you are the owner and composer of the project content and images.
Cooler Master and it's event partners may use your case mod and images with or without any pseudo (pen) name credit.
Cooler Master and it's event partners has the perpetual right to use your case mod and case mod images. We may use your case mod and case mod images locally, nationally or globally and such use may be made in any media, information or communication platform currently in use or later developed.
Cooler Master reserves the right in its sole discretion to alter contest rules at any time.
Prizes are non-exchangeable, non-transferable, and not redeemable for cash or other prizes.
Cooler Master retains the right to substitute the prize with another prize of similar value in the event that the original prize offered is no longer available.
Usage or reprint of your case mod and case mod images will be at sole discretion of Cooler Master.
In case of dispute, Case Mod World Series 2017 will be governed by Taiwan (R.O.C.) law and entrants to the Case Mod World Series submit to the exclusive jurisdiction of the Taiwan (R.O.C.) courts.
By participating, you conclusively agree to be bound by these contest rules and conditions. This is an irrevocable condition of entry.
Contestants who do not agree to the above terms and conditions should not enter this contest.
You will see a message at at the bottom of the registration form confirming that your registration has successfully completed. If you have any problems or questions, please contact us at mod@coolermaster.com.tw.
Am I qualified to take part in the competition without completing the registration?
No. You must complete registration during the period of February 3 – March 31, 2017 to take part in the competition.
Can I create multiple entries in the same category, for example two entries in Tower Mod?
Yes. However, each mod is considered an individual entry by the judges and public voting, meaning you must register each project separately.
Am I allowed to take part in both categories?
Yes. You can create multiple entries for both categories, as long as each mod meets the requirements for each qualification and is registered separately.
Why am I required to submit so much information during registration?
The information you provide during registration is used to create your worklog and final submission gallery pages on the Case Mod World Series website and to verify your identity if you win a prize. Your pages can be updated during the event with additional content by contacting us at mod@coolermaster.com.tw.
What if my contact information changes after I register?
If your contact information should change after registration, you can update it via email at mod@coolermaster.com.tw.
What information does my gallery page contain?
A gallery contains uploaded photos of your final submission, a description of your mod project (written by you), a link to your project worklog, the hardware spec list of your submission, and links to your social media pages (optional).
Why do I need a gallery page?
Your gallery page is the public face of your final project. This is where the world will see what all your hard work has amounted to. Once you accomplish the final submission steps, your gallery will undergo a public vote and review by our panel of judges.
Why isn't my gallery updating?
The photo and data you submit for the final submission gallery will be updated as soon as we can depending on the amount of entries we get. If you want to know more, please contact us here or via email at mod@coolermaster.com.tw.
How do I submit my mod for final review?
Upload five photos (High Quality Photos) and the required information on the Final Submission tab of the Case Mod World Series website.
Once you complete the final submission process, your project will be considered completed and you will be unable to make further changes that will affect the judges' final decision.
What is public voting for?
Public voting is for the purpose of deciding the winners of the People's Choice Award. All mods will be open to public vote through the final submissions gallery page.
Voters can cast an unlimited number of votes, but only one per mod.
What if I started my mod before registration opened?
If you started a project prior to the contest, the start date cannot be earlier than July 1st, 2016. Any mods that are found to have been started/shared prior to July 1st, 2016 will be automatically disqualified.
When is the project completion deadline?
All final submissions are due by May 3, 2017 at 11:59pm (GMT -11).
Do I need to send my mod to Cooler Master to complete my final submission entry?
No. You need only to have completed your worklog and completed the final submissions process to officially complete your project and be considered for a prize.
Are there any age or location restrictions?
Location: No. The competition is open to everyone worldwide.
Age: Yes. You must be considered a legal adult in your country of residence.
All winners will be announced May 19th, 2017.
Judges will score each mod based on predetermined criteria. The mods with the highest scores will win prizes. The People's Choice Award prizes will be determined by public voting through the final project gallery. See Rules for more details.
How many winners will be announced this year (2017)?
Tower Build Award: 1st, 2nd & 3rd place.
Scratch Build Award: 1st, 2nd & 3rd place.
25th Anniversary Award: One Winner.
People's Choice Award: One Winner.
How do I enter for the 25th anniversary Award?
The 25th Anniversary Award is a bonus prize this year with specific qualifying factors. Please see the "About – Mod Categories" section under for specific information about this award.
Can I also enter a Case Mod World Series that is held in my own country?
Yes, in fact we encourage you to enter both the global and regional edition!
To celebrate Cooler Master's 25th Anniversary, we've upped the ante to bring even more cash and prizes than last year and as well as adding a bonus 25th Anniversary Award. A total of 8 winners will be walking away with fantastic prizes from us and our great sponsors.
*Prize is limited to product series availability in winner's home country.
Bit-Tech is the UK's leading online destination for tech enthusiasts of all types and has one of the most active modding forums in the world. Representing the site is professional modder Dave Alcock. Dave has been modding for a number of years and has done a number of sponsored mods as well as competed in many modding competitions.
Tantricmodz is one of the premier custom PC providers in Philippines. Its founder and lead, Mhike Samsin, first got into modding over 10 years ago as a hobby. His shop and services were formed out of the belief that modding is more than just a niche interest for expert level PC builders. Mhike has won numerous case modding competitions, including the Case Mod World Series with Starscr3am in 2012 and Sabertooth PC in 2013/2014.
Modders-Inc was created as place where people could go to learn and talk about PC modding. They are also responsible for hosting the annual US Case Modding Championship at Quakecon. The founder, Dewayne Carel, has been modding for more than 15 years. He has worked on projects with top case manufacturers and game producers to create numerous custom PCs.
Anthony Leather is quite possibly the most prestigious modding journalist in the world. He's been modding for almost 20 years, has run Bit-Tech's modding section for the last nine years, writes about modding for both Forbes and Custom PC magazine, and is responsible for having built possibly the world's smallest, most powerful, fully water-cooled Intel X99 system.
Crit TV evolved from a desire to present unfiltered information about gaming, hardware, and the culture surrounding them in an entertaining way. Its founder, Logan Hale, has always worked to push the boundaries of hardware focused entertainment and never apologizes for it. Collectively the team has a background in every step of the tech manufacturing process.
TweakTown was founded in 1999 with the goal of providing trustworthy content about tech, computing, hardware, and gaming in the form of news, reviews, and editorials on a daily basis. Representing the group will be its founder, Cameron Wilmot, and senior editor, Anthony Garreffa, who both have had a keen interest in modding for many years.
Founded in 2014, UNILAD is one of the biggest global platforms for breaking news and shareable content. Launched in January 2016, UNILAD Tech is the fastest growing and most engaged dedicated tech publisher in the world. The Tech division is being represented by Tomi Adebayo who has over six years of expertise in Software Engineering and Computing.
Creativity and originality in modding techniques used.
Attention to detail in case construction and aesthetic design.
Quality of aesthetic presentation in reference to use of color, shape, and craftsmanship.
Overall uniqueness in comparison to entries presented this year.
All project worklogs can be found on the Cooler Master Forum. Click the button below to see the progress of all the entries in this year's competition.
Come meet us and join fellow modders, enthusiast and World Series participants at an event near you. We'll be running modding workshops and meetups to bring your modding skills to the next level.
Modding is about community both online and offline. To celebrate and promote the close knit offline communities of modders around the world, the Case Mod World Series will feature multiple region based competitions this year. All Case Mod World Series entries will be automatically entered into their regional edition in addition to the global competition where applicable.
All regional entries must adhere to the same rules, terms, and conditions of the global Case Mod World Series event to be qualified for regional winner selection. Each regional edition will have an separate prize pool in addition to the world wide prizes. Regional edition winners will be announced after the conclusion of the Case Mod World Series global competition.


One thing that all successful positive cash flow property investors will tell you is that investing in the residential property market is a markedly different business than it is when buying your own home.
There are altogether different reasons why people choose to buy a property for themselves and buying a house or houses with the intention of creating a positive cash flow investment portfolio.
Investing in a residential property should be seen as purely an investment activity with the whole concept or idea being that you want to make a profit or regular income by spending a lot less on the property than you earn from it.
But buying an investment property can be a risky business because you are making such a large commitment of money in the one area. This is a completely different scenario from say investing in managed funds or shares whereby the risks of losing such a large sum of money at the one time is almost too insignificant to mention.
One of the biggest mistakes that new property investors make is to choose a home based on emotion.
They see a property that they like, think of it as a wonderful place that any young couple with a family would love to live in, lose their calm attitude and rush in to buy it without real thought behind the original reasons for buying an investment property in the first place.
They become distracted with imagining what it would look like with blue curtains and a white picket fence when their only real concern or attraction should be with the property as an investment.
What happens then is that when they either rent the house out to tenants or sell it as a wrap property, they have become attached to it and that can lead to unnecessary problems. Instead of looking at the new tenants or owners with an eye of whether or not they can afford payments on the property, they are looking purely at whether or not they deem them to 'be good enough' to reside in the property.
So it really is important to keep calm and not to rush into any positive cash flow property investment without doing your homework first. The short term potential for your income and the long term potential for your capital growth should really be your primary concerns, not whether or not the neighbours are nice!


text:
\section{Introduction}

An on-site repulsive Hubbard interaction cannot, by itself, lead to attractive pairing except through the Kohn-Luttinger mechanism~\cite{raghu1,kohn1}. However, the on-site Hubbard $U$ is not the only local interaction present. When multiple orbitals on each site are considered, then additional atomic interactions are possible, such as the interorbital repulsion $U^\prime$ and Hund's $J$ interaction~\cite{dagotto1}. It is natural to ask whether such additional local atomic interactions can lead to attractive pairing states and thus provide a local mechanism for superconductivity~\cite{spalek1,han1,klejnberg1,hoshino1,werner1,vafek1}.

A recent work addressing this question through dynamical mean field theory (DMFT) finds that the Hund's interaction can stabilize spin triplet pairing states with electrons residing on different orbitals~\cite{hoshino1}. From a DMFT perspective, the authors invoke the effect of ``spin freezing'' to rationalize their results~\cite{werner1,hoshino1}. Furthermore, the DMFT study finds that the superconducting state is most stable when the Hund's spin coupling is strongly anisotropic. This hints at the importance of spin-orbit coupling which was treated only in a phenomenological manner via an anisotropic spin coupling.

Vafek and Chubukov also examined this problem from a weak coupling perspective~\cite{vafek1}. Motivated by iron-based superconductors, they studied a two dimensional problem with two orbitals per site: $d_{xz}$ and $d_{yz}$. They considered the most general normal state Hamiltonian including local spin-orbit interactions and extracted the local Cooper pairs. Their central finding was that when the renormalized, low energy, $J>U^\prime$, an attractive spin triplet, orbital singlet pairing channel with $A_{1g}$ symmetry emerges. Such a renormalization is consistent with the numerical finding of such a pairing state in DMFT~\cite{hoshino1}. This novel pairing channel only exhibits a pairing instability at the Fermi surface \textit{in the presence of spin-orbit coupling}. Henceforth, we refer to this state as the Vafek-Chubukov (VC) state.

Thus, the theoretical work so far suggests a new mechanism for spin triplet superconductivity based on local \textit{interorbital} atomic interactions, which necessarily require a multi-orbital system, and the presence of on-site spin-orbit coupling. This motivates two directions for further study: (1) What happens when non-local spin-orbit coupling is included? (2) Are these pairing channels unique to the two orbital system in tetragonal crystal symmetry, or can they be realized in octahedral systems including all three $T_{2g}$ orbitals?

In this work, we first revisit the two orbital system and confirm the existence of the VC state which survives in the weak coupling limit when on-site spin-orbit coupling is present. We then add non-local spin-orbit coupling which stabilizes additional pairing channels belonging to non-trivial irreducible representations. We discuss the conditions under which these states are stable. Next, we study the three orbital system with the $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals in an octahedral crystal environment. In this limit, the pairing channel analogous to the VC state is found to become \textit{repulsive} when higher energy bands are not allowed to cross the Fermi energy. The energetics of other pairing channels are also critically affected by the number of bands allowed to cross the Fermi energy.

This article is organized as follows. In Section~\ref{sec:normalH}, we use symmetry methods~\cite{vafek1,fernandes1} to construct the most general normal state Hamiltonian and add local atomic interactions to it to construct the full model. In Section~\ref{sect:tetra}, we study the pairing channels of the two orbital case in tetragonal crystal symmetry and discuss the crucial role played by the form of spin-orbit coupling in determining which channels can be stabilized, as well as the effects of allowing only one band to cross the Fermi energy. In Section~\ref{sect:octa}, we turn to the three orbital case in octahedral crystal symmetry and highlight how constraining bands to not cross the Fermi energy also prevents certain pairing channels from becoming attractive. Section~\ref{sec:summary} contains a summary and further discussions.

\section{Model} \label{sec:normalH}

\subsection{In $D_{4h}$ crystal symmetry}

We first derive the normal state Hamiltonian for the case of two orbitals on each site, $d_{xz}$ and $d_{yz}$. We require the system to have $D_{4h}$ point group symmetry. The most general form of the normal state Hamiltonian $\hat{H}_N$ may be written as:
\begin{equation}
\hat{H}_N=\sum_{\alpha,\beta}c^\dag_{\alpha}\mathcal{H}_{\alpha\beta}c_{\beta},
\label{eq:1}
\end{equation}
in which $\alpha,\beta$ index both orbital and spin quantum numbers. In general, $\mathcal{H}$ may be expressed as a linear combination of direct products of Pauli matrices $\sigma_\mu,\mu=0,...,3$ parametrizing operators in spin space and Pauli matrices $\tau_\mu,\mu=0,...,3$ parametrizing operators in orbital space:
\begin{equation}
\mathcal{H}=\sum_{\mu\nu}f_{\mu\nu}(\vec{k})\sigma_\mu\otimes\tau_\nu,
\label{eq:2}
\end{equation}
in which $f_{\mu\nu}(\vec{k})$ represent general $\vec{k}$ dependent coefficients. In orbital space, $\tau_0$ has $A_{1g}$ symmetry, $\tau_1$ has $B_{2g}$ symmetry, $\tau_2$ has $A_{2g}$ symmetry, and $\tau_3$ has $B_{1g}$ symmetry. In spin space, $\sigma_0$ is $A_{1g}$, $\sigma_1$ and $\sigma_2$ form an $E_g$ set, and $\sigma_3$ is $A_{2g}$.

In general, all possible combinations of $\mu$ and $\nu$ should be considered. The expression can be substantially simplified by requiring the normal state Hamiltonian to have time reversal and parity symmetry. Note that $\sigma_\mu$ and $\tau_\nu$ are all even under parity such that the direct product $\sigma_\mu\otimes\tau_\nu$ is always even. The coefficient $f_{\mu\nu}(\vec{k})$ must contain only even powers of $\vec{k}$ in order for each term in Eq.~\ref{eq:2} to be overall even under parity.

Hence, parity symmetry requires $f_{\mu\nu}(\vec{k})$ to be even under time reversal; $\sigma_\mu\otimes\tau_\nu$ must then be even under time reversal for each term in Eq.~\ref{eq:2} to be overall even under time reversal. $\mathcal{H}$ can now be conveniently divided into two components:
\begin{equation}
\mathcal{H}=\mathcal{H}_0+\mathcal{H}_{s.o.},
\label{eq:3}
\end{equation}
in which $\mathcal{H}_0$ contains terms with $\tau_\nu$ which are even under time reversal ($\nu=0,1,3$) and $\mathcal{H}_{s.o.}$ contains terms with $\tau_\nu$ which are odd under time reversal ($\nu=2$). The reason for the chosen subscripts will become clear momentarily.

Because the coefficients $f_{\mu\nu}$ and the $\tau_\nu$ appearing in $\mathcal{H}_0$ are by construction even under time reversal, the spin operator is necessarily $\sigma_0$. Thus, $\mathcal{H}$ represents the ``bare'', non-spin-orbit coupled part of the normal state Hamiltonian. We may express $\mathcal{H}_0$ as:
\begin{equation}
\mathcal{H}_0=\sigma_0\otimes[f_{00}\tau_0+f_{01}\tau_1+f_{03}\tau_3].
\label{eq:4}
\end{equation}
The remaining task is to determine the form of the coefficients $f_{0\nu}$. This is accomplished by requiring each term to have $A_{1g}$ symmetry overall. This yields:
\begin{align}
\mathcal{H}_0=\sigma_0\otimes [&(A+Bk_\perp^2+Ck_z^2)\tau_0 \nonumber \\
&+D k_x k_y \tau_1 +E(k_x^2-k_y^2)\tau_3].
\label{eq:5}
\end{align}
in which $k_\perp^2\equiv k_x^2+k_y^2$ and $A,B,C,D$ and $E$ represent material dependent parameters independent of $\vec{k}$.

Following the same line of reasoning, $\mathcal{H}_{s.o.}$ can also be determined. The spin matrices involved here must be $\sigma_3$, $\sigma_2$, and $\sigma_3$ for each term to be even under time reversal. Thus, $\mathcal{H}_{s.o.}$ describes spin-orbit coupling interactions and can be expressed as:
\begin{equation}
\mathcal{H}_{s.o.}=\mathcal{H}_{s.o.}^\Gamma+\mathcal{H}_{s.o.}^{\vec{k}},
\label{eq:6}
\end{equation}
where:
\begin{equation}
\mathcal{H}_{s.o.}^\Gamma=\alpha\sigma_3 \otimes \tau_2,
\label{eq:7}
\end{equation}
is the local spin-orbit interaction containing $\vec{k}$-independent terms, and:
\begin{align}
\mathcal{H}_{s.o.}^{\vec{k}}=\gamma(k_x k_z \sigma_1\otimes\tau_2+k_y k_z \sigma_2\otimes\tau_2),
\label{eq:8}
\end{align}
represents all non-local spin-orbit interactions. $\alpha$ and $\gamma$ are material dependent parameters independent of $\vec{k}$.

\subsection{In $O_h$ crystal symmetry}

The same steps may be repeated to obtain the normal state Hamiltonian for the three orbital case in crystals possessing $O_h$ point group symmetry. The main difference is that because we work with three orbitals, the orbital space is now parametrized by the nine Gell-Mann matrices $\lambda_\mu,\mu=0,...,8$. Specific forms of the Gell-Mann matrices used and our choice of indices to label the orbitals can be found in Appendix~\ref{appendix}. The symmetry classifications of all matrices under the $O_h$ point group are as follows: $\lambda_0$ has $A_{1g}$ symmetry, $\{\lambda_1,\lambda_2,\lambda_3\}$ have $T_{2g}$ symmetry, $\{\lambda_4,\lambda_5,\lambda_6\}$ have $T_{1g}$ symmetry, and $\{\lambda_7,\lambda_8\}$ have $E_{g}$ symmetry. For the spin Pauli matrices, $\sigma_0$ has $A_{1g}$ symmetry and $\{\sigma_1,\sigma_2,\sigma_3\}$ have $T_{1g}$ symmetry.

The net effect is to restore the symmetry between the $z$ direction and the $x$ and $y$ directions. We simply state the final result here:
\begin{widetext}
\begin{align}
\mathcal{H}_0^{O_h}=\ &\sigma_0\otimes \left[(A+Bk^2)\lambda_0+E(k_x k_y\lambda_1+k_y k_z \lambda_2+k_x k_z \lambda_3)+F((k_x^2-k_y^2)\lambda_7+(2k_z^2-k_x^2-k_y^2)\lambda_8/\sqrt{3})\right], \\
\mathcal{H}_{s.o.}^{\Gamma,O_h}=\ &\alpha(\sigma_3 \otimes \lambda_4 -\sigma_1\otimes \lambda_5 + \sigma_2 \otimes \lambda_6), \\
\mathcal{H}_{s.o.}^{\vec{k},O_h}=\ &\lambda\left[k_xk_z(\sigma_1\otimes\lambda_4-\sigma_3\otimes\lambda_5)+k_yk_z(\sigma_2\otimes\lambda_4+\sigma_3\otimes\lambda_6)+k_xk_y(\sigma_1\otimes\lambda_6-\sigma_2\otimes\lambda_5)\right] \nonumber \\
 &+\gamma\left[-(k_x^2-k_y^2)(\sigma_1\otimes\lambda_5+\sigma_2\otimes\lambda_6)+(1/3)(2k_z^2-k_x^2-k_y^2)(2\sigma_3\otimes\lambda_4+\sigma_1\otimes\lambda_5-\sigma_2\otimes\lambda_6)\right],
\label{eq:9}
\end{align}
\end{widetext}
in which $k^2\equiv k_x^2+k_y^2+k_z^2$. These replace the corresponding quantities in Eq.~\ref{eq:3} and Eq.~\ref{eq:6}.

\begin{comment}
\subsection{Eigenenergies of $H_N$ at $\Gamma$} \label{sec:localevals}

To simplify the analysis of the eigenspectrum of the normal state Hamiltonian at $\vec{k}=0$, we reference the energies of the $d_{xz}$ and $d_{yz}$ orbitals to zero and set the energy of the $d_{xy}$ orbital to $\Delta$. This corresponds to choosing $A=\Delta/3$ and $G=-\Delta/\sqrt{3}$ in Eq.~\ref{eq:5}. We further set $\alpha=\beta$ in Eq.~\ref{eq:7}, i.e. letting the anisotropy in the $z$ direction enter only through the bare orbital energies and not through the spin-orbit coupling. The normal state Hamiltonian $\mathcal{H}$ therefore simplifies to:
\begin{equation}
\mathcal{H}=\alpha(\sigma_3 \otimes \lambda_4 -\sigma_1\otimes \lambda_5 + \sigma_2 \otimes \lambda_6)+\Delta \sigma_0\otimes \begin{pmatrix}0&0&0\\0&0&0 \\0&0&1 \end{pmatrix}.
\label{eq:10}
\end{equation}
The eigenvalues of this Hamiltonian are:
\begin{align}
E_1&=-\alpha, \nonumber\\
E_2&=\frac{1}{2}(\Delta+\alpha-\sqrt{\Delta^2-2\Delta\alpha+9\alpha^2}), \nonumber\\
E_3&=\frac{1}{2}(\Delta+\alpha+\sqrt{\Delta^2-2\Delta\alpha+9\alpha^2}).
\label{eq:11}
\end{align}
They are plotted as a function of $\Delta$ in Fig.~\ref{fig:1}. Each eigenvalue is doubly degenerate. In the perfectly cubic limit ($\Delta\rightarrow 0$), there is a fourfold degenerate set of states and a twofold degenerate set of states separated by an energy $\Delta E=3\alpha$. Note that even as we move away from the perfectly cubic limit by tuning $\Delta$ away from $0$, the four states ($E_1$ and $E_2$) at energy $E=-\alpha$ in the cubic limit, while splitting off into two energies themselves, always remain well separated from the two remaining states ($E_3$). We will make use of this separation in the energies of the two submanifolds in the next section.

\begin{figure}[ht]
\includegraphics[width=\columnwidth]{Fig1.eps}
\caption{(Color online) Eigenvalues (Eqs.~\ref{eq:11}) of the local normal state Hamiltonian (Eq.~\ref{eq:10}). Each eigenvalue branch is doubly degenerate. At $\Delta\rightarrow 0$ (the cubic limit), there is a fourfold degenerate set of states at $E=-\alpha$ and a twofold degenerate set of states at $E=2\alpha$. These two sets of states remain well separated in energy as $\Delta$ is tuned away from zero.}
\label{fig:1}
\end{figure}

\end{comment}

\subsection{Adding local atomic interactions} \label{sect:channels}

Finally, we add local atomic interactions $\hat{V}$ to the normal state Hamiltonian so that the total Hamiltonian is:
\begin{equation}
\hat{H}=\hat{H}_N+\hat{V},
\label{eq:12}
\end{equation}
in which~\cite{dagotto1}:
\begin{align}
\hat{V}=&\ \frac{U}{2}\sum_{i\gamma,\sigma\neq\sigma^\prime}n_{i\gamma\sigma}n_{i\gamma\sigma^\prime}+\frac{U^\prime}{2} \sum_{i\sigma,\sigma^\prime,\gamma\neq\gamma^\prime}n_{i\gamma\sigma}n_{i\gamma^\prime\sigma^\prime} \nonumber \\
&+\frac{J}{2}\sum_{i\sigma,\sigma^\prime,\gamma\neq\gamma^\prime}c_{i\gamma\sigma}^\dag c_{i\gamma^\prime \sigma^\prime}^\dag c_{i\gamma\sigma^\prime}c_{i\gamma^\prime\sigma} \nonumber \\
&+\frac{J}{2}\sum_{i\sigma\neq\sigma^\prime,\gamma\neq\gamma^\prime}c_{i\gamma\sigma}^\dag c_{i\gamma\sigma^\prime}^\dag c_{i\gamma^\prime\sigma^\prime}c_{i\gamma^\prime\sigma}.
\label{eq:13}
\end{align}
Here, $i$ is the site index, $\gamma,\gamma^\prime$ are orbital indices, and $\sigma,\sigma^\prime$ are spin indices. The first term describes repulsion between antiparallel electrons on the same orbital, parametrized by $U>0$. It is expected to be the dominant energy scale. The second term describes repulsion among electrons on different orbitals, parametrized by $U^\prime>0$. The third and fourth terms respectively represent the Hund exchange interaction and the Hund ``pair hopping'' interaction, both parametrized by $J>0$. The goal is to extract the energies and forms of the local Cooper pairs that arise from these local atomic interactions compatible with the relevant normal state Hamiltonian.

\begin{comment}
We work in the eigenbasis of the \textit{local} normal state Hamiltonian ($\mathcal{H}_0+\mathcal{H}_{s.o.}^\Gamma$). Let $d_p^\dag/d_p,p=1,...6$ be the creation/annihilation operators for the six local eigenstates as described in Section~\ref{sec:localevals}. Let $p=1,2,3,4$ describe the four states in the lower energy manifold with energies $E_1$ and $E_2$, and $p=5,6$ describe the two states in the higher energy manifold with energy $E_3$. Each operator $c_{\gamma\sigma}^\dag/c_{\gamma\sigma}$ in the original orbital-spin basis may be expressed as a linear combination of the $d$ operators:
\begin{align}
c_{\gamma\sigma}=\sum_{p=1}^6 a_{\gamma\sigma,p}d_p \nonumber \\
c_{\gamma\sigma}^\dag=\sum_{p=1}^6 a_{\gamma\sigma,p}^* d_p^\dag,
\label{eq:14}
\end{align}
with appropriate expansion coefficients $a_{\gamma\sigma,p}$. Using Eqs.~\ref{eq:14}, $\hat{V}$ may be expressed entirely in terms of the $d$ operators:
\begin{equation}
\hat{V}=\sum_{i,j,k,l=1}^6 A_{i,j,k,l}(U,U^\prime,J)d_i^\dag d_j^\dag d_k d_l.
\label{eq:15}
\end{equation}

We now make use of the crucial observation in Section~\ref{sec:localevals} that the four lower energy states and two higher energy states are well separated in energy. This assumes the spin-orbit coupling strength, characterized here by $\beta$, to be the dominant energy scale. Under this assumption, we neglect any tunneling matrix elements that connect the two manifolds. I.e., let $\mathcal{P}$ be the projection operator which projects out all such elements in Eq.~\ref{eq:15}; then:
\begin{equation}
\mathcal{P}^{-1}\hat{V}\mathcal{P}=V_2+V_4,
\label{eq:16}
\end{equation}
in which $\hat{V}_2$ represents all terms within the higher energy manifold, and $\hat{V}_4$ represents all terms within the lower energy energy manifold. The local Cooper pairs and their pairing interactions within each manifold can be extracted by analyzing the terms in $\hat{V}_2$ and $\hat{V}_4$ independently. By ignoring any inter-manifold coupling and treating the manifolds independently from one another, we are effectively allowing only one of the manifolds (representing bands once momentum dependence is restored) to cross the Fermi energy. As we will show in Section~\ref{sect:octa}, this has important effects on the energetics of the allowed pairing channels in the octahedral limit. The number of bands actually crossing the Fermi energy depends on the specific material being studied.
\end{comment}

\section{Tetragonal system with two orbitals}\label{sect:tetra}

We first examine the two orbital system with tetragonal point group symmetry considered in Ref.~\onlinecite{vafek1}. Without spin-orbit coupling, six pairing channels emerge. Their pairing interactions, symmetries, and gap structures are summarized in Table~\ref{table:1}.

\begin{center}
\begin{table}[ht]
\begin{tabular}{|c|c|c|c|}
\hline
 & Energy & Irrep. & Structure \\
\hline\hline
1.\ &$U^\prime -J$ & $A_{1g}$ & $\tau_2\otimes\sigma_3\sigma_2$  \\
2.\ &$U^\prime -J$ & $E_g$    & $\tau_2\otimes\sigma_1\sigma_2$  \\
3.\ &$U^\prime -J$ & $E_g$    & $\tau_2\otimes\sigma_2\sigma_2$  \\
4.\ &$U        +J$ & $A_{1g}$ & $\tau_0\otimes\sigma_0\sigma_2$ \\
5.\ &$U^\prime +J$ & $B_{2g}$ & $\tau_1\otimes\sigma_0\sigma_2$  \\
6.\ &$U        -J$ & $B_{1g}$ & $\tau_3\otimes\sigma_0\sigma_2$  \\
\hline
\end{tabular}
\caption{Local Cooper pair channels due to atomic interactions in the absence of spin-orbit coupling for the two orbital system in tetragonal crystal symmetry.  Channels 1, 2, and 3 are the novel ``spin triplet, orbital singlet'' pairs that can be stabilized when $J>U^\prime$. Channel 1 is the VC state.}
\label{table:1}
\end{table}
\end{center}

We focus first on Channel 1 whose Cooper pair takes the form $\tau_2\otimes\sigma_3(i\sigma_2)$ (where $\tau_\mu$ describes the pairing in orbital space and $\sigma_\nu(i\sigma_2)$ describes the pairing in spin space). Channel 1 represents a pair with $A_{1g}$ symmetry because $\tau_2$ and $\sigma_3$ both have $A_{2g}$ symmetry and $A_{2g}\otimes A_{2g}=A_{1g}$. Channel 1 corresponds to the VC channel. Crucially, its pairing interaction may become attractive when $J>U^\prime$.

What are the conditions that must be satisfied for a given pairing channel to develop a non-zero $T_c$ once spin-orbit coupling is turned on? In the weak coupling limit, a superconducting channel will lead to a pairing instability only if it involves \textit{intraband pairing}. The intuition for this can be seen by the following argument. Begin in the limit when all orbitals have the same kinetic energy $\epsilon(\vec{k})$. In this case, different orbitals have the same Fermi surface, just like spin up and spin down electrons in conventional $s$-wave superconductors in the absence of a Zeeman field. Consequently, an orbitally anti-symmetric Cooper pair will be stable in the weak coupling limit in the same way a spin singlet state is stable. Now consider adding some energy splitting between the different orbitals -- this will look like a Zeeman field in the spin analogy. Once this splitting becomes of the order of the gap, the weak coupling state will no longer survive. In general, different bands will be split off from one another, thus induced intraband pairing is required to avoid the obstacle to orbitally anitisymmetric superconductivity described above.

A theoretical formalism for determining whether a pairing channel develops intraband pairing as required for a pairing instability in the weak coupling instability was developed in Refs.~\onlinecite{ramires1} and~\onlinecite{ramires2}. The key result was the introduction of the concept of ``superconducting fitness'', quantities which may be calculated to determine whether intraband and/or interband pairing are present. The relevant quantity for determining whether intraband pairing is present is:
\begin{equation}
F_A(\vec{k})(i\sigma_2)=\mathcal{H}(\vec{k})\Delta(\vec{k})+\Delta(\vec{k})\mathcal{H}^*(-\vec{k}),
\label{eq:fitness1}
\end{equation}
where $\Delta(\vec{k})$ is the gap matrix of interest (last column in Table~\ref{table:1}) and $\mathcal{H}(\vec{k})$ is the normal state Hamiltonian matrix. \textit{If $F_A(\vec{k})(i\sigma_2)\neq 0$, then an intraband pairing component exists}. For the VC state with $\Delta(\vec{k})=\tau_2\otimes\sigma_3(i\sigma_2)$, all kinetic energy terms of the normal state Hamiltonian do not contribute to $F_A(\vec{k})(i\sigma_2)$. However, the local spin-orbit interaction $\alpha\tau_2\otimes\sigma_3$ guarantees an non-zero $F_A(\vec{k})(i\sigma_2)$:
\begin{align}
F_A(\vec{k})(i\sigma_2)\sim&\ (\tau_2\otimes\sigma_3)(\tau_2\otimes\sigma_3(i\sigma_2)) \nonumber \\
&+(\tau_2\otimes\sigma_3(i\sigma_2))(\tau_2\otimes\sigma_3)^* \nonumber \\
\sim&\ (\tau_2\otimes\sigma_3)(\tau_2\otimes\sigma_3(i\sigma_2)) \nonumber \\
&-(\tau_2\otimes\sigma_3(i\sigma_2))(\tau_2\otimes\sigma_3) \nonumber \\
\sim&\ \tau_0\otimes i\sigma_2 - \tau_0 \otimes (-i\sigma_2) \nonumber \\
\sim&\ 2\tau_0 \otimes i\sigma_2 \neq 0.
\label{eq:fitness2}
\end{align}
Thus, spin-orbit coupling is essential for the VC state to develop an intraband pairing component and hence possess a pairing instability at a finite temperature. The kinetic energy terms of the normal state Hamiltonian can be shown to lead to interband pairing. Their existence hence suppresses the $T_c$, but they cannot make $T_c$ go to zero as long as spin-orbit coupling is present. The same result was obtained in Ref.~\onlinecite{ramires2}.

\subsection{Gap equations for the VC state} \label{projectionconsequence}

Note that Channel 4 in Table~\ref{table:1} also has $A_{1g}$ symmetry and is therefore allowed to mix with the VC channel. In fact, the intraband pairing component acquired by the VC channel upon turning on spin-orbit coupling arises precisely due to mixing with Channel 4. Channel 4 is an $s$-wave spin singlet channel and has a repulsive pairing interaction $U+J$. The key result obtained by Vafek and Chubukov in Ref.~\onlinecite{vafek1} is that mixing of the VC channel with Channel 4 does \textit{not} suppress this pairing channel. However, implicit in their work was the assumption that both bands cross the Fermi energy and are relevant for pairing. They do not consider pairing restricted to a single band. We show that in the single band limit, the two channels compete and the resulting pairing channel becomes repulsive.

Let $\Delta_{vc}$ and $\Delta_{ss}$ be the order parameters associated with the VC superconducting channel and the on-site spin singlet repulsive channel, respectively. The result above can be seen most readily from the coupled gap equations obeyed by $\Delta_{vc}$ and $\Delta_{ss}$ when only the local spin-orbit coupling is included in the normal state Hamiltonian:
\begin{align}
-\frac{\Delta_{vc}}{g_{vc}}&=\sum_{\vec{k}}[f(\epsilon_+) (\Delta_{vc}+\Delta_{ss})+f(\epsilon_-) (\Delta_{vc}-\Delta_{ss})]\nonumber \\
-\frac{\Delta_{ss}}{g_{vc}}&=\sum_{\vec{k}}[f(\epsilon_+) (\Delta_{vc}+\Delta_{ss})+f(\epsilon_-) (\Delta_{ss}-\Delta_{vc})],
\label{eq:gapeqs1}
\end{align}
where $g_{vc}=U^\prime-J$ and $g_{ss}=U+J$, $f(E)=\tanh(\beta E/2)/E$, and $\epsilon_+$/$\epsilon_-$ are the energies associated with the two bands (they are $\vec{k}$-independent since we are ignoring the $\vec{k}$-dependent terms in the normal state Hamiltonian). In spite of the strongly repulsive $\Delta_{ss}$ channel, Vafek and Chubukov have solved these equations when both bands cross the chemical potential and find that a pairing instability exists for the state with both $\Delta_{vc}$ and $\Delta_{ss}$ non-zero.

Now consider the case when only one band is relevant. Projecting out one of the bands is equivalent to taking $\epsilon_-\rightarrow \infty$. In this limit, $f(\epsilon_-)\rightarrow 0$. Hence, the coupled gap equations reduce to:
\begin{align}
-\frac{\Delta_{vc}}{g_{vc}}&=\sum_{\vec{k}} f(\epsilon_+) (\Delta_{vc}+\Delta_{ss})\nonumber \\
-\frac{\Delta_{ss}}{g_{vc}}&=\sum_{\vec{k}} f(\epsilon_+) (\Delta_{vc}+\Delta_{ss}).
\label{eq:gapeqs2}
\end{align}
It immediately follows that $\Delta_{vs}/g_{vs}=\Delta_{ss}/g_{ss}$. Thus, there is only one distinct gap whose $T_c$ is governed by the manifestly repulsive interaction $g_{ss}+g_{vc}=U+U^\prime$. Hence, we conclude that the result of Vafek and Chubukov arises only in the limit when both bands cross the Fermi energy.

\subsection{The $E_g$ channels} \label{egchann}

In addition to the VC channel, Channels 2 and 3, which comprise a degenerate pair of $E_g$ symmetry, can also be attractive when $J>U^\prime$. These channels have the form $\tau_2\otimes\sigma_1(i\sigma_2)$ and $\tau_2\otimes\sigma_2(i\sigma_2)$ and are hence also orbital singlet, spin triplet states. Note that unlike the VC channel, there are no other competing, repulsive channels of $E_g$ symmetry. Hence, their pairing interaction always remains $U^\prime-J$. Evaluation of the superconducting fitness function $F_A(\vec{k})(i\sigma_2)$ is therefore sufficient to determine their stability.

For these states, the on-site spin-orbit coupling interaction is unable to induce intraband pairing. However, non-local spin-orbit interactions of the form in Eq.~\ref{eq:8} can induce intraband pairing in these states. Once these interactions are included, we find that the non-trivial, \textit{even parity} $E_g$ channels are generically stable states exhibiting spin triplet, orbital singlet superconductivity. Such terms were neglected in the treatment in Ref.~\onlinecite{vafek1} because they considered a two dimensional limit. Experimental evidence suggests that the $E_g$ pairing channel is realized for URu$_2$Si$_2$~\cite{schemm1,yano1,kasahara1,ikeda1}. Here, multiple orbitals are indeed relevant to the electronic bands near the chemical potential~\cite{ikeda1}, suggesting that the argument presented here may provide a viable explanation for the stability of this state.

The remaining three channels in Table~\ref{table:1} are all clearly repulsive in their interactions (assuming $U>J$ for Channel 6).

\section{Octahedral system with three orbitals}\label{sect:octa}

We now consider the octahedral case when all three of the $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals are included. We work near the $\Gamma$ point and assume that the on-site spin-orbit interaction $\mathcal{H}_{s.o.}^{\Gamma,O_h}$, parametrized by $\alpha$, is the dominant energy scale. Under this assumption, the eigenstates of the normal state Hamiltonian split into a twofold degenerate higher energy manifold (pseudospin $j=1/2$) and a fourfold degenerate lower energy manifold (pseudospin $j=3/2$).

Initially, we consider the local pairing channels that emerge after restricting to the $j=3/2$ manifold by projecting out the $j=1/2$ manifold. Under these assumptions, six pairing channels emerge from the local atomic interactions. Their energies and symmetries are summarized in Table~\ref{table:2}. We label these six channels with letters A to F to avoid confusion with the channels of the two orbital case of Table~\ref{table:1}.

\begin{center}
\begin{table}[ht]
\begin{tabular}{|c|c|c|}
\hline
 & Energy & Irrep. \\
\hline\hline
A.\ & $2U/3+U^\prime/3+J$ & $A_{1g}$ \\
B.\ & $U^\prime -J/3$     & $T_{2g}$ \\
C.\ & $U^\prime -J/3$     & $T_{2g}$ \\
D.\ & $U^\prime -J/3$     & $T_{2g}$ \\
E.\ & $U/3+2U^\prime/3-J$ & $E_g$    \\
F.\ & $U/3+2U^\prime/3-J$ & $E_g$    \\
\hline
\end{tabular}
\caption{Local Cooper pair channels due to atomic interactions in three orbital octahedral system. Channel A is the state analogous to the VC channel in the two orbital case.}
\label{table:2}
\end{table}
\end{center}

\subsection{What happens to the VC state?}

In the octahedral, three orbital case, the state analogous to the VC state has the form $\vec{\lambda}\cdot\vec{\sigma}(i\sigma_2)$ where we define $\vec{\lambda}=(\lambda_6,\lambda_5,\lambda_4)^T$ and $\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)^T$. At first, it may be tempting to associate the the VC analog with one of the $T_{2g}$ channels found (Channels B, C, and D), as these have energies $U^\prime-J/3$ and can therefore be potentially attractive. However, this cannot be the case because the VC state and its three orbital analog has $A_{1g}$ symmetry. The VC analog must instead be associated with Channel A. Channel A however contains a strongly repulsive contribution $2U/3$ in its pairing interaction and cannot be attractive.

Analogous to the discussion for the tetragonal two orbital case in Section~\ref{projectionconsequence}, this discrepancy can be understood as a result of projecting out the higher energy $j=1/2$ subspace. Without spin-orbit coupling, the VC analog state with pairing interaction $g_{1,A_{1g}}=U^\prime-J$ and the on-site spin singlet state ($\sim\lambda_0 \otimes \sigma_0(i\sigma_2)$) with pairing interaction $g_{0,A_{1g}}=U+2J$ do not mix (all pairing channels in the absence of spin-orbit coupling are summarized in Table~\ref{table:3}). Under the ``single manifold'' limit, the VC analog state forms a linear combination with the on-site spin singlet state and acquires the energy $U$ in its pairing interaction to create a repulsive pairing state. The on-site spin singlet state also belongs to $A_{1g}$ and is allowed to mix with the VC analog state.

Again, this argument can be made more rigorous by deriving the coupled linearized gap equations for the order parameters of the two channels. Let the combined gap matrix of the system be:
\begin{equation}
\hat{\Delta}=\hat{\Delta}_{0,A_{1g}}+\hat{\Delta}_{1,A_{1g}},
\label{eq:cubic1}
\end{equation}
where $\hat{\Delta}_{0,A_{1g}}=\Delta_{0,A_{1g}}\lambda_0 \otimes \sigma_0(i\sigma_2)$ and $\hat{\Delta}_{1,A_{1g}}=\Delta_{1,A_{1g}} \vec{\lambda}\cdot\vec{\sigma}(i\sigma_2)$. $\Delta_{0,A_{1g}}$ and $\Delta_{1,A_{1g}}$ are the gaps for the on-site spin singlet state and the VC analog state, respectively. The linearized gap equations can be most conveniently obtained through the following expression for the free energy~\cite{agterberg1,ho1}:
\begin{equation}
\mathcal{F}=\frac{1}{2}\sum_{i=0}^1 \frac{1}{g_i}\textrm{Tr}[\hat{\Delta}_i^\dag \hat{\Delta}_i]-\frac{\beta}{2}\sum_{\vec{k},\omega_n} \sum_{\ell=1}^\infty \frac{1}{\ell} \textrm{Tr}[\hat{\Delta}\hat{\tilde{G}} \hat{\Delta}^\dag \hat{G}]^\ell,
\label{eq:cubic2}
\end{equation}
where $\beta=1/k_B T$, $\hat{G}$ and $\hat{\tilde{G}}$ are Green's function matrices associated with the normal state Hamiltonian $\hat{H}_N$, defined by $(i\omega_n - \hat{H}_N)^{-1} \hat{G} = 1$ and $(i\omega_n + \hat{H}_N^T)^{-1} \hat{\tilde{G}} = 1$, and $\omega_n$ are the fermionic Matsubara frequencies $(2n+1)\pi/\beta$. For simplicity, we ignore momentum dependence and take $\hat{H}_N=\epsilon \lambda_0 \sigma_0 + \alpha \vec{\lambda}\cdot\vec{\sigma}$.

\begin{center}
\begin{table}[ht]
\begin{tabular}{|c|c|c|c|}
\hline
Irrep. & Spin structure & Energy & Gap structure \\
\hline\hline
$A_{1g}$ & Singlet  & $U+2J$       & $\lambda_0 \sigma_0(i\sigma_2)$     \\
         & Triplet  & $U^\prime-J$ & $\vec{\lambda}\cdot\vec{\sigma}(i\sigma_2)$ \\
\hline
$T_{2g}$ & Singlet	& $U^\prime+J$ & $\lambda_1 \sigma_0(i\sigma_2)$	\\
         &        	&              & $\lambda_2 \sigma_0(i\sigma_2)$	\\
         &        	&              & $\lambda_3 \sigma_0(i\sigma_2)$	\\
         & Triplet	& $U^\prime-J$ & $(\lambda_6\sigma_2+\lambda_5\sigma_1)(i\sigma_2)$	\\
         & 					& 						 & $(\lambda_6\sigma_3+\lambda_4\sigma_1)(i\sigma_2)$	\\
         & 					& 						 & $(\lambda_5\sigma_3+\lambda_4\sigma_2)(i\sigma_2)$	\\
\hline
$E_{g}$  & Singlet	& $U-J$        & $\lambda_7 \sigma_0(i\sigma_2)$	\\
         &        	&              & $\lambda_8 \sigma_0(i\sigma_2)$	\\
         & Triplet	& $U^\prime-J$ & $(\lambda_6\sigma_1-\lambda_5\sigma_2)(i\sigma_2)$	\\
         & 					& 						 & $(\lambda_6\sigma_1+\lambda_5\sigma_2-2\lambda_4\sigma_3)(i\sigma_2)$ \\
\hline
$T_{1g}$ & Triplet	& $U^\prime-J$ & $(\lambda_6\sigma_2-\lambda_5\sigma_1)(i\sigma_2)$	\\
         & 					& 						 & $(\lambda_6\sigma_3-\lambda_4\sigma_1)(i\sigma_2)$	\\
         & 					& 						 & $(\lambda_5\sigma_3-\lambda_4\sigma_2)(i\sigma_2)$	\\
\hline
\end{tabular}
\caption{All fifteen local Cooper pair channels \textit{without spin-orbit coupling}. We define $\vec{\lambda}=(\lambda_6,\lambda_5,\lambda_4)^T$ and $\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)^T$. All spin triplet channels have pairing interaction $U^\prime-J$ without spin-orbit coupling. The pairing interactions are derived from the atomic interactions of Eq.~\ref{eq:13}. When spin-orbit coupling is present and the single-manifold approximation is made, channels belonging to the same irreducible representations mix to form the channels in Table~\ref{table:2}.}
\label{table:3}
\end{table}
\end{center}

We take only the $\ell=1$ contribution in the summation over $\ell$ in Eq.~\ref{eq:cubic2}. The two coupled linearized gap equations then follow directly from $\partial \mathcal{F}/\partial \Delta_{i,A_{1g}}=0$ for $\Delta_{0,A_{1g}}$ and $\Delta_{1,A_{1g}}$:
\begin{align}
\frac{6\Delta_{0,A_{1g}}}{g_{0,A_{1g}}}&=\sum_{\vec{k}}[-2(\Delta_{0,A_{1g}}-\Delta_{1,A_{1g}})f(\epsilon_4) \nonumber \\
&-(\Delta_{0,A_{1g}}+2\Delta_{1,A_{1g}})f(\epsilon_2)] \nonumber \\
\frac{12\Delta_{1,A_{1g}}}{g_{1,A_{1g}}}&=\sum_{\vec{k}}[2(\Delta_{0,A_{1g}}-\Delta_{1,A_{1g}})f(\epsilon_4) \nonumber \\
&-2(\Delta_{0,A_{1g}}+2\Delta_{1,A_{1g}})f(\epsilon_2)],
\label{eq:cubic3}
\end{align}
where $\epsilon_4=\epsilon-\alpha$ and $\epsilon_2=\epsilon+2\alpha$ are, respectively, the normal state eigenenergies associated with the $j=3/2$ and $j=1/2$ manifolds. The approximation of projecting out the $j=1/2$ manifold associated with energy $\epsilon_2$ then amounts to taking $\epsilon_2\rightarrow\infty$. In this limit, $f(\epsilon_2)\rightarrow 0$ and the second terms on the right hand sides of Eqs.~\ref{eq:cubic3} can be discarded. Eqs.~\ref{eq:cubic3} become:
\begin{align}
\frac{6\Delta_{0,A_{1g}}}{g_{0,A_{1g}}}&=-\sum_{\vec{k}}2(\Delta_{0,A_{1g}}-\Delta_{1,A_{1g}})f(\epsilon_4) \nonumber \\
\frac{12\Delta_{1,A_{1g}}}{g_{1,A_{1g}}}&=\sum_{\vec{k}}2(\Delta_{0,A_{1g}}-\Delta_{1,A_{1g}})f(\epsilon_4).
\label{eq:cubic4}
\end{align}
Hence, $2\Delta_{1,A_{1g}}/g_{1,A_{1g}}=-\Delta_{0,A_{1g}}/g_{0,A_{1g}}$ implying that only one distinct gap remains after projecting out the upper manifold. This gap obeys the following gap equation:
\begin{equation}
\Delta_{0,A_{1g}} = -\frac{1}{3}\sum_{\vec{k}}(2g_{0,A_{1g}}+g_{1,A_{1g}})\frac{\Delta_{0,A_{1g}}}{2\epsilon_4}\tanh(\beta \epsilon_4/2),
\label{eq:cubic5}
\end{equation}
and is governed by the pairing interaction $(2g_{0,A_{1g}}+g_{1,A_{1g}})/3=2U/3+U^\prime/3+J$, exactly the energy found for the channel of $A_{1g}$ symmetry in Table~\ref{table:2}.

We conclude that the analog of the VC state in the cubic, three orbital case will not have a pairing instability upon projection onto the $j=3/2$ manifold due to mixing with the on-site spin singlet channel. We showed in Section~\ref{projectionconsequence} that under a similar projection onto a single band in the two orbital case, the VC state (Channel 1 of Table~\ref{table:1}) also becomes repulsive with interaction $U+U^\prime$ due to mixing with the on-site spin singlet channel (Channel 4). It becomes energetically stable only when both bands are allowed to cross the Fermi energy. By analogy, the VC analog state in the three orbital case becomes attractive when bands associated with both the $j=1/2$ and $j=3/2$ manifolds are allowed to cross the Fermi energy.

\subsection{Other pairing channels and their stability}

In addition to the $A_{1g}$ pairing channel, we also find channels belonging to a $T_{2g}$ triplet (Channels B-D) and an $E_g$ doublet (Channels E and F). The pairing interactions of these channels can also be explained by mixing between spin triplet and spin singlet states of the zero spin-orbit coupling limit upon projection onto to the lower energy manifold.

To illustrate this for the $T_{2g}$ channels, take $\hat{\Delta}_{0,T_{2g}}=\Delta_{0,T_{2g}}\lambda_1\otimes\sigma_0(i\sigma_2)$ as the spin singlet channel and $\hat{\Delta}_{1,T_{2g}}=\Delta_{1,T_{2g}}(\lambda_6\otimes\sigma_2+\lambda_5\otimes\sigma_1)(i\sigma_2)$ as the spin triplet channel. With respect to the atomic interactions, these two channels have interaction energies $g_{0,T_{2g}}=U^\prime+J$ and $g_{1,T_{2g}}=U^\prime-J$, respectively (Table~\ref{table:3}). Following the same procedure as for the $A_{1g}$ case, the linearized gap equations for $\Delta_{0,T_{2g}}$ and $\Delta_{1,T_{2g}}$ can be derived after projecting out the higher energy manifold:
\begin{align}
\frac{4\Delta_{0,T_{2g}}}{g_{0,T_{2g}}}&=-\sum_{\vec{k}}\frac{2}{3}(\Delta_{0,T_{2g}}-2\Delta_{1,T_{2g}})f(\epsilon_4) \nonumber \\
\frac{8\Delta_{1,T_{2g}}}{g_{1,T_{2g}}}&=\sum_{\vec{k}}\frac{4}{3}(\Delta_{0,T_{2g}}-2\Delta_{1,T_{2g}})f(\epsilon_4).
\label{eq:cubic6}
\end{align}
We obtain $\Delta_{1,T_{2g}}/g_{1,T_{2g}}=-\Delta_{0,T_{2g}}/g_{0,T_{2g}}$ again implying that there is only one distinct gap obeying the gap equation:
\begin{equation}
\Delta_{0,T_{2g}}=-\frac{1}{3}\sum_{\vec{k}}(g_{0,T_{2g}}+2g_{1,T_{2g}})\frac{\Delta_{0,T_{2g}}}{2\epsilon_4}\tanh(\beta \epsilon_4/2),
\label{eq:cubic7}
\end{equation}
governed by the pairing interaction $(g_{0,T_{2g}}+2g_{1,T_{2g}})/3=U^\prime-J/3$, in agreement with the energy found for the $T_{2g}$ channels.

For the $E_g$ channels, we can similarly take $\hat{\Delta}_{0,E_g}=\Delta_{0,E_g}\lambda_7\otimes\sigma_0(i\sigma_2)$ as the spin singlet channel and $\hat{\Delta}_{1,T_{2g}}=\Delta_{1,E_g}(\lambda_6\otimes\sigma_1-\lambda_5\otimes\sigma_2)(i\sigma_2)$ as the spin triplet channel. The two gaps then obey:
\begin{align}
\frac{4\Delta_{0,E_g}}{g_{0,E_g}}&=-\sum_{\vec{k}}\frac{2}{3}(\Delta_{0,E_g}+2\Delta_{1,E_g})f(\epsilon_4) \nonumber \\
\frac{8\Delta_{1,E_g}}{g_{1,E_g}}&=-\sum_{\vec{k}}\frac{4}{3}(\Delta_{0,E_g}+2\Delta_{1,E_g})f(\epsilon_4),
\label{eq:cubic8}
\end{align}
with $g_{0,E_g}=U-J$ and $g_{1,E_g}=U^\prime-J$. These equations reduce to:
\begin{equation}
\Delta_{0,E_g}=-\frac{1}{3}\sum_{\vec{k}}(g_{0,E_g}+2g_{1,E_g})\frac{\Delta_{0,E_g}}{2\epsilon_4}\tanh(\beta \epsilon_4/2).
\label{eq:cubic9}
\end{equation}
Thus, the $E_g$ channels are governed by the pairing interaction $(g_{0,E_g}+2g_{1,E_g})/3=U/3+2U^\prime/3-J$.

We can speculate on the conditions under which these other orbital singlet, spin triplet channels belonging to non-trivial degenerate irreducible representations can develop a finite $T_c$. Again, by analogy with the work done in Ref.~\onlinecite{vafek1}, if no bands are projected out, then the suppression of the pairing channels due to mixing with repulsive spin singlet channels of the same symmetry may be avoided. The $T_{2g}$ channels (Channels B-D) are of particular interest because even after projecting out the higher energy manifold, the pairing interaction can never acquire contributions from the Hubbard $U$ energy (the spin singlet channels of $T_{2g}$ symmetry do not involve intraorbital pairing).
Finally, in principle there remains a set of spin triplet channels of $T_{1g}$ symmetry (the last set of states in Table~\ref{table:3}), but their pairing remains purely interband for the symmetry allowed normal state Hamiltonian and therefore cannot generate a finite $T_c$ in the weak coupling limit.

\section{Summary and discussions} \label{sec:summary}

In this article, we studied the possible local pairing channels that arise from atomic interactions in tetragonal lattices with $d_{xz}$ and $d_{yz}$ orbitals on each site, and in octahedral lattices with $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals on each site. In the tetragonal, two orbital case we confirmed the existence of the VC state of $A_{1g}$ symmetry. The VC state is only stable when both bands are allowed to cross the chemical potential. In addition, a degenerate pair of states of $E_g$ symmetry also exist and they are stable regardless of how many bands cross the chemical potential. All three states have pairing interaction $U^\prime-J$ and can become attractive when $J>U^\prime.$ Such on-site Cooper pairs are necessarily even parity.

Spin-orbit coupling plays a pivotal role in the realization of such novel ``spin triplet, orbital singlet'' superconducting states. Not only is its presence or absence important, but the form of spin-orbit coupling (here, whether it is local or non-local) also dictates whether the additional spin triplet pairing channels with non-trivial symmetry develop intraband pairing and can survive in the weak coupling regime. In particular, the stabilization of the channels with $E_g$ symmetry by including non-local spin-orbit interactions introduces another layer of novelty to the potential superconducting states. Superconducting channels belonging to degenerate, non-trivial $E_g$ symmetry have been proposed to exist in heavy fermion systems such URu$_2$Si$_2$ in which the pairs are even under parity~\cite{schemm1,yano1,kasahara1}.

It is interesting to note that in the strong coupling DMFT study of Ref.~\onlinecite{hoshino1}, anisotropy in the spin interactions was found to play a critical role in stabilizing the superconducting state. Another cause of spin anisotropy is of course single particle spin-orbit coupling. Thus, both strong and weak coupling studies point to spin-orbit coupling as a key partner to the Hund's interaction in this superconducting mechanism. It is worthwhile to extend the DMFT work of Ref.~\onlinecite{hoshino1} to include spin-orbit interactions in a more rigorous fashion. Moreover, non-local spin-orbit interactions must also be considered in light of our results.

As in the two orbital tetragonal case, in the cubic, three orbital case, the analog of the VC state can only be stable when all bands are allowed to cross the chemical potential. This is also true for the $E_g$ and $T_{2g}$ pairing states in the cubic case, unlike in the tetragonal case in which the $E_g$ channel is generically stable (once $J>U^\prime$). When the $j=1/2$ manifold is projected out, the mixing between spin triplet and spin singlet states belonging to the same irreducible representations prevents the existence of channels with interaction $U^\prime-J$. Hence, in the search for materials which may realize such spin triplet, orbital singlet Cooper pairs such as the VC state, the number of bands crossing the Fermi energy must be considered.

From a theoretical perspective, the full consequences of including orbital degrees of freedom in superconductivity are still not fully understood. Recent works continue to reveal new features such as emergent Bogoliubov Fermi surfaces~\cite{agterberg1} and topological non-trivial nodes in the gap function~\cite{brydon1,timm1}. From an experimental perspective, a natural next step would be to predict the experimental signatures of such novel spin triplet superconducting states in measurements of the spin susceptibility and Knight shift~\cite{abrikosov1,anderson1,frigeri1,yu1}. In addition to URu$_2$Si$_2$, Sr$_2$RuO$_4$~\cite{ishida1,mackenzie1} and SrTiO$_3$ for example are materials in which spin-orbit coupling is expected to be an important energy scale~\cite{haverkort1,swartz1}. It is also known that the bands associated with $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals all cross the Fermi energy for these materials. It will be of interest to combine our results with more realistic microscopic descriptions of these materials.

\section*{Acknowledgments}

We thank S. Raghu, Y. Yu, P. Brydon, A. Ramires, and M. Sigrist for useful discussions. A. C. was supported by the DOE Office of Basic Energy Sciences, contract DEAC02- 76SF00515 and a NSERC PGS-D Scholarship. D. F. A. was supported by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant No. GBMF4302.

redpajama_set_name: RedPajamaArXiv


'Quest' is an online group community led by Jeffrey Davis at Tracking Wonder at the end of each year for the purpose of creating inspiring and meaningful business goals for the year ahead. Jeffrey uses techniques of 'Business Artistry' to push beyond the boundaries of normal thinking, which opens up new avenues for creativity, impact, and purpose. This blog article is a summary of four different articles I wrote as part of the Quest 2018 process.
Last year I participated in Quest for the first time with Jeffrey Davis' Tracking Wonder team. Quest is a refreshing, innovative approach to goal-setting that takes you on a free, four-week journey of self-reflection and strategizing to develop fresh perspective and ideas for the year ahead.
Jeffrey invites entrepreneurial visionaries from different fields and backgrounds to engage in discussion on different topics related to small business and entrepreneurship. The visionaries then ask the Quest participants to answer poignant questions that stimulate innovative thinking and reflection. The participants then blog, vlog or journal their responses on their different social media platforms and (if they want to) share them in a private Quest Facebook Group where the motto is Doing it Together (DIT) beats Doing it Yourself (DIY).
This is what I love most about Quest; the sense of community and camaraderie of being with a group of like-minded people, supporting and encouraging each other to work towards their 'best year'. You can find different responses to this year's Quest across social media platforms using the hashtags #BestYear and #WeQuest.
To get the ball rolling, Jeffrey sent a number of questions to reflect upon to imagine what our 'best year' could look like in 2018. Before launching into visioning the year ahead, I felt the need to reflect on everything that unfolded in 2017. Sometimes we can get to the end of the year and feel as though we didn't achieve as much as we wanted. But when we take the time to carefully reflect on what did unfold, it can be uplifting and empowering to document, acknowledge and celebrate just how much we did do. So I'm using Jeffrey's great questions to celebrate what I managed to accomplish in 2017 – to ignite the fires of creativity, passion, and purpose for 2018.
How did you show up for your best work?
In 2017 I decided that if I was going to show up and do my best work, it meant that I would have to step out of the shadows and get my face, message and work out there in front of people. This was an incredibly scary step for me. I'd never put myself out there in front of the public eye on social media before. Even though in my previous corporate career I was used to running workshops, meetings, doing presentations, and presenting at international conferences, I'd always been quite private on social media. So opening myself up to public scrutiny through being visible on social media felt like a big, scary step.
But at the same time, I was getting tired of inaction and inertia. It felt like I wasn't moving my coaching business forward in the way I wanted and needed to. I was in a pattern of getting frozen in perfectionism, waiting to 'get it right' before showing up. So, in January this year, I committed to the motto of "Imperfect action beats perfect inaction" and as scary as it was, I put my first video blog out into the public eye on my Facebook business page. It was well received and it gave me the courage and optimism to continue 'vlogging' on a regular basis.
What kinds of distinct activities did you engage in? What did you make or create as part of your best work?
I joined Lisa Marie Pepe and Divya Parekh and a team of nine other girls in writing and publishing the first ever book in the series of The Art of Unlearning: Conscious Choices for Empowered Living, which became an International Bestseller! This was such an uplifting experience and never in my wildest dreams did I think I would ever be a contributor to a 'bestselling' book. This whole experience gave me a boost of self-confidence and allowed me to tap into a nourishing and empowering sisterhood of like-minded friends.
I became a member of Jeffrey Davis' Writing Den and became (I think) so much more skillful at writing. Thanks to Jeffrey I learned much about creating engaging written content, the importance of consistency and routine, and his four pillars of Consistency, Signature Ideas, Well-Positioned Voice and Broadcasting. This helped improve the quality and professionalism of my Whispering Heart blog.
I learned how to use graphic design tools (Canva and GetStencil) for creating engaging and visually appealing graphics for my course materials and social media posts.
I developed high-quality workbooks and materials for my signature coaching program The Professional Freedom Formula.
I refined and improved my signature coaching program and pilot tested it with two wonderful women who gave me glowing feedback.
I learned how to use content scheduling tools Buffer, Tailwind and CoSchedule to ensure a more consistent presence on my social media platforms Twitter, Pinterest, Facebook, YouTube and Instagram.
I commissioned Black Dahlia Press to do a completely new website for me, including new logo design and branding colours, which I love. It felt so good to finally feel proud of my website.
I learned how to use WordPress from scratch and can now make any changes needed to my website when and where needed (that feels like huge progress to me).
I set up a payment system online for my customers and clients to be able to buy from me via my website. This also felt like a huge step forward in professionalism.
Who did you engage and how did your work positively impact them?
I worked with some lovely new clients testing out my new and improved signature coaching program. According to their feedback, The Professional Freedom Formula program helped them get a very clear and deep understanding of their authentic nature, desires, and strengths, which allowed them to find alignment with their professional path and purpose. Both reported feeling a sense of clarity and connection to themselves that they'd never felt before.
I worked with some wonderful women in The Art of Unlearning and together we learned the joy of empowering and uplifting others.
I started my series 'Conversations That Matter' which is something I've wanted to do for a long time. I recorded two discussions with two amazing women and have a list of willing interviewees lined up for the coming year.
I joined Business Women Australia here in Perth, Australia and found a supportive and encouraging network of wonderful, like-minded women to collaborate with.
I met some amazing new people through my social media channels and have developed some wonderful new contacts.
I participated in my very first TeleSummit and was interviewed by the lovely Dr. Julie Wilkening on the topic of The Art of Purposeful Living for her Healthy Habits and Productivity Summit.
What did you do differently that stretched you?
The year 2017 was a year of breaking through barriers. I dared to step up and put myself 'out there'. I dared to undertake 'imperfect action' in order to stick to my commitment to consistency and regularity, as opposed to perfection and procrastination. I proactively sought out collaborations that flourished and helped me step into my authentic self-confidence and power.
What 1, 2, or 3 big goals did you reach?
I developed a brand new website that I love.
I co-wrote our book The Art of Unlearning.
I started video blogging regularly and am no longer afraid to put my face and voice out there in the public eye.
I developed strategies to produce and schedule high-value content on a regular basis.
In the process, what challenges did your 'best self' meet and how?
This year my main challenge felt like time-constraint. Being a mother to two young boys while trying to develop and run my own business is not an easy task. Plus, we moved from South Korea (where we had been living for 4 years) back to Perth Australia. A big international move, getting our family moved and set up into our new home and community was also no small feat. Maintaining continuity in my business with all the personal upheaval this year has been challenging. But I met the challenge and feel happy with where I am today.
What skill set or craft did you learn or improve upon?
I improved upon the art of consistency and routine in terms of content production.
I improved upon the art of imperfect action.
I improved the art of collaboration (or as Jeffrey says "Doing it Together (DIT) instead of Doing it Yourself (DIY)".
And I improved the art of juggling motherhood with small business.
What 1 habit did you add, adjust, or drop?
I stopped spending so much time on social media and kept my attention focused on my milestones. I was much more focused and aligned in general. I said 'no' to more things, which is a new skill for me.
And overall, how did you feel throughout the year when you engaged with your best work?
When I'm engaged in my best work I feel alive, excited, inspired, impactful and GRATEFUL.
Jeffrey suggests the following exercise for 'dreaming big' to get the creative juices flowing.
Imagine your best possible self at the end of a year from now.
Imagine you are in December 2018 looking back upon how you have shown up for 2017. See yourself in a specific place as if you were looking back upon the year: maybe a favorite chair, a deck or balcony, a mountainside. As your best self "looks back" upon the year (2018), reflect upon and write in intimate detail your response to these questions.
In 2018 we were much more settled as a family. There were no more international moves from one side of the world to the other, and we all felt increasingly settled into our new home, schools, environment, communities. This meant that I was able to settle into a routine of consistency and structure.
I showed up with regularity, consistency, conviction, inspiration, and a desire to serve and impact others positively.
I launched my first-ever group program for The Professional Freedom Formula (which until now has been a private mentorship program only) and had more than twenty people in the group. I mastered the art of group coaching.
I completed meaningful collaborations with Business Women Australia, including presentations and workshops on the topic of Personal & Professional Freedom, and initiated collaborations with other coaches/teachers/healers in the industry, to bring more value to the people I work with.
I did presentations for graduating high school students on what it means to find a joyful and meaningful career. Many of them enrolled in my Joyful Career Academy program and found their joyful professional path and purpose.
I had some of my written pieces accepted into major publications such as Huffington Post and Elephant Journal.
I proactively grew my email list and social media platforms which allowed a steady stream of clients to move into my group programs and one-on-one mentoring.
I engaged with other professional women, with other coaches/healers, with high school students and other like-minded people walking the path to greater authenticity and alignment in heart-body-mind-spirit. Their engagement with me allowed them to dig deeper to find their personal truth and to develop a map of inspired action that allows them to live it.
I stepped up in confidence and visibility. My energy was magnetic, drawing the right people and situations to me.
I launched my signature group program The Professional Freedom Formula.
I took my program to high schools to help graduates make meaningful career choices aligned with their authentic nature and heart.
In the process, what challenges did your best self meet and how?
I came up against time constraints and nay-sayers but mastered the art of keeping my energy aligned with my vision and intention.
I improved upon my skill of group coaching and teaching.
I added the habit of consistency, structure, and routine in my daily work which helped create more space and productivity in my working hours.
Overall, how did you feel throughout the year when you engaged in your best work?
When engaged with my best work, I felt aligned, impactful, valuable and joyful.
How about you? What will your #BestYear look like and how will you develop goals that inspire and excite you?
If you'd like to join my upcoming DREAM BIG IN 2018 email challenge I'd love to have you join us to make 2018 a great year!
Ah, 'purpose', my favourite topic in the world.
It took me more than fifteen years to figure out exactly what mine is, but once I found my purpose, all those years of struggle, chronic illness, and frustration with life became a thing of the past. I believe that purpose is something so fundamental to our being, so central to our sense of self-worth and zest for life, that I've dedicated my coaching business to helping people find theirs.
And this is certainly how I've experienced life. Once I found my 'big why', my life turned around for the better. Our 'big why' is the thing that we want to dedicate our time and energy, creativity and full potential, and our heart and soul to. It's our 'raison d'etre' as the French call it, or 'Your reason for getting out of bed in the morning.' And I love seeing the transformation that happens in my clients' lives when they finally find their purpose. Here's an overview of what I believe to be the essential qualities of your 'big why'.
I write about all this and more in my free E-book PATHFINDING: How to Find Your Unique Path and Purpose.
The visionaries we spent time with in this week's Quest 2018discussion were Katie Dalebout, host of the 'Let it Out' podcast and Caroline Miller, bestselling author of 'Getting Grit'. The topic of discussion was 'Finding purpose vs. Testing purpose'.
Katie Daelebout hosts the podcast Let It Out and is author of the Amazon best-selling book of the same name, Let It Out: A Journey Through Journaling. Her unfiltered and authentic writing style helps others cultivate more self-awareness in their lives. Katie's work has been featured in countless publications including Daily Mail, Women's Health, Refinery29, Teen Vogue, The Telegraph, and Glamour.
When we get clarity on what we want to create and it's for the greater good of humanity, then that vision can happen more easily. When we make space to show up this way, the universe will fill it with our desires, but we have to make space. How are you going to make space in 2018 to create what you really want that will be for the greater good of humanity? What ritual might help your mindset make space?
Katie brings up a very important point, which is asking yourself the question "What do I want my work to be in service to?" or "How does my work benefit the greater good for humanity?". When we shift our focus from "How much money can I make?" or "What can I get out of this?", to "How can I personally make a positive difference in the world?", you propel yourself and your work into a new level where 'synchronicity' shows up on a regular basis. Katie calls this "the universe filling the space with your desires".
How are you going to make space in 2018 to create what you really want that will be for the greater good of humanity?
As Katie suggests, ritual is central to creating that space. 'Ritual' meaning something you make time for every day, to the point where it becomes such a natural and nourishing thing to do that you can't bear to skip a day. Ritual is a conscious habit that we choose that uplifts and aligns us.
On the days when I manage to do this, it sets my whole day up in a positive vibe. And on the days when I wake up too late or something comes up and I'm not able to do it (one of my boys waking up before me, for example), my day usually feels a lot harder and more chaotic.
In 2018, I commit to continuing this ritual to pursue it with even more rigour.
If you choose to tackle harder goals on a daily basis, imagine how you could amplify the positive impact you want to have on the world. It's often said that "You can't keep what you don't give away.'" What will you give to other people through your best work in 2018 that will positively impact them so that you might keep it, as well? What do you need to do in 2018 to ensure that you live without unnecessary regrets and have that kind of fulfilling purpose and impact on others?
Caroline's quote "You can't keep what you don't give away" reminds me of the similar quote "We teach what we most need to learn." And in my case, this is certainly true. The one thing that I struggled with until my mid-thirties was the ability to live in alignment with my own personal truth. Once I made the commitment to be true to myself, things started shifting for the better. My health improved, my relationships improved and my whole life improved. But it's not a decision that we make once and move on from. Living this way requires continuous choices each and every day.
In my coaching business, I work with people to help them find their own personal truth and then to find ways to live more in alignment with that truth, so that they too can experience improvements in all areas of life. So when I'm teaching this, I'm also re-learning it myself. Or I'm "giving it away in order to keep it" as Caroline said.
What do you need to do in 2018 to ensure that you live without unnecessary regrets and have that kind of fulfilling purpose and impact on others?
I need to show up with consistency and impact. I need to show up with inspiration and vision. I need to help people see and engage with the vision that represents the full expression of their life.
This means that I need to plan and execute a number of meaningful programs, activities, and engagements that allow me to fully express myself and my work.
I need to stretch myself out of my comfort zone and "tackle harder goals" so that I can test the limits of what I'm truly capable of.
And most importantly I need to make continuous choices each day to live in alignment with my personal truth, while helping others live theirs.
What about you? How will you find purpose and test purpose in 2018?
I love this week's Roundtable Discussion in Quest 2018 with Jeffrey Davis at Tracking Wonder #WeQuest #BestYear.
This week's roundtable conversation was with two visionary leaders who know this terrain well, Ishita Gupta and Charlie Gilkey.
Both visionaries Ishita and Charlie shared their experiences in their formative years that allowed them to develop their gifts of creativity and productivity. Ishita grew up with a twin sister and describes herself as "always curious", asking many questions, often ones that would get her into trouble. Her insatiable curiosity has served her well, together with her innate ability to dream big and to develop systems and structures that help her "get stuff done". Charlie was a Boy Scout and later an Eagle Scout and went on to work in the Military, and his strength has always been an ability to be resourceful, productive and finding ways to be more efficient. He has an innate ability to look at how other people do things and to improve on them, to continuously improve his own efficiency and productivity.
My defining moment of the need to develop 'grit'.
I remember it well. I was in Year 5 (aged 10) and we were all given the task of researching the River Nile and putting all our findings into a wall poster with pictures and information. I wasn't particularly fascinated by the topic and found it very hard to make myself sit down to work on it. As a result, my poster was poorly done and not very visually appealing at all.
At the end of the project, all our parents were invited to school to have a look at the posters, which were put up all around the walls of our classroom. When my parents came in and asked me where my poster was, I felt embarrassed to show them. When I pointed out my very plain poster in between all the other colourful and vibrant posters that other students had done, my parents were horrified. "Is that yours?" they asked while I hung my head and nodded.
On the way home in the car, my parents told me how disappointed they were with me and how embarrassed they were to see my poster hanging there. I felt so small and humiliated, and like most children, I desperately want to please my parents. So I made a commitment in that moment to do much better next time.
From then on, I did everything I could to develop 'grit', which has become a popular term since Angela Duckworth popularised it in her TED Talk on True Grit: Can Perserverance Be Taught? It's often defined as "Passion and perseverance for long-term goals". I did whatever it took, hours of devotion, research or studying, to make sure I never let down my parents again. And I developed long-term over-achiever tendencies as a result, which served me very well, earning me a University Degree with Honours and a Ph.D., and a successful career as a consulting process engineer for almost 15 years. But it also eventually became my own downfall, causing me to suffer from burnout in my mid-thirties and a need to completely switch careers.
And while this week's Roundtable discussion acknowledges the critical importance of "Getting stuff done", it also asks the question of how we can create space in our lives to "Dream stuff up".
Creating 'white space' in my life is something I've been forced to learn to do. My earlier over-achieving tendencies of pushing through all kinds of discomfort and pain not only led to chronic fatigue and illness, but also to a complete disconnect from my own heart and soul. If we want to do our best work, it's imperative that we find the time to unplug and drop into silence on a regular basis, or at the very least to sit in quiet reflection.
Creating white space in our lives for quiet reflection, dreaming, and stillness allows us to reconnect to our hearts and our full creative potential.
Writing Prompt #1: Ishita Gupta – Creative Entrepreneur helping people unleash their full potential.
Since quitting my consulting engineering career in 2017, I've spent the past 6+ years living overseas for my husband's work while raising our two young boys. I've been "squeezing my business startup into the cracks of motherhood", and while I've managed to do a lot in that time, I haven't had the time and energy to really 'give it my all' in my business. And now I feel beyond ready to do so. My boys are now 6 and 3 and so I feel like those highly intense first years are easing up and there's more time for 'me' in my life now.
What I currently desire most is to be more impactful in my work.
What I would love most is to be able to stand up in front of large crowds of high school students who are contemplating which professional field to go into, and to help them make a career choice that is aligned with their heart, their natural gifts, their true nature and their passion. Rather than being overly influenced by those in their environment who would have them pursue a career for money, status, or other inauthentic reasons. I chose my career path for mostly the wrong reasons and ended up burned out and miserable as a result. So I feel passionate about helping young people avoid the heartache that I went through, to find their authentic and joyful professional path and purpose.
And if that showed up on my doorstep tomorrow, I would be one very happy person.
I want to feel excited about the projects I'm working on and the people I'm working with. I make time in my life for the things that make me feel alive.
I want to feel connected into a web of like-minded, passionate, soulful, creative entrepreneurs and people whose mission is to uplift and inspire those around them. I want to feel nurtured, encouraged and supported to be my best self.
I want to know that my work matters and that I can make a positive impact on people's lives through the work I do.
I want to feel as though I'm important and invaluable the people in my life, including my clients.
I want to feel nourished by the work I do, emotionally and intellectually. I make time in my life for things that nourish and uplift my heart and soul.
I want my work to bring me great joy and satisfaction.
I want to feel magic in my life. As I tune into and pay attention to the synchronicities, pleasant 'coincidences' and grace that's at work in my life, I make way for more of it to flow to me.
I want to feel financially abundant and free.
What about you? What will be your words for 2018?
Quest 2018 – Can Competition Be Healthy in Entrepreneurship?
Quest Topic #4: Can Competition be Healthy in Entrepreneurship?
It's hard to believe we're already in the last week of our 4-week Quest journey with Jeffrey Davis at Tracking Wonder. This is the second year I'm participating and once again it's been full of rich, stimulating ideas, discussions, and reflections. #WeQuest #BestYear #DoingItTogether.
It's school holidays and I'm home with my two young children attempting to squeeze this week's response into the cracks of their needs and demands. With my husband offshore for work for the Christmas and New Year period, it's a challenging time of single parenting for me. However, I managed to listen to this week's wonderful roundtable discussion in the car on the way back from a 5-day holiday down at a seaside holiday village in Western Australia.
This week's topic is intriguing and very relevant: Competition versus Building Community. Responses from different Quest participants can be found across social media using the hashtag #CompetitionCommunity. The discussion between Dorie Clark, Marketing & Strategy Consultant and Jonathon Fields of the Good Life Project explored the different aspects of competition in entrepreneurship and the need to build a loyal community of people who resonate with your brand and message.
On the topic of competition, both Dorie and Jonathon discussed how they've never been motivated by competition with others, but rather by an intrinsic drive to always improve on their own internal standards. Both have achieved wonderful things in their careers due to this desire to continuously improve themselves and their work. I think I've operated in a similar way over the course of my life.
Competition is healthy to a point; it helps us look around at others to see where we can strive for more ourselves. It gives us the impulse and motivation to push ourselves harder, to see what we're truly capable of. It becomes unhealthy when the sole motivation is to always be 'better than' everyone else, leaving no room for cooperation, collaboration and 'win-win' solutions. I love these wise words from Franklin Roosevelt.
"Competition has been shown to be useful up to a certain point and no further.
True power and possibility come from shifting our focus from simply 'outdoing' others to generating collaborative, win-win solutions in contribution to a vision larger than ourselves.
It's been shown again and again in different 'happiness' studies that those who focus their energy and efforts on something larger than themselves feel more fulfilled and genuinely happy than those who focus their efforts simply on increasing their own power or status. Both Dorie Clarke and Jonathon Fields have a very clear vision for a future that they're working towards and this vision drives their efforts and communities in a way that leaves them feeling impactful, fulfilled and satisfied.
There is a lot of 'competing' out there particularly in the coaching world that I'm now part of; people trying to protect the communities they've worked hard to build and protect their own ideas and intellectual property. While I can certainly appreciate that it's important to protect ourselves, last year I noticed that things started shifting for me personally once I stopped looking at fellow coaches as 'competitors' and started looking at them as potential opportunities for collaboration, and vehicles for bringing more resources, tools, and resources to the people I'm here to serve.
Once we can move past 'competing' to focusing our efforts instead on creating collaborative opportunities to serve our people more effectively, we open the doors to even greater impact and possibility.
Jeffrey Davis does this brilliantly in the way he sets up his whole annual Quest collaboration. And when we 'light another candle' by embracing our competitors and their work and providing opportunities for their greater impact and visibility, we expand our own networks, our friendships and our own ability to serve and impact others more genuinely and effectively.
In the discussion of competition, the topic of envy came up and how the emotion can be used as a signpost for what we desire more of in our lives. Rather than feeling bad about our feelings of envy towards others, we can use them to uncover more about our own genuine desires.
– You become an 'expert' of sorts on the topic you're researching and you're able to share your learnings and insights with others to help them.
Again, Jeffrey does this brilliantly in Quest. Another good example is Chris Guillebeau's book Born For This: How to Find the Work You Were Meant To Do in which he travels the world interviewing people who have made the decision to go for their dreams, despite all kinds of difficult (sometimes seemingly impossible) obstacles they have up against them. In the process, he learns so much, establishes a broad network of like-minded people doing great things in the world and brings all this knowledge and insight to his followers and his readers. I'm currently pondering how I could do something along those lines to expand my network of like-minded dreamers.
Both visionaries Dorie Clarke and Jonathon Fields discussed the importance of building our own communities of people with shared values and beliefs, particularly as entrepreneurs.
We become the best version of ourselves when we find the people and communities who support us to be ourselves and reach for the stars. #relationships #community.
To get an invitation, you have to give an invitation.
– Empower, uplift and inspire each other.
I'm going to continue the series this year and consider inviting some people I really admire and look up to. I'm still pondering who that might be.
3) Expand my network of supportive friends and colleagues.
Doing it Together (DIT) Beats Doing it Alone (DIY).
Thank you Jeffrey for providing such thought-provoking topics again this week.
Alicia Anderson's beautiful post Inviting Change.
Julie Peatt Cassaday shares her thoughtful response here: Week 4 Quest 2018 Competition versus Community Building.
Thanks so much for journeying with me!


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