Iceland. The country's name might seem to say it all. But in reality, only about 10 percent of this island nation is covered with ice. The other 90 percent encompasses cooled lava flows and cliff-lined fjords, craters and bubbling hot springs, boulder fields and hardy, persistent plant life.
Forged from volcanoes and glaciers, Iceland's rugged beauty is captivating. Photographer Feodor Pitcairn and geophysicist and poet Ari Trausti Guðmundsson reveal this land of fire and ice in the new exhibition Primordial Landscapes: Iceland Revealed, opening July 2. Primordial Landscapes looks at some of the Earth's most dramatic geological processes using themes of Fire, Ice, and Transformation.
Iceland is one of the most geologically active places on Earth. Volcanic eruptions and subsurface geothermal activity produce hot springs, geysers, and steam plumes. Arctic ice and snow, rain, and glacial melt combine to form powerful rivers and waterfalls that carve the Earth's surface. After active lava flows cool, life slowly and persistently begins to take hold. To see the Icelandic landscape is to see many of our planet's most fundamental processes at work.
This steam plume is found in the highlands of the Torfajökull volcanic system, which contain big, powerful geothermal fields. Geothermal fields are subsurface reservoirs of the Earth's heat. On the surface, they produce visible hot springs, geysers, and steam plumes. Image by Feo Pitcairn Fine Art.
A new landscape of lakes, scree (broken rock) slopes, hills, and large boulders is revealed after a glacier retreats. Soon vegetation settles in, and the land is scattered with hardy flowers, grass, moss, and lichen. Glaciers in Iceland—and throughout the Arctic—are vanishing due to a rapidly warming climate. Image by Feo Pitcairn Fine Art.
In addition to the photography, Primordial Landscapes: Iceland Revealed also features poetry by geophysicist and writer Ari Trausti Guðmundsson, translated from the original Icelandic. Excerpts are printed large scale on the exhibition gallery walls, and a soundscape of Guðmundsson reciting his work in Icelandic is paired with English translations projected onto the walls. The exhibit's soundscape also includes wind, cracking ice, geysers, and other environmental sounds recorded in Iceland. In addition, visitors can get a close-up view of volcanic rock samples and see several examples of Iceland's extremely hardy plant life from the NMNH Botany Department's collection.
NMNH lighting specialist Virginia Croskey created an immersive lighting display for the exhibition to allow visitors to experience a taste of the aurora borealis, or northern lights. After failing to catch a glimpse of the northern lights on a trip to Norway, Croskey saw them earlier this year while visiting Iceland.
The aurora borealis, or "northern lights," come from the Earth's magnetic field being continuously hit by electrically charged particles from the Sun. They are transported toward the magnetic poles, where they collide with nitrogen and oxygen molecules in the upper atmosphere. These molecules absorb the energy and then release it as visible light. Image by Feo Pitcairn Fine Art.
Primordial Landscapes: Iceland Revealed parallels the United States' 2015-2017 chairmanship of the Arctic Council, the governing body responsible for coordinating policy affecting Arctic nations and indigenous peoples living within the Arctic Circle. The Museum will be celebrating the U.S. chairmanship with public programming and research symposia directed by the Smithsonian's Arctic Studies Center.
A syllid polychaete. Photo by Alexander Semenov.
The United Nations establishes international days to promote awareness on global issues and to commemorate important events. Other groups declare a day for much less serious reasons, for example, the delicious Strawberry Shortcake Day or the playful Go Fly a Kite Day. Depending on the calendar, there is a "Day" for nearly everything you can imagine, except …..polychaetes.
But, what is a polychaete?
A tomopterid polychaete. Photo by Karen Osborn, Smithsonian Institution.
Polychaetes are segmented worms (I heard those eews!) that are found mostly in saltwater. There are about 8,000 named species in the world and at least twice that many still to be named. They are very important animals in the food webs that are critical to keeping our oceans healthy --- many are even breathtakingly beautiful.
Polychaetes are found in all marine habitats from the deepest oceans to the shallowest tide pool, and even in freshwater habitats. They are often the most numerous animals in a square meter of sea floor mud. They range in size from less than a millimeter as adults to several meters long. They are found in the fossil record back through the Cambrian. They build reefs from their tubes, turn over the seafloor much like their earthworm cousins, and are a major part of marine food webs. Most have hydrostatic skeleton, which means instead of their muscles working on hard structures like bone or cartilage, they work on pressure from water – think about a water balloon pushing and pulling against that constricted water. Bottom line, they are essential and fascinating members of our world.
Serpula uschakovi, a serpulid polychaete. Photo by Alexander Semenov.
So, if Gummy Worms get a day, we decided it was time that polychaetes get one too. We chose to commemorate a great polychaete scientist, Kristian Fauchald, a dedicated scientist who spent 36 years at the Smithsonian. Kristian passed away in April, and we decided to commemorate him on what would have been his 80th birthday – July 1, 2015.
Our goal is to share the beauty, interest, and importance of polychaetes with the world on this day in honor of Kristian. To do that, we are encouraging a global conversation with a large network of activities and experts talking to the public at museums, aquaria, and science education centers around the world.
Find an event near you and tell us how you're celebrating by using the hashtag #InternationalPolychaeteDay. Browse incredible images of polychaetes on our Ocean Portal, explore current research from the Department of Invertebrate Zoology, and join us for a full day of free activities to celebrate poylchaetes!
Orchids are starting to bloom in their greenhouses at Smithsonian Gardens.
January is the nation's capital is dreary, but the Museum will be bursting with color. We're filling an entire hall with live flowers for our biannual orchid exhibit, Orchids: Interlocking Science and Beauty, by Smithsonian Gardens and the United States Botanic Garden. The show, which runs January 24th through April 26th, examines how new ideas, technologies, and inventions change the way we study, protect, and enjoy orchids. Just consider the greenhouse, which allows us to grow exotic plants even in the dead of winter.
Orchids provide splashes of brilliant magenta, bright white, and delicate pink.
I took a field trip out to Smithsonian Gardens just to see how the orchids were coming along, and got a tour of the greenhouses from Tom Mirenda, Orchid Collection Specialist. It was the perfect way to spend a drizzly, freezing morning. I asked Tom which orchids were his favorites, which turned out to be an impossible question to answer. His favorites are constantly shifting to whichever plant is "behaving" at the moment. The day I went to visit, that was Angraecum sesquipedale, also known as Darwin's orchid. In 1862, Charles Darwin famously hypothesized that the large, white, star-shaped orchid was pollinated by a moth with a proboscis long enough to reach the nectar at the bottom of the flower's 10-16 inchlong spur. Twenty-one years after his death, scientists found the moth with the absurdly long proboscis, Xanthopan morganii praedicta.
Angraecum sesquipedale, the Darwin Orchid.
Some of the orchids in the greenhouses were rare species with fascinating back-stories. Others were hybrids familiar to anyone who's ever bought a corsage. All were lovely, and will give our dinosaurs some competition for this winter's coolest museum attraction.
These Psychopsis Butterfly 'Mountain' orchids look a bit like butterflies.
Vanda Manuvadee 'Pramate' tucked among the other orchids.
Speaking of "cool," how do we get the orchids from their greenhouses at Smithsonian Gardens to the Museum in the bitter cold? With a protective layer of shrink wrap, of course! When the orchids are ready to bloom, we put them on rolling carts, wrap each cart in plastic, load them onto heated trucks, and hustle them over to the Museum. Because this exhibition will be open for a few months, we'll need to repeat the process many times with new flowers as they bloom.
Twin buds, just waiting to bloom.
Mele Kalikimaka with Cake from Anthropology!
Every year a little magic happens at our Department's holiday party. If you were around last year, you might have seen what I mean: the amazing #anthrocakes that archaeologist Eric Hollinger creates for the party every year. I think you'll agree it's "Tut"-ally amazing!
King Tut's Tomb. Edible sculpture by Eric Hollinger and Lauren Sieg, 2013. Photo Credit: Meghan Mulkerin.
He's created everything from a delicious caramel recreation of the Cliff Dwellings at Mesa Verde, to a giant chocolate Aztec calendar stone that he carved painstakingly with a nail! You can see more of his past creations here; don't miss the chocolate terracotta army from the tomb of the first Emperor of China!
The cake is a big secret every year until the day of the party. So what did he make this year? A giant chocolate fondue volcano of Hawaiian-inspired archaeological goodness! This year's cake was inspired by the archaeological site, Pu`uhonua O Hōnaunau. I've visted the site before, and I can attest to the accuracy of the sea turtles on the beach in this cake! They swim right up on the beach there, and the entire site is fascinating.
Hawaii Cake, edible sculpture by Eric Hollinger, Archaeologist, Department of Anthropology at NMNH, 2014. Photo Credit: Meghan Mulkerin.
So what is so important about this site, you ask? Well, do you remember that scene from Disney's Hunchback of Notre Dame, when Quasimodo yells "Sanctuary!" from the top of the church, claiming refuge for Esmeralda? In old Hawaii, if people needed a safe refuge, they'd make their way to Pu`uhonua O Hōnaunau! A puuhonua was a place of refuge for anyone who had broken a law. Did I mention that the punishment for breaking a law in old Hawaii was death? Laws, or kapu, were central to Hawaiian society, and governed many different aspects of behavior. So if you'd transgressed, your only shot was to hightail it to a puuhonua, and hope you did not get caught on the way there.
You don't want to mess with these guys. Ki'i, guardians of the Kapu. Photo Credit: Eric Wienke, Flickr.
While this year's cake was not an exact recreation of an archaeological site, it drew inspiration from Hawaii's natural beauty and cultural heritage. It features a Hawaiian volcano with flowing (chocolate) lava, a waterfall, tropical flowers, palm trees, sharks, turtles, beaches and ocean, as well as elements of Hawaiian cultural sites. The cultural features included tapa cloth, tiki statues, huts, taro fields, outrigger canoe with sail, and stacked stone walls.
From top left, clockwise: Huts and taro field with macademia nut wall; flowing chocolate lava with coconut syrup waterfall; statue; modeling chocolate orchid; fondant palm trees and sour ropes vines; gummy turtle and chocolate covered macademia nut rocks; outrigger canoe with sail. Hawaii-inspired cake 2014, edible sculpture by Eric Hollinger. Photo Credit: Meghan Mulkerin.
The volcano was inspired by Kiluawa's recent eruptions. It was about 2.5 feet high and was built around a chocolate fountain. An underlaying structure of recycled ethofoam packing blocks and cardboard were erected to form the main body of the volcano without it becoming too heavy to move.
The volcano under construction using recycled ethofoam. Photo Credit: Eric Hollinger.
One challenge was to redirect the flow of chocolate from the uppermost part of the fountain out onto the mountain and back again to the bottom reservoir where it could be warmed again and pumped back to the top. This was accomplished by forming aluminum foil channels to guide the flow in to paths around and back to the bottom. The foil was hidden under a coating of melted chocolate. Once finished, party-goers were able to dip fresh cut pineapple, papaya and banana into the lava as a fondue!
Delighted party-goers partake in the fondue! Photo Credit: Melissa Powell.
The mountain was covered with a total of 11 red velvet cakes which were left mostly exposed to show the surface. The outer crust of the cake, the color, and the interior spongy bubbly nature of the cake mimicked what much of the Hawaiian lava actually looks like when it has cooled.
Cake under construction. Photo Credit: Eric Hollinger.
Hawaii just wouldn't be Hawaii without flowers, so Eric took a class on flower making! The flowers are made of a specialty modeling chocolate which is cut, shaped and then painted to look like real flowers. Some of the flowers Eric made were actual violets from his yard that were coated with sugar for preservation!
Handmade edible flowers created by Eric Hollinger and Beyrl Loveland. Photo Credit: Meghan Mulkerin.
Eric used many sources to assist him in crafting the cultural objects you see on the cake. Books on Hawaiian carvings and Hawaiian canoes were studied and Curator Adrienne Kaeppler advised on the project to make sure the canoe and statue were accurate.
Oldest exisiting Hawaiian outrigger canoe in the world, given to the Smithsonian by Queen Kapi'olani in 1887. Catalog No. E160416. Photo Credit: Hawaiian Treasures NMNH Exhibit.
The canoe was carved from solid chocolate and a Pocky, chocolate covered biscuit stick, was used as a mast. It was very hard to rig the distinctive Hawaiian sail since it was made of a flexible printable icing sheet, like the kind used to print photos onto a cake. The tapa were printed onto this same material. Tapa is made from Mulberry bark pounded into a cloth and dyed. Dr. Adrienne Kaeppler, Curator of Oceanic Ethnology, has a long term project on tapa cloth, which you can read about here.
Edible tapa cloth printed on sheet icing using digital photos of real tapa from NMNH's collections. Photo Credit: Eric Hollinger.
The tiki statue was carved from solid chocolate using a real statue as a model. The huts were made using pretzel sticks and chocolate to form a frame and large shredded wheat biscuits were cut in half to form the thatch of the roof. Taro, a main staple of the Hawaiian diet, was featured in a wet field made of molten sugar and fondant taro plants. Palm trees were made of large pretzels and had fondant fronds attached at the top. The wall, which encloses the royal grounds at the actual archaeological site, was made of chocolate covering macadamia nuts.
Royal grounds of the cake, complete with huts and taro fields. Photo Credit: Eric Hollinger.
By: Meghan Mulkerin, Arctic Studies Program Coordinator and Social Media Manager, and Eric Hollinger, Archaeologist in NMNH's Repatriation Office, Department of Anthropology.
Meghan Mulkerin and Eric Hollinger enjoying this year's cake. Let us know what you think! Photo Credit: Hollinger and Mulkerin.


text:
COVID-19 Fundraiser for Community of Indian Armed Forces
Community of Indian Armed Forces
Updated on July 20th-
Thanks to all of you for contributing to this campaign. We were able to raise over INR 73 Lakhs to fund the procurement and shipment of MOGS from Australia to India. Here is a glimpse of the product:
https://www.youtube.com/watch?v=3kXkj2nXaag
We have received the required clearances with customs and other authorities at both ends.
The equipment is prepping for transit and should be handed over within the next couple of months.
Once again, we deeply appreciate your support in getting this project to execution. Thank you!
Updated Jun 17th -
We have 4 days left! Some of our AIT alumni will match $ to $ of any donation that you will contribute.
Once donated, let the campaign organizer know and the amount will be matched till the campaign hits the $100K mark. We are very close to our target amount and we're on the final lap.
Your contribution NOW, whatever be the amount, will have double the impact! Click and donate at http://m-lp.co/y4d.
Updated Jun 15th - Only $15K needed to meet target! Fundraiser has been extended till June 20th
Updated - 9th June 2021
6 more days to go, we are now $40K (approx INR 30 L) short of our goal to procure the 2nd MOGS. In addition we have exciting news about a collaboration with a campaign run by Mr Ashish Mitra, CEO of Endoluminal Sciences(ELS) Pty Ltd. For details see the Updates tab.
Revising the target to $120K( approx 90L)
We are short by $50K to procure the second MOGS system available with the vendor.
MOGS are Military Medical Oxygen Generators. These FDA approved, modular oxygen generation systems are rugged but lightweight and portable. They can be lifted by helicopter to remote locations, connects to electricity and starts working. It is rugged and this unit can fill up to 10 oxygen cylinders at once. So field areas can regularly get oxygen cylinders filled from this unit. It can be deployed in the base and serve a large number of forward areas.
Help us raise $50K more in these 10 days! Contribute and share - http://m-lp.co/y4d.
We are raising INR 90L to augment the efforts of the Indian Army in fighting against the COVID-19 pandemic. These funds will be utilised for procuring medical equipment to be donated to facilities run by the armed forces/in aid of the armed forces community. Facilities will be identified in conjunction with the nodal agency coordinating the COVID-19 relief effort for the Indian Army.Our initial target is to donate:
2-3 oxygen plants
50-100 oxygen concentrators
50 BIPAPs and ventilators.
We will keep this list updated as the situation evolves.
We are a group of alumni from the Army Institute of Technology (Pune) and owe our success to our alma mater and the Indian Armed Forces community. Today as India faces the brunt of the second wave of COVID-19, we want to mobilise our resources to support recovery. Time to give back to our community!
We are collaborating with the Y4D Foundation.
Y4D Foundation, an NGO with prominent AIT Alumni, has a pan India presence through its wide network of volunteer chapters across the country. Y4D was setup in 2015 and has worked with a wide range of communities. Apart from its steady focus on causes such as youth, women empowerment, environment initiatives etc, the foundation also gets engaged in projects as the situation demands. COVID-19 pandemic support, flood, drought relief being some of them. Volunteers present across 17 states have impacted 237437+ people so far and counting.
Together we hope to collaborate efforts to arrange supplies as required to relief efforts for the defence community.
For further questions reach out to us at support@y4d.ngo.
What is the problem we are solving?
In the past few months, the COVID-19 situation in India has spiralled out of control. While things seem to be settling down very slowly, the problem will continue to shift to smaller towns. The long term effects of this wave will continue impacting the community in terms of the support and care needed.
It is important that there is sufficient capacity built in to manage this scale. This will require as much help as possible right now.
Ground reality will continue to evolve over the coming months. It is important for support to reach the right places at the hour of need. This becomes challenging due to various factors in the supply chain.
How are we solving this?
There is a near term need for speedy access to supplies such as oxygen concentrators, oxygen generation plants and medical equipment (e.g. bipaps/ventilators). We are setting up this fund to
Augment capacity as needed - Coordinate with facilities operated by (or in the aid of) the defence community that need supplies and procure materials as may be required. (70-80% of funds generated)
Create oxygen concentrator banks in select cities to provide assistance to veterans/servicemen on a need basis. (20-30% of funds generated)
Y4D will contribute any excess funds as of 15th August 2021 to the Army Central Welfare fund for continued support activities.
Who is assisting in this process?
A core group of AIT Alumni across the globe will
Provide a governance framework
Assist Y4D in procurement of supplies
Liaise with concerned authorities to understand needs
Y4D Foundation will
Manage the funds raised, purchase material and deploy as required
Provide regular updates on deployment of materials
How will you provide materials to hospitals/clinics run by defence forces? - AIT Alumni and Y4D will coordinate with defence authorities handling COVID-19 relief to understand their ask and will deploy materials. Other NGOs/donors have similarly donated hospitals/oxygen plants/oxygen concentrators to defence hospitals/clinics.
Which cities will you setup the oxygen concentrator banks? - Oxygen concentrator banks will be setup in Tier 2 cities with large community presence. e.g. Jammu, Meerut, Jaipur etc.
What will you do with the concentrators in the banks post utilisation?- The concentrators will be handed over to the nearest defence hospital/clinic by December 31st 2021.
Are we providing cash to the beneficiaries? - No cash will be provided. Aid will be provided only in the form of supplies. Any excess funds as of 15th Aug 2021, will be directly transferred to the Army Central Welfare Fund.
Why is this specific NGO selected? - Y4D has AIT alumni on its advisory committee who have proven experience in running non-profit initiatives nationally. Y4D is FCRA approved and has a national presence.
Why not any other NGO working for COVID relief? - Larger NGOs already have their priorities defined and are executing on that mandate. We selected an NGO which would help channel these funds to assist the defence community specifically. Y4D has AIT Alumni on the advisory board representing our objective.
What are the tax benefits mentioned in the campaign? – All INR funds donated to this fundraiser will be issued an 80G tax exemption certificate from Y4D which is approved to do so. Non INR funds will not be applicable for this exemption.
How do I get information of how this campaign is operating? – Write in with your specific queries to support@y4d.ngo. Once you donate, you will also get regular updates from Milaap each time we post status updates on this campaign.
Who comprises the core group of AIT Alumni working on this initiative?- The alumni core group comprises a mix of business executives and entrepreneurs from across the AIT batches right from 1998 onwards. Specific core group details are available on request from support@y4d.ngo or the AIT Alumni office. Below are the linkedin profiles of select members of this group -
Rohit Mediratta, Kapil Dahiya , Jyotika Bisht, Namrata Sharma, Shashikant Tomar, Nupur Arora , Ankush Tiwari, Vijender Yadav, Nitin Nair, Vandana Balakrishna, Alok Arora, Vivek Jain, Pallavi Tyagi, Tamanjit Bindra, Amit Chaubey, Garima Sangwan

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
\label{sec:introduction}
  Nematics, regarded as a sort of liquid crystal in a more general context,
  now forms a wide range of phases of matter in crystalline solids.
  The ``electronic nematic state'' was first proposed in a doped Mott insulator as a consequence of the melting of stripes, aiming to understand the origin of high-$T_c$ phase in cuprates~\cite{Kivelson1998_Nature_393_550}.
  More recent example is the nematicity of electronic wave functions induced by the orbital ordering
  in iron-based superconductors, possibly dominating the stability of the superconductivity
  ~\cite{Laad2011_PhysRevB_84_054530,Onari2012_PhysRevLett_109_137001,Fernandes2014_natphys_10_97}.
  When defined on a crystal lattice, the nematics of charges and orbitals,
  and also of spins are all described by the quadrupolar order parameter representing the symmetry of their wave function.
\par
  In insulating quantum magnets~\cite{Blume1969_JApplPhys_40_1249,Andreev1984_SovPhysJETP_60_267},
  nematic phases appear when the spin moments break their rotational symmetry
  and form a wave function in the shape of rod- or disk-like director which collectively align in space.
  The search of spin nematics has been a challenge, since proposals relevant to experiments are provided only in limited numbers of systems;
  in a layered solid $\,^3$He~\cite{Ishida1997_PhysRevLett_79_3451},
  in an artificially designed optical lattices~\cite{Kimura2005_PhysRevLett_94_110403,Forges2014_PhysRevLett_113_200402},
  and in a quantum spin-1/2 magnets near the saturation field~\cite{Svistov2011_JETPLett_93_21,Kohama2019_PNAS_116_10686,Skoulatos2019_PhysRevB_100_014405}.
\par
  There had been some reasons that the quantum spin systems cannot easily become a good platform of spin nematics.
  In materials, the electrons carry spin-1/2, which is a dipole by itself,
  and to form a quadrupole which is a rank-2 tensor, we need at least spin-1 with three different $\mathcal{S}^{z}$-levels
  (see Eq.~(\ref{eq:quadrupole_operator})).
  There are two ways to construct spin-1 from spin-1/2.
  One is to use the Hund's coupling between spin-1/2 in different orbitals on the same site,
  which will generate \textit{site}-nematics.
  The other is to efficiently compose spin-1 from two spin-1/2's on neighboring sites by the interaction,
  in which case the \textit{bond}-nematic is formed.
  In the former case, a quantum spin-1 bilinear-biquadratic (BLBQ) model
  is known to host a spin nematic phase for a large biquadratic interaction,
  $(\boldsymbol{S}_{i}\cdot \boldsymbol{S}_{j})^{2}$,
  where $\boldsymbol{S}_{i}$ is a spin-1 operator~\cite{Blume1969_JApplPhys_40_1249,Chen1971_PhysRevLett_27_1383,
  	Papanicolaou1988_NuclPhysB_305_367,Tanaka2001_JPhysAMathGen_34_304,
    Harada2002_PhysRevB_65_052403,Tsunetsugu2006_JPhysSocJpn_75_083701,
    Lauchli2006_PhysRevLett_97_087205,Bhattacharjee2006_PhysRevB_74_092406,
    Lauchli2006_PhysRevB_74_144426,Harada2007_JPhysSocJpn_76_013703,
    Li2007_PhysRevB_75_104420,Tsunetsugu2007_JPhysCondensMatter_19_145248,
    Stoudenmire2009_PhysRevB_79_214436,Toth2012_PhysRevB_85_140403,
    Niesen2017_PhysRevB_95_180404,Niesen2017_SciPostPhys_3_030,
    Niesen2018_PhysRevB_97_245146}.
  The biquadratic interaction works as a strong quantum fluctuation exchanging the spin-1 pairs,
  and kills the anti-symmetric (dipolar) component.
  Since the biquadratic interaction is generally much smaller than the Heisenberg (bilinear)
  interaction~\cite{Tanaka2018_JPhysSocJpn_87_023702},
  this spin nematics is hardly realized in materials.
  To have the latter bond nematics in a spin-1/2 model~\cite{Andreev1984_SovPhysJETP_60_267}, often a very high magnetic field and a frustration effect are required;
  In a fully polarized spin-1/2 state, the standard lowest energy excitation is an $S^{z}=-1$ magnon.
  However, if there are good reasons to suppress the kinetics of magnon,
  e.g. the frustration effect on a $J_1$--$J_2$ square lattice model or a ring exchange model,
  the lowest excitation is replaced by the multi-magnons propagating together~\cite{
    Momoi2000_PhysRevB_62_15067,Momoi2005_ProgTheorPhysSuppl_159_72,
    Shannon2006_PhysRevLett_96_027213,Momoi2006_PhysRevLett_97_257204,
    Hikihara2008_JPhysSocJpn_77_014709,
    Hikihara2008_PhysRevB_78_144404,Sudan2009_PhysRevB_80_140402,
    Zhitomirsky2010_EurophysLett_92_37001,Momoi2012_PhysRevLett_108_057206,Wang2018_PhysRevLett_120_247201}.
  For example, the bounded two-magnons consisting of two spin-1/2's pointing downward form
  a quadrupole by definition, and condense into a spin nematic phase near the saturation field
  in spin-1/2 ladders~\cite{Hikihara2008_PhysRevB_78_144404,Sudan2009_PhysRevB_80_140402}.
  Whereas in practice, it is hard to realize such a high field in experiments.
\par
  Recently, a double-layered spin-1/2 dimer system is proposed as a platform of spin nematics in a \textit{zero magnetic field}~\cite{Yokoyama2018_PhysRevB_97_180404}.
  When the spin-1/2's are antiferromagnetically coupled within the dimer,
  they form a singlet, and the inter-dimer Heisenberg interactions work as chemical potential
  and dope the $\mathcal{S}=1$ triplets.
  By the additional spin-1/2-four-body inter-dimer interaction,
  a biquadratic interaction between these doped $\mathcal{S}=1$'s is generated
  and two different types of spin nematic phases appear next to the singlet phase.
  The model based on the ferromagnetically coupled spin-1/2 dimers
  have $\mathcal{S}=1$ triplet on a dimer-bond by construction,
  and are also found to host a small window of spin nematic phase, when the second order perturbative
  inter-dimer exchange coupling becomes relevant~\cite{Hikihara2019_PhysRevB_100_214414}.
  These works show that the spin nematics of the same type as that of the spin-1 BLBQ model
  is available in the spin-1/2 dimer systems by introducing the inter-dimer four-body exchanges
  which kill the anti-symmetric part of the spin-1/2 wave function, like the biquadratic interaction does on spin-1.
\par
  In the present paper, we replace the spin-1/2 dimer in Ref.~\onlinecite{Yokoyama2018_PhysRevB_97_180404}
  with spin-1 dimer,
  and deal with the two-dimensionally stacked spin-1 dimer forming a bilayer triangular lattice.
  We show that there are abundant types of magnetic and nonmagnetic long range ordered phases
  that are described by the quadrupolar as well as dipolar order parameters defined on a dimer-bond.
  Considering a strong intra-dimer antiferromagnetic and biquadratic coupling,
  where maximally $\mathcal{S}=1$ moment ($\mathcal{S}=1$ is the on-bond spin-1 throughout the paper)
  is generated in each dimer out of two spin-1's,
  the original dimerized spin-1's degrees of freedom is transformed to an $\mathcal{S}=1$ hard-core boson.
  The inter-dimer Heisenberg spin exchange interaction is then transformed to the kinetic and pair fluctuation effect
  of these bosons as well as the magnetic exchange between $\mathcal{S}=1$'s.
  When the former fluctuation effect dominates, the $\mathcal{S}=1$ bosons lose its dipolar moment and
  condense, partially occupying the dimers as quadrupoles which are called FQ-BEC phases.
  Besides, for a parameter region of nearly decoupled dimers,
  we find a spin nematic phase; we call ``spin nematics'' the phase based on the bound pairs of $S=1$,
  typically found in spin-1 BLBQ model.
  In a dimer-based system, spin nematics can be realized when the bosons are nearly fully packed on dimers.
  These quadrupolar phases are classified by the types of low-lying tower-of-state excitations
  in the energy spectrum that tell us how they break the symmetry.
  The richness of the phase diagram is possibly because of the large fluctuation of spin-1 moments allowed by the
  larger spin space compared to spin-1/2's, since the former affords larger entanglement in constructing $\mathcal{S}=1$.
  \par
  The technical aspect of the present work is that,
  despite a seeming difficulty in increasing the degrees of freedom of the spin moments from spin-1/2 to 1,
  one can treat it within a low-energy approximation using the same bosonic model
  as the spin-1/2 dimer case~\cite{Yokoyama2018_PhysRevB_97_180404}.
  Aside from the transformation of the spin-1/2 model being exact,
  the only difference is how to map the interaction parameters of spin models to those of the bosonic model.
  The direct motivation of dealing with spin-1 dimers instead of spin-1/2 dimers is to explain
  the intriguing nonmagnetic phase of Ba$_{3}M$Ru$_{2}$O$_{9}$ ($M=$ Zn, Ca, etc.)~\cite{Terasaki2017_JPhysSocJpn_86_033702,Yamamoto2018_JPhysCondensMatter_30_355801}, which will be finally discussed.
\section{Model Hamiltonian}
  We consider a system consisting of dimers of two spin-1's.
  As shown in Fig.~\ref{fig:lattice}(a),
  the dimers stack parallelly and form a double-layered triangular lattice.
  The Hamiltonian is given as
        \begin{align}
            \label{eq:spin_hamiltonian}
            \mathcal{H}
        & = \mathcal{H}_{\text{intra}} + \mathcal{H}_{\text{inter}}, \notag \\
        &    \mathcal{H}_{\text{intra}}
         = \sum_{i = 1}^{N}
                \left[
                    J \boldsymbol{S}_{i_{1}} \cdot \boldsymbol{S}_{i_{2}}
                + B \left( \boldsymbol{S}_{i_{1}} \cdot \boldsymbol{S}_{i_{2}} \right)^{2}
                \right], \notag \\
        &    \mathcal{H}_{\text{inter}}
         = \sum_{\braket{i, j}} \sum_{\gamma = 1, 2}
                \left(
                    J' \boldsymbol{S}_{i_{\gamma}} \cdot \boldsymbol{S}_{j_{\gamma}}
                + J''\boldsymbol{S}_{i_{\gamma}} \cdot \boldsymbol{S}_{j_{\bar{\gamma}}}
                \right).
        \end{align}
  where $\boldsymbol{S}_{i_{\gamma}}$ is the spin-1 operator of $\gamma$-th site on a $i$-th dimer.
  Here, we descriminate $\boldsymbol{S}_{i_{\gamma}}$ defined on a lattice site
  from $\boldsymbol{\mathcal{S}}_{i}$ which is spin-1 defined on a dimer-bond introduced the next section.
  The summation $\braket{i, j}$ is taken over all the neighboring pairs of dimers,
  and $\bar{1} = 2$ and $\bar{2} = 1$.
  $J$~$(> 0)$ and $B$~$(> 0)$ denote the antiferromagnetic Heisenberg and the biquadratic interactions,
  respectively,
  $J'$ and $J''$ are the inter-dimer Heisenberg interactions (Fig.~\ref{fig:lattice}(b)),
  and $N$ is the number of dimers.
\par
  In \S.~\ref{sec:boson}, we analyze the Hamiltonian Eq.~(\ref{eq:spin_hamiltonian}) by transforming it
  to the effective model of the $\mathcal{S}=1$ bosons via the perturbation theory,
  and then solving it by the numerical exact diagonalization (ED) on a finite cluster.
  The magnetic properties of the model is described by the $\mathcal{S}=1$ carried by the hard-core boson.
\par
  To disclose the details of the phase diagram, particularly to fix the existence of the long range order,
  we analyze the structure of the low lying excited states, namely a tower of states in \S.~\ref{sec:lro}.
  There, the spin-1 bosonic description is found to be not enough to understand several different classes of
  phases that form a quadrupolar ordering, which are classified as different types of ``nematic'' phases.
  We therefore introduce different operators defined on a dimer-bond other than $\boldsymbol{\mathcal{S}}_{i}$.
  To avoid confusion, we separate \S.~\ref{sec:lro} from \S.~\ref{sec:boson} where the latter deals
  fully with the $\mathcal{S}=1$ bosonic description.
\par
  The classification of the above mentioned different types of phases in comparison with
  the previously known phases based on quadrupolar moments is given in \S.~\ref{sec:discussions},
  followed by a brief summary in \S.~\ref{sec:summary}.

\section{Effective \texorpdfstring{$\mathcal{S}=1$}{S=1} bosonic model}
\label{sec:boson}
\subsection{Derivation of the effective Hamiltonian of bosons}
\label{sec:derivation}
\subsubsection{Low-energy states of spin-1 dimers}
\label{subsec:lowenergy_states}
  Let us first consider an isolated dimer consisting of two spin-1 interacting via the BLBQ interactions
  $\mathcal{H}_{\text{BLBQ}} = J \boldsymbol{S}_{1} \cdot \boldsymbol{S}_{2}
   + B \left( \boldsymbol{S}_{1} \cdot \boldsymbol{S}_{2} \right)^{2}$.
  The energy eigenstates of $\mathcal{H}_{\text{BLBQ}}$ are classified into singlet~($s$), triplets~($t$), and quintets~($q$),
  and their energies are given as $e(s) = -2J+4B$, $e(t) = -J+B$, and $e(q) = J+B$, respectively.
  Figure~\ref{fig:lattice}(c) shows these energy levels as a function of $B/J$.
  At small $B/J$, the Heisenberg interaction is dominant and the lowest energy state is a singlet.
  This singlet state is replaced by the triplet state when $B/J > 1/3$,
  while the quintet cannot have lower energy than the triplet and remain as the excited states.
\par
  As a starting point of the perturbation,
  we take $\mathcal{H}_{\text{inter}} = 0$,
  where the ground state is the product state of the singlets on the isolated dimers for $B/J < 1/3$,
  and that of triplets for $B/J>1/3$.
  In introducing $\mathcal{H}_{\text{inter}} \ne 0$,
  we consider the processes up to second order in $J'/J$ and $J''/J$,
  so that the effective interactions between two adjacent dimers appear mainly in the result.
  The energies of the disconnected two dimers with $\alpha$- and $\beta$-multiplets $E(\alpha, \beta)$ are shown in Fig.~\ref{fig:lattice}(d).
  One can see that the states including quintets are higher in energy than the states without quintets when $B/J < 2/3$.
  Therefore, based on the natural assumption that $B/J$ is small enough, we construct the effective Hamiltonian for the low-energy manifold of states including only singlets and triplets.
\par
  The first order process contributes to the energy correction of singlet and triplet states, as well as
  to the exchange of triplet and singlet on the neighboring two dimers.
  Within the second order perturbation processes between two adjacent dimers,
  the intermediate excited states have at least one quintet
  as shown in the examples of the processes;
  in Fig.~\ref{fig:lattice}(e),
  the two-dimer state $\ket{s, t_{0}}$ returns to the same state through the excited states $\ket{t_{0}, q_{0}}$,
  $\ket{t_{+1}, q_{-1}}$ and $\ket{t_{-1}, q_{+1}}$, where $\ket{s}$ is the singlet state,
  and $\ket{t_{\mu}}$ and $\ket{q_{\mu}}$ are the triplet and  the quintet states with $S^{z} = \mu$, respectively.
  In the processes shown in Fig.~\ref{fig:lattice}(f),
  $\ket{t_{+1}, t_{-1}}$, the two-dimer states with $S^{z} = +1$ and $S^{z} = -1$ triplet dimers,
  mixes with $\ket{t_{-1}, t_{+1}}$ via the three excited states
  $\ket{s_{0}, q_{0}}$, $\ket{q_{0}, s_{0}}$ and $\ket{q_{0}, q_{0}}$.
\par
  The low-energy basis can be described in the spin-1 hard-core bosonic language.
  The singlet corresponds to the vacuum,
  and the triplets are the bosons which are not allowed to doubly occupy a dimer.
  This kind of treatment is equivalent to the bond-operator approach,
  developed for the spin-1/2 dimer systems~\cite{Chubukov1989_JETPLett_49_129,Sachdev1990_PhysRevB_41_9323},
  and later applied to spin-1 dimer systems~\cite{Brenig2001_PhysRevB_64_214413,Wang2000_PhysRevB_61_4019} and also to general spin-$S$ dimers~\cite{Kumar2010_PhysRevB_82_054404}.
  We choose the time-reversal invariant form of the basis set $\left\{ \ket{t_{i, \alpha}} \right\}$ described as,
        \begin{align}
            \ket{t_{i, x}}
        & = \dfrac{\mathrm{i}}{2}
                    \left( \ket{+1, 0} - \ket{0, +1} - \ket{0, -1} + \ket{-1, 0} \right), \notag \\
            \ket{t_{i, y}}
        & = \dfrac{1}{2}
                    \left( \ket{+1, 0} - \ket{0, +1} + \ket{0, -1} - \ket{-1, 0} \right), \notag \\
            \ket{t_{i, z}}
        & = - \dfrac{\mathrm{i}}{\sqrt{2}} \left( \ket{+1, -1} - \ket{-1, +1} \right),
        \label{eq:triplets}
        \end{align}
  where the dimer states on the r.h.s. described as $\ket{S_{i_{1}}^{z}, S_{i_{2}}^{z}}$ are those classified by the $S^{z}$-values of the two spins forming a dimer.
  The details of the bond-operator approach
  and the description of the original spin operators using the bosonic operators are shown in Appendix~\ref{subsec:appendix_bondoperator}.
		\begin{figure}
			\centering
			\includegraphics[width=86mm]{lattice.pdf}
			\caption{(a) Schematic illustration of the spin-1 dimer model on a triangular lattice.
							(b) Intra- and inter-dimer interactions, where $J>0$, $J'$, $J''$ are the Heisenberg exchanges, and $B$ is the biquadratic exchange.
							(c) Eigenenergy levels of $\mathcal{H}_{\text{intra}}$ of an isolated dimer.
								$s$, $t$, and $q$ denote the singlet, triplet, and quintet states of the dimer,respectively.
							(d) Eigenenergy levels of $\mathcal{H}_{\text{intra}}$ of two isolated dimers without the inter-dimer $J'$ and $J''$.
							(e), (f) Examples of the second order perturbation processes.
							Processes in panel (e) returns to the original state $\ket{s, t_{0}}$
							and those in panel (f) exchange the $\mathcal{S}^{z} = +1$ and $\mathcal{S}^{z} = -1$ triplets.
							They are the origins of the biquadratic interactions between two triplets
							$\left( \boldsymbol{\mathcal{S}}_{i} \cdot \boldsymbol{\mathcal{S}}_{j} \right)^{2}$.
			}
			\label{fig:lattice}
		\end{figure}
\subsubsection{Effective Hamiltonian}
\label{subsec:effective_hamiltonian}
  The triplet state with $\alpha$-component $\ket{t_{i, \alpha}}$ at site-$i$ is expressed as $b_{i, \alpha}^{\dag} \ket{0}$,
  where $\ket{0}$ is the singlet state and $b_{i, \alpha}^{\dag}$ is the creation operator of a boson representing that triplet.
  Using this bosonic operator, the effective Hamiltonian $\mathcal{H}_{\text{eff}}$ up to second order in $J'/J$ and $J''/J$ is given as
        \begin{align}
            \label{eq:boson_hamiltonian}
            \mathcal{H}_{\text{eff}}
        & = E_{0} + \mathcal{H}_{\mu} + \mathcal{H}_{t} + \mathcal{H}_{P} + \mathcal{H}_{V} + \mathcal{H}_{\mathcal{J}} + \mathcal{H}_{\mathcal{B}} + \mathcal{H}_{\text{3body}}, \notag \\
            \mathcal{H}_{\mu}
        & = - \mu \sum_{i = 1}^{N} n_{i} \notag \\
            \mathcal{H}_{t}
        & = t \sum_{\braket{i, j}} \sum_{\alpha = x, y, z}
                    b_{i, \alpha}^{\dag} b_{j, \alpha} + \text{h.c.}, \notag \\
            \mathcal{H}_{P}
        & = P \sum_{\braket{i, j}} \sum_{\alpha = x, y, z}
                    b_{i, \alpha}^{\dag} b_{j, \alpha}^{\dag} + \text{h.c.}, \notag \\
            \mathcal{H}_{V}
        & = V \sum_{\braket{i, j}} n_{i} n_{j}, \notag \\
            \mathcal{H}_{\mathcal{J}}
        & = \mathcal{J} \sum_{\braket{i, j}}
                \boldsymbol{\mathcal{S}}_{i} \cdot \boldsymbol{\mathcal{S}}_{j} n_{i} n_{j}, \notag \\
            \mathcal{H}_{\mathcal{B}}
        & = \mathcal{B} \sum_{\braket{i, j}}
                    \left( \boldsymbol{\mathcal{S}}_{i} \cdot \boldsymbol{\mathcal{S}}_{j} \right)^{2} n_{i} n_{j}.
        \end{align}
  Here, $n_{i} = \sum_{\alpha} b_{i, \alpha}^{\dag} b_{i, \alpha}$ is the number operator,
  and the hard-core condition $n_{i} = 0$ or $1$ is imposed on the number operator.
  The spin-1 operator of $i$-th boson is expressed by $\boldsymbol{\mathcal{S}}_{i}$,
  where $\mathcal{S}_{i}^{\alpha} = - \mathrm{i} \sum_{\beta, \gamma} \varepsilon_{\alpha\beta\gamma} b_{i, \beta}^{\dag} b_{i, \gamma}$
  and $\varepsilon_{\alpha\beta\gamma}$ is the Levi--Civita symbol.
  We note that the Hamiltonian keeps the $\mathrm{SU}(2)$ symmetry of triplets~\cite{Lecheminant2006_PhysRevB_74_224426,Totsuka2012_PhysRevB_86_014435,Yokoyama2018_PhysRevB_97_180404},
  as far as the magnetic field is not applied~\cite{Nikuni2000_PhysRevLett_84_5868}.
\par
  The parameters included in $\mathcal{H}_{\text{eff}}$ is described by the original interaction parameters in Eq.~(\ref{eq:spin_hamiltonian}) as
        \begin{align}
            \label{eq:boson_coefficients}
            &    E_{0} = \left( -2J + 4B \right) N, \notag \\
            &    \mu = - J + 3B
                            + \dfrac{20z}{27 \left( J - B\right)} \left( J' - J'' \right)^{2}, \notag \\
            &    t = \dfrac{4}{3} \left( J' - J''\right), \
                P = - \dfrac{4}{3} \left( J' - J''\right), \notag \\
            &    V = \left[ \dfrac{40}{27 \left( J - B \right)} - \dfrac{8}{9 \left( J + 3B \right)}
                                        - \dfrac{2}{9J}
                        \right] \left( J' - J'' \right)^{2}, \notag \\
            &    \mathcal{J} = \dfrac{J' + J''}{2}
                            + \left[ - \dfrac{4}{3 \left( J + 3B \right)} + \dfrac{1}{12J} \right] \left( J' - J'' \right)^{2}, \notag \\
            &    \mathcal{B} = \left[ - \dfrac{4}{9 \left( J + 3B \right)} - \dfrac{1}{144J} \right] \left( J' - J'' \right)^{2},
        \end{align}
  where $z$ is the coordination number.
  One can immediately see that $\mu, t, P$ and $\mathcal{J}$-terms include the terms that originate from the first order process,
  whereas $V$ and $\mathcal{B}$-terms do not.
\par
  There are some processes at the second order level where the three dimers take part in,
  which we denote as $\mathcal{H}_{\text{3body}}$ in Eq.~(\ref{eq:boson_hamiltonian}).
  We numerically evaluate the effects of $\mathcal{H}_{\text{3body}}$ on the effective Hamiltonian
  by comparing the energies of the ground states of the original Hamiltonian $\mathcal{H}$ (Eq.~(\ref{eq:spin_hamiltonian})),
  and of the effective Hamiltonian $\mathcal{H}_{\text{eff}}$ (Eq.~(\ref{eq:boson_hamiltonian}))
  with and without $\mathcal{H}_{\text{3body}}$ in a small cluster,
  finding that it does not play a significant role.
  We thus discard this $\mathcal{H}_{\text{3body}}$ term in the following for simplicity.
  The details of the evaluation of the effective model is shown in the Appendix~\ref{subsec:appendix_evaluation}.
  We further show that even the other second order terms included in Eq.~(\ref{eq:boson_hamiltonian}),
  do not contribute much to the majority of phases we deal with.
  The way how the inter-dimer interactions work thus turns out to be simple.
\subsubsection{Physical quantities}
\label{subsec:physical_quantitles}
  For the analysis of the effective model, we calculate the following properties that characterize the ground state.
  The boson density per dimer is denoted as $\braket{n_{t}}=N^{-1}\sum_{i=1}^{N} \braket{n_{i}}$,
  and its structure factor is given as
        \begin{align}
            N(\boldsymbol{k})
        = \dfrac{1}{N} \sum_{i, j = 1}^{N} \braket{n_{i}n_{j}}
                \mathrm{e}^{\mathrm{i}\boldsymbol{k} \cdot \left( \boldsymbol{r}_{i} - \boldsymbol{r}_{j} \right)}.
        \end{align}
\par
  The magnetic properties are examined by the spin and quadrupole structure factors
        \begin{align}
            \mathcal{S}(\boldsymbol{k})
        &    = \dfrac{1}{N} \sum_{i, j = 1}^{N}
                    \braket{\boldsymbol{\mathcal{S}}_{i} \cdot \boldsymbol{\mathcal{S}}_{j} n_{i}n_{j}}
                    \mathrm{e}^{\mathrm{i}\boldsymbol{k} \cdot \left( \boldsymbol{r}_{i} - \boldsymbol{r}_{j} \right)}, \label{eq:spin_structure_factor} \\
            \mathcal{Q}(\boldsymbol{k})
        &    = \dfrac{1}{N} \sum_{i, j = 1}^{N}
                    \braket{\boldsymbol{\mathcal{Q}}_{i} \cdot \boldsymbol{\mathcal{Q}}_{j} n_{i}n_{j}}
                    \mathrm{e}^{\mathrm{i}\boldsymbol{k} \cdot \left( \boldsymbol{r}_{i} - \boldsymbol{r}_{j} \right)}, \label{eq:quadrupole_structure_factor}
        \end{align}
  where $\boldsymbol{\mathcal{Q}}_{i}$ is the 5-component vector representation of quadrupole operator of spin-1 bosons defined as
        \begin{align}
            \boldsymbol{\mathcal{Q}}_{i}
        = \begin{pmatrix}
                \mathcal{Q}_{i}^{x^{2}-y^{2}} \\ \mathcal{Q}_{i}^{3z^{2}-r^{2}} \\
                \mathcal{Q}_{i}^{xy} \\ \mathcal{Q}_{i}^{yz} \\ \mathcal{Q}_{i}^{zx} \\
            \end{pmatrix}
        = \begin{pmatrix}
                \left( \mathcal{S}_{i}^{x} \right)^{2} - \left( \mathcal{S}_{i}^{y} \right)^{2} \\
                \dfrac{1}{\sqrt{3}}
                    \left[ 3 \left( \mathcal{S}_{i}^{z} \right)^{2}
                                - \mathcal{S} \left( \mathcal{S} + 1 \right) \right] \\
                \mathcal{S}_{i}^{x} \mathcal{S}_{i}^{y} + \mathcal{S}_{i}^{y} \mathcal{S}_{i}^{x} \\
                \mathcal{S}_{i}^{y} \mathcal{S}_{i}^{z} + \mathcal{S}_{i}^{z} \mathcal{S}_{i}^{y} \\
                \mathcal{S}_{i}^{z} \mathcal{S}_{i}^{x} + \mathcal{S}_{i}^{x} \mathcal{S}_{i}^{z} \\
            \end{pmatrix}.
            \label{eq:quadrupole_operator}
        \end{align}
\par
  In a system with spin-1 defined on each site, typically represented by the spin-1 BLBQ models,
  the quadrupole operator $\boldsymbol{\mathcal{Q}}_{i}$ is the on-``site'' operator.
  For a system with spin-1/2 per site, the quadrupole operator is defined on a bond instead, since one needs to prepare a spin-1 from
  two spin-1/2's~\cite{Shannon2006_PhysRevLett_96_027213,Penc2011_IFM_Chap13_SpinNematics}.
  In the present case, the two sites forming a dimer each hosts spin-1 operators, $\boldsymbol{S}_{i_{1}}$ and $\boldsymbol{S}_{i_{2}}$,
  and $\boldsymbol{\mathcal{S}}_{i}$, which is defined on a dimer-bond is a composition of these two spin-1's.
  For this reason, one can also define another quadrupole operator on dimer-bond as
        \begin{align}
            Q_{i_{12}}^{\alpha\beta}
        = S_{i_{1}}^{\alpha} S_{i_{2}}^{\beta} + S_{i_{1}}^{\beta} S_{i_{2}}^{\alpha}
            - \dfrac{2}{3} \left( \boldsymbol{S}_{i_{1}} \cdot \boldsymbol{S}_{i_{2}} \right) \delta_{\alpha\beta}.
        \end{align}
  Then, one finds that $\boldsymbol{Q}_{i_{12}}$ and $\boldsymbol{\mathcal{Q}}_{i}$ are equivalent in terms of our triplet states, namely,
        \begin{align}
            \braket{t_{\alpha}|Q_{i_{12}}^{\mu\nu}|t_{\beta}}
        = \braket{t_{\alpha}|\mathcal{Q}_{i}^{\mu\nu}|t_{\beta}}
        \end{align}
  holds for $\alpha, \beta, \mu, \nu = x, y, z$.
  In the same manner, the spin operator inside $i$-th spin-1 dimer defined as
        \begin{align}
            S_{i_{12}}^{\alpha} = S_{i_{1}}^{\alpha} + S_{i_{2}}^{\alpha}
        \end{align}
  works in the same way as $\mathcal{S}_{i}^{\alpha}$ for the triplet states, i.e.,
        \begin{align}
            \braket{t_{\alpha}|S_{i_{12}}^{\mu}|t_{\beta}}
        = \braket{t_{\alpha}|\mathcal{S}_{i}^{\mu}|t_{\beta}}
        \end{align}
  holds for $\alpha, \beta, \mu = x, y, z$.
\par
  Unlike the spin-1 BLBQ models~\cite{Lauchli2006_PhysRevLett_97_087205},
  the number of spin-1 bosons per dimer is not fixed in our Hamiltonian.
  However, one can consider the spin-1 BLBQ model as the $\braket{n_{t}}=1$-limiting case of our model since the two models share the same definition, Eq.~(\ref{eq:quadrupole_operator}).
  One can thus make use of the analysis applied to the spin-1 BLBQ model~\cite{Lauchli2006_PhysRevLett_97_087205};
  there are so-called $\mathrm{SU}(3)$-points in the BLBQ model,
  where the three components of $\boldsymbol{\mathcal{S}}$ and the five components of $\boldsymbol{\mathcal{Q}}$
  equivalently form the eight elements of the $\mathrm{SU}(3)$ Lie algebra.
  Exactly at this point the transition between the magnetic and the spin nematic phases is known to take place.
  Numerically, this transition is identified by the point where $\mathcal{S}(\boldsymbol{k})$ and $\bar{\mathcal{Q}}(\boldsymbol{k})\equiv (3/5)\mathcal{Q}(\boldsymbol{k})$
  take the same values. We thus use this normalized value $\bar{\mathcal{Q}}(\boldsymbol{k})$ to determine the phase transitions between the magnetic and the quadrupolar states.
    \begin{figure}
      \centering
      \includegraphics[width=86mm]{phasediagram_2ndorder.pdf}
      \caption{
                Ground state phase diagram of the spin-1 dimer triangular lattice at (a) $B/J = 0.2$ and (b) $0.4$,
                obtained by the numerical diagonalization of the effective model with $N = 12$.
		            Filled and open circles represent the first and second order phase transitions,
		            where the transition between FM-BEC and FQ-BEC-2 is weakly first order.
                FM, AFM, FQ represent the ferromagnetic, antiferromagnetic and ferro-quadrupolar phases,
                and $\braket{n_{t}}\approx 1$ and $\lesssim 0.9$ are the solid and BEC states of bosons.
                Colors in the phase diagram are the density plot of the triplet number $\braket{n_{t}}$.
                The small $J'$, $J''$ region marked with red square in (b) encloses SN phase (see Fig.~\ref{fig:smallphased}(b)).
                (c) Phase diagram on the plane of $J'/J$ and $B/J$, whose fixed $B/J$-lines correspond to
                the $J'=J''$ line of the phase diagrams in panels (a) and (b).
      }
      \label{fig:phasediagram_2ndorder}
    \end{figure}
    \begin{figure}
      \centering
      \includegraphics[width=86mm]{physicalquantities_bb020.pdf}
      \caption{
                $J'/J$ dependences of the physical quantities at $B/J = 0.2$
                when (a)--(c) $J''/J = -0.2$ and (d)--(f) $J''/J = +0.1$.
                (a), (d) Total energies $e_{\text{all}}$ and the contributions from
                major terms in the effective Hamiltonian, $e_{t}, e_{P}, e_{\mathcal{J}}$ and $e_{\mathcal{B}}$.
                (b), (e) Triplet densities $\braket{n_{t}}$.
                (c), (f) Spin (Eq.~(\ref{eq:spin_structure_factor})) and quadrupole (Eq.~(\ref{eq:quadrupole_structure_factor}))
                structure factors at $\Gamma$, $\mathrm{K}$, $\mathrm{M}$-points in the reciprocal space.
                Quadrupole structure factors denoted as $\bar{\mathcal{Q}}(\boldsymbol{k})$ are normalized
                to be compared with the spin structure factors $\mathcal{S}(\boldsymbol{k})$.
      }
      \label{fig:physicalquantities_bb020}
   \end{figure}
\subsection{Results of the \texorpdfstring{$\mathcal{S}=1$}{S=1} bosonic model}
\label{sec:results}
\subsubsection{Phase diagram}
\label{subsec:phase_diagram}
  We numerically diagonalize $\mathcal{H}_{\text{eff}}$ on the $N = 12$ triangular lattice ($z = 6$) under the periodic boundary condition.
  The phase diagrams on the plane of $J'/J$ and $J''/J$ at $B/J = 0.2$ and $0.4$ are shown
  in Figs.~\ref{fig:phasediagram_2ndorder}(a) and ~\ref{fig:phasediagram_2ndorder}(b).
\par
  The phase diagram is divided into four parts in overall.
  When $J'\sim J'' >0$, the antiferromagnetic phases with $\braket{n_{t}}\approx 1$ (AFM-solid) and $\braket{n_{t}}\lesssim 0.9$ (AFM-BEC) are stabilized by the antiferromagnetic interaction,
  $\mathcal{J}>0$, between bosons occupying the neighboring dimers.
  On the opposite part of the phase diagram, $J'\sim J' <0$, the ferromagnetic phase with $\braket{n_{t}}\approx 1$
  (FM-solid) is realized for the similar reasons.
  When $J'-J'' < 0$ and $J'-J'' > 0$,
  two different types of ferroquadrupolar (FQ) phases, FQ-BEC-1 and FQ-BEC-2
  appear over a wide parameter region.
  Throughout both of the phase diagrams, we see no particular features of bosons,
  i.e., $N(\boldsymbol{k})$ takes the maximum value at $\Gamma$-point,
  which indicates that bosons distribute uniformly in space and
  does not show any translational symmetry breaking long range order.
\subsubsection{\texorpdfstring{$J' = J''$}{J' = J''} line}
\label{subsec:diagonal}
  The starting point is $J'=J''=0$, at which the ground state is the product state of the isolated-dimer state.
  As one can see from Eq.~(\ref{eq:boson_coefficients}), most of the parameters, namely $t,P,V,\mathcal{B}$
  are the linear or the square functions of $(J'-J'')$.
  Therefore, these parameters remain zero exactly at $J'=J''$, namely the inter-dimer interactions cancel out
  because of the geometrical frustration effect.
\par
  In fact, when $B/J = 0.2$, the singlet product state, namely $\braket{n_{t}}=0$,
  remains a ground state along this line.
  The endpoint of this singlet phase is evaluated in the following manner;
  when $J' = J''$, the effective Hamiltonian consists only of two terms
        \begin{align}
            \mathcal{H}_{J'=J''}
        = -\mu \sum_{i = 1}^{N} n_{i}
        + \mathcal{J}\sum_{\braket{i, j}} \boldsymbol{\mathcal{S}}_{i} \cdot \boldsymbol{\mathcal{S}}_{j} n_{i} n_{j},
        \end{align}
  with $\mu = -J + 3B$ and $\mathcal{J} = J'$.
  Regardless of its sign, $\mathcal{J}$ works as an effective attractive interaction between bosons,
  since it is energetically favorable to occupy the neighboring pairs of dimers with triplets to gain the magnetic interaction energy.
  Then, there is a first order transition between the $\braket{n_{t}}=0$-singlet and the $\braket{n_{t}}=1$-FM or AFM solid phases.
  The phase boundary can be obtained by comparing their energies, $E_{0}(N)$ and $E_{1}(N)$, where there is a relationship,
        \begin{align}
            E_{1}(N) = E_{0}(N) -\mu N + 3N e_{\text{bond}}.
        \end{align}
  Here, $e_{\text{bond}}$ is evaluated as the bond-energy of the ground state of the spin-1 triangular lattice Heisenberg model
  $J'\sum_{\braket{i, j}}\boldsymbol{S}_{i} \cdot\boldsymbol{S}_{j}$ for $N=12$.
  Figure~\ref{fig:phasediagram_2ndorder}(c) shows the resultant phase diagram on the plane of $J'=J''$ and $B$ with $J=1$.
  The singlet phase corresponding to the straight line in Fig.~\ref{fig:phasediagram_2ndorder}(a)
  shrinks toward smaller $J'=J''$ value with increasing $B/J$, and disappears at $B/J=1/3$.
  For $B/J>1/3$, $\braket{n_{t}}=1$ is realized throughout the whole $J'=J''$ line.
\subsubsection{Ferromagnetic and antiferromagnetic phases}
\label{subsec:magnetic_phases}
  The FM and AFM phases extend from the endpoints of the $J' = J''$-singlet phase discussed above.
  In Fig.~\ref{fig:physicalquantities_bb020},
  we show the total energy $e_{\text{all}}$ and the contributions from each terms, $e_{t}$, $e_{P}$, $e_{\mathcal{J}}$, and $e_{\mathcal{B}}$,
  the boson density, $\braket{n_{t}}$, and the values of the structure factors at $\Gamma$, $\mathrm{K}$ and $\mathrm{M}$ points of the Brillouin zone.
  We vary $J'/J$ along the fixed $J''/J=-0.2$ and $0.1$ lines.
  In the former case, a jump in the physical quantity is found at the transition from the FM-solid to the FQ-BEC-2 phase.
  Compared to other phases, the FM-solid phase has a large energy gain of $e_{\mathcal{J}}$, indicating that the magnetic interaction $\mathcal{J}$ is responsible for stabilizing the FM-solid.
  Indeed, $\mathcal{S}(\boldsymbol{k})$ shows a peak at the $\Gamma$-point in this phase
  while the other $\mathcal{S}(\boldsymbol{k})$ and $\mathcal{Q}(\boldsymbol{k})$ remain small.
  When we vary $J'/J$ along $J''/J=0.1$ (Figs.~\ref{fig:physicalquantities_bb020}(d)--(f)),
  $\braket{n_{t}}\lesssim 0.55$, and the system remains a BEC.
  The transitions along this line are of second order.
  At $J'/J\gtrsim 0.3$, $\mathcal{S}(\boldsymbol{k})$ at $\mathrm{K}$-point starts to overwhelm $\mathcal{Q}(\boldsymbol{k})$ at the $\Gamma$-point
  which we recognize as the AFM-BEC phase, following the treatment in Ref.[~\onlinecite{Lauchli2006_PhysRevLett_97_087205}] (see the last part of \S.~\ref{subsec:physical_quantitles}).
  The phase boundaries in Fig.~\ref{fig:phasediagram_2ndorder} are classified into first and second order ones (filled and open circles)
  according to this analysis.
\subsubsection{FQ phases}
\label{subsec:spinnematic_phases}
  In the phase diagram, there are two different ferroquadrupolar phases, FQ-BEC-1 extending at $J''>J'$ and FQ-BEC-2 at $J''<J'$.
  As we see in Figs.~\ref{fig:physicalquantities_bb020}(e) and \ref{fig:physicalquantities_bb020}(f),
  $\braket{n_{t}}$ and $\mathcal{Q}(\boldsymbol{k})$ both once decrease down to zero at the boundary of the two phases
  where the singlet state appears, which marks the second order transition.
  In such a case, the order parameters of the two phases should differ.
  In fact, although $\mathcal{Q}(\boldsymbol{k})$ at $\Gamma$-point is dominant in both phases, only in the FQ-BEC-2 phase
  $\mathcal{Q}(\boldsymbol{k})$ at $\mathrm{K}$-point takes as large value as well.
\par
  Figures~\ref{fig:correlation_circle}(a) and \ref{fig:correlation_circle}(b) show
  the two-point quadrupole correlations between site-1 and site-$j$,
  $\braket{\boldsymbol{\mathcal{Q}}_{1}\cdot\boldsymbol{\mathcal{Q}}_{j}}$,
  in FQ-BEC-1 and FQ-BEC-2 phases.
  The former correlation develops uniformly in space, whereas in the latter,
  there is apparently a growth of correlation in the period of twice the lattice spacing in all three directions.
  This three-sublattice-like structure of quadrupole moments corresponds to the peak of $\mathcal{Q}(\boldsymbol{k})$ at the $\mathrm{K}$-point.
  Figures~\ref{fig:correlation_circle}(c) and \ref{fig:correlation_circle}(d)
  are the two-point correlation of bosons, $\braket{n_{1}n_{j}}$, which are both uniform in space.
  This indicates that the three-sublattice structure of the quadrupolar moment in the FQ-BEC-2 is not because of
  the modulated the bosonic distribution but originates purely from the correlation between the spin degrees of freedom $\boldsymbol{S}_{i}$;
  the nearest-neighbor quadrupolar correlation is suppressed,
  while the next nearest-neighboring correlations are ferroic.
    \begin{figure}
      \centering
      \includegraphics[width=86mm]{correlation_circle.pdf}
      \caption{Spatial correlation functions for $(J'/J, J''/J)= (-0.2, +0.2)$ in FQ-1
               and $(+0.2, -0.2)$ in FQ-2 phases at $B/J = 0.2$;
               the quadrupolar correlations in (a), (b) $\braket{\boldsymbol{\mathcal{Q}}_{1} \cdot \boldsymbol{\mathcal{Q}}_{j}}$
               and the boson-boson correlation in (c), (d) $\braket{n_{1}n_{j}}$.
               Areas of the circles are proportional to the amplitude of correlations,
               $\left| \braket{\boldsymbol{\mathcal{Q}}_{1} \cdot \boldsymbol{\mathcal{Q}}_{j}} \right|$
               or $\left|\braket{n_{1}n_{j}}\right|$.
               Red and blue circles in (a), (b) correspond to the signs of $\braket{\boldsymbol{\mathcal{Q}}_{1} \cdot \boldsymbol{\mathcal{Q}}_{j}}$, positive and negative, respectively.
      }
      \label{fig:correlation_circle}
    \end{figure}
\subsubsection{Case of \texorpdfstring{$B/J = 0.4$}{B/J = 0.4}}
\label{subsec:results_bb040}
  We now focus on the case of $B/J = 0.4$, where $\mu$ takes a positive value.
  The singlet phase no longer exists and the triplet product state realized at $J'=J''=0$ immediately transforms to either
  of the phases we discussed earlier when the inter-dimer interactions become finite.
  Figure~\ref{fig:physicalquantities_bb040} shows the $J'/J$ dependences of energies,
  boson density, and the structure factors to be compared with Fig.~\ref{fig:physicalquantities_bb020}.
  The first order transitions separating the FM-solid from FQ phases are observed.
  The boson density in the FQ-BEC phase remains quite stable at around $\braket{n_{t}}\approx 0.55$,
  indicating that the nature of the BEC phases does not change much with $B/J$.
 \begin{figure}
 	\centering
 	\includegraphics[width=86mm]{physicalquantities_bb040.pdf}
 	\caption{
 		$J'/J$ dependence of the physical quantities at $B/J = 0.4$
 		when (a)--(c) $J''/J = -0.1$ and (d)--(f) $J''/J = +0.2$.
 		(a), (d) Total energies and partial energies of some terms of the effective Hamiltonian.
 		(b), (e) Triplet densities.
 		(c), (f) Spin (Eq.~(\ref{eq:spin_structure_factor})) and quadrupole (Eq.~(\ref{eq:quadrupole_structure_factor})) structure factors of some points in the reciprocal space.
 		The qudrupole structure factors are normalized to $\bar{\mathcal{Q}}(\boldsymbol{k})$
 		to be compared with the spin structure factors $\mathcal{S}(\boldsymbol{k})$.
 	}
 	\label{fig:physicalquantities_bb040}
 \end{figure}
\subsubsection{The orders of perturbation}
\label{subsec:perturbation_order}
  The interaction parameters in Eq.~(\ref{eq:boson_coefficients}) include the first and second order terms.
  Among them, $V$ and $\mathcal{B}$ disappear when we neglect the second order terms.
  To see how much the second order terms contribute to the determination of the phase diagram,
  we perform the numerical diagonalization by limiting the parameter values to those up to the first order in $J'/J$ and $J''/J$ with $N=12$.
  Figures~\ref{fig:phasediagram_1storder}(a) and ~\ref{fig:phasediagram_1storder}(b) show the phase diagrams at $B/J = 0.2$ and $0.4$.
  The phase diagrams are in good agreement with those in Figs.~\ref{fig:phasediagram_2ndorder}(a) and (b),
  indicating that $V$ and $\mathcal{B}$ do not play a major role in the five representative phases,
  FM-solid, AFM-solid/BEC, FQ-BEC-1 and FQ-BEC-2.
    \begin{figure}
      \centering
      \includegraphics[width=86mm]{phasediagram_1storder.pdf}
      \caption{
                (a), (b) Phase diagrams of the effective model up to the first order in $J'/J$ and $J''/J$
                at (a) $B/J = 0.2$ and (b) $B/J = 0.4$,
                obtained by the analysis of the results of the numerical diagonalization on the $N = 12$ cluster.
            }
      \label{fig:phasediagram_1storder}
    \end{figure}
\section{Long range orders}
\label{sec:lro}
We so far disclosed the overall magnetic properties of the system in the $\mathcal{S}=1$ bosonic description.
Since the phase diagrams is obtained by correlation functions in a small finite cluster,
the nature of the collective ground state, particularly of the FQ phases, should be explicitly determined.
For that purpose, we perform the thick-restart Lanczos method~\cite{Wu2000_SIAMJMatrixAnalAppl_22_602}
and disclose the scaling properties of the low-lying levels of the exact spectra.
Figures~\ref{fig:tos}(a)--\ref{fig:tos}(c) show the ones from the FQ-BEC-1, FQ-BEC-2 and AFM-solid phases, respectively.
For the (a) FQ-BEC-1 and (c) AFM-solid phases, we clearly find quasidegenerate joint states (QDJS)~\cite{Anderson1952_PhysRev_86_694,Bernu1992_PhysRevLett_69_2590,Bernu1994_PhysRevB_50_10048} that indicate the existence of SU(2) symmetry broken
long range order marked with solid lines.

The QDJS found in Fig.~\ref{fig:tos} (a) consists of $S=0,1,2,3,\cdots$ states
with only one level in each $S$-sector all belonging to $\Gamma$-point,
characterizing the excitation of the U(1) uniaxial rotator.
They clearly differ from the one known for the spin-1 nematic phase~\cite{Penc2011_IFM_Chap13_SpinNematics}
(cf. see Fig.~\ref{fig:smallphased}(a)).
This FQ-BEC-1 phase is identified as the one found in the $S=1/2$ spin ladder system~\cite{Totsuka2012_PhysRevB_86_014435}
denoted as F-nematic phase.
In this phase, if we shift from the bosonic picture to the original pair of spin-$\boldsymbol{S}_{i_{\gamma}}$ forming a dimer,
one can redescribe state in terms of different operators defined on a dimer;
the staggered operator of the two spin and the vector chiral operator
\begin{align}
\label{eq:q}
	\boldsymbol{q}_{i}
	& = \dfrac{1}{2}\left( \boldsymbol{S}_{i_{1}}- \boldsymbol{S}_{i_{2}} \right), \\
\label{eq:p}
	\boldsymbol{p}_{i}
	& = \boldsymbol{S}_{i_{1}} \times \boldsymbol{S}_{i_{2}}
\end{align}
following Eq.~(15) in Ref.~\onlinecite{Totsuka2012_PhysRevB_86_014435},
which is related to our $\mathcal{S}=1$ bosonic operator as
$b_{i,\alpha}\propto q_{i}^{\alpha} - \mathrm{i} p_{i}^{\alpha}$ and $b_{i,\alpha}^{\dagger}\propto q_{i}^{\alpha} + \mathrm{i}p_{i}^{\alpha}$.
The structural factor of $\boldsymbol{q}_{i}$ and $\boldsymbol{p}_{i}$ are
\begin{align}
\label{eq:staggered_spin_structurefactor}
	\mathcal{N}(\boldsymbol{k})
& = \dfrac{1}{N} \sum_{i,j=1}^{N}
		\braket{\boldsymbol{q}_{i}\cdot\boldsymbol{q}_{j}}
		\mathrm{e}^{\mathrm{i}\boldsymbol{k}(\boldsymbol{r}_{i}- \boldsymbol{r}_{j})}, \\
	\mathcal{C}(\boldsymbol{k})
& = \dfrac{1}{N} \sum_{i,j=1}^{N}
		\braket{\boldsymbol{p}_{i}\cdot\boldsymbol{p}_{j}}
		\mathrm{e}^{\mathrm{i}\boldsymbol{k}(\boldsymbol{r}_{i}-\boldsymbol{r}_{j})}.
\end{align}
Since the quadrupolar moment on a dimer-bond is rewritten in the form
\begin{align}
  \mathcal{Q}_{i}^{\alpha\beta}
 = - \left\{
       \frac{3}{4}
         \left[ \left( q_{i}^{\alpha} q_{i}^{\beta} + p_{i}^{\alpha} p_{i}^{\beta} \right)
              + \left( q_{i}^{\beta} q_{i}^{\alpha} + p_{i}^{\beta} p_{i}^{\alpha} \right) \right]
     - \frac{2}{3} \delta_{\alpha\beta}
     \right\} n_{i},
\end{align}
$\braket{\mathcal{Q}_{i}} \ne 0$ indicates that at least
$\braket{q_{i}} \ne 0$ or $\braket{p_{i}} \ne 0$ is fulfilled.
In this description, the F-nematic phase is the ferroic order of staggered operator $\braket{\boldsymbol{q}_{i}\cdot\boldsymbol{q}_{j}} > 0$ and $\braket{p_{i}}=0$, breaking SU(2) down to U(1).
This means that the small moments induced on these two spins are always forming an antiparallel state
and \textit{keep the dimer unit always nonmagnetic},
whereas these small moments form a ferromagnetic long range order inside the upper and lower 2D layers.
We confirmed by comparing with the $N=9$ QDJS that the tower of states collapses to the ground state as $1/N$.
\par
Figure~\ref{fig:tos}(d) shows the boson density $\braket{n_{t}}$ and structural factors
when $J'/J$ is varied from FQ-BEC-1 ($J'/J<0.15$), FQ-BEC-2 ($0.15<J'/J< 0.25$) to AFM-BEC phases.
Large $\mathcal{N}(\boldsymbol{k})$ at the $\Gamma$-point characterizes FQ-BEC-1,
$\mathcal{C}(\boldsymbol{k})$ as well as $\mathcal{N}(\boldsymbol{k})$ at the $\mathrm{K}$-point take large values in FQ-BEC-2
and particularly $\mathcal{C}(\boldsymbol{k})$ differenciates FQ-BEC-2 from AFM phase.
\par
We now go back to Fig.~\ref{fig:tos}(b) for FQ-BEC-2 with a QDJS-like structure.
The states between two broken lines may form a set of ground state manifold with $(2S+1)$-degeneracy
for each $S$-sector, in consistency with the biaxial rotator or a quantum top
as found in the 120$^\circ$-N\'{e}el ordering in the spin-1/2 triangular lattice antiferromagnet~\cite{Bernu1992_PhysRevLett_69_2590,Bernu1994_PhysRevB_50_10048}.
However, these low lying levels are not well separated from the states above them.
It can be contrasted from the AFM-solid phase with distinct $(2S+1)$-degenerate levels forming QDJS
which is a clear indication of 120$^\circ$-N\'{e}el ordering of ${\mathcal S}=1$ moments.
In FQ-BEC-2, one can also focus on the lowests level in each sector near the lower broken lines;
their $S=0,3,6$ levels at $\Gamma$-point and $S=1,2,4,5$ at $\mathrm{K}$-point are
symmetric under the symmetry action when putting a uniaxial rotator on a triangular three sublattices,
suggesting the $p$-nematic type of property~\cite{Lauchli2005_PhysRevLett_95_137206}.

Again refering to the phase diagram of the spin ladder system in Ref.~\onlinecite{Totsuka2012_PhysRevB_86_014435},
there is a similar phase for $J'>0$ called ``NAF (N\'{e}el AF)-dominant'' characterized by antiferroic $\braket{\boldsymbol{q}_{i}\cdot\boldsymbol{q}_{j}} < 0$ and $\braket{p_{i}}=0$.
Similarly, one may expect some sort of 120$^\circ$-long range order
of $\braket{q_{i}}$ compatible with the triangular lattice geometry,
which may be a combined 120$^\circ$-order of $\boldsymbol{S}_{i_{\gamma}}$ in each layer,
while keeping the dimer nonmagnetic.
However, in FQ-BEC-2 both $\boldsymbol{q}_{i}$ and $\boldsymbol{p}_{i}$ show large correlation (Fig.~\ref{fig:tos}(d),
which is not a simple 120$^\circ$-$\braket{q_{i}}$-ordering.
These results are in agreement with the QDJS, that encloses both the 120$^\circ$-Ne\'el and the $p$-nematic properties.
One possibility is that the in-plane 120$^\circ$-N\'{e}el ordering of small $\boldsymbol{S}_{i_{\gamma}}$ exists
but is not fully face to face between layers.
Whereas, considering the fact that 120$^\circ$-N\'{e}el ordering is relatively subtle due to small moment
even a standard spin-1/2 triangular lattice Heisenberg model~\cite{Mezzacapo2010_NewJPhys_12_103039},
such kind of ordering may not be stable as a long range order in our system
where the activated local magnetic moment is expected to be small, and may even be masked by the
intra-dimer quantum fluctuation, in which case the pure $p$-nematic ordering may be stabilized.
\par
Besides these two FQ-phases we find a conventional spin nematic (SN) phase at $B/J \gtrsim 1/3$
in a small region of $J'/J$ and $J''/J$, which can only be
detected by the Anderson tower analysis.
Figure~\ref{fig:smallphased}(a) shows the QDJS consisting of $S=0,2,4,\cdots$,
which indicates the SU(2)-symmetry-broken SN on a triangular lattice~\cite{Penc2011_IFM_Chap13_SpinNematics}.
The phase boundary between the FQ-BEC phase and SN is thus detected as the crossing of the
$S=1$ and $S=2$ lowest excited states (see Appendix~\ref{sec:appendix_smalljj}, Fig.~\ref{fig:app_sn}(a)).
The structure factors (Fig.~\ref{fig:app_sn}(b)) indeed indicate that
inside the SN phase $\braket{\mathcal{Q}(\boldsymbol{k})}$ at $\Gamma$-point overwhelms
$\mathcal{N}(\boldsymbol{k})$ at the $\Gamma$-point,
and vise versa for the FQ-BEC-1 phase at $J'/J\lesssim -0.015$.
The boson density in the SN phase is $\braket{n_{t}} \sim 1$ (Fig.~\ref{fig:app_sn}(d)),
consistent with the SN phase of a spin-1 BLBQ model.
When $B/J$ is large the lowest energy state of each dimer is a triplet, so that the bosons are fully occupied
and the weak inter-dimer coupling works to exchange these triplets and form a SN.
    \begin{figure*
      \centering
       \includegraphics[width=172mm]{tos.pdf}
      \caption{Low-energy excited states as a function of $S(S+1)$
                for (a) FQ-BEC-1, (b) FQ-BEC-2, (c) AFM-solid phases for $N = 12$ cluster.
                Filled circles and triangles are the $\Gamma$ and $K$ points of the Brillouin zone
                and open circles are from the other $\boldsymbol{k}$-points.
                Those following the solid line are tower of states indicating the long range order.
                (d) Structure factors at $\Gamma$, $\mathrm{K}$-points,
                including $\mathcal{N}(\boldsymbol{k})$ and $\mathcal{C}(\boldsymbol{k})$ which gives the information on the
                details of the magnetic properties of the dimerized spin-$\boldsymbol{S}_{i_{\gamma}}$.
                 $J'/J$ is varied from FQ-BEC-1 ($J'/J<0.15$),
                 FQ-BEC-2 ($0.15<J'/J< 0.25$) to AFM-BEC phases.
            }
      \label{fig:tos}
    \end{figure*}
    \begin{figure}
      \centering
      \includegraphics[width=86mm]{smallphased.pdf}
      \caption{ (a) Low-energy excited states as a function of $S(S+1)$ at $J'/J=0.01$ and $J''/J=-0.01$, $B/J = 0.4$.
                (b) Phase diagram at small $J'/J$ and $J''/J$ and $B/J = 0.4$ determined by the Anderson tower analysis,
                which is the magnification of the region indicated by red square
                at the center part of Fig.~\ref{fig:phasediagram_2ndorder}(c).
                SN phase is found for $J'\sim -J'$, where the lowest excited state is $S=2$.
                The boundaries are given both from the crossing of $S=1,2$ levels (filled circle)
                and from the structural factor (open circle).
		            Physical quantities along the fixed $J''/J=0.01$ line in the phase diagram
                is given in Fig.~\ref{fig:app_sn}.
            }
      \label{fig:smallphased}
    \end{figure}
\section{Discussion}
\label{sec:discussions}
\subsection{Origin of quadurupolar moments on a dimer-bond}
\label{subsec:origin_quadrupolar}
  In the FQ-BEC-1 and FQ-BEC-2 phases of our $\mathcal{S}=1$ effective model (Eq.~(\ref{eq:boson_hamiltonian})),
  the pair-creation and annihilation term, $\mathcal{H}_{P}$ seems to play a major role as
  indicated by the large energy gain $e_{P}$.
  To clarify the role of $P$, we choose the parameter $J'=-J''$ to exclude the contribution from $\mathcal{J}$.
  Then, Eq.~(\ref{eq:boson_hamiltonian}) at the first order level in $J'$ and $J''$ is reduced to
        \begin{align}
            \mathcal{H}_{\text{quad}}
        & = - \mu \sum_{i = 1}^{N} n_{i}
                + \sum_{\braket{i, j},\alpha}
                        \left[
                            \left(
                                t b_{i, \alpha}^{\dag} b_{j, \alpha}
                            + P b_{i, \alpha}^{\dag} b_{j, \alpha}^{\dag}
                            \right) + \text{h.c.}
                        \right],
            \label{eq:boson_quadratic}
        \end{align}
   which is transformed via
  $b_{\boldsymbol{k}, \alpha}^{\dag} = \dfrac{1}{\sqrt{N}} \sum_{i = 1}^{N} b_{i, \alpha}^{\dag} \mathrm{e}^{-\mathrm{i}\boldsymbol{k}\cdot\boldsymbol{r}_{i}}$ to
        \begin{align}
            \label{eq:boson_quadratic_FT}
            \mathcal{H}_{\text{quad}}
        & = \dfrac{1}{2} \sum_{\boldsymbol{k}, \alpha}
                \left[
                    \left( t \eta_{\boldsymbol{k}} - \mu \right)
                    \left(
                        b_{\boldsymbol{k}, \alpha}^{\dag} b_{\boldsymbol{k}, \alpha}
                    + b_{\boldsymbol{k}, \alpha} b_{\boldsymbol{k}, \alpha}^{\dag}
                    \right)
                \right. \notag \\
        &\qquad
                \left.
                + P \eta_{\boldsymbol{k}}
                    \left(
                        b_{\boldsymbol{k}, \alpha}^{\dag} b_{-\boldsymbol{k}, \alpha}^{\dag}
                    + b_{-\boldsymbol{k}, \alpha} b_{\boldsymbol{k}, \alpha}
                    \right)
                \right]
            + \text{const.},
        \end{align}
  where $\eta_{\boldsymbol{k}}
        = 2 \big( \cos{k_{x}}
                + \cos{( \dfrac{k_{x} + \sqrt{3} k_{y}}{2})}
                + \cos{( \dfrac{k_{x} - \sqrt{3} k_{y}}{2} )}
                \big)$.
  Then, using the Bogoliubov transformation
        \begin{align}
            \begin{pmatrix} \beta_{\boldsymbol{k}} \\ \beta_{-\boldsymbol{k}}^{\dag} \\ \end{pmatrix}
        = \begin{pmatrix}
                \cosh{\theta} & \sinh{\theta} \\
                \sinh{\theta} & \cosh{\theta} \\
            \end{pmatrix}
            \begin{pmatrix} b_{\boldsymbol{k}} \\ b_{-\boldsymbol{k}}^{\dag} \\ \end{pmatrix}
        \end{align}
  with $\tanh{2\theta} = P\eta_{\boldsymbol{k}}/(t\eta_{\boldsymbol{k}} - \mu)$,
  the Hamiltonian can be diagonalized as
        \begin{align}
            \mathcal{H}_{\text{quad}}
        = \sum_{\boldsymbol{k}, \alpha}
                \varepsilon_{\boldsymbol{k}}
                \left(
                    \beta_{\boldsymbol{k}, \alpha}^{\dag} \beta_{\boldsymbol{k}, \alpha}
                + \beta_{\boldsymbol{k}, \alpha} \beta_{\boldsymbol{k}, \alpha}^{\dag}
                \right)
            + \text{const.},
        \end{align}
  where the particle-hole symmetric energy bands are obtained as
        \begin{align}
            \varepsilon_{\boldsymbol{k}}
        = \pm \dfrac{1}{2}
                \sqrt{\left(t\eta_{\boldsymbol{k}} - \mu\right)^{2}-\left(P\eta_{\boldsymbol{k}}\right)^{2}}.
        \end{align}
  Figure~\ref{fig:boson_energyband}(a) and \ref{fig:boson_energyband}(b) show
  $\varepsilon_{\boldsymbol{k}}/J$ for $J'-J'' < 0$ and $J'-J'' > 0$, respectively, at $B/J = 0.2$.
  When the bottom of the band touches the zero level,
  the instability takes place and the $\beta_{\boldsymbol{k}, \alpha}$-bosons of that wave number condense and form a BEC phase.
  This happens by increasing $J'=-J''$ only up to $\left| J'-J'' \right| \sim 0.05$,
  which is consistent with the numerical analysis that the singlet product state immediately gives way to the FQ phases in the $J'=-J''$ direction.
  The wavenumber at which the $\varepsilon_{\boldsymbol{k}}$ takes the minimum is the $\Gamma$-point when $J'-J'' < 0$,
  whereas it is the $\mathrm{K}$-point for $J'-J''>0$.
  The former is the usual uniform FQ ordering,
  and the latter explains well the particular three-sublattice-like structure of
  the quadrupole correlations in FQ-BEC-2 phase we saw in Fig.~\ref{fig:correlation_circle}(b).
    \begin{figure}
      \centering
      \includegraphics[width=86mm]{boson_energyband.pdf}
      \caption{
                Energy bands of the eigenstates of the quadratic Hamiltonian (Eq.~(\ref{eq:boson_quadratic}))
                for (a) $J'-J'' < 0$ and (b) $J'-J'' > 0$.
                We set $B/J = 0.2$,
                and use the parameters $t$, $P$ and $\mu$ defined in Eq.~(\ref{eq:boson_coefficients}),
                at the first order.
            }
      \label{fig:boson_energyband}
    \end{figure}
\subsection{Classification of ``nematic'' phases}
  Conventionally, a typical spin nematic (SN) phase in spin-1 system
  is identified by the absense of net local-sublattice magnetic ordering and remaining quadrupolar ordering.
  Typically in spin-1 BLBQ models written in the form,
         $ J\boldsymbol{S}_{i} \cdot \boldsymbol{S}_{j}
        + B \left( \boldsymbol{S}_{i} \cdot \boldsymbol{S}_{j} \right)^{2}
        =  (J-B/2) \boldsymbol{S}_{i} \cdot \boldsymbol{S}_{j}
        + (B/2) \boldsymbol{Q}_{i} \cdot \boldsymbol{Q}_{j}
        + \text{const.}$,
    the dipolar (magnetic) and quadrupolar orders compete with each other,
    and the latter appears when $|B| \gtrsim |J|$ where the magnetic ordering is suppressed.
    The QDJS is formed by $S=0,2,4\cdots$, indicating that the SU(2) symmetry is
    broken by the binding of two-magnon pairs.
\par
  Another series of spin nematics is a magnon-pair condensation in the spin-1/2 models~\cite{Zhitomirsky2010_EurophysLett_92_37001}.
  Starting from the fully poralized ferromagnetic phase such as those in a high magnetic field,
  two-magnon instability overwhelms the one-magnon one when there is a frustration effect that prohibits
  the kinetic motion of a single magnon.
  The bound pair of magnons propagate together, which is is by definition a quadrupolar order parameter itself written as
  $\braket{b_{i\downarrow}^\dag b_{j\downarrow}^\dag}=\braket{S_{i}^{-}S_{j}^{-}} = \mathcal{Q}\mathrm{e}^{\mathrm{i} 2\theta}\neq 0$.
  This bond-nematics is realized when the bond is almost fully occupied by bosons;
  the adjacent bosons carrying ${\mathcal S}^z=1$ and -1 exchange or flucuate in pairs and form a quadrupole,
  and accordingly the ${\mathcal S}^z=-1$ propage in space.
  It is equivalent to the spin-2 BEC in cold atoms formed by $S=0$ and 2 sectors~\cite{Ueda2002_PhysRevA_65_063602}.
  For a particular model that excludes the $S=1$ sector out of the low energy subspace,
  this kind of spin nematics can be found at zero field,
  which is characterized by the QDJS of bound-bosons similar to the spin-1 BLBQ case
  \cite{Shannon2006_PhysRevLett_96_027213,Momoi2012_PhysRevLett_108_057206}.
\par
  These two types of ordering are categolized as $n$-type spin nematics, and are only possible when
  the $\mathcal{S}=1$ bosons can be bounded by a strong quantum flucuation,
  mostly in a strong magnetic field that helps to increase the population of $\mathcal{S}=1$ bosons.
\par
  The other known class of spin nematic phase is the $p$-type nematic found in spin-1/2 systems~\cite{Lauchli2005_PhysRevLett_95_137206}.
  The vector chirarity of two spins $\boldsymbol{p}_{i}=\boldsymbol{S}_{i_{1}}\times\boldsymbol{S}_{i_{2}}$ defined on a bond ($\boldsymbol{p}_{i}$ in Eq.~(\ref{eq:p})) condense and form a nematic order,
  again suppressing the sublattice magnetic ordering.
  This time the SU(2) symmetry is broken down to U(1) of a uniaxial rotator,
  and the QDJS consisting of $S=0,1,2,\cdots$ indicates the BEC from a one-magnon instability.
\par
  In a 1D and 2D spin dimer systems, the quadrupolar moment is defined on a dimer-bond, and depending on the choice of
  the parameters, one can control the number of bosons from zero to one.
  When $\braket{n_{t}} =1$ the same situation as the first spin-1 type of SN is realized, whereas
  for $\braket{n_{t}} <1$, a BEC type of quadrupolar ordering is realized which is FQ-BEC-1 or F-nematic.
  In our FQ phases, a \textit{single} magnon condenses by the hopping and pair creation and annihilations,
  breaking the SU(2) symmetry down to U(1),
  which is an analogue of the spinor BEC in cold atoms~\cite{Ho1998_PhysRevLett_81_742},
  and shares the same property as the $p$-type nematics.
  Since the dimer remains always nonmagnetic,
  it can be regarded as some sort of ``nematic'' order in terms of $\mathcal{S}=1$ boson.
  When separately looking at upper/lower-layer carrying small moments, they form a ferromagnetic
  sublattice long range order,
  whereas the intra-dimer quantum fluctuation kills the moment inside the dimer,
  which should be regarded as a different phase from the mean-field type of
  inter-layer AF and intra-layer ferromagnetic ordering
  of the full spin moments observed in the ferromagnetic dimer model~\cite{Hikihara2019_PhysRevB_100_214414}.
\par
  We saw in \S.\ref{sec:lro} that the FQ-BEC-2 phase has a strong chirality correlation
  and the possible 120$^\circ$ in-plane magnetic long range correlation of small moments, which are masked
  and the dimer unit remains nonmagnetic.
  There are two possibilities;
  In a mean-field analysis these two are incompatible, whereas our treatment treating
  the full quantum many body effect may allow for a new possibility that the two types of order may coexist.
  The other possibility is the absense of sublattice magnetic ordering that may stabilize the $p$-type spin nematics.
  The Anderson tower still does not allow for the separation of the behavior of the biaxial or uniaxial rotator,
  namely the full breaking of SU(2) or the breaking only down to U(1)
  within the present study.
\subsection{Exchange of \texorpdfstring{$\mathcal{S}=1$}{S=1} moments}
\label{subsec:exchange_spin1}
  Previously, in a spin-1/2 dimer system~\cite{Yokoyama2018_PhysRevB_97_180404,Tanaka2018_JPhysSocJpn_87_023702},
  we showed that the origin of the spin nematic phase is the inter-dimer
  ring exchange interactions that permutate the four spin-1/2 along the twisted path as,
  $(1,2,3,4) \rightarrow (2,3,4,1)$, which is shown in Fig.~\ref{fig:origin}(a).
  In that case, the two spin-1/2's on a dimer form an $\mathcal{S}_{i}=1$ triplet, and the ring exchange interaction
  exchanges the spin-1's on neighboring dimers, $(\mathcal{S}_i^{z}, \mathcal{S}_{j}^{z})=(+1,-1)$ with $(-1,+1)$-states (see Fig.~\ref{fig:origin}(b)), and suppress the dipolar ordering.
  This plays the same role as the biquadratic interaction, $\mathcal{B}(\boldsymbol{\mathcal{S}}_{i}\cdot\boldsymbol{\mathcal{S}}_{j})^{2}$,
  and when all the dimers are filled with a triplet, $\mathcal{S}_i=1$, the system is reduced to the BLBQ model.
\par
  In the ferromagnetically coupled spin-1/2 dimer model~\cite{Hikihara2019_PhysRevB_100_214414},
  it is shown that the exchange interaction, $J'$ and $J''$ ($J_{\parallel}$ and $J_{\times}$ in their notation),
  operated twice at the second order perturbation is important to stabilize the spin nematic phase.
  As shown schematically in Fig.~\ref{fig:origin}(a), this works in the same manner as the ring exchange interaction,
  and generates an effective biquadratic term~\cite{Hikihara2010_PhysRevB_81_064432}.
  However, this time they need a larger $J''$ as their spin nematics need to compete with the stable ferromagnetic phase.
\par
  In our spin-1 dimer, the pair-creation and annihilation term results in an off-diagonal pair condensation of up and down spin-1's
  via the processes shown in Fig.~\ref{fig:origin}(c).
  These processes, when performed twice, will give the same effect as the
  biquadratic interaction in Fig.~\ref{fig:origin}(b).
  The advantage here is that it is a first order process and can be more easily realized than $\mathcal{B}$ or the ring exchange processes.
\par
  The second order perturbation process in the spin-1/2 dimer systems giving the effective biquadratic interactions can also be understood as the pair-creation and annihilation process,
  while not discussed in the spin-1 bosonic language there~\cite{Hikihara2010_PhysRevB_81_064432,Hikihara2019_PhysRevB_100_214414}.
  Nevertheless, the pair fluctuation effect in the spin-1 dimer systems is stronger than that of spin-1/2 dimer systems;
  One can see from Eq.~(\ref{eq:triplets}) that the triplet states $\mathcal{S}=1$
  consists of twice as large dimer-spin terms
  than the triplets formed by two spin-1/2's, which means that the entanglent between the two $\mathcal{S}=1$ can be
  easily enhanced.
  We consider this to be the origin of a variety of quadrupolar phases in our system.
        \begin{figure}
            \centering
            \includegraphics[width=86mm]{origin.pdf}
            \caption{
                Three different types of fluctuations that contribute to the formation of the spin nematics.
                Single and double arrows represent the spin-1/2 and spin-1, respectively.
                (a) In spin-1/2 dimer system, ring exchange interaction that permutates spin-1/2's as
                $(1,2,3,4) \leftrightarrow (2,3,4,1)$ in the upper panel
                (see Refs.~\onlinecite{Yokoyama2018_PhysRevB_97_180404,Tanaka2018_JPhysSocJpn_87_023702})
                and the second order perturbation terms operated twice, $(J''\boldsymbol{s}_{i}\cdot \boldsymbol{s}_{j})^2$, with $\boldsymbol{s}_{i}$ the spin-1/2 operator
                discussed in Refs.~\onlinecite{Hikihara2010_PhysRevB_81_064432,Hikihara2019_PhysRevB_100_214414},
                work in the similar manner.
                (b) Fluctuation between on-bond spin-1's that are equivalent to those of panel (a).
                (c) In our spin-1 dimer system, the pair creation and anihilation term ($P$) plays a major role
                which originates from the first order in $J'$ and $J''$.
            }
            \label{fig:origin}
        \end{figure}
\section{Summary and perspective}
\label{sec:summary}
  In conclusion, we found various types of quadrupolar ordering in a spin-1 dimer model forming a triangular lattice bilayer.
  In the decoupled dimer limit, the dimer-bond host singlet at small $B/J$ and triplet at larger $B/J$.
  Including the inter-dimer Heisenberg exchange terms $J'$ and $J''$, perturbatively up to second order,
  we derived an effective hard-core bosonic model describing the triplet on a dimer-bond,
  which reproduces the low-energy properties of the original model.
  The major part of the bosonic Hamiltonian consists of the hopping ($t$) and pair-creation and annihilation ($P$) of bosons,
  as well as the chemical potential ($\mu$) and the antiferromagnetic exchange interaction ($\mathcal{J}$).
  The bosons are doped by $\mu$ and the $t$-term contributes to the formation of BEC.
  The ferromagnetic (FM) and antiferromagnetic (AFM) phases appear due to $\mathcal{J}<0$ and $\mathcal{J}>0$.
  When $t$ and $P$ are dominant at $J' \sim -J''$
  the FQ-BEC-1 and FQ-BEC-2 phases are observed, which are the condensation of quadrupolar moments on a dimer-bond,
  with $\sim 0.55$ bosons per dimer, similarly to the anisotropic superfluidity in cold atoms\cite{Ho1998_PhysRevLett_81_742}.
  Also when $B/J\gtrsim 1/3$, the typical spin-1 nematic phase is found near the inter-dimer decoupling ($J', J''\sim 0$) region,
  since each dimer is occupied by a spin-1 boson which exchange by $P$.
\par
  Our results are widely applied to the bilayer quantum spin dimer systems,
  since the interactions appear in Eq.~(\ref{eq:spin_hamiltonian}), are all standard ones that are derived
  naturally from the strong coupling perturbation theory of Mott insulator;
  the Heisenberg exchange interactions, $J$, $J'$ and $J''$,
  and the biquadratic intra-dimer interaction $B$.
  The value of $B$ is reported to be relatively larger than it has been believed before~\cite{Tanaka2018_JPhysSocJpn_87_023702}.
\par
  The FQ-BEC-2 phases discussed here might be exotic in the sense that the distribution of $S=1$-bosons are uniform in space,
  whereas the quadrupolar correlation may develop a particular spatial modulation.
  It differs from the SN of the $S=1$ BLBQ model and from the multi-magnon-bound states of the $S=1/2$ models in high fields.
  We consider that the magnetic long range ordering is possibly absent, and the $p$-nematic type of correlation develops toward the long range ordering.
\par
  We finally discuss the relevance with the actual material.
  In a family of Ba$_3M$Ru$_{2}$O$_{9}$ ($M=$ Ca, Sr, Co, Ni, Cu, Zn)~\cite{Darriet1976_JSolidStateChem_19_213,
  	Lightfoot1990_JSolidStateChem_89_174,Rijssenbeek1999_JSolidStateChem_146_65,
  	Beran2015_SolidStateSci_50_58,Terasaki2017_JPhysSocJpn_86_033702,Yamamoto2018_JPhysCondensMatter_30_355801}
  the two face-shared RuO$_{6}$ octahedra form a dimer which is stacked along the two-dimensional triangular lattice
  in the same way as our model Fig.~\ref{fig:lattice}(a).
  The Ru$^{5+}$ carries possibly $S=1$~\cite{Terasaki_privatecommun}
  and the material is well described by our Hamiltonian, Eq.~(\ref{eq:spin_hamiltonian}).
  Intriguing magnetic properties were reported;
  for a Zn-compound, the uniform susceptibility is strongly suppressed down to 37~mK,
  a much lower temperature than the value of $J\sim 150-200$~K~\cite{Terasaki2017_JPhysSocJpn_86_033702,Yamamoto2018_JPhysCondensMatter_30_355801}.
  In a Co-, Ni- or Cu-compound with shorter inter-dimer distances, namely having larger $|J'/J|$ and $|J''/J|$,
  the system undergoes an antiferromagnetic transition at $T_{\mathrm{N}}\sim 100$~K~\cite{Lightfoot1990_JSolidStateChem_89_174,Rijssenbeek1999_JSolidStateChem_146_65}.
  The Ca- and Sr-compound with the longer inter-dimer distances contrarily favor a standard nonmagnetic singlet state~\cite{Darriet1976_JSolidStateChem_19_213}.
  Usually, $J$, $J'$ and $J''$ are antiferromagnetic ones, and if we increase the inter-dimer interactions
  from the center of the phase diagram in Eq.~(\ref{fig:phasediagram_2ndorder}) toward the upper right direction,
  the ground state transforms from singlet, FQ-BEC-2, and to an antiferromagnetic phase, in good agreement with the
  experimental observation, (Sr, Ca)$\rightarrow$(Zn)$\rightarrow$(Co, Ni, Cu).
  Since the conventional SN phases were all found next to the fully polarized ferromagnetic/antiferromagnetic phase,
  the exotic nonmagnetic phase in the Zn-compound was not really connected to the quadrupolar ordering.
  Our series of studies on spin dimer systems~\cite{Yokoyama2018_PhysRevB_97_180404}
  point out that SN and other quadrupolar phases of spin-1 can be found next to the spin-0 singlet phase.
  Since this kind of symmetry breaking is not directly detected from the standard susceptibility measurements,
  the way to identify them experimentally should be discussed in the next step for the
  clarification of the unexplored nature of the spin dimer materials.
\begin{acknowledgments}
  We thank Keisuke Totsuka, Tsutomu Momoi and Toshiya Hikihara for helpful discussions,
  Yuto Yokoyama for advice in the early stage of this work,
  Masataka Kawano for helpful comments, and Ichiro Terasaki for fruitful communications.
  This work is supported by JSPS KAKENHI Grants No. JP17K05533, No. JP18H01173, No. JP17K05497,
  No. JP17H02916.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
CII whitepaper: No place for another regulator to control misleading advertisements
Suggests co-regulation between ASCI and government
In the wake of concerns raised by Department of Consumer Affairs (DCA) of the Ministry of Consumer Affairs, Food and Public Distribution, about a need for stronger government intervention in case of misleading advertisements, CII National Committee on Marketing has released a whitepaper on "Self regulation in advertising in India – a critical evaluation." The paper takes an independent view of the concerns regarding misleading advertisements and critically evaluates the role and responsibilities of all stake holders – regulators, industry, activists and consumers.
To construct an independent view, CII has involved KPMG in India, Amarchand Mangaldas and AZB & Partners as knowledge partners. The concluded analysis has further been substantiated through discussions with various stakeholders – advertisers like HUL, ITC and Godrej Consumer Products Ltd., etc., media companies such as Star India and The Hindu and members of the civil society.
According to the whitepaper, the solution to the problems posed by misleading advertisements is not to add another legislation, in form of an Administrative Authority as proposed by the DCA, to the existing basket of laws. It advocates co-regulation between Advertising Standards Council of India (ASCI) and other regulators such as DCA and Ministry of Information and Broadcasting (MIB) to enforce compliance currently vested with ASCI but without any punitive powers. The paper recommends only in cases of non-compliance of the Consumer Complaints Council's (CCC) decisions should the matter be referred to the parent regulatory body for further actions.
The whitepaper states that ASCI is committed to the cause of self regulation in advertising and protection of interests of consumers through initiatives such as National Advertising Monitoring Service (NAMS) for monitoring almost all TV and print ads to ferret out misleading ads; and bi-monthly CCC meetings versus monthly ones for speedier resolutions of consumer complaints, the processing time for which is around 4-6 weeks. "We are also evaluating a procedure called 'Suspension pending investigation' for highly objectionable advertisements wherein a commercial will be withdrawn till CCC takes a final decision," explained Sam Balsara, past chairman, ASCI & chairman and managing director, Madison World.
Besides co-regulation and suspension, the paper also suggests some areas of improvements for ASCI including: Mandatory membership of ASCI for all industry players; integrating ASCI code into statutory provisions; and expanding coverage of ASCI code to digital and social media.
According to Balsara, one of the key challenges facing ASCI is the lack of awareness amongst stakeholder specially the consumers. The whitepaper, thus, recommends building awareness about ASCI's role and code amongst the stakeholders through actively leveraging various media vehicles.
The paper further stresses on an incessant drive to improve the complaints handling system with an emphasis on continuous review and improvements to the system. This will revitalize ASCI as a more efficient and transparent self-regulatory organization.
azb & partners
sam balsara
Advertisements in newspapers garner most trust: ...
ASCI issues Covid-19 advertising advisory
ASCI investigates liquor brand extension advertising...
Subhash Kamath
Blog: ASCI's Subhash Kamath responds to open letter ...


text:
from __future__ import unicode_literals

from django.conf import settings
from django.db import migrations, models
import django.db.models.deletion

class Migration(migrations.Migration):

    initial = True

    dependencies = [migrations.swappable_dependency(settings.AUTH_USER_MODEL)]

    operations = [
        migrations.CreateModel(
            name="Profile",
            fields=[
                (
                    "id",
                    models.AutoField(
                        auto_created=True,
                        primary_key=True,
                        serialize=False,
                        verbose_name="ID",
                    ),
                ),
                ("name", models.TextField(blank=True, null=True)),
                ("image", models.CharField(max_length=100, null=True)),
                ("image_small", models.CharField(max_length=100, null=True)),
                ("image_medium", models.CharField(max_length=100, null=True)),
                (
                    "user",
                    models.OneToOneField(
                        on_delete=django.db.models.deletion.CASCADE,
                        to=settings.AUTH_USER_MODEL,
                    ),
                ),
            ],
        )
    ]

redpajama_set_name: RedPajamaGithub