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TV ShowsStarTrek: The Original Series
Reliquary: The Letters (Magnificent Obsession Series: 6)
By: KayCee1951
Having formed a bond that transcended distance, culture, and logic over the remainder of a lifetime, his mother had another full, rich life, outside the sphere of being the wife of a Vulcan, outside of being his mother. See my profile for story order
Rated: Fiction K - English - Sci-Fi - Spock, Amanda G. - Words: 4,918 - Reviews: 1 - Favs: 3 - Follows: 1 - Published: 11/2/2019 - Status: Complete - id: 13423855
Reliquary: The Letters
by KayCee1951
Once, he had a most startling dream. She was trying to tell him something but he could not hear her…until there came a time when he began listening….
From a place and time far away, he felt a tug on his sleeve.
At the Federation Council Complex, the morning had been a constant stream of reviews and consultations in preparation for upcoming treaty negotiations between two minor planets vying for a place within the safety of the Federation. These days, the tensions between the Klingons and the Romulans made strange bedfellows. Systems that had remained staunchly against any association with what was considered Federation arrogance and interference were nowadays lining up to hitch their rigs to the Federation wagon train. The small systems in the sectors sandwiched between both Imperial domains occupied much of the time and energies of the council.
It was amidst the bureaucratic entanglements of the embassy office to which Spock was assigned that Admiral Leonard McCoy inserted himself. When Spock was given a message from the doctor to meet him for a drink, he immediately began to steel himself and briefly entertained the thought of begging off. He had been looking forward to a visit with his oldest living friend but had not intended that it be mid point in a particularly long and tedious day. The situation between them being what it was, he decided not to shun the invitation.
McCoy was waiting for him at a table well to the back of the bar. For a Human of 118, the doctor was in remarkably good condition. His habit of sinking deeper into that Georgia accent with each passing year was what made him seem ancient. The doctor excelled at being curmudgeonly, an admiral's rank affording him the excuse to be more so, and he visited it upon the Vulcan at every opportunity and with particular relish for the past twenty years.
Giving full attention to the amber colored drink on the table, McCoy looked up from the still-full glass only after Spock settled himself in the chair on the other side of the table.
"Spock. How's the red tape these days?"
"The status remains unchanged." Spock studied McCoy's face and understood the meeting was unlikely to be just a 'Hi! How are ya?' call.
"I'm sorry to call you away from your diplomatic duties Spock…but when I found out you planned on making a night of it at the office…and I have to leave on this damnable inspection tour in the morning…"
"No need to explain, Doctor." The waiter passed by, and noticing the new arrival at the table, asked for his drink order. "Altair water, please."
When the waiter left, Spock entwined the fingers of both hands and placed them ceremoniously on the table, waiting for McCoy.
The doctor peered at him without moving his head from its bent position over the drink glass. He took a sip and said, "I, um, have something that belongs to you."
With great deliberation, McCoy pulled a box from under his chair. Its surface was coated plainly with ruddy enamel and trimmed in brass. He placed it on the table, and pushed it gingerly toward his old friend as if not wanting to let it go.
Atop the box and bound to it with a shank of azure ribbon was a small card that read, "For Spock."
Tuesday, February 27, 2345...
Nearly five days passed before Spock took the box back to his apartment. It had set on the credenza behind his desk long enough to become an object of curiosity to anyone who frequented the office. The definitely Earthly, deco art thing stood out among the spartan Vulcan relics.
Alone, in the darkened apartment, as an ancient melody he had selected softly filled the air around him, Spock watched through the window as the rain nearly obscured the view of San Francisco. He could see only the lights from other windows that seemed to flicker between the drops.
Taking the pin from the antique latch he lifted the lid, releasing a distinct odor of age from inside the trinket box where he found a variety of envelopes neatly stacked.
While the music found its way inside him, he turned over the envelope on top to find a broken seal of wax which had been embossed with a familiar symbol. Unfolding the pages within, he heard his mother's voice in the delicately hand scribed words:
{April 9, 2272
Dear Lauren,
I apologize for the time between our meeting and this letter. The life of an ambassador's wife is not always a tranquil one, especially when we are on Earth. It is a wonder I managed the trip to New Orleans with my cousin Sara at all. However, that is not an excuse.
That I have been remiss in thanking you for your hospitality during our visit is unforgivable. But now, the arrival of the rose bushes makes me even more remorseful that I have not responded to your kindness.
I have begun this letter many times. With each attempt I struggled with what to say and how to say it. I have finally decided just to say what is in my heart and hope it is enough…
When Sara and I planned our outing to your beautiful city, I had no intention of intruding upon your life. I 'am' writing a journal on derivations of the French language and thought it both logical and prudent when Sara suggested we make a trip to New Orleans to study the origins of Creole French. Oh, I know I could do that with the voluminous research compiled over several centuries. But nothing substantive has been added in the last hundred years or so; and I wanted to get a sense of how the language is viewed in the 23rd century…and Sara, the force of nature she is, had made arrangements before I could form second thoughts.
The plan was to spend three days talking to residents in the area whose family lineage dated back to the time when Creole French would have still been in use. On very short notice, a linguistics professor from Tulane assisted us in choosing the families we would visit and interview.
So we set off for New Orleans, with my husband's blessing. At the time, he and his staff were embroiled in diplomatic negotiations and not inclined to protest when Sara came to steal me away.
When I saw Ducheaux House on the list, I did not know who resided there until the day of our visit. I suppose I should have known.
I am grateful for your kindness and understanding. Your daughter is obviously a reflection of her mother's grace and dignity.
I am so sorry that I was the one who enlightened you about her emotional ties to my son. It was unintentional, for I assumed you knew. Poor Sara, she was taken quite by surprise, a thing not easily accomplished. But then, I should have remembered how carefully guarded Christine is on the subject. I fear the awareness of others causes her distress but she cannot shield it from me – I know both the joys and the burdens. When I saw her a few months ago at an official function, I knew her feelings for Spock had not diminished. If I could offer her solace, I would; I don't think anyone can.
We do not know what prompted Spock to decide the mastery of Kolinahr was the answer for him. I begged him not to go. His father was against it. I think Christine feels some responsibility. I don't pretend to know all that transpired between them during the time they served together on the Enterprise. Most of the mission is classified and the crew, being a very tight-knit family, would not share with outsiders. So, be patient with her.
It has been more than a year since he left; and I fear that when I see my son again, he will be drastically, and forever, changed. I believe Christine fears this as well. But she is true to what I saw in her that first time we met - strong, resilient, capable, and determined. She will survive, my dear; I think she will be the stronger for it.
Sarek and I return to Vulcan within the week. I always enjoy my time on Earth but miss my home and will be happy to return to my garden. The rose bushes you sent will be a beautiful addition to it. I will be happy to report to you on their progress. Although I am quite illogically proud of my own garden amid the arid landscape of Vulcan, it is dwarfed by the beauty adorning the grounds of your home. Along with the fascinating collection of Creole language volumes in your library, the peace and tranquility of your garden walk is something I will remember for a very long time.
I hope to be able to visit again, if the opportunity arises, the next time we travel to Earth.
With that, I thank you again and bid you 'adieu.'}
Little would have astounded the Vulcan in Spock more than the very personal nature of his mother's words to a stranger after an accidental meeting.
The letter was signed, 'With the hope of continued friendship, Amanda Grayson.'
McCoy had followed his presentation of the trinket box with a short tale of how he came to be in possession of it. "The Corp of Engineers has been working on Galadriel III..."
Spock studied the workmanship. The lid was inlaid with pink mother of pearl, faux jade, and abalone in an architectural design distinctive to early 20th century Earth. He had seen a similar box in the display window where he had purchased the rudimentary makings of a mnemonic memory circuit around the corner from the 21st Street Mission, in 1930's New York City.
"I am aware of the activity on Galadriel III."
"I know that, Spock," McCoy grumbled. "What you don't know and what I'm trying to tell you is – while they were working on what was left of the medical compound, they found Christine's foot locker under the rubble of what used to be her quarters."
On the surface, the relationship between Spock and McCoy would have appeared to be the same as always to the casual observer. McCoy's genuine agitation and the edge in his voice was evidence that things had not been right between them for a long time.
When McCoy was satisfied he had shut the Vulcan up, he continued. "She would have known she only had a short time…her affairs were in order, the locker was addressed to me…she didn't have time to take it with her when the relief team bugged out..."
There had been no logical reason for McCoy to elaborate on the seven days that followed the relief team's hasty retreat from Galadriel III…Christine's collapse into a coma and her death a few days later. But elaborate he did, and for the next five minutes.
Hoping to have elicited even a small measure of personal investiture from Spock, McCoy had left the bar sullen, disappointed, and wondering why he had expected anything more.
Gone was the Spock he knew, replaced by a Vulcan he wasn't certain he wanted to know. Believing he had no one to blame but himself, McCoy was not sure what else he could do.
The two following letters were tucked inside the pages of a small clothbound book; its binder was tattered at the spine and corners, the pages barely held together with glue that had not been manufactured on Earth for three centuries.
{April 11, 2272
Dearest Amanda,
You need not have been concerned about your visit, or the delay in writing the letter I received today.
Please do not chide yourself for telling me how Christine feels about your son. I was taken by surprise, as you say Sara was; but I have had time for reflection. I can't say that I understand it but, at least, now I know what Christine is 'not' saying. I suppose she didn't think I would have a proper frame of reference. Looking back, she may have been right. In the future I will know which subjects are the ones composed of eggshells.
Although I am not sure how in so short a time you were able to understand my daughter better than I did, you are right about her. When she went out into the blackness of space looking for Roger Korby, I cursed him. When she stayed on the Enterprise, I cursed the ship. I cursed the ship as I had cursed Roger because I thought they changed her, derailed her from the life she should have had.
I realize now that Christine simply grew up into a strong independent woman while I wasn't there. My husband has been trying to tell me that for a while now; and he was right. If he knew, he might never let me live it down.
The little volume of Creole proverbs accompanying this letter is a gift for you. If you remember, I had wanted to show it to you when you were here. I finally located it sandwiched between two larger books and wedged into the back of the bookcase. I must remember to dust there more often. You will find page 10 marked. When we were discussing the proverbs, I told you most of them don't make sense, but I think you will find this one does. You will know why when you read it. If the proverb is true, and I believe it is, your son must be someone very special, as I have finally realized Roger was special.
It is probably a good thing that parents do not choose their children's mates. I fear we would bollox the job.
I hope, with all my heart, your son returns to you. But you will have to forgive me if I hope that he does not return to my daughter's life.
Thank you for sharing with me. You will never know how much it has helped me become close to my daughter again. For that, I will be forever grateful. I too wish we could meet again.
Your new friend,
Lauren Ducheaux-Chapel
P.S. I would very much like to read the journal when it is complete. If there is anything else I can add, please don't hesitate to let me know. I will be happy to contribute what I can.
The proverb marked in the small book would have made little sense to the Spock of decades past. He had learned much in what was still the first third of his life – lost much and gained more in the losing.
How he did long to hear Amanda's voice again and tell her what he had learned.
{November 18, 2272
My Dear Lauren,
"Di moin qui vous laimein, ma di cous qui vous ye." Tell me whom you love, and I will tell you who you are.
Merci. Vousavezplus de compréhension quevousle savez,cher ami. Maintenantc'est moiquiserai éternellement reconnaissant.
The rose bushes are thriving. I have enclosed a chip with images of my garden, and the view from the veranda at sunset; so very different from Earth, it is still quite serene and beautiful.
You will also find in this package the first chapter of my journal. I have not completed the portion on the Creole language as yet, but will send that as soon as it is finished. I would be grateful for your opinion.
I am truly happy you have been able to come to terms with your daughter's choices. Heaven knows, I am still trying to come to terms with my son's.
I will be traveling with Sarek to a conference in two weeks; then we will be on Earth for a month. It will be Christmastime. I would love to see the foyer of Ducheaux House adorned the way you described it but our schedule is grueling and I will not be able to get away. I thought if you were going to be in San Francisco we might lunch, somewhere away from the official scene.
And yes, you are forgiven. I understand your wish to shield Christine. No matter their age or independent lives, we are still their mothers. I wish it was as simple as wishing. If it were, we would neither of us wonder about what might have been.
Amanda}
Spock remembered the roses – intense color and a light, memorable scent. In answer to his query about the new addition to her garden, his mother had replied, "Experimental – from a lovely garden I once visited on Earth."
The roses bloomed in the garden still, tended by the third wife of Sarek.
The next few days would be crucial in the affairs of the Federation Council. In spite of his ability to go days without sleep, his duties required him to be at peak performance.
However, he would make time for one more letter before retiring. Setting aside the book of Creole proverbs, he opened the next envelope.
The first chapter of the journal is wonderful, very rich in the flavor of the French language. Your comparison to the romance languages of Rigelian and Midosian antiquity is one I had not considered. Bravo! I found myself wanting to learn more about the two languages that are so similar to French in many ways. Even in this, the 23rd century, I am astounded by the similarities in our cultures and those found on other worlds. I begin to understand the allure of galactic travel.
I think Christine misses that and the Enterprise. I don't know if it's the excitement, the camaraderie, or just being on a starship.
Patterson and I went to space dock to watch the ship come in almost two years ago now. The scars on her hull, the blast marks from God only knows what. I don't think I want to know the details. I know Christine received commendations and as much as I'm ashamed to say it, I'm afraid to ask how she earned them.
But she is a doctor now, settled in her residency. We have no idea what her plans are beyond that. We hope she does and that they include staying Earthbound for some time.
My teaching duties are calling me, so I will end this letter with a thank you for the first chapter of your journal. Teaching music and taking care of this wonderful old anachronism of a house doesn't make for a tranquil life either. But I would not trade my life for that of anyone else.
I look forward to lunch, Lauren}
Sunday, March 4, 2345...
Four days passed before Spock was able to justify the appropriate amount of time to sit again in his chair by the window. The rain had stopped mid day on Thursday but the damp cold remained. Spock barely tolerated the cold but welcomed the rain.
The box sat on the table next to the chair, just as he had left it. He read the correspondence in succession, just as they had been arranged in the chest.
Over the years, Amanda wrote of her travels through the galaxy and of Vulcan philosophy. Lauren wrote of her family history and provided insight for Amanda's journal. Always, the letters made reference to a mutual interest in horticulture and on rare occasions, made mention of their children. Mixed here and there through the years, short notes enhanced the colors of the portrait of three women, who, on the surface, seemed to have a fragile commonality, but who met regularly for luncheon or coffee, whenever Amanda was on Earth.
Sara's notes, the fascinating Bohemian that she was, were rife with intrigue, cryptic notes with times and places on Earth – small villages, out of the way garden cafes.
Lauren's note to Amanda upon hearing of Spock's death read simply, 'I grieve with thee;' and, upon his rebirth, "I rejoice with thee.'
There were no notes or letters in the trinket box that corresponded with the date of Sara's death, leaving Spock to wonder about how the two remaining friends honored the one who was gone, something with which Spock was personally familiar.
Having formed a bond that transcended distance, culture, and logic over the remainder of a lifetime, his mother had another full, rich life, outside the sphere of being the wife of a Vulcan, outside of being his mother.
One envelope was left in the red box.
His mind knew exactly how long he'd been sitting there but he resisted the urge to calculate it precisely. Dawn would come soon enough and he did not want to mark the time before the spell would be broken and he would, yielding to logic and reality, return to the world beyond his memories.
Spock had hoped the last letter would have been from Amanda. But Christine, whose wisdom and insight he had come to respect and appreciate had left this one, out of chronological sequence, for him to read last – one letter from her mother to his.
{December 9, 2293
Much has happened since I saw you only two days ago when we shared the sadness of Jim Kirk's death. He was so prominent in both our children's lives – so important to them. The memorial was a fitting tribute to a life devoted to space exploration, to his crew, and to his ship.
I would not have understood it half so well had it not been for Christine's service and my friendship with you.
But I digress. First, I have to tell you that I have finally met Spock.
I can only imagine the shocked expression that must have been on my face when I opened the kitchen door and found your son standing on my back porch, my daughter at his side as if it was the most natural thing in the world. He looked so completely Vulcan out of uniform. I suppose that may have unnerved me more than finding him there.
True to his nature, Patterson took it all in stride and filled the vacuum left by my initial reaction – one of the many reasons I married him.
Spock was formal and all that I expected of a son of Sarek in his manner, in the preciseness and diplomacy of his words.
Just as I attempted to deliver the Vulcan greeting as you taught me, he extended his hand. As I took it, Amanda, I looked into his eyes and saw, not the son of Sarek, but my dearest friend looking back at me.
Lauren}
Thursday, April 27, 2345...
When Spock met with McCoy after his return from the inspection tour nearly two months later, it was he who showed up at the doctor's door.
"Spock..?" McCoy's surprise at this unheralded visit was unmistakably etched in every wrinkle on his face, but he recovered enough to say, "Come in."
"Thank you, Leonard."
McCoy went rigid, reeling from the further shock at the Vulcan's use of his given name for only the third or fourth time in nearly eighty years. "How about a drink, and I don't mean Altair water. I brought back a case of Capellan liqueur…was just about to pour myself a snifter. It has to breathe, like brandy."
"I will have a small glass, thank you."
The Vulcan was full of surprises tonight. The doctor managed a broad grin until he noticed the object cradled under his Vulcan friend's arm. The grin disintegrated into a question mark.
Pouring the thick liquid into two glasses, McCoy's mood changed to the solemnity he read on Spock's face. For all his protestations about not understanding the pointy-eared hobgoblin, he understood Spock's facial expressions as clearly as he understood his own – sometimes better.
"So, Spock, what brings you here in the middle of a workday?" His eyes remained transfixed on the box as Spock pulled it from under his arm and presented it to him.
"I have something that belongs to you."
"Spock, I can't take this…she left it for you."
Without additional comment, Spock pushed the thing at McCoy.
Opening the box, McCoy stared at the object inside, seconds elapsing before he spoke.
"I gave this to her..." He cradled the ancient artifact, a sculptured representation of a caduceus, in his hand, rubbing across its distressed surface with his thumb. "It was the subject of some fierce debates…" His smile was one of fond remembrance.
"Yes…of course you do…this was what was in the box?"
"Yes." Spock resisted making a point of McCoy's predilection for stating the obvious.
"Then she clearly meant it for you." With a measurable reluctance, he offered the artifact to Spock.
His hand raised to reject the offering, Spock softened his tone. "I have removed that which was left for me. What remains was meant for you."
"Why…why didn't she just leave it in the foot locker with my name on it?" he pondered, the question posed to himself as much as to the sober Vulcan standing in front of him.
"I know only that she wished it to be returned to you."
Turning away from his old sparring partner, McCoy walked to the window. Fixing his attention not on the view of the city but on the object he held in his hand, he reflected, "You think I'm angry with you for letting her go."
There was no response from Spock.
McCoy forged on, not wanting to let the opportunity to make amends pass again. "I was never angry with you…I've been angry with them, all of them – Jim for disappearing without a trace and not even a body to cry over, at Pavel for dying on my table…at Christine for being able to leave you to go traipsing off to Galadriel III even though she knew her time was running out…"
Spock was suddenly next to him at the window. Outside, it was raining again, the raindrops on the trans-alum panes making soft thumping sounds.
"A common Human ailment, Spock. Anger at the ones who left…left us alone to carry on. But more than anger at them, I've been mad at myself for not being able to save them." He wrapped his fingers around the caduceus and clutched it tightly in his fist.
"You had no means of saving them. Their lives…and their deaths were out of your control."
"I know. I know. But knowing and believing are two different animals…it's something I've never been able to shake. And, maybe I have been angry at you, all these years. You loved them all in your own Vulcan way…but you've found a way to go on without them."
"It is not the Vulcan way to mourn the dead unless the life was wa…"
"Damnit Spock! Forget the Vulcan way! What about your way? I've always been afraid to ask…because I can't even imagine she would have agreed…" McCoy looked into Spock's eyes, as if trying to locate his soul. "She is in there…somewhere…isn't she? You couldn't get to Jim or Pavel, but you were with her when she…when her body died."
His stare demanded a response but the Vulcan's silence was answer enough and provided the doctor a tenuous solace.
Staying just long enough to finish the drink McCoy had poured for him, Spock announced his intent to leave and turned to egress the apartment.
Before he reached the door, McCoy called out after him, "Thank you Spock, for bringing this to me. It means a lot."
Spock's simple nod in the affirmative was accompanied by an invitation to dinner on Tuesday.
"Sure, Spock, I'd like that."
And not to let the perfect opportunity go by to settle back into a familiar but not recently used repertoire, he added, "But don't think that when I am on my deathbed you can visit me. I've been inside your head, Spock, and it's not the place I intend to spend eternity."
With only the slightest hint of a clandestine smile, Spock replied, "Doctor, that chilling thought never entered my mind."
In another time and place….
Sector 3, Alpha Quadrant
Spock the Elder stood on a rocky hillside overlooking the construction of a new Vulcan home world when he felt a second tug on his sleeve. Looking up at him was the beaming bright smile of a Human child that had set him remembering.
When she tugged on his sleeve a third time, Spock the Elder gently asked, "Angelique, class has been dismissed; should you not be on your way home? Your parents will be concerned."
Angelique flashed him another innocent smile and, with a conspiratorial look, said, "Sir, we want to know more about Miri and the other children. What happened to them after the Enterprise left the planet?"
Looking over at the group of Vulcan children of varying ages he noted their unsuccessful attempts to appear disinterested.
Taking the child's hand, and heading down the hillside with the group of Vulcan boys and girls following closely behind, Spock the Elder told Angelique the rest of the story.
Once, he had a most disturbing dream. They were trying to tell him something but he could not hear them…until there came a time when he began listening.
He was listening still.
Third Place Winner – STARFLEET Writing Contest 2011


This indicator was originally developed by Vitali Apirine (Stocks & Commodities , V.34:9 (August, 2016): "The Middle-High-Low Moving Average"). You can use it to create a crossover trading system in conjuction with EMA/SMA. Good luck!


A new D2 newsletter hit the stands today. If you are still not getting the newsletter please go subscribe. Lots of material in the newsletter. One of the topics is that we re-cover the changes we are making due to Adobe's threats. No new changes to report. What I posted before on the subject is what we are doing. Overall, the press and more importantly the customers who are posting their opinions on websites and blogs, are overwhelming stating their disappointment with the situation. Some nut jobs of course come out and predict deep conspiracy theories, but those seem to be pretty rare. In general, the simplest explanation is usually right. We are trying to provide functionality that are customers are asking for, Adobe is asking us not to do that. Nothing deeper at play than that.
Slides from our presentations at WinHEC went live this week. Links to slides on printing and color are here.


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A Ordem dos Clérigos Regulares ou Teatinos (Latim: Ordo Clericorum Regularium vulgo Theatinorum), também designada no Brasil Ordem de São Caetano, é uma ordem religiosa masculina católica fundada no dia 14 de setembro de 1524 por Caetano de Thiene, Bonifácio de Colli, Paulo Consiglieri e João Pedro Carafa, bispo de Chieti (cujo nome latino é Theate, do qual provém o gentílico teatino).

O Cardeal Carafa seria mais tarde eleito Papa Paulo IV.

A Igreja Católica vivia o período da chamada Contrarreforma ou Reforma Católica.

Alguns teatinos foram proclamados santos ou beatos pela Igreja Católica: São Caetano de Thiene, Santo André Avellino, São José Maria Tomasi e os beatos Giovanni Marinoni e Paolo Burali d'Arezzo.

Ver também
 Ordem religiosa
 São Caetano

Ligações externas
 Página oficial no Brasil

Teatinos

redpajama_set_name: RedPajamaWikipedia


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Reese Witherspoon embraces clean beauty with Biossance
The Biossance brand has chosen American actress and producer Reese Witherspoon as its new global ambassador. The star of "The Morning Show" joins a long list of celebrities who are committed to a more skin-friendly and environmentally-friendly beauty.
While she's not launching her own brand of cosmetics, Reese Witherspoon is turning to clean beauty by becoming the global ambassador of the brand Biossance, as Women's Wear Daily (WWD) crevealed. And she's signed up for a period of five years.
Beyond the status of brand ambassador, the American actress will have a role focusing on education, a key value for Biossance, as its president Catherine Gore underlines in an interview with the specialized site. Reese Witherspoon will share information about the products and their development, but also produce content; something she is already adept at with her company Hello Sunshine (which produces a podcast, newsletter, website, book club, etc.) celebrating women.
Known for her roles in "Cruel Intentions," "Legally Blonde", and "Wild," Reese Witherspoon has also been seen in television series such as "Big Little Lies" and "The Morning Show."
Freemans extends successful media deal with iProspect UK
Tesco to revamp F&F in H2 as it moves creative agency
Covent Garden launches campaign to brighten up dull winter

redpajama_set_name: RedPajamaCommonCrawl


(384 BC – 322 BC) A Greek philosopher, scientist and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology. Together with Plato and Socrates, Aristotle is one of the most important founding figures in Western philosophy.


MLS# 1358160 - Built by Highland Homes - CONST. COMPLETED Mar 31 ~ Boerne ISD! I10 access, Main St Boerne 5 min., RIM La Cantera 10 min, this one level stone & brick home features WHITE kitchen cabinets with granite counter tops and 7x22 tile flooring throughout. Level homesite, no steps. Oversized Master Bedroom, 3 beds, 2.5 baths and large study w window & closet. 13 ft ceilings throughout. A MUST SEE!
Listing provided courtesy of Ben Caballero of Highland Homes Realty.
© 2019 San Antonio Board of REALTORS® All rights reserved. IDX information is provided exclusively for consumers' personal, non-commercial use and may not be used for any purpose other than to identify prospective properties consumers may be interested in purchasing. Information is deemed reliable but is not guaranteed accurate by the MLS or NewHomePrograms.com LLC. Last updated 2019-04-22T20:32:21.11.


Tyler is still using the ECMO machine at 100% which is perfectly fine. Remember, that gives his body precious time to regroup and heal without having to work. The nurses gave him a bath and he did move his arms in a "purposeful" manner. (Lord, I hope he was polite.) But, that was a great sign. Otherwise we are holding steady with no sign of infections.
Keep the thoughts and prayers coming.
Keep the humor going. Loving the giggles you are providing while going thru all of the details on Tyler.
He's young and strong and has a great group of friends that acted quickly to start saving him! Many prayers are helping the doctors and nurses in the continuing efforts.
Try to take care of yourself while others are taking care of Tyler. He'll need you in top shape when he's awake. Much love and prayers to you all, and I'm asking all my friends and family that have passed on to make sure Trey stays in fight mode!!


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text:
Richardson Oliveira dos Santos, mais conhecido como Richardson (Rio de Janeiro, 30 de janeiro de 1976), é um ex-futebolista brasileiro que atuava como meia.

Um dos jogadores pior utilizados pelos técnicos vascaínos, pois era um meio de campo habilidoso com a perna canhota, que chegou ao Vasco em 1992, ainda como juvenil, vindo do Paduano-RJ. Em 1995 fez sua primeira partida com a camisa cruzmaltina, mas o time era muito fraco e não pode mostrar todo seu futebol, apesar de ter feito excelentes partidas. Sua facilidade em conduzir a bola com a perna esquerda era tanta que foi lançado nos profissionais com apenas 19 anos. A falta de experiência também foi um fator para que o canhotinho não despontasse no futebol. O Vasco passava, em 1995, uma fase de transição, lançando, além de Richardson, jogadores como: Pedrinho, Brener, Juninho (que acabara de chegar do Sport), Bill, Cristiano, Fabrício Eduardo e Frazão. Além destes jovens, o time ainda havia perdido jogadores que foram importantes na conquista do tricampeonato carioca (1992/93/94) como: William, Jorge Luís, Alexandre Torres, Jardel e Dener (falecido). Um time muito novo, e diferente do antigo, que acabou não dando certo e fazendo uma péssima campanha no Carioca e no Brasileiro. Em 1996 o apoiador praticamente não jogou, apenas 3 partidas. Talvez para preservar a jovem promessa. No ano seguinte foi emprestado, junto com Gian, para o América-RN onde se destacou e acabou retornando ao Vasco no início da temporada seguinte. Sua volta ficou marcada pelo golaço marcado contra o Botafogo no Rio-SP 98, quando deixou dois zagueiros pra trás em uma jogada fantástica e tocou na saída do goleiro. Era para ter sido o jogo da afirmação do meia, mas não para Antônio Lopes. O time havia mantido a base campeã brasileira em 97, base esta que contava com Ramón, Pedrinho e Juninho Pernambucano, além do recém chegado Vágner. Isto praticamente fechou as portas do clube para Richardson, que acabou se transferindo no ano seguinte para a Matonense. Depois rodou pelo Brasil e disputou o último Brasileirão da série B (2007) pelo Avaí.

Títulos
Vasco da Gama
 Campeonato Carioca: 1998
 Copa Libertadores da América: 1998

Paysandu
 Campeonato Paraense: 2000

Paulista
 Campeonato Paulista – Série A2: 2001
 Campeonato Brasileiro – Série C: 2001

Ligações externas

 Ogol
 Sambafoot
 "Lembra dele no Vasco?" - Richardson

Naturais da cidade do Rio de Janeiro
Futebolistas do Rio de Janeiro (estado)
Futebolistas do Club de Regatas Vasco da Gama
Futebolistas do América Futebol Clube (Rio Grande do Norte)
Futebolistas da Sociedade Esportiva Matonense
Futebolistas do Paysandu Sport Club
Futebolistas do Paulista Futebol Clube
Futebolistas do Mogi Mirim Esporte Clube
Futebolistas do Club Sport Marítimo
Futebolistas do Sport Clube de Mirandela
Futebolistas do Grupo Desportivo de Chaves
Futebolistas do América Futebol Clube (São José do Rio Preto)
Futebolistas do Clube Atlético Metropolitano
Futebolistas do Joinville Esporte Clube
Futebolistas do Avaí Futebol Clube
Futebolistas do Canoas Sport Club
Futebolistas do Clube Recreativo e Atlético Catalano


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text:
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text:
\section{Introduction}

Groundbreaking experimental progress
\cite{bloc08,blat12,spie14,taba15,haef15,haef17,nogu14,bosh14,moul12,phil11,note1}
in the field of trapped ultracold ions and neutral atoms, in
particular the unprecedented control of interpaticle interactions and the attainment of ultracold temperatures,
offer these systems as prime resources for experimental realization of the emergent exciting concept of a  quantum
space-time crystal (QSTC). Inspired by the relativistic 3+1-dimensions analogy \cite{wilc12},
\textcolor{black}{
the QSTC idea extends translational symmetry breaking (SB) to encompass both the spatial and time dimensions.}
Indeed, the original QSTC proposal
\cite{wilc12,li12} motivated an abundance of scientific discussion, commentary, and exploration
\cite{brun13,brun13.2,nozi13,wilc13,li13,wata15,sach15,sond16,naya16,yao16,fazi17}.

The original QSTC was proposed in the form of crystalline spatial-particle-density arrangements \cite{li12}, or
other solitonic-type (charge-density-wave) formations \cite{wilc12} revolving around a ring-shaped ultracold trap
without dispersion or damping.
Although significant experimental progress has been reported toward this goal \cite{haef15,haef17}, formation of a
QSTC in this experimental configuration is yet to be demonstrated. At the same time, experimental progress for a
``weaker class'' \cite{gibn17} of discrete-time-crystals \cite{sach15,sond16,naya16,yao16,fazi17} limited
exclusively to the time domain has been reported \cite{monr17,luki17}, employing time-periodically-driven spin
systems. Contributing to this state of affairs are
\textcolor{black}{limitations}
of earlier theoretical treatments of the QSTC
that were discussed extensively in previous commentary \cite{brun13,brun13.2,nozi13,wata15}, e.g., limiting oneself
to mean-field (MF) dynamics \cite{wilc12}, or considering solely the energetics of states with good total angular
momenta which (as a matter of principle) have uniform spatial densities \cite{li12,li12.2}. To throw further light
on the nature and properties of QSTCs, it is imperative that a formulation and implementation of appropriate
many-body trial wave functions for the QSTC on a ring be advanced. The sought-after trial wave functions should
explore for a finite system of $N$ particles the interplay \cite{yann07} between the mean-field symmetry-broken
states, which are not eigenstates of the total angular momentum $\hat{L}$, and the exact symmetry-preserving (good
total-angular-momentum) states.

Here, we introduce such trial wave functions and analyze their spectra and combined spatially dispersionless and
temporally undamped evolution,
\textcolor{black}{which are the defining characteristics of a QSTC.}
Contrasting with these findings,
previous beyond-mean-field theoretical studies \cite{degu12,kavo17,cede11,carr16,schm17} that investigated spatial
solitonic formations in finite boson systems in one dimension or on a ring have revealed drastically different
behaviors, such as increasing dispersion with time accompanied by a revival at the initial position of the propagated
inhomogeneous wave packet \cite{robi15}.

We employ a beyond-mean-field methodology of symmetry restoration via projection techniques, introduced by us
previously \cite{yann02,yann02.2,yann03.2,yann03,yann04,yann04.2,yann06,yann07,yann09,yann11}
for two-dimensional semiconductor quantum dots (with and without an applied magnetic field ${\bf B}$). The
multilevel symmetry-breaking and symmetry-restoration approach which we persue provides a
complete theoretical framework for treating symmetry breaking aspects in
finite systems, without reference to the $N \rightarrow \infty$ limit. Indeed this approach originated, and is
widely employed, in nuclear physics and chemistry
\cite{yann07,peie57,ringbook,lowe55,robl02,bend03,doba07,sun16,ring00,naza12}.

The paper is organized as follows. In Section II we introduce and illustrate the hierarchical, multilevel methodology
that we use for the construction of the trial wave functions for the microscopic many-body Hamiltonian of few
ultracold ring-confined interacting particles. Following a short synopsis of the method, we discuss first in section
II A the mean-field, broken-symmetry state, and subsequently in section II B a beyond-mean-field level is outlined,
entailing symmetry-restoration via the use of an angular momentum projection technique.
\textcolor{black}{This results}
in many-body
stationary-state good-angular-momentum solutions of the microscopic Hamiltonian. These (projected) symmetry-restored
states show uniform particle density around the ring. However, simultaneously they posses hidden crystalline symmetries
which can be revealed through the analysis of the corresponding conditional probability densities. In section III C
we complete our exposition of the construction of the QSTC trial wave functions by analyzing the properties of
superpositions of pairs of the above-noted symmetry-restored (projected) stationary states [see Eq.\ (\ref{pwm})] that
are favored to mix in the vicinity of crossings of Aharonov-Bohm-type spectra of ground-state energies versus
applied magnetic-field (Fig.\ \ref{fspec}). The particle density corresponding to such superposed wavefunctions reveals
crystalline structure on the ring.  Numerical solutions using the trial wave functions are illustrated and analyzed
for the case of few (even and odd in number) Coulomb-repelling fermionic ions, and for a lump of contact-interacting
attractive bosons. When evolved with the microscopic many-body Hamiltonian, these trial functions exhibit, for both
the fermionic repelling ions and attracting bosons, undamped and non-dispersive space and time crystalline periodic
evolution -- that is, they exhibit breaking of both the space and time symmetries.

Section III is devoted to further elaboration
\textcolor{black}{on}
three main topics. In section III A we discuss the symmetry
properties of the symmetry-restored (projected) wave functions and the selection rules for their ``magic'' angular
momenta. Section III B analyzes the properties of the initial wave packets and their associated time evolution, and
section III C comments on the relation between the constructed trial functions (in particular  the aforementioned
symmetry-restored stationary states) and the wavefunctions obtained through exact-diagonalization
[configuration-interaction (CI)] solutions of the microscopic many-body Hamiltonian.

We conclude in section IV with a summary of our work, including a brief listing of recent progress achieved in
developing experimental techniques for preparation and measurement of ring-confined ultracold particles. The
Appendices give tables of numerical results (rotational energies for different magic angular momenta, and moments of
inertia) for the systems investigated in the paper, as well as explicit expressions for the conditional probability
distribution and single particle density.

\section{The interplay between symmetry-broken and symmetry-preserving states: Group-theoretical formulation}
\label{sbsp}

In connection with the QSTC, we consider three levels of many-body trial wave functions: (1) A Slater determinant for
localized fermions (or permanent for localized bosons) on the ring. We denote this wave function by $\Psi^{\rm SB}$; it
corresponds to the unrestricted Hartree-Fock (UHF), or Gross-Pitaevskii (GP), mean-field step \cite{yann07}
that exhibits symmetry breaking of the space degrees of freedom. $\Psi^{\rm SB}$ does not preserve the total
angular momentum. Out of the three levels in the hierarchical scheme (see below), it is the trial wave function
closest to the familiar concept of a classical Wigner crystal \cite{giulbook}.
(2) A stationary multideterminantal (multipermanental)
wave function $\Phi^{\rm PROJ}_L$ characterized by a good total angular momentum $\hbar L$, which is generated
by applying a projection operator ${\cal P}_L$ (see below) on
$\Psi^{\rm SB}$. This step goes beyond the MF approximation and restores (as required) the quantum many-body
Hamiltonian symmetries in the stationary-state solutions. Unlike $\Psi^{\rm SB}$, $\Phi^{\rm PROJ}_L$ exhibits an
azimuthally uniform single-particle density [SPD, $\rho({\bf r},t)$], which is also time-independent (stationary).
Previously, we referred to such projected wave functions $\Phi^{\rm PROJ}_L$ as quantum rotating Wigner
molecules \cite{yann09}.
\textcolor{black}{
(3) Coupling between the stationary states (brought about by a perturbation which we term in the following as
``the pinning agent'') results in a superposition of two projected wave functions with different angular momenta
$L_1$ and $L_2$, leading to formation of a pinned Wigner molecule (PWM), i.e.,}
\begin{equation}
\Phi^{\rm PWM}(L_1,L_2; t=0) =
\alpha \Phi^{\rm PROJ}_{L_1} + \beta e^{i\phi(t=0)}\Phi^{\rm PROJ}_{L_2},
\label{pwm}
\end{equation}
where $\phi(t=0)$ can be set to zero without loss of generality, and $\alpha^2+\beta^2=1$.
\textcolor{black}{
In the following we illustrate the case of $\alpha=\beta =1/\sqrt{2}$ (the physics of the PWM maintains for other
choices of the mixing coefficients).}
For selected magic (see below) $L_1$ and $L_2$, $\Phi^{\rm PWM}(L_1,L_2; t=0)$ represent a special family of quantal
wave packets with broken azimuthal symmetry.
Consequently their corresponding $\rho({\bf r},t=0)$ are not uniform, forming
instead a crystal-like particle density pattern, with $kN$, $k=1,2,3,\ldots$ possible peaks for $N$ fermionic ions
and $1,2,3,\ldots$ possible peaks for $N$ attractive bosons. When the pinning agent is lifted, the
$\Phi^{\rm PWM}(L_1,L_2;t)$ evolve in time undamped according to the exact many-body quantum Hamiltonian dynamics,
i.e., the phase $\phi$ will vary as $\phi(t)=(E_2-E_1) t/\hbar$, and the associated $\rho({\bf r},t)$ will
oscillate at any given space point with a time period $T=\tau/n$; $\tau=2\pi\hbar/|E_1-E_2|$, $E_{1(2)}$
being the energies of the stationary states $\Phi^{\rm PROJ}_{L_i}$ with $i=1,2$, respectively.
For either statistics (fermions or bosons), $n = N$ for repelling ions and $n = 1$ for attractive particles;
as aformentioned, here we discuss explicitly Coulomb-repelling fermionic ions and contact-interacting attractive
bosons. Such undamped and dispersionless periodic time variation is not possible for the MF Hartree-Fock (or
Gross-Pitaevskii) wave packet $\Psi^{\rm SB}$, because it contains all the possible angular momenta when expanded
in the complete basis set of the stationary wave functions $\Phi^{\rm PROJ}_L$. Additionally, the MF wave functions
lose \cite{grif76} their single-determinant (single-permanent) character under the exact time evolution.

The many-body Hamiltonian of $N$ identical particles in a ring-type trap threaded by a constant
magnetic field ${\bf B}$ is
\begin{align}
\begin{split}
{\cal H}= \sum_{i=1}^N & \biggl( \frac{({\bf p}_i-\eta {\bf A}_i)^2}{2M} +
\frac{(r_i-R)^2}{2l_0^2/(\hbar \omega_0)} \biggr) + \sum_{i < j} V( r_{ij}),
\end{split}
\label{hmb}
\end{align}
where ${\bf A}({\bf r})={\bf B}{\times}{\bf r}/2$ is the vector potential in the symmetric gauge,
$r=\sqrt{x^2+y^2}$, $\omega_0$ is the frequency of the trap, $R=\sqrt{X^2+Y^2}$ is the ring radius,
the oscillator length $l_0=\sqrt{\hbar/(M\omega_0)}$, and $r_{ij}=|{\rm r}_i-{\rm r}_j|$. ${\bf B}$ can be the
familiar magnetic field in the case of charged ions (when $\eta=e/c$), or a synthetic one in the case
of ultracold neutral atoms \cite{spie14}.

\textcolor{black}{
\subsection{First level: The mean-field-ansatz, symmetry-broken crystalline state}}
\label{fl}

\textcolor{black}{
In view of the strong inter-particle interactions (large values of the parameters $R_W$ and $R_\delta$, see below)
associated with the proposed experimental realizations of the QSTC's \cite{wilc12,li12}, resulting, for ultracold
quasi-1D-ring-trapped particles, in formation
of a Wigner crystal (in the case of Coulomb repelling ultracold ions) or a lump (in the case of contact
attracting bosons), we construct here the symmetry-broken initial state via the use of an ansatz. This ansatz explores
the localized nature of the $N$ space orbitals (corresponding to the $N$ ring-trapped particles) from which the
single broken-symmetry determinant (permanent) is formed in the corresponding fermion (boson) systems.
Such ansatz proved most adequate to approximate broken-symmetry mean-field solutions (reflecting individual
particle localization) in previous studies of fermions and bosons in harmonically confined quantum dots and other 2D
systems \cite{yann07,maki83}.}

We describe each particle localized at position ${\bf R}_j$ as a displaced Gaussian function
\begin{align}
u({\bf r}, {\bf R}_j) =\frac{1}{\sqrt{\pi} \lambda} \exp \biggl( -\frac{({\bf r}-{\bf R}_j)^2}{2\lambda^2}
-i \varphi({\bf r},{\bf R}_j; B) \biggr),
\label{uorb}
\end{align}
with $\lambda=\sqrt{\hbar/(M\Omega)}$; $\Omega=\sqrt{\omega_0^2+\omega_c^2/4}$ where
$\omega_c=\eta B/M$ is the cyclotron frequency. The phase in Eq.\ (\ref{uorb}) is due to the gauge
invariance of magnetic translations \cite{landbook,peie33}) and is given by $\varphi({\bf r},{\bf R}_j; B)=
(xY_j-yX_j)/(2l_B^2)$, with $l_B=\sqrt{\hbar /(\eta B)}$ being the magnetic length. For simplicity, in the
following we provide examples for only three cases: (i) that of $N$ fully polarized fermionic ions with odd
$N$, (ii) that of $N$ fully polarized fermionic ions with even $N$, and (iii) that of $N$ spinless bosons interacting
via an attractive contact potential. As will be shown explicitly, case (ii) presents different characteristics compared
to case (i).

In the case of ultracold ions repelling each other via the Coulomb interaction, we take the
${\bf R}_j=R_{\rm eq} e^{2\pi (j-1) i/N}$, $j=1,2,\ldots,N$ to coincide
with the equilibrium positions (forming a regular polygon) of $N$ classical charges inside the annular
confinement specified in Eq.\ (\ref{hmb}). Then $R_{eq}$ $(> R)$ is given by the real solution of the
cubic equation $aw^3+bw^2+d=0$, where $a=1$, $b=-R$, $d=-l_0^3 R_W S_N/4$, with the Wigner parameter
(the ratio between the characteristic interparticle repulsion and the kinetic zero-point energy of the
ring-confined particle), $R_W=e^2/(l_0 \hbar \omega_0)$ \cite{yann07} and $S_N=\sum_{j=2}^N 1/\sin[(j-1)\pi/N]$.
Then the corresponding MF wave function, $\Psi^{\rm SB}$, is the determinant formed by the $N$ orbitals
$u({\bf r}_i, {\bf R}_j)$.

In the case of $N$ ultracold neutral bosons attracting each other with a contact interaction
$-g \delta({\bf r}_i-{\bf r}_j)$,
\textcolor{black}
{the atoms are localized at the same}
position, and thus $R_{\rm eq}=R$ and
$R_j=R e^{i\theta_0}$, $j=1,2,\ldots,N$. Then the MF wave function, $\Psi^{\rm SB}$, is the product (permanent) of
the orbitals $u({\bf r}_i, R e^{i\theta_0})$. The parameter corresponding to $R_W$ is given here by
$R_\delta =  g M/\hbar^2$.\\

\subsection{Second level (beyond mean field): The projected, symmetry-restored stationary state}
\label{sl}

A stationary many-body state that preserves the total angular momentum, as well as the rotational symmetry of the
annular trap, can be projected out of the symmetry-broken $\Psi^{\rm SB}$ by applying the projector operator
${\cal P}_L$,
\begin{align}
{\cal P}_L = \frac{1}{2\pi} \int_0^{2\pi} e^{i \gamma (L-\hat{L})} d\gamma,
\label{prjop}
\end{align}
where $\hat{L}=\sum_{i=1}^N \hat{l}_i$, $i=1,2,\ldots,N$, and $\hbar \hat{L}$ is the total angular-momentum
operator. Then the projected many-body state is given by
\begin{align}
\Phi^{\rm PROJ}_L =
\frac{1}{2\pi} \int_0^{2\pi} d\gamma \Psi^{\rm SB}(\gamma) e^{i\gamma L}.
\label{prj}
\end{align}

${\cal P}_L$ is analogous to the projector operators used in chemistry for molecular orbitals governed by point
group symmetries \cite{yann03,wolbbook,cottbook,hamebook}. Such projection operators are constructed
through a summation over the characters of the point group \cite{yann03,cottbook,hamebook};
the phases $e^{i\gamma L}$ are the characters of the rotational group in two dimensions \cite{yann03,hamebook}
and the operator $e^{-i \gamma \hat{L}}$ is the corresponding generator of 2D rotations. Alternatively, Eq.\
(\ref{prj}) may be viewed as a linear superposition of all the (energy-degenerate) symmetry-broken states
$\Psi^{\rm SB}(\gamma)$, azimuthally rotated by $\gamma$. Due to the rotational symmetry, the coefficients of
this superposition, i.e., the phases $e^{i\gamma L}$, can be determined a priori, without the
need to diagonalize a Hamiltonian matrix.

The projected energies, associated with the stationary wave functions $\Phi^{\rm PROJ}_L$, are given by
\begin{align}
E^{\rm PROJ}(L)=\int_0^{2\pi} h(\gamma) e^{i\gamma L}d\gamma \bigg/
\int_0^{2\pi} n(\gamma)e^{i\gamma L} d\gamma,
\label{eprj}
\end{align}
where
\begin{equation}
h(\gamma)=\langle \Psi^{\rm SB}(0)|{\cal H}| \Psi^{\rm SB}(\gamma) \rangle,
\label{hme}
\end{equation}
and the norm overlap
\begin{equation}
n(\gamma)=\langle \Psi^{\rm SB}(0)| \Psi^{\rm SB}(\gamma) \rangle
\label{nov}
\end{equation}
enforces proper normalization of $\Phi^{\rm PROJ}_L$. Note that the original double integration reduces to a
single integration over $\gamma$ because ${\cal P}_L^2={\cal P}_L$, $[{\cal P}_L,{\cal H}]=0$.
\textcolor{black}{
We note that the unresticted HF energies for the ansatz determinant (or permanent), $\Psi^{\rm SB}$, before projection
are simply given by
\begin{equation}
E_{\rm UHF}=h(0)/n(0).
\label{euhf}
\end{equation}
}
\begin{figure}[t]
\centering\includegraphics[width=8.0cm]{qtc_fig1.pdf}
\caption
Aharonov-Bohm-type, quantum-rigid-rotor energy spectrum [second term in the right-hand-side of Eq.\ (\ref{spec})]
as a function of the magnetic flux through the
ring, $\Phi/\Phi_0$, for the symmetry-restored (stationary) states $\Phi^{\rm PROJ}_L$ for $N=7$ fermionic ions. The
remaining parameters are: Wigner parameter $R_W=1000$, ring radius $R=200 l_0$, and oscillator strength $l_0=50$ nm.
According to Table \ref{t1} in the Appendix \ref{appa}, the parameter $C_R$ in Eq.\ (\ref{spec}) was taken equal to
$1.7847 \times 10^{-6} \hbar \omega_0$. Each curve is labeled with the corresponding magic total angular momentum
$L_m$. The circles highlight several energy-crossing points most susceptible to symmetry breaking.
Energies in units of $10^{-4} \hbar \omega_0$.
}
\label{fspec}
\end{figure}

We have carried
\textcolor{black}{out}
numerical calculations to determine the rotational spectrum of the $\Phi^{\rm PROJ}_L$'s.
For the calculation of $h(\gamma)$ and $n(\gamma)$, we use the rules for determinants composed of nonorthogonal
orbitals; see, e.g., Ref.\ \cite{hurlbook}. Similar rules apply for permanents. The numerical calculations
are facilitated by the fact that the one-body and two-body matrix elements between the orbitals
$u({\bf r}, {\bf R}_j)$ have closed analytic expressions \cite{ishi03,boys50,yannpub}.

In all three cases [(i) odd number of repelling fermions, (ii) even number of repelling fermions, and (iii)
attractive bosons], and for all values of $N \leq 10$, large localization parameters $R_W \geq 200$ and
$R_\delta \geq 50$, and large ratios $R/l_0 \geq 40$ that we studied, we found that indeed the numerically calculated
energies of the $\Phi^{\rm PROJ}_L$'s according to Eq.\ (\ref{eprj}) (see, e.g., Tables \ref{t1}, \ref{t2}, and
\ref{t3} in Appendix \ref{appa}) can be well-fitted by that of an Aharonov-Bohm-type
spectrum associated with a quantum many-body rigid rotor (see also \cite{li12,robi15}), i.e.,
\begin{align}
E^{\rm PROJ}(L) \approx V_{\rm int} + C_R (L - N \Phi/\Phi_0)^2.
\label{spec}
\end{align}

\textcolor{black}{
$V_{\rm int}$ approximates the ground-state energy of the few-particle system and takes different values for
different many-body wave functions.}

\textcolor{black}{The numerically determined coefficient $C_R$ is essentially a constant (see below).}
$\Phi=\pi R_{\rm eq}^2 B$ is the magnetic flux through the ring and $\Phi_0=h/\eta$ is the magnetic flux quantum. The
values of the angular momenta $L$ are not arbitrary. Because of the crystalline symmetries, as well as the symmetric
or antisymmetric behavior under particle exchange, they are given by proper sequences of magic angular momenta $L_m$
(see section \ref{symm} below for further discussion). In particular, all values of angular momenta are allowed for
the case of attractive bosons, i.e., $L_m=0,\pm 1,\pm 2,\ldots$. For the case of fully polarized repelling fermions
with $N$ odd or spinless repelling bosons with any $N$, the allowed angular momenta are restricted to the sequence
$L_m=kN$, with $k=0,\pm 1,\pm 2,\ldots$.
For the case of fully polarized repelling fermions with $N$ even, the allowed angular momenta are given by a different
sequence $L_m=(k+1/2)N$, with $k=0,\pm 1,\pm 2,\ldots$.

\textcolor{black}
{Due to the very large values of $R_W$ and $R_\delta$}, the value of $C_R$ is very close to that of a
classical rigid rotor, corresponding to $N$ point particles in their equilibrium configuration inside the annular
confinement, i.e., $C_R \approx C_R^{\rm cl}=\hbar^2/[2 {\cal I}(R_{\rm eq})]$, with inertia moment
${\cal I}(R_{\rm eq})=NM R_{\rm eq}^2$. As typical examples, in Table \ref{t1}, Table \ref{t2}, and Table \ref{t3}
of Appendix \ref{appa}, we list calculated energies according to Eq.\ (\ref{eprj}) for an odd number $N=7$ and
an even number $N=8$ of fermionic ions, as well as for $N=10$ attractive bosons, respectively; $R_W=1000$
for the repelling ions and $R_\delta=50$ for the attractive bosons. The ratio $\tilde{f} \equiv C_R/C_R^{\rm cl}
\approx 1$ for all $L < 165$ for ions, and for all $L \leq 30$ for attractive bosons.
\textcolor{black}
{As aforementioned, the rigid-rotor-type spectrum in Eq.\ (\ref{spec}) was explored earlier in the
QSTC literature \cite{li12,robi15}; however, by itself it does not lead to the derivation of appropriate QSTC wave
functions. The demonstrated agreement between the microscopically calculated rotational part of the spectrum
[Eq.\ (\ref{eprj})] and the analytic second term in Eq.\ (\ref{spec}) expected for a QSTC \cite{li12,robi15} validates
the expressions $\Phi^{\rm PROJ}_L$ introduced in Eq.\ (\ref{prj}) as proper trial wave functions for the QSTC.}

\textcolor{black}{
We showed previously that the limit of a quantum rigid rotor for a system of strongly interacting particles can
also be reached in external confinements with geometries other than the ring geometry. In particular, the rigid-rotor
limit for $R_W=200$ and in a fully two-dimensional parabolic confinement was demonstrated for two electrons in Ref.\
\cite{yann00} using exact many-body wave functions and for a few electrons in Ref.\ \cite{yann04} using the
same ansatz as in Eq.\ (\ref{prj}) here. In this universal rigid-rotor limit, the rotational part of the spectra is
naturally similar. However, the presence of strong many-body correlations (which result from the
beyond-mean-field, multi-determinant, or multi-permanent, nature of the wave function) is reflected in the actual
numerical values of the first term, $V_{\rm int}$, in Eq.\ (\ref{spec}).
}

\textcolor{black}{
In our scheme, which allows for an expanded variational freedom by employing unrestricted orbitals at the mean-field
single-determinant, or single-permanent, level (i.e., a different orbital for each particle), the ground-state
energy is lowered at every step (see in particular Fig.\ 1 in Ref.\ \cite{yann07}): restricted HF (symmetry conserving)
$\rightarrow$ unrestricted HF (symmetry breaking) $\rightarrow$ symmetry restoration via projection techniques (an
example of the full scheme can be seen in Fig.\ 5 of Ref.\ \cite{yann07}). This scheme for repelling bosons
translates as: symmetry-conserving Gross-Pitaevskii $\rightarrow$ symmetry-breaking ansatz of unrestricted
permanent $\rightarrow$ symmetry restoration via projection techniques. We note that allowing the single orbital of the
Gross-Pitaevskii equation to break the azimuthal symmetry of the ring leads \cite{kana08,kana09,kana10} to
higher-energy solitonic Bose-Einstein condensate branches in the rotational part of the spectrum, which are sharply
different from the QSTC wave functions introduced in this paper.
}

\textcolor{black}{
As a specific example of the lowering of the ground-state energy in our scheme, we report that the energy for the
unrestricted ansatz determinant [see Eq.\ (\ref{euhf})] in the case of $N=8$ ultracold ions on a ring of
radius $R=200l_0$ with $R_W=1000$ (case described in Table \ref{t2} of Appendix A) is $E_{\rm UHF}=118.1771$
$\hbar \omega_0$, while the corresponding restored-symmetry ground state has indeed a lower energy $V_{\rm int}=
117.9271$ $\hbar \omega_0$. This lowering of the total energy is immense compared to the quantum of the
rotational motion $C_R^{\rm cl}=1.5614 \times 10^{-6} \hbar \omega_0$ (see caption of TABLE \ref{t2} in Appendix A).}

An illustrative case of the
\textcolor{black}{rigid-rotor rotational spectra encoded in the second term}
in Eq.\ (\ref{spec}) is displayed in Fig.\ \ref{fspec}.
A main feature of these spectra are the crossing points (several of them encircled) between pairs of curves
with different $L_m$'s. The crossings define special magnetic-field values, $\tilde{\Phi}/\Phi_0=(L_1+L_2)/(2N)$,
in the neighborhood of which the system is particularly susceptible to symmetry breaking via the
intermixing of two angular momenta and the ensuing generation of the PWM wave packets [see Eq.\ (\ref{pwm})].

Because the symmetry-restored (projected) wave function $\Phi^{\rm PROJ}$ [Eq.\ (\ref{prj})] preserves the
group-theoretical requirements of the continuous 2D rotational group, its single-particle density is azimuthally
uniform. However, the crystalline order of the original MF (symmetry-broken) wave function $\Psi^{\rm SB}$ is not
destroyed in the symmetry-restoration step; instead, it mutates into a hidden order, which however
can be revealed via the conditional probability distribution (CPD) (density-density correlation function).
The CPD is  given by
\begin{align}
& {\cal D}({\bf r},{\bf r}_0) = \langle \Phi^{\rm PROJ}_L|\sum_{i\neq j}
\delta({\bf r}_i,{\bf r}) \delta({\bf r}_j,{\bf r}_0)|\Phi^{\rm PROJ}_L \rangle.
\label{cpd}
\end{align}
The CPD provides the probabilty of finding a particle in position ${\bf r}$ assuming that another one
is located at the fixed point ${\bf r}_0$. Substitution of the expression [Eq.\ (\ref{prj})] that
defines $\Phi^{\rm PROJ}_L$, yields for ${\cal D}({\bf r},{\bf r}_0)$ a double integral over the azimuthal angles
$\gamma_1$ and $\gamma_2$; this integral expression is given in the Appendix \ref{appb}.

\begin{figure}[t]
\centering\includegraphics[width=8cm]{qtc_fig2.pdf}
\caption
(a) CPD of the symmetry-restored stationary (beyond MF) state $\Phi^{\rm PROJ}_L$ for $N=5$ fermionic ions along the
perimeter of the ring (at a radius $R_{\rm eq}$). The arrow at $\theta_0=1.1\pi$ denotes the fixed point
${\bf r}_0=R_{\rm eq} e^{i \theta_0}=R_{\rm eq} e^{1.1 i \pi}$. Note the $2\pi/5$ angle between the nearest-neighbor
humps and between the arrow and the two adjacent humps. Other parameters are: magic angular momentum
$L=L_m=N$, $R_W=200$, $R=40 l_0$, $l_0=50$ nm,
and $\Phi/\Phi_0=0.8$. There is no hump at the fixed point. CPD in units of $10^{-2}/(2\pi \lambda^4)$.
(b) SPD of the original MF state $\Psi^{\rm SB}(\gamma=0.1\pi)$ (a determinant) for $N=5$ fermionic ions,
exhibiting explicitly symmetry breaking. Other parameters are $R=40 l_0$, $l_0=50$ nm.
SPD in units of $10^{-1}/(2\pi \lambda^2)$. In contrast to the symmetry-broken MF SPD in (b), the SPD of the
symmetry-restored, beyond-mean-field $\Phi^{\rm PROJ}_L$ is azimuthally uniform; see black dashed line in Fig.\
\ref{fdens}(a). Azimuthal angle $\theta$ in units of $\pi$.
}
\label{fcpd}
\end{figure}

Fig.\ \ref{fcpd}(a) displays an illustrative example of the hidden order in the symmetry-restored wave function
$\Phi^{\rm PROJ}_L$. The CPD in Fig.\ \ref{fcpd}(a) exhibits well localized features; it contrasts with the uniform
horizontal black dashed lines in Figs.\ \ref{fdens}(a) and \ref{fdens}(b) which describe $\rho({\bf r},t)$'s of
$\Phi^{\rm PROJ}_L$ along the perimeter of the ring trap (at a radius $R_{\rm eq}$).
\textcolor{black}{
We stress that the fixed point ${\bf r}_0$ in the CPD is arbitrary, i.e., the four peaks in the CPD in Fig.\
\ref{fcpd}(a) readjust to a different choice of $\theta_0$ so that the relative distance between them and the arrow
remain unchanged.}
Fig.\ \ref{fcpd}(b) displays the SPD of the original state $\Psi^{\rm SB}(\gamma=0.1\pi)$
(a determinant) for $N=5$ fermionic ions, exhibiting explicitly the symmetry breaking at the mean-field level.

\begin{figure*}[t]
\centering\includegraphics[width=15cm]{qtc_fig3.pdf}
\caption
Snapshots of undamped inhomogeneous single-particle densities rotating around the ring that were calculated with
the wave packet $\Phi^{\rm PWM}(L_1,L_2; t)$. (a) $N=5$ fermionic ions and magic angular momenta $L_1=0$, $L_2=N$.
The SPDs are shown for the time instances $t_j=j\tau/4$, with $j=0$, $j=2$, and $j=4$
(the $j$'s label the curves); $\tau= 2\pi \hbar /(|E^{\rm PROJ}(L_2)-E^{\rm PROJ}(L_1)|)$.
Other parameters: $\alpha=\beta=1/\sqrt{2}$, $R_W=200$,
$R=60 l_0$, $l_0=50$ nm, and $\Phi/\Phi_0=1.8$.
(b) $N=7$ attractive bosons and magic angular momenta $L_1=0$, $L_2=1$. The SPDs are shown for the time instances
$t_j=j\tau/4$, $j=0,1,\ldots,4$. Other parameters: $\alpha=\beta=1/\sqrt{2}$, $R_\delta=50$, $R=40 l_0$, $l_0=1$
$\mu$m, and $\Phi/\Phi_0=6.4$. The black horizontal dashed lines represent the uniform density of either one of the
stationary states $\Phi^{\rm PROJ}_{L_m}$ (with $L_m=L_1$ or $L_m=L_2$) that contribute to the nonstationary wave
packet $\Phi^{\rm PWM}(L_1,L_2; t)$.
(c) $N=5$ fermionic ions and magic angular momenta $L_1=0$, $L_2=2N$ (higher-harmonic of the QSTC).  $t_j=j\tau/4$,
with $j=1$ and $j=2$.
Other parameters as in (a). For repelling ions, panels (a) and (c), the period is $T=\tau/N$; for attractive bosons,
the period is $T=\tau$. The SPDs are in units of $10^{-2}/(2\pi\lambda^2)$. Azimuthal angle $\theta$
in units of $\pi$.}
\label{fdens}
\end{figure*}

\subsection{Third level (beyond mean field): Periodic time evolution of the spatially inhomogeneous
$\rho({\bf r},t)$ associated with the wavepacket $\Phi^{\rm PWM}(L_1,L_2; t=0)$}
\label{tl}

As aforementioned, the two-state wave packet in Eq.\ (\ref{pwm}) is not an eigenstate of the total angular
momentum, and thus it is not a stationary state when the pinning agent is lifted; such a pinning agent could be
implemented, for example, as a distortion of the circular geometry of the trap confinement, or as a modulation of
the trap potential in the azimuthal direction along the ring \cite{haef17}. (A sudden variation of the magnetic
field can also transform an eigenstate $\Phi^{\rm PROJ}_L(B_1)$ at a given $B_1$ value to a superposition of
$\Phi^{\rm PROJ}_L(B_2)$ states at another $B_2$ value \cite{yannpub}.) The resulting time
evolution is associated with a time-dependent phase $\phi(t)$ as discussed previously. Here we will show
explicitly that $\phi(t)$ represents an undamped rotation of spatially inhomogeneous $\rho({\bf r},t)$'s
around the ring, so that the many-body $\Phi^{\rm PWM}(L_1,L_2; t)$ exhibit the desired behavior of a QSTC. The
successful theoretical identification and experimentally implemented superposition of two appropriate many-body
spin eigenstates of the Ising Hamiltonian (resulting in a ``spin Schr\"{o}dinger-Cat'' state) were
keys to the emulation of the ``weaker class'' of discrete time crystals \cite{sond16,naya16,monr17,fazi17}.

The $\rho({\bf r},t)$ of $\Phi^{\rm PWM}(L_1,L_2; t)$ is defined as
\begin{align}
\rho({\bf r};t) = \langle \Phi^{\rm PWM}(L_1,L_2; t) | \sum_{i=1}^N \delta({\bf r}_i-{\bf r}) |
\Phi^{\rm PWM}(L_1,L_2; t) \rangle.
\label{rho}
\end{align}

\textcolor{black}{As in the case of the CPD,}
$\rho({\bf r};t)$ entails a double integral over the azimuthal angles $\gamma_1$ and $\gamma_2$;
the lengthy expression is given in Appendix \ref{appc}.

Fig.\ \ref{fdens} displays the periodic time evolution of $\rho({\bf r},t)$'s for two illustrative
$\Phi^{\rm PWM}(L_1,L_2; t)$ cases, one for $N=5$ Coulomb repelling fermionic ions [Fig.\ \ref{fdens}(a)] with
$L_2-L_1=N$ and the other for $N=7$ neutral bosons with $L_2-L_1=1$ [Fig.\ \ref{fdens}(b)] interacting via an
attractive contact interaction. The $\rho({\bf r},t)$'s were calculated at times $t_j=j\tau/4$, where
$\tau = 2 \pi \hbar /(|E^{\rm PROJ}(L_2)-E^{\rm PROJ}(L_1)|)$;
\textcolor{black}{the actual used}
$j$'s label the
$\rho({\bf r},t)$ curves. The number of humps exhibited by the PWM $\rho({\bf r},t)$'s in Fig.\ \ref{fdens}(a)
and Fig.\ \ref{fdens}(b) is equal to that in the original MF densities, i.e., $N$ for the repelling-fermions PWM
and one for the attractive-bosons lump. The period of the PWM $\rho({\bf r},t)$'s is $T=\tau/N$ for repelling ions and
$T=\tau$ for attractive bosons.

Finally, Fig.\ \ref{fdens}(c) demonstrates a different state of matter, i.e., multi-harmonic excitations of the QSTC
exhibiting a multiple number of density humps, i.e., $k N$ and $k$ (with $k=2,3,\ldots$),
corresponding to $\Phi^{\rm PWM}(L_1,L_2; t)$'s with $L_2-L_1=k N$ for repelling
fermions and with $L_2-L_1 =k$ for attractive bosons, respectively.

\textcolor{black}{
We note that the PWM broken-symmetry state introduced here to describe a QSTC has an energy intermediate between
$E_1$ and $E_2$ because $\alpha^2+\beta^2=1$ (i.e., $E^{\rm PWM}=\alpha^2 E_1 + \beta^2 E_2$). In particular, at the
crossing point of the two parabolas (where $E_1=E_2=E_{\rm cross}$), one has always $E^{\rm PWM}=E_{\rm cross}$. This
contrasts with the behavior of the energy of the non-crystalline states studied in Refs.\ \cite{kana08,kana09,kana10},
which lies always well above the crossing point.}

\section{Discussion}
\label{disc}

\subsection{Symmetries of the trial wave functions, magic angular momenta, and rigidity}
\label{symm}

\textcolor{black}{
Despite the fact that the trial wave functions in Eq.\ (\ref{prj}) are a
good approximation to the rotational-symmetry-preserving
many-body eigenstates, they do embody and reflect in an optimum way the crystalline point-group symmetries (familiar
from bulk crystals).}
Specifically, the $C_N$ point-group symmetry of the ``classical'' crystal, which is accounted for
through the kernel of symmetry-broken MF determinants (or permanents) $\Psi^{\rm SB}$, is reflected in the fact that
the trial wave functions $\Phi^{\rm PROJ}_L$ are identically zero except for a subset of {\it magic\/} angular
momenta $L_m$.

In the case of $N$ repelling particles, the magic total angular momenta can be determined by considering the
point-group symmetry operator $\hat{R}(2\pi/N) \equiv \exp (-i 2\pi {\hat L} /N)$ that rotates on the ring
simultaneously the localized particles by an angle $2\pi/N$.
\textcolor{black}{
In connection to the state $\Phi^{\rm PROJ}_L$, the operator $\hat{R}(2\pi/N)$ can be invoked in two different ways,
namely either by applying it on the ``intrinsic'' part $\Psi^{\rm SB}$ or the ``external'' phase factor
$\exp(i \gamma L)$ (see Ch. 4-2c Ref.\ \cite{bomo98}).}
One gets in the case of fermions
\begin{equation}
\hat{R}(2\pi/N) \Phi^{\rm PROJ}_L = (-1)^{N-1} \Phi^{\rm PROJ}_L,
\label{rot1}
\end{equation}
from the first alternative and
\begin{equation}
\hat{R}(2\pi/N) \Phi^{\rm PROJ}_L = \exp(-2\pi L i/N) \Phi^{\rm PROJ}_L,
\label{rot2}
\end{equation}
from the second alternative. The $(-1)^{N-1}$ factor in Eq.\ (\ref{rot1}) results from the fact that the $2\pi/N$
rotation is equivalent to exchanging $N-1$ rows in the  $\Psi^{\rm SB}$ determinant. Now if
$\Phi^{\rm PROJ}_L \neq 0$, the only way that Eqs.\ (\ref{rot1}) and (\ref{rot2}) can be simultaneously true is
if the condition $\exp (2\pi L i/N)=(-1)^{N-1}$ is fulfilled. This leads to the following sequence of magic angular
momenta,
\begin{equation}
L_m = k N; \;\;\; k=0,\pm 1, \pm 2, \pm 3, \ldots,
\label{mag1}
\end{equation}
for $N$ odd, and
\begin{equation}
L_m = (k + \frac{1}{2}) N; \;\;\; k=0, \pm 1,\pm 2,\pm 3, \ldots,
\label{mag2}
\end{equation}
for $N$ even.

Because a permanent is symmetric under the interchange of two rows, the corresponding magic $L_m$'s for
spinless bosons are given by the sequence in Eq.\ (\ref{mag1}) for both odd and even numbers of localized bosons.

Regarding the numerical aspects, the fact that $\Phi^{\rm PROJ}_L$ is zero for non-magic $L$ values results in the
\textcolor{black}{vanishing}
(within machine precision) of the normalization factor $\int_0^{2\pi} n(\gamma)e^{i\gamma L} d\gamma$
in Eq.\ (\ref{eprj}). As a result only the physically meaningful energies associated with
magic angular momenta are given in Table \ref{t1}, Table \ref{t2}, and Table \ref{t3} of Appendix \ref{appa}.

\textcolor{black}{
We stress that the properties and physics associated with magic-angular-momentum yrast states are well known in the
literature of 2D quantum dots \cite{yann07,ruan95,maks96,seki96,maks00}.
Of immediate relevance to this paper is the enhanced energy stabilization that they acquire
in their neighborhood (thus characterized often as ``cusp'' states) in the regime of strong interactions (i.e., for
large $R_W$ or $R_\delta$). This energy stabilization can be explicitly seen in Fig.\ 15 of Ref.\ \cite{yann07}, where
the triplet state correponds to the fully polarized case for two electrons with magic angular momenta
$L_m=(2k+1)$, $k=0,\pm 1, \pm 2, \ldots$. The fact that large energy gaps do develop between the
magic-angular-momentum rotational yrast states and the other (excited) states is also well established in the QSTC
literature; for the case of ultracold ions on a ring, see Refs.\ \cite{li12,robi15}, and for the case
of the bosonic lump, see Ref.\ \cite{wilc12}.}

In this paper, we consider fully polarized fermions only, that is cases when $S=S_z=N/2$, where $S$ is the
total spin and $S_z$ is its projection. Consideration with our methodology of the other spin values $S_z < N/2$
is straightforward; it requires, however, restoration of both the total spin ${\bf S}^2$
and the total angular momentum. An explicit example for $N=3$ fermions is discussed in Ref.\ \cite{yann03}.

In addition to the magic angular momenta,
the properties of the original crystalline structure
\textcolor{black}{built-in}
in $\Psi^{\rm SB}$ are reflected in the high-degree of
rigidity exhibited by the symmetry-preserving $\Phi^{\rm PROJ}_L$. As demonstrated previously, the SPD of
$\Phi^{\rm PROJ}_L$ is uniform, but the CPD of
$\Phi^{\rm PROJ}_L$ reveals the now hidden crystalline structure of $N$ strongly repelling  particles on the ring.
The rigidity of $\Phi^{\rm PROJ}_L$ is manifested in that the CPDs have the same $N$-hump shape and are
independent of the actual value of the magic angular momenta, as well as of the fermion or boson statistics and
of whether the number $N$ of fermions is odd or even.
This rigidity is a consequence of the strong two-body interaction and cannot be found in many-body
wave functions associated with weak interparticle interactions.

\subsection{Initial wave packets and associated time evolution}

The focus of this paper is the construction of a symmetry-preserving wave function $\Phi^{\rm PROJ}_L$ associated with
the finite-crystal symmetry-broken determinant (or permanent) $\Psi^{\rm SB}$. However, it is instructive to inquire
about the reverse process, that is how to represent the symmetry-broken crystal as a superposition in the
complete basis formed by the symmetry-preserving $\Phi^{\rm PROJ}_L$'s. Indeed one can write the expansion
\begin{equation}
\Psi^{\rm SB} = \sum_L {\cal C}_L \Phi^{\rm PROJ}_L,
\label{sbe}
\end{equation}
where [using Eq.\ (\ref{prj})] the expansion coefficients are given by
\begin{equation}
{\cal C}_L = \frac{1}{2\pi} \int_0^{2\pi} d\gamma e^{-i\gamma L} n(\gamma),
\label{sbc}
\end{equation}
and the norm overlap $n(\gamma)$ was defined in Eq.\ (\ref{nov}). Of course the index $L$ runs over the
appropriate sequence of magic angular momenta as discussed in Section \ref{symm}.

Eqs.\ (\ref{sbe}) and (\ref{sbc}) illustrate the fact that with respect to the exact many-body (linear)
Schr\"{o}dinger equation the symmetry-broken-crystal wave
function $\Psi^{\rm SB}$ is a wave packet and not a stationary eigenstate. This is also in general true for all
SB mean-field solutions, whether they are solutions of the unresticted Hartree-Fock equations in the case of confined
electrons (e.g., in quantum dots \cite{yann07}), or they are the familiar solitonic solutions of the
Gross-Pitaevskii equations in onedimensional bosonic systems. Due to the rapid experimental control,
the latter cases are currently attracting a
lot of attention. Indeed motivated by experiments that suggest the need to go beyond-mean field dynamics, the number
of related theoretical investigations has burgeoned \cite{degu12,kavo17,cede11,carr16,schm17}.

These theoretical studies investigate how an initial state approximating a solution of the nonlinear
Gross-Pitaevskii equation evolves in time under the
exact many-body Hamiltonian. For both the cases of dark \cite{degu12,kavo17} (a hole in matter density)
and bright \cite{cede11,carr16,schm17} solitons (an excess in matter density, like the case of the lump considered
in this paper), these studies are finding a ``universal'' behavior of dispersion in space, decay in time, and
time revival
\textcolor{black}{at the initial position}
of the soliton. This behavior can be understood by taking into consideration the expansion in Eq.\ (\ref{sbe}).
In fact, each energy mode in
\textcolor{black}{Eq.\ (\ref{sbe})}
will evolve in time according to its own phase $\exp(-iE_L t/\hbar)$,
and the interaction between all of them results in a decay-type behavior. Moreover, the initial occupation
amplitudes ${\cal C}_L^2$ of the different modes are unequal,
\textcolor{black}{and as a result}
probability flows from the
higher occupied modes to those with lower initial occupations, which leads to a dispersive behavior.
However, because the system is finite, there exists a Poincar\'{e} period, and the system will eventually
experience a revival \cite{bocc57,schu78}.

For achieving a QSTC, we propose here a different initial wave packet, i.e., a two-mode one with equal weights, as
specified in Eq.\ (\ref{pwm}). As explicitly demonstrated through numerical claculations, such a two-mode initial wave
packet preserves at all times, and without damping, the spatial and temporal periodicities
expected from the classical finite crystal. We note that the consideration of two-mode Schr\"{o}dinger-cat
states is a key element in the theory of the discrete time crystal, where the focus is to enable a sloshing
behavior between these specialized paired states by minimizing interactions to the rest of the system. In fact,
our symmetry-preserving trial functions $\Phi^{\rm PROJ}_L$ [see Eq.\ (\ref{prj})] can be viewed as a more
complex class of Schr\"{o}dinger-cat states. This analogy is straightforward for the case of the mirror superposition
used in Ref.\ \cite{sach15}, which can be reproduced from expression (\ref{prj}) as a limiting case by using
only two angles $\gamma_1=0$ and $\gamma_2=\pi$.

\subsection{Relation to configuration-interaction (CI) wave functions}

As mentioned previously, the symmetry-preserving trial functions $\Phi^{\rm PROJ}_L$ are identically zero
for values of $L$ different from the magic angular momenta $L_m$; see section \ref{symm}. Naturally the
exact many-body spectrum has a plethora of additional states with good $L$, which however cannot be reached with the
approach in this paper. Indeed this approach is tuned to extracting from the complete spectrum only the ground
states that correspond to non-vibrating classical finite crystal arrangements. The remaining many-body states can be
reached by using the CI approach, which is in principle an exact methodology when converged; the CI is often
referred to as exact diagonalization (EXD). The CI approach is computationally expensive, but comparisons between
the symmetry-restored trial functions and the CI wave functions have been used by us to demonstrate the numerical
accuracy of the symmetry-restored wave functions, as well as to clarify their special place in the whole spectrum,
namely that for particular magnetic-field ranges they can become the global ground state, as is the case with
the Aharonov-Bohm spectrum in Fig.\ \ref{fspec} of the present paper. Higher-in-energy CI
solutions with different $L$ (and also with $L=L_m$) do incorporate vibrational and other type of internal
excitations, and as a result a superposition of two random CI states with good $L_1$ and $L_2$ will not necessarily
exhibit the crystalline single-particle-density structure of exactly $N$ humps.

Systematic comparisons between symmetry-restored states and CI wave functions have been carried out by us
previously for the case of a few electrons confined in parabolic quantum dots. Although the external confining
potential and particle species in parabolic QDs are different from the case of the ring traps considered here, the
symmetry properties of the many-body wave functions are universal. Thus the analysis presented in our previous QD
studies can be used to gain further insights to the results for the QSTC presented in section \ref{sbsp}.
In particular, Fig.\ 6 and Fig.\ 7 of Ref.\ \cite{yann11} offer an explicit illlustration of the fact
that $N$-humped crystalline SPD structures arise only when both $L_1$ and $L_2$ coincide with magic angular
momenta and the associated CI wave functions correspond to global ground states in some range of magnetic
fields (see Fig.\ 5 and section III in Ref.\ \cite{yann11}).

In our previous studies of QDs, excellent agreement was found between the total energies of symmetry-restored trial
functions and the corresponding CI energies for both the cases with or without an applied magnetic field, as
testified by the many reported direct numerical comparisons. We mention here a few specific examples, i.e.,
Table III and IV in Ref.\ \cite{yann03}, Table IV in Ref.\ \cite{yann03.2}, and Fig.\ 4 in Ref.\ \cite{yann09}.

Such systematic numerical comparisons between symmetry-restored and CI wave functions for the case of
ring-trapped ultracold ions and neutral bosons are outside the scope of the present paper. However, they will
be reported in subsequent publications \cite{yannpub}, including the case away from the quasi-1D regime
(when the dependence on the ring width becomes important).

\section{Conclusions}
\label{concl}

The discussion \cite{wilc13,li13,wata15}, motivated by the criticism \cite{brun13,brun13.2,nozi13,wata15} of the
original \cite{wilc12,li12} QSTC proposals (which were based on ground states), spurred speculations about
non-equilibrium low-lying states as possible instruments for describing QSTCs.
For $N$ rotating particles on a ring, and using the theory of
symmetry breaking and symmetry restoration via projection techniques \cite{yann07}, this
paper succeeded in explicitly uncovering the existence of low-lying states with QSTC behavior, by introducing
beyond-MF appropriate trial many-body wave functions (see Fig.\ \ref{fdens}). Along with its conceptual and
methodological significance, we propose to focus experimental attention on selected applied magnetic field values
where the Aharonov-Bohm-type spectra corresponding to different magic angular momenta are most susceptible to mixing
(Fig.\ \ref{fspec}), resulting in rotating pinned-Wigner-molecule many-body states found here to exhibit QSTC
behavior. This constructive platform fills an apparent gap in the quest for ultracold ring-confined ions or
neutral-atom QSTCs.

We recall that although the original proposals for the quantum space-time crystal \cite{wilc12,li12} suggested
realization of the concept through the use of ultracold few ring-trapped particles, this is yet to be achieved
experimentally. Nevertheless, for the case of ultracold ions, several publications have reported significant progress
in controlling aspects of a quantum rotor on a ring. In particular, the ability
to generate and control symmetry-breaking through pinning of the rotating ion crystal has been demonstrated
by using up to 15 $^{40}$Ca$^+$ ions in a ring with a microfabricated
silicon surface Paul trap \cite{haef17}, or 3 $^{40}$Ca$^+$ ions in a 2D ring-type configuration in a linear
Paul trap \cite{nogu14}. To fully inplement and control the QSTC trial functions presented in this paper, the ions
need to be cooled
\textcolor{black}{down to near}
the ground states. In this respect, Ref.\ \cite{haef17} has achieved temperatures
$\sim$ 3 mK (for a trap with a radius of $\sim$ 60 $\mu$m), while Ref.\ \cite{nogu14} reported temperatures
in the nanometer range (for an effective ring radius in the 6 to 8
\textcolor{black}{micrometer range).}
It is expected that cooling techniques
and procedures will be further optimized and will be successful in the near future in producing near-ground-state
temperatures, as is exemplified by a very recent publication \cite{che17}.

An essential requirement, met by our theory, is that it is imperative that the proposed beyond-mean-field many-body
trial wave functions (i.e., beyond the UHF or GP treatments) for predicting proper quantum space-time-crystal
behavior of particles moving on a ring will be based on solutions to the
interacting particles Schr\"{o}dinger equation that posses good angular momenta, as well as exhibit (hidden)
ordering that reflects an underlying finite crystalline symmetry. This is achieved in our theory
through the first two construction stages, namely, the unrestricted Hartree-Fock solution followed by an
angular momentum projection, yielding the function $\Phi^{\rm PROJ}_L$ [Eq.\ (\ref{prj})]. It is then proposed by us
that these projected and stationary many-body wave functions are susceptible to mixing,
\textcolor{black}{see Eq.\ (\ref{pwm}),}
favored to occur in the vicinity of crossings of Aharonov-Bohm-type spectra of ground-state energies {\it vs.\/}
applied magnetic field (see circles in Fig.\ \ref{fspec}). This mixing results in non-stationary low-lying states that,
when evolved with the many-body Hamiltonian, yield undamped and non-dispersing periodic oscillations in both space and
time.

\begin{acknowledgments}
Work supported by the Air Force Office of Scientic Research under Award No. FA9550-15-1-0519.
Calculations were carried out at the GATECH Center for Computational Materials Science.
\end{acknowledgments}

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