Hi Penny, I just found your reviews of my books. Thank you so much! I'm glad you enjoyed them. Thanks for spreading the word!
It's a cute one, Erik. Hope your library has it.
Oh this one is on my list. I LOVE your activity. Have a great week.
I think the activity would be appealing to kids. Heck, I think I would love making a puddle map! You have a great week, too!
Aw! This book looks so cute, Penny! And I love the language – it would be so fun to read aloud! Thanks so much for sharing this one!
Yes, perfect read-a-loud and one to study for fun language.
How adorable is that cover! A must have for every little kid as they love puddles!
Putting this one on my to-read list!
Hope you enjoy it, Buffy. I think you'll find it charming!
Pugs, pigs and puddles, Oh Yes!
Pretty perfect, huh? Thanks for stopping by Joanna.
Normally I shy away from alliterated characters, but Percy is adorably illustrated and the excerpt left me desperate for more. Thanks, Penny.
That's so funny that you shy away from alliterated characters…I am drawn to them. That's one of the many reasons I like PPBF…there is such a nice variety of reviewers and I learn about books I probably wouldn't have picked up if it weren't for my PPBF pals. Hope you enjoy the book, Joanne.
Love it! "deeper-than-a-doggy puddles" Hilarious! I'd some rumblings about this book. Now I've got to get it. Thanks!
That is one of the cutest lines, although there are a slew of cute lines in this book! Hope you enjoy reading it.
I see pugs everywhere (not at all like seeing dead people) but never would have thought of this untapped market and fun story! Since I like Kim's other books, this is on my have-to-read-it list.
Ha! You're hilarious, Wendy! Isn't Kim great? Such a talented writer. Hope you find this one soon.
This sounds adorable. I will definitely check it out. Thanks for the review.
I do find it adorable, Rosi. I hope you found it. I'm of the opinion that it's one of those that everyone will enjoy.
Pug and pigs? Punascious mother? Oh, yeah! Putting it on hold now. Thanks, Penny!
Oh goodie! I'm glad you were able to put it on hold! Hope you liked it.
I'm so glad you found it at the library and loved it, too. I thought you would 😉 And the resolution is great!
What a very entertaining story with great rhyming and a wonderful plot. Kids will really have fun with this story. You know how to pick them!
Thank you, Pat. I do read a lot of books before deciding one is "worthy" of PPBF. I imagine you do the same before you pick one because you know how to pick them too 🙂 That's probably my favorite thing about PPBF….I get to read a lot of PBs in order to make my choice.
Penny, This sounds wonderfully cute. What kid wouldn't love a book about a pug who likes to splash around in puddles? Great review!
I agree. I substitute in kindergarten quite a bit and I can see them wanting to hear this one over and over again.
This sounds like my kind of book! Thanks for sharing!
Oh dear – a pig pug conflict? This sounds cute. Like you, I just love IF IT"S SNOWY…!
If you love IF IT'S SNOWY, Cathy, I'm betting you'll be taken by this one, too.


The discord between Turkey and Greece has grown deeper and wider over time, over a series of seemingly vital issues, which have at times brought the two countries to the brink of war. Yet in 1999 the two countries opened a dialogue on non-sensitive issues such as trade, the environment and tourism.
The causes of the current rapprochement progress are explored in this book in relation both to the international environment which is increasingly conducive to this progress, and the significant domestic changes that both Greece and Turkey have experienced since the end of the Cold War. This book confronts each of these important dimensions by addressing issues of continuity and change in Greek-Turkish relations.
Mustafa Aydin is Associate Professor of International Relations, Ankara University. Besides his authored books, he is editor of the Turkish Yearbook of International Relations, and the Review of International Affairs; and is currently co-editing two volumes on different aspects of Turkish foreign policy.
Kostas Ifantis is Assistant Professor of International Relations at the University of Athens; and Research Fellow at the Hellenic Foundation for European and Foreign Policy. His publications include: Theory and Reform in the European Union, and NATO and the New Security Paradigm: Power, Strategy, Order and the Transatlantic Link.


text:
Our keynote, feature articles, and conscious conversations
Global unity is our evolutionary destiny. This is what we believe at Kosmos, and why we continue.
By Joni Carley and Daniel Perell
Our Keynote authors represent the Coalition for Global Citizenship (CGC2030). The Coalition, based at the United Nations Headquarters in New York, advocates that governments, public and private sectors, and civil society, all work from a basis of values of just and fair treatment, inclusion and cooperation.
By Kurt Johnson
Of course, the question was—after a century and a half of "Social Darwinism" claiming that it was all about competition, from the lowest to the highest—could our world now dominated by the shark-tank rubrics of business, economics, and politics adjust to this astounding news? Could it realize that because of this basic blunder about Darwin's message, we had, in fact, ended up with dystopia instead of utopia?
By Wahinkpe Topa (Four Arrows), aka Donald Trent Jacobs
...witnessing brave communities who have managed to hold onto Indigenous language and ways, and continue their advocacy for the Earth, is important. We can appreciate the Indigenous experience unfolding today as a model for what is happening to all of us.
By Andrew Petrisin
Westphalian Sovereignty was created to end the Thirty Years' War, but is it adaptable enough to underpin our current geopolitical system? It might be time for an update.
By Michael Foley
Farmers in urban civilizations have always been subject to powers beyond them. Indeed, there is ample evidence that urban civilizations were invariably built on the conquest and subjection of farming cultures. Our current food industry grew out of the defeat of farmers' efforts in the late nineteenth century to win fair prices for their production.
By Richard Barrett
The largest impediment to improving the well-being of humanity is not in finding the funding to implement the Sustainable Development Goals; the largest impediment is the unwillingness of the part of the heads of state, senior government politicians, and business leaders to embrace higher order human values.
By David Bollier
I believe the commons paradigm can help us develop a new social and cultural vision, and new strategies for practical change. Paradoxically enough, redirecting our attention away from conventional politics and policy may offer the most promising possibilities for developing a transformational vision.
By Judy Rodgers, Gayatri Naraine, Rhonda Fabian
Judy Rodgers | ...I think decisions about things like governance or institutions source from our awareness. We can't build something if it's outside the scope of our awareness. We can't design a true commons if we're in a very limited consciousness.
By Skye Fitzgerald
Kosmos Journal | In 2015, your team produced 50 Feet from Syria, a film focused on the civilian impact of the Syrian conflict. Your new film, Lifeboat, bears witness to refugees desperate enough to risk their lives in rubber boats leaving Libya. What was different for you personally about making those two films?
By Wendell Berry and Allen White
How do we cultivate a sense of place in an industrialized, globalizing world? Writer and farmer Wendell Berry discusses the role of agrarian values in nurturing communitarian consciousness with Tellus Senior Fellow Allen White. *** Allen White | You have [...]

redpajama_set_name: RedPajamaCommonCrawl


Hypnotic, beautifully written, this mesmerizing novel explores the corrosive effect of evil – and how painful psychological truths long buried within a family can corrupt the present and, through courage and understanding, lead to healing and renewal.
Hypnotic, beautifully written, this mesmerizing novel by "an extremely gifted writer" (San Francisco Chronicle) explores the corrosive effect of evil–and how painful psychological truths long buried within a family can corrupt the present and, through courage and understanding, lead to healing and renewal. "Like Scheherezade in the grip of a fever dream, Kathryn Harrison . . . has written one of those rare books, in language of unparalleled beauty, that affirm the holiness of life," said Shirley Ann Grau, about Poison. And the same can be said about Envy.
Will leans out of the driver's-side window toward his wife. "It's not too late to change your mind," he says.
Her dark glasses show him the houses on their side of the block, greatly reduced and warped by the convexity of each lens. The fancy wrought-iron bars on their neighbor's windows, the bright plastic backboard of the Little Tikes basketball hoop one door down, the white climbing rose, suddenly and profusely in bloom, on the trellis by their own mailbox: it's as if he were studying one of those jewel-like miniatures painted in Persia during the sixteenth century; the longer Will looks, the more tiny details he finds.
"Did you remember to bring pictures?" Carole asks.
He points to an envelope on the seat beside him. "I mentioned the pool at the hotel?"
Envy was a disappointment after The Binding Chair. Call me middle-aged but there are just so many subjects that I find more interesting than other people's sexual fantasies. However, I can see that Envy would be of appeal to many. As always, we provide an excerpt so that you can browse the book for yourself and decide if it's right for you. (Reviewed by BookBrowse Review Team).
Writers and critics who complain of the shrinking audience for literary fiction argue -- validly -- that huge advances and advertising budgets for blockbusters reduce the resources left to promote serious novels. But the chances of good literary fiction finding an audience are also damaged when books such as Kathryn Harrison's Envy are published and passed off as worthy. Ten pages of Envy are enough to make you yearn for the juiciest trash novel you can find; 50 will have you dreaming of box-top recipes, road maps, computer instructions -- anything with a purpose.
Harrison's measured, matter-of-fact prose gives each perverse twist of her pulpish plot a nasty kick, taking readers into the heart of Will's deep sadness and out the other side. An unsparing examination of the turbulent depths beneath anunsuspecting hero's most unexceptionable-seeming fantasies, and a life patently too normal to be true.
Starred Review. Will's profession as an analyst seems too convenient - allowing Harrison to analyze her own novel through the voice of her main character - but this is a pardonable flaw in a book so juicy and intelligent.
Harrison's dialogue is electrifying, the sophistication of her psychology is mesmerizing, and her characters, so astutely drawn, are bewitching.
Brilliant, lucid language, considered plot, satisfying ending.


"We continued working on math concepts for the placement exam, including linear and quadratic equations, exponents, fractions, etc. "
"Today the student and I reviewed activity 6.2 and a part of activity 6.3, which covered ratios and rate measurements. We also discussed graph plotting, slope calculations and rotations on graph."
"Today I worked with two students. Student #1 worked very quickly and completed the entire assigned packet. Student #2 is still working on the packet but did a great job. Both students performed extremely well in terms of behavior and content."
"I worked with the student on his subtraction facts for 12's today. We worked with the fact families and I also had him see the relationships. He did an excellent job with his 11 facts from last time and completed his assessment paper, getting 100%. I assigned for him to practice his flash cards and complete a practice paper working with his 12 subtraction facts. He did an awesome job today!"
"Today we mostly focused on the student completing his missing homework and his review for an upcoming test. I recommended that he should keep a planner and write all of his assignments down and make a habit of checking it everyday. His father agreed, and said they were making an effort to encourage him to utilize his agenda. Other than that, the student did well when it came to solving problems."


We would like to express our heartfelt sympathy to all the victims and their families affected by the Tohoku Pacific Offshore Earthquake and the resulting Tsunami.
For supporting the sufferers the Japanese Homoeopathic Foundation has set up an account for donation.
The full amount of your donation will be devoted for relief efforts through homoeopathy and reconstruction assistance of the affected area.
We appreciate your understanding that a bank transfer fee will be required at your end.
through homoeopathy and reconstruction assistance of the affected area.
Please note that it is not refundable once you made the donation.
the costs incurred managing and administering our relief activities.
Please contact us if you have any question about the donation.


ROME, Sept. 17 - The two faces of Italy - one that is increasingly secular and the other that is still deeply influenced by the church - collided this week in a noisy electoral debate that left one thing clear: Italy, it can safely be said, will not legalize gay marriage any time soon.
The possibility was, in fact, never on the table. But Romano Prodi, the leader of the center-left who is fighting to regain the job of prime minister, ruled it out anyway, after coming under attack by the church and conservative politicians when he said he would support certain rights for unmarried couples, including gay ones.
"I never talked about marriage or adoption," he told reporters this week. "There is a will to misconstrue my thinking." Mr. Prodi said that, as a Roman Catholic with several relatives who were priests, the entire fight caused him "very much suffering."
The fight began last weekend when Mr. Prodi wrote a letter to a gay-rights group saying that, if he won Italy's general elections next spring, he would support granting equal rights to unmarried couples.
Italy does not recognize common-law unions, a position that causes complications recognized on the right and left as the number of unmarried couples has tripled, by one estimate, in the last 25 years to more than half a million.
Mr. Prodi's proposal would allow for rights of inheritance, for visiting a sick partner, making medical decisions and extension of private medical insurance.
But, for critics, the proposal would also imply a de facto recognition of homosexual unions, a delicate topic to the church, particularly since Spain, another overwhelmingly Catholic country, legalized gay marriage this summer.
The official Vatican newspaper, L'Osservatore Romano, criticized Mr. Prodi, saying he had acted in "search of votes" in an effort to "relativize and ideologize the reality of the family." Avvenire, the newspaper reflecting the views of Italian bishops, warned Mr. Prodi, "Hands off the family."
Many commentators in the Italian media drew parallels between the church's quick and strong reaction this week with another fight on morality this year, in which the Vatican intervened with a well-organized campaign to defeat a June referendum that sought to overturn a law that imposes strict rules on assisted fertility.
Sandro Magister, an author and specialist on the Vatican for the magazine L'Espresso, said he did not think the Vatican had organized any major "intervention" against Mr. Prodi's proposals. But he said the strong reaction showed a church ready, as in the referendum fight, to inject itself into politics where principle was concerned. He said he detected a slightly more "intense" willingness to do so under the new pope, Benedict XVI, who spoke publicly against the referendum.
"For the hierarchy of the church, the important thing is to defend marriage between men and women as the only form of marriage," Mr. Magister said, explaining that the church feared Mr. Prodi's proposals could create a de facto form of marriage.
Mr. Prodi strongly denied that was the case, saying that a secular state had the obligation to extend basic protections to all its citizens, gay or not.
Sergio Lo Giudice, the president of Arcigay, the gay rights group Mr. Prodi had written his letter to, said he also did not see the law as creating legal gay unions and said his group would continue to push for gay marriage. Nonetheless, in a nation where homosexuality remains relatively discreet, he said Mr. Prodi's proposals were a "victory" for gay people, although only last week he was criticizing Mr. Prodi for his silence on common-law unions. "This is the first time that gays and lesbians will be represented by someone who could win the election," he said. Gay people, he said, "feel more like citizens."
As in the fight over the fertility referendum, some cracks, perhaps surprising ones, have occurred in the usual party lines. At least one cardinal, though a relatively liberal one, sided with Mr. Prodi publicly.


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2023 CISAA/COSSA/LOSSA Championships
COSSA, LOSSA and CISAA secondary schools from across the province will compete for the opportunity to represent their school at the OFSAA Nordic Skiing Championship, 22-24 Feb at Lakefield College School.


The employees forms you've built thus far in Design view and Layout view look fairly plain. They use default fonts and a background color that's inherited from the color you have defined in Windows for three-dimensional (3-D) objects (sometimes called the Button Face color). In this section, you'll learn about additional enhancements you can make to your form's design. To follow along with the rest of this chapter, you can use either the form you created in Design view, the form you created in Layout view, or one of the example forms saved in the database. We used the form we created using Design view for the remainder of this chapter. Open your form in Design view to continue with the next examples.
Access 2007 comes with two drawing tools, the Line tool and the Rectangle tool, that you can use to enhance the appearance of your forms. You can add lines to separate parts of your form visually. Rectangles are useful for surrounding and setting off a group of controls on a form.
On your employees form, it might be helpful to add a line to separate the primary information about the employee in the first column from personal information in the second column. To make sufficient room for the line, you should move the controls in the first column to the left. The easiest way to do this is to switch to Design view, select all the affected controls and labels, and then move them as a group. Start by clicking the top ruler just above the right edge of the controls, and then drag inside the ruler toward the left until the selection indicator touches all the controls in the left column. (If you can't see the rulers, be sure that the Ruler button is selected in the Show/Hide group on the Arrange tab.) Release the mouse button, and all the controls and labels in the left column will be selected. To be sure you move all these controls as a group, click the Group button in the Control Layout group on the Arrange tab. Access shows you that the controls are now grouped by placing a rectangular line around all the controls. Rest the mouse pointer on the square in the upper-left corner of the group so that the pointer changes to a pointer with a crosshairs shape (see Figure 12–34), and slide the entire group left a bit.
Figure 12–34: You can group a set of controls and then move them together.
Next, on the Design tab, in the Controls group, click the Line tool. To draw your line, click near the top of the form between the two columns, about two grid rows below the top edge, and drag toward the form's bottom edge. If the line isn't exactly vertical, you can drag the bottom end left or right to adjust it. You can also set its Width property to 0, in the property sheet to make it perfectly vertical. (As you might imagine, setting the Height property to 0, makes the line horizontal.) Use the Line Thickness button in the Controls group on the Design tab to make the line a little thicker if you want. (Or, change the Border Width property in the property sheet.) Click the button, and choose the thickness you want. Your form should now look similar to the one shown in Figure 12–35.
Figure 12–35: Use the Line tool to draw a line on a form; use the Border Thickness button to adjust the line width.
When drawing a line on your form, you can make your line exactly horizontal or exactly vertical if you hold down the Shift key as you click and draw the line.
You can add emphasis to the form by drawing a rectangle around all the controls. To do this, you might first need to move all the controls down and to the right a bit and make the Detail section slightly wider and taller. First, expand your form by about 0.5 inch across and down. Click the Select All button in the Controls group on the Design tab, and then drag all the controls so that you have about 0.25 inch of space around all the edges. (This might seem like too much space, but we'll use the extra space to have some fun later.) Select the Rectangle tool in the Controls group, click where you want to place one corner of the rectangle, and drag to the intended location of the opposite corner. When you draw a rectangle around all the controls, your form will look similar to the one shown in Figure 12–36.
Figure 12–36: Use the Rectangle tool to place a rectangle with a default etched look on the employees form.
Note that the rectangle control actually covers and is on top of all the other controls. However, because the default rectangle is transparent with an etched special effect, you can see the other controls through the rectangle. (You might need to click the form or another control so that you can see the etched look of the rectangle.) If you prefer a solid rectangle, you can select the rectangle control and then use the Fill/Back Color button in the Font group on the Design tab to select the color you want. (A light gray will work best.) When you add a solid control like this after you've created other controls, the solid control will cover the previous controls. You can select the control and click Send To Back in the Position group on the Arrange tab to reveal the covered controls and keep the solid control in the background.
Go ahead and make the rectangle a solid light gray and send it to the back. Now switch to Form view, and see how your form looks up to this point. Your employees form should look similar to the one shown in Figure 12–37.
Figure 12–37: The employees form in Form view has a line and a solid rectangle added.
You can also use color and special effects to highlight objects on your form. For example, you can make all the controls appear to "float" on a raised surface on the form. To do so, switch to Design view, and select the rectangle you just created. Click the arrow on the Special Effect button in the Controls group on the Design tab, and then click Special Effect: Raised to change the rectangle from Etched to Raised. Your form in Form view will look similar to the one shown in Figure 12–38.
Figure 12–38: The rectangle behind the controls appears raised above the surface of the form background.
Next, switch to Design view if you changed to Form view, select the Rectangle tool again, and set Back Color to dark gray and Special Effect to Sunken using the buttons in the Font and Controls groups. Draw a second rectangle so that it forms a border about halfway between the edge of the first rectangle and the edge of the grid. Click Send To Back in the Position group on the Arrange tab to send this latest rectangle to the background. Switch to Form view to see the result. The first gray rectangle now appears to float on the form, surrounded by a "moat" of dark gray, as shown in Figure 12–39.
Figure 12–39: The first light gray rectangle appears to float on the form using special effects.
You can find this form saved as frmXmplEmployee4 in the sample database.
Although you can certainly pick from a broad palette of colors for any design object, you might want to design your forms so that they always inherit colors from the options the user has set in the Windows Display dialog box. In fact, if you select the Detail section of the employees form you've been building, open the Property Sheet window, and find the Back Color property, you'll find the property set to System Button Face. This special option tells Access to use the color set in Windows for button faces and other 3-D objects. In any color property, Access lists 31 system color names that you can use to set colors in your forms and controls to match those set in Windows objects. All these color names begin with the word System.
If you select any of the colors under Access Theme Colors, Access enters the name of the theme or color option on the property line. If you click one of the colors under Standard Colors, Access sets a hexadecimal value in the property that represents the red, green, and blue (RGB) color value-#000000 for Black, #FF0000 for Red, #00FF00 for Green, #0000FF for Blue, and #FFFFFF for White, for example. See Article 5, "Color Names and Codes," on the companion CD for a list of common color hexadecimal equivalents.
Under Recent Colors on the color palette, Access displays up to 10 recently used colors. Click one of these to enter the code or name for that color in the color property. If you want to create your own custom color, you can click More Colors to open the Colors dialog box. In the Colors dialog box, you can choose from a wider selection of colors on the Standard tab or define a custom color on the Custom tab by adjusting individual RGB values or selecting the color you want from a color rainbow palette.
Another way you can enhance the appearance of your forms is by varying the fonts and font sizes you use. When you select any control that can display text or data, Access makes font, font size, and font attribute controls available in the Font group on the Design tab so that you can easily change how the text in that control looks. Click the arrow next to the Font Name combo box to open a list of all the available fonts, as shown in Figure 12–40. Select the font you want for the control.
Figure 12–40: This is a partial list of fonts available to you in the Font Name combo box.
The font list shows all fonts currently installed on your computer. Use the Fonts folder in Windows Control Panel to add or remove fonts. A double-T icon next to the font name in the list indicates a TrueType font that is suitable for both screen display and printing. A printer icon next to the font name indicates a font designed for your printer but that might not look exactly the same when displayed on your screen. A font with no icon indicates a font designed for your screen; a screen font might look different when you print it.
If you want to add some variety, you can use bold or italic type in a few key places. In this case, select all the labels on the form, and select a serif font such as Times New Roman.
You can add a label to the header of the form to display a title such as Employees. You can either do this manually or use a button on the Ribbon designed specifically for this task. If you'd like Access to do most of the work, click the Title button in the Controls group on the Design tab. Access opens the header and footer section, places a new label in the header section, and sets the caption to the name of your form. Double-click the new label to select the existing text, type Employees in the label, and press Enter.
To do this same task manually, you need to first open the header and footer of the form by clicking the Form Header/Footer button in the Show/Hide group on the Arrange tab. Grab the bottom edge of the footer, and close it up so that it has zero height. Expand the header to give yourself some room to work. Choose the Label tool in the Controls group on the Design tab, draw a label about 1.5 inches wide and 0.5 inch high, type the word Employees in the label, and press Enter.
Set the label in the header to the Tahoma font (a sans serif font), bold, italic, and 18 points in size. Double-click one of the sizing boxes to size the control to fit, and drag the right edge to the right if all the letters don't show in the label. You can see a portion of this work under way in Figure 12–41.
Figure 12–41: Adjust the size of the label to fit the form header title.
Copy the label you just created to the Clipboard, and paste it in the header.
Change the font color of the pasted label to white, and then click the Send To Back button in the Position group on the Arrange tab.
Turn off the Snap To Grid command if it is on in the Control Layout group on the Arrange tab, and use the arrow keys to move the white label so that it is slightly lower and to the right of the first label.
Click the Form Header bar to select that section, and set the background color to light gray to provide some contrast.
Figure 12–42: The employees form now has a title and some different fonts for variety.
A form with too many fonts or font sizes will look busy and jumbled. In general, you should use only two or three fonts per form. Use one font and font size for most bound data displayed in controls. Make label text bold or colored for emphasis. Select a second font for controls in the headers and perhaps a third (at most) for information in the footers.


If you've done all of this and you still have not received your refund yet, please contact us at mim@mirelle.co.za.
We only replace items if they are defective or damaged. If you need to exchange it for the same item, send us an email at mim@mirelle.co.za and send your item to: Mirelle Enterprises (Pty) Ltd, 7 Cleevewood Park, 3 Camp Road, Gillitts, KwaZulu-Natal, 3610, South Africa.
You'll be responsible for paying for your own shipping costs for returning your item. Shipping costs are non-refundable. If you receive a refund, the cost of return shipping will be deducted from your refund.


text:
Mindfulness Characters
This is a great place to get a custom Mindfulness Calligraphy Wall Scroll!
niàn
념
念 is the simplest way to write "mindfulness" in Chinese, Japanese Kanji, and old Korean Hanja.
念 can be defined these ways: To read; to study (a degree course); to read aloud; to miss somebody (keeping them in your mind); idea; remembrance; sense; thought; feeling; desire; concern; attention; recollection; memory; to think on/about; reflect; repeat, intone; a moment.
Obviously, the context in which the character is used determines which definition or meaning is perceived. As a single character, it's open and perhaps ambiguous. Thus, it can be read with any or all of these meanings.
念 is used in a Buddhist context (often written as 正念 or "right mindfulness") with similar meanings of thought and contemplation.
In Japanese, this character is sometimes used as the name "Nen."
See Also: Buddhism | Enlightenment
7. Right Mindfulness / Right Memory / Perfect Mindfulness
Samyak Smriti / Samyak Smrti / Samma Sati
zhèng niàn
sei nen
정념
正念 is one of the Noble Eightfold Paths of Buddhism. Right Mindfulness, along with Right Effort and Right Concentration, constitute the path to Concentration or Perfect Thought.
Right Mindfulness is about remaining focused on one's body, feelings, mind, and mental qualities. It's also about being ardent, aware, and mindful, and supposes that you've already put aside worldly desire and aversion.
Monk Bhikkhu Bodhi described this as "The mind is deliberately kept at the level of bare attention, a detached observation of what is happening within us and around us in the present moment." When practicing right mindfulness, the mind is trained to remain in the present, open, quiet, and alert, contemplating the present event.
Another definition: Ongoing mindfulness of body, feelings, thinking, and objects of thought.
This term is exclusively used by devout Buddhists. It is not a common term, and is remains an unknown concept to most Japanese and Chinese people.
See Also: Buddhism | Enlightenment | Noble Eightfold Path
jué xīn
kesshin
결심
決心 is a Chinese, Japanese, and Korean word that holds the dictionary definition of determination but literally means determined heart.
The first character means to determine or determined.
The second character means heart, mind, or soul, so you can imagine that this form of determination partially means putting your heart into something. It can also be translated as resolve, resolution, or decision (as in a decision made and followed).
See Also: Devotion | Tenacious | Passion | Dedication | Will-Power
Devotion / Dedication / Attentive / Focused
zhuān xīn
sen shin
전심
專心 makes a word that means "paying attention with your heart."
It's often translated as "dedication," as in "be absorbed in" or "concentrate one's efforts." It's also used to mean "with the single mind," "whole-heartedly," "paying attention," "undivided attention," "concentration (-ed)," "engrossed," "devotionally (listening/watching)," and/or "attentive."
The first character means "for a particular person, occasion, or purpose," "focused on one single thing," "concentrated," and sometimes, "special."
The second character means "heart" or "mind" by itself.
My favorite translation, which comes from the Oxford Advanced Chinese/English Dictionary, is, "wholehearted devotion."
If it seems like the meaning of this word is quite open, you are correct. The context in which the word is used matters a lot. It can mean different things depending on how you use it. This makes it kind of nice as you can decide what this means to you (within some limits). This is always positive in meaning, so even if a Chinese person reads it differently than you, it will still have a good meaning.
In Japanese, they tend to use a variation of the second character which has one less stroke. If you want your calligraphy written this Japanese form, please click on the Kanji shown to the right instead of the button above. Note: Japanese and Chinese people will recognize either form.
See Also: Faith | Devotion
Heart / Mind / Spirit
xīn
심
心 would often be translated as "heart."
However, because it was believed in Chinese culture for thousands of years that your consciousness and thoughts came from the big red organ in the middle of your chest, it also means "mind" or "spirit" and sometimes even "soul."
In Korean, beyond heart, mind, and spirit, this character can mean moral, nature, mind, affections, intentions, core, and center. In fact, it is used in Chinese to mean "center" as well but only with another character in front of it. For instance, "medical center" or even "shopping center." Separately and alone, it will not be read with that "center" meaning unless thought of as "the center of your soul."
Mindfulness 念 nen niàn / nian4 / nian nien
7. Right Mindfulness
Right Memory
Perfect Mindfulness 正念 sei nen / seinen zhèng niàn
zheng4 nian4
zheng nian
zhengnian cheng nien
chengnien
Determination 決心
决心 kesshin / keshin jué xīn / jue2 xin1 / jue xin / juexin chüeh hsin / chüehhsin
Focused 專心 / 専心 / 耑心
专心 sen shin / senshin zhuān xīn
zhuan1 xin1
zhuan xin
zhuanxin chuan hsin
chuanhsin
Spirit 心 kokoro xīn / xin1 / xin hsin


text:
\section{Introduction}
\label{intro}

The recent progress achieved by quantum simulators in carefully
controlling the dynamics of an increasing number of qubits has enticed
further theoretical studies on the coherent time evolution of quantum
correlations and energy exchanges of finite-size quantum systems (see,
e.g., Refs.~\cite{RV-21, Dziarmaga-10, PSSV-11, GAN-14, NC-book}).
On the one hand, a deeper understanding of the decoherence dynamics and
spreading of entanglement correlations is of fundamental importance,
both for quantum-information purposes and for the improvement of
energy conversion in complex networks~\cite{NC-book, LCCLCN-13}.  On
the other hand, the study of the energy storage among distinguished
parts of a quantum system, exploring different geometric schemes, is
relevant for quantum-thermodynamical purposes~\cite{GHRRS-16, VA-16},
as well as for the efficiency optimization of recently developed
quantum batteries~\cite{CPV-18}. In experiments with ultracold atoms,
such quantities can be also used to monitor the time evolution of a
finite amount of interacting qubits~\cite{SFSTT-20, BDZ-08}.

However, in any experimental setup, one is interested in bringing to
light the properties of the system under study, carefully minimizing
the perturbations ascribed to the devices employed to probe it.  As
the superposition principle drives disjoint parts of the same network
to be entangled during the time evolution, a collective framework in
which the system and the probing apparatus are treated on the same
footing is often necessary to correctly describe the evolution of the
overall setup~\cite{Zurek-03}.  In this regime, it is interesting to
find out how much information we may retrieve by considering the
coherent collective dynamics of the system and the probing apparatus
as a whole, but only looking at the quantum devices used to probe the
system under analysis.
Among the various schemes of composite systems that can be useful to
address this kind of issues, we mention the so-called {\em central
  spin} models, which have attracted a great deal of attention in the
recent years, both theoretically and experimentally (see,
e.g., Refs.~\cite{Zurek-82, HB-04, CPZ-05, QSLZS-06, RCGMF-07, CFP-07,
  BS-07, CP-08, Zurek-09, DQZ-11, NDD-12, SND-16, V-18, FCV-19, RV-19,
  ASSSCD-19, VCPPC-20, LSSWY-21, AM-21, NSC-21, RV-21}).

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.8\columnwidth, clip]{quenchsunburst.pdf}
  \caption{Skecth of the setup studied in this paper.  For times $t<0$
    (left side), the system is initialized and left in the
    ground-state configuration where ancillary spins ($\mathcal{A}$,
    red dots) belonging to the probing apparatus are initially
    decoupled from the Ising ring qubits ($\mathcal{I}$, blue dots).
    At time $t=0$, the ring and the external spins are suddenly
    coupled together (thick red lines) by quenching the Hamiltonian
    parameter $\kappa$ governing the $\mathcal{I}$-$\mathcal{A}$
    interaction, thus unitarily driving the global system out of
    equilibrium (right side).  In the figure, we set $L=16$, $n=8$,
    and $b=2$, see text.}
  \label{fig_quenchsunburst}
\end{figure}

In this work we focus on an alternative lattice configuration, in the
class of the {\em sunburst} quantum models (for a sketch see
Fig.~\ref{fig_quenchsunburst}), in which many independent spins are
longitudinally coupled to a quantum Ising ring.  Specifically, we
consider a set of $L$ spin-1/2 quantum objects (qubits) disposed in a
ring geometry (blue dots), interacting through nearest-neighbor
Ising-like interactions.  We probe the system by means of $n$
equally-spaced qubits (being $b$ their interspace distance),
completely independent of each other, to which we will refer in the
following as the {\em ancillary} spins or simply the {\em external}
spins (red dots).  A thorough analysis of the equilibrium ground-state
properties of this apparatus has been recently performed in
Ref.~\cite{FRV-22}.  Here we are interested in investigating the
out-of-equilibrium dynamics triggered by a sudden quench of an
external parameter $\kappa$, which couples the system under study with
the monitoring devices.  When the monitoring devices act as a small
perturbation, leading to {\em soft} quenches, the out-of-equilibrium
quantum evolution is studied within dynamic finite-size scaling
frameworks, which turn out to be particularly effective to infer
dynamic scaling laws for finite-size systems~\cite{PRV-18, RV-21}.  In
particular, two different dynamic FSS limits can be defined at large
sizes, either by keeping the number $n$ of external qubits fixed or by
maintaining their interspace distance $b$ constant.  We have carried
out a careful FSS study of both limits, when the ring system is close
to either continuous or first-order quantum transitions (CQTs or
FOQTs, respectively) of the Ising ring. Moreover, we have considered
quenches in the Ising-disordered phase.  In any case, the dependence
of observables on $n$ can be always taken into account by a proper
rescaling of the parameter $\kappa$ by a $\sqrt{n}$ prefactor.  We
also discuss the many-body dynamics arising from {\em hard} quenches,
when the ancillary system introduces a relatively large amount of
energy with respect to the energy differences of the lowest levels of
the Ising ring. Finally, we address the role of a nearest-neighbor
coupling between the external qubits.

The paper is organized as follows. In Sec.~\ref{sec_sunburst_model} we
describe the setup of the model under analysis: the lattice
Hamiltonian, the quench protocol, and the observables addressed during
time evolution. In the next two sections we put forward a dynamic FSS
framework that holds for systems close to a CQT (Sec.~\ref{sec_CQT})
and to a FOQT (Sec.~\ref{sec_FOQT}), and support our theory with
numerical results based on exact diagonalization.  In
Sec.~\ref{sec_disordered_phase} we show some results at fixed $b$ in
the disordered phase.  In the last part of this work, we focus on hard
quenches, comparing our results with the ones
obtained when consecutive monitoring qubits also interact among
themselves (Sec.~\ref{sec_fixed_ring}).  Finally, our conclusions are
drawn in Sec.~\ref{sec_summary}.

\section{System setup}
\label{sec_sunburst_model}

\subsection{The lattice Hamiltonian}

We are going to address some properties of the unitary
out-of-equilibrium quantum dynamics of an Ising ring coupled to a set
of external qubits along the longitudinal axis, which is ruled by the
following global lattice Hamiltonian:
\begin{equation}
  \hat{H} = \hat{H}_{\mathcal I} + \hat{H}_{\mathcal A} + \hat{H}_{\mathcal IA}\,.
  \label{hamiltonian}
\end{equation}
The first term $\hat{H}_{\mathcal I}$ describes a standard
ferromagnetic Ising ring (${\mathcal I}$) in a transverse field, with
$L$ sites,
\begin{equation}
  \hat{H}_{\mathcal I} = -J\sum_{i=1}^L
  \hat{\sigma}^{(1)}_i \hat{\sigma}^{(1)}_{i+1}
  - g \sum_{i=1}^{L}\hat{\sigma}^{(3)}_i\,,
  \label{label_H_I}
\end{equation}
where $\hat{\sigma}^{(k)}_i$ (with $k=1,2,3$) denote the spin-1/2
Pauli matrices on the $i$th the lattice site, while
$\hat{\sigma}^{(k)}_{L+1}=\hat{\sigma}^{(k)}_1$, since we are
considering periodic boundary conditions (PBC).  In the following we
set $\hbar = 1$, $J=1$ as the energy scale, $a=1$ as the lattice
spacing, and take $g>0$ without loss of generality.

The Ising model is a paradigmatic quantum lattice system where
fundamental issues related to the physics of quantum many-body
networks can be thoroughly investigated, given our extended knowledge
of its quantum phases and correlations, both in the thermodynamic and
FSS limit~\cite{Sachdev-book, RV-21}.  The ground state exhibits a
zero-temperature CQT at $g=g_{\mathcal I}$ belonging to the
$(d+1)$-dimensional Ising universality class, separating a
ferromagnetic phase ($g<g_{\mathcal I}$) from a disordered phase
($g>g_{\mathcal I}$).  In the one-dimensional case, the quantum phase
transition is located at $g_{\mathcal I}=1$.  Close to the quantum
critical point (QCP), the system develops long-range correlations
$\xi$, diverging as $\xi\sim\abs{g - g_{\mathcal I}}^{-\nu}$, where
$\nu=y_g^{-1}=1$ and $y_g$ is the renormalization group (RG) exponent
associated with the difference $g-g_{\mathcal I}$.  The gap between
the ground-state energy and the first excited state
$\Delta_{\mathcal I}$ is instead suppressed as $\Delta_{\mathcal I}\sim\xi^{-z}$
close to the CQT (here $z$ denotes the dynamic critical exponent, $z=1$).
Consequently, for finite-size systems the gap is power-law suppressed
as $\Delta_{\mathcal I}(g=g_{\mathcal I}, L)=\pi/(2L)+O(L^{-2})$
at the QCP, whereas it vanishes exponentially at a FOQT when the
boundary conditions do not favor any of the phases
$\Delta_{\mathcal I}(g<g_{\mathcal I}, L)\approx e^{-c(g) \, L}$, as in the
case of PBC~\cite{CNPV-14, PRV-20}.

The second term $\hat{H}_{\mathcal A}$ in Eq.~(\ref{hamiltonian})
characterizes the energy of $n$ independent ancillary ($\mathcal{A}$)
spin-1/2 systems (qubits), acting as quantum monitoring devices for
the ring system under analysis,
\begin{equation}
  \hat{H}_{\mathcal A} = - \frac{\delta}{2}
  \sum_{i=1}^{n} \hat{\Sigma}^{(3)}_{i}\,,
  \label{hamiltonian_HA}
\end{equation}
where $\hat{\Sigma}^{(k)}_i$s  (with $k=1,2,3$)
still represent Pauli matrices on the
$i$th monitoring spin.  The coupling $\delta$ parameterizes the energy
variation defined by the flip of a single qubit, which is the
elementary excitation within the ensemble of the $n$ independent
qubits.

The last term $\hat{H}_{\mathcal IA}$ couples together the ring system
with the ancillary qubits along the longitudinal direction. In our
analysis, the number $n$ of equally-spaced spins is always
commensurate with the length $L$ of the Ising ring, so that each pair
of consecutive qubits is separated by a fixed distance $b=L/n$. It
reads as follows
\begin{equation}
  \hat{H}_{\mathcal IA} = - \kappa \sum_{i=1}^{n}
  \hat{\sigma}^{(1)}_{x=ib} \, \hat{\Sigma}^{(1)}_{i}\,,
  \label{hamiltonian_pairing}
\end{equation}
where $\kappa$ parameterizes the intensity of the coupling.

The model is characterized by a $\mathbb{Z}_2$ symmetry under
a global spin flip of both the ring and the ancillary spins,
identified by the parity operator
\begin{equation}
  \hat{P} = \bigg( \bigotimes_{i=1}^L\hat{\sigma}_i^{(3)}\bigg)
  \bigg( \bigotimes_{j=1}^n\hat{\Sigma}^{(3)}_j \bigg),
\end{equation}
which acts on the Pauli operators as
\begin{subequations}
  \begin{eqnarray}
  \hat{P} \, \hat{\sigma}^{(1)}_i \! & \to -\hat{\sigma}^{(1)}_i, \qquad \qquad
  \hat{P} \, \hat{\Sigma}^{(1)}_i \! & \to -\hat{\Sigma}^{(1)}_i \,,\\
  \hat{P} \, \hat{\sigma}^{(2)}_i \! & \to -\hat{\sigma}^{(2)}_i, \qquad \qquad
  \hat{P} \, \hat{\Sigma}^{(2)}_i \! & \to -\hat{\Sigma}^{(2)}_i \,,\\
  \hat{P} \, \hat{\sigma}^{(3)}_i \! & \to  \hat{\sigma}^{(3)}_i,
  \qquad \qquad \ \ \,
  \hat{P} \, \hat{\Sigma}^{(3)}_i \! & \to  \hat{\Sigma}^{(3)}_i\,.
\end{eqnarray}
\end{subequations}
We point out that the global Hamiltonian~(\ref{hamiltonian}) is left
invariant under a change in the sign of $\delta$ and/or $\kappa$,
after a proper redefinition of the Pauli matrices. Thus, without loss
of generality, in our analysis we will restrict ourselves to the cases
$\delta, \kappa \geq0$.

As mentioned in the introduction, the overall setup realizes the same
lattice configuration as the sunburst quantum Ising model, whose
equilibrium ground-state properties have been addressed in
Ref.~\cite{FRV-22}.  In a time-dependent perspective, the equilibrium
limit corresponds to an adiabatically slow dynamics for finite-size
systems.
In contrast, here we are going to focus on the out-of-equilibrium
decoherence properties and energy exchanges from the point of view of
the ancillary spins, aiming at perceiving the capability of these
probes to retrieve information on the Ising ring system under study,
and in particular its equilibrium quantum phases before the quench.
To this purpose, we consider two different dynamic FSS limits, depending
on the parameters we decide to maintain fixed with increasing the
lattice size $L$ of the ring~\cite{FRV-22}:
\begin{enumerate}
\item We keep the number $n=L/b$ of ancillary spins fixed while
  increasing $L$, thus the spins probing the ring get diluted in the
  FSS limit.  The bulk properties of the Ising ring system (such as
  the location of the QCP or the equilibrium phase diagram) are left
  unchanged by the presence of the probing devices.

\item We keep $b$ fixed, so that the number of external qubits
  increases linearly with the ring size $L$.  In this case, the qubits
  are expected to modify the bulk properties of the Ising ring (for
  example, the location of the ferromagnet-to-paramagnet QCP).
\end{enumerate}

\subsection{The quench protocol}
\label{sec_quench_protocol}

We now outline the sudden-quench protocol considered in the present
work (see Fig.~\ref{fig_quenchsunburst}). The Ising ring and the
ancillary spins are initially decoupled, i.e., with reference to
Eq.~(\ref{hamiltonian}), we start with $\kappa=0$ and choose the
initial state $\ket{\Psi_0}$ as the ground state of $\hat{H}_{\mathcal
  I}+\hat{H}_{\mathcal A}$:
\begin{equation}
  \ket{\Psi_0} = \ket{\psi_0} \otimes
  \bigg( \bigotimes_{i=1}^n\ket{\uparrow}_i \! \bigg)\,,
  \label{eq_groundstate_before_quench}
\end{equation}
where $\ket{\psi_0}$ is the ground state of the Ising ring and
$\otimes_{i=1}^n\ket{\uparrow}_i$ is the vector in the Hilbert space
$\mathcal{H}_{\mathcal A} = (\mathbb{C}^2)^n$ with all the $n$
auxiliary qubits pointing upward along the $(3)$-direction. At $t=0$
the system is suddenly driven out of equilibrium by quenching the
parameter $\kappa$ to a finite value: $0 \rightarrow \kappa$, such
that $\ket{\Psi_0}$ is no longer a Hamiltonian eigenstate.  Thus for
$t>0$ the global system evolves nontrivially in real time, according
to the usual Schr{\"o}dinger equation,
\begin{equation}
  \ket{\Psi(t)} = e^{-i\hat{H}t}\ket{\Psi_0}\,,
  \label{eq_schroedinger}
\end{equation}
where $\hat{H}$ is the total Hamiltonian reported in
Eq.~(\ref{hamiltonian}).

In the following discussion, we distinguish between two types of
sudden quench: a {\em soft} quench is related to a tiny change of the
parameter $\kappa$ (decreasing with $L$), so that the system stays
close to a quantum transition and thus excites only critical
low-energy modes. In contrast, a {\em hard} quench is not limited by
the above condition and typically involves the injection into the
system of an extensive amount of energy, in such a way that
also {\em high-energy} excitations are involved.

\subsection{Decoherence function and energy exchanges}

To characterize the scaling properties of the lattice model after the
quench, in the different phases and regimes considered, we focus on
the coherence and the energetic properties exhibited by the $n$
ancillary spins (subsystem $\mathcal{A}$).

First of all, we define the decoherence function $D_{\mathcal A}$
as~\cite{RV-21}
\begin{equation}
  D_{\mathcal A}(t) = 1 -
  \Tr_{\mathcal A}\big[ \hat{\rho}_{\mathcal A}^2(t)\big]\,,
  \label{def_decoherence}
\end{equation}
where $\hat{\rho}_{\mathcal A}\equiv\Tr_{\mathcal
  I}\big[\ket{\Psi}\bra{\Psi}\big]$ is the reduced density matrix of
the external qubits and $\Tr_{\mathcal{I}/\mathcal{A}}$ represents,
respectively, the trace over the Hilbert space of the ring system
($\mathcal{I}$) or that of the probing spins ($\mathcal{A}$).  This
observable quantifies the coherence properties of $\mathcal{A}$ and is
intimately related to the bipartite entanglement shared by the two
distinguished subsystems $\mathcal{I}$ (blue dots in
Fig.~\ref{fig_quenchsunburst}) and $\mathcal{A}$ (red dots in
Fig.~\ref{fig_quenchsunburst})~\footnote{More precisely, the
decoherence function $D$ is strictly connected with the second-order
R{\'e}nyi entropy, $S^{(2)}_{\mathcal A}(t) = - \log \, \Tr_{\mathcal
  A} \big[ \hat{\rho}_{\mathcal A}^2(t)\big]$, which is an
entanglement monotone.}.  Namely, it ranges from $D_{\mathcal A}=0$,
for a pure state, to $D_{\mathcal A} \simeq 1$, for a maximally
entangled state (in the latter case, deviations from the unit value
are due to finite Hilbert spaces $\mathcal{H}_{\mathcal A}$).  By
means of the Schmidt decomposition, it is easy to prove that, when the
global state $\ket{\Psi}$ is pure (equivalently, it can be expressed
in terms of a single wave function), the decoherence of the system is
equal to that of the $n$ external spins, i.e. $D_{\mathcal A} =
D_{\mathcal I}\equiv D$.  Here we deal with global pure states, being
our global system isolated, therefore from now on we will drop the
subscript ${}_{\mathcal A}$ (or ${}_{\mathcal I}$).

To address the energy-related and quantum thermodynamic properties of
the half-integer monitoring spins, we first define the work statistics
distribution $P(W)$ associated with the quantum quench presented in
Sec.~\ref{sec_quench_protocol}~\cite{TLH-07, CHT-11, RV-21}
\begin{equation}
  P(W) = \sum_{m, n} \delta \big[ W - (E_{f, n} - E_{i, m}) \big]
  \, \abs{\braket{n_f}{m_i}}^2 \, p_{i, m}\,,
\end{equation}
where $\{\ket{m_{i}},\ket{n_{f}}\}$ and $\{E_{i, m}, E_{f, n}\}$ are
the eigenstates and the eigenvalues of, respectively, the starting and
the final Hamiltonian (before and after the quench), and
$p_{i,m}=e^{-\beta E_{i,m}}/Z$ is the probability of finding the
initial state in the eigenstate $\ket{m_i}$ of the beginning
Hamiltonian, with $\beta\equiv1/T$ the inverse temperature and $Z={\rm
  Tr} \big[ e^{-\beta \hat H} \big]$ the partition function. At $T=0$
we have $p_{i,0}=1$ and $p_{i,\alpha}=0, \ \forall \alpha>0$.  The
momenta of the work distribution are defined by the integral
\begin{equation}
  \expval{W^m} \equiv  \int \dd{W} P(W) \, W^m\,.
\end{equation}
In particular, the first two momenta $\expval{W}, \expval{W^2}$ can be
easily addressed by evaluating a static expectation value on the
ground state (without the complete knowledge of the whole spectrum),
as they have an extremely simple expression in terms of
$\hat{H}_{\mathcal IA}$:
\begin{equation}
  \expval{W^m} = \mel{\Psi_0}{\hat{H}_{\mathcal IA}^m}{\Psi_0}\,,
  \quad \text{if} \ m=1, 2\,.
  \label{eq_momentum_W}
\end{equation}
The above relations do not hold for higher momenta of the
distribution, with $m>2$.  However, given the ground state
$\ket{\Psi_0}$ of Eq.~(\ref{eq_groundstate_before_quench}), one can
easily obtain two simple exact relations for the first two work
momenta done on the whole system at the quantum quench:
\begin{equation}
  \expval{W}=0\,,\qquad \expval{W^2}=n\kappa^2\,.
  \label{workav}
\end{equation}

After quenching at $t=0$, the energy of the global sunburst system is
conserved along the evolution for $t>0$. However, we have significant
exchange of energy between the two subsystems. We define the
energy-exchange distribution (i.e. the quantum heat) of the qubits
associated with two different energy measurements of the subsystem
$\mathcal{A}$, the first one at $t=0$ and the next one at a generic
time $t>0$~\cite{CHT-11, RV-21},
\begin{equation}
  P_{\mathcal A}(U, t) \equiv \sum_{n, \alpha}
  \delta\big[U - (E_{\mathcal{A}_\alpha}-E_{\mathcal{A}_0})\big] \,
  \abs{\!\mel{n, \alpha}{e^{-i\hat{H}t}}{\Psi_0}}^2\,,
  \label{def_heat_distribution}
\end{equation}
where $n$ and $\alpha$ represent, respectively, the energy levels of
the ring system and the ones associated with the external spins before
the quench.  In particular, a simple relation holds for the first
momentum $\expval{U}$ of the distribution, i.e.
\begin{equation}
  \expval{U}(t) = \mel{\Psi(t)}{\hat{H}_{\mathcal A}}{\Psi(t)}
  - \mel{\Psi_0}{\hat{H}_{\mathcal A}}{\Psi_0}\,.
  \label{def_prob_distribution_U}
\end{equation}
We remark that the above observable quantifies the average energy
stored into the set of $n$ half-integer spins, as soon as the
$\mathcal{I}$ and $\mathcal{A}$ subsystems are decoupled at a given
time $t$.  With this respect, the quantum quench protocol we address
can be seen, in a different perspective, as the refilling protocol of
$n$ discharged quantum batteries connected to a charger Ising ring
system.

\section{Quantum quenches at the CQT}
\label{sec_CQT}

\subsection{Finite number of qubits in the FSS regime}
\label{CQT_finite_n_FSSregime}

In this section, we present the dynamic FSS framework derived for soft
quenches: at $t = 0$, the system is suddenly driven out of equilibrium
by switching on the coupling $\kappa$, such that the global system
remains close to a QCP ($g\approx g_{\mathcal I}$) and the pairing
Hamiltonian $\hat{H}_{\mathcal IA}$ acts as a small perturbation of
$\hat{H}_{\mathcal I}+\hat{H}_{\mathcal A}$.  We begin our analysis by
keeping the number $n$ of external qubits fixed in the large-$L$ limit
[FSS limit (i)].

As a starting point, we recap some recent FSS results
put forward for the equilibrium ground-state properties for the
sunburst quantum Ising model~\cite{FRV-22}, in which $\kappa$ is
different from zero and does not depend on time, and then introduce
our scaling ansatz required to characterize the various dynamic
regimes, basing our arguments on FSS and RG arguments.
The natural scaling variable associated with the transverse field
strength $g$ is defined analogously as in the standard quantum Ising
model close to the CQT, therefore it reads:
\begin{subequations}
  \label{def_WAK}
\begin{equation}
  G = (g - g_{\mathcal I}) \, L^{y_g}, \qquad y_g=1\,,
  \label{def_W}
\end{equation}
where $y_g=1/\nu$ is its RG dimension ($\nu=1$ in one dimension).  On
the basis of energy scales considerations, a common ansatz to provide
us with another working scaling variable is given by the ratio of the
two relevant low-energy scales $\delta$, which characterizes the
energy gap of the ancillary qubits close to the critical point, and
$\Delta_{\mathcal I}(g_{\mathcal I}, L)$, which represents the
finite-size gap of the Ising ring at the QCP.  Therefore we consider
the scaling variable
\begin{equation}
  A = \frac{\delta}{\Delta_{\mathcal I}(g_{\mathcal I},L)}
  \sim \delta \, L^z, \qquad z=1\,,
  \label{def_A}
\end{equation}
where $z$ is the dynamic critical exponent.  The scaling variable
associated with the quench parameter $\kappa$ is expected to play a
role similar to that of a quench generated by Ising symmetry-breaking
defects $\sum_{i=1}^n\hat{D}_i=-\kappa
\sum_{i=1}^n\hat{\sigma}_{x=ib}^{(1)}$ at a CQT. For a finite number
$n$ of external spins, the RG dimension of the Hamiltonian parameter
$\kappa$ at the Ising fixed point is given by
$y_\kappa=(2+z-d-\eta)/2=7/8$~\cite{FRV-22a}. Therefore, we define the
corresponding scaling variable as~\cite{FRV-22}
\begin{equation}
  K = \kappa \, L^{y_\kappa}, \qquad y_\kappa=7/8\,.
  \label{def_K}
\end{equation}
\end{subequations}

Then, the FSS limit is obtained by taking the large-size limit ($L \to
\infty$), while keeping constant the values of the above mentioned
scaling variables $G, A$, and $K$ in Eqs.~(\ref{def_WAK}), therefore
requiring $\delta \sim L^{-z}$ and $\kappa \sim L^{-y_\kappa}$.  For
instance, a natural ansatz for the scaling behavior of the equilibrium
ground-state decoherence function $D= 1- {\rm Tr}_{\mathcal A} \, [
  \hat \rho^2_{\mathcal A}]$ in the FSS limit and close to the CQT of
the Ising ring is given by the scaling equation~\cite{FRV-22}
\begin{equation}
  D(n, g, \delta, \kappa, L) \approx \mathcal{D}(n, G, A, K)\,,
  \label{FSS_static_decoherence}
\end{equation}
with $D(n,g,\delta,0,L)=\mathcal{D}(n, G, A, 0) = 0$.  Analogous
scaling ansatzes can be put forward for other quantities and
observables, such as the gap of the global system, the two-point
correlation function of the order parameter, as well as its
susceptibility, the correlation length, etc \ldots~\cite{FRV-22}.  These
have been carefully verified by means of extensive numerical
simulations for finite-size systems in which all the Hamiltonian
parameters $g$, $\delta$, and $\kappa$ have been properly rescaled
with $L$ according to the above definitions of scaling
variables~\cite{FRV-22}.

We now generalize the above picture to a dynamic scenario, induced by
a sudden switch on of the coupling between the ancillary and Ising
subsystems, from zero to a nonzero value $\kappa$. We consider a soft
quench, in such a way that the coupling parameter $\kappa$ after the
quench decreases with $L$, keeping the scaling variables $K$ fixed.
In general, dynamic behaviors exhibiting nontrivial time dependences
are expected to require another scaling variable associated with the
time variable $t$, defined as~\cite{RV-21}
\begin{equation}
  \Theta = t \, \Delta_{\mathcal I}(g_{\mathcal I},L) \sim t \, L^{-z},
  \qquad z=1\,.
  \label{def_Theta_scaling_variable}
\end{equation}
The FSS limit for $L \to +\infty$ is obtained by keeping constant $G,
A, K$ in Eqs.~(\ref{def_WAK}), and also the time variable $\Theta$ in
Eq.~(\ref{def_Theta_scaling_variable}).  Therefore, in this
circumstance, we conjecture that observables should generally satisfy
universal relations, depending on their scaling dimensions and
arguments.
For instance, a natural ansatz for the scaling behavior of the
decoherence function $D$ of Eq.~(\ref{def_decoherence}), in the FSS
limit, is given by the scaling equation
\begin{equation}
  D(n, g, \delta, \kappa, t, L) \approx \mathcal{D}(n, G, A, K,
  \Theta)\,,
  \label{FSS_decoherence}
\end{equation}
which generalizes Eq.~(\ref{FSS_static_decoherence}) to the dynamic
case, by simply adding a further dependence on the time scaling
variable $\Theta$.  We conjecture an analogous scaling ansatz for the
average energy $\expval{U}$ stored into the auxiliary qubits
[cf.~Eq.~(\ref{def_prob_distribution_U})]:
\begin{equation}
  \expval{U}(n, g, \delta, \kappa, t, L)
  \approx L^{-z} \mathcal{E}(n, G, A, K,\Theta)\,,
  \label{FSS_energy}
\end{equation}
assuming that further analytical contributions are
suppressed~\cite{RV-21}.

As we shall see, the approach to the asymptotic dynamic FSS when
keeping $n$ finite is generally characterized by leading $O(L^{-7/4})$
corrections.  They arise from the analytic expansion of the non-linear
scaling field associated with the quench parameter $\kappa$, i.e.,
$u_\kappa\approx \kappa + c\kappa^3+...$ (where the quadratic term
vanishes, due to the symmetric properties of the Ising ring parameter
$\kappa$), whose third-order term gives rise to corrections decaying
as $O(L^{-7/4})$~\cite{FRV-22}. We also mention that also subleading
$O(L^{-2})$ corrections are generally present in one-dimensional
systems with PBC, while systems with boundaries are generally subject
to $O(L^{-1})$ corrections~\cite{CPV-14}.

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.95\columnwidth, clip]{CQTscaling.pdf}
  \caption{Analysis of the decoherence function $D$ (top panels) and
    of the rescaled energy stored in the external qubits $\expval{U}
    /\Delta_{\mathcal I}$ (bottom panels) in the FSS limit keeping the
    number $n$ of external spins fixed.  Left panels: the quantities
    versus the rescaled time variable $\Theta$.  The different curves
    are obtained for various Ising-ring lengths up to $L=16$ (and
    fixed $n=2$, $G=0$, $A=1$, $K=3$), starting from the initial state
    $\ket{\Psi_0}$ of Eq.~(\ref{eq_groundstate_before_quench}).  Right
    panels: the corresponding scaling corrections for fixed
    $\Theta=0.6$ and $2.3$, respectively) are shown to be consistent
    with a decay $\sim L^{-7/4}$ (straight lines are drawn to guide to
    the eye).}
  \label{fig_CQT}
\end{figure}

To verify the above dynamic FSS relations~(\ref{FSS_decoherence})
and~(\ref{FSS_energy}) at the CQT, in Fig.~\ref{fig_CQT} we present
some numerical results obtained by means of exact diagonalization
and $4^{\text{th}}$-order
Runge-Kutta techniques to find the coherent time evolution by
integrating the Schr{\"o}dinger equation in
Eq.~(\ref{eq_schroedinger})~\footnote{We always used a discrete time
step of the order of $\delta t= (1 \divisionsymbol 3) \times 10^{-3}$
We have checked that the interval $\delta t$ is sufficiently small to
allow us to neglect systematic errors arising from the discretization
of the temporal evolution.}.  As visible from the nice data collapse
of the various curves obtained for different ring lengths $L$, the
presented results are in agreement with our dynamic FSS theory.

\subsection{FSS in the large-$n$ limit}
\label{sec_largen}

It is also interesting to note that, for sufficiently large $n$, the
dependence on the number of ancillary spins $n$ can be absorbed into a
redefinition of the quench parameter $K$, for sufficiently large $n$:
\begin{equation}
  K^\prime = \sqrt{n} \, K\,.
  \label{def_sqrtn_K}
\end{equation}
Note the large-$n$ limit considered here should be performed after the
$L\to\infty$ limit.  In particular, the quantum model under
investigation exhibits appreciable hints of a $\sqrt{n}$-law already
for relatively small values of $n \sim 3 \divisionsymbol 5$, as
reported in Fig.~\ref{fig_sqrtn_CQT} for the decoherence function $D$.
The asymptotic regime appears to be rapidly approached, with $O(1/n)$
scaling corrections.  The square-root dependence on $n$ of the quench
parameter $K^\prime$ turns out to be the result of the collective
(independent) behavior of the ancillary spins, which are indirectly
coupled only through the Ising ring. Other observables we have
obtained by means of exact diagonalization techniques, as the ratio
$\expval{U}/\Delta_{\mathcal I}(g_{\mathcal I}, L)$, obey the same
large-$n$ FSS relation introduced for $K^\prime$ (not shown).

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.95\columnwidth, clip]{CQTmanyn.pdf}
  \caption{Top panel: the decoherence function $D$ versus $\Theta$,
    for several values of $n$ at the largest lattice sizes at our
    disposal, for fixed $G=0$, $A=1$, $K^\prime=0.5$.  We observe data
    collapse with increasing $n$, within scaling corrections that get
    asymptotically suppressed as $n^{-1}$, as shown for $\Theta=1$
    (bottom left panel) and $\Theta = 1.5$ (bottom right panel). We
    emphasize that each point in the bottom panels is obtained, for
    fixed $n$, by means of large-size extrapolations ($L \to \infty$,
    at fixed $n$), assuming $\sim L^{-7/4}$ scaling
    corrections. Straight lines are drawn to guide to the eye.}
  \label{fig_sqrtn_CQT}
\end{figure}

\subsection{Qubits at fixed distance $b$ in the FSS regime}
\label{sec_bfixed}

We carry on our discussion by considering the case of soft quenches in
which the external spins are placed at fixed distance $b$ in proximity
of a CQT, thus in the large-$n$ limit [FSS limit (ii)].  As working
hypothesis, we follow the reasoning for the equilibrium
scenario~\cite{FRV-22} and suppose the theory put forward in the
previous section for large $n$ [in particular, the $\sqrt{n}$-law of
  Eq.~(\ref{def_sqrtn_K})] to survive also in this different FSS
limit.  The most natural guess for the scaling variable associated
with the quench strength $\kappa$ is given in terms of the new scaling
variable $\widetilde{K}$, obtained by replacing $\sqrt{n}\to \sqrt{L/b}$
in the definition of $K'$, cf. Eq.~(\ref{def_sqrtn_K}), i.e.
\begin{equation}
  \widetilde{K} \equiv \sqrt{n} \, K = \kappa \, \frac{L^{11/8}}{b^{1/2}}\,.
  \label{def_widetilde_K}
\end{equation}

According to this hypothesis, the previous dynamic FSS relations,
cf. Eqs.~(\ref{FSS_decoherence}) and~(\ref{FSS_energy}), should be
modified by removing the $n$ dependence in the corresponding scaling
functions and substituting $K$ with $\widetilde{K}$.  For example, for
the decoherence function $D$, we conjecture that
\begin{equation}
  D(b, g, \delta, \kappa, t, L)
  \approx \mathcal{D}(A, \widetilde{K},\Theta)\,.
  \label{FSS_decoherence_2}
\end{equation}
Analogous relations are expected to hold also for other
observables, as for the energy stored into the monitoring spins
$\expval{U}$.
Note that the dependence of $D$ on $n$ (which is an extensive energy
by definition) is completely encapsulated in the attenuation of the
quench intensity $K$ (being $\widetilde{K}$ fixed in the FSS regime)
by a factor $\sqrt{n}\sim \sqrt{L}$.

The above conjecture has been verified numerically in
Fig.~\ref{fig_CQT_b_scaling}, for a fixed distance $b=3$ and for $G=0,
A=1$, and $\widetilde{K}=0.5$.  In this figure the decoherence
function $D$, together with the ratio $\expval{U}/\Delta_{\mathcal
  I}$, show a nice scaling in terms of the variable $\Theta$, thus
confirming our hypothesis.  We mention that results for $b=3$ and $4$
are consistent with each other, within scaling corrections that decay
roughly as $\sim L^{-1}$. Such scaling corrections may be related to
the $O(1/n)$ corrections observed in the large-$n$ limit considered in
Sec.~\ref{sec_largen}, since $n\sim L$ at fixed $b$. To leave the
figure easily readable, in the left panels we report the time behavior
only for the $b=3$ data, however it is worth mentioning that the
scaling of the $b=4$ data with increasing lattice sizes is very
similar to that shown for $b=3$, at all times $\Theta<4$.

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.95\columnwidth, clip]{CQTbscaling.pdf}
  \caption{Scaling of the decoherence function $D$ and the
    RG-invariant ratio $\expval{U}/\Delta_{\mathcal I}$ versus
    $\Theta$ (top and bottom left panels, respectively), for fixed
    $b=3$ and $G=0$, $A=1$, $\widetilde{K} = 0.5$.  Right panels: the
    corresponding data for $b=3$ and $b=4$ (not shown on the left), at
    fixed $\Theta=3$ and $2$ respectively, are consistent with each
    other, within scaling corrections suppressed as $\sim
    L^{-1}$. Straight lines are drawn to guide the eye.}
  \label{fig_CQT_b_scaling}
\end{figure}

\subsection{Qubits at fixed distance $b$ and fixed $\delta$}
\label{sec_CQT_finite_delta}

We now consider the dynamics of the quantum Ising ring model, with
many independent spins attached, in the hard-quench limit: at $t = 0$,
the system is suddenly driven out of equilibrium by switching on the
coupling $\kappa$ from zero to a fixed finite value, in such a way
that an extensive amount of energy is injected.  We fix the spacing
distance parameter $b$ and the excitation energy $\delta$, examining
different ring lengths $L$. Note also that when $\delta>0$ the energy
scale of the ancillary system is much larger than the gap of the
critical Ising ring. In this case, the dynamic FSS frameworks
presented in the previous section do not hold anymore, since the hard
quench drives the system out of the critical domain. Therefore,
the resulting dynamics should be little sensitive to the
value $g$ of the parameter controlling the Ising criticality.

In Fig.~\ref{fig_manyn_fixed_delta} we show results for fixed
$g=\delta=1$ and for a quench intensity $\kappa:0\to 0.5$. Differently
from the FSS regime analyzed in the previous section, the energy
stored per unit qubit is $\sim O(1)$, thus scaling extensively with
the number of ancillary qubits $n$ (see the peak in the top panel).
For small times $t\lesssim0.2$, we also observe a linear growth with
$n$ in the decoherence function $D$, for lattice sizes up to $L=15$.

\begin{figure}[!t]
    \centering
    \includegraphics[width=0.95\columnwidth, clip]{manyn1g.pdf}
    \caption{Top panel: the decoherence function $D$ versus
      $t$. Different curves are for various values of $n$ and $L$,
      while fixing $b=3$, setting $g=\delta=1$, and quenching $\kappa$
      from $0$ to $0.5$. We note that $D$ generally increases with $n$
      and, for $t\ll1$, its growth is proportional to the number $n$
      of qubits attached to the ring system, as shown in the inset on
      the left panel that zooms in on the domain $t\in[0, 0.5]$ (note
      that on the y-axis we show $D/n$). Bottom panel: the energy stored per
      unit qubit $\expval{U}/n$ versus the time $t$.  The first peak
      at $t\approx2$ is stationary with increasing $n$, consistently
      with an extensive scaling of the energy.}
    \label{fig_manyn_fixed_delta}
\end{figure}

\section{Quantum quenches along the FOQT line}
\label{sec_FOQT}

\subsection{Finite number of qubits in the FSS regime}
\label{FOQT_FSS}

We now analyze the model in proximity of the FOQT line ($g<g_{\mathcal
  I}$). For the sake of simplicity, we start from the case with a
fixed number $n$ of ancillary spins [FSS limit (i)].  We stress again
that, in the FSS and thermodynamic limit, a finite number of external
spins cannot change the bulk properties of the ring system, and in
particular its equilibrium quantum phases.  To begin with, we present
the scaling variables employed across this section to derive dynamic
FSS relations for quantum quenches at FOQTs.

Still relying our FSS arguments on the basis of the relevant energy
scales associated with each subsystem, we can first introduce the
analogue of the scaling variables~(\ref{def_A}) and~(\ref{def_K}) for
systems close to FOQTs, in the static case.  Namely, we define the
variable~\cite{FRV-22, RV-21}
\begin{subequations}
\begin{equation}
  Y_\delta = \frac{\delta}{\Delta_{\mathcal I}(g, L)}\,,
  \label{def_Y_delta}
\end{equation}
as the ratio between the energy of the elementary excitation of a
single qubit and the gap of the finite-size Ising ring
$\Delta_{\mathcal I}(g, L)$ at size $L$ and for a transverse field
$g<g_{\mathcal I}$.  Similarly to Ising models at the FOQT driven by a
longitudinal magnetic field, we may also introduce the scaling
variable $Y_\kappa$ associated with the variable
$\kappa$,~\cite{FRV-22, CNPV-14}
\begin{equation}
  Y_\kappa = \frac{\kappa}{\Delta_{\mathcal I}(g, L)}\,,
  \label{def_Y_kappa}
\end{equation}
\end{subequations}
which determines the {\em softness} of the quantum quench.  To extend
the FSS framework to the dynamic scenario, a further scaling variable
associated to the time $t$ is required, analogously as was done in
Eq.~(\ref{def_Theta_scaling_variable}).  Namely,~\cite{PRV-18}
\begin{equation}
  \Theta = t \; \Delta_{\mathcal I}(g, L)\,,
  \label{def_Gamma}
\end{equation}
where $\Delta_{\mathcal I}(g,L)$ now denotes the gap of the Ising-ring
system for $g<g_{\mathcal I}$.  Therefore, differently from CQTs in
the FSS limit, data collapse of observables at FOQTs is expected to be
recovered at time scales that substantially depend on the way in which
the gap $\Delta_{\mathcal I}$ closes with $L$: generally, these may
range from power-law to exponentially large times, according to the
different boundary conditions~\cite{RV-21}.  In our specific case
with a PBC geometry, we expect an exponential scaling
$\Delta_{\mathcal I}(g<g_{\mathcal I},L) \propto g^L$.

Relations analogous to those presented at the CQTs are still formally
valid in proximity of FOQTs, after properly replacing the
corresponding scaling variables.  For example, the dynamic FSS
relations related to the decoherence function $D$
[cf.~Eq.~(\ref{FSS_decoherence}), for systems close to the CQT] is now
expected to scale as
\begin{subequations}
\label{FSS_FOQT}
\begin{equation}
  D(n, g, \delta, \kappa, t, L) \approx
  \mathcal{D}(n, g, Y_\delta, Y_\kappa,\Theta)\,,
\end{equation}
while the energy stored into the auxiliary qubits
[cf.~Eq.~(\ref{FSS_energy}) at the CQT]
should read
\begin{equation}
  \expval{U}(n, g, \delta, \kappa, t, L)
  \approx \Delta_{\mathcal I}(g, L) \;\,
  \mathcal{E}(n, g, Y_\delta, Y_\kappa, \Theta)\,.
\end{equation}
\end{subequations}
The analogous counterparts for static conditions with a constant and
finite $\kappa$ (i.e., without time dependence induced by changes of
the parameter $\kappa$) have been numerically verified in
Ref.~\cite{FRV-22}.  Here we rather focus on numerically testing
our above dynamic FSS framework for soft quenches close to the FOQT
line $g<g_{\mathcal I}$.

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.95\columnwidth, clip]{FOQTnscaling.pdf}
  \caption{Top left panel: the decoherence function $D$ in terms of
    the rescaled time $\Theta$ at the FOQT for $g=0.9$, with
    $Y_\delta=3, Y_\kappa=1.5$, and for fixed $n=2$.  Bottom left
    panel: the ratio $\expval{U}/\Delta_{\mathcal I}$ versus $\Theta$.
    In the corresponding right panels we report the same data as a
    function of $L$, for fixed $\Theta = 1.25$ and $0.75$, showing
    that scaling corrections are substantially consistent with a decay
    $\sim L^{-2}$.  Straight lines are drawn to guide the eye.}
  \label{fig_FOQT_nfixed}
\end{figure}

Specifically, the dynamic FSS laws in Eqs.~(\ref{FSS_FOQT}) for
systems close to FOQTs are supported by the data in
Fig.~\ref{fig_FOQT_nfixed}, which shows an excellent data collapse for
the various curves at different ring sizes $L$, and for rescaled times
smaller than $\Theta<2.8$ (left panels).  Scaling corrections appear
to be suppressed roughly as $\sim L^{-2}$, with increasing $L$ (right
panels).  We have checked that the above FSS predictions are verified
numerically also for a different value of $g=0.8$, by considering the
case for fixed $n=2$ and ring sizes up to $L=12$ (not shown).

Also at FOQTs we observed that the dependence of observables on the
number of ancillary spins could be absorbed into a redefinition of the
variable $Y_\kappa$ as
\begin{equation}
  Y^\prime_\kappa = \sqrt{n} \, Y_\kappa\,.
  \label{def_Y_kappa_prime}
\end{equation}
To this extent, in Fig.~\ref{fig_manyn_FOQT} we present result for the
decoherence function $D$ for the largest lattice sizes at our
disposal, for several values of $n$.  For fixed $\Theta = 0.5$ (bottom
left) and $\Theta = 1$ (bottom right panel) we observe a power-law
convergence in the number of external spins as $\sim
n^{-1}$~\footnote{For each $n$, we have performed large-size
extrapolations assuming $L^{-2}$ scaling corrections, by fitting an
ansatz function $D(L)=a + b/L^{2}$}.

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.95\columnwidth, clip]{manynFOQT.pdf}
  \caption{Top panel: the decoherence function $D$ versus $\Theta$,
    for several values of $n=1, \ldots ,5$ for the largest lattice
    sizes available with our exact diagonalization methods.  Scaling
    corrections at $\Theta=0.5$ (bottom left panel) and $1$ (bottom
    right panel) are consistent with a decay $\sim L^{-1}$.  Straight
    lines are drawn to guide the eye.  Here we stay close to the FOQT
    at $g=0.9$, fixing $Y_\delta=3$ and $Y_\kappa^\prime=1.5$.}
  \label{fig_manyn_FOQT}
\end{figure}
\subsection{Qubits at fixed distance $b$ in the FSS regime}
\label{sec_FOQT_bfixed}

Following the results carried out in the previous section, we now
extend our dynamic FSS framework at FOQTs to the domain when $b$ is
fixed, thus the number of quantum probes increases with the ring size
$L$ [FSS limit (ii)]. In this limit, analogously to what was done in
Eq.~(\ref{def_widetilde_K}) close to a CQT, we recognize as a relevant
scaling variable for the quench intensity the ratio~\cite{FRV-22}
\begin{equation}
  \widetilde{Y}_K = \sqrt{n} \,
  Y_K = \frac{\kappa L^{1/2}}{\Delta_{\mathcal I}(g, L) \, b^{1/2}}\,.
  \label{def_K_tilde}
\end{equation}
With reference to the previous section, dynamic FSS relations can be
obtained by replacing $Y_K$ with $\widetilde{Y}_K$, e.g., in
Eqs.~(\ref{FSS_FOQT}).  Our scaling hypotheses are checked numerically
in Fig.~\ref{fig_scaling_FOQT_2b} for lattice sizes up to $L=14$,
within scaling corrections that appear to decay as $\sim L^{-1}$,
which should be again related to the $O(1/n)$ corrections found in the
large-$n$ limit at fixed $n$.

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.95\columnwidth, clip]{FOQT2b.pdf}
  \caption{Top panels: the decoherence function $D$ versus the
    rescaled time $\Theta$ (left) and the inverse size (right). Bottom
    panels: same as in the top panels, but for the stored energy
    $\expval{U}$ rescaled with the gap $\Delta_{\mathcal I}(g=0.9, L)$
    of the finite-size Ising ring.  The scaling corrections shown in
    the right panels, evaluated for the top and bottom panels
    respectively at $\Theta=0.6$ and $1$, are in good agreement with a
    decay $L^{-1}$ (straight lines are drawn to guide the eye).  We
    fix $g=0.9$, $b=2$, and $Y_\delta=3$, $\widetilde{Y}_\kappa=2$.
    However, similarly to Fig.~\ref{fig_CQT_b_scaling}, in the insets
    we also show some data associated with a different value of $b=3$,
    to emphasize the independence of the observables on $b$ in the
    large $L$ limit, while keeping $\widetilde{Y}_\kappa=\kappa
    L^{1/2}/[b^{1/2}\Delta_{\mathcal I}(g, L)]$ fixed.}
    \label{fig_scaling_FOQT_2b}
\end{figure}

We skip the discussion of the hard-quench limit starting from the
ferromagnetic phase with $g<g_{\mathcal I}$, since we do not find any
relevant difference to that found when performing the quench from the
QCP.  In the zero-temperature equilibrium phase diagram of the
sunburst quantum Ising model, it was indeed conjectured that, when $b$
is fixed, the critical line characterizing the breaking of the
$\mathbb{Z}_2$ global symmetry shifts to higher values
$g_c(\kappa)>g_{\mathcal I}$ for any $\kappa \neq 0$~\cite{FRV-22}.
Since the hard-quench protocol we consider always
pushes the system away from criticality into the ordered phase, we do
not find any remarkable difference between quenches performed when
$g<g_{\mathcal I}$ or $g=g_{\mathcal I}$.

\section{Quantum quenches from the disordered phase}
\label{sec_disordered_phase}

Let us now present some numerical results for the dynamics of our
lattice system, when it is driven out of equilibrium from the disordered
phase ($g>g_{\mathcal I}$), fixing the distance $b$ between two
consecutive external spins.  Guided by the $\sqrt{n}$-behavior put
forward in the dynamic FSS at CQTs and FOQTs, and by the equilibrium
properties of the sunburst quantum Ising model~\cite{FRV-22}, we first
address the dynamics of the system in the disordered phase with
rescaled quench intensity $\kappa: 0\to 1/\sqrt{n}$.

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.95\columnwidth, clip]{disorderedsqrtn.pdf}
  \caption{Results for quantum quenches starting from the disordered
    phase.  We keep $b=3, g=1.5, \delta=1$ fixed, and rescale the
    intensity of the quench with $n$, such that
    $\kappa:0\to1/\sqrt{n}$. Top left panel: the decoherence function
    $D$ versus $t$.  Bottom left panel: the energy $\expval{U}\propto
    n$ stored into the ancillary qubits versus $t$.  Scaling
    corrections, shown in the corresponding right panels for $t=1$,
    are consistent with a decay $\sim L^{-1}$ (straight lines are
    drawn to guide the eye).}
  \label{fig_disordersqrtn}
\end{figure}

In Fig.~\ref{fig_disordersqrtn} we present results for fixed $g=1.5$
and $\delta=1$.  We remember the reader that the typical fluctuations
associated with the global work done on the overall system at the
quench scale as $\sqrt{\expval{W^2}}=\kappa \sqrt{n}$, so that it
remains constant in this limit.  Therefore, in
this regime, the coupling between the ring and the external spins
decreases as the lattice size increases. Our results suggest that, for
small times $t\lesssim 1$, the peak of both the decoherence function
$D$ and the energy stored into the probes $\expval{U}$ approach a
constant behavior for large $L$ (note that the latter quantity
increases linearly with the lattice size $\expval{U}\propto n \langle
\hat \Sigma^{(3)} \rangle$).
We also present some results for fixed quench intensity $\kappa$ and $g=1.5$
in Fig.~\ref{fig_disordered_kfixed}. As expected, in this second case,
the decoherence function $D$ generally increases with the number $n$
of external spins (at small times it depends on $n$ as $D\propto n$,
as shown in the correspondent inset) and the first peak of $\expval{U}$
scales extensively with the number of ancillaries.

\begin{figure}
    \centering
    \includegraphics[width=0.95\columnwidth, clip]{disorderedbdixedkfixed.pdf}
    \caption{Results for quantum quenches starting from the disordered
      phase ($g=1.5$). We maintain fixed $b=3$, $\delta=1$ and perform
      a quench for $\kappa:0\to1$. The decoherence function generally increases
      with increasing the number of ancillary spins $n$. The inset shows that
      such function increases linearly with $n$ at small times $t\lesssim0.2$.
      Bottom panel: The maximum energy per unit spin encapsulated
      in the external qubits is roughly independent of $n$.}
    \label{fig_disordered_kfixed}
\end{figure}

\section{Quenches with Ising ring of fixed length}
\label{sec_fixed_ring}

Finally, we analyze some examples of hard quenches when the length $L$
of the Ising ring is fixed, but the number of external probes vary,
which is perhaps the most relevant case for quantum-metrology purposes
or for setting a hypothetical quantum-battery device.  To this extent,
the focus is on the role of $n$ in determining the dynamics of
observables (in particular the energy $\expval{U}$) when the ancillary
qubits which probe the ring are independent from each other. To remark
this fact, we compare our results with the analogous ones obtained by
coupling consecutive nearest-neighbor external qubits along the
transverse direction.  As an exploratory study, we consider the
following extension of the ancillary Hamiltonian~(\ref{hamiltonian_HA})
\begin{equation}
  \hat{H}^\prime_{\mathcal A} = \hat{H}_{\mathcal A} - J_\Sigma
  \sum_{i=1}^{n} \hat{\Sigma}^{(3)}_{i} \hat{\Sigma}^{(3)}_{i+1},
  \label{eq_H_p_kinetic_ancillary}
\end{equation}
where PBC are intended in the second term, thus preserving the
translational invariance of the lattice model, modulo $b$.  In the
following we present some results for the case $J_\Sigma=1$.

Note that the pre-quench state $\ket{\Psi_0}$ in
Eq.~(\ref{eq_groundstate_before_quench}) is still the ground state of
the global system with $\kappa=0$, independently of the presence of
the new interaction term of Eq.~(\ref{eq_H_p_kinetic_ancillary}).  Of
course, the post-quench unitary dynamics will change, and thus also
the average energy stored into the qubits:
\begin{equation}
  \expval{U} = \mel{\Psi(t)}{\hat{H}^\prime_{\mathcal A}}{\Psi(t)}
  - \mel{\Psi_0}{\hat{H}^\prime_{\mathcal A}}{\Psi_0}\,.
  \label{eq_heat_kinetic}
\end{equation}

\begin{figure}[!t]
  \centering
  \includegraphics[width=0.95\columnwidth, clip]{quenchL12.pdf}
  \caption{The average energy per unit spin stored into an external
    qubit versus the time $t$, for both cases of independent external
    spins ({\em free} in the legend) and of external spins which
    interact according to Eq.~(\ref{eq_H_p_kinetic_ancillary}) ({\em
      coupled} in the legend). Here we consider $L=12$ and
    vary $n$ as indicated in the legend, fix $\delta=1$, and perform a
    quench of $\kappa$ from $0$ to $1$ for several values of $g$.
    There is no striking difference between the two quenches shown at
    $g=0.8$ and $1$: the {\em free} and the {\em coupled} model can
    absorb roughly the same amount of energy
    $\expval{U}/n\approx0.5$. The capability of energy absorption
    worsen in both models for $g=1.5$ (note that the $y$ scale in the
    last graph is different from the upper panels).  }
  \label{fig_L_fixed}
\end{figure}

In Fig.~\ref{fig_L_fixed} we present some results for fixed
$\delta=1$, quench intensity $\kappa:0\to 1$ and ring length $L=12$
(which offers several combinations of $b$, commensurate to the ring
length) to be investigated. Looking at the peaks in the energy stored
for all the three values of the coupling $g$ we have considered, each
one representing a different phase of the Ising ring, we first note an
extensive scaling of the energy with the number $n$ of qubits: In
particular, the maximum energy stored into a single spin is roughly
the same for $g=0.8$ and $1$, $\expval{U}/n \approx 0.45
\divisionsymbol 0.50$, whereas it appears strongly suppressed to
$\expval{U}/n \approx 0.08$ for both models, when $g=1.5$).  However,
it is also worth noting that the charging of the qubits that are {\em
  coupled} is generally significantly faster than that obtained when
the external qubits are independent from each other.  This last fact
holds for all values of $g$ shown in Fig.~\ref{fig_L_fixed}, as well
as the extensiveness with $n$ of the energy stored $\expval{U}$. We
have not found evidence of striking differences between the energy
stored into the probing apparatus of the two models. However, we
should say that we have not done an extended study of the optimal
parameters in this respect.

\section{Summary and conclusions}
\label{sec_summary}

We have studied the post-quench unitary dynamics of an ensemble of
quantum spin-1/2 objects arranged in a sunburst geometry: namely, at a
given reference time, some of the $L$ spins of a transverse-field
quantum Ising ring are suddenly and locally coupled to a set of $n$
independent external qubits ($n \leq L$), along the longitudinal
direction, in such a way to maintain a residual translation invariance
and the Ising $\mathbb{Z}_2$ symmetry.  The system exhibits various
quantum phases separated by first order (FOQTs) or continuous (CQTs)
quantum transitions, which are controlled by the various Hamiltonian
parameters as the Ising transverse field $g$, the energy scale
$\delta$ of the external qubits, and the interaction strength $\kappa$
between the two subsystems~\cite{FRV-22}.  This scenario represents
a useful playground where to study the mutual interplay and the
properties of different subparts of a complex quantum many-body
system, such as the onset of decoherence or the exchange of heat and
work between them.  In particular, the set of $n$ qubits can be
interpreted as a probing apparatus ($\mathcal{A}$) for the set of $L$
spins composing the Ising ring ($\mathcal{I}$).

By using proper dynamic FSS frameworks~\cite{RV-21} we show that, at
the Ising CQT ($g = g_{\mathcal I}$) or along its FOQT line
($|g| < g_{\mathcal I}$), peculiar dynamic scaling regimes may appear.
To this purpose, we explicitly consider two different large-size limits
where dynamic FSS laws emerge: we allow $L\to\infty$ while fixing either
the number $n$ of external qubits, or their nearest-neighbor interspace
distance $b$.  For a sufficiently large number $n$ of external spins,
the dependence of observables on the number $n$ of qubits can be
reabsorbed into a redefinition of the quench parameter, by replacing
the scaling variable associated with $\kappa$ with $\sqrt{n} \,
\kappa$.

We also consider quenches within the Ising-disordered phase ($g>
g_{\mathcal I}$), addressing the role of the $\sqrt{n}$-behavior out
of the FSS framework. We observe that, by maintaining $\kappa\sqrt{n}$
fixed, which represents the typical energy scale of work done on the
whole system [in particular $\sqrt{\expval{W^2}}=\kappa\sqrt{n}$,
  cf. Eq.~(\ref{workav})], several quantities that are expected to
scale extensively with $n$ approach a constant value in the large
$L$ limit.  In other words, our results suggest that even beyond the
FSS regime, there is a fair tradeoff between the {\em linear}
enlargement of both the Ising ring and ancillary spins (they both
increase with $L$ when $b$ is fixed) and the $\sqrt{n}$-reduction of
the quench intensity.

Finally, we compare our results to those obtained when also a
nearest-neighbor coupling among the external qubits is taken into
account, maintaining the Ising-ring length fixed.  For quenches with
$\kappa:0\to1$, no substantial differences emerge between the two
lattice models: although the recharging of the {\em coupled} model is
faster than that of the {\em free} system, the two models can roughly
store the same amount of energy. However, our study was only
exploratory, we have not done a through optimization study of the
model parameters.

Our dynamic framework has been carefully tested by means of numerical
exact diagonalization techniques on systems with only $\sim O(20)$
quantum spins.  Moreover, results for the time behavior of the
decoherence function are strictly connected with the experimentally
measurable R{\'e}nyi entropy~\cite{Islam-etal-2015,
  Brydges-etal-2019}, while the average energy stored into the
external qubits may represent a central thermodynamic quantity to be
monitored in the study of quantum-battery devices~\cite{CPV-18}.
The reduced number of spins in our simulated systems suggests that
quantum simulators with existing technology should be able to access
our predictions, e.g., by using trapped ions~\cite{Debnath-etal-16},
ultracold atoms~\cite{Labuhn-etal-16, SFSTT-20}, or superconducting
qubits~\cite{Salathe-etal-15, Cervera-18}.

\section*{References} %

redpajama_set_name: RedPajamaArXiv


Karen Kubicko has a great way of explaining the natural response we have as spirits being human. Remembering past lives, and acknowledging your intuition are major components in navigating your life today.
Trust is the most important thing to have when it comes to remembering a past life. If you want to remember a past life, you will need to trust the information that you receive.
Trust the information since it all comes from within you.
I compare it to the same as trusting your own intuition. Messages from your intuition (your higher self, your spirit guide, sixth sense, your conscience) come in six main ways.
Second Program Completed: Two More Professional Psychic Mediums are Certified.
As the second (one-year) program for 2012 finalizes, two more women are certified as professional psychic mediums: Vivian Harris of Dayton, Ohio, and Susan Williams of Crooksville, Ohio. They have made a major decision in their lives to expand who they are and how they envision their future in the metaphysical world.
Harris and Williams have participated in and completed the requirements for full certification in Off the Beaten Path Intuitive Art's Program for Professional Psychic/Mediums. The program was implemented and facilitated by Reverend Melissa Leath in 2011, and structured around her latest book, Psychic Integrity, The Respected Practice of Modern-Day Mystics (Balboa Press, division of Hay House Publishing).
The preparation of the program was in the making for three years as Leath pulled over 30 years of experience into four levels of study: (1) Spiritual Awakening, covering grounding and foundational concepts, development meditation, and self-exploration; (2) Psychic Awareness, understanding earth vibrations, energy exercises, working with auras and energy fields; (3) Mental Mediumship, direct vibrational messages, public presentation and polishing, getting ready for responsibility and gallery demonstration; and, (4) Professional Psychic-Medium, understanding and incorporating your gift into service, creating your own reality, living your spiritual understanding, and preparing for professional work with the public.
The program included not only four workshops throughout the year, but also had mandatory accumulation of student study points. This was in the form of volunteer outreach in the field of metaphysics, verifiable attendance in other classes and workshops with reports, research and essays, critiques of other practitioners and experiences, practice readings, and notes on meditations. The year was full of purposeful study, so it was a specific plan of work in the spiritual and metaphysical fields.
Vivian Harris comes from a holistic health background with a specialty in Reiki healing and desire to assist people with their emotional and holistic needs. She has participated in many intuitive development classes and other holistic workshops, and is continuing unfoldment of her psychic and medium gifts. She is also a Divine Grace Healing facilitator.
Susan Williams has worked with spirit energy and those who have passed over for her entire life, and has recently decided to pursue this area of expertise more diligently. She has participated in many workshops and classes in the metaphysical and holistic areas. Her interest is around grief counseling and verification of life after death. She is following her interest in volunteer hospice serve, after being involved as an emergency medical specialist providing care for the sick or injured in emergency medical settings. Williams also works closely with psychic energies of animals and pets.
Thanks Shaheen for these great reminders to always clear your space and own energy field. We all have felt strange around certain places, or people. It's usually just picking up stale or negatively charged energy left over from other things. It is a great practice to clear your own home/office and personal energy daily. it makes a world of difference.
You are a vibrating, microcosmic Universe. A multifaceted being made of energy, and intention. You are not just your body… You are a corona of light and frequency. Some call this the Aura… The rainbow layers of color that surround you. I call this your energetic story!
Your energetic story is composed of your thoughts, ideas and beliefs about the world, and how you fit into it. On this energetic level you are pushing and pulling things into your life. Interacting with the people and places around you.
Are you stuck in a community or family of strong religious beliefs?
But, do you still wish you could connect with more enlightened people? People who think outside the box?
Do you wish you could study metaphysics and spiritual concepts without anyone knowing? Or wish you could develop your psychic or medium gifts?
I applaud their journey into the world of the unknown. I have been on that journey a great portion of my life — although, I am known for doing it. So for them to do this is secret, is so much fun to watch!
As they pursue their studies and report via their woo-woo blog, I will be watching and reading and supporting from my own blog point. And I'll be sharing some of their posts as a re-blog here. So pay attention!
Good luck, Sisters! We await your reports!
ABOUT THE PROGRAM: Competion Completion of the entire program is necessary in order to become certified as a psychic professional developed for Off the Beaten Path Intuitive Arts by Rev. Melissa Leath. Because each stage is designed to challenge and inspire you to your own level of integrity, and builds on each stage, all four levels must be taken in order to become certified. You will be introduced into a specific model of psychic and mediumship professionalism which you can fully integrate into your specialty and expand into a broader vision for your life's work.
** Pre-Requisite: You must have used your psychic or medium gifts with the public for at least one year to benefit from this program. It is not meant to be a beginner's development class.
Gather Student Study Points by volunteer outreach in various centers, bookstores, or metaphysical centers, etc.
Short essay on what you will use your certification for.
* Want to learn about "cold readings"?
When I started this blog, a couple of days ago, I figured I would take time to go back and start from the beginning. I imagined I would begin with my first paranormal experiences then work my way to the present, writing them down one by one. I foolishly assumed more time would pass before having another instance of the supernatural kind. Well, I was wrong. Today has already taught me something about myself.


Enjoy a slide show full of candids from this morning's party in the park. Weather was in our favor and the children clearly had a blast! So nice to see them playing together in a setting other than the classroom. Thank you for all of your contributions to the party. It was a huge success! Happy Easter and we will see you all next Wednesday.
Here you will find little snippets of our days together. When my children were in preschool and would come home I would always ask, "So, what did you do today?". All they ever said was, "Played." That didn't tell me much. I hope this blog will be a place to share what we have done and help you ask more directed questions of your child. You will see first hand just what we do after you kiss your little one goodbye in the morning. It will also be a great place to visit and see just how much your child has grown and learned throughout the year.