A great way to make the backyard more attractive and add extra leisure space is by laying a stamped concrete patio. One of the stamped concrete purposes are the appearances more attractive like many other paver materials, such as natural stone, ashlar, granite, or brick and get the look with an affordable alternative, and looks very realistic at the same time. Stamped concrete enables to create a smooth, continuous surface, and at the same time more stable than pavers since it's one solid mass rather than multiple bricks or stones. The Stamped concrete can easily add the recoupable value of the properties.
The price for stamped concrete patio project is dependent on factors like labor needs, concrete patterns, and detailing. One of the largest factors is the size of the area. The type of colors and patterns may also play a factor in the cost of the job. A stamped concrete patio with one base color will likely cost less than a stamped concrete surface with multiple accent colors.
Basic stamped concrete costs between $8 and $12 per square foot, but more involved projects can be as high as $18 per square foot. This cost guide will determine the average cost of laying a stamped concrete patio. In general, for 1000 square feet of stamped concrete can cost approximately $5000 US.
The detail of cost breakdown can be approached in 2 ways by an itemized bill, or one based on cost per square foot. The contractor chooses largely depends on their preference, but typically, you can expect that smaller projects will get itemization. Most contractors charge a set rate based on the overall square footage of the space.
Excavation: If your land needs excavation or significant grading prior to building the patio, expect total costs to increase by 50%.
Basic: Stamped concrete installation with one simple pattern (ex: herringbone or ashlar) and one color. Cost: $8-$12 per sq.ft.
Mid-range: Customize stamped concrete plus borders, or a second pattern or color. Cost: $12-$18 per sq.ft.
High-end: Highly detailed stamped concrete including multiple patterns and colors. Can include specialty patterns and hand-coloring made to mimic natural stone color variations. Cost: $18+ per sq.ft.
In addition to that, there are an additional enhancement and improvement costs. This includes paint and hand-detailing. For painting and sealing stamped concrete can give it even more visual appeal. The typical cost of this is $0.70-$0.86 per sq. ft.. Whereas for hand-detailing, by using detailing tools in addition to the basic stamps can add richness to your patio's appearance. Detailing is done while the concrete is still wet so that custom patterns and textures can be easily carved. This normally costs $225.
The cost of stamped concrete varies broadly, depending on the prices for materials and labor in your local market and the complexity of the job. The most inexpensive stamped concrete uses just one pattern and color, while the more expensive installations feature multi-pattern designs with special coloring effects, such as hand-applied stain accents.


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Klansman gang significantly degraded — Anderson
KINGSTON, Jamaica — Commissioner of Police, Major General Antony Anderson, says the Klansman gang is significantly degraded from where it was as the police continues operations against it.
Speaking at a Jamaica Constabulary Force virtual press conference this afternoon, Anderson said the trial for the gang is expected to begin in January.
He noted that 60 alleged members of the gang are currently in police custody.
Anderson the police continues to investigate the gang, particularly its source of funding.
In November, a woman believed to be the spouse of reputed leader of the Klansman Gang, Tesha Miller, was arrested during a targeted operation. She was charged with money laundering and possession of criminal property.

redpajama_set_name: RedPajamaCommonCrawl


Can the Arts loosen Europe's borders?
In the summer season of 2011, the Athens cultural centre About ran a conducting course under the leadership of the artistic director of the Camerata Europaea, Maria Makraki. During the seminar event, which will be repeated in Greece in 2012, Maria Makraki spoke to Stella Tziva about the course, the Camerata Europaea and the current situation in Greece.
C. N.: Could you tell us something about the profile, activities and aims of the Camerata Europaea?
M.M.: The Camerata Europaea is a musical ensemble, which was founded in 2007 and is made up of outstanding musicians from different member states of the European Union. Their concerts are aimed at introducing a wider audience to works by contemporary European composers. The idea behind concert series such as Music Plus, Spiritual Europe, Polyphonic Europe and Trio Europe is to develop representative programmes that bridge European countries whilst combining the traditional with the modern and the familiar with the unfamiliar, in this way awakening the interest of the audience for the richness of contemporary European culture.
C. N.: You have directed orchestras in Great Britain, Germany, Switzerland, Poland, Rumania, Greece and in Ukraine. What is the nature of the relationship between the conductor and the orchestra musicians? Is successful collaboration only possible if the 'chemistry' between the conductor and the orchestra is right?
- Why do you think there are so few female conductors?
That's slowly changing. You can now find women who are studying conducting at all levels.
- Have you experienced discrimination personally?
No, not really. Of course, you always have to convince people of your abilities. The first minutes are always crucial, but after that a person's gender is no longer relevant. There will always be conservative orchestras and conservative musicians. But conducting is about communicating a concept, and women do that equally as well as men. As in any other profession, a conductor can have a great deal of assertiveness, charisma, musicality and authority.
- In this respect, does it make a difference whether you are standing in front of your own orchestra or an unfamiliar one?
Yes, there is a different. With your own orchestra you feel more liberated, more able to experiment and test, even cross boundaries. A pleasant but powerful musical atmosphere can be conveyed to the musicians to create a special – sometime even bombastic – tone-entity.
- What is your personal strategy for asserting yourself in male dominated hierarchies?


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Microsoft launches Windows 8 this week – what to look out for
By Robert Brand
Yes folks it's finally here, the week that will see Microsoft launch Windows 8, spearheading its massive 2012 product launch schedule. It's keystone of the 'Three Screens & Cloud' play and this is where we see how it's all going to shake out.
The pre-orders for Surface have started and even run dry in some regions, invites to the various launch events have gone out and now it's time to put all the cards on the table. With such a wide ranging spruce-up of the entire product line from Xbox to Windows Phone, we're in for a wild ride. Windows 8 will be at the centre of these launches, not just because it's Windows but because with it we will see a whole host of services light up to enable the entire eco system.
The launch of Windows 8 affects us Windows Phone users deeply. Both are now locked together by their common core and more importantly by their set of backend services like Xbox Music and Video. It's the set of services behind Windows 8 that will enable the overarching dream of three screens and the cloud to solidify. How well Microsoft can pull PC, console and phone together will rest on their ability to leverage cloud technologies to help enable a seamless experience across them all. What are we waiting for?
Will we see the Windows Phone 8 and Windows 8 sync client?
We have seen early snaps of the Windows 8 Metro style synchronisation app and it's my hope that during launch we will finally hear more about this key piece of integration. At present we're using the Zune desktop client which provides music management, playback, syncing and converting of music and video files. With the dedicated apps for music and video on Windows 8, will be see only a rudimentary client to sync photos and videos back and forth or will it be more fleshed out, allowing conversion and other capabilities?
While it would be great to hear more about this tie up during the Windows 8 launch, they may hold that one back for the full launch of Windows Phone 8 on the 29th. After all, there is so much we do know about the next version of the mobile OS having a few surprised might be a good thing eh?
SmartGlass
Some of that technology is already in place, we can control the Xbox from our phone or with Windows 8. What we haven't seen is the "glass" experience in final form. That's using your smartphone or tablet to fully navigate around the menus in Internet explorer. If IE on the console is to be in any way useful this experience will need to be top notch. Will they pull off a smooth and seamless effect when using it? I'd expect to see an updated SmartGlass app for Windows 8 on launch day to finally enable this. The potential to see Windows Phone 7 and 8 clients is there too but again this could be held off for the full reveal on the 29th. Either way, an updated Windows 8 app will show where this technology is heading to.
Xbox Music and Video Services
We're about to see the Zune Music service dusted off and given a new lick of paint for the Windows 8 launch. Recently the lights were switched off over at www.zune.net (opens in new tab) and key features of Xbox Music were outlined but we have still yet to see them fully. The Xbox Music website is still only a holding page and while the broad stroke of the enhanced service are there we still could see much more get revealed.
Xbox, Windows and Windows Phone will all share Video and Music services. In many ways how these line up and play out will determine how well this eco system will feel like to use. Jumping between a movie on the 360, phone and computer or syncing playlists and songs to various devices, all of this will add to the feeling of harmony when using these disparate devices. We'd like to see some demos of how well this will all fit together. Will the combination of SmartGlass and synchronised starting and stopping of cloud based music and video files be enough to pull it all it? Could be, eyes will firmly be on these coming demos.
Fresh new start
Windows 8 is likely the most radical release of Windows ever, the changes run not only through the UI itself but in the devices and services that accompany it. The launch represents the start of a series of launches across their entire product line up and should point the way to what we're likely to see when Windows Phone 8 launches the following week. By the end of this month we're finally going to see how Microsoft's 'Epic Year' is going to pan out.
Excited about Windows 8 or just keen to see it out of the way so we can get to the meat and see Windows Phone launch? What are you keen to see with the launch of Windows 8? Tell us know you feel in those comments below.
Robert Brand
Modern Ui
Xbox One Smartglass
Groove Music
Wish I had a time machine! Maybe Microsoft will invent one!
I bet u an apple troll will post here replying to u, about Macs having a time machine. I can read the future.
And if Microsoft made a time machine, apple would make one, then say Microsoft copied theirs by function of going back and forth through time, and would take them to patent court asking for cease and assist order. Even though they both copied Doc Brown anyways.
Ballmer would just go back in time and file the patent before Apple. :)
Err, i think wat he was refering was the backup software/hardware for Mac - Time Machine. I know he maybe trolling but i think u got on the wrong thing as well, lol...
peace n cheers :)
ps: i'm an iMac user but i love my WP, i do like the new W8 interface but i beliv both hav their perks n all depends on every1's working style, let's not blast the canons to each others :)
No no. I was not trolling at all. I just used that word time machine in the general sense of the word.
Apple is a fruit. While iphone is junk. :-D
Microsoft is really bringing it this Fall. I'm excited for W8 just as much as I am WP8.
+1 already had WP8 phone and W8 tablet on the shopping list, and now the beefed up xbox integration actually has my head spinning. Time to lie down in a dark room for a while
I thought we were going to be able to pre-order Windows Phone 8 Nokias and HTCs from AT&T yesterday? What happened to that?
From best buy. And it is sort of gone. It won't pull up in the search, but if you have the link, it works. So you can still preorder it. :-)
Patience my friend patience...
Does anyone know when the AT&T pre-order will be available? I ask because I want to dump my Nokia 900 whiel its still selling for greater than 200 to take some of the sting out of the new 920's full price I'm gonna have to pay...
Were supposed to be able to from Best Buy only for AT&T that is but I don't know what happened with them taking the 920 off.
I wish the Windows RT was less like the iPad and more like the windows 86x
It is less like the iPad.
Do you make that statement because of the price?
Windows Surface PRO will have the full Windows 8 OS installed on it, although at a greater (unannounced) price.
Really, I cannot think of a tablet OS that is less like the iPad than Windows 8. What do you mean exactly? RB
Windows RT depends on the app store like the iPad, while the 86x can run most of the programs I bought
get an Acer iconia W7 tab and it will be. That's the great thing about Windows is you have CHOICE.
The start menu they show, how does one lable the different section of the start menu?I have been running windows 8 from sdk, to beta to rtm. and I see no way of labeling the groups in a start menu.is that coming with the full release?
From the start screen, click the negative "-" magnifier in the lower right side of the screen, the groups will become small, right click the one you want, options at the bottom of the screen will read, "Name Group", left click and type what you want
The option has been there since Release Preview or even earlier.
To label Start Screen groups, you need use Symantec Zoom (pinch on touch screen) then select a group and an option will appear at the bottom of the screen. Have fun!
You meant Semantic Zoom I presume.
Haha I knew how do it since it launched beta. That most basic think to do
All I can say is, finally tablets worth buying...
This week is killing me...
I WANT TO HOLD THE SURFACE IN MY HAND.. I WILL BE AT MSFT STORE THURSDAY AT SOUTH COAST PLAZA IN COSTA MESA IM BUYING IT DONT STOP ME...OR MAYBE WAIT A LITTLE BIT FOR BLACK FRIDAY.. SHIT I DONT KNOW WHAT TO DO.. LUMIA 920 IS DUE TO COME OUT TOO...OH PLEASE...
My head hurts from all your yelling, but I get your excitement.
Here's your answer: buy both of them. I am. :-D
I'm fearing that Microsoft will drop the ball on the Smartglass tech. The Windows Phone app called "Xbox Companion" has been around for almost 2 years now I think, which lets you control your xbox from your phone. Since then not much has been announced in regards to the Xbox Smarglass (which I believe will just be a renaming and upgraded version of the Xbox Companion).
Xbox Smartglass is note quite an App but a technology/API to be used by game developers to release game companion apps for WP, iOS and android. Unfortunately so far the only confirmed SmartGlass app is "Dance Central 3 DJ" coming this fall. And yes, the Xbox Companion app has been around for years, it features a game and app launcher and a simple controller (d-pad and ABXY buttons). Meanhwile the new Xbox Smartglass app on Windows 8, which mostly serves the same purpose, features a silly "swipe" controller instead of the classic d-pad controls, which were actually easier to use.
Silly swipe controler is better than ABXY when using IE on the console? I'd have to say I cannot agree there. Using the great keyboard on the smartphone for many reasons could be a real winner too. This is not tech aimed just at games. RB
I never claimed swipe was better for anything. I prefer the arrow buttons in the old Companion App to navigate the dashboards. It is simply more reliable. (Both the classic companion and the smartglass companion feature ABXY buttons, that is not the difference). As for the new Xbox IE - last time I checked the swiping in the Win8 SmartGlass App made the cursor in IE move only like a pixel per swipe - it was totally unusable.
Dance Central 3 (which released last week) is the first game to include SmartGlass functionality, which you can take advantage of now with Windows 8 RTM. SmartGlass users can add songs to a playlist in the game's "Party Mode", adding a Karaoke-parlor-like vibe to the whole thing... people can choose their songs on the SmartGlass while waiting for their turn to dance.
There's no Dance Central 3 companion app available in the Windows 8 Store yet. And the standard Xbox SmartGlass App doesn't offer Dance Central 3 DJ functionality.
I really welcome this change from Microsoft. It's kinda refreshing.
I'm one of the category that can't wait for this launch to be passed. I'm really looking forward to WP8 & all this windows 8 talk is just not exciting to me. Sorry just not a fan of Windows 8, live tiles on my laptop don't do it for me. But live tiles on my phone I love.
I agree with you. though I've been using Win8 on one laptop and this rig now I must admit that at first I didn't like it now however I could never go back to using Win7..
it's so interesting how everybody use to hate the new look way back when they started the event 4 months ago and now they love it. It's good Microsoft is now showing tv commercial of windows 8 surface and windows 8. I hope they can make a great tv commercial for windows phone 8
This is pissing me off that that we really haven't heard much about where the Ativ S is going to be, because that's the phone I'm getting and im on T-Mobile..
It'll be cool if you could utilize Smartglass to turn your phone into a fully functional controller. Though I haven't figured that part out yet...
I am missing one important thing - since it looks like the XBox console is not ready to replace my Microsoft Media Center, I would like SmartGlass to do a "Play to MediaCenter" instead of to XBox.
There is no way I will have an XBox in my living room until MS ships a 100% silent one. And then I would want TV recording as Media Center does before I would consider it to replace my current Media Center.


As part of andculture's "You Minded" campaign for Elizabethtown College and in collaboration with Jacob Hopkins, MAKE produced various films that show how students at Etown are not just a number, but individuals who deserve the best education with transparent and affordable pricing.
These films were shot on our RED Epic Dragon. Since it was a pretty run and gun shoot we shot as much footage in natural light while supplementing it with an LED light or black flag. It was meant to feel a little bit moody and edgy.


Italy's outgoing Prime Minister Mario Monti is due to reveal whether he intends to run for office in elections called for February.
Norway's foreign minister has urged the UK to assess the advantages of staying in the European Union, rather than consider leaving.
Russia has urged South Sudan to identify and punish those responsible for shooting down a UN helicopter and killing its four Russian crew.
The helicopter was apparently shot down on Friday while on a reconnaissance mission in eastern Jonglei state.
Pope Benedict XVI has pardoned his former butler, Paolo Gabriele, who is serving an 18-month jail sentence for stealing confidential papers.


The applicant is requesting approval of a zoning ordinance map amendment, rezone, and preliminary plat application to allow for the development of 11 lots on 118 acres.
Location: The site is located on the west side of SH-55 across from Shadow Valley Golf Course.
Request for Approval of Minutes and Minute Entries from the January 17, 2018 Commission Meeting.
Construction change order under the 2017 Microseal contract held by Boswell Paving Solutions to include pavement markings in the contract.
Staff requests approval of the Cooperative Agreement with FHWA for the SMART Arterial Management Project.
Staff recommends the ACHD Commission approve the 2018 MD-Piggyback Purchase Agreement with MetroQuip sweepers for the amount of $616,740.00 for the purchase of two mechanical sweepers to be converted to a four (4) year lease with approximate annual payments of $154,200.00.
Confirm the Impact Fee Advance Balance of $13.7M.
Condemnation of Right-Of-Way for Cloverdale Road, Ustick Road to McMillan Road - Parcel 1 3429 North Cloverdale Road, Boise, Idaho.


and instantly I thought: daffodils for Easter!
I wanted the flowers to speak for themselves with the sentiment on the interior for this card. In Shimmery White cardstock I cut a shape with the Rectangle Stitched Framelits (and used a card base in the American size to match).
Then I used the three inks in this week's challenge to colour in the image from the You're Inspiring stamp set using my Aqua painter. Popped it up on dimensionals and stamped the interior of the card with Happy Easter.
Hey presto! Gorgeous Easter card at the ready. Thanks to GDP for the challenge!
Thanks so much, Brian! I love the quiet joy of Easter and wanted to transfer that into a card.


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You can pre-order this beautiful $365 Metroid statue starting today
By Michael McWhertor@MikeMcWhertor Jul 29, 2014, 10:30pm EDT
Source First 4 Figures
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First 4 Figures, a maker of fine collectibles, opened pre-orders for one of its best-looking statues today, a one-quarter scale Samus Aran wearing her light suit from GameCube game Metroid Prime 2: Echoes. The price tag for this LED-lighted work isn't cheap: $364.99.
The Samus statue, which stands 20 inches tall, is the first in a new series of Metroid Master Arts resin statues from First 4 Figures. The company has already released statues of Samus wearing her gravity suit, varia suit, phazon suit and zero suit, as well as recreations of her gunship.
The light suit Samus is available in two options, the less expensive standard version ($339.99) and the "exclusive" light up version ($364.99). The standard version is limited to 2,500 pieces and the exclusive version is limited to 750 pieces. Both are expected to ship in the first quarter of 2015.
Here's a full look at the statue.


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Robbie Henshaw With The Midland Warriors Rugby Club
Intersport Elverys are the official retail partner of the IRFU. In a highly anticipated year for Irish rugby, we teamed up with Canterbury, and an Irish rugby international, for a memorable, rugby themed day.
In the heart of Ireland, and with Buccaneers Rugby Club as the backdrop, Moate based Midland Warriors RFC were the stars for the day. Meeting with Graham and Kate Ross, who set up Midland Warriors, Intersport Elverys and the Ross's share a passion for local, grassroots rugby.
Along with the kids from Midland Warriors, another star was on show. Local and national rugby icon, Robbie Henshaw was on site to lend his expertise for the day.
From a pep talk to the pitch, and sporting the Official World Cup training range, Robbie took the young athletes on an exuberant training session. Flagged by coaches and spectators, the Irish centre put the Midland Warriors to work.
With adrenaline levels easing Robbie sat down with the Ross's to chat about all things rugby. Their interchangeable rugby experiences were conversed in the stands of Buccaneer's Rugby Club. An easy going, free flowing conversation ensued.
Their passion for rugby in their daily lives was evident when Kate and Graham described how their love of the sport led them to set up Midland Warriors.
Robbie's passion for the sport began at a young age. His commitment to the game, like the Ross's, has led him to be the calibre of player that he is today.
Talking about the ins and outs of everyday life in Midland Warriors, the Ross's spoke to Robbie of what they do to grow the club in the community.
Robbie spoke about his everyday life as a professional rugby player. He described how he grows as a player, by making sacrifices and by committing each day to rugby. From video sessions to gym sessions, training days to set plays, the life of a professional athlete takes passion, grit and determination.
In an extremely important few months for Irish rugby, Robbie, Graham and Kate spoke about their respective preparations for the World Cup. Pre-season camp for the Irish Rugby team started in Carton House on June 17th. The World Cup warm-up games commence in August. Meanwhile, Graham and Kate are building in their preparations for the World Cup. As Robbie and his teammates tog out for their first game, the Ross's will be busy hosting events in their rugby club (KO for Ireland's opener, vs Scotland, is at 8:45am).
That ended our day with Robbie, The Ross's, and Midland Warriors. Thanks to all who took part in the content capture day.
As the Official Retailer Partner for Irish Rugby, here's hoping Robbie and Co. bring home the Webb Ellis trophy this Autumn. Shop Irish Rugby HERE.


Members may use the rinks whenever there is no match and rinks are not booked for competitions.
There are now only mixed matches against local Clubs and those from further a field are usually played on Wednesday and Saturday afternoons.
Members wishing to play should enter their names on the team sheets displayed on the notice board. Tea and biscuits are usually provided during the afternoons but two-course meals may be provided when the teams travel a significant distance.
Two League teams play in the Brighton, Hove & District League on most Tuesdays during the season commencing at 18.00.
There is a Club mixed drive on Mondays starting at 2pm. Please arrive by 1.45pm to put your name down and be ready to play at 2pm.
Club competitions are held between May and September. They include singles, handicapped singles and pairs. Some members also enter county and national competitions.
Club competitions are held between May and September. Some members also enter county and national competitions. The ladies also play in the Brighton, Hove District Triples League.


Ben Affleck will not direct "The Batman" but he will still be a producer and the star of the film.
It seems as if Affleck wants to focus on being the best batman in history; and to do so, he will have to five up the director hat and work only in front of the camera. Although, there are rumours that Warner Bros would not give him full control and the creative freedom he demanded to direct this highly anticipated film.
Do you think he is making the right decision?


This Marie Curie mug is part of a biographical series profiling unique scientists & inventors. This mug includes an image of Marie Curie and a short biography.
This Charles Darwin mug is part of a biographical series profiling unique scientists & inventors. This mug includes an image of Charles Darwin and a short biography.
This Albert Einstein mug is part of a biographical series profiling unique scientists & inventors. This mug includes an image of Albert Einstein and a short biography.
The Mary Jackson mug is part of a biographical series profiling unique individuals who shaped the Science & Physics history of the United States. An image of Mary Jackson and a short biography are features of this history mug.
The Katherine Johnson mug is part of a biographical series profiling unique individuals who shaped the Science & Physics history of the United States. An image of Katherine Johnson and a short biography are features of this history mug.
The Guglielmo Marconi mug is part of a biographical series profiling unique individuals. An image of Guglielmo Marconi and a short biography are features of this history mug.
This Isaac Newton mug is part of a biographical series profiling unique scientists & inventors. This mug includes an image of Isaac Newton and a short biography.
The Silk Moths mug is part of a biographical series profiling unique scientific categories and includes an image of silk moths and a short description.
The Nikola Tesla mug is part of a biographical series profiling unique individuals and includes an image of Nikola Tesla and a short biography.
The Dorothy Vaughan mug is part of a biographical series profiling unique individuals who shaped the Science & Physics history of the United States. An image of Dorothy Vaughan and a short biography are features of this history mug.
This Eli Whitney mug is part of our collection of Civil War biographical mugs. An image of Eli Whitney and a short biography are features of this history mug.


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James Madison Campaign Issues
May 30, 2019 · The Human Rights Campaign is America's largest civil rights organization working to achieve lesbian, gay, bisexual, transgender and queer equality.
Quotes on the Second Amendment: "As civil rulers, not having their duty to the people before them, may attempt to tyrannize, and as the military forces which must be occasionally raised to defend our country, might pervert their power to the injury of their fellow citizens, the people are confirmed by the article in their right to keep and bear their private arms."
James Madison, his position on France; supported June 1781 pro-France Peace instructions, the disclosure of responses to French insults ended the issue.
Did Christopher Columbus Go To School Learn about immigration on Ellis Island in this interactive, virtual tour. Facts about immigration, pictures of Ellis Island, oral histories, and videos help explain the immigration process to kids. President Donald Trump waded into a new statue controversy on Tuesday night when he called out people who want to tear down statues of Christopher. Columbus
But there's a bigger issue that none of the three House members. was suggested as an additional cause for impeachment, but.
Other groups are urging a convention to approve an amendment to address problems caused by the Supreme Court's Citizen United decision on campaign finance. Benjamin Franklin, James Madison and.
In what was virtually a door-to-door campaign, unheard of in 18th century America, Madison managed to narrowly win a House seat against future President James Monroe. out of the bill presented a.
An archive of presidential campaign commercials from 1952 to the present, organized by year, type, and issue, with teacher resources and playlists by experts.
The conservative lawyer James Bopp. profit law firm. The Madison Center shares characteristics with the ostensibly nonpolitical "social welfare" groups that are spending millions in undisclosed.
The Founding Fathers, US Constitution, Constitution Amendments, Bill of Rights, Declaration of Independence, Articles Of Confederation. Constitution IQ Quiz, Constitution Day Materials, Constitution Bookstore, Pocket Constitution Books. Fascinating Facts.
Candidates and issues. Deciding not to run for reelection, Pres. Thomas Jefferson unofficially anointed James Madison, his secretary of state and fellow.
Coming off the heels of a 2018 election that saw the Sunshine State. Just a few of the issue successes we have long.
James Madison was one. He is to political science what Albert Einstein. Linus Pauling won a Nobel Prize for Chemistry in 1954, which gave him an international platform and megaphone to campaign.
A supporter of a strong, centralized government, Wilson joined the likes of James Madison and another Pennsylvania. Matt Riffe is a museum programs demonstrator at the National Constitution Center.
Trump has blocked House committees' subpoenas and other efforts to dig into the Trump-Russia issue, insisting Mueller's.
Library > Bill of Rights • History • That's Not What They Meant by Wayne LaPierre • The Second Amendment History: A Drafting and Ratification of the Bill of Rights in the Colonial Period: As heirs to the majestic constitutional history of England, the intellectual and political leaders of the new Colonies intended nothing less than to incorporate into their new government the laws and.
Apr 04, 2019 · For the first time in 14 years, LeBron James isn't going to the playoffs. Not only that, but he isn't going to finish the season, as the Los Angeles Lakers on.
Apr 16, 2019. James Madison (1751-1836) was a founding father of the United States and. Britain, which had grown tumultuous over the issue of British taxation. Madison's War" and would not allow their militias to join the campaign.
Hand-wringing over the responsibilities of social media companies and users for the perceived deterioration of political discourse online continues, but, with Trump's election. In Federalist Paper.
When Congressman James Madison started pursuing a young vivacious widow named. While he was rarely off the campaign trail and gave multiple media interviews a week, Melania was rarely seen on the.
Why, then, should our support for political causes not also be secret before Election Day, simply because such support is financial? OUR VIEW: 'Dark money' flows into the fall campaign To paraphrase.
May 03, 2019 · An anti-racism activist says she was punched by supporters of Tommy Robinson during ugly scenes on his campaign trail yesterday. Fights broke.
We live in a time when the people in our country are divided over many social, political and economic issues. We all try to understand. saw a similar situation in his young country. James Madison,
This list is subject to change. Please check back often for candidate and link updates. View full candidate list below. View candidates by state: All 2018 candidates: (To search by state in the drop-down field below, please type the full state name rather than the postal abbreviation.)
Oct 29, 2009 · James Madison (1751-1836) was a founding father of the United States and the fourth American president, serving in office from 1809 to 1817. An advocate for.
May 06, 2016 · James Brien Comey, Jr. is a traitor. James Brien Comey, Jr. is the (former) Director of the (FBI) Federal Bureau of Investigation. James Comey has continually refused to comment on the ongoing investigation into the use of Hillary Clinton's private email server.
Jan 08, 2010 · From George Washington's uncontested run for president to the divisive campaigns of 2016, see an overview of all the presidential elections in U.S. history.
James Madison, the chief author of the Bill of Rights and the First. Other articles in Americans prominently involved with First Amendment issues, Presidents. James. In a hard-fought 1788 campaign for a seat in the First Congress, Madison.
American Revolution Map Battles The American Revolution – (Battles) The Glorious Cause for American Independence On any afternoon, I could easily fight the American Revolution, the Civil War. surround sound noise to my imagined battles. He had never seen the American continent, and rarely set foot outside London. But his private papers reveal that he closely tracked the war's
Mar 20, 2017. James Madison, architect and chief lobbyist for the constitution, made it clear. Campaigns target all their money and attention to few states.
In a republic, an official set of fundamental laws, like the U.S. Constitution and Bill of Rights, prohibits the government from limiting or taking away certain "inalienable" rights of the people, even if that government was freely chosen by a majority of the people.In a pure democracy, the voting majority has almost limitless power over the minority.
Oct 2, 2018. America Is Living James Madison's Nightmare. From Our October 2018 Issue. During the election of 1912, the progressive populists Theodore Roosevelt and Woodrow Wilson insisted that the president derived his.
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\subsection{#1}}
\newcommand{\mathfrak}{\mathfrak}
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\newcommand*\D{\mathop{}\!\mathrm{d}}

\newcommand{\begin{equation}}{\begin{equation}}
\newcommand{\end{equation}}{\end{equation}}

\usepackage[capitalise]{cleveref}

\begin{document}

\title{Weyl nodal surfaces}

\author{O\u{g}uz T\"{u}rker}
\email{oguz.tuerker@tu-dresden.de}
\author{Sergej Moroz}
\email{sergej.moroz@tum.de}
\affiliation{ Department of Physics, Technical University of Munich, 85748 Garching, Germany}

\begin{abstract}
We consider three-dimensional fermionic band theories that exhibit Weyl nodal surfaces defined as two-band degeneracies that form closed surfaces in the Brillouin zone. We demonstrate that topology ensures robustness of these objects under small perturbations of a Hamiltonian. This topological robustness is illustrated in several four-band models that exhibit nodal surfaces protected by unitary or anti-unitary symmetries.  Surface states and Nielsen-Ninomiya doubling of nodal surfaces are also investigated.
\end{abstract}

\maketitle

\section{Introduction} \label{intro}

The advent of topological insulators in the last decade deepened  our understanding of interplay of topology and symmetries in band insulators \cite{Hasan2010, Qi2011}. This work culminated in the development of the ten-fold way classification of non-interacting gapped topological phases \cite{Schnyder2008} and the emergence of new symmetry protected topological phases of matter.

In last years the main interest in the field shifted towards systems with band degeneracies \cite{Turner2013, Burkov2015, Armitage2017}. In three dimensions the simplest and most well-studied are Weyl (semi)metals which are distinguished by isolated point-like two-band degeneracies in the Brillouin zone (BZ). Although in condensed matter physics Weyl points appeared first long time ago in the superfluid A phase of $^3$He \cite{volovik1992exotic, volovikbook}, only recently Weyl (semi)metals were discovered experimentally \cite{Xu2015, Lv2015}. As long as inversion or time-reversal symmetry is broken, Weyl points appear generically. They are topological defects and cannot be gapped out individually but must be destroyed only via pair-wise annihilation \cite{Nielsen1981}. Weyl (semi)metals exhibit robust phenomena such as chiral anomaly \cite{Son2013}, anomalous Hall effect \cite{Yang2011} and zero-energy Fermi arc surface states \cite{Wan2011}. More recently Weyl loop (semi)metals \cite{Burkov2011}, where two-band degeneracies take place on closed one-dimensional manifolds also attracted considerable attention. In contrast to Weyl points, nodal loops are not generic, but require some symmetry (such as chiral sub-lattice symmetry) to protect them. The defining feature of a Weyl loop is a non-trivial $\pi$ Berry phase along any closed contour that links with it. In the semimetal regime, any boundary surface, where the loop projects non-trivially, supports drumhead states \cite{Burkov2011, Kim2015}.

One can make a step further and consider three-dimensional translation-invariant fermionic systems with Weyl nodal surfaces, where two bands touch each other on two-dimensional surfaces in the BZ.   Recently nodal surfaces were predicted to appear in quasi-one-dimensional crystals \cite{QiFeng2016},  graphene networks \cite{Zhong2016},   multi-band superconductors with broken time-reversal symmetry \cite{Agterberg2017, Timm2017} and also were found within the ten-fold way classification of gapless inversion-enriched systems \cite{Bzdusek2017}.
Given a model with a nodal surface it is natural to ask if the nodal structure is robust under certain class of small perturbations of the Hamiltonian. Generically a perturbation can (i) open a gap everywhere and fully destroy the nodal object (ii) gap it out partially leaving behind nodal loops and/or points \footnote{One can imagine a scenario where a closed nodal surface is partially gapped to a nodal surface with boundaries. This case is not discussed in this paper.}, (iii) preserve the nodal surface and not open a gap anywhere on it. As we show in this paper the degree of robustness (i)-(iii) is determined by topology of the perturbed system. To see how it works, consider first a system which has only one nodal surface (tuned for convenience to the Fermi level), but no other gapless objects at the Fermi level such as additional Fermi surfaces, nodal loops or points. Now we enclose the nodal surface of the original model by a lower-dimensional ($d_\text{m}<3$) manifold in the BZ (see Fig. \ref{fig1}).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{manifolds.pdf}
\caption{Examples of (a) zero-, (b) one- and (c) two-dimensional enclosing manifolds (blue) around nodal surfaces (orange). Although technically a zero-dimensional manifold does not enclose the nodal surface with slight abuse of notation we still refer to it here as an enclosing manifold.}\label{fig1}
\end{center}
\end{figure}
 By construction, all bands are gapped on the enclosing manifold. Imagine now that in the original model we can define a topological invariant (such as Chern number, winding number, $\mathbb{Z}_2$ invariant, etc) on this enclosing manifold. In this paper such invariant will be denoted as $c_{d_{\text{m}}}$, where the subscript specifies the dimension of the enclosing manifold.  Importantly, the invariant does not change under a certain class of perturbation of the Hamiltonian and as we show now this has implications for the degree of robustness of the nodal surface under this class of perturbations. Previously, this set of ideas was introduced for determining the robustness of Weyl nodal loops \cite{Fang2015, Fang2016}.

 First, we discuss topological invariants defined on a point ($d_\text{m}=0$, Fig. \ref{fig1} a) enclosing manifold. As long as such invariants are different for point manifolds placed inside and outside of the nodal surface in the BZ, the nodal surface cannot be gapped. The reason for that is the following:
notice that on any imaginary one-dimensional trajectory in the BZ that connects the inner and outer zero-dimensional manifolds  there should be a point, where the energy gap closes allowing the topological invariant to change. In turn this ensures that no gap can open at any point of the nodal surface.

 Second, consider non-trivial topological invariants defined on enclosing manifolds of dimension $d_\text{m}>0$ (Fig. \ref{fig1} b and c). These invariants do not fully protect the nodal surface, but guarantee that it cannot be fully gapped out by perturbations. Generically,   a nodal line (nodal points) should survive in the perturbed system if $d_\text{m}=1$ ($d_\text{m}=2$). We can explain it by following similar arguments as before: in the unperturbed model compute the topological invariant on two enclosing manifolds, one being inside and another outside the nodal surface. Since the topological invariant on the interior manifold  is necessarily trivial (the manifold can be shrunk to a point without encountering a band), a non-trivial invariant on the exterior manifold necessarily implies a non-trivial difference of the interior and exterior invariants which in turn guarantees a lower-dimensional nodal object in between the two enclosing manifolds in the perturbed system.

We emphasize that arguments presented above also imply robustness of general nodal surfaces, not necessarily tuned to the Fermi level. In a general case it is useful to formulate the above arguments in energy-momentum space, as detailed in Appendix \ref{App1}. Our analysis thus does not reduce to previous studies of  robustness of Fermi surfaces \cite{volovikbook, horava2005, Zhao2013, Matsuura2013}.

To apply these ideas in practice, in this paper we investigate nodal surfaces protected by different mechanisms: (i) a nodal surface can be protected by a global internal symmetry. In this case any two bands that carry different quantum numbers with respect to the symmetry generically intersect on a two-dimensional surface in momentum space. Since in the presence of the symmetry the two bands cannot be hybridized, the nodal surface is protected against any small symmetry-preserving perturbations. (ii) in spirit of ten-fold way, Weyl nodal surfaces can be protected by anti-unitary symmetries. These nodal surfaces have already appeared in the literature \cite{Agterberg2017, Bzdusek2017, Timm2017} and here we investigate their robustness.
We analyze simple models that exhibit nodal surfaces protected by both mechanisms mentioned above.
The leitmotif of our construction is the following:  we start from double-degenerate nodal points or nodal lines and split them in energy.
The resulting continuum four-band models together with minimal lattice extensions thereof are used to analyze the physics of nodal surfaces and investigate their robustness. In particular, we identify symmetries that protect the nodal surface and construct appropriate topological invariants. In addition, we investigate Nielsen-Ninomiya doubling of nodal surfaces in the BZ and look for surface states and topological invariants that protect these states.

\section{Nodal surfaces protected by unitary $U(1)$ symmetry}  \label{nodalU1}
In this section we construct and investigate non-interacting fermionic four-band models that exhibit nodal surfaces protected by an internal unitary $U(1)$ symmetry.
In examples discussed here this symmetry is generated by the axial $\gamma^5$ matrix. More physical realization of a global internal $U(1)$ symmetry might be the conservation of a single component of the spin operator which can happen in a constant magnetic field if the spin-orbit coupling is weak.

\subsection{Nodal sphere} \label{nss}
Given the set of four-by-four Dirac matrices $\gamma^\mu$ that satisfy the Clifford algebra $\{\gamma^\mu, \gamma^{\nu} \}=2 \eta^{\mu\nu}$ with the Minkowski metric $\eta^{\mu\nu}$ and $\mu, \nu=0,x,y,z$, we first define $\alpha^i=\gamma^0 \gamma^i$ and $\gamma^5=\mathrm{i}\gamma^0 \gamma^x \gamma^y \gamma^z$. Now
consider the Hamiltonian, which describes a massless Dirac fermion perturbed by a $\gamma^5$ term
\begin{equation} \label{mod1}
\mathcal{H}(\mathbf{k})=k_i \alpha^i- \lambda \gamma^5,
\end{equation}
where $k_i$ is the crystal momentum and $\lambda \in \mathbb{R}$. Since $[\gamma^5, \mathcal{H}]=0$, the model has an internal $U(1)$ symmetry \footnote{In high energy physics this is known as the axial symmetry.} generated by the matrix $\gamma^5$.  In this paper we use the chiral representation of the Dirac matrices resulting in $\alpha^i=\sigma^{z}\otimes\tau^{i}$ and $\gamma^{5}=-\sigma^{z} \otimes \tau^{0}$. The last term in  \cref{mod1} preserves time-reversal, but breaks the inversion symmetry. It splits the Dirac point at $\mathbf{k}=0$ into a pair of Weyl points of opposite chirality by separating them in energy by $2\lambda$. This model is in DIII symmetry class of the ten-fold way classification \cite{Schnyder2008}.  It appeared in the context of studies of the chiral magnetic effect \cite{Fukushima2008}.
The energy spectrum
\begin{equation}
E(\mathbf{k})=\pm |\mathbf{k}|\pm \lambda
\end{equation}
exhibits a band degeneracy at the Fermi level $E=0$ on a sphere defined by $|\mathbf{k}|=|\lambda|$.  The nodal surface is protected by the $\gamma^{5}$ symmetry against perturbations since two bands that cross each other have different $\gamma^{5}$ eigenvalues and cannot be hybridized. Under a generic $\gamma^5$-symmetric perturbation the sphere will deform and move away from $E=0$, but will not be gapped out. On the other hand, by adding a mass term $\sim\gamma^0= \sigma^x\otimes \tau^0$ to the Hamiltonian, given by \cref{mod1}, the symmetry is broken and the nodal sphere disappears. This behavior can be understood using topology, as we will explain in the following.

On a zero-dimensional ($d_m=0$) enclosing manifold (Fig. \ref{fig1} a) an integer-valued topological invariant tied to the $\gamma^5$ symmetry can be defined  as  the $\gamma^5$ quantum number of the lower band that generates the nodal sphere (in our model this is the  occupied band with the highest energy). Given a normalized Bloch state $|u(\mathbf{k)}\rangle$ of this band, we define
\begin{equation} \label{cof}
c_0(\mathbf{k})=\langle u(\mathbf{k})| \gamma^5 | u(\mathbf{k}) \rangle.
\end{equation}
If this band is degenerate in energy, the $\gamma^5$ quantum numbers of individual bands should be summed, thus $c_{0}\in\mathbb{Z}$.
 In the presence of a nodal surface one can define a topological invariant $\Delta c_0=\frac{1}{2}[c_{0}(\mathbf{k}_{\text{in}})-c_{0}(\mathbf{k}_{\text{out}})]$, where the momentum $\mathbf{k}_{\text{in(out)}}$ is  located anywhere inside (outside) the nodal surface. Since the occupied band with the highest energy has opposite $\gamma^5$ quantum numbers outside and inside the nodal surface, the difference $\Delta c_0$ is non-trivial for the model given by \cref{mod1}. As discussed in Sec. \ref{intro},  if we think of a general $\gamma^5$-symmetric system with a nodal sphere, we cannot gap out a nodal sphere with an infinitesimal $\gamma^5$-symmetry preserving term, if  $\Delta c_0\in \mathbb{Z} \setminus  \{ 0\}$. Consequently we conclude that $\Delta c_0$ is a $\mathbb{Z}$-valued topological charge.

In addition, an integer-valued topological invariant can be defined on a two-dimensional ($d_m=2$) manifold enclosing the nodal sphere (Fig. \ref{fig1} c). This is the Chern number of the lower band that generates the nodal sphere (the occupied band with the highest energy)\begin{equation} \label{Chc2}
c_2(S)= \frac{\mathrm{i}}{2\pi} \oiint_{S} \D \mathbf{k} \, \nabla_{\mathbf{k}} \times \mathcal{A}(\mathbf{k}),
\end{equation}
where the Berry connection $\mathcal{A}(\mathbf{k})=\langle u (\mathbf{k})| \nabla_{\mathbf{k}}u (\mathbf{k})\rangle$. In the model, given by \cref{mod1}, this Chern number is non-trivial but does not change as the radius of the enclosing manifolds crosses the nodal sphere, i.e. $c^{\text{in}}_2=c^{\text{out}}_2$. Since the difference of the Chern numbers \cref{Chc2} inside and outside the nodal sphere is zero, we conclude that the nodal surface in this model has no robustness with respect to a generic perturbation that breaks $\gamma^5$ symmetry. Such perturbation, as for example the mass term $\sim\gamma^0= \sigma^x\otimes \tau^0$, fully gaps out the nodal surface.

We now investigate the surface states of the model given by \cref{mod1}.
Imagine that the system has a boundary  at $z=0$ and fills only a half of the space $z>0$. Due to $\gamma^5$ symmetry the Hamiltonian splits into two decoupled Weyl blocks. Consequently, one expects two degenerate Fermi arcs starting at the nodal surface that are protected by $\gamma^5$ symmetry. This can be demonstrated analytically by  first introducing the following boundary condition
$
\sigma^0 \otimes \tau^y \psi=\psi
$
that preserves helicity of excitations. In the spirit of \cite{Witten2016}, we extend the boundary condition also into the bulk. Solving the Schr\"odinger equation  $\mathcal{H}\psi_{\mp}=E_{\mp}\psi_{\mp}$, where $\mathcal{H}$ is given by  \cref{mod1} gives the surface states $ \psi_{-}=(0,\phi_{-} )^T$ and  $\psi_{+}=( \phi_+,0)^T$ with
\begin{equation}
\begin{split}
&\phi_{\mp}=\exp(\mathrm{i}k_{x}x+\mathrm{i}k_{y}y-\kappa z)\phi^0,
\end{split}
\end{equation}
and
\begin{equation}
	E_{\mp}=\mp(k_{y}+\lambda),
\end{equation}
where $\phi^0$ is a constant spinor that satisfies $ \tau^y \phi^0=\phi^0$ and $\kappa =-k_{x}$.
For $\kappa>0$ these solutions are normalizable, and localized close to the boundary. If we set $E=0$, we find a doubly-degenerate Fermi arc at the surface BZ  at $(k_x<0, k_y=-\lambda)$. It starts at ($k_{x}=0$), i.e. the boundary of the projection of the nodal surface on the surface BZ (see \cref{fig: fig2}).  $\psi_{+}$ and $\psi_{-}$   have opposite $\gamma^5$ eigenvalues and counter-propagate along the $y$ axis.
These counter-propagating surface modes are robust under $\gamma^5$-symmetric perturbations $\mathcal{H}_{\text{imp}}$ since $\langle\psi_{-}|\mathcal{H}_{\text{imp}}|\psi_{+}\rangle=0$.

After calculating the  surface states, we investigate their topologically stability. We have seen that each sub-block of  \cref{mod1} produces its
own surface state with a particular $\gamma_{5}$ quantum number.
As long as $\gamma_{5}$ quantum number is conserved,  states with different quantum numbers cannot influence each other.
Thus we have to check the topological properties of each subsystem
with a definite $\gamma_{5}$ eigenvalue, separately. However, those subsystems are just Weyl Hamiltonians \emph{displaced}
in energy. Therefore, what we are actually interested in is, the topology
of \emph{Fermi surfaces} of the  Weyl sub-systems. The appropriate topological invariant for the blocks is  $\gamma^5$-Chern number, which is given by
\begin{equation}
c_2^{(\ell)}(S)=\sum_{n\in \operatorname{occupied}}\frac{\mathrm{i}}{2\pi} \oiint _{S} \D \mathbf{k}\langle\nabla_{\mathbf{k}}u_{n,\ell}(\mathbf{k})|\times|\nabla_{\mathbf{k}}u_{n,\ell}(\mathbf{k})\rangle\,\label{eq: g5cn}
\end{equation}
where $\ell$ denotes the eigenvalue of $\gamma^{5}$ matrix.
 We emphasize that $c_2^{(\ell)}$ characterises the stability of the surfaces states under $\gamma^{5}$ preserving infinitesimal perturbation, on the other hand the Chern number of the  upper filled band \cref{Chc2} characterizes the stability of the nodal surface under generic infinitesimal perturbations. For our model, the absolute value of the $\gamma^{5}$-Chern numbers is unity (zero) for enclosing manifolds of radius larger (smaller) than the radius of the nodal sphere.

The continuum model \cref{mod1} cannot be used in the full BZ since it violates its periodicity. We conclude this section with a discussion of a minimal lattice realization of this model. The  lattice Hamiltonian is given by
\begin{equation} \label{mod1latt}
\begin{split}
\mathcal{H}(\mathbf{k})=&-\Big([2-\cos k_y-\cos k_z]+2 t [\cos k_x-\cos k_0] \Big)\alpha^x \\
&-2 t \sin k_y \alpha^y -2 t \sin k_z \alpha^z- \lambda \gamma^5,
\end{split}
\end{equation}
where $t\in\mathbb{R}$.
The model originates from the minimal lattice model of Weyl semimetals introduced in  \cite{Yang2011} and discussed extensively in \cite{McCormick2017}. As sketched in Fig. \ref{figdoub2}, the lattice model has two pairs of Weyl points located at $\mathbf{k}=(\pm k_0, 0, 0)$. Due to the $U(1)$ symmetry only Weyl points of the same $\gamma^5$  quantum number can be connected to each other. Hence the Nielsen-Ninomiya theorem must be applied in the two $\gamma^5$ sectors separately and nodal spheres appear necessarily in pairs.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{figdoub2.png}
\caption{Schematic spectrum of the lattice model \cref{mod1latt} for $t, \, \lambda>0$: Red and blue bands have $\gamma^5$ quantum number $+1$ and $-1$ and cannot be hybridized and connect to each other.}\label{figdoub2}
\end{center}
\end{figure}

Since there are two nodal surfaces in the minimal lattice realization, it is natural to expect that the Fermi arcs that we found above will connect the two nodal surfaces projected on the boundary BZ (see Fig. \ref{fig: fig2} a). This is indeed what one finds by diagonalizing numerically the lattice Hamiltonian in a slab geometry (see Fig. \ref{fig: fig2} b). The Fermi arcs are protected by the $\gamma^5$ symmetry since by adding a mass term $\sim \sigma^x\otimes \tau^0$ to the lattice Hamiltonian \cref{mod1latt}, the Fermi arcs hybridize and gap out.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{5dg3.png}
\caption{Fermi arcs in a slab geometry: (a)A degenerate pair of counter-propagating surface Fermi arcs (cyan) on the upper boundary connecting the projections of nodal Fermi surfaces. For simplicity the Fermi arcs on the lower boundary are not displayed; (b) Fermi arcs (red) in the energy spectrum of the Hamiltonian \cref{mod1latt}. The counter-propagating  surface states has opposite $\gamma^5$ quantum numbers and cannot be hybridized by symmetry-preserving perturbations.  }\label{fig: fig2}
\end{center}
\end{figure}

\subsection{Nodal torus} \label{ntu}
We start from a four-band model with the effective low-energy Hamiltonian
\begin{equation} \label{mod2}
\mathcal{H}(\mathbf{k})=(k_{\perp}-k_0) \alpha^x+ k_z \alpha^y - \lambda \gamma^5,
\end{equation}
with $k_{\perp}=\sqrt{k_{x} ^{2}+k_{y} ^{2}}$. This model has the $\gamma^5$ symmetry. The last term in  \cref{mod2} splits a doubly-degenerate (Dirac) loop of radius $k_0$ at $k_z=0$ into a pair of Weyl loops by separating them in energy by an amount of $2\lambda$.
The dispersion relation for this system is given by
\begin{equation}
E(\mathbf{k})=\pm \sqrt{(k_{\perp}-k_{0})^{2}+k_{z} ^{2}}\pm\lambda,
\end{equation}
and exhibits a band degeneracy at $E=0$ with $\sqrt{(k_{\perp}-k_{0})^{2}+k_{z} ^{2}}=\lambda$ which for $\lambda<k_0$ forms a torus in momentum space.

In a close analogy to the nodal sphere discussed in Sec. \ref{nss},  the nodal torus is protected by the unitary $\gamma^5$ symmetry. The appropriate $\mathbb{Z}$-valued topological invariant defined on a zero-dimensional enclosing manifold is given by  \cref{cof}. For the model \cref{mod2} the difference of the invariants defined inside and outside the nodal torus is non-trivial which ensures its full robustness under  infinitesimal  $\gamma^5$-preserving perturbations.

In addition to the $\gamma^5$ symmetry, the model, given by  \cref{mod2}, is also invariant under the combination of inversion $P$ and  time reversal $T$, i.e. $PT$\footnote{\label{fn1}. In addition to the $PT$ symmetry, the nodal tori models \cref{mod2} and \cref{mod4} are also invariant a unitary mirror ($z\to -z$) symmetry, but we do not discuss its consequences in this paper.},  which implies the following constraint on the   Hamiltonian
\begin{equation}
U_{PT} \mathcal{H}^{*}(\mathbf{k})U_{PT}^{-1}=\mathcal{H}(\mathbf{k}),
\end{equation}
where $U_{PT}=\alpha^x$.
Analogously to the previous section, we can treat the diagonal sub-blocks of this system separately. The sub-blocks give rise nodal loops  lying at energies $\pm\lambda$,
which are protected by the $PT$ symmetry. The topological invariant that characterizes the stability  of such nodal loops is the Berry phase which is quantized in  $PT$-symmetric systems\footnote{If we break $PT$ symmetry, the block Berry phase can be  changed continuously.}\cite{rui2017}. In our model \cref{mod2} we find $\pi$ Berry phase for the blocks.
In addition, the Berry curvature is forced to be  zero for every non-singular point in the BZ. As a result, $\pi$ Berry phase guarantees the existence of nodal loops that
 are stable   under infinitesimal perturbations that preserve  $PT$ and $\gamma^5$ symmetries. On the other hand, the $PT$ symmetry is not required for the stability of the nodal torus.  In fact, if we break this symmetry without breaking $\gamma^5$ symmetry, the nodal torus will not be destroyed, but the nodal loops lying at   energies $\pm\lambda$ will be gapped. Hence we can refer to $PT$ symmetry as an accidental symmetry.

In order to illustrate the arguments given above in a concrete example, we consider a minimal lattice model that hosts a nodal torus. Its Hamiltonian is given by
\begin{equation} \label{mo21latt}
\begin{split}
\mathcal{H}(\mathbf{k})=&-\Big(6t_1-2 t_2 [\cos k_x+ \cos k_y+ \cos k_z] \Big)\alpha^x \\
&-2 t_2 \sin k_z \alpha^y- \lambda \gamma^5,
\end{split}
\end{equation}
where $t_{1,2}$ are constants which fix the radius of the torus. This Hamiltonian  is motivated by the lattice model of a nodal loop semimetal investigated in \cite{Wang2017}. It is worth pointing out that this lattice model has just  one nodal torus in the BZ  (no Nielsen-Ninomiya doubling).
The bulk energy spectrum of this Hamiltonian is shown in  \cref{fig: nodalloops}-b.
Furthermore, in a slab geometry (as a result of the existence of the nodal loops in  the bulk energy spectrum) we find  drumhead surface states that have the same energy as the nodal loops as shown in \cref{fig: nodalloops}-a. A $\gamma^5$-invariant perturbation that breaks the $PT$ symmetry destroys the nodal loops, but not the nodal torus (see \cref{fig: nodalloops}-c,d).

\begin{figure}[h]
	\includegraphics[width=0.47\textwidth]{nodalloops}
	\caption{\label{fig: nodalloops} Energy spectrum of the model \eqref{mod2} in the (a) slab and (b) infinite geometry. The drumhead surface states in (a)  lie at the energy levels of the nodal loops. In (b) the crossings of the red and black bands  at the Fermi level form a nodal torus. The panels (c) and (d) display the energy spectrum of the model \eqref{mod2} perturbed by a term that breaks accidental $PT$ symmetry, but preserves $\gamma^5$ symmetry.}
\end{figure}

\section{Nodal surfaces protected by anti-unitary symmetry} \label{nsaU}
Here we turn to four-band models that exhibit nodal surfaces which are protected by anti-unitary symmetries. These objects appeared in the recent literature \cite{Agterberg2017,  Bzdusek2017, Timm2017} and formed the starting point of our investigation.

\subsection{Nodal sphere} \label{nst}
Consider a four-band model with the Hamiltonian
\begin{equation} \label{infW}
\mathcal{H}(\mathbf{k})= k_i \tilde \alpha^i- \lambda \gamma^5-\delta \gamma^0,
\end{equation}
where  $\tilde\alpha^i=\sigma^0\otimes \tau^i$, $\gamma^5=-\sigma^z\otimes \tau^0$ and $\gamma^0=\sigma^x\otimes \tau^0$.
The model consists of a pair of Weyl points of the same chirality which are split in energy by the last two terms in  \cref{infW}. Notice that the Hamiltonian has no unitary $U(1)$ symmetry apart from the particle number conservation.
Even in the absence of a unitary symmetry, the spectrum of the Hamiltonian \cref{infW}
\begin{equation}
E(\mathbf{k})=\pm |\mathbf{k}|\pm \sqrt{\lambda^2+\delta^2}
\end{equation}
contains a band degeneracy at $E=0$ which  is located on a sphere of radius $|\mathbf{k}|=\sqrt{\lambda^2+\delta^2}$.

We will argue now that in this model the nodal surface is protected by an anti-unitary combination of inversion $P$ and particle-hole $C$ symmetry \cite{Kobayashi2014, Zhao2016} and construct a $\mathbb{Z}_2$-valued topological invariant tied to this symmetry \cite{Agterberg2017, Bzdusek2017}.
To this end consider a class of Hamiltonians of the form
\begin{equation} \label{PCH}
\mathcal{H}(\mathbf{k})=a_i (\mathbf{k}) \tilde \alpha^i+ b_i (\mathbf{k})\tilde\beta^i
\end{equation}
with $\tilde\alpha^i=\sigma^0\otimes \tau^i$ and $\tilde\beta^i=\sigma^i\otimes \tau^0$ and $\mathbf{a}$ and $\mathbf{b}$ being real functions of $\mathbf{k}$. The model \cref{infW} is a special case of \cref{PCH} since $\gamma^0 =\tilde \beta^x$ and $\gamma^5=-\tilde \beta^z$. It will turn out important in the following that the Hamiltonian \cref{PCH} can be unitary rotated to an antisymmetric form \cite{Agterberg2017, Bzdusek2017} to be denoted by $\bar{\mathcal{H}}$.   The spectrum of \cref{PCH}
\begin{equation}
E(\mathbf{k})=\pm|\mathbf{a}(\mathbf{k})|\pm|\mathbf{b}(\mathbf{k})|
\end{equation}
supports a zero-energy nodal surface at $|\mathbf{a}(\mathbf{k})|=|\mathbf{b}(\mathbf{k})|$.
A generic model \cref{PCH} breaks inversion $P$, time-reversal $T$ and particle-hole $C$ symmetries. Nevertheless, it is invariant under the anti-unitary $PC$ symmetry which acts on the Hamiltonian as
\begin{equation}
U_{PC} \mathcal{H}^{*}(\mathbf{k})U_{PC}^{-1}=-\mathcal{H}(\mathbf{k}).
\end{equation}
In our representation $U_{PC}=\sigma^y\otimes \tau^y$ is real. Note that $(PC)^2=U_{PC}U_{PC}^*=+\mathbb{1}$. Due to this symmetry the nodal surface is fixed to the Fermi level and thus necessarily coincides with the Fermi surfaces of the two touching bands. It was found in \cite{Kobayashi2014, Zhao2016} that three-dimensional Fermi surfaces with $PC$ symmetry that squares to unity  have topological $\mathbb{Z}_2$ classification. In our formulation, the topological invariant defined on a zero-dimensional enclosing manifold can be calculated as the sign of the Pfaffian of the antisymmetric form of the Hamiltonian $\bar{\mathcal{H}}$
\begin{equation} \label{pfc}
c_0(\mathbf{k})=\text{sgn}\text{Pf} \, \bar{\mathcal{H}}(\mathbf{k}).
\end{equation}

Since in our representation the Hamiltonian \cref{PCH} is not antisymmetric, in order to compute the invariant, we must first unitary rotate it to the antisymmetric form
$
\bar{ \mathcal{H}}= \Omega \mathcal{H} \Omega^\dagger.
$
The resulting Pfaffian
\begin{equation} \label{Pfeq}
\text{Pf} \, \bar {\mathcal{H}}=\mathbf{b}^2(\mathbf{k})-\mathbf{a}^2(\mathbf{k})
\end{equation}
is real and vanishes on the nodal surface.
In particular, for the model \cref{infW} the Pfaffian has a simple zero on the nodal sphere and thus it has opposite signs inside and outside of it. Following the arguments of Sec. \ref{intro}, the non-trivial difference of the Pfaffians makes the nodal sphere fully robust with respect to small $PC$-invariant perturbations. We notice that although the choice of $\Omega$ and the sign of the resulting $c_0(\mathbf{k})$ is not unique, this ambiguity does not affect the difference of the topological invariants.

We discuss now the Chern number invariant \cref{Chc2} in the context of the model \cref{infW}.
 For an enclosing manifold located inside (outside) the nodal sphere the band Chern number is non-trivial and in addition $c^{\text{in}}_2=-c^{\text{out}}_2$. The non-trivial difference of the band Chern numbers $\Delta c_2=c^{\text{in}}_2-c^{\text{out}}_2$ ensures that a generic small perturbation cannot fully gap out the nodal sphere, but has to leave in the band structure at least a pair of Weyl points close to the Fermi level. Thus contrary to the nodal objects from Sec. \ref{nodalU1}, the nodal sphere discussed here has certain robustness with respect to arbitrary perturbations of the Hamiltonian.

In the presence of a spatial boundary, for $\lambda=\delta=0$ in  \cref{infW} a pair of same-chirality Weyl points at the Fermi level gives rise to a pair of chiral co-propagating zero-energy Fermi arcs. As the Weyl points are split in energy  for $\lambda, \, \delta \ne 0$, the Fermi arcs survive but must start at the Fermi surfaces. These surface states are robust against arbitrary small perturbations thanks to the the total Chern number
\begin{equation} \label{Chern}
\text{total Chern number}=\frac{i}{2\pi} \int_{S^2} d \mathbf{k} \, \text{tr} \mathcal{F}
\end{equation}
which is non-trivial on an enclosing manifold outside the Fermi surfaces. Here the Berry curvature two-form $\mathcal{F}=d \mathcal{A}+\mathcal{A}\wedge \mathcal{A}$ is defined in terms of the non-abelian Berry connection $\mathcal{A}^{ab}=\langle u^a(\mathbf{k})| d u^b(\mathbf{k}) \rangle$, where $a,b$ label only occupied bands. On an enclosing manifold living inside the nodal sphere the total Chern number \cref{Chern} vanishes and thus the nodal sphere represents a locus of the total Berry flux. It is an inflated double Weyl monopole \cite{Agterberg2017}.

Due to the Nielsen-Ninomiya theorem the nodal spheres protected by the anti-unitary $PC$ symmetry should always appear in pairs. To demonstrate it explicitly we considered the minimal lattice Hamiltonian
\begin{equation} \label{latdoub}
\begin{split}
\mathcal{H}(\mathbf{k})=&-\Big([2-\cos k_y-\cos k_z]+2 t [\cos k_x-\cos k_0] \Big)\tilde \alpha^x \\
&-2 t \sin k_y \tilde \alpha^y -2 t \sin k_z \tilde \alpha^z-\delta \gamma^0- \lambda \gamma^5.
\end{split}
\end{equation}
and determined numerically its energy spectrum in the bulk.
In addition, we examined the energy spectrum of this lattice Hamiltonian in a slab geometry. As expected, it contains a pair of chiral zero-energy Fermi arcs which are robust with respect to arbitrary perturbations of the Hamiltonian.

\subsection{Nodal torus} \label{ntt}
Finally we construct and analyze a four-band model that exhibits a nodal torus that is protected by an anti-unitary symmetry.  The model is defined by the real Hamiltonian
\begin{equation} \label{mod4}
\mathcal{H}(\mathbf{k})=(k_{\perp}-k_0) \tilde \alpha^x+ k_z \tilde \alpha^z - \lambda \gamma^5-\delta \gamma^0
\end{equation}
with $k_{\perp}=\sqrt{k_{x} ^{2}+k_{y} ^{2}}$. Apart from the particle number symmetry, there is no $U(1)$ symmetry. Nevertheless the energy spectrum
\begin{equation}
E(\mathbf{k})=\pm \sqrt{(k_{\perp}-k_{0})^{2}+k_{z} ^{2}}\pm\sqrt{\lambda^2+\delta^2}
\end{equation}
has a zero-energy band degeneracy at $\sqrt{(k_{\perp}-k_{0})^{2}+k_{z} ^{2}}=\sqrt{\lambda^2+\delta^2}$ forming a torus in momentum space for $\sqrt{\lambda^2+\delta^2}<k_0$. One can view  the nodal torus as an inflated double Weyl loop \cite{Agterberg2017}.

Since the model \cref{mod4} falls into the class of $PC$-symmetric Hamiltonians \cref{PCH}, similar to the nodal sphere discussed in section \ref{nst}, one can define the Pfaffian $\mathbb{Z}_2$ invariant \cref{pfc}. By evaluating the Pfaffian \cref{Pfeq} in the model \cref{mod4}, one finds that the difference of the Pfaffian invariant inside and outside the nodal torus is non-trivial and thus this object cannot be gapped out by small $PC$-invariant perturbations of the Hamiltonian.

Incidentally, the model \cref{mod4} enjoys more symmetries. Similar to the nodal torus model discussed in \cref{ntu}, this system has the anti-unitary $PT$ symmetry. In this case the Hamiltonian is real and $PT$ acts on the Hamiltonian as
\begin{equation} \label{PTsym}
\mathcal{H}^{*}({\mathbf{k}})=\mathcal{H}({\mathbf{k}}).
\end{equation}
In general, this symmetry implies that the Chern number \cref{Chern} is zero in this symmetry class.
In addition, this symmetry together with the anti-unitary $PC$ symmetry implies the unitary chiral sublattice (anti)symmetry of the model \cref{mod4}
\begin{equation}
\{\mathcal{H}(\mathbf{k}), U_S \}=0, \qquad U_S=\sigma^y \otimes \tau^y
\end{equation}
and gives rise to a $\mathbb{Z}_2$ winding number topological invariant introduced in \cite{Bzdusek2017}. In order to construct this invariant, it is convenient to perform a unitary rotation $\Omega$ which diagonalizes the chiral sublattice symmetry operator $S\to\Omega^\dagger S \Omega=\sigma^z\otimes \tau^0$. This transformation brings the Hamiltonian into the block off-diagonal form
\begin{equation} \label{chH}
\mathcal{H}(\mathbf{k})=\left(
\begin{array}{cc}
 0 & h(\mathbf{k}) \\
 {h}^\dagger(\mathbf{k}) & 0
\end{array}
\right),
\end{equation}
Here due to the $PT$ symmetry the block Hamiltonian $h$ is real.
To proceed, it is useful to define a flattened Hamiltonian which for a general non-interacting fermionic system with $n$ filled and $m$ empty bands reads
\begin{equation}
Q(\mathbf{k})=U(\mathbf{k})\left(
\begin{array}{cc}
 \mathbb{1}_{m\times m} & 0 \\
0 & -\mathbb{1}_{n\times n}
\end{array}
\right)
U^{\dagger}(\mathbf{k}),
\end{equation}
where the unitary matrix $U(\mathbf{k})$ diagonalizes the Hamiltonian
\begin{equation}
\begin{split}
&U^{\dagger}(\mathbf{k}) \mathcal{H}(\mathbf{k}) U(\mathbf{k})= \\
&=\text{diag}(\underbrace{\epsilon_{m+n}(\mathbf{k}), \dots, \epsilon_{n+1}(\mathbf{k})}_{\text{empty bands}}, \underbrace{\epsilon_{n}(\mathbf{k}), \dots, \epsilon_{1}(\mathbf{k})}_{\text{filled bands}}).
\end{split}
\end{equation}
The flattened Hamiltonian $Q(\mathbf{k})$ is well-defined only away from band degeneracies. In the presence of the chiral sublattice symmetry $m=n$ and the flattened Hamiltonian is block off-diagonal
\begin{equation} \label{chq}
Q(\mathbf{k})=\left(
\begin{array}{cc}
 0 & q(\mathbf{k}) \\
q^\dagger(\mathbf{k}) & 0
\end{array}
\right)
\end{equation}
in any basis which diagonalizes the operator $S$.
Note that in general $q(\mathbf{k})\in \text{U}(n)$ since $Q^2=\mathbb{1}$. The additional reality condition \cref{PTsym} that follows from the $PT$ symmetry implies that the flattened block $q(\mathbf{k})\in \text{O}(n)$.
In particular, for the four-band model \cref{mod4} one has $q(\mathbf{k})\in \text{O}(2)$.
Now  on any closed one-dimensional manifold that does not intersect the torus in the BZ one can compute the topological winding number \cite{Bzdusek2017} as the homotopy equivalence class of mappings $S^1\to SO(2)$. We define the winding number as
\begin{equation}
c_1=-\frac{i}{4\pi}\oint ds \text{Tr} \, \big[ q^{T}\sigma^y \partial_s q \big],
\end{equation}
where $s$ is a coordinate that parametrizes the one-dimensional enclosing manifold. The invariant
 is integer-valued since the homotopy group $\pi_1[SO(2)]=\mathbb{Z}$ \footnote{Since $\pi_1[SO(n)]=\mathbb{Z}_2$ for $n>2$, the winding number is only $\mathbb{Z}_2$-valued for models with more than four bands \cite{Bzdusek2017}.}. It is instructive now to evaluate the winding number invariant for the nodal torus model \cref{mod4}. To this end we consider an enclosing $S^1$ manifold of radius $k_S$ that is positioned in the $x-z$ plane and is centered at the momentum $\mathbf{k}=(k_0, 0, 0)$. A straightforward calculation reveals that if $k_S>\sqrt{\lambda^2+\delta^2}$, i.e., the manifold encloses the torus from outside and links with it, the absolute value of the winding number $c^{\text{out}}_1$ is equal to unity. On the other hand for $k_S<\sqrt{\lambda^2+\delta^2}$, the winding number $c^{\text{in}}_1$ vanishes since in this case the enclosing manifold can be shrunk to a point without crossing energy bands. Following general arguments from Sec. \ref{intro}, the non-trivial difference of the winding numbers ensures that the nodal torus cannot be fully gapped out by small perturbations that are invariant under $PC$ and $PT$ symmetries. Note however that in the present case this prediction has little practical value because the nodal torus is fully protected against any small $PC$-symmetric perturbation by the Pfaffian invariant.
It would be interesting to find a model protected by the non-trivial difference of a topological invariant defined on one-dimensional enclosing manifolds, where a nodal surface is gapped out to nodal loop(s).

We observe that similar to the model discussed in Sec. \ref{ntu}, the present model does not support robust zero-energy surface states. The pair of zero-energy drumhead surface states present in the limit of the double-Weyl loop ($\lambda=\delta=0$) is shifted to a finite energy by the last two terms in the Hamiltonian \cref{mod4}. One arrives to the same conclusion by diagonalizing the lattice the Hamiltonian
\begin{equation} \label{mo21latta}
\begin{split}
\mathcal{H}(\mathbf{k})=&-\Big(6t_1-2 t_2 \left[ \cos k_x+ \cos k_y+ \cos k_z \right] \Big)\tilde\alpha^x \\
&-2 t_2 \sin k_z \tilde\alpha^z- \lambda \gamma^5-\delta \gamma^0
\end{split}
\end{equation}
in a slab geometry.

\section{Conclusion and outlook}
In this paper we investigated the physics of three-dimensional fermionic band models that exhibit two-dimensional Weyl nodal surfaces. We argued that the robustness of these nodal objects is ensured by topology. Specifically, we showed that the degree of robustness is determined by the dimensionality of gapped enclosing manifolds, where topological invariants are evaluated.
We demonstrated this in several four-band toy models that exhibit nodal surfaces protected by unitary or anti-unitary symmetry. It would be interesting to study the effects of interactions and disorder on nodal surfaces, investigate transport properties and find realistic models of materials where these ideas can be applied.

\section*{Acknowledgments:}
We acknowledge fruitful discussions with Barry Bradlyn, Anton Burkov, Tom\'a\v s Bzdu\v sek, Titus Neupert and Grigori Volovik. Our work is supported by the Emmy Noether Programme of  German Research Foundation (DFG) under grant No. MO 3013/1-1.

redpajama_set_name: RedPajamaArXiv


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Our real enemies are the people who make us feel so good that we are slowly, but inexorably, pulled down into the quicksand of smugness and self-satisfaction. Five great enemies to peace inhabit with us -- avarice, ambition, envy, anger and pride. If those enemies were to be banished, we should infallibly enjoy perpetual peace.


text:
Microsoft to cut 18,000 jobs worldwide
Tribune — July 19, 2014 1 comment
US firm to lay off thousands of staff, including nearly half of Nokia employees who joined in April after acquisition.
Microsoft has announced the biggest layoffs in its 39-year history, outlining plans to cut 18,000 jobs worldwive in a move that marks the chief executive's sharpest pivot yet away from his predecessor's drive for the company to make its own devices.
Although some job losses had been expected ever since Microsoft acquired Nokia's mobile-device unit, the number, which amounts to 14 percent of the Microsoft workforce, is about twice the total analysts had estimated.
The cuts announced on Thursday will include about 12,500 jobs associated with the Nokia unit – nearly half of the 28,000 employees Microsoft brought on board in April through the acquisition, the AP news agency reported.
When the cuts are complete, the company will still have about 10,000 more employees than before the Nokia acquisition, with an overall headcount of 109,000.
In a public email to employees, chief executive Satya Nadella, who was appointed in February, said the changes were needed for the company to "become more agile and move faster".
The move also pushes Nokia to focus solely on the Windows Phone operating system.
Nadella is clearly backing away from former chief executive Steve Ballmer's strategy of getting Microsoft to make its own smartphones and tablets.
"He's making a pretty serious game-changing strategy move away from hardware," said Michael Turits, managing director of equity research for financial services company Raymond James and Associates.
Nadella indicated that Microsoft will largely abandon low-price Nokia Asha phones – which work on their own non-Windows operating system – and reverse a strategically questionable move by Nokia in February to launch a line of phones called "X" that supported rival Google's Android platform.
"To win in the higher price tiers, we will focus on breakthrough innovation that expresses and enlivens Microsoft's digital work and digital life experiences," Nadella said in the memo.
Severance payments
About 1,350 Seattle-area workers around Microsoft's Redmond, Washington, headquarters were laid off on Thursday, as were 1,800 workers in Hungary and nearly 370 in San Diego.
The Nokia cuts include 1,100 jobs in Finland.
In addition, the company is closing its Xbox Entertainment Studios, a two-year-old venture based in Santa Monica, California, that produced original video content but was seen as separate from Microsoft's core business.
Microsoft expects charges of $1.1bn to $1.6bn over the next four quarters, largely for severance payments.
The move puts the company on track to meet the target it set in September, when it announced the Nokia purchase, of saving $600m in annual costs within 18 months after the deal closed.
The move dwarfs Microsoft's previous biggest job cut, when it eliminated about 5,800 jobs in 2009.
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text:
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We hear the debate about grass-fed or grain-fed a lot, but a strikingly large number of Australian red meat consumers don't really know what either of those terms mean. They have both been in 'vogue' products with ebs and flows in popularity and both seem to have a fairly good wrap. But we thought we would help you understand the real difference, so next time you're at the butcher buying meat you can make a decision that suits you.
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How it impacts the animal: Grass-fed comes with a lovely mental image of a cow happily roaming a paddock for its entire life – and this is pretty accurate. The animals are likely to get more exercise than grain-fed because they roam, however it takes a longer time to reach market weight, so are an older animal.
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What it means: Grain-fed beef means that cattle must be fed a grain based diet for over 60 days. This is often the last 60 days of its life as the grain based diet can increase weight rapidly, helping the animal to reach market weight faster. The grains fed are usually what, barley or sorghum.
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