prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How can players skip dialogue on the quest map?

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Context 1: Hey folks! Here is the shiny new Changelog thread  We're including the archived patch notes from the old forums, so that they are preserved for anyone that would like to reference back to them  We will continue to update this thread with new notes as the patches are released 
Chat is now accessible from the quest board, upgrade screen, and many other menus 
Tapping on objects and menus may reveal helpful hints about that object 
Team PI is now colored red if lower than recommended for the quest 
Many text fixes and consistency improvements 
• A new Basic Catalyst found in Special Events is used in every recipe!
Several heroes have received improvements to their base stats 
The abilities of all Champions have increased in effectiveness 
A new Critical Boost buff has been introduced 
Iron Fist and Spiderman now have the ability to Armor Break with their Critical Hits 
Deadpool’s ability to Regenerate is more powerful, but only triggers once per fight 
Scarlet Witch now has a chance to trigger Nullify off of any Critical Hit 
Juggernaut and Rhino now have a layer of Armor 
Punisher and Winter Solider now may also trigger Fury in addition to Bleed 
Colossus now further increases his base Armor with the Armor Up ability 
Thor and Ronan no longer Armor Break; instead, base stats and Stun durations have improved 
We reduced the effectiveness of the Revive items in order to give away more as rewards 
A bonus of 50% for using ISO-8 matching your Champion’s Class can now be previewed on the Upgrade screen 
It’s now possible sell Champions in exchange for ISO-8 and Gold  The amount received increases proportionately to the Rank and Level of the sold Champion 
-You can now skip dialogue on the quest map by pressing ‘SKIP’ 
-Added a ‘Quit’ button directly on the quest interface 
-The Back button on the Top Bar now returns the player to the Home screen 
-Various game balance and cosmetic improvements to the available quests 
• PVP energy has been replaced with Hero Stamina  Each Hero has their own Stamina values, meaning the more Heroes you have the more you can play in PVP 
• Each Hero has 1 Stamina and takes 2 hours to recharge 
We have removed the Next Quest button for a much more favorable and flavorful approach to teaching and informing people about Marvel : Contest of Champions  In the Main Menu(Bottom Right Corner) you will now see an image of the Collector showing you what the best or recommended actions that you should preform  This can be anything from opening Crystals, Continuing a Quest, Ranking Up Champions if the difficulty is too hard, Tips where to obtain items, and Playing Versus/Arenas 
• Adjusted the PI calculation for Power Burn and Power Drain abilities to improve accuracy 
• Significantly increased the Power Burn multiplier as well as the amount of Power burned  Prior to this change, Vision's Special Attack damage output was far below the curve  Vision's Special Damage is distinct from other heroes in that the dependency on opponents' Power levels cause the damage dealt to be highly variable, and sometimes quite low; however, when striking an opponent with high Power levels, Vision has the potential to deal very high amounts of direct, Armor-ignoring damage 
• Slightly adjusted the Armor Break trigger to be less punishing to opponents with the Armor Up ability without sacrificing PI or damage output 
• Slightly increased his base Health and, in turn, the amount of Health recovered by Regeneration  This improvement is reflected by an increase to PI of about 1% 
• Slightly reduced the damage from Bleeding, but slightly increased the amount of Power drained by E M P  Arrow to compensate  This added utility strengthens the choice between whether to offensively Bleed the enemy or defensively drain their Power  These changes may modify PI by +/-1% 
• Slightly reduced the frequency of Nullify for basic attacks, but slightly increased the chance a Special Attack is critical  Chaotic Bombardment no longer has a chance to critical, and instead has a 100% chance to Nullify the target  This is less punishing to opponents with beneficial effects, while providing a more reliable source of Nullify  Overall, her PI has decreased by about 2% 
• Decreased base Health and Attack by 2% each to bring his PI in line with other Champions without compromising Special Attack effectiveness 
• Slightly increased base Health by 2% to bring his PI in line with other Champions  This change may result in a PI increase of up to 1% 
• Fixed a bug with her Bleed ability scaling incorrectly  This has no effect on PI 
• User's on iPhone 4 devices will no longer encounter a progression blocker after fighting Iron Man in the tutorial 
• Fixed an issue where player's Hero would disappear after using a special move 
• Fixed an issue where very rarely a character would lose all functionality when dashing 
• Added additional Network support to better diagnose disconnects  The game should resolve and recover much more gracefully than in previous updates 
• Adjusted some of the touch sensitivity while fighting  Heroes moves should feel more responsive  This is something that is going to be an ongoing process  Please let us know how you think it feels 
• Fixed various issues with Chat 
• We have updated Open GL versions/drivers for iOS devices that support Open GL 3 0 
• User's will no longer receive delayed Game Center notifications  This caused some weirdness to occur while opening Crystals in the Crystal Vault 
• The Crystal Vault has received another polish pass and should now feel much more responsive, thank you for all your feedback on this feature!
• Many more minor bug fixes were included in this update 
• Special Attack 1 base damage increased by +25% Attack Rating 
• Heavy Attack base Power gained reduced to 63 points 
We recently improved the functionality of Heavy Attacks, so they’re easier to use  Their base Power has been reduced to normal levels – previously, they generated Power at a higher rate to compensate for their difficult execution  Special Attacks have been adjusted to give the unlucky recipients more of a fighting chance  These changes bring these attacks in line with existing damage-to-power ratios 
*NOTE: Special Attacks only generate Power for the target struck, not for the user; this prevents infinite loops and helps serve as a comeback mechanic 
Versus Crystal prizes have been adjusted due to the Champion Stamina changes 
Arena Crystal prizes have been increased to help balance the adjustments to the Versus Crystal 
Payouts have significantly increased when receiving a duplicate Champion with a Star rating of two or more  The boosted amount increases based on Star rating  We apologize for any inconvenience caused by delivering each reward individually, and are working to get a fix to you as soon as possible  In the meantime, using the “Skip” button avoids the inconvenience 
• We fixed a bug where finding a new match could cost a player Units 
• Spending Units to find a new opponent will now return opponents with lower ratings 
• Chapters 3 and 4 of Act 2 Story Quests are now available  A mysterious opponent awaits you at the end of Act 2!
*NOTE: This caused some players' progress to reset for a brief time, but that issue should now be corrected 
• Event Quest difficulty has been adjusted to match Catalyst availability 
• Rank-Up Recipes have been adjusted to be more accessible across all ranks 
• Bosses for the Monday through Saturday Daily Events now have a small chance to drop a Class Catalyst  This is in addition to the drop chance from Chests 
• Ambush Rates have been adjusted on all Event Quests 
• Increased Catalyst drops for the Collector Free-For-All Event Quest 
• Alpha Catalysts now have a chance to drop from Chests in Medium and Hard difficulties of The Collector Free-For-All event 
• The unobtainable chest in Act 1, Chapter 1, Quest 6 has been removed from the Battlerealm 
Increased the amount of Gold awarded by the Arena Crystal 
Slightly reduced the cost to level-up a 3-Star Champion at Rank 1 to cleanly align with ISO-8 chunk values 
Fixed a bug with Billion-Dollar Punch not triggering Armor Break 
• Duplicate 2-Star, 3-Star, and 4-Star Champions now awaken a brand new ability unique to that Champion in addition to the rare ISO8 they currently give  Duplicates thereafter continue to level-up this ability to make it stronger  When a Champion is awakened, their Stars turn bright and glow, making them easy to identify (and look pretty cool too)  These new abilities can be quite powerful, so please fight responsibly!
• Various other improvements, including rank and level information for opponents, find match options in team select, and animation tuning 
• There is now a chance to encounter the elusive Treasure Adaptoid, who divulges his hoard of ISO8 and Gold to those able to defeat him in battle 
• Class Relationships can be viewed by tapping “Enemy Classes” before entering a quest, and preview the number of enemies in that quest for each class type 
• You can also now see rewards for completion and exploration on the Edit Team screen 
• Opponents are more aware of the distance between you and them, improving their interaction with knockback effects, such as that from Heavy Attacks 
Mutant Champions are now effective against Skill Champions 
• The high Special Attack damage and regenerative abilities of Mutant Champions are effective against Skill Champions, which typically rely on Bleed damage from their weaponry  We think of this relationship as if the X-Gene grants Mutant Champions superpowers that evolved to be stronger than Champions that are merely “Skilled” 
Skill Champions are now effective against Science Champions 
• While scientists fiddle in their cute little laboratories to create flasks full of serums to turn even frail young men into super-soldiers, Skill Champions were just born that way baby  Often donning sharp weaponry to make their opponents Bleed, Skill Champions enjoy watching the high base attributes of Science Champions just melt away 
Cosmic Champions are now effective against Tech Champions 
• Tech Champions construct durable robots and thick suits of Armor to outlast their opponents in battles of tank-the-nuke   which gives Cosmic Champions extra time to build up stacks of beneficial effects to overrun Tech Champions using their peculiar alien enhancements 
• Tech Champions are still effective against Mutant Champions 
Tech Champions typically excel at Armor, Resistance, and Power manipulation, which is effective against the high Special Attack damage of Mutant Champions  Think of the robotic Sentinels adapting for tactical advantages in the war against Mutantkind!
Science Champions are still effective against Mystic Champions 
• Science Champions – a Class of behemoths like Hulk and super-soldiers like Captain America – typically have above average base attributes like Health, Attack, and Armor  These raw stats cannot be affected by pesky Mystics and their removal abilities: Nullify and Purge 
Mystic Champions are still effective against Cosmic Champions 
• Cosmic Champions explore strange new beneficial effects to seek out new power and new abilities, to boldly take their attributes where no class has gone before  Well, not if Mystic Champions – who are fully capable of stripping Cosmic Champions of their beneficial effects – have anything to say about it! Maybe it’s the Mystic Agenda to protect the secrets of the universe?
These changes ensure that having a Class Bonus always gives you the advantage it promises, as it now also reflects ability trends for a particular Class  Please keep in mind that these are generalizations, and some Champions abilities may not always strictly align with these relationships  Learn more about Champions’ abilities by viewing their profiles and tapping on features for detailed information 
• When you attack someone, you charge up their Power in addition to yours  This meant they would reach a full three bars while you only reached one and a half  We've reduced the amount defenders receive such that you'll be at two bars when they're at three  This change maintains the underdog functionality to give defenders a chance to comeback while being less punishing to players earning high Combos 
New damage types for attacks now play a larger role in the abilities of Champions  For example, some heroes power-up by successfully blocking magical damage, while others’ abilities may harm anyone that makes physical contact with them 
New Resistances and Immunities have found their way to the Battlerealm  Some heroes are completely immune to specific status effects based on either lore from the comics or logic  For example, the android Vision has no blood, and is therefore fully immune to Bleed conditions  We’ve also strengthened the effectiveness of certain status effects, so be careful who you choose to bring into battle! Could you guess who might be immune to the new “Poison” condition?
• Poison: Inflicts damage over time and reduces healing and regeneration effectiveness 
• Unstoppable: A buff to shrug off the impact from attacks, but still take the damage 
• Weakness: A debuff that reduces Attack attributes 
• Heal Block: Fully prevents the target from gaining health in any way 
• Power Lock: Seals the target, preventing them from gaining any Power 
• When fighting, you may notice that many status effects are now able to stack  This also changes how certain beneficial “buffs” and detrimental “debuffs” interact with one another  For example, it's now possible to have both Armor Up and Armor Break effects on you simultaneously  Let the tug-o-war begin, and may the strongest effects win!
• Black Bolt's Corkscrew: +25% damage, but at the cost of minor recoil damage 
• Punisher's “Wrath” has been replaced by "Payback"  Payback deals additional damage based on the total damage dealt to Frank 
• Colossus' “Unbreakable” now deal bonus damage based on his armor level at the time of activation 
• All of Black Panther’s special attacks now deal bonus damage based on the number of Bleeds on the target 
• Spider-Man’s Web-Slinger now has a chance to inflict Weakness 
• Vision’s Physical Disruption: Added a minor Power Burn effect due to “his” use of his Infrared Beam  “He” also now purges all status effects while phasing through the ground 
• Scarlet Witch: Increased the Critical Hit Chance for Hex Bolt and Hex Sphere 
• Many knockback effects have been adjusted to improve consistency 
We’ve tested the Signature Abilities quite extensively before releasing them, but there have been a few abilities that we have been keeping an eye on  We’ve compared our notes with the feedback you’ve been sending us and are making some balance changes to them  Thanks for your feedback!
Slightly reduced the frequency and duration of Juggernaut’s “Unstoppable” ability 
• He was indeed a bit too   unstoppable  We’ve toned down the frequency this ability triggers, as well as reduced the duration it’s active for when it does trigger  We feel Juggernaut is still a powerful Champion despite these revisions  Take care!
Slightly reduced the starting values of Wolverine's “Cellular Regeneration” 
• We found that Cellular Regeneration was too strong at lower levels where fewer counters to Regeneration exist 
Re-scaled Gamora's “Assassination” to start higher but scale slower 
• At lower levels, Special Attacks were used too infrequently, giving this powerful ability little visibility  We’ve adjusted the scaling to better match Special Attack usage at all levels 
Increased the frequency that Black Bolt’s “Provocation” triggers 
• Due to the varying Critical Hit rates across all Champions, in some cases Provocation would trigger rarely or not at all within a fight  We’ve increased the frequency to ensure you’ll see it every match – but especially so against opponents with high Critical Hit rates 
We’ll continue to follow the effect of these new abilities on gameplay  Please keep your feedback coming!
Hey everyone! We have been hard at work on improving the game and have prepared a big update inspired in part by your great community feedback  Please keep letting us know what you think!
• Fixed many Dash, Medium, Heavy and Special Attacks missing or failing to execute 
• Added Alliances and a new Alliance Crystal 
• Rocket Raccoon and Unstoppable Colossus join The Contest 
• Temporary Boosts to Attack, Health, and XP are now available from the Alliance Crystal 
• Rewards for completing and exploring Chapters and Acts  Earn a guaranteed 3-Star hero crystal for each fully explored Act! This is retroactive, just complete any quest to claim them 
• A new Fight Menu combines The Arenas, Story Quests and Event Quest menus 
• Updated Summoner Profiles with new information  Inspect other players’ Profiles and brag about your achievements!
• A list of blocked users has been added to Chat windows  The option to unblock these users is found in this new menu  The power is in your hands now!
• We fixed Dash and Medium Attack issues for many heroes that sometimes missed or did not activate 
• We fixed issues to Drax and Colossus Light and Medium Attacks where they would not connect 
• Fixed an issue where the camera would stop moving after a level 3 special sequence 
• Fixed an issue where the player’s heavy attack would get stuck in charge even after the player has released input 
• Fixed a rare bug where Champions were still able to deal damage after they died, resulting in tied fights 
Form Alliances with your Friends!
What is better than playing? Playing with your friends! Create a new Alliance or join an existing one through the new Alliance Menu 
• Invite other players to your Alliance 
• Search for an Alliance by name or join a Recommended Alliance 
• Receive rewards for entering your first Alliance 
• Alliance News Feed  The news feed celebrates your Alliance member’s achievements 
• Alliance Chat  Chat with other members of your Alliance in a private channel all to yourself 
• Help Allies  Players can ask for help when out of Energy or Stamina  Alliance members help each other as much as they can to earn Loyalty Points  Loyalty points have a daily limit to how many can be earned 
• Alliance Crystal  Access a new Alliance Crystal while part of any Alliance  Use new Loyalty Points for purchasing Alliance Crystals 
He may start out slow, but watch out for his immense power at high ranks!
• Adjusted the range of many Heavy Attacks, including Hulk and Drax, to ensure they correctly connect with enemies 
• Many Special Attacks, including those for Wolverine, Iron Fist, Winter Soldier, Punisher, Black Panther, and many others have had their range adjusted to ensure they correctly connect with enemies even if activated immediately after a combo that knocked the enemy back 
• Payback and Unbreakable now display their maximum potential damage bonus 
• Added detailed descriptions for Bleed Immunity and Poison Immunity 
• Gamora: We’ve adjusted the scaling of her base Special Attack damage to ensure they scale up more similarly to other heroes  This also makes Gamora more reliant on her high Bleed damage, and improves the chances of opponents able to deal with her high Bleed 
Vital Strike and Jade Assassin damage decreased by 10% 
Godslayer damage increased by 10% 
• Magik: Rewind is a game-changer for Magik that allows her to go up against foes like Gamora and Rewind off big Critical Hits and Bleed damage; however, the frequency of Rewind triggering was too low to be there when she needed it 
Increased the likelihood Rewind triggers by +20% at all levels 
Rewind now heals over one second instead of instantly 
Fixed a bug allowing Magik to break out of an enemy combo using Rewind  It now only removes Status Effects 
• Hulk: Given the riskiness of losing Health in certain game modes, Hulk’s anger-management provided too little help too late in the game  We’ve increased the Attack boost to ensure he’s appropriately scary in all game modes – as long as he’s angry!
Increased Hulk Rage by +20% Attack at all ability levels 
Arc Overload no longer causes Armor Break when it expires 
• Vision: Added Poison Immunity to our robot friend 
Arena tuning is an ongoing process  The team is continually making adjustments to Arenas to improve the experience 
Ultron has infected The Contest!
Many new Champions join the battle against Ultron 
Quest through the new Ultron’s Assault Event 
Wield new power with Summoner Masteries 
Grow your Friend’s List with the new Social Hub 
Team up with your Alliance in new Events, Arenas, and more!
Filter and sort your Stash 
Fights have been optimized for performance improvements on all devices 
Users can now filter through the items in their Stash 
Fixed several issues where Hero Rating would fluctuate 
Fixed a bug with Rhino and Juggernaut having 11-20% more Armor than intended 
Fixed a bug with Rocket Raccoon’s Dash attack being slower than intended 
Added a confirmation popup when spending Units on stamina recharges and unlocking arenas 
Regeneration no longer displays green Health values if you’re at full Health 
Several new improvements to how status effects are displayed 
AI opponents are no longer able to perform one unavoidable attack in response to a Special Attack 3 
A new and improved look for all Health Potions in the Battlerealm 
All Revive Potions now revive your Champions with +10% more Health 
We’re adding so many new Champions, they could form their own Alliance!
Some of your favourite heroes of the Marvel Cinematic Universe join The Contest!
Summoner Mastery is on the horizon!
Masteries provide beneficial effects for your Champions 
Access Masteries through your Summoner Profile 
Earn Mastery Points when you level up 
Choose your Masteries wisely and strategically customize your benefits 
Recover your points to try a new specialization as often as you’d like 
Keep an eye on in-game messaging for more information 
The daily loyalty limit has been set to refresh at 08:00UTC for all players 
A timer has been added to show when the daily loyalty limit resets 
Loyalty balance is now displayed in the Alliance menus 
Ask for Versus help with a single tap on the ‘Help’ icon in Team Select 
New Alliance Events are coming very soon!
Work together with your Alliance to complete objectives and receive rewards!
Muster your might, Alliance Arenas will soon open their gates!
Competing in Alliance Arenas shares your points across your whole Alliance; work together to reach milestones and top ranks!
Work together to amass a huge score, and defeat your competition in classic Arena combat! No slackers here either - if you don’t contribute to win the competition, you’re not eligible for the goods!
All social features (Chat, Mail, and Friends) can now be accessed through the new Social Hub 
Search for and add friends, and send private messages to Summoners on your Friends List 
Redesigned chat and mail screens 
Take on other Summoners’ top Champions for bragging rights and prizes in 1-on-1 Duels!
A new series of special Ultron quests are available, starting with the first Chapter  Fight back against Ultron’s infection alongside the Summoner, and team up with some of Marvel’s finest! New quests unlock each week!
The Spider-Man Champion gate has been removed from Act 1, Chapter 1, Quest 5 
• Fixed an issue where chat snapped to the most recent message 
• Fixed several issues where Hero Rating would fluctuate 
• Various improvements to the Summoner Mastery screens and descriptions 
• Increased the ISO8 awarded by duplicate 2-Star Champions 
Quest through the new single-player campaign, Ant-Man’s Adventure!
In addition to Ant-Man and Yellowjacket feuding throughout the Battlerealm, additional new Champions will be joining The Contest!
Access more Masteries in the new Utility Mastery tree!
Please note, these changes may result in a loss of Hero Rating as incorrect effects are restored back to normal levels 
Improved and polished combat mechanics to reduce the amount of stutters and lost input 
Fixed and optimized rendering related issues with Metal enabled devices 
Team up with Ant-Man, and put a stop to Yellowjacket’s mysterious mission!
All Alliance Quests only last for a specified amount of time, defeat the boss with your Alliance before it expires!
New Prestige System - A dynamic difficulty and score setting that adjusts as you and your Alliance succeed in harder quests  The better you do and the tougher your Alliance is, the higher the prestige  The higher the prestige, the better the rewards!
Choose your teams carefully as Champions within Alliance Quests cannot be used in other Story or Event Quests 
Act 4 has been released! Play Chapter 1 now!
Summoner level maximum has been increased to level 60!
5-Star Champions are coΩming to The Contest! These are the most powerful Champions yet!
Additional improvements have been made to the UI, Versus Arenas, Synergy Bonuses, the Stash & Items Store 
Act 4 - Chapter 1 released!
New challenges - more path variation and features to challenge the strongest Summoners!
Greater challenge means greater rewards! Earn 4 Star Crystals and Mastery Points!
The Summoner Level cap has been increased by ten levels to level 60!
Champion Items will be coming soon! These allow you to apply items and buffs to a specific Champion, keep an eye out for updates on these new Champion Items!
Synergy Bonuses have updated iconography and the calculation has been updated to a distinct, additive bonus - What you see is what you get!
Alliance class distribution is now displayed on team select - Choose the right class!
Your Catalysts now have their own inventory, and will no longer appear in the Upgrade Item inventory 
The Stash is now separated into three tabs: Catalysts, Rewards and ISO, allowing you to sort and view your Stash much faster!
The UI flow for both Quests and Arenas have been greatly improved  You can now skip through fight victory and reward animations!
Here is the rundown of patch 5 1 0, filled with various bug fixes and optimizations  The important ones to note are below 
New Champions, new theme, and a new arena!
To celebrate our one year anniversary AND the holidays, we’ll be running a special event quest! Battle through the history of The Contest, and test your mettle against familiar faces both old and new!
A special reward will be available to those who master every quest!
Our Anniversary Celebration will be happening very soon; stay tuned for more info!
More Act 4 quests are coming very soon!
Opponents in Story Quests now have the ability to use their Special 3 attack! Note that we are not changing previous quest opponents to have this special attack (Act 1-3, Proving Grounds, Realm of Legends will not change); this will be in effect starting with the soon-to-be-released Act 4 content 
As with our previous major build releases (3 0’s Ultron, 4 0’s Ant Man, and 5 0’s Battlerealm), the Contest has been reskinned with a new theme!
The Road to Knowhere map is here! Fight in a new level inspired by Guardians of the Galaxy!
A new button in your Alliance Chat to take you directly to Alliance Quests!
You can now collect Catalyst Fragments in Event Quests, Proving Grounds, and Alliance Quests; these can be pieced together into a Catalyst!
Selling Items is now a thing! Sell any items in your inventory for gold!
Level 3 and Level 4 Health Potions have arrived! These are powerful instruments to help you tackle all the new Act 4 content!
Over 400 bugs were fixed in this patch!
This patch is a fix for the missing Champions during the Special 3 animation on Android devices 
This issue occurred during our upload process to the Google Play Store  This was an odd edge case scenario that we could not have caught during our internal tests, as it began appearing once we uploaded to the Google Play Store  This hotfix will be out by tomorrow, and will put Android at version 6 0 1  As this issue does not occur on iOS devices, iOS will remain at version 6 0 
3:30pm PST: We have started slow-rolling this patch out to Android devices, beginning with about 20% of users  We expect this to be available for 100% of users within the next 24 hours 
We have a few new Champions that you will see within the next couple of months (including one of my personal favorites)!
Over 200 total bugs squashed in this patch!
An artifact left over from the early days of the contest was Black Panther’s ability to gain a Critical Hit Rate boost during Special 3 attacks  As many might know, Critical Hits aren’t possible during a Special 3 anymore, making this effect   unhelpful  We’ve switched it out with a new ability to stack up even more Bleed effects on the opponent based on how many Bleeds are already active 
Example: The opponent has 4 stacks (instances) of Bleed on them when you launch a Special 3  With this new ability, you have a chance to add an additional 0 - 4 more stacks (instances) of Bleed 
Previously, a bug existed that allowed champions with Evade to continue to dodge Black Widow’s attacks, even if her Signature Ability was maxed out  This issue has been fixed 
Captain America WW2 has started to become outpaced by his non-WW2 counterpart and while we want the two to feel different and each have their own specific uses, we also want to ensure they are kept within range of each other in terms of balance  To accomplish this, we’ve given WW2 Cap the ability to Stun with his Special 1 and Special 3 attacks, but kept his Bleed on Special 2 the same, giving him options during combat against non-bleeding champions 
A bug that prevented Daredevil from triggering Armor Breaks from Heavy Attacks has been fixed and is now working as intended 
Against non-bleeding champions: Critical Hits have a chance to Armor Break on Special Attacks 
Increase range of Signature to 25% from 20% 
Many players found Elektra’s signature ability lacked enough opportunities to use it  To remedy this, we’ve increased the range from 20% to 25%  Additionally, to help make Elektra unique from other skill champions, we’ve given her the ability to deal with naturally Bleed Immune champions  Note: This Armor Break only applies to champions naturally immune to bleed, such as Colossus and Ultron, and not to champions granted Bleed Immunity from Local or Link Nodes 
Guillotine’s Bleed effect used to have a chance to activate from any given attack, meaning that it had to be kept quite weak to compensate for the frequency of triggers  We’ve made the switch to have her Bleed behave closer to existing champions, and in doing so have boosted the strength of the Bleed and have allowed it to stack 
Norman Osborn overloads the Arc Reactor in his chest if Health drops below 10%, granting a large burst of power, with (18% - 48% ) Armor, Regeneration, and Power Gain  After that, his suit burns out and cannot trigger Armor Up, Armor Break or Stun and loses all base Armor 
Many players didn’t like Iron Patriot’s old signature ability, feeling that due to the lack of Regeneration, it was considerably weaker than Iron Man’s  While we agreed, we didn’t want to just copy and paste his signature ability, but rather give him his own unique twist on the ability  This “all or nothing version” feels more like Norman Osborn, pushing his suit to the limit to get a larger boost but at the cost of damaging the suit  The addition of Power Gain allows Iron Patriot a large attack before the suit burns out, if timed correctly 
Heavy Attacks: 90% chance to Stagger the enemy for 8 seconds  A Staggered enemy cannot gain their next beneficial effect 
All versions of Juggernaut, even those who haven’t been awakened, now gain the 2 second Unstoppable ability at the start of the fight when they hit Rank 2 
We wanted to add some new functionality to Juggernaut, while also keeping him true to his Mystic class assignment  To accomplish this, we added this “buff smasher” effect which keeps an opponent from gaining their next beneficial effect  Additionally, we wanted to make non-awakened versions of Juggernaut more fun to play, without adding more power to the awakened variations  As a result, we gave all versions of Juggernaut the ability to become Unstoppable at the start of the fight 
While many players liked the new functionality of Star-Lord’s Element Gun effect, they found it to be a little too random, specifically when it would Heal Block a champion incapable of Healing  We’ve now added in some contingencies that will make Heal Block appear less unless the opposing champion shows that he / she can Heal during the fight  This includes both activated healing effects, such as Wolverine or Ultron’s Heal, or passive healing effects gained from Masteries, such as Salve or Willpower 
It’s been a bit weird that Bucky wasn’t friends with his most famous friend  Well, he is now  This affects 3 Star and above versions 
We’ve increased the overall speed of this attack, allowing quick players to use this ability after a four or five hit combo 
It seems the Marvel’s have gotten tired of their beams being dodged so easily and have decided to angle it a little better, increasing the overall range of the attack and making it harder to dodge away from  We’ve also increased the speed of both special attacks to allow them to better flow into combat 
In order to allow this attack to better flow in combat, we’ve shaved off a few frames from the beginning, allowing players to chain this attack into 4 and 5 hit combos 
Alliance Wars have arrived! It’s Alliance versus Alliance in a war for Battlerealm supremacy!
Enter the NEW Loyalty Store to buy Alliance Potions, Mastery items, or other EXCLUSIVE items 
Gain Power back from Special Attacks, enhance or defend against Special Attacks, OR gain a temporary Arena Point Boosts with hoards of new Summoner Boost items!
Additional changes and improvements are listed below 
This patch will be released February 24th 
A new area of the Battlerealm has been opened! Compete with your Alliance-mates for pride, glory, and PRIZES!
Matchmake to find a rival Alliance, then combine strategy and teamwork to dominate them 
Setup the ultimate defensive team to fortify your Battlerealm, then take your offensive team on the assault!
Watch your War Rating skyrocket as your Alliance works together to defeat rivals!
Load up on Crystal Shards, Loyalty, and brand new exclusive rewards!
Note that this will be slow-rolled to Alliances in phases, similar to the introduction of Alliance Quests (to ensure server stability and gather your feedback on the new mode)  Expect tuning changes throughout these phases, as well as into Season 1 
Use Loyalty instead of Units to obtain items for Alliance Quests & Wars!
Items will rotate daily, similar to how the Mastery cores in the current Store change 
Store contents will be randomly chosen from a pool of categories/items; a select few items will be persistent and always be available for purchase 
A 5-Star version of Unstoppable Colossus will be available in the Loyalty Store (keep in mind, this is an expensive Champion due to his exclusivity; this will require winning quite a few Alliance Wars and saving up!) 
This is accessible from the “Store” section of the pop-down menu, and will be available at a later date after the initial 7 0 launch; there will be advance notice through forums and in-game before we release the Loyalty Store 
New Summoner Boosts have arrived in the Loyalty Store; NEW Boost types, purchasable with Loyalty Points 
Class specific Boosts, such as Mystic Champions restoring power after using Special Attacks 2 and 3, or Skill Champions boosting their Special Attack Damage 
Defensive Boosts, where your Champions take reduced incoming Special 3 Attack Damage 
Gain a temporary Arena Point boost with new Arena Boost items!
Fixed an issue where, after Parrying certain Champion’s Special Attacks, your Champion would be stuck in a blocking state until the Special Attack finished 
Fixed an issue where 90s Cyclops’ Armor Breaks would not remove Armor Ups 
Fixed an issue with Scarlet Witch’s Signature Ability proc rate (previously, the % chance displayed did not match in-game functionality; this is now fixed) 
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The Alliance Crystal now has a purchase limit that resets daily 
Permanently increased the Alliance Crystal’s points in Summoner Advancement (from 30 to 300) 
Updates to Champion Special Attack animations, flow, and timing 
7 0 1 will be released within the next few days 
A celebration message is sent to the War Room when an Alliance War battlegroup is cleared 
Players can now tap directly on another node icon while the tile info popup is open (previously, the popup had to be closed before selecting another node) 
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The “Place Your Defenders” text now disappears much faster after tapping on the screen 
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Resolved an issue with Class Masteries (specifically Mystic Dispersion) not functioning 
The Juggernaut issue with his linked nodes not appearing in Act 4, Chapter 3, Quest 3 (4 3 3) has been fixed 
Fixed a crash that occurs when a player who is not in an Alliance enters Alliance Wars through an outside link 
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Resolved ~20 various rare crashes and additional minor issues in different game modes 
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Tuned and updated many Champion Special Attack animations to improve timing and combat flow  Please see the expanded forum post HERE for a full list 
Fixed She-Hulk’s Special Attacks being marked as a projectile (allowing Daredevil to evade them) 
Fixed an issue where the player would be stuck in place after parrying Captain America’s Special 1 
Fixed an issue where chaining 2 medium attacks into Old Man Logan’s Special 2 would cause the first 2 strikes to miss opponents 
Fixed an issue with Daredevil or Spider-man missing with a dash attack if Vision charges a heavy attack during the dash 
Fixed an issue where some hidden information in Alliance Wars was visible 
Fixed a display issue where Defender Placement percentage was not displaying all placed Alliance members 
Resolved minor issue with the total Alliance’s score being displayed on the War Progress widget (now only displays the score of the battlegroup being viewed) 
Multiple minor Alliance War issues have also been fixed in this patch 
Fixed a display issue where Shard amounts provided by defeating a boss displayed as double 
Fixed a display issue where opponent PI values would display differently between the map, prefight screen, and in combat 
Boss power is now correctly displayed after removing Global and Linked boosts 
Fixed an issue where a player in Alliance Quests would lose input ability on the quest board after sleeping the device 
Fixed an issue where a player enters Alliance Quests and gets stuck after viewing the linked node or buff node tutorial 
Fixed an issue where sending an Alliance invite to a player would cause the “Add Friend” button to become greyed out 
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Context 2: Paper Info

Title: Nuclear Liquid-Gas Transition in the Strong Coupling Regime of Lattice QCD
Publish Date: 28 Mar 2023
Author List: J Kim (from Institute for Advanced Simulation (IAS-4), Forschungszentrum Jülich), P Pattanaik (from Fakultät für Physik, Bielefeld University), W Unger (from Fakultät für Physik, Bielefeld University)

Figure

FIG  1 Typical 2-dimension configuration at β = 1 0, at non-zero quark mass, temperature, chemical potential The black dots are monomers, the blue lines are dimers, the red arrows are baryon loop segments (or triplets g b + f b = ±3 if adjacent to a non-trivial plaquette), and the green squares are plaquette occupations ±1 The actual configurations are 3+1-dimensional 
FIG 2 Chiral susceptibility on a 2 4 volume for various quark masses, as a function of the bare anisotropy γ (with aT = γ 2 /2), analytic results from enumeration compared to numerical data from simulations via the worm algorithm 
FIG 3 Various observables in the µB-T plane on a 2 4 volume at amq = 0 1 The back-bending of the first order transition at temperatures below aT = 0 5 in all observables is an artifact of the small volume, and vanishes in the thermodynamic limit The temperature aT = 1/2 corresponds to the isotropic lattice here 
FIG  4  The chiral condensate (left) and the baryon density (right) for quark mass m = 1 5 as a function of the chemical potential and for various temperatures 
FIG  7  ∆f at amq = 0 2 as a function of chemical potential and β the on a 6 3 × 4 lattice
FIG  8  Baryon mass from ∆E as a function of the quark mass amq, and contributions from different dual variables: monomers, dimers and baryon segments 
FIG  9  Baryon density for volume 4 3 × 8 in the full µB − mq plane, illustrating the strong quark mass dependence of the onset to nuclear matter 
FIG  10 Baryonic observables on various volumes in the first order region amq = 1 5 Vertical bands indicate the mean and error of the nuclear transition 
FIG  12  Left: Extrapolation of the pseudo-critical values of µB for the various volumes into the thermodynamic limit Right: Critical baryon chemical potential for different quark masses The first order transition region is shown in blue, the crossover region is shown in red and the range for critical end point is marked in black 
FIG  17  Nuclear interaction scaled with baryon mass As the quark mass increases, it tends to zero 
FIG  18  Critical baryon chemical potential and baryon mass from different approaches 
Parameters for the Monte Carlo runs to determine the nuclear transition at strong coupling, with statistics after thermalization 

abstract

The nuclear liquid-gas transition from a gas of hadrons to a nuclear phase cannot be determined numerically from conventional lattice QCD due to the severe sign problem at large values of the baryon chemical potential  In the strong coupling regime of lattice QCD with staggered quarks, the dual formulation is suitable to address the nuclear liquid gas transition 
We determine this first order transition at low temperatures and as a function of the quark mass and the inverse gauge coupling β  We also determine the baryon mass and discuss the nuclear interactions as a function of the quark mass, and compare to mean field results  It is known from experiments that at low temperatures, there is a phase transition between dilute hadron gas and dense nuclear matter as the baryon chemical potential increases 
This transition is of first order and terminates at about T c = 16 MeV in a critical end point  The value of the chemical potential µ 1st B at zero temperature is given roughly by the baryon mass m B , where the difference of µ 1st B −m B is due to nuclear interactions  For a review on nuclear interactions see  
As the nuclear force between baryons to form nuclear matter is due to the residual strong interactions between quarks and gluons, it should be accurately described by QCD  We choose to study the nuclear transition and nuclear interaction via lattice QCD , with its Lagrangian being a function of the quark mass and the inverse gauge coupling 
In order to understand the nature of the transition, it is helpful to study its dependence on these parameters  However, at finite baryon density, lattice QCD has the infamous sign problem which does not allow us to perform direct Monte Carlo simulations on the lattice  Various methods have been proposed to overcome the numerical sign problem, but they are either limited to µ B /T 3 or can not yet address full QCD in 3+1 dimensions in the whole µ B − T plane , in particular the nuclear transition is out of reach 
An alternative method is to study lattice QCD via the strong coupling expansion  There are two established effective theories for lattice QCD based on this: (1) the 3-dim  effective theory for Wilson fermions in terms of Polyakov loops, arising from a joint strong coupling and hopping parameter expansion , the dual representation for staggered fermions in 3+1 dimensions, with dual degrees of freedom describing mesons and baryons 
Both effective theories have their limitations: is limited to rather heavy quarks (but is valid for large values of β) whereas ( ) is limited to the strong coupling regime β 1 (but is valid for any quark mass)  We study lattice QCD in the dual formulation, both at infinite bare gauge coupling, β = 0, and at leading order of the strong coupling expansion in the regime β < 1, which is far from the continuum limit 
But since strong coupling lattice QCD shares important features with QCD, such as confinement, and chiral symmetry breaking and its restoration at the chiral transition temperature, and a nuclear liquid gas transition, we may get insights into the mechanisms, in particular as the dual variables give more information in terms of its world lines, as compared to the usual fermion determinant that depends on the gauge variables 
To establish a region of overlap of both effective theories, we have chosen to perform the Monte Carlo simulations in the dual formulation extending to rather large quark masses  This paper is organized as follows: in the first part we explain the dual formulation in the strong coupling regime, in the second part we provide analytic results based on exact enumeration and mean field theory, in the third part we explain the setup of our Monte Carlo simulations and present result on the m q -and β-dependence of the nuclear transition 
Since the strong coupling regime does not have a well defined lattice spacing, we also determine the baryon mass am B to set the parameters of the grand-canonical partition function, aT and aµ B , in units of am B   We conclude by discussing the resulting nuclear interactions, and compare our findings with other results 

Staggered action of strong coupling QCD and its dual representation

In the strong coupling regime, the gauge integration is performed first, followed by the Grassmann integration to obtain a dual formulation  This was pioneered for the strong coupling limit in and has been extended by one of us to include gauge corrections   The sign problem is mild in the strong coupling limit and still under control for β < 1, where we can apply sign reweighting 
The dual degrees of freedom are color-singlet mesons and baryons, which are point-like in the strong coupling limit, and become extended about a lattice spacing by incorporating leading order gauge corrections  The partition function of lattice QCD is given by where DU is the Haar measure, U ∈ SU(3) are the gauge fields on the lattice links (x, μ) and { χx , χ x } are the unrooted staggered fermions at the lattice sites x 
The gauge action S G [U] is given by the Wilson plaquette action and the staggered fermion action S F [ χ, χ, U] is: where the gauge action depends on the inverse gauge coupling β = 2Nc g 2 and the fermion action depends on the quark chemical potential aµ q which favors quarks in the positive temporal direction, and the bare quark mass am q  
First we consider the strong coupling limit where the inverse gauge coupling β=0 and hence the gauge action S G [U] drops out from the partition function in this limit  The gauge integration is over terms depending only on the individual links (x, μ) so the partition function factorizes into a product of one-link integrals and we can write it as:
with z(x, μ) the one-link gauge integral that can be eval-uated from invariant integration, as discussed in , where we write the one-link integral in terms of new hadronic variables: Only terms of the form (M (x)M (y)) k x, μ (with k x,μ called dimers which count the number of meson hoppings) and B(y)B(x) and B(x)B(y) (called baryon links) are present in the solution of the one-link integral 
The sites x and y = x + μ are adjacent lattice sites  It remains to perform the Grassmann integral of the fermion fields χ, χ  This requires to expand the exponential containing the quark mass in Eq  (4) (left), which results in the terms (2am q M (x)) nx (with n x called monomers)  To obtain non-vanishing results, at every site, the 2N c Grassman variables χ x,i and χx,i have to appear exactly once, resulting in the Grassmann constraint (GC):
where n x is the number of monomers, k x,μ is the number of dimers and the baryons form self-avoiding loops x,μ , which due to the constraint cannot coexist with monomers or dimers  With this, we obtain an exact rewriting of the partition function Eq  ( ) for N c = 3, in terms of integer-valued dual degrees of freedom {n, k, }:
where the sum over valid configurations has to respect the constraint (GC)  The first term in the partition function is the contribution from dimers and the second term is the contribution from monomers  The weight factor w( ) for each baryon loop depends on the baryon chemical potential µ B = 3µ q and induces a sign factor σ( ) which depends on the geometry of :
Here, ω is the winding number of the loop   The total sign factor σ( ) ∈ {±1} is explicitly calculated for every configuration  We apply sign reweighting as the dual formulation has a mild sign problem: baryons are non-relativistic and usually have loop geometries that have a positive signs  The dual partition function of the strong coupling limit is simulated with the worm algorithm (see Section III A) and the sign problem is essentially solved in this limit 

Extension to finite β

The leading order gauge corrections O(β) to the strong coupling limit are obtained by expanding the Wilson gauge action Eq  ( ) before integrating out the gauge links  A formal expression is obtained by changing the order of integration (first gauge links, then Grassmann-valued fermions) within the QCD partition function:
With this the O (β) partition function is The challenge in computing Z (1) is to address the SU(N c ) integrals that receive contributions from the elementary plaquette U P   Link integration no longer factorizes, however the tr[U P ] can be decomposed before integration: Integrals of the type J ij with two open color indices -as compared to link integration at strong coupling -have been derived from generating functions
for either J = 0 or for G = U(N c )   The SU(3) result was discussed in , in terms of the dual variables, neglecting rotation and reflection symmetries, there are 19 distinct diagrams to be considered  The resulting partition function, valid to O(β), is with q P ∈ {0, ±1}, and the site weights w x → ŵx , bond weights w b → ŵb and baryon loop weights w → ŵ receive modifications compared to the strong coupling limit Eq  ( ) for sites and bonds adjacent to an excited plaquette q P = 1 
The weights are given in , and are rederived for any gauge group in   The configurations {n, k, , q p } must satisfy at each site x the constraint inherited from Grassmann integration: which is the modified version of Eq  ( ) with q x = 1 if located at the corner of an excited plaquette q p = 0, otherwise q x = 0 
A more general expression that we obtained via group theory and is valid to higher orders of the strong coupling expansion is discussed in terms of tensor networks   A typical 2-dimensional configuration that arises at β = 1 in the Monte Carlo simulations is given in Fig    Note that if a baryon loop enters a non-trivial plaquette, one quark is separated from the two other quarks, resulting in the baryon being extended object, rather being point-like in the strong coupling limit 
The O(β) partition function has been used in the chiral limit to study the full µ B − T plane via reweighting from the strong coupling ensemble  Whereas the second order chiral transition for small values of the aµ B decreased up to the tri-critical point, the first order nuclear transition was invariant: aµ 1st B 1 78(1) at zero temperature has no β-dependence 
For the ratio T (µ B = 0)/µ 1st B (T 0) we found the values 0 787 for β = 0 and 0 529 β = 1, which should be compared to T c / 0 165 for full QCD   However, since reweighting cannot be fully trusted across a first order boundary, direct simulations at nonzero β are necessary  The Monte Carlo technique to update plaquette variables is discussed in Section III A 
In this section, we provide analytic results from exact enumeration for small volumes, and mean field results based on the 1/d expansion, valid in the thermodynamic limit
Context 3: \section{Introduction}

In spite of being an attractive material with excellent electronic properties \cite{ahcn09},  practical applications of graphene as in conventional semiconductor devices are still questionable due to its gapless nature  In particular, the ON/OFF current ratio is low while the saturation of current is poor in pristine graphene transistors \cite{schw10}  Many efforts of bandgap engineering in graphene \cite{yhan07,khar11,lher13,jbai10,zhan09} have been made to solve these issues  The pioneer technique proposed \cite{yhan07} is to cut 2D graphene sheets into 1D narrow nanoribons  In 2D graphene sheets, some options as Bernal-stacking of graphene on hexagonal boron nitride substrate \cite{khar11}, nitrogen-doped graphene \cite{lher13}, graphene nanomesh lattice \cite{jbai10,berr13} and Bernal-stacking bilayer graphene \cite{zhan09} have been explored  However, the possibility to open a sizable bandgap in graphene as large as those of standard semiconductors is still very unlikely  In particular, it requires a very good control of lattice geometry and edge disorder in narrow graphene nanoribbons (GNRs) \cite{quer08} and in graphene nanomesh lattices \cite{hung13}, while the bandgap opening in bilayer graphene by a perpendicular electric field may not be large enough for realistic applications \cite{fior09}  Other methods should be further verified by experiments 

\begin{figure}[!t]
\centering
\includegraphics[width=2 8in]{Fig01 pdf}
\caption{Schematic of unstrained/strained graphene junctions investigated in this work }
\label{fig_sim1}
\end{figure}

On the other hand, graphene was experimentally demonstrated to be able to sustain a much larger strain than conventional semiconductors, making it a promising candidate for flexible electronics (see in a recent review \cite{shar13})  Indeed, strain engineering has been suggested to be an alternative approach to modulating efficiently the electronic properties of graphene nanomaterials  In particular, the bandgap has periodic oscillations in the armchair GNRs \cite{ylu210} while the spin polarization at the ribbon edges (and also the bandgap) can be modulated by the strain in the zigzag cases  In 2D graphene sheets, a finite gap can open under large strains, otherwise, it may remain close to zero but the Dirac points are displaced \cite{cocc10,per209,pere09,huan10}  Many interesting electrical, optical, and magnetic properties induced by strain in graphene have been also explored, e g  see in \cite{bunc07,pere09,kuma12,per010,pell10,guin10,tlow10,zhai11} 

Besides, local strain is a good option to improve the electrical performance of graphene devices \cite{pere09,ylu010,fuji10,juan11,baha13}  For instance, it has been shown to enhance the ON current in a GNR tunneling FET \cite{ylu010} and to fortify the transport gap in GNR strained junctions \cite{baha13}  In a recent work \cite{hung14}, we have investigated the effects of uniaxial strain on the transport in 2D unstrained/strained graphene junctions and found that due to the strain-induced shift of Dirac points, a significant conduction gap of a few hundreds meV can open with a small strain of a few percent  This type of strained junction was then demonstrated to be an excellent candidate to improve the electronic operation of graphene transistors  It hence motivates us to further investigate the properties of this conduction gap so as to optimize the performance of graphene devices  On the one hand, the effects of strain should be, in principle, dependent on its applied direction  On the other hand, because the appearance of conduction gap is a consequence of the shift of Dirac points along the $k_y$-axis, it is predicted that this gap should also depend on the transport direction  Note that here, Oy (Ox) - axis is assumed to be perpendicular (parallel) to the transport direction  The effects of both strain and transport directions will be clarified systematically in the current work 

\section{Model and calculations}

In this work, the $\pi$-orbital tight binding model constructed in \cite{per209} is used to investigate the electronic transport through the graphene strained junctions schematized in Fig  1  The Hamiltonian is ${H_{tb}} = \sum\nolimits_{nm} {{t_{nm}}c_n^\dag {c_m}}$ where $t_{nm}$ is the hopping energy between nearest neighbor \emph{n}th and \emph{m}th atoms  The application of a uniaxial strain of angle $\theta$ causes the following changes in the $C-C$ bond vectors:
\begin{eqnarray}
 {{\vec r}_{nm}}\left( \sigma \right) &=& \left\{ {1 + {M_s}\left( \sigma, \theta \right)} \right\}{{\vec r}_{nm}}\left( 0 \right) \\
 {M_s}\left( \sigma, \theta \right) &=& \sigma \left[ {\begin{array}{*{20}{c}}
 {{{\cos }^2}\theta  - \gamma {{\sin }^2}\theta }&{\left( {1 + \gamma } \right)\sin \theta \cos \theta }\\
 {\left( {1 + \gamma } \right)\sin \theta \cos \theta }&{{{\sin }^2}\theta  - \gamma {{\cos }^2}\theta }
 \end{array}} \right] \nonumber
\end{eqnarray}
where $\sigma$ represents the strain and $\gamma \simeq 0 165$ is the Poisson ratio \cite{blak70}  The hopping parameters are defined as $t_{nm} \left( \sigma \right) = t_0 \exp\left[-3 37\left(r_{nm} \left( \sigma \right) /r_0 - 1\right)\right]$, where the hopping energy $t_0 = -2 7$ $eV$ and the bond length $r_{nm} \left( 0 \right) \equiv r_0 = 0 142$ $nm$ in the unstrained case  Therefore, there are three different hoping parameters $t_{1,2,3}$ corresponding to three bond vectors ${\vec r}_{1,2,3}$, respectively, in the strained graphene part of the structure (see Fig  1)  Here, we assume a 1D profile of applied strain, i e , the strain tensor is a function of position along the transport direction Ox while it is constant along the Oy-axis  The transport direction, $\phi$, and strain direction, $\theta$, are determined as schematized in Fig  1  Based on this tight binding model, two methods described below can be used to investigate the conduction gap of the considered strained junctions 

\textbf{Green's function calculations } First, we split the graphene sheet into the smallest possible unit cells periodically repeated along the Ox/Oy directions with the indices $p/q$, respectively (similarly, see the details in \cite{hung12})  The tight-binding Hamiltonian can therefore be expressed in the following form:
\begin{eqnarray}
{H_{tb}} = \sum\limits_{p,q} {\left( {{H_{p,q}} + \sum\limits_{{p_1},{q_1}} {{H_{p,q \to p_1,q_1}}} } \right)}
\end{eqnarray}
where $H_{p,q}$ is the Hamiltonian of cell $\{p,q\}$, and $H_{p,q \to p_1,q_1}$ denotes the coupling of cell $\{p,q\}$ to its nearest neighbor cell $\{p_1,q_1\}$  We then Fourier transform the operators in Eq  (2) as follows:
\begin{eqnarray}
 {c_{p,q}} = \frac{1}{{\sqrt {{M_{cell}}} }}\sum\limits_{{\kappa_y}} {{e^{i{q\kappa_y}}}} {{\hat c}_{p,{\kappa_y}}},
\end{eqnarray}
where $M_{cell}$ is the number of unit cells and $\kappa_y \equiv k_y L_y$ with the size $L_y$ of unit cells along the Oy direction  The Hamiltonian (2) is finally rewritten as a sum of $\kappa_y$-dependent 1D-components:
\begin{eqnarray}
{H_{tb}} &=& \sum\limits_{{\kappa_y}} {\hat H\left( {{\kappa_y}} \right)} \\
\hat H\left( {{\kappa_y}} \right) &=& \sum\limits_p {{{\hat H}_{p \to p - 1}}\left( {{\kappa_y}} \right) + {{\hat H}_p}\left( {{\kappa_y}} \right) + {{\hat H}_{p \to p + 1}}}\left( {{\kappa_y}} \right) \nonumber
\end{eqnarray}
With this Hamiltonian form, the Green's function formalism can be easily applied to compute transport quantities in the graphene strained junction with different transport directions  In particular, the conductance at zero temperature is determined as:
\begin{eqnarray}
 \mathcal{G} \left( \epsilon \right) = \frac{{e^2 W}}{{\pi h L_y}}\int\limits_{BZ} {d{\kappa_y} \mathcal{T}\left( {\epsilon, {\kappa_y}} \right)}
\end{eqnarray}
where $\mathcal{T}\left( {\epsilon,{\kappa_y}} \right)$ is the transmission probability computed from the Green's functions  The integration over $\kappa_y$ is performed in the whole first Brillouin zone  As in ref  \cite{hung13}, the gap of conductance (conduction gap) is then measured from the obtained data of conductance 

\textbf{Bandstructure analyses } To determine the conduction gap of strained junctions, we find that another simple way based on the analysis of graphene bandstructures could be efficiently used  It is described as follows  Since the conductance is computed from Eq  (5), the appearance of conduction gap is essentially governed by the gaps of transmission probability, which is determined from the energy gaps in the unstrained and strained graphene sections  These energy gaps can be defined directly from the graphene bandstructures  Therefore, our calculation has two steps, similar to that in \cite{hung14}  From the graphene bandstructures obtained using the tight-binding Hamiltonian above, we first look for the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ for a given $\kappa_y$ of two graphene sections  The maximum of these energy gaps determines the gap $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ of transmission probability through the junction  Finally, the conduction gap $E_{cond gap}$ is obtained by looking for the minimum value of $E_{junc}^{gap}\left( {{\kappa_y}} \right)$ when varying $\kappa_y$ in the whole Brillouin zone 

In particular, the energy bands of strained graphene are given by
\begin{eqnarray}
 E\left( {\vec k} \right) =  \pm \left| {{t_1}{e^{i\vec k{{\vec a}_1}}} + {t_2}{e^{i\vec k{{\vec a}_2}}} + {t_3}} \right|
\end{eqnarray}
where the plus/minus sign corresponds to the conduction/valence bands, respectively  For a given direction $\phi$ of transport, in principle, the vectors $\vec L_{x,y}$ defining the sizes of unit cell along the Ox and Oy directions, respectively, can be always expressed as ${\vec L_x} = {n_1}{\vec a_1} + {n_2}{\vec a_2}$ and ${\vec L_y} = {m_1}{\vec a_1} + {m_2}{\vec a_2}$ with $\cos \phi = \frac{{{{\vec L}_x}\vec L_x^0}}{{{L_x}L_x^0}}$ and $\sin \phi = \frac{{{{\vec L}_x}\vec L_y^0}}{{{L_x}L_y^0}}$ while $\vec L_{x,y}^0 = {\vec a_1} \pm {\vec a_2}$  Note that $n_{1,2}$ and $m_{1,2}$ are integers while $\frac{{{m_1}}}{{{m_2}}} =  - \frac{{{n_1} + 2{n_2}}}{{{n_2} + 2{n_1}}}$, i e , ${\vec L_{x}} {\vec L_{y}} = 0$  In other words, we have the following expressions
\begin{eqnarray}
{{{\vec a}_1} = \frac{{ - {m_2}{{\vec L}_x} + {n_2}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}};\,\,\,{{\vec a}_2} = \frac{{{m_1}{{\vec L}_x} - {n_1}{{\vec L}_y}}}{{{n_2}{m_1} - {n_1}{m_2}}}}
\end{eqnarray}
On this basis, the energy bands can be rewritten in terms of $\kappa_{x, y} = \vec k \vec L_{x,y} \left( { \equiv {k_{x,y}}{L_{x,y}}} \right)$ by substituting Eqs  (7) into Eq  (6)  This new form of energy bands is finally used to compute the conduction gap of strained junctions 

As a simple example, in the case of $\phi = 0$ (armchair direction), we calculate the conduction gap as follows  First, Eq  (6) is rewritten in the form
\begin{eqnarray}
 E_{\phi = 0}\left( {\vec \kappa} \right) =  \pm \left| {{t_1}{e^{i\kappa_y/2}} + {t_2}{e^{ - i\kappa_y/2}} + {t_3}{e^{ - i\kappa_x/2}}} \right|
\end{eqnarray}
with the vectors $\vec L_{x,y} \equiv \vec L_{x,y}^0$  Using this new form, the energy gap of strained graphene for a given $\kappa_y$ is determined as
\begin{equation}
{E_{strain}^{gap}}\left( {{\kappa_y}} \right) = 2 \left| {\sqrt {{{\left( {{t_1} - {t_2}} \right)}^2} + 4{t_1}{t_2}{{\cos }^2}\frac{{{\kappa_y}}}{2}}  + {t_3}} \right|
\end{equation}
while ${E_{unstrain}^{gap}}\left( {{\kappa_y}} \right)$ is given by the same formula with $t_1$ = $t_2$ = $t_3$ $\equiv$ $t_0$  The gap of transmission probability through the junction is then determined as ${E_{junc}^{gap}}\left( {{\kappa_y}} \right) = \max \left[ {E_{unstrain}^{gap}\left( {{\kappa_y}} \right),E_{strain}^{gap}\left( {{\kappa_y}} \right)} \right]$ and, finally, the conduction gap is given by ${E_{cond gap}} = \min \left[ {E_{junc}^{gap}\left( {{\kappa_y}} \right)} \right]$ for $\kappa_y$ in the whole Brillouin zone 

We would like to notice that the Green's function calculations and the banstructure analyses give the same results of conduction gap in the junctions where the transition region between unstrained and strained graphene sections is long enough, i e , larger than about 5 to 6 nm  In the case of short length, as discussed in \cite{baha13,hung14}, this transition zone can have significant effects on the transmission between propagating states beyond the energy gaps and hence can slightly enlarge the gap of conductance, compared to the results obtained from the bandstructure calculations 

\section{Results and discussion}

\begin{figure}[!t]
\centering
\includegraphics[width=3 0in]{Fig02 pdf}
\caption{Dependence of graphene bandgap (in the unit of eV) on the applied strain and its direction: tensile (a) and compressive (b)  The radius from the central point indicates the strain strength ranging from 0 (center) to 30 $\%$ (edge of maps) while the graphene lattice is superimposed to show visibly the strain direction  The orange circle corresponds to the strains of $\sigma = 23 \%$ }
\label{fig_sim2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3 4in]{Fig03 pdf}
\caption{Conductance ($G_0 = e^2W/hL_y$) as a function of energy in graphene strained junctions for $\sigma = 4 \%$ with different strain directions  The transport along the armchair direction ($\phi = 0$) is considered  The data obtained in a uniformly strained graphene is displayed for the comparison }
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5 8in]{Fig04 pdf}
\caption{Local density of states (left panels) and corresponding transmission coefficient (right panels) for three different wave-vectors $k_y$ obtained in an unstrained/strained graphene junction of $\sigma = 4 \%$, and $\theta \equiv \phi = 0$  On the top is a schematic of graphene bandedges illustrating the strain-induced shift of Dirac points along the $k_y$-direction }
\label{fig_sim4}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=5 6in]{Fig05 pdf}
\caption{Maps of conduction gap in unstrained/strained graphene junctions: tensile (a,c) and compressive cases (b,d)  The transport is along the armchair $\phi = 0$ (a,b) and zigzag $\phi = 30^\circ$ directions (c,d)  The strain strength ranges from 0 (center) to 6 $\%$ (edge of maps) in all cases }
\label{fig_sim4}
\end{figure*}
First, we re-examine the formation of the bandgap of graphene under a uniaxial strain  From Eq  (9), it is shown that a strain-induced finite-bandgap appears only if ${E_{strain}^{gap}}\left( {{\kappa_y}} \right) > 0$ for all $k_y$ in the first Brillouin zone, i e , ${k _y} \in \left[ { - \frac{\pi}{L_y}, \frac{\pi}{L_y}} \right]$, otherwise, the bandgap remains zero  Hence, the condition for the bandgap to be finite is either
\begin{equation*}
 \left| {{t_1} - {t_2}} \right| > \left| {{t_3}} \right|\,\,\,\,\,{\rm{OR}}\,\,\,\,\,\left| {{t_3}} \right| > \left| {{t_1} + {t_2}} \right|
\end{equation*}
and the corresponding values of bandgap are
\begin{equation*}
 {E_{gap}} = 2\left( {\left| {{t_1} - {t_2}} \right| - \left| {{t_3}} \right|} \right)\,\,\,\,\,{\rm{OR}}\,\,\,\,\,2\left( {\left| {{t_3}} \right| - \left| {{t_1} + {t_2}} \right|} \right)
\end{equation*}
This result was actually reported in \cite{per209,hase06}  We remind as displayed in Fig  2(a) that a finite bandgap opens only for strain larger than $\sim 23 \%$ and the zigzag (not armchair) is the preferred direction for bandgap opening under a tensile strain \cite{per209}  We extend our investigation to the case of compressive strain and find  (see in Fig  2(b)) that (i) the same gap threshold of $\sigma \simeq 23 \%$ is observed but (ii) the preferred direction to open the gap under a compressive strain is the armchair, not the zigzag as the case of tensile strain  This implies that the properties of graphene bandstructure at low energy should be qualitatively the same when applying strains of $\left\{ {\sigma ,\theta } \right\}$ and of $\left\{ {-\sigma ,\theta + 90^\circ} \right\}$  This feature can be understood by considering, for example, strains of $\left\{ {\sigma , \theta = 0} \right\}$ and of $\left\{ {-\sigma , \theta = 90^\circ} \right\}$  Indeed, these strains result in the same qualitative changes on the bond-lengths, i e , an increased bond-length $r_3$ and reduced bond-lengths $r_{1,2}$  However, for the same strain strength, because of the exponential dependence of hoping energies on the bond-lengths, the compressive strain generally induces a larger bandgap than the tensile one, as can be seen when comparing the data displayed in Figs  2(a) and 2(b)  To conclude, we would like to emphasize that a large strain is necessary to open a bandgap in graphene  This could be an issue for practical applications, compared to the use of graphene strained junctions explored in \cite{hung14} 

We now go to explore the properties of conduction gap in the graphene strained junctions  In Fig  3, we display the conductance as a function of energy computed from Eq  (5) using the Green's function technique  As discussed above, a small strain of a few percent (e g , 4 $\%$ here) can not change the gapless character of graphene, i e , there is no gap of conductance in the case of uniformly strained graphene  However, similar to that reported in \cite{hung14}, a significant conduction-gap of a few hundreds meV can open in the unstrained/strained graphene junctions  The appearance of this conduction gap, as mentioned previously, is due to the strain-induced shift of Dirac points and is explained as follows  Actually, the strain causes the lattice deformation and can result in the deformation of graphene bandstructure  Therefore, the bandedges as a function of wave-vector $k_y$ in unstrained and strained graphene can be illustrated schematically as in the top panel of Fig  4  As one can see, the shift of Dirac points leads to the situation where there is no value of $\kappa_y$, for which the energy gaps $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are simultaneously equal to zero  This means that the transmission probability always shows a finite gap for any $\kappa_y$  For instance, the energy gap is zero (or small) in the unstrained (resp  strained) graphene section but finite in the strained (resp  unstrained) one in the vicinity of Dirac point $k_y = K_{unstrain}$ (resp  $K_{strain}$)  Accordingly, as illustrated in the pictures of LDOS in the left panels of Fig  4 and confirmed in the corresponding transmissions in the right panels, clear gaps of transmission are still obtained  Far from these values of $k_y$, $E_{unstrain}^{gap}\left( {{\kappa_y}} \right)$ and $E_{strain}^{gap}\left( {{\kappa_y}} \right)$ are both finite (e g , see the LDOS plotted for $k_y = K_{gap}$) and hence a finite gap of transmission also occurs  On this basis, a finite gap of conductance is achieved  More important, Fig  3 shows that besides the strength of strain, the strain effect is also strongly dependent on the applied direction  For instance, the conduction gap takes the values of $\sim$ 295, 172 and 323 meV for $\theta = 0$, $30^\circ$ and $90^\circ$, respectively 

Below, we will discuss the properties of the conduction gap with respect to the strain, its applied direction, and the direction of transport  Note that due to the lattice symmetry, the transport directions $\phi$ and $\phi + 60^\circ$ are equivalent while the applied strain of angle $\theta$ is identical to that of $\theta + 180^\circ$  Hence, the data obtained for $\phi$ ranging from $-30^\circ$ to $30^\circ$ and $\theta  \in \left[ {0^\circ ,180^\circ } \right]$ covers the properties of conduction gap in all possible cases 

In Fig  5, we present the maps of conduction gap with respect to the strain and its applied direction in two particular cases: the transport is either along the armchair ($\phi = 0$) or the zigzag ($\phi = 30^\circ$) directions  Both tensile and compressive strains are considered  Let us first discuss the results obtained in the armchair case  Figs  5(a,b) show that (i) a large conduction gap up to about 500 meV can open with a strain of 6 $\%$ and (ii) again the conduction gap is strongly $\theta$-dependent, in particular, its peaks occur at $\theta = 0$ or $90^\circ$ while the gap is zero at $\theta \approx 47^\circ$ and $133^\circ$ for tensile strain and at $\theta \approx 43^\circ$ and $137^\circ$ for compressive strain  In principle, the conduction gap is larger if the shift of Dirac points in the $\kappa_y$-axis is larger, as discussed above about Figs  3-4  We notice that the strain-induced shifts can be different for the six Dirac points of graphene \cite{kitt12} and the gap is zero when there is any Dirac point observed at the same $\kappa_y$ in the two graphene sections  From Eq  (9), we find that the Dirac points are determined by the following equations:
\begin{eqnarray*}
  {\cos}\frac{\kappa_y}{2} &=& \pm \frac{1}{2}\sqrt{\frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{{t_1}{t_2}}}}, \\
  \cos \frac{{\kappa_x}}{2} &=& \frac{{{t_1} + {t_2}}}{{\left| {{t_3}} \right|}}\cos \frac{{\kappa_y}}{2},\,\,\,\sin \frac{{\kappa_x}}{2} = \frac{{{t_2} - {t_1}}}{{\left| {{t_3}} \right|}}\sin \frac{{\kappa_y}}{2},
\end{eqnarray*}
which simplify into ${\cos}\frac{\kappa_y}{2} = \pm \frac{1}{2}$ and, respectively, $\cos \left( {\frac{{{\kappa _x}}}{2}} \right) = \mp 1$ in the unstrained case  Hence, the zero conduction gap is obtained if
\begin{equation*}
  \frac{{t_3^2 - {{\left( {{t_1} - {t_2}} \right)}^2}}}{{4{t_1}{t_2}}} = \frac{1}{4}
\end{equation*}
Additionally, it is observed that the effects of a strain $\{\sigma,\theta\}$ are qualitatively similar to those of a strain $\{-\sigma,\theta+90^\circ\}$, i e , the peaks and zero values of conduction gap are obtained at the same $\theta$ in these two situations  To understand this, we analyze the strain matrix $M_s \left(\sigma,\theta\right)$ and find that in the case of small strains studied here, there is an approximate relationship between the bond lengths under these two strains, given by \[{r \left( \sigma, \theta \right)} - {r \left( -\sigma, \theta + 90^\circ\right)} \simeq \sigma \left( {1 - \gamma } \right) r_0,\] which is $\theta$-independent for all \emph{C-C} bond vectors  It implies that there is a fixed ratio between the hopping energies $t_i \left( \sigma, \theta \right)$ and $t_i \left( -\sigma, \theta + 90^\circ\right)$ and hence there is the similar shift of Dirac points in these two cases 
\begin{figure}[!t]
\centering
\includegraphics[width=3 4in]{Fig06 pdf}
\caption{Map showing the dependence of conduction gap on the directions ($\theta,\phi$) for $\sigma = 4 \%$  The top is a diagram illustrating the rotation of Dirac points in the \emph{k}-space with the change in the transport direction $\phi$ }
\label{fig_sim6}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5 5in]{Fig07 pdf}
\caption{Maps of conduction gap obtained in tensile/compressive strained junctions  The transport along the armchair/zigzag directions is considered in (a,b)/(c,d), respectively  The strains $\sigma_c = -2 \%$ and $\sigma_t = 2 \%$ are applied in (a,c) while $\sigma_c = -1 \%$ and $\sigma_t = 3 \%$ in (b,d) }
\label{fig_sim4}
\end{figure*}

We now go to analyze the properties of conduction gap shown in Figs  5(c,d) where the transport is along the zigzag direction $\phi = 30^\circ$  In fact, the conduction gap in this case can reach a value as high as that of the case of $\phi = 0$ but has different $\theta$-dependence  In particular, the conduction gap has peaks at $\theta \approx 47^\circ$ and $133^\circ$ for tensile strain and at $\theta \approx 43^\circ$ and $137^\circ$ for compressive strain, where it is zero in the case of $\phi = 0$  It is also equal to zero at $\theta = 0$ and $\theta = 90^\circ$ where the peaks of conduction gap occur in the latter case of $\phi = 0$  The relationship between these two transport directions can be explained as follows  On the one hand, based on the analyses above for $\phi = 0$, we find that for a given strength of strain, a maximum shift of Dirac points along the $k_y$-axis corresponds to a minimum along the $k_x$-one and vice versa when varying the strain direction $\theta$  On the other hand, as schematized in the top of Fig  6 below, the change in the transport direction results in the rotation of the first Brillouin zone, i e , the $k_x$ (resp  $k_y$) axis in the case of $\phi = 30^\circ$ is identical to the $k_y$ (resp  $k_x$) axis in the case of $\phi = 0$  These two features explain essentially the opposite $\theta$-dependence of conduction gap for $\phi = 30^\circ$, compared to the case of $\phi = 0$ as mentioned  Again, we found the same qualitative behavior of conduction gap when applying the strains of $\{\sigma,\theta\}$ and $\{-\sigma,\theta+90^\circ\}$ 

Next, we investigate the conduction gap with respect to different transport directions $\phi$  We display a ($\theta,\phi$)-map of conduction gap for $\sigma = 4 \%$ in Fig  6 and, in the top, an additional diagram illustrating the rotation of Dirac points in the $k-$space with the change in the transport direction  It is clearly shown that (i) a similar scale of conduction gap is obtained for all different transport directions, (ii) there is a smooth and continuous shift of $E_{cond gap}-\theta$ behavior when varying $\phi$, and (iii) the same behavior of $E_{cond gap}$ is also observed when comparing the two transport directions of $\phi$ and $\phi+30^\circ$, similarly to the comparison above between $\phi = 0^\circ$ and $30^\circ$  The data plotted in Fig  6 additionally shows that $E_{cond gap}$ takes the same value in both cases of $\{\phi,\theta\}$ and $\{-\phi,-\theta\}$ with a remark that the strains of $-\theta$ and $180^\circ-\theta$ are identical  Moreover, the values of $\theta$ and $\phi$, for which the conduction gap has a peak or is equal to zero, have an almost linear relationship  In particular, the relationship for conduction gap peaks is approximately given by $\theta = \theta_A - \eta_s \phi$  For tensile strains, $\eta_s$ takes the values of $\sim 1 5667$ and $1 4333$ for $\theta_A = 0$ and $90^\circ$, respectively  On the opposite, it is about $1 4333$ and $1 5667$ for $\theta_A = 0$ and $90^\circ$, respectively, for compressive strain cases  All these features are consequences of the rotation of Dirac points in the $k$-space with respect to the transport direction $\phi$ as illustrated in the diagram on the top and the lattice symmetry of graphene 

Finally, we investigate other junctions based on compressive and tensile strained graphene sections  The idea is that in this type of strained junction, the shifts of Dirac points are different in two graphene sections of different strains, which offers the possibilities to use smaller strains to achieve a similar conduction gap, compared to the case of unstrained/strained junction  In Fig  7, we display the maps of conduction gap with respect to the directions of compressive ($\theta_c$) and tensile ($\theta_t$) strains in two cases of transport direction $\phi = 0$ (armchair) and $30^\circ$ (zigzag) for given strain strengths  Indeed, as seen in Fig  7(a,b), with smaller strains $\left\{ {{\sigma _c},{\sigma _t}} \right\} = \left\{ { - 2\% ,2\% } \right\}$ or $\left\{ { - 1\% ,3\% } \right\}$, similar conduction gap of about 310 meV can be achieved (see Figs  7(a,b)) while it requires a strain of 4 $\%$ in the unstrained/strained junctions discussed above  However, since the shift of Dirac points is strongly dependent on the direction of applied strains and the transport direction, the properties of conduction gap are more complicated than in the latter case  In particular, our calculations show that the preferred transport directions to achieve a large conduction gap are close to the armchair one  Otherwise, the conduction gap is generally smaller, similarly to the data for $\phi = 30^\circ$ compared to $\phi = 0$, as shown in Fig  7  Additionally, it is shown that the preferred directions of applied strains in the case of $\phi = 0$ are close to ${\theta _c} \equiv {\theta _t} = 0$ or $90^\circ$ 

\section{Conclusion}

Based on the tight binding calculations, we have investigated the effects of uniaxial strain on the transport properties of graphene strained junctions and discuss systematically the possibilities of achieving a large conduction gap with respect to the strain, its applied direction and the transport direction  It has been shown that due to the strain-induced deformation of graphene lattice and hence of graphene bandstructure, a finite conduction gap higher than 500 meV can be achieved for a strain of only 6 $\%$  Moreover, as a consequence of the shift of Dirac points along the $k_y$-axis, the conduction gap is strongly dependent not only on the strain strength but also on the direction of applied strain and the transport direction  A full picture of these properties of conduction gap has been presented and explained  The study hence could be a good guide for the use of this type of unstrained/strained graphene junction in electronic applications 

\textbf{\textit{Acknowledgment }} This research in Hanoi is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103 02-1012 42  We also acknowledges the French ANR for financial support under the projects NANOSIM-GRAPHENE (Grant no  ANR-09-NANO-016) and MIGRAQUEL (Grant no  ANR-10-BLAN-0304) 





ground_truth: Players can skip dialogue on the quest map by pressing the 'SKIP' button.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the main focus of the research paper?

Read the following text and answer briefly.

Context 2:  The main purpose is to compare our Monte Carlo results to these analytic predictions 

Exact enumeration

To establish that our Monte Carlo simulations indeed sample the partition functions Eq  ( ) and Eq  ( ), we have obtained analytic results on a 2 4 volume at strong coupling, and at finite beta in two dimensions on a 4 × 4 volume, comparing O (β) and O β 2 truncations  Our strategy to obtain an exact enumeration of the partition function Z is to enumerate plaquette configurations first, then fixing the fermion fluxes which together with the gauge fluxes that are induced by the plaquettes form a singlet, a triplet or anti-triplet, i e  on a given bond b, g b + f b ∈ {−3, 0, 3}, and last we perform the monomerdimer enumeration on the available sites not saturated by fermions yet by a depth-first algorithm  
At strong coupling, with no plaquettes, g b = 0 and f b are baryonic fluxes  All observables that can be written in terms of derivatives of log(z), such as the baryon density, the chiral condensate, the energy density, and also the average sign, are shown in Fig 

Expectations from mean field theory

Another analytical method to study strong coupling lattice QCD is the mean field approach, where the partition function is expanded in 1 d (d is the spatial dimension) and then a Hubbard-Stratonovich transformation performed   After this procedure, the free energy is a function of temperature T , the chiral condensate σ and chemical potential µ B :
here E[m] is one-dimensional quark excitation energy which is a function of the quark mass m = am q   For N c = 3 and d = 3 we determined the minimum of the free energy with respect to the chiral condensate  This gives us the equilibrium chiral condensate as a function of (T, m, µ B )  The chiral condensate and the baryon density as a function of the baryon chemical potential in lattice units aµ B and for various temperatures at quark mass m = 1 5 is shown in Fig    We have determined the critical temperature to be aT c = 0 23 , which is characterized by an infinite slope of the chiral condensate 
For lower temperatures, there is a clear discontinuity of the chiral con-densate, separating the low density phase from the high density phase  For temperatures above and in the vicinity of aT c the chiral condensate and baryon density has no discontinuity but rapidly changes, corresponding to a crossover transition 
With this method, the phase diagram is plotted for different quark masses in Fig    The second order phase transition in the chiral limit is plotted in solid blue line, the dotted lines show the first order phase transition for different quark masses and the solid red line indicates the critical end point for the different quark masses 
Mean field theory also gives an expression for the pion mass am π and the baryon mass am B : The mean field baryon mass for N c = 3, d = 3 is also plotted in red in Fig    Whereas the baryon mass is around N c in the chiral limit (am B 3 12 for N c = 3), it approximately doubles at m = 3 5 (am B 6 28) which corresponds to the pion mass am π = 4 45, i e  m π /m B = 0 708 
Hence, at around bare mass m = 3 5, the valence quark mass of the baryon corresponds roughly to 1/3 of the chiral limit value of the baryon mass  The first Monte Carlo simulations that could extend in the µ B − T plane was the MDP algorithm , but it required the introduction of the worm algorithm to make substantial progress 
First studies of the worm algorithm applied to the strong coupling limit QCD (with gauge group U(3)) are , and for gauge group SU   Monte Carlo simulations to extend the worm to incorporate leading order corrections were first proposed in   We will shortly review the setup of or Monte Carlo strategy for the nuclear transition, with an emphasis on the challenges to address large quark masses 

Strong Coupling

Without any further resummation, there is a mild sign problem in the dual formulation of lattice QCD in the strong coupling limit  When the average sign σ is not too small (close to zero), it implies that most of the configurations have a positive weight thus allowing us to perform sign reweighting strategies 
In Fig  , ∆f is plotted as a function of the baryon chemical potential and the quark masses  It is seen that ∆f is close to zero for most cases except near the critical chemical potential and for small quark masses, but never exceeds 5 × 10 −4   Hence sign reweighting can be performed in the full parameter space 
The result that the sign problem becomes even milder when increasing the mass is related to the fact that larger critical chemical potentials result in a larger fraction of static baryons (spatial baryon hoppings become rare)  FIG    ∆F at strong coupling as a function of chemical potential and quark mass on a 6 3 × 8 
The sign problem becomes milder as the quark mass increases 

Finite β

All runs at finite β have been obtained for N τ = 4, which corresponds to a moderately low temperature aT = 0 25 compared to the value of the chiral transition aT 1 54  Those simulations were too expensive to attempt N τ = 8 runs, in particular as a higher statistics was required  The spatial volumes are 4 3 , 6 3 and 8 3  
For β values are from 0 0 to 1 0 with step size 0 1, and for am q values from 0 00 to 1 00 with step size 0 01  The values of aµ were chosen close to the nuclear transition, the scanning range is shifted to large values as am q increases  At small quark masses the scanning range is from aµ = 0 4 to 1 0 and for the large quark masses, it is from 0 6 to 1 2 with step size 0 01 
The statistics used for are 15 × 10 4 measurements and between measurement, 40 × N 3 s worm updates 

Residual sign problem

Although it is possible to resum the sign problem at strong coupling with a resummation of baryon and pion world lines, this is not possible when including gauge corrections  In order to compare both sign problems, we kept the original dual formulation to monitor the severity of the sign problem  This is done via the relation
between the average sign σ and the difference of the free energy density ∆f between the full ensemble f and of the sign-quenched ensemble f ||  

Nuclear interactions

We have found that aµ 1st B is very different from the baryon mass  This must be due to strong attractive interactions of nucleons  In contrast to continuum physics, in the strong coupling limit there is no pion exchange due to the Grassmann constraint  Instead, nucleons are point like and hard core repulsive 
However, the pion bath, which is modified by the presence of static baryons, results in an attractive interaction  In , this has been analyzed in the chiral limit using the snake algorithm, and it has been found that the attractive force is of entropic origin  Here, we do not quantify the nuclear interaction via the nuclear potential, but via the difference between critical baryon chemical potential and baryon mass, in units baryon mass, as shown in Fig  , given the am B as measured in Section III C 
This compares better to the 3dim  effective theory  The nuclear interaction is maximal and more than 40% in the chiral limit, which is related to pions being massless: the modification of the pion bath is maximal  We clearly find that the nuclear interaction decreases drastically and almost linearly until it almost approaches zero at about am q = 2 0, corresponding to a pion mass am π = 3 36, see Section II B  The large error bars for larger quark masses, that are due to the subtraction of almost same magnitudes, makes it difficult to extract a non-zero nuclear interaction at the largest quark masses 
In this work, we have determined the baryon mass and the nuclear transition via Monte Carlo: the worm algorithm based on the dual formulation, at finite β equipped with additional updates  All those numerical results and various analytic expressions are summarized in Fig    We find that as the quark mass becomes large, spatial mesons hoppings (i e 
spatial dimers) become rare, which makes this 3+1-dimensional system closer to 1dim  QCD   Also, both the baryon mass and the baryon chemical potential obtained in our dual representation, i e  for staggered fermions, approaches the baryon mass of the 3-dim  effective theory which is based on Wilson fermions 
Another comparison that summarizes the validity of the mean field approach discussed in Section II B is shown in Fig    It is evident that mean field theory has strong deviations for small quark masses, but this discrepancy becomes smaller for larger quark masses  The extension of the study of the nuclear transition to finite inverse gauge coupling β is summarized in Fig  , which shows the β-dependence of aµ c B for various quark masses 
For all quark masses ranging from am q = 0 to am q = 1 0, there is only a very weak β-dependence, confirming the expectation from mean field theory   This works was restricted to isotropic lattices ξ = a/a t = 1, i e  we performed simulations at fixed temperature  Non-isotropic lattices are necessary to vary the temperature at fixed values of β 
This requires to include two bare anisotropies, γ for the fermionic action and γ G for the gauge action  Finite β has only been studied by us in the chiral limit   Clearly, it is interesting to study the location of the nuclear critical point also including higher order gauge corrections and at finite quark mass 
Simulations including O(β 2 ) are under preparation 



ground_truth: Nuclear liquid-gas transition in lattice QCD.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does the conduction gap depend on the strain direction?

Read the following text and answer briefly.

Context 4: 

He was comfortably re-elected in Clutha-Southland in the 2011 election but opted to run as a party-list candidate in 2014  

The pairing of John Key as leader of the National Party and English as his deputy has been compared to that of Bob Hawke and Paul Keating (in Australia) and Tony Blair and Gordon Brown (in the UK) 

English acceded to the role of Finance Minister in the continuing wake of the financial crisis  In response to New Zealand's rising debt, English made budget deficit-reduction his main priority  His first budget outlined three focuses in New Zealand's financial recovery: "improving the business environment and removing roadblocks to growth; investment in productive infrastructure; and improving the way government works"  One of his first acts was creating the National Infrastructure Unit, charged with formulating a plan for infrastructure projects and investments  He commissioned a government-wide spending review, with an aim to reducing government expenditure—with the exceptions of a two-year stimulus package and long-term increases on infrastructure spending 

In April 2011, the Opposition criticised English for suggesting that New Zealand businesses could use New Zealand's low wages to help it compete with Australia  The National Government campaigned for re-election in 2011 on its economic record  The Government boasted growth for five consecutive quarters up to mid-2010, totalling 1 6% of real GDP 

Strong growth resulted in a surplus of $473 million for the 2015/16 financial year, projected to rise to $8 5 billion by 2020/21  In his 2016 Economic and Fiscal Update address, English stated that reducing debt and tackling the costs of the 2016 Kaikōura earthquake were higher priorities than reducing rates of tax 

Allowances issue
In 2009, the media, including TVNZ and TV3 revealed that English was receiving about NZ$900 a week as part of a living allowance for ministers, to live in his own NZ$1 2 million Wellington home  At the time, English also received $276,200 in his annual salary as Deputy Prime Minister  It was also revealed other ministers with homes in the capital city were also claiming accommodation allowances  On 3 August 2009, Prime Minister John Key started a review of the housing allowances claimed by cabinet ministers  English subsequently paid back $12,000 and only claimed about $24,000 a year in living allowances  The Auditor-General's office said in September 2009 that they were making "preliminary enquiries" into parliamentary housing expenses in response to a letter of complaint from Progressive party leader Jim Anderton  Two days later English stated that he would no longer take up any housing allowance and had paid back all the allowance he had received since the November 2008 election 

Prime Minister (2016–2017)

John Key resigned on 12 December, and endorsed English as his successor in the resulting leadership election  Following the drop-out of both Judith Collins and Jonathan Coleman from the leadership election, English was sworn in as the 39th Prime Minister of New Zealand on 12 December 2016 

English appointed his first cabinet on 18 December  In a reshuffle, he appointed Steven Joyce to succeed him as Finance Minister, while most ministerial portfolios remained the same 

In February 2017, English did not attend Waitangi Day commemorations at the historic treaty grounds, reportedly in response to the Ngāpuhi iwi's decision to stop the Prime Minister from speaking at the marae  Ngāpuhi have protested the Government's negotiation of the Trans Pacific Partnership Agreement (TPPA), which the iwi believe infringes upon Māori sovereignty, and thus does not adhere to the Treaty of Waitangi  English had been invited to attend in an official capacity; his non-attendance was criticised by a Ngāpuhi elder and Opposition leader Andrew Little 

In his first overseas trip as Prime Minister, English travelled to Europe to discuss trade ties, including a prospective New Zealand–European Union free trade agreement  He first travelled to London on 13 January 2017 to meet British Prime Minister Theresa May  Discussing trade relations, English said the two nations were "natural partners" and would "continue to forge ties" after the UK's withdrawal from the EU  He also arranged to meet with London Mayor Sadiq Khan, Belgian Prime Minister Charles Michel and German Chancellor Angela Merkel   In a meeting with Merkel, English received crucial backing from Germany for a trade deal with the EU  On 16 January, English stated that his government would continue to promote TPPA, despite the United States' decision to withdraw from the agreement  He explained that Southeast Asian countries would now be treated as a priority in negotiations—he also asserted that the United States was ceding influence to China by its rejection of the trade pact 

At a press conference at the Beehive on 1 February 2017, English announced that the 2017 general election would be held on 23 September  The Prime Minister later confirmed that his party would approach ACT, United Future and the Māori Party if confidence and supply agreements were required to form a government following the election  In his second cabinet reshuffle on 24 April, English appointed Gerry Brownlee as his new Foreign Affairs Minister; he also promoted Nikki Kaye to the portfolio of Education Minister, and moved Mark Mitchell into the cabinet to become Defence Minister  The reshuffle was perceived as an election preparation 

On 13 February 2017, English welcomed Australian Prime Minister Malcolm Turnbull to Wellington  The two leaders reaffirmed their shared trade agenda, and discussed changes to the Australian citizenship pathway which will affect permanent residents originating from New Zealand 

On 19 June, it was reported that Todd Barclay, who succeeded English as MP for Clutha-Southland, had clandestinely recorded one of his employee's conversations the previous year, and that John Key's leaders' budget was used to pay a confidential settlement after the employee resigned  English admitted that he had been aware of the illegal recording and the settlement, and thus implicated in the scandal 

During the 2017 National campaign launch, English introduced a $379 million social investment package including digital learning academies for high school students, more resources for mathematics, and boosting support for teaching second languages in schools, and maintaining National Standards in the school curriculum  Prime Minister English also sought to defend National's financial management and economic track record and claimed that the opposition Labour Party would raise taxes  Early opinion polling had forecast a poor showing in the election for the Labour Party, but in early August 37-year-old Jacinda Ardern took over as Labour leader and seemingly energised younger voters 

At the 2017 general election, National won the largest share of the party vote (44 4%) and the largest number of seats (56) in the House Representatives  However, National lacked enough seats to govern alone due to two of the party's support partners, the Māori Party and United Future, losing their parliamentary seats  In response, English stated that the party would be entering into talks to form a coalition with New Zealand First  Following talks with the two largest parties, New Zealand First entered a coalition arrangement with the Labour Party  English was succeeded as prime minister by Jacinda Ardern on 26 October 

Opposition (2017–2018)

Leader of the Opposition
English was re-elected as National Party leader on 24 October 2017  At the time of his re-election, English announced his intention to stay on as leader until the next general election  On 13 February 2018, however, he stood down as National Party leader due to personal reasons, and instructed the party to put into motion the processes to elect a new leader  He also retired from Parliament  English's resignation followed weeks of speculation that he would step aside for a new leader  On 27 February, he was succeeded as party leader by Simon Bridges as the result of the leadership election held that day 

Post-premiership 
In 2018, English joined the board of Australian conglomerate, Wesfarmers  English serves in Chairmanships of Mount Cook Alpine Salmon, Impact Lab Ltd and Manawanui Support Ltd  He is also a director of The Instillery, Centre for Independent Studies and The Todd Corporation Limited, and is a member of the Impact Advisory Group of Macquarie Infrastructure and Real Assets 

Political and social views

English is regarded as more socially conservative than his predecessor, John Key  He has stated his opposition to voluntary euthanasia and physician-assisted suicide, same-sex civil unions, and the decriminalisation of prostitution  As Prime Minister he opposed any "liberalisation" of abortion law 

In 2004, English voted against a bill to establish civil unions for both same-sex and opposite-sex couples  In 2005, he voted for the Marriage (Gender Clarification) Amendment Bill, which would have amended the Marriage Act to define marriage as only between a man and a woman  English voted against the Marriage (Definition of Marriage) Amendment Bill, a bill that legalised same-sex marriage in New Zealand  However, in December 2016 he stated, "I'd probably vote differently now on the gay marriage issue  I don't think that gay marriage is a threat to anyone else's marriage" 

In 2009, English voted against the Misuse of Drugs (Medicinal Cannabis) Amendment Bill, a bill aimed at amending the Misuse of Drugs Act so that cannabis could be used for medical purposes 

Personal life 
English met his future wife, Mary Scanlon, at university  She was studying medicine at the time, and became a general practitioner  Both her parents were immigrants, her father being Samoan and her mother Italian, born on the island of Stromboli  They have six children: a daughter and five sons 

English is a practising Roman Catholic, but has stated that he considers his religious beliefs personal and thus separate from politics 

In June 2002, English took part in TV3's Fight For Life, a celebrity boxing fundraiser to raise money for the Yellow Ribbon anti-youth-suicide campaign, influenced by the death of a teenage nephew in 1997  He lost a split decision to former university colleague Ted Clarke 

Honours
In the 2018 Queen's Birthday Honours, English was appointed a Knight Companion of the New Zealand Order of Merit, for services of over 27 years to the State 



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