Urdd Athletics Competition On Tuesday, 26th of June, 19 pupils travelled from Bro Tegid went to Eirias Park to complete in 14 activities against six other counties. We are very proud of the pupuls who reached this round and it was a pleasure to see them all giving it their best and showing plenty of determination. Thanks to this, they all broke their personal records whether that in time or distance. We can announce that 5 activities saw us end up on the podium. This is the highest total the school have ever achieved and we are immensely proud of everyone especially as we were challenging against schools 3 times larger than Bro Tegid! Here are the full results: 1st Llew Jones - javellin 1st Saran Jones - javellin 2nd Aron Jones - javellin 2nd boys relay year 3&4 3rd girls relay year 3&4 Massive congratulations to everyone!
School Sports Evening We had a special night in our yearly sports evening with a crowd full of parents, friends and former pupils in attendance which made for a relaxing atmosphere. It was a pleasure to see everyone taking part with huge smiles on their faces. We hope that the adults weren't in too much pain after the exciting tug of war contest!!! The yellow team were champions this year with red in 2nd, blue in 3rd and green in 4th. Well done everyone!
Ambassadors of the Year Congratulations to Cara and Ioan who were rewarded in the Gwynedd and Anglesey Sports Ceremony in the Gallery, in Caernarfon on the 12th. Both have worked very hard this year encouraging sport in the school and hosting various activities to the pupils and showed plenty of enthusiasm and dedication in their roles within the school. We're very proud of them and their achievements and thank them for being such special Ambassadors.
Llanfyllin Compeition On Saturday, 14th July, two teams from Bro Tegid went to Llanfyllin to compete in their 5 a side football competition. Our Year 5 and 6 team finished 3rd in the group after winning three, drawing one and losing two. A great effort, well done everyone! Year 3 and 4 fought their way all the way to the final after winning three and drawing four games. The final was very close against Ysgol Dyffryn Banw. Dyffryn Banw went ahead but the Bro Tegid boys showed a lot of determination and responded well with Lewys equalising. In added time, Steffan scored with a bullet of an effort into the back of the net and securing the win for the lads! A huge congratulations to this talented teams.
We had a mega busy week in school before closing for the summer with plenty of fun and various activities going on throughout the week. Rafting - as always one of the year highlights is when year 5 and 6 go rafting and it didn't disappoint. They all enjoyed the experience and excitement. A bonus were the valueable photos that captured the day!!! Swel's Poetry Competition - our theme this year was 'pirates' and from reading all the pupils' poems, it was apparent that they all enjoyed the work! Chief Poet Elwyn Edwards or as we all fondly know him Swel, was full of praise of the high standard of poems that were full of comparisons unique and effective. Anya Roberts and Cara Jones were the winners of the best sentence with Rachel Edwards creating the best verse. But winner of Poet of the Year was Hannah Griffith. Hanna's poem stood out from the rest and Swel was so pleased with the standard and choice of wise words. Congratulations Hanna and thanks again to Swel this year. Visit from Teleri Davies - it was a pleasure to welcome Teleri back to the school to show us her Wales cap and shirt, after representing our country in the Womens Six Nations against Scotland back in February. We are so proud of Teleri's successes and her proud achievement especially as a former pupil of the school. It was interesting to listen to her experiences, her rugby career and of course a report of the game from her perspective. Congratulations again to Teleri and thank you for her time. All the best for the future. Dad v Mum's Game! - It was that time of year again with the dads and mums of the schools taking on the pupils in football and netball matches. We had an exciting and fun day with everyone enjoying the games. The red team came out on top in both tournaments and more importantly the pupils won the important game which was winning against the adults!!! The girls netball won 4-3 against the mums and the boys won 8-2 against the dads. We thank everyone who came to take part and contribute to the day. SO58 Competition - KS2 pupils had a great opportunity to design a t-shirt for Tim Williams, Anni May from reception's dad to sell as part of his popular business Spirit of 58. The designs were fantastic and Tim decided to reward 9 pupils in total but the design that came out on top and the idea that will be on a t-shirt and sold at the Eisteddfod down in Cardiff this wear was from Mared Rowlands with the slogan "dad is football mad!". Congraulations to Mared, what a special prize and thanks to Tim for his generosity and time towards the pupils. Saying goodbye to Year 6 - it came the time to say goodbye to 19 of our special pupils. We have been very fortunate to have such a great year 6 who have been a brilliant example to everyone. Its been a pleasure to teach them and see them flourishing over the years. Thank you to you all - we wish you all the best for the future.
Summer Reading Challenge Annually, the Summer Reading Challenge is open to all primary school pupils as they have to visit the library three times during the summer holidays and pick various books to read. We are delighted that 80 pupils took part and completed the challenge! Brilliant everybody and keep it up!
Bro Tegid's Macmillan Coffee Morning We had a hugely successful Macmillan Coffee Morning yet again this year at the school. We raised £465 for this important charity and this will be going towards our local centres here in North Wales. A big well done to the School Council for their thorough work and to those who donated towards the raffle and gave prizes, to those who were busy making them delicious cakes and to all that came to support. Thanks also to the pupils for supporting so well. We are extremely grateful to you all!
Urdd Football Meirionnydd Congratulations to our two teams who competed in the Urdd Meirionnydd Football tournament on Thursday, 18th October. We are proud of their successes and the girls individual performances. Once again, there was plenty of excitement and competitive football with fantastic effective teamwork. The blue team reached the final with Ysgol y Traeth the opponents and what a game this was! The effort and enthusiasm by the girls was exceptional and they were determined to succeed and through plenty of endevour they scored the only goal of the game in dramatic fashion with only three seconds left to play when Hana Williams netted and crowning Bro Tegid as champions. They will now represent Meirionnydd in the National Competition in Aberystwyth in May. Congratulations and good luck girls!
Gwynedd Netball Competition Following their success through Meirionnydd before halfterm, Bro Tegid's team travelled to Porthmadog to take on Gwynedd's largest schools in the Sports for Life competition. There as always was high playing standards and it was a pleasure to see every member of the squad performing so well and bringing more pride to the school. Bro Tegid had the perfect start as they went on to win their first five games before losing to Ysgol y Gelli and a draw against Bro Idris. With these results, Bro Tegid finished in second place behind Gelli and they will now go on to take on the rest of north Wales' counties in January. Congratulations to every one of you and good luck!
Operation Christmas Child Once again this year there was plenty of interest as we supported the charity Operation Christmas Child. We collected a total of 163 boxes between all the schools in the area! Everyone's donations go to make a huge change in the lives of the children that these boxes will be going to. Thanks to everyone for their incredible support.
Urdd Gymnastics The regional Urdd Gymnastics competition took place in Blaenau Ffestiniog on the 22nd of November. Annually, the standard is unbelieveably high and it was great to see it so high this year. Alaw Roberts came 2nd in the Individual and the Bro Tegid group came 1st and due to this and representing Meirionnydd, they now progress to the National competition in Aberystwyth in February! We'd like to say a big thank you to Delyth Jones for teaching the girls to such an impressive level. Congratulations and good luck to you in Aber.
Urdd Swimming Gala Congratulations to these three pupils from Ysgol Bro Tegid who had success in the Urdd Regional Swimming Gala. The three finished in the top three in their specific races which is a fantastic achievement for them all.
Bridging Generations Since September, Year 1 pupils and residents from Awel Y Coleg have been meeting up on a weekly basis to take part in the Bridging Generations project Some of the children have developed close relationships with the residents and it's apparent how happy they are to see each other every week. Before the start of the project, the residents and pupils created a timetable and everyone had imput by creating a timetable that was fun and appropriate for everyone. They have been creative with clay, having a 'Keep Moving' session with Aunty Olwen in charge and a singing session with Aunty Millie plus much more. By now, some of the Cysgod y Coleg residents have joined us and we look forward to continuing these weekly activities for the rest of the school year.
Sportshall Meirionnydd The time came once again for us to compete in the annual Sportshall competition which is indoor athletics. Dolgellau was the location for the first round for Meirionnydd schools. The standards were high yet again this year. Ysgol Bro Tegid came out on top with Ysgol O M Edwards closely in second. The times for both schools on the track were fantastic but Bro Tegid's throwing and jumping skills were superb as they won four out of the five activities. Congratulations to both schools and good luck to them in the next round as they take on the larger schools of Gwynedd on the 22nd of February in Bangor.
Sportshall Gwynedd Athletics After their success in Meirionnydd, it was time for our talented Athletics team to head to Bangor to compete against the best from Gwynedd's largest schools in the Sportshall competition. Annually the competition created plenty of excitement and an electric atmosphere with the standard raising every year. This year, three records were broken with the three from Bro Tegid with the Speed Bounce and two times in the relay races. The attitudes of the 9 pupils were superb and they showed incredible concerntration skills and eager to do their best. Ysgol Bro Tegid were winners with Ysgol O.M. Edwards in second - a huge honour for Bala that there is plenty of talent yn Penllyn! This is Bro Tegid's final opportunity to compete in the competition and it was the perfect ending for us especially as it was our 7th successive win! A brilliant achievement and a massive congratulations to everyone involved. Well done!
Young Cooks of Bala! Once again this year, the enthusiastic cooks of Ysgol Bro Tegid and Ysgol Beuno Sant were competing in the Urdd's cooking competition CogUrdd. They had to prepare vegetable kebabs and pesto without nuts infront of the beady eye of judge Mrs Catrin Morris. Congratulations to them all for doing so well. The winners were: Ysgol Bro Tegid - 1st for Alys, joint 2nd for Lili and Hana. Thank you to them all for taking part and for the fantastic support from their parents and to Mrs Morris for judging.
Urdd Regional Eisteddfod We have a very successful Regional Eisteddfod with 9 different items competing. Kaytlin came 2nd in the recitation for Learners Year 5 and 6, the Twovoice party and recitation group also came 2nd. The folk dancing group Year 3 and 4 and the folk dancing group Year 5 and 6 came 2nd and everyone had plenty of praise for their efforts. Representing Ysgol Bro Tegid in the National Eisteddfod in Cardiff in May will be Caio with the individual singing for Year 2 and under, Madi in the Year 6 and under Hip Hop dancing, unison party and choir. We are so proud with everyone's success and commitment from everyone involved and wish them all well!
More about the Eisteddfod! We had even more success in the Urdd's Arts and Crafts. Congratulations to Jac Owen, Becca Roberts, Tili Galdo, Mared Rowlands, Cadi Llywela, Crystal Kimberley and to Noa, Maxli, Sebi and Thomas' group for coming 1st and their work will be going down to the Nationals in Cardiff to be judged against the best in Wales. Good luck to you all! Work from 28 pupils succeeded to come in the top three and you can see the full results here: http://digwyddiadau.urdd.org/files/8915/5430/7027/Canlyniadau_Celf_Rhanbarth_2019.pdf A big thank you to the volunteers (Rita, Liz, Diane, Dorothy Ann and Helen) for their time and support with the with the amazing amount of work with the children.
Meirionnydd Tag Rugby On the 5th of April Year 4, 5 and 6 girls took part in the tag rugby competition held on the 3G, with an International player called Gwenllian Pyrs refereeing. There were 3 teams from the school (Y Gwyniaid, Y Gleision a'r Gwylltiaid) taking part and it was a pleasure to see each team playing together particularly well and enjoying themselves. The Gwylltiaid reached the final in the 'The Plate 'competition. Y Gleision went to the semi-final in 'The Cup' and Y Gwyniaid went one further and won the main competition. The final was a very exciting and close game against Ysgol Bro Tryweryn, with Elin Jones scoring a try in the final moments to ensure the victory for the Gwyniaid. They will now represent Meirionnydd in the National Urdd competition in Aberystwyth in May. Huge congratulations and good luck to you all!
More about the Eisteddfod! We had even more success in the Urdd's Arts and Crafts. Congratulations to Jac Owen, Becca Roberts, Tili Galdo, Mared Rowlands, Cadi Llywela, Crystal Kimberley and to Noa, Maxli, Sebi and Thomas' group for coming 1st and their work will be going down to the Nationals in Cardiff to be judged against the best in Wales. Good luck to you all! Work from 28 pupils succeeded to come in the top three and you can see the full results here: http://digwyddiadau.urdd.org/files/8915/5430/7027/Canlyniadau_Celf_ Rhanbarth_2019.pdf A big thank you to the volunteers (Rita, Liz, Diane, Dorothy Ann and Helen) for their time and support with the with the amazing amount of work with the children.
Meirionnydd Tag Rugby On the 5th of April Year 4, 5 and 6 girls took part in the tag rugby competition held on the 3G, with an International player called Gwenllian Pyrs refereeing. There were 3 teams from the school (Y Gwyniaid, Y Gleision a'r Gwylltiaid) taking part and it was a pleasure to see each team playing together particularly well and enjoying themselves. The Gwylltiaid reached the final in the 'The Plate 'competition. Y Gleision went to the semi- final in 'The Cup' and Y Gwyniaid went one further and won the main competition. The final was a very exciting and close game against Ysgol Bro Tryweryn, with Elin Jones scoring a try in the final moments to ensure the victory for the Gwyniaid. They will now represent Meirionnydd in the National Urdd competition in Aberystwyth in May. Huge congratulations and good luck to you all!


via the seller/listing: "This guitar is a ONE OF A KIND custom made ESP "KH2 Vintage" (manufactured this year, and played once) Giants guitar. It was used once by Kirk Hammett to play the National Anthem before the Giants vs. Dodgers game in May 2013. The guitar was auctioned off during the game, and after much bidding, I won. I called ESP guitars and they said that the guitar itself (before adding things that made it one of a kind) would normally sell for $7000 but they said its worth much more than that because of the Giants emblem, bands autographs, and being played by Kirk; there will never be another one like it. They couldn't find it by serial # K1117502 because it was a one of a kind, but they did find it by Model- KH-2 "Vintage"-BLK. Comes with case and 2 letters of authenticity, one from ESP Guitars and one from the Giants. Has a 3 way switch..."


text:
The Betrayal	of the Kurds
Cengiz Yar for Rolling Stone
Home Politics Politics Features
On the ground in northern Syria in the perilous weeks after Trump's military withdrawal
Jason Motlagh
On the morning of October 12th, Hevrin Khalaf, a rising young Kurdish political leader, rode along the M4 highway in northern Syria. Seated in the back of a bulletproof Toyota SUV, she rushed past the battle-scarred villages of her homeland, now three days into a brutal military assault from Turkey, made possible by Donald Trump's decision to pull U.S. troops from the region. She was on her way to Raqqa, the short-lived capital of the Islamic State and the largest and most heavily damaged city in Kurdish-held territory. Raqqa was beginning to recover, and Khalaf was headed to one of her frequent political meetings there. In 2018, she had helped found the Future Syria Party (FSP), with the lofty goal of advancing pluralism and democracy across Syria's sectarian fault lines.
Khalaf, 35, a Syrian Kurd with long brown hair and a dimpled chin, believed a new kind of politics was needed. Kurds, Christians, Turkmens, and Arabs alike had suffered under the Syrian regime and ISIS' reign of terror, and she wanted to unify historically fractious communities fed up with violence. "The aim of our party is unity and brotherhood for everybody who lives on this land," she said in February. "We will engage in dialogue with all people to restore the spirit of tolerance in the region."
Khalaf lived in Derik, a working-class city of cinder-block buildings, about six miles from the Turkish border. It was part of the Kurdish-led semi-autonomous region in northeast Syria known as Rojava, an area that became a symbol of hope for the long-persecuted minority, 35 million strong, divided among four hostile countries. Rojava champions diversity, self-rule, and women's rights. It is the "most inclusive governance structure in the most diverse region of Syria," says Nicholas Heras, the Middle East security fellow at the Center for a New American Security.
The democratic experiment could not have emerged under more adverse circumstances. Syrian dictator Bashar al-Assad's regime is determined to reclaim every inch of territory lost during the country's nine-year civil war, and it has made steady gains, with backing from Russia and Iran. The Kurds and Assad had maintained a de facto non-aggression pact for most of the conflict, allowing the Kurds to focus on their own affairs until the ISIS threat emerged. Aided by U.S. air power and special forces, Kurdish fighters cleared the militants from one-third of the country, at a cost of more than 11,000 lives.
The hard-won calm was shattered in early October. Khalaf's car sped through a countryside once again under bombardment. Less than a week earlier, following a phone call with Turkish President Recep Tayyip Erdogan, Trump ordered all U.S. military forces to withdraw from Syria, effectively greenlighting the attack on the Kurds. Turkey had already launched limited military operations along the border in 2016 and 2018 but had been held in check by the American presence. They viewed the Kurds' main force, the YPG, as an extension of the banned Kurdistan Workers' Party, which it considers a terrorist group. Turkey's declared intention is to create a 20-mile-deep, 300-mile-long "safe zone" in Kurdish lands, to buffer Turkish territory and resettle millions of Syrian war refugees. But people fear a more malevolent goal. "It has to be understood," says Kurdish journalist Mohammed A. Salih, a Ph.D. student at the University of Pennsylvania, "as a deliberate act of setting up the Kurds and religious minorities in northeastern Syria for ethnic cleansing."
Khalaf had brushed aside warnings not to travel. "She was braver than any man I know," says her friend and party member, Samira Abdel Aziz. On the highway near the town of Tal Abyad, Khalaf's vehicle was ambushed by Ahrar al-Sharqiya, a Turkish-backed militia. A video posted online shows the Toyota askew off-road, riddled with bullets. The driver lies dead, facedown in the dirt as militants crowd around him. "Another fleeing pig has been liquidated by the hands of the National Army," one man gloats. A female voice is briefly audible in the background. These are the last words ever heard from Khalaf.
Khalaf advocated for tolerance and democracy. "Hevrin didn't believe violence would work," her mother says. "She never touched a bullet."
Photograph by Cengiz Yar for Rolling Stone
I entered Syria 10 days into the Turkish assault, coming overland from Iraq, taking the only border crossing still open, at Semelka, a pontoon bridge across the Tigris River. Vans packed with people pass us on their way out, sagging close to the ground. We drive fast, past shuttered shops and billboards of martyred YPG fighters, eager to get off the dark road. ISIS sleeper cells and militias had been setting up fake checkpoints, and Turkish shells were killing and maiming at random.
Under fire from both Republicans and Democrats for his careless decision, Trump had dispatched Vice President Mike Pence to broker a temporary cease-fire, but artillery barrages, drone attacks, and ground incursions continued, making a mockery of Trump's pledge to "obliterate" the Turkish economy with sanctions if it did anything "off-limits." To date, several hundred civilians have been killed and more than 300,000 people displaced, most of them pushed deeper into Kurdish-held territory, where schools are being converted into shelters. Another 15,000 and counting have become refugees in Iraq.
I visit Khalaf's family compound in Derik. Relatives and friends grieve in silence and drink bitter coffee beneath posters of her face. Wearing an all-black abaya festooned with a YPG flag pin, Khalaf's mother, Souad Mohammad, takes phone calls from media outlets in Europe. "I hope the blood of my daughter will unite all of Kurdistan — Kurds, Arabs, Christians — and I hope the whole nation will live in peace," Mohammad says, her anger building in a crescendo that fills the room. "Hevrin sacrificed her soul for this. Sehid namirin! Sehid namirin! Sehid namirin!" Martyrs never die.
Mohammad shows me a wall-to-wall banner featuring her two brothers and first-born daughter — all of them guerrillas who died fighting for Kurdish autonomy in Turkey. Khalaf's image had been added to the martyrs' gallery, but her mother insists she was a different breed: "Hevrin did not believe violence would work. She never touched a bullet."
Khalaf's promise did not go unnoticed by U.S. officials. She and other leaders were tasked with the "necessary and often thankless task of building stability after ISIS that the U.S. had required of them," says Heras. On October 3rd, State Department representatives had visited the FSP in Raqqa, says party president Ibrahim al-Qaftan, and assured them that their party would have a role in international talks over Syria's future. (The State Department did not respond to repeated requests for comment.)
As secretary-general of the Future Syria Party, Khalaf kept a dizzying schedule and met frequently with Arab tribal leaders to build trust and resolve disputes, says al-Qaftan, an Arab. She also led workshops for victims of domestic abuse and tutored students in math, leaving little time for herself.
"She was concerned with bigger things," says Steristan Haji, a friend since childhood. Never married, Khalaf lived in the same room she grew up in, with few indulgences: scented candles, dark chocolate, poetry. Haji hands me a half-finished volume by Kurdish writer Loran Khatib about a woman's struggle for respect in a conservative, male-centric culture. "Hevrin always said we must raise our voice," says Haji.
Marwan Khalif, a fighter with the Kurds, was killed by Turkish-backed militants who were clearing the way for the Syrian Arab refugees to be resettled in Kurdish lands.
According to an autopsy report, Khalaf sustained multiple gunshots and fractures to her leg and skull; flesh was ripped out of her scalp, a sign she had been dragged by force before execution. Turkish state-linked media immediately hailed a "successful operation" to "neutralize" a politician affiliated with a "terrorist" group. Khalaf, her driver, an aide, and at least eight more unarmed civilians were murdered that day on the same stretch of road. Amnesty International decried the summary killings and other indiscriminate attacks on residential areas as war crimes.
Khalaf's mother says Khalaf's face was mutilated beyond recognition, her neck, ears, and wrists stripped of jewelry, like they wanted to "erase" her. "She was not against Turkey; she was not against anyone," Mohammad says. "How could the U.S. let Erdogan do this?"
Since before taking office, Trump had expressed an aversion to open-ended wars in Afghanistan and Iraq that have cost tens of thousands of lives and trillions of dollars. But for a businessman with a cynically transactional nature, supporting the Kurds made good sense: a low-cost, high-reward investment (the U.S. had about 1,000 special forces on the ground) that held stability across a vast swath of Syria while keeping Assad's regime, and its Russian and Iranian backers, in check.
"You are leaving us to be slaughtered," Gen. Mazloum Kobane Abdi, commander of the Kurdish-led Syrian Democratic Forces (SDF), told the top American diplomat in northern Syria, William Roebuck, in a meeting the day Khalaf was killed. If the U.S. did not intervene, he warned, he would be forced to cut a deal with Russia and Assad, for protection from Turkey. Ten days later, Russian President Vladimir Putin and Erdogan announced a pact that saw Russian and Turkish forces take joint control of Kurdish-held -territory. Seemingly on a whim, Trump had managed to upend the dream of Rojava; swing the balance of power in Syria decisively in favor of arch foes Assad, Russia, and Iran; and risk giving a second wind to ISIS.
The sudden shift and ignominy that followed — Russian flags raised at vacant U.S. outposts, Kurds pelting American convoys with rocks and garbage — has rankled active-duty special-forces operatives. "The guys I've spoken with are very disappointed," says a veteran of the campaign against ISIS. "The Kurds are the most motivated and capable partner forces I've ever served with — they fought to the death — and they've gotten the short end of the stick again," he says, noting the long history of American betrayal of the Kurds, including when President George H.W. Bush called on them to rise up against Saddam Hussein in 1991, only to stand by as they were massacred.
"The consequences of such unreliability from the Oval will reverberate well beyond Syria," tweeted Brett McGurk, who served as the special presidential envoy to combat ISIS from 2015 to 2018. "The value of an American handshake is depreciating. Trump today said we could 'crush ISIS again' if it regenerated. With who? What allies would sign up? Who would fight on his assurances?"
U.S. officials estimate there are as many as 18,000 ISIS members still at large in Iraq and Syria; another 12,000 are being held in Kurdish prisons, where control is now in jeopardy. A Kurdish intelligence official tells me that all ISIS prisoners have been relocated farther south from the Turkish-occupied zone. "We are protecting not just for Europe but all the world," he says, "but our biggest priority is fighting against Turkey and the defense of our nation." Half the prison guards have been pulled away to fight since the invasion began, he says, and more than 100 fighters have already escaped, along with 785 sympathizers.
ISIS is brazenly exploiting the vacuum. In Qamishli, a bomb detonated in the city center the day I left, the second attack in as many weeks attributed to ISIS. On November 11th, three explosions claimed six more lives. That same day, an Armenian priest and his father were murdered on a highway south of the city. ISIS took credit. According to the Rojava Information Center, ISIS is targeting people who work with Kurdish-led authorities in Deir Ezzour province, claiming over 100 killings since the Turkish invasion. Lina Abdulwahid, another female leader of the Future Syria Party, was shot on November 17th but survived.
Children play outside Khalaf's home in Derik, six miles from the Turkish border, well within shelling range. A spate of videos online shows Kurdish captives being tortured and executed by Turkish-backed militants.
Meanwhile, Trump has backtracked on his call for a full withdrawal and redeployed U.S. troops to secure oil fields around Deir Ezzour. Despite his claims of "massive amounts of oil," Syria's reserves amount to less than one percent of global output, and experts say it would require many years and billions in investment to rehabilitate the infrastructure. And then there's the fact that the oil belongs to Syria. Senior U.S. officials have tried to downplay the oil gambit, but Trump has been publicly stating that troops remain "only for the oil." Some 500 to 600 U.S. forces will stay indefinitely with a hazy mandate to protect oil fields, not our besieged Kurdish allies who sacrificed so much in the name of pluralism.
In November, a leaked memo from Roebuck, the American envoy in northern Syria, slammed the Trump administration for not doing enough to stop the Turkish invasion, which "represents an intention-laced effort at ethnic cleansing." House Democrats and Republicans are trying to impose sanctions on Turkey, but the legislation would have to pass a Republican-controlled Senate and then be signed by the president. Were there any doubt on where Trump stands, on November 13th he honored Erdogan with a White House reception and thanked him for the job he's done in Syria, declaring himself "a big fan."
The road from Qamishli to Iraq is grim. "Rojava is our nation," says our chain-smoking driver, Abdulrahman, waving a tattooed arm across the horizon. "We are not refugees on this land — this is our land!" But the scenes we pass say otherwise. Some SDF checkpoints are totally empty, torn flags whipping in the wind; others are guarded by a few holdouts. Oil pumps creak on the gray horizon, the air tinged with acrid smoke from makeshift refineries. Flatbed trucks full of fleeing families queue at gas stations. Those without transport wait roadside for a ride, clutching what they can carry: infants, bedrolls, plastic bags of clothes.
As we move along, Abdulrahman concedes it may be time to move his family to Iraq — or "pick up my gun again" — fearful that the Syrian regime will expand its presence and enact reprisals. The previous night, Russia and Turkey announced a deal to carry out joint ground patrols in northeast Syria and allow Syrian troops to move back into border areas from which they'd been absent for years.
Juliet Nicola recovers from shrapnel wounds to her back and stomach in a hospital in Qamishli, the de facto capital of Rojava. "She may never walk again," says her husband, Fadi. Their home and shop were destroyed by Turkish shelling.
Until Rojava, Kurds had lived as second-class citizens in Syria, their language and culture suppressed. Back in 2008, Abdulrahman spent 14 months in prison for lighting a bonfire on the Naw Ruz holiday, disregarding a regime prohibition against celebrations. When fighting broke out in his native Aleppo, he joined a Kurdish protection unit and fought more than three years to beat back jihadist and Assad forces, losing some 30 friends along the way. "Nowhere is safe for the Kurds," he says. "If the racist regime comes back, they won't allow us to breathe."
Erdogan intends to fill Kurdish areas with 2 million Syrian Arabs who are currently refugees in Turkey. Proxy militias are clearing the way, deploying cruel shock tactics. Turkish-backed fighters have been accused of attacking civilians with white phosphorus, a highly flammable chemical that eats flesh. A spate of videos has emerged showing militants torturing Kurdish captives and mutilating dead bodies. In one recording, a man goes door-to-door in the border town Serekaniye, shouting "kill the pigs, kill the infidels" as he spray-paints vacant homes to differentiate which belonged to Kurds, Christians, and Muslim Arabs. He pledges to burn the first, loot the second, and leave the third untouched.
"Turkey is planting the seeds of future civil strife between Arabs and Kurds," says Hassan Hassan, director of the nonstate actors program at the Center for Global Policy. He suspects that radical militias like Ahrar al-Sharqiya, the group that killed Khalaf, have been put forward as the "face" of the campaign to aggravate sectarian tensions. "It is a sinister agenda."
When we reach the border, Abdulrahman bids us a quick goodbye, eager to get back to his wife and children in Qamishli. He named his baby girl Hevrin; she was born two days after the politician was slain. I ask why he chose her name. "Because she gave us dignity," he says.
Over the border in Iraqi Kurdistan, refugee camps are swelling. At the entrance to Bardarash, a vast U.N.-run camp, Amin Mohammad and his family stand amid the swarm waiting for their housing assignment. Mohammad's brother was killed by a Turkish shell in Serekaniye on the first day of the Turkish offensive, while riding his motorcycle to the market to buy food. Their family home was also leveled, and for the past 14 days they had moved from village to village, sleeping rough by the road and scrounging for food. "We are so tired," Mohammad says. "I can't think about the future of Rojava, just how to find rice to eat."
Bardarash, a U.N.-run camp just over the border in Iraq, has been swelling with thousands of refugees since the invasion.
For the next half-hour they wander through a maze of white canvas tents searching for their new home. They trudge past gaggles of scruffy children and the flat stares of families, until they spot a tent spray-painted with their number, C-652. Daylight is fading fast, and everyone hurries to set up bedding on the concrete slab, which is flooded from a broken outhouse pipe.
"I wish I had died in my home, not here," says Mohammad's wife, Hukmiya. "The Americans sold us out. May the same thing happen to Erdogan and Trump — one day they should live under a tent."
Just across the lane, Selva Sedo cradles her nine-month-old daughter, Nivar. Originally from Raqqa, Sedo moved to the Kurdish enclave of Afrin in 2014 to escape the fighting, and had to move again when Turkey occupied the town in early 2018. She was living in Serekaniye with her husband until Turkish shelling forced her to leave once more. "The war is following me," she deadpans.
Sedo's parents are still in Afrin. Most of its former 200,000 Kurdish residents have left, but those who remain face harassment and looting from Syrian Arab militias involved in Turkey's current operation. "I'm so sick of fighting," she says. "I want to leave Syria, [and] Kurdistan. I'll go anywhere outside the Middle East." After a pause, she asks. "Do you know anyone who can help me get to Europe?"
On my last afternoon in Syria, Souad Mohammad takes me out to the martyrs' cemetery where Khalaf is buried. It sits on the outskirts of Derik, about three miles from the Turkish border, well within shelling range. An airstrike near the cemetery five days earlier forced the family to cancel a visit.
Amin Mohammad's family made it to Bardarash after two weeks on the road. Their home was leveled, and his brother killed by a shell on the first day of the assault. "I can't think about the future of Rojava," Mohammad says, "just how to find rice to eat."
The road out of town is empty of cars and people. With few words between us, we pass a vacant camp that used to house scores of Yazidis, the religious minority that Kurdish fighters rallied to save from an ISIS campaign of systemic slaughter as much of the world stood by and did nothing.
We pull up to the cemetery gate. More than 1,200 fallen are interred on the grounds, some 700 of whom perished in the battle against ISIS. We pass row after row of framed portraits of the dead, an even mix of women and men in camo fatigues, some stern and some smiling, until we arrive at Khalaf's grave, draped with plastic flowers and a sash reading martyrs never die.
Mohammad sinks to her knees. "Oh, my dear, oh, my dear. How much I miss you. You gave your life for peace, your blood for brotherhood. Where are human rights? Where is the world?"
Nearby, a weary group of YPG fighters takes turns digging nine fresh graves for comrades killed in recent clashes at the border. Behind them, a walled expanse longer than a football field sprawls flat and unbroken, waiting for tomorrow's casualties.
In This Article: Kurds,	long reads,	Syria,	Turkey
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text:
\section{Introduction}

Since Wigner's pioneering work in the 1950s, random matrix theory (RMT) has developed into an important field dedicated to the statistical investigation of complex systems. From nuclear and particle physics~\cite{BFFMPW1981,MRW2010,SV1993,Verbaarschot1994,GW1997,Verbaarschot2011,Akemann2017} to finance~\cite{LCBP1999, PGRAS1999,MS1999}, from classical and quantum information theory~\cite{TV2004,CD2011,KP2010c,ZS2001,SZ2004,BL2002,Majumdar2011,KP2011a,KSA2017} to transport phenomena in mesoscopic systems~\cite{Beenakker1997,Beenakker2015}, RMT has found widespread application in a broad range of areas~\cite{Mehta2004,GMW1998,Handbook2011,Forrester2010,Haake2010}. One of the fascinating aspects of RMT is its prediction of certain universal features which are found to be shared by a wide variety of completely unrelated systems. It was conjectured in a seminal work by Bohigas {\it et al.} that the local fluctuation properties of the spectra of quantum chaotic systems coincide with those of random matrices~\cite{BGS1984} and there has been overwhelming evidence in favor of this conjecture~\cite{BGS1984,Mehta2004,GMW1998,Handbook2011,Forrester2010,Haake2010}. In contrast, as was asserted by Berry and Tabor, integrable systems conform to Poissonian spectral fluctuations~\cite{BT1977}.

Dyson's three fold classification scheme involves three invariant classes of random matrices, which apply depending on the time-reversal and rotational symmetries exhibited by a system~\cite{Dyson1962,Mehta2004,GMW1998,Forrester2010,Haake2010,Handbook2011}. In the classical Gaussian case, these are Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble (GSE). Additionally, there are instances when a symmetry is \emph{partially broken} and correspondingly the spectra exhibit statistics intermediate between two invariant classes. Such systems are modeled by crossover random matrix ensembles which interpolate between two symmetry classes as a certain symmetry-breaking or crossover parameter is varied~\cite{Dyson1972,PM1983,MP1983,PS1991,Pandey1981,Guhr1996,LH1991,LZS1991,FNH1999,NF2003,KP2009,KP2011,CK2013,CDK2014,Kumar2019,Mehta2004,Forrester2010,SMW2017}. The intermediate cases are realized when the crossover parameter assumes values between its two extremes. This crossover parameter can often be related to some physical quantity when modeling a quantum chaotic system. For instance, application of a weak magnetic field on a mesoscopic system (e.g., a quantum dot) leads to a partial time-reversal invariance violation~\cite{Beenakker1997}. Correspondingly, along with the change in spectral behavior, one also observes the effect on electronic transport properties leading to phenomena such as magnetoconductance~\cite{Beenakker1997,FP1995,BHMH2006,BC2007,KP2010a,KP2010b}. In this case, the crossover parameter governing the orthogonal to unitary transition can be related to the magnetic flux. Another prominent example of such crossovers is the one involving Poisson to RMT spectral fluctuations. It is used to model transition from integrability to chaos and has been used in several contexts~\cite{Brody1973,BR1984,Izrailev1990,MNS1994,RDP1999,PH2010,CFIM2012,CDK2014,LLA2015,BG2016,KRBL2010,CARK2012,IORH2013,RCCKLR2017,CK2013}. Interestingly, crossover ensembles also find applications in problems where the above-mentioned symmetries do not have any direct meaning, for example, in multiple channel wireless communication where the crossover is governed by a certain signal-fading parameter~\cite{KP2010c}.

Nearest-neighbor spacing distribution (NNSD) is an important and widely studied statistical measure to quantify the local fluctuation properties of a given spectrum. As a matter of fact, the agreement of NNSD with the RMT result or the Poissonian statistics is considered as a litmus test to decide whether a system under consideration is chaotic or integrable. The NNSD expressions for the classical Gaussian ensembles based on $2\times 2$ matrices, famously known as \emph{Wigner surmise}, approximate the corresponding exact results for large matrices very closely and therefore are of immense usefulness in studying spectra of complex systems. Similarly, for the crossover ensembles, several Wigner-surmise-like results exist for the spacing distribution and apply to large spectra once the crossover parameter is suitably scaled~\cite{Brody1973,BR1984,HMF1988,Izrailev1989,CIM1991,AM1996,KS1999,SBW2012,SLMW2017,SMW2017}. Unfortunately, the empirical calculation of NNSD for a general system involves the difficult task of unfolding the spectrum which requires knowledge of the density of states to a very good accuracy, which is not always possible to acquire. Consequently, one seeks alternatives to NNSD which can be used to quantify the local fluctuation behavior without the need for unfolding.

The ratio $r_n=s_n/s_{n-1}$ of two consecutive level spacings $s_n$ and $s_{n-1}$ is a quantity which is independent of the local density of states and has become quite popular in recent times. Here, the spacing is calculated as $s_{n}= x_{n+1}-x_{n}$, with $\{x_1,x_2,...\}$ being the ordered set of levels. The ratio of adjacent spacings was introduced by Oganesyan and Huse to study the spectral statistics of a one-dimensional tight-binding model of strongly interacting spinless fermions in a random potential~\cite{OH2007}. It was shown that spectral statistics of finite-sized samples show a crossover from GOE statistics in the diffusive regime at weak random potential to Poisson statistics in the localized regime at strong randomness. For their analysis, Oganesyan and Huse considered not $r_n$ but the related quantity $\tilde{r}_{n}=\min(s_{n},s_{n-1})/\max(s_{n},s_{n-1}) = \min(r_{n} ,1/r_{n})$. In Refs.~\cite{ABGR2013, ABGVV2013}, Atas {\it et al.} derived the probability density function (PDF) of the adjacent spacing ratio for the classical Gaussian ensembles and additionally for the Poissonian case which applies to spectra of integrable systems. Although these results are based on $3\times 3$ random matrices, similar to the Wigner surmise, they have been found to describe the fluctuation behavior of large spectra to a reasonable accuracy.

Due to the above-indicated advantage of the ratio distribution over the spacing distribution, it has gradually found widespread application in analyzing spectra of complex systems. This includes systems ranging from spin chains~\cite{OPH2009,PH2010,CFIM2012,CDK2014,LLA2015,BG2016}, Bose-Hubbard model~\cite{KRBL2010,CARK2012}, many-body quasiperiodic system~\cite{IORH2013}, molecular resonances~\cite{RCCKLR2017}, and fermions or bosons described by embedded random matrix ensembles~\cite{CK2013}. One of the key aspects considered in these works has been the Poisson-GOE crossover, for which a detailed analysis has been done by Kota and co-workers for the ratio distribution in Refs.~\cite{CK2013,CDK2014}. An empirical formula for ratio distribution in this crossover has also been provided in Ref.~\cite{CK2013}. The ratio distribution has also been used to examine the quantum chaos transition in a two-site Sachdev-Ye-Kitaev model, which is dual to an eternal traversable wormhole~\cite{GNRV2019}. Furthermore, exact distribution for the spacing ratio has been derived in Ref.~\cite{TKS2018} for distinguishing between localized and random states in quantum chaotic systems. Higher-order spacing ratios have been also explored in ~Refs. \cite{ABGVV2013,TBS2018,BTS2018} and utilized for studying the spectral fluctuations in complex quantum systems as well as those observed in empirical correlation matrices constructed using stock market and atmospheric pressure data. In addition, very recently, a phenomenological formula for the ratio distribution has been proposed in Ref.~\cite{CR2019} for several symmetry crossovers.

In this paper, we focus on an ensemble interpolating between GOE and GUE, and derive exact distribution for the ratio of consecutive level spacings $r$ as well as for $\tilde{r}=\min(r,1/r)$ using a $3\times 3$ random matrix model. Moreover, we give exact results for the corresponding averages $\langle r \rangle$ and $\langle \tilde{r} \rangle$. Similar to the Wigner surmise, our result can be applied to large dimension cases and to non-Gaussian random matrix ensembles once the crossover parameter is properly scaled. We substantiate this claim by considering interpolating ensembles of Gaussian and Wishart-Laguerre ensembles comprising large matrices. In quantifying the level fluctuation behavior of an arbitrary quantum chaotic system, our result can be applied by relating the crossover parameter of the RMT model to the time-reversal symmetry-breaking parameter of the given system. We demonstrate this by investigating the time-reversal symmetry breaking in a quantum kicked rotor (QKR) system~\cite{CCFI1979}.

The rest of this paper is organized as follows. In Sec.~\ref{MMG}, we present the random matrix model used to capture the GOE-GUE crossover. In Sec.~\ref{rDist}, we derive exact results for the PDFs of $r$ and $\tilde{r}$, and the corresponding averages. In Sec.~\ref{scaling}, we examine the scaling behavior of the crossover parameter for Gaussian ensembles in large dimension cases. Section~\ref{ALE} deals with the investigation of orthogonal to unitary crossover in the Wishart-Laguerre ensemble of random matrices. In Sec.~\ref{QKR}, we apply our analytical result for studying the time-reversal symmetry breaking in a quantum kicked rotor system. We conclude with a brief summary in Sec.~\ref{discsum}.

\section{Matrix model for the GOE-GUE crossover}
\label{MMG}

We use the random matrix model proposed by Pandey and Mehta for GOE-GUE crossover~\cite{PM1983, MP1983}:
\begin{equation}
\label{PMmatrix1}
H=\sqrt{1-\alpha^2}\,H_1+\alpha H_2.
\end{equation}
In this equation, $H_1$ and $H_2$ are real-symmetric and complex-Hermitian random matrices with probability densities
\begin{equation}
\mathcal{P}(H_\beta)\propto \exp\left(-\frac{1}{4v^2}\tr H_\beta^2\right),~~~\beta=1,2,
\end{equation}
respectively. The parameter $v$ appearing above fixes the scale of the problem. Clearly, $H_1$ and $H_2$ belong to the GOE and GUE, respectively. The variances of the diagonal and off-diagonal parts (both real and imaginary for GUE) of the Gaussian matrix elements in either case are $2v^2$ and $v^2$, respectively. The matrix model in Eq.~\eqref{PMmatrix1} interpolates between GOE and GUE as the crossover parameter $\alpha$ is varied from 0 to 1. Other equivalent forms for the above matrix model are also possible. For example, Lenz and co-workers have considered the random matrix model~\cite{LH1991,LZS1991}
\begin{equation}
\label{PMmatrix2}
H=\frac{1}{\sqrt{1+\lambda^2}} H_1+\frac{\lambda}{\sqrt{1+\lambda^2}} H_2.
\end{equation}
The relation between parameters $\alpha$ and $\lambda$ can be readily established to be $\alpha=\lambda/\sqrt{1+\lambda^2}$ or $\lambda=\alpha/\sqrt{1-\alpha^2}$. The parameter $\lambda$ gives GOE and GUE in the limits $0$ and $\infty$, respectively. The joint eigenvalue density for the matrix model in Eq.~\eqref{PMmatrix1} is used in the next section to obtain an exact result for the distribution of ratio of consecutive level spacings.

\section{Exact ratio distribution for $3$-dimensional Gaussian matrices}
\label{rDist}
Considering the three-dimensional case, the joint probability density function for the orthogonal-unitary crossover in the Gaussian ensemble is given by~\cite{PM1983, MP1983}
\begin{align}
\label{jpdf}
\nonumber
&P(x_1,x_2,x_3)= \frac{1}{48\sqrt{2}\,\pi v^6(1-\alpha^2)^{3/2}} \\
\nonumber
&\times  [f(x_1-x_2)-f(x_1-x_3)+f(x_2-x_3)]\\
&\times(x_1-x_2)(x_2-x_3)(x_1-x_3)\,e^{-\frac{1}{4v^2}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})},
\end{align}
where
\begin{equation}
f(u)=\erf\left[\left(\frac{1-\alpha^2}{8\alpha^2 v^2}\right)^{1/2}u\right],
\end{equation}
with $\erf(z)=(2/\sqrt{\pi})\int_0^z e^{-t^2}dt$ being the error function.
The limits $\alpha\to0$ and $\alpha\to 1$ in Eq.~\eqref{jpdf} lead to the joint probability densities of eigenvalues for GOE and GUE, respectively. We have
\begin{align}
\nonumber
P_\text{GOE}(x_1,x_2,x_3)=\frac{1}{48\sqrt{2}\pi v^6}\,e^{-\frac{1}{4v^2}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})}\\
\times|(x_1-x_2)(x_1-x_3)(x_2-x_3)|,
\end{align}
\begin{align}
\nonumber
P_\text{GUE}(x_1,x_2,x_3)=\frac{1}{768\pi^{3/2}v^9}\,e^{-\frac{1}{4v^2}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})}\\
\times[(x_1-x_2)(x_1-x_3)(x_2-x_3)]^2.
\end{align}

For the purpose of computing the desired ratio distribution, we order the eigenvalues such that $x_1\leq x_2\leq x_3$. The ratio of consecutive level spacings is then given by
\begin{equation}
r=\frac{x_3-x_2}{x_2-x_1}.
\end{equation}
The probability density function of $r$ can be obtained as
\begin{align}
\label{pr0}
\nonumber
p(r)=3! \int_{-\infty}^{\infty}dx_2 \int_{-\infty}^{x_2} dx_1\int_{x_2}^{\infty} dx_3 \,P(x_1,x_2,x_3)\\
\times \delta\left(r-\frac{x_3-x_2}{x_2-x_1}\right).
\end{align}
We proceed similarly to Refs.~\cite{ABGR2013,ABGVV2013} and implement the change of variables $(x_1,x_2,x_3)\rightarrow(x,x_2,y)$ with $x$ = $x_2-x_1$, $y$ = $x_3-x_2$. This gives us
\begin{align}
\nonumber
p(r)& =6 \int_{-\infty}^{\infty}dx_2\int_0^\infty dx\int_0^\infty dy \,P(x_2-x,x_2,x_2+y)\,\\ \nonumber&\times\delta\left(r-\frac{y}{x}\right).
\end{align}
On using Eq.~\eqref{jpdf}, we find that in the above expression, the variable $x_2$ appears only in the exponential factor and therefore the integral over it can be trivially performed, leading us to
\begin{align}
\label{integral}
 \nonumber
&p(r)=\frac{1}{4\sqrt{6\pi}\, v^5(1-\alpha^2)^{3/2}}\int_{0}^{\infty}dx \,\int_{0}^{\infty}dy\,x\, y\,(x+y)\\
&\times[f(x)+f(y)-f(x+y)]\,e^{-\frac{1}{6v^2}(x^2+xy+y^2)}\,\delta\Big(r-\frac{y}{x}\Big).
\end{align}
From the above equation, the joint density of consecutive spacings $x$ and $y$ can be read as
\begin{align}
\label{jpdsp}
\nonumber
&\widehat{P}(x,y)=\frac{1}{4\sqrt{6\pi}\, v^5(1-\alpha^2)^{3/2}}\,x\, y\,(x+y)\\
&\times[f(x)+f(y)-f(x+y)]\,e^{-\frac{1}{6v^2}(x^2+xy+y^2)},
\end{align}
which is symmetric in $x$ and $y$. Now, using the result $\delta(r-y/x)= x\,\delta(y-rx)$, we get rid of the $y$ integral from Eq.~\eqref{integral}  and are left with an integration on $x$ only,
\begin{align}
\nonumber
&p(r)=\frac{r(r+1)}{4\sqrt{6\pi}\, v^5(1-\alpha^2)^{3/2}}\int_{0}^{\infty} dx\,x^4\\
&\times[f(x)+f(rx)-f(r+rx)]\,e^{-\frac{x^2}{6v^2} (r^2+r+1)}.
\end{align}
The above expression involves three integrals of identical structure which offers an exact result,
\begin{align}
\label{getazeta}
\nonumber
g(\eta,\zeta)& := \frac{4\sqrt{\pi}}{v^5}\int_{0}^{\infty}\, dx\,x^4\,\exp\left(-\frac{\eta^2 x^2}{v^2}\right)  \erf\left(\frac{\zeta x}{v}\right)\\
&= \frac{\zeta(5\eta^2+3\zeta^{2})}{\eta^4(\eta^2+\zeta^{2})^2}+\frac{3}{\eta^5}\arctan\!\left(\frac{\zeta}{\eta}\right).
\end{align}
Using the above integration result, we obtain the final expression for the probability density function for the ratio of two consecutive level spacings as
\begin{equation}
\label{pr}
p(r)= \frac{r(r+1)}{16\sqrt{6}\,\pi(1-\alpha^2)^{3/2}}
\left[g(a,b)+g(a,br)-g(a,br+b)\right],
\end{equation}
where $g(\eta,\zeta)$ is as given in Eq.~\eqref{getazeta} and
\begin{eqnarray}
\label{ab}
a=\sqrt{\frac{r^2+r+1}{6}},~
b=\sqrt{\frac{1-\alpha^2}{8\alpha^2}}.
\end{eqnarray}
We note that the function $g(\eta,\zeta)$, and hence $p(r)$, is independent of $v$. This is expected, as ratio distribution has to be independent of the global scale of the spectrum. This can be seen from Eq.~\eqref{pr0} also by considering the scaling $x \rightarrow vx$. The three ``arctan" terms in $p(r)$ as in Eq.~\eqref{pr} may be combined to yield a single term, giving us, overall,
\begin{align}
\label{pr1}
\nonumber
p(r)&= \frac{r(r+1)}{16\sqrt{6}\,\pi(1-\alpha^2)^{3/2}}
\Bigg[\frac{b(5a^2+3b^2)}{a^4(a^2+b^2)^2}\\
\nonumber
&+\frac{br(5a^2+3b^2r^2)}{a^4(a^2+b^2r^2)^2}-\frac{b(r+1)(5a^2+3b^2(r+1)^2)}{a^4(a^2+b^2(r+1)^2)^2}\\
&+\frac{3}{a^5}\arctan\left(\frac{b^3r(r+1)}{a^3+ab^2(r^2+r+1)}\right)\Bigg].
\end{align}
Equations~\eqref{pr} and~\eqref{pr1} may be written in terms of the parameter $\lambda$ of the matrix model given in Eq.~\eqref{PMmatrix2} by replacing $1/(1-\alpha^2)^{3/2}$ by $(1+\lambda^2)^{3/2}$ and $b$ in Eq.~\eqref{ab} by $1/(\sqrt{8}\,\lambda)$.
In the limit $\alpha\rightarrow 0$, we have $b\rightarrow\infty$, and therefore $g(a,b)=g(a,br)=g(a,br+b)=3\pi/(2a^5)$. Consequently, we obtain the result for GOE~\cite{ABGR2013,ABGVV2013}:
\begin{equation}
\label{pGOE}
p_\mathrm{GOE}(r)=\frac{27}{8}\frac{r(r+1)}{(r^2+r+1)^{5/2}}.
\end{equation}
Now, for $\zeta\rightarrow0$, we obtain $g(\eta,\zeta)=8\zeta/\eta^6-8\zeta^3/\eta^8+\mathcal{O}(\zeta^5)$ by invoking the expansion of $\arctan z$ about $z=0$. Hence, in the limit of $\alpha\rightarrow 1$ or $b\rightarrow 0$, we have
\begin{align}
\nonumber
&g(a,b)+g(a,br)-g(a,br+b)
=\frac{8b}{a^6}-\frac{8b^3}{a^8}+\frac{8br}{a^6}\\
\nonumber
&-\frac{8b^3r^3}{a^8}-\frac{8(r+1)b}{a^6}+\frac{8(r+1)^3b^3}{a^8}+\mathcal{O}(b^5)\\
\nonumber
&=\frac{24r(r+1)b^3}{a^8}+\mathcal{O}(b^5)\\
&=\frac{24r(r+1)(1-\alpha^2)^{3/2}}{8^{3/2}a^8}+\mathcal{O}\big((1-\alpha^2)^{5/2}\big).
\end{align}
This leads to GUE result in the limit $\alpha\rightarrow1$~\cite{ABGR2013,ABGVV2013}:
\begin{equation}
\label{pGUE}
p_\mathrm{GUE}(r)=\frac{81\sqrt{3}}{4\pi}\frac{r^2(r+1)^2}{(r^2+r+1)^4}.
\end{equation}
\begin{figure}[h!]
\centering
\includegraphics[width=0.95\linewidth]{Fig1.pdf}\\
\caption{Probability density functions (a) $p(r)$ and (b) $\tilde{p}(\tilde{r})$, as in Eqs.~\eqref{pr1}, and~\eqref{prtilde} for different $\alpha$ values. The corresponding average values $\langle r \rangle$ and $\langle \tilde{r} \rangle$ are also mentioned in each case.}
\label{fig1}
\end{figure}
\begin{figure*}[ht!]
\centering
\includegraphics[width=0.9\linewidth]{Fig2.pdf}
\caption{Comparison between analytical results (solid lines) as in Eqs.~\eqref{pr1}, and~\eqref{prtilde} and numerical simulations (symbols) in the $3\times 3$ case for $p(r)$: (a) $\alpha=0.2$, (b) $\alpha=0.7$, and for $\tilde{p}(\tilde{r})$: (c) $\alpha=0.3$, (d) $\alpha=0.6$.}
\label{fig2}
\end{figure*}

The PDF $p(r)$ of the ratio $r$ can also be used to obtain the expression for the PDF of $\tilde{r}=\min(r,1/r)$. We have
\begin{equation}
\label{prtilde}
\widetilde{p}(\tilde{r})=\left[p(\tilde{r})+\frac{1}{\tilde{r}^2}p\left(\frac{1}{\tilde{r}}\right)\right]\Theta(1-\tilde{r}),
\end{equation}
where $\Theta(z)$ is the Heaviside step function.
As seen in Eq.~\eqref{jpdsp}, the joint density $\widehat{P}(x,y)$ of consecutive spacings is symmetric under the exchange for the $3\times 3$ Gaussian ensemble. This implies that the distributions of $r$ and $1/r$ are identical, i.e., $p(r)=p(1/r)/r^2$. Therefore, in this case, $\widetilde{p}(\tilde{r})=2p(\tilde{r})\Theta(1-\tilde{r})$, as also argued in Ref.~\cite{ABGR2013}. The same holds for all classical random matrix ensembles in the bulk of the spectrum in the large dimension limit and therefore it is also expected to apply for spectra of chaotic systems.

Along with the distributions of $r$ and $\tilde{r}$, the respective averages $\langle r \rangle$ and $\langle \tilde{r} \rangle$ have been found to be of importance in analyzing the crossover behavior. For instance, in Refs.~\cite{OH2007,PH2010,KRBL2010,CARK2012,CFIM2012,IORH2013,CK2013,CDK2014,LLA2015,BG2016,RCCKLR2017,GNRV2019}, the variation in these averages has been studied as certain physical parameters are varied. The averages have also been used to decide a critical value of the scaled transition parameter signifying the extent of symmetry crossover~\cite{CDK2014}. This kind of statistics is particularly useful in real complex systems where one is unaware of the full analytical solution of crossover from one symmetry class to another. In the present case, it is possible to obtain exact analytical results for the averages $\langle r \rangle$ and $\langle \tilde{r} \rangle$ using the corresponding PDFs. We have
\begin{align}
\nonumber
&\langle r \rangle = \int\limits_{0}^{\infty} r\, p(r) dr  \\
\nonumber
 &= \frac{9\sqrt{3}\,\alpha}{2 \pi (3+\alpha^{2})} -\frac{3}{4}+\frac{5+\alpha^{2}}{\pi (1-\alpha^{2})}\arctan\left(\frac{3-\alpha^{2}}{2\sqrt{3}\,\alpha}\right)\\
&  -\frac{7 + 5 \alpha^{2}}{2 \pi(1-\alpha^2)}\arctan(\frac{\alpha}{\sqrt{3}}).
\end{align}
\begin{align}
\nonumber
&\langle \tilde{r} \rangle = \int_0^\infty \tilde{r}\,\tilde{p}(\tilde{r}) d\tilde{r} = 2\int\limits_{0}^{1}  \tilde{r} \,p(\tilde{r}) d\tilde{r} \\
\nonumber
&=\frac{4 \left(2+\alpha ^2\right)}{\pi  \left(1-\alpha ^2\right)}\arctan\left(\frac{\sqrt{3}(1+\alpha ^2)}{2 \alpha }\right)\\
\nonumber
&-\frac{4 \sqrt{3}}{\pi  \left(1-\alpha ^2\right)^{3/2}} \arctan\left(\frac{\left(1-\alpha ^2\right)^{3/2}}{\alpha (3+\alpha ^2)}\right)\\
&-\frac{17+7 \alpha ^2}{\pi  \left(1-\alpha ^2\right)}\arctan\left(\frac{\alpha }{\sqrt{3}}\right)-\frac{1}{\pi}\arctan\left(\sqrt{3}\, \alpha \right).
\end{align}
Similar to the distributions, these quantities can also be written as functions of the equivalent symmetry-breaking parameter $\lambda$. For $\alpha\to 0^+$ and $\alpha \to 1$, the above expressions yield the average values pertaining to GOE and GUE~\cite{ABGR2013}, respectively,
\begin{align}
\nonumber
&\expval{r}_\mathrm{GOE}=\frac{7}{4}=1.75,\\
\nonumber
&\expval{r}_\mathrm{GUE}=\frac{27 \sqrt{3}}{8 \pi }-\frac{1}{2}\approx 1.3607;\\
\nonumber
&\expval{\tilde{r}}_\mathrm{GOE}=4-2\sqrt{3} \approx0.5359,\\
&\expval{\tilde{r}}_\mathrm{GUE}=\frac{2 \sqrt{3}}{\pi }-\frac{1}{2}\approx 0.6027.
\end{align}
In Fig.~\ref{fig1}, we plot the results given in Eqs.~\eqref{pr1} and~\eqref{prtilde} for probability density functions $p(r)$ and $\widetilde{p}(\tilde{r})$ using four $\alpha$ values. The $\alpha=0.01$ and $0.99$ curves are close to GOE and GUE results, respectively, while the $\alpha=0.22$ and $0.40$ curves depict intermediate situations. In Fig.~\ref{fig2}, we consider $\alpha=0.2,0.7$ for $p(r)$ and $\alpha=0.3,0.6$ for $\tilde{p}(\tilde{r})$ and compare analytical-results-based plots (solid curves) with numerical results (overlaid symbols) obtained from the Monte Carlo simulation involving 50000 matrices of dimension $3 \times 3$ following the matrix model of Eq.~(\ref{PMmatrix1}). We find excellent matches as expected.

\begin{figure*}[!ht]
\centering
\includegraphics[width=0.95\linewidth]{Fig3.pdf}
\caption{Behavior of the ratio distribution $p(r)$ computed using the middle part (red stairs), edges (blue stairs), and the entire spectrum (green stairs) for the GOE-GUE crossover: (a) $\lambda=0$, (b) $\lambda=0.02$, (c) $\lambda=0.04$, (d) $\lambda=0.05$, (e) $\lambda=0.06$, (f) $\lambda=0.08$. The extremes corresponding to the GOE and GUE cases are depicted using dotted gray and dashed orange curves, respectively.}
\label{fig3}
\end{figure*}

\begin{figure}[!tbp]
\centering
\includegraphics[width=0.9\linewidth]{Fig4.pdf}
\caption{Scaling behavior of the effective transition parameter $\lambda_\text{eff}$ in the GOE-GUE crossover. The dashed gray line is a fit based on the data points occurring in the linear regime.}
\label{FigScalingG}
\end{figure}

\section{Scaling of $\lambda$ for large $N$ Gaussian ensemble}
\label{scaling}

For the invariant cases, i.e., $\lambda=0 \,(\alpha=0)$ and $\lambda\to\infty \,(\alpha\to 1)$, it has been shown that the Wigner-surmise-like results, given by Eqs.~\eqref{pGOE} and~\eqref{pGUE}, for the ratio distribution work even for large matrix dimension $N$~\cite{ABGR2013,ABGVV2013}. However, for the intermediate cases, we must scale the crossover parameter $\alpha$ or $\lambda$ appropriately so that Eq.~\eqref{pr1} can be used for large $N$ also. As a matter of fact, it is known that the transition rate depends on the local density of states or the level density~\cite{PM1983, MP1983,Pandey1981,FKPT1988,Guhr1996,KP2009,KP2011,SBW2012}. For large $N$, the matrix model in Eq.~(1) leads to the the Wigner's semicircular level density,
\begin{equation}
R_1(x)=\begin{cases}\frac{1}{\pi}\sqrt{2N-x^2}, & -\sqrt{2N}\le x\le \sqrt{2N},\\
0, & \text{otherwise},
\end{cases}
\end{equation}
for the choice $v^2=1/[2(1+\alpha^2)]=(1+\lambda^2)/[2(1+2\lambda^2)]$. It should be noted that the mean level spacing is given by $1/R_1(x)$.
The effective crossover parameter for small $\alpha$ is $\alpha_\text{eff}\sim \alpha R_1(x)$ and equivalently, for small $\lambda$, we have $\lambda_\text{eff}\sim \lambda R_1(x)$. This implies that the transition rate is faster in the center of the semicircle. Therefore, although the ratio of consecutive spacings is independent of the local density of states, for the crossover ensemble, the fluctuation behavior of the part of spectrum near the center is closer to GUE than that near the edges. We verify this by considering 1000 matrices of size $N=1000$ using the matrix model~\eqref{PMmatrix2} and numerically obtain the ratio distribution independently using (i) 100 eigenvalues in the center of the spectrum, (ii) 100 eigenvalues comprising 50 and 50 from the left and right edges of the spectrum, and (iii) the entire spectrum. These are displayed in Fig.~\ref{fig3} using histogram stairs of colors red, blue, and green, respectively. The extremes corresponding to the GOE and GUE cases based on Eqs.~\eqref{pGOE} and~\eqref{pGUE} are shown using dotted gray and dashed orange curves, respectively. For $\lambda=0$, all histograms fall on the GOE curve, indicating that everywhere in the spectrum, the fluctuations conform to the orthogonal symmetry. Similarly, for higher values of $\lambda$, e.g., 0.08, the histograms more or less coincide with the GUE curve, implying that the crossover is almost complete. However, for intermediate cases, the red histogram (spectrum bulk) is closer to the GUE curve than the blue histogram (spectrum edges). This effect can be seen prominently in $\lambda=0.02$ and 0.04 cases. We also see that for these intermediate cases, the green histogram, which is based on the entire spectrum, is closer to the red histogram. Therefore, the overall behavior is dominated by the bulk and, as an approximation, we may use the scaling $\lambda_\text{eff}\sim \lambda R_1(0)\sim \sqrt{N} \lambda$ and consider the entire spectrum to examine the local fluctuations. To justify this proposition, based on the matrix model~\eqref{PMmatrix2}, we numerically obtain ratio distributions for matrix dimensions $N=100, 500,$ and 1000 along with several values of $\lambda$. These empirical distributions are then fitted with Eq.~\eqref{pr1} to obtain the effective crossover parameter $\lambda_\text{eff}$. The plots of $\lambda_\text{eff}$ vs $\sqrt{N} \lambda$, as shown in Fig.~\ref{FigScalingG}, do exhibit linear behavior for small $\lambda$ values and fall nearly on each other, thereby supporting our deduction above. This scaling was also found to hold in the NNSD expression for the Poisson-GOE crossover studied in Ref.~\cite{LZS1991}. Scaling behavior has also been investigated in Ref.~\cite{CDK2014} in the context of ratio distribution.

\begin{figure*}
\centering
\includegraphics[width=0.9\linewidth]{Fig5.pdf}
\caption{(a),(c) Averages and (b),(d) KL divergences for $r$ and $\tilde{r}$, respectively, in GOE-GUE crossover plotted against the effective transition parameter $\sqrt{N}\lambda$. The numerical simulations have been performed using the matrix model in Eq.~\eqref{PMmatrix2} with matrix dimensions 100, 500, and 1000.}
\label{GaussAvKLD}
\end{figure*}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.9\linewidth]{Fig6.pdf}
\caption{Ratio distribution $p(r)$ for (a) Laguerre orthogonal ($\beta=1$) and (b) Laguerre unitary ($\beta=2$) ensembles. (c),(d) The respective plots for $\tilde{p}(\tilde{r})$. The dashed and dotted lines are based on $3\times 3$ analytical results for Laguerre and Gaussian ensembles, respectively. The symbols depict the simulation results for Laguerre ensembles with matrix dimensions as indicated.}
\label{FigRL}
\end{figure*}
\begin{figure}[!tbp]
\centering
\includegraphics[width=0.9\linewidth]{Fig7.pdf}
\caption{Scaling behavior of the effective transition parameter $\lambda_\text{eff}$ in the LOE-LUE crossover. The dashed gray line is a fit based on the points occurring in the linear regime.}
\label{FigScalingL}
\end{figure}
As mentioned above, the variation of the averages $\expval{r}$ and $\expval{\tilde{r}}$ as a function of the scaled parameter has been used in earlier works~\cite{OH2007,PH2010,KRBL2010,CARK2012,CFIM2012,IORH2013,CK2013,CDK2014,LLA2015,BG2016,RCCKLR2017,GNRV2019} to visualize the crossover in spectral fluctuations. It has also been used to determine a threshold value of the scaled transition parameter above which the crossover can be deemed very nearly complete. However, since one should be careful in drawing conclusions based solely on the behavior of the average, we examine the Kullback-Leibler divergence (KLD)~\cite{KL1951} to ensure that the crossover behavior of the average is consistent with that of the full probability density function. Kullback-Leibler divergence, also called the relative entropy, serves as a measure for comparing two distributions. We use the symmetrized KLD between two PDFs, $p_1(x)$ and $p_2(x)$, which is defined as
\begin{align}
\label{KLD}
\nonumber
D_\mathrm{KL}(p_1||p_2)&=\int p_1(x)\ln\left(\frac{p_1(x)}{p_2(x)}\right)dx\\
&+\int p_2(x)\ln\left(\frac{p_2(x)}{p_1(x)}\right)dx,
\end{align}
where the integrals are over the domain on which the distributions are defined. We examine below how the empirical PDFs and the corresponding averages display crossover from GOE to GUE as the scaled transition parameter $\sqrt{N}\lambda$ is varied. We consider both $D_\mathrm{GOE}\equiv D_\mathrm{KL}(p_\mathrm{emp}||p_\mathrm{GOE})$ and $D_\mathrm{GUE}\equiv D_\mathrm{KL}(p_\mathrm{emp}||p_\mathrm{GUE})$, where $p_\mathrm{emp}$ is the empirical PDF obtained using numerical simulation. Similarly, we consider $\widetilde{D}_\mathrm{GOE}\equiv D_\mathrm{KL}(\tilde{p}_\mathrm{emp}||\tilde{p}_\mathrm{GOE})$ and $\tilde{D}_\mathrm{GUE}\equiv D_\mathrm{KL}(\tilde{p}_\mathrm{emp}||\tilde{p}_\mathrm{GUE})$. It can be calculated that $D_\mathrm{KL}(p_\mathrm{GOE}||p_\mathrm{GUE})=D_\mathrm{KL}(\tilde{p}_\mathrm{GOE}||\tilde{p}_\mathrm{GUE})\approx 0.1137$. Therefore, for the GOE-GUE crossover, $D_\mathrm{GOE}$ and $\widetilde{D}_\mathrm{GOE}$ should move away from zero to this value, while $D_\mathrm{GUE}$ and $\widetilde{D}_\mathrm{GUE}$ should approach zero starting close to this value. In practice, since the $p_\mathrm{emp}$ is obtained numerically, we use the discretized version of Eq.~\eqref{KLD}, i.e.,
\begin{align}
\label{KLD2}
\nonumber
D_\mathrm{KL}(p_1||p_2)&\approx\sum_i p_1(x_i)\ln\left(\frac{p_1(x_i)}{p_2(x_i)}\right)\Delta x\\
&+\sum_i p_2(x_i)\ln\left(\frac{p_2(x_i)}{p_1(x_i)}\right)\Delta x.
\end{align}
Moreover, for $r$, we consider the domain $[0,30]$ with bin width $\Delta r=0.06$ for extracting the numerical PDFs as the full domain $[0,\infty)$ cannot be sampled. It should be noted that for $r\in[0,30]$, theoretically we obtain $D_\mathrm{KL}(p_\mathrm{GOE}||p_\mathrm{GUE})\approx 0.1091<0.1137$. For $\tilde{r}$, we sample the full domain $[0,1]$ with $\Delta \tilde{r}=0.002$. Further, we add very small numerical values to both $p_1(x)$ and $p_2(x)$ in Eq.~\eqref{KLD2} to avoid incidences such as division by 0 or logarithm of 0. Finally, for each value of $N$, the number of matrices used for simulation was kept so as to ensure about 150000 data points for obtaining the PDFs and averages. Although the calculated values for KLD do vary based on these small numerical details, the overall behavior remains unaffected.

In Fig.~\ref{GaussAvKLD}, we show the crossover curves for the averages and the KLDs with respect to the scaled crossover parameter for $N=100, 500,$ and $1000$. It can be seen that averages do behave in harmony with the KLDs and that the GOE-GUE crossover is almost complete for $\sqrt{N}\lambda \approx 1.5$. We also note from Fig.~\ref{FigScalingG} that up to roughly this value, there is a linear relationship between $\lambda_\text{eff}$ and $\sqrt{N}\lambda$.

\section{Application to Laguerre ensemble}
\label{ALE}

In this section, we discuss the orthogonal to unitary crossover in the Laguerre ensemble, which is also referred to as the Wishart or Wishart-Laguerre ensemble. The Laguerre ensemble appears naturally in several contexts, for example, in mathematical statistics, classical and quantum information theory~\cite{TV2004, CD2011,KP2010c,ZS2001,SZ2004,BL2002,Majumdar2011,KP2011a,KSA2017}, quantum chromodynamics~\cite{SV1993,Verbaarschot1994,GW1997,Verbaarschot2011,Akemann2017}, econophysics~\cite{LCBP1999, PGRAS1999, MS1999}, etc. We study the Laguerre ensemble below in connection with the universality of the ratio distribution obtained in the preceding section since it constitutes a very important non-Gaussian random matrix ensemble. We should also point out that very recently, in Ref.~\cite{BTS2018}, the ratio distribution was used to investigate the spectra of empirical correlation matrices for which the Laguerre ensemble serves as a natural random matrix model.

Consider ($N\times M$)-dimensional real and complex Ginibre matrices $A_1$ and $A_2$, respectively, from the distributions
\begin{equation}
P_\beta(A_\beta)\propto \exp(-\frac{\beta}{2}\tr A_\beta A_\beta^\dag);~~\beta=1,2,
\end{equation}
where $\dag$ represents conjugate transpose and may be replaced by only transpose ($T$) for $\beta=1$.
Considering $N\le M$, the Laguerre orthogonal ensemble (LOE) and Laguerre unitary ensemble (LUE) comprise the Wishart matrices $A_1A_1^T$ and $A_2A_2^\dag$, respectively. The distribution of the ratio of two consecutive level spacings is known from Ref.~\cite{ABGVV2013} for a $3\times 3$ matrix model of Laguerre ensemble in the invariant cases with Laguerre weight $e^{-\beta x/2}$. These correspond to the Wishart matrices $A_1A_1^T$ with $N=3, M=4$, and $A_2A_2^\dag$ with $N=M=3$, and read
\begin{equation}
\label{pLOE}
p_\mathrm{LOE}(r)=32\frac{\left(r^2+r\right)}{(r+2)^5},
\end{equation}
\begin{equation}
\label{pLUE}
p_\mathrm{LUE}(r)=420\frac{ \left(r^2+r\right)^2}{(r+2)^8}.
\end{equation}
 In the limits $r\to0$ and $r\to\infty$, the asymptotic behaviors of these functions are $r^{\beta}$ and $r^{-\beta-2}$, respectively, which coincide with the corresponding asymptotic behaviors of the GOE and GUE results in Eqs.~\eqref{pGOE} and \eqref{pGUE}. However, the overall shapes of these distributions based on the $3\times 3$ cases of Laguerre and Gaussian ensembles are very distinct. Furthermore, it may be shown that the joint density of consecutive spacings for the $3\times 3$ model of the Laguerre case does not have the symmetry as exhibited by the corresponding Gaussian case, as in Eq.~\eqref{jpdsp}. Therefore, the probability density $\tilde{p}(\tilde{r})$ in this case does not reduce to $2p(\tilde{r})\Theta(1-\tilde{r})$ and has to be calculated using~\eqref{prtilde}.

For large $N$, we find that the observed ratio distributions for the Laguerre ensemble are closely approximated by the Gaussian ensemble results instead of the above two equations. The same holds for the restricted distribution $\tilde{p}(\tilde{r})$. This is demonstrated in Fig.~\ref{FigRL} where we plot the ratio distributions obtained from numerical simulation based on the above-described Wishart-Laguerre matrix model. For comparison, we also plot the analytical expressions from Eqs.~\eqref{pGOE},~\eqref{pGUE},~\eqref{pLOE}, and~\eqref{pLUE}. We can see that for $N=500$, the empirical ratio distributions for the Laguerre ensemble are well described by the Gaussian $3\times3$ results, thereby confirming their universality. This is similar to the Wigner surmise based on $2\times2$ matrices from the Gaussian ensemble. The underlying reason is that in the bulk of the spectrum of a large class of random matrix ensembles, the correlation functions are governed by the sine kernel~\cite{Mehta2004,Forrester2010,DKMVZ1999,PG2001,GP2002,DGKV2007}. The exact distribution of the nearest neighbor can then be expressed in terms of Fredholm eigenvalues of this kernel, which in turn is very well approximated by the Wigner surmise~\cite{Mehta2004}. A similar explanation holds behind the universality of the ratio distribution, as argued in Ref.~\cite{ABGR2013}.

We now analyze the LOE-LUE crossover. The matrix model that is implemented is
\begin{equation}
\label{wishart}
W=AA^\dag,
\end{equation}
where
\begin{equation}
A=\frac{1}{\sqrt{1+\lambda^2}}A_1+\frac{\lambda}{\sqrt{1+\lambda^2}}A_2.
\end{equation}
The limits $\lambda\to0$ and $\lambda\to\infty$ produce LOE and LUE, respectively.
We focus on the large $N=M$ case, for which the level density for the Wishart random matrix $W$ is given by
\begin{equation}
R_1(x)=\begin{cases}\frac{1}{\pi}\sqrt{\frac{2N-x}{x}}, & 0\le x\le 2N,\\
0, & \text{otherwise}.
\end{cases}
\end{equation}
For the above LOE to LUE crossover ensemble, the effective transition parameter is known to be $\lambda_\text{eff}\sim\lambda\sqrt{x}R_1(x)\sim \lambda\sqrt{2N-x}$~\cite{KP2009,KP2011,footnote}, which depends on the location in the spectrum. However, away from the right edge, we have $\lambda_\text{eff}\sim\sqrt{N}\lambda$ and we expect this scaling to work for the entire spectrum collectively, similar to the Gaussian case. We verify this with the aid of matrices generated using Eq.~\eqref{wishart} for $N=500,1000,1500$ and varying $\lambda$ values. The empirical ratio distributions are fitted with the formula in Eq.~\eqref{pr1} to obtain the effective $\lambda_\text{eff}$ and then plotted against $\sqrt{N}\lambda$ in Fig.~\ref{FigScalingL}. The data points almost fall on a common line for small values of $\lambda$, which indicates the validity of the above-mentioned scaling for LOE-LUE crossover.

We also investigate the behavior of the averages and KL divergences in the present case similar to the Gaussian ensemble. These are shown in Fig.~\ref{LagAvKLD} and we find that by the value $\sqrt{N}\lambda\approx 1.5$, the crossover is nearly achieved. It should be noted that $\lambda_\text{eff}$ and $\sqrt{N}\lambda$ behave linearly with respect to each other up to this value, as observed in Fig.~\ref{FigScalingL}.
\begin{figure*}
\centering
\includegraphics[width=0.9\linewidth]{Fig8.pdf}
\caption{(a),(c) Averages and (b),(d) KL divergences for $r$ and $\tilde{r}$, respectively, in LOE-LUE crossover plotted against the effective transition parameter $\sqrt{N}\lambda$. The numerical simulations are based on the matrix model in Eq.~\eqref{wishart} for matrix dimensions 500, 1000, and 1500.}
\label{LagAvKLD}
\end{figure*}

\section{Symmetry crossover in quantum kicked rotor}
\label{QKR}
\begin{figure}[!tbp]
\centering
\includegraphics[width=0.9\linewidth]{Fig9.pdf}
\caption{Dependence of the effective crossover parameter $\lambda_\text{eff}$ on the time-reversal violation parameter $\gamma$ of QKR system.The dashed gray line is a fit based on the points occurring in the linear regime.}
\label{FigscalingQKR}
\end{figure}
\begin{figure*}[!ht]
\centering
\subfloat{\includegraphics[width=0.9\linewidth]{Fig10.pdf}}
\caption{(a),(c) Averages and (b),(d) KL divergences for $r$ and $\tilde{r}$, respectively, in the quantum kicked rotor system plotted against the effective transition parameter $N^{3/2}\gamma$ for Floquet operator dimensions 51, 101, and 151.}
\label{KRAvKLD}
\end{figure*}

The quantum kicked rotor (QKR) was introduced to serve as a simple yet significant model to aid investigations into quantum chaos~\cite{CCFI1979}. Since then, QKR and its variants have been extensively used in several contexts~\cite{CCFI1979,Izrailev1986,Casati1990,Izrailev1990,PRS1993,SP1997,AL2001,SL2015,PKP2017,KPP2019,JPP2019}. The Hamiltonian for the kicked rotor system is given by
\begin{equation}
\mathcal{H}=\frac{(p+\gamma)^2}{2}+\alpha \cos(\theta+\theta_0)\sum_{n=-\infty}^\infty \delta(t-n).
\end{equation}
It describes a particle restricted to a ring and receiving position-dependent periodic kicks. In the above equation, $p$ is the momentum operator, $\theta$ is the position operator, and $\alpha$ is the stochasticity parameter. The role of the parameters $\gamma$ is to imitate the effect of time-reversal breaking, while $\theta_0$ facilitates the parity violation. The periodic kicks at integer time instants ($t=n$) are provided by the Dirac comb.
The associated quantum dynamics may be described by the discrete time evolution operator (Floquet operator) $U$ by imposing the torus boundary condition on the phase space. Considering the finite-dimensional model, we obtain the evolution operator in position basis as~\cite{Casati1990,Izrailev1986}
\begin{align}
U_{mn} &= \frac{1}{N} \exp[-i \alpha \cos(\frac{2 \pi m}{N}+\theta_{0})]\\ \nonumber& \times \sum_{l=-N'}^{N'}\exp [-i\left(\frac{l^2}{2}-\gamma l+ \frac{2\pi l(m-n)}{N}\right)].
\end{align}
In the above summation, $N'$ is $ (N-1)/2 $ with $N$ odd and $m,n$ take the values $-N',-N'+1,...,N'$. A high degree of chaos is necessary for the spectral fluctuation properties of the QKR to correspond to classical RMT ensembles~\cite{Casati1990, Izrailev1986, Izrailev1990}. This can be achieved by assigning the parameter $\alpha$ a very high value. For studying the violation of time-reversal invariance, we set $\theta_0=\pi/(2N)$ (fully broken parity symmetry) and then vary $\gamma$ gradually. This leads the eigenangle (or eigenphase) spectrum of $U$ to exhibit a transition from orthogonal to unitary class, which has been shown to be described very well using the crossover from circular orthogonal ensemble (COE) to circular unitary ensemble (CUE)~\cite{PS1991,PRS1993,SP1997}. In the following, we examine the crossover in the distribution of the ratio of two consecutive eigenangles and dependence of the transition parameter $\lambda_\text{eff}$ on the time-reversal violation parameter $\gamma$.

For our analysis, we consider $N = 51, 101,151$ and generate the corresponding ensembles of $U$ matrices by varying the stochasticity parameter $\alpha$ in the neighborhood of 20000. These matrices are diagonalized to obtain the eigenvalues which are of the form of $e^{i\phi_j}$, with $\phi_j$ being the eigenangles. The eigenangles are uniformly distributed in $[-\pi,\pi)$ and therefore the level density is $R_1(\phi)=N/(2\pi)$. It is also known based on a semiclassical analysis that the effective transition parameter $\lambda_\text{eff}$ for this system is proportional to $\gamma N^{3/2}$ \cite{PRS1993, SP1997}. This is verified in Fig.~\ref{FigscalingQKR} where we plot $\lambda_\text{eff}$ versus $N^{3/2}\gamma$. The $\lambda_\text{eff}$ values have been obtained by fitting the numerically obtained ratio distributions with Eq.~\eqref{pr1}. We observe linear behavior for small values of $\gamma$, which confirms the above relationship between $\lambda_\text{eff}$ and $\gamma$.

Finally, we examine the transition behavior of the averages and KL divergences with the scaled crossover parameter. The corresponding results are shown in Fig.~\ref{KRAvKLD} and we find that the crossover is nearly complete for $N^{3/2}\gamma\approx 7$. Moreover, as already seen in Fig.~\ref{FigscalingQKR}, up to this value the scaled crossover parameter $N^{3/2}$ varies linearly with $\lambda_\text{eff}$.

\section{Summary and Outlook}
\label{discsum}

For analyzing the spectral fluctuations of complex physical systems, distributions of nearest-neighbor spacing and their ratio are two widely used measures. The latter is convenient to apply since it circumvents the procedure of unfolding the spectrum. For the invariant classes, the ratio distributions were recently derived. These results are analogous to the Wigner surmise for the nearest-neighbor spacing distributions. For the crossover ensembles, such as one describing a gradual time-reversal breaking, similar results are known for the nearest-neighbor spacing distribution. However, such results corresponding to the ratio of spacings have been missing. In this work, we aimed to fill this gap by deriving the Wigner-surmise-like result for ratio distribution and the corresponding average in the orthogonal to unitary symmetry crossover. These results were verified using Monte Carlo simulations. We also examined the proper scaling of the transition parameter which is needed to handle large size spectra by studying the Gaussian and Laguerre crossover ensembles. Additionally, we investigated the effect of time-reversal symmetry breaking in the spectrum of quantum kicked rotor by examining the behavior of the ratio distribution of eigenangles and the associated scaling of the transition parameter.

Aside from the orthogonal-unitary crossover ensemble, there are several random matrix models which are useful in exploring other kinds of spectral transitions. Although phenomenological formulas for the ratio distribution are available for some of these, it would be of interest to see if exact analytical results can be obtained.

\acknowledgements
A.S. acknowledges DST-INSPIRE for providing the research fellowship (Grant No. IF170612). S.K.~acknowledges the support by Grant No. EMR/2016/000823 provided by SERB, DST, Government of India. The authors are also grateful to the anonymous reviewers for their constructive remarks.

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text:
Q: How to load and unload linux drivers that are built into kernel I want to load and unload linux drivers in the device terminal,and I have two options but I do not want to do the first one

*

*Build driver as a module
CONFIG_DRIVER = m
and I can use rmmod and modprobe to unload and load device driver.

*Build device driver into kernel itself
CONFIG_DRIVER = Y
I want to follow the 2nd option but I do not know how to unload and load the device driver, can the open source community please help me out here !

A: It's easy as that. You find a device and driver which you want to unbind. For example, on my Intel Minnownboard (v1) I have PCH UDC controller (a PCI device):
% lspci -nk
...
02:02.4 0c03: 8086:8808 (rev 02)
        Subsystem: 1cc8:0001
        Kernel driver in use: pch_udc

Now I know necessary bits:

*

*bus on which the device is located: PCI

*device name: 0000:02:02.4 (note that lspci gives reduced PCI address, i.e. without domain or i.o.w. BDF, while driver expects domain:BDF)

*driver name: pch_udc
Take altogether we can unbind the device:
% echo 0000:02:02.4 > /sys/bus/pci/drivers/pch_udc/unbind
[ 3042.531872] configfs-gadget 0000:02:02.4: unregistering UDC driver [g1]
[ 3042.540979] udc 0000:02:02.4: releasing '0000:02:02.4'

You may bind it again. Simple use bind node in the same folder.
The feature appeared more than 15 years ago and here is the article on LWN that explains it.

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