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New York (State)
The organization New York (State) represents an institution, an association, or corporate body that is associated with resources found in Biddle Law Library - University of Pennsylvania Law School.
The Resource New York (State)
32 Items by the Organization New York (State)
A digest of statutes and decisions relating to the Board of supervisors of the county of New York
Acts of assembly, passed in the province of New-York, from 1691, to 1718
Annotated consolidated laws of the state of New York : as amended to January 1, 1910 : containing also the federal and state constitutions with notes of Board of Statutory Consolidation, tables of laws and index, edited by Clarence F. Birdseye, Robert C. Cumming and Frank B. Gilbert
Charter of the city of Brooklyn : passed June 28, 1873 : as subsequently amended : with the charter of April 17, 1854, and the amendments thereto, and other laws relating to said city : also, the ordinances of the Common council of the city of Brooklyn, as codified and revised and adopted Dec.10, 1877, (comp. by Wm. G. Bishop, city clerk.)
Digest of the laws of the State of New York : comprising the Revised statutes and statutes of general interest in force on January 1, 1874, by Joseph D. Fay
General laws of New York : containing all amendments to the close of the session of 1899, compiled and edited by Edward L. Heydecker
Inferior criminal courts act of the city of New York, annotated, by W. Bruce Cobb
Laws and ordinances of New Netherland, 1638-1674, compiled and translated from the original Dutch records in the Office of the Secretary of State, Albany, N.Y., by E.B. O'Callaghan
Laws of the state of New York : containing all the acts passed at the 28th and 29th sessions of the Legislature, 1804-5 and 1806
Laws of the state of New-York : passed by the legislature of said state, at their last meeting of the eighth session
Laws of the state of New-York : revised and passed at the thirty-sixth session of the Legislature, with marginal notes and references, furnished by the revisors, William P. Van Ness & John Woodworth
Laws of the state of New-York, of a general nature : passed, from 1828 to 1841, inclusive : all acts relating to the same subject, classified and brought together under their appropriate heads, selected and arranged, with references to judicial decisions, by a counsellor at law
Proposed Constitution of the State of New York : as amended and revised, adopted by the Convention September 10, 1915, with explanatory abstracts and form for submission to the voters, pursuant to Chapter 668, Laws of 1915, prepared under the direction of Francis M. Hugo, Secretary of State
Reducing realty taxes : with remedies for illegal and excessive assessments and methods of proving realty values
Revised charter of the city of Elmira with the several acts applicable to the city and officers thereof
Statutes at large of the state of New York : comprising the revised statutes, as they existed on the 1st day of January, 1867, ed. by John W. Edmonds
Statutes of the state of New York of a public and general character : passed from 1829 to 1851, both inclusive : with notes and references to judicial decisions, and the constitution of 1846, compiled and arranged by Samuel Blatchford ; with a copious index by Clarence A. Seward
The Constitution of the state of New York : adopted November 6, 1894 as amended and in force January 1, 1909
The charter of the city of Buffalo as amended from time to time since adoption in 1891 and as now in force, comp. and annotated by Henry Adsit Bull
The code of civil procedure of the State of New York : being chapter 448 of the Laws of 1876, as amended by chapter 416 and 422 of the Laws of 1877
The code of procedure of the state of New-York : as amended by the Legislature, by an act passed April 11, 1849, printed under the direction of the secretary of state
The codified Negotiable instruments law of the state of New York. : Laws of New York, 1897, chapter 612. Being chapter L of the general laws ... with an appendix containing New York interest laws., Ed. by James W. Eaton ... and by H. Noyes Greene
The constitution of the state of New York : adopted November 3, 1846 : as amended and in force January 1, 1887, prepared from the original in office of Secretary of State, under direction of Frederick Cook, Secretary of State
The new revision of the statutes of the state of New York : [part II] an act relating to property, and other matters connected with private rights ..., examined, reviewed, and amended, and reported at the session commencing January 7, 1879, by Samuel S. Edick, Edwin Hicks, Charles Hughes, special committee of the Senate, (electronic resource)
The new revision of the statutes of the state of New York : report of the commissioners and accompanying bill, with appendices and an index, transmitted to the Legislature at the one hundredth session, commencing January 2, 1877, by Montgomery H. Throop, Sullivan Caverno, James Emott, (electronic resource)
The revised statutes of the state of New York : as altered by subsequent legislation: together with the other statutory provisions of a general and permanent nature, (except the code of civil procedure, the code of criminal procedure, and the penal code,) passed from the year 1778 to February 1, 1889, and now in force : arranged in connection with the same or kindred subjects in the revised statutes : to which are added an analysis of the entire work : references to judicial decision upon the different enactments : explanatory notes : a full and complete alphabetical index : and an indexed table of the statutes contained in the work, edited by Montgomery H. Throop
The revised statutes of the state of New York : as altered by subsequent legislation; together with the unrepealed statutory provisions of a general nature, passed from the time of the revision to the close of the session of the Legislature of 1875, edited by George W. Cothran
The revised statutes of the state of New York, together with all the other general statutes, (except the civil, criminal and penal codes) as amended and in force on January 1, 1896
The revised statutes of the state of New-York : passed during the years one thousand eight hundred and twenty-seven, and one thousand eight hundred and twenty-eight : to which are added, certain former acts which have not been revised, printed and published under the direction of the revisers, appointed for that purpose
The statute law of the state of New York : comprising the revised statutes and all other laws of general interest, in force January 1, 1881, arranged alphabetically according to subjects
Uniform commercial code, effective Sept. 27, 1964 : a guide to fillings [under Article 9: Secured transactions] New York State, [by] John P. Lomenzo, Secretary of State
Context of New York (State)
Annotated consolidated laws of the state of New York : as amended to January 1, 1910 : containing also the federal and state constitutions with notes of Board of Statutory Consolidation, tables of laws and index
The code of procedure of the state of New-York : as amended by the Legislature, by an act passed April 11, 1849
The constitution of the state of New York : adopted November 3, 1846 : as amended and in force January 1, 1887
The new revision of the statutes of the state of New York : [part II] an act relating to property, and other matters connected with private rights ...
The new revision of the statutes of the state of New York : report of the commissioners and accompanying bill, with appendices and an index, transmitted to the Legislature at the one hundredth session, commencing January 2, 1877
The revised statutes of the state of New York : as altered by subsequent legislation: together with the other statutory provisions of a general and permanent nature, (except the code of civil procedure, the code of criminal procedure, and the penal code,) passed from the year 1778 to February 1, 1889, and now in force : arranged in connection with the same or kindred subjects in the revised statutes : to which are added an analysis of the entire work : references to judicial decision upon the different enactments : explanatory notes : a full and complete alphabetical index : and an indexed table of the statutes contained in the work
The revised statutes of the state of New York : as altered by subsequent legislation; together with the unrepealed statutory provisions of a general nature, passed from the time of the revision to the close of the session of the Legislature of 1875
The revised statutes of the state of New-York : passed during the years one thousand eight hundred and twenty-seven, and one thousand eight hundred and twenty-eight : to which are added, certain former acts which have not been revised
Digest of the laws of the State of New York : comprising the Revised statutes and statutes of general interest in force on January 1, 1874
General laws of New York : containing all amendments to the close of the session of 1899
Laws and ordinances of New Netherland, 1638-1674
Laws of the state of New-York : revised and passed at the thirty-sixth session of the Legislature
Laws of the state of New-York, of a general nature : passed, from 1828 to 1841, inclusive : all acts relating to the same subject, classified and brought together under their appropriate heads, selected and arranged, with references to judicial decisions
Proposed Constitution of the State of New York : as amended and revised, adopted by the Convention September 10, 1915, with explanatory abstracts and form for submission to the voters, pursuant to Chapter 668, Laws of 1915
Statutes at large of the state of New York : comprising the revised statutes, as they existed on the 1st day of January, 1867
Statutes of the state of New York of a public and general character : passed from 1829 to 1851, both inclusive : with notes and references to judicial decisions, and the constitution of 1846
Inferior criminal courts act of the city of New York, annotated
The charter of the city of Buffalo as amended from time to time since adoption in 1891 and as now in force
Charter of the city of Brooklyn : passed June 28, 1873 : as subsequently amended : with the charter of April 17, 1854, and the amendments thereto, and other laws relating to said city : also, the ordinances of the Common council of the city of Brooklyn, as codified and revised and adopted Dec.10, 1877
The codified Negotiable instruments law of the state of New York. : Laws of New York, 1897, chapter 612. Being chapter L of the general laws ... with an appendix containing New York interest laws.
Uniform commercial code, effective Sept. 27, 1964 : a guide to fillings [under Article 9: Secured transactions] New York State
Enacting jurisdiction of

 Data from New York (State) - Biddle Law Library - University of Pennsylvania Law School

Data Citation of the Organization New York (State)


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DOLE FOOD COMPANY RECOGNIZED BY U.S. SECRETARY OF STATE FOR CORPORATE SOCIAL RESPONSIBILITY PROGRAMS IN THE PHILIPPINES
Westlake Village, CA - September 05, 2008
Dole Food Company announced today its was chosen as a finalist for the U.S. Secretary of State's prestigious 2008 Award for Corporate Excellence (ACE.) Dole is one of eleven U.S. companies operating abroad chosen as finalists, selected from more than sixty nominations submitted by U.S. Ambassadors around the world.
As an ACE finalist, the U.S. Department of State has recognized Dole's operations in the Philippines as an exemplary corporate citizen. Dole Philippines is a leader in environmental stewardship, with education, training, environmental protection and health programs benefiting Dole employees and the communities in which they live.
Among other contributions, Dole recently planted its millionth tree under the Company's "Chairs-for-Trees" program, increasing the number of desks and chairs available for local schools and, in doing so, implementing recycling and reforestation initiatives. The desk-chair combinations are made from Dole's recycled wooden pallets, and in exchange beneficiaries plant seedlings.
"Public diplomacy and long-term community investment have been at the heart of Dole's core principles for decades," said David DeLorenzo, president and chief executive officer of Dole Food Company. "In the rural areas in which we operate around the world, Dole is working and building communities, even where others are often fighting."
About the Award for Corporate Excellence (ACE):
The U.S. Secretary of State has awarded ACE since 1999 to recognize U.S. businesses operating abroad for advancing good corporate citizenship, innovation, and democratic principles. This year marks the 10th anniversary of the ACE program. Secretary of State Rice will present the winners for 2008 on November 6th.
About Dole Food Company:
Dole Food Company, Inc., with 2007 revenues of $6.9 billion, is the world's largest producer and marketer of high-quality fresh fruit and fresh vegetables. Dole markets a growing line of packaged foods and frozen fruit and is a produce industry leader in nutrition education and research. For more information on the company's CSR efforts, please visit www.dolecsr.com.

redpajama_set_name: RedPajamaCommonCrawl


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Newly Diagnosed With a Blood Cancer
Ann W
We've just launched a brand new patient information booklet for people who have been recently diagnosed with a blood cancer or related condition.
In our new booklet we provide many of the answers to questions that may arise when someone has just been given a diagnosis, to help them to cope and understand their condition. We also include details of other organisations to contact for further help or information.
We spoke to many patients and their families and health professionals to find out what information would be most helpful. They told us that a booklet which takes them through the initial stages after being given a diagnosis would be useful.
Receiving feedback on our publications is invaluable and helps us to create better resources for people who have been affected by blood cancers. If you would like to be involved in reviewing and commenting on our publications please contact our Patient Information team at patientinfo@beatingboodcancers.org.uk
To download or order copies of 'Newly Diagnosed with a Blood Cancer' or any of our other patient information booklets please visit: beatingbloodcancers.org.uk/order-patient-information
Ann Woodham - Patient Information & Support Manager


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Full Committee Hearing
The Consumer Financial Protection Bureau's Semi-Annual Report to Congress
Date: Thursday, December 15, 2022 Time: 10:00 AM Location: Dirksen Senate Office Building 538
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THE COMMITTEE ON BANKING, HOUSING, AND URBAN AFFAIRS will meet in OPEN SESSION, HYBRID FORMAT to conduct a hearing entitled, "The Consumer Financial Protection Bureau's Semi-Annual Report to Congress." The witness will be: The Honorable Rohit Chopra, Director, Consumer Financial Protection Bureau.
Pursuant to guidance from the CDC and OAP, Senate office buildings are not open to the public other than official business visitors and credentialed press at this time. Accordingly, in-person visitors cannot be accommodated at this briefing. We encourage the public to utilize the Committee's livestream of the hearing, available on the website at http://www.banking.senate.gov.
All hearings are webcast live and will not be available until the hearing starts. Individuals with disabilities who require an auxiliary aid or service, including closed captioning service for webcast hearings, should contact the committee clerk at 202-224-7391 at least three business days in advance of the hearing date.
Member Statements
Chairman Sherrod Brown (D - OH)
Download Statement
Ranking Member Patrick J. Toomey (R - PA)
The Honorable Rohit Chopra


I don't know how many times I'll be able to do Tanka Tuesdays in a row, but I am going to get a couple up this month. I will continue to post something every Monday for you, and I will try to make it a haiku. And yes, I exaggerate a little in this poem, but I was speaking of formal dresses, which I have worn 5 times in my 31 years. Just for kicks, here I am dressed up on my 31st birthday. You can't see my bottom half, but that's alright because I was wearing jeans anyway.
So pretty and inside also. Beautiful Spirit. Praise God !!


Makenro Challenge is a fangame based on a franchise called Makeruna! Makendou. It is a simple arcade-style game still in development that focuses on the boss battle from the original SNES game. It will feature several modes besides the original and highscores for each one.
Keep checking for development news about this game by visiting our website or following us on social networks.


Beautiful Studio Condo for rent at "The Bond"situated in the heart of downtown Miami's Brickell Financial District. 24 hr security. Luxurious amenities. Concealed built in Bosch microwave with convection oven, Bosch Stainless dishwasher with integrated design to cabinetry, luxury porcelain flooring, base and walls in wet areas.Very close to malls and restaurants. Very easy to show!


Mickey Mouse has been shaping our lives and winning our hearts for 90 years, and we think an icon like him deserves to celebrate his anniversary longer than just one day. And we're not the only ones who think that—just announced today, Disney is opening an interactive art exhibit in New York in honor of the 90th anniversary of Mickey Mouse. Titled Mickey: The True Original Exhibition, the exhibit is an immersive, pop-up art exhibition inspired by Mickey's status as a "true original" and his global impact on popular culture and art.
Running from November 8, 2018 – February 10, 2019, Mickey: The True Original Exhibition will be open Tuesday-Sunday from 10 a.m. to 8 p.m. at 60 10th Avenue, New York. Tickets are available for $38 per person at disney.com/MickeyTrueOriginal.
With 16,000 square-foot of interactive experiences, you can experience the magic of Mickey through original work from renowned and emerging contemporary artists. Curated by designer Darren Romanelli, the exhibit will feature exclusive original artwork from contemporary artists including Kenny Scharf, Amanda Ross-Ho and Shinique Smith, with more artists to be announced in the coming weeks. The experience will captivate the senses to surprise and delight guests, evoking the inspiration, creativity, and true original spirit of Mickey Mouse. The exhibit will span from Mickey's beginning in black and white in Steamboat Willie to the technicolor world of more than 100 cartoon shorts. Get ready to step into an unforgettable journey unlike which you have ever experienced—all to celebrate Mickey!
Funko 90th Anniversary Mickey Mouse Pops!
In celebration of Mickey Mouse's 90th Birthday, Funko will release new Pop! figures in August. Keep an eye out for Steamboat Willie, Apprentice Mickey, Firefighter Mickey, Conductor Mickey, Brave Little Tailor and Band Concert Mickey. Pre-order the Pop's! at Barnes & Noble HERE for $9.95 each.
As classic Disney fans, we're obsessed with all things Steamboat Willie—which is why we're loving the latest LEGO Ideas submission.
Available today from Raph Cosentino, a new print based on a series of paintings he did for a show in Japan curated by Mark Nagata back in 2010-2011 where the artist mashed up Disney characters with Japanese monsters; Mickey Mouse + Gojira ( Gojira is the Japanese pronunciation of Godzilla ) = Mojira.
With this print Cosentino wanted to create a parody of the classic "pie-eyed Mickey" stock cartoon posters from the thirties.
The print will be available HERE.
A hand-drawn, black-and-white Mickey Mouse animated short film will debut June 11 at the Annecy Animation Festival in Annecy, France, Walt Disney Animation Studios announced Tuesday.
The vintage-feeling short, which is called "Get a Horse," features the voice of Walt Disney as Mickey Mouse.
The Burbank studio described the short as "never before seen," but a studio spokeswoman, when asked if the short is newly made, refused to comment.
The enigmatic cartoon finds Mickey, his cheerful gal pal Minnie Mouse and their pals Horace Horsecollar and Clarabelle Cow enjoying a pastoral musical wagon ride—until Peg-Leg Pete shows up and tries to run them off the rode!
Presenting this "lost" Mickey short at Annecy will be filmmakers Lauren MacMullan (Wreck-It Ralph, Avatar: The Last Airbender, The Simpsons), Dorothy McKim (Prep & Landing: Naughty vs. Nice, The Ballad of Nessie, Meet the Robinsons) and legendary Disney artist and animator Eric Goldberg.
The next feature from Disney Animation, "Frozen," opens Nov. 27.


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Latin America's International Projection: Trends and Trajectories for 2012
Gian Luca Gardini, Mar 29 2012, 1580 views
Three crucial political trends are likely to characterise the global projection of Latin America in 2012 and the years to come: the rise of Brazil and its consequences for the region in global affairs; the mushrooming of regional projects supposedly aimed at promoting Latin American unity; and the summit exercises in which Latin America engages with its major international partners, the US, the EU, and China.
Brazil is the most powerful country in Latin America. It is not only an emerging economy but it is widely perceived as a rising political power in the global sphere as well.However its leadership credentials in Latin America and beyond are mixed and the impact of its rise upon the rest of the continent is not conclusive yet. In economic and military terms Brazil has all it takes to be a regional leader and a would-be global player. It has recently overtaken the UK as the world 6th largest economy and it is expanding its military capabilities at a sustained rate. Also, its successful social policies are now being exported to other countries in Latin America and Africa. Brazil is more and more active within international organisations. It leads the UN mission in Haiti, plays a key role in WTO negotiations, is bidding for a permanent seat in the UN Security Council, and sponsors the formation of new international aggregations such as the BRIC, IBSA (India, Brazil, South Africa) or UNASUR, the Union of South American Nations.
Yet some caveats affect Brazil's power status. A coin has two sides and the other side of leadership is followership. Brazil has been struggling to drive the agenda in Latin America and secure other Latin American countries' support for its own visions and leading role. In the words of Andres Malamud, Brazil risks becoming 'a leader without followers'. However things may change rapidly. With rising economic and political weight, Brazil will have more and more to offer in terms of incentives or penalties to its interlocutors. A different question is whether or not Brazil is in fact willing and ready to take up a leading role in Latin America. So far Brasilia has not been keen to pay the cost, either political or economic, associated with leadership and has favoured a consensual management style often avoiding taking sides or even a position. With power comes responsibility, but Brazil appears reluctant to tie up its hands in the region, possibly to the detriment of its increasingly global interests and positioning. Regionally, Latin American countries have proven themselves to be reluctant followers and Brazil has behaved as a reluctant leader.
Globally, Brazilian presence and capacity of projection is still relatively limited in spite of its rapid ascendance. According to data collected and elaborated by the Real Instituto Elcano, Brazil is only 25th in the ranking of countries with the most extensive interests and economic, social and cultural influence globally. The Elcano Global Presence Index shows that in the 5 areas analysed (economy, defence and security, migration and tourism, science and culture, and development aid), Brazil does not feature in the top 10 countries in any category. Countries such as Ireland, Turkey, Malaysia, Austria, Mexico, Sweden, India, and South Korea all fare better than Brazil. While these league table may be subjective and controversial, the fact remains that Brazil's contribution to international governance is still relatively small; for instance Brazil contributes to the IMF or the WTO budget as much as Belgium. This is not to say that Brazil does not have the potential to become a global power; only that GDP alone does not generate either leadership or followership. 2012 will provide some indications about Brazilian orientation towards one or the other.
Latin American regionalism is mushrooming, not necessarily flourishing. A plethora of regional projects and schemes characterise the current panorama. Yet the rationale for many of them is often very different if not divergent or outright incompatible. Some of the existing projects are the legacy of the neoliberal model; espousing a reduced role for the state, economic opening, and the trade liberalisation that had informed national and regional politics throughout the 1990s. Expressions of this model include the creation of the North American Free Trade Area (NAFTA) between Mexico, the US and Canada and the Common Market of the South (MERCOSUR), which is composed of Argentina, Brazil, Paraguay and Uruguay, as well as the revival of the Andean Community (currently made of Colombia, Ecuador, Peru and Bolivia).
By the early 2000s there was a realisation that the neoliberal model had failed to tackle the traditional problems of poverty, inequality and dependence on external markets. This raised questions in the region regarding social issues and redistribution of wealth, the role of the state as engine of development, endogenous and equitable growth, and greater autonomy in international affairs. In 2004 Venezuelan President Hugo Chavez inaugurated the Bolivarian Alliance for the Americas (ALBA), consisting of Venezuela, Cuba, Bolivia, Ecuador, Nicaragua and a few Caribbean small islands. ALBA rejects free trade in favor of a trade strategy that borders on bartering. The discourse is highly ideological and anti-US and its propositions are quite controversial and divisive.
In 2008, Brazil launched its own project, the Union of South American Nations (UNASUR). This brings together all twelve South American countries, defining South America, as opposed to Latin America, as the region of reference. This is a Brazilian attempt to spread its geostrategic vision and influence, aiming at coordinating regional decisions on strategic matters like regional security as well as social issues including health and education. An economic dimension is to be added by the end of the decade. Also MERCOSUR adapted to the new mantra and created a regional parliament and a structural fund to address democratic participation and economic asymmetries. In 2011 the Community of Latin American and Caribbean States (CELAC) was born. This brings together all 33 Latin American and Caribbean countries including Cuba, with the aim of providing a unitary bloc for Latin America's global interactions. There is no economic dimension; its singular purpose is political coordination in order to strengthen Latin America's position on the world stage.
At the turn of the second decade of the 21st Century, a new trend seems to be taking shape: a sort of counterrevolution to last decade's revolt against neo-liberalism. The sidelining of economic issues to give central stage to political cooperation seems to have lost sight of the real possibilities of integration. Emphasis on social issues has often been empty rhetoric. The search for unachievable political unity may ultimately jeopardise Latin America's economic interests. This is a possible reading of the 2010 launch of the Pacific Alliance (Mexico, Peru, Chile and Colombia), a forum trying to reemphasise economic interests and the potential of the Asian markets. The recent creation of the Latin American Integrated Market (MILA) also points towards less ambitious but more economically focused projects. MILA is a self-contained sectorial scheme that integrates the stock exchange markets in Bogota, Lima and Santiago. MERCOSUR too now discusses the opportunity of loosening ties among its members and allowing them to conclude individually free trade agreements with relevant third parties. 2012 and the years to come will be a probing test for Latin American regionalism and will provide an indication of whether regionalism is deepening or in fact being diluted.
Initially, major summits with key international partners were to be a central tenet of Latin America's 2012 agenda. This may not be the case. The Summit of the Americas, where the US and Canada meet all Latin American countries, except Cuba, will take place in Cartagena, Colombia, in April. So far signs have not been encouraging. The ALBA countries have threatened to boycott the summit over the exclusion of Cuba imposed by the US. From a legal point of view, the US position is correct. The 2001 Inter-American Democratic Charter and the Quebec City Declaration explicitly state that non-democratic states cannot participate in the Summit. Cuba at present is not a democracy, regardless of all other considerations and preferences. The political question is a different one. ALBA countries seek for a more plural and autonomous continent, free from US interference. The persisting US embargo on Cuba is at least anachronistic and clearly driven by ideological and domestic pressures. For these reasons, at the previous Summit in Trinidad and Tobago in 2009, no consensus was reached on the final declaration. A similar outcome may cool US-Latin American relations or, more likely, reinforce the US preference for dealing with Latin American countries on a bilateral basis.
The scheduled 2012 EU-Latin America Summit is not going to take place due to the creation of CELAC. It will be replaced by the inaugural EU-CELAC Summit in January 2013, where for the first time Latin America will act as a bloc in its global projection. The summit will be a propitious occasion to reassess the whole bi-regional process of institutionalised interactions between the EU and Latin America in the light of changed circumstances. Traditionally the EU has dominated the agenda and paid for most of the expenses, with the two aspects reinforcing one another. With the on-going debt and financial crises in Europe, and the economic and political rise of Latin America, the latter may better disposed to pay a bigger share of the bill and take a more assertive role in the defining the agenda. Also, with the fast changing panorama of Latin American regionalism-where new projects are fast replacing traditional ones like the Andean Community that have lost legitimacy as well as relevance-the EU may want to reconsider its very articulate but also very expensive and quite cumbersome inter-regional approach to Latin America. If leaders on both sides of the Atlantic are willing to acknowledge the changed circumstances, the EU-CELAC summit may open the path to a significant evolution in EU-Latin American relations. 2012 is crucial for the preparatory works leading to the summit.
Finally, with a much less glamorous but perhaps more effective approach, China will hold its annual summit with Latin America. The Chinese model has two advantages: summits are annual, which gives a certain continuity and adaptability to the process, and negotiations are quite focused and pragmatic with the business sector directly involved and tangible results more readily obtainable than in summits with the US or the EU. Whether big political questions are indeed openly debated is a different matter altogether. The Chinese presence in Latin America is growing fast and the UN Economic Commission for Latin America and the Caribbean predicts that by 2015 China will overtake Europe as Latin America's second largest trading partner. While China may offer significant advantages to Latin America in the short-run, especially in terms of export markets and foreign direct investments, these benefits may be outweighed by the corrosive long-term trajectory. Historically, economic dependence has led to political allegiance and Chinese products may soon affect Latin American production and global market shares. China may turn out to be an El Dorado or a new form of dependency for Latin America. The quality and vision of Latin American leaders will be decisive for determining which outcome wins out.
In the second decade of the 21st century, Latin America is in its best international position ever. The continent is more present and assertive in world affairs, its economic indicators are all positive, and its political weight is on the rise. Yet, the way in which Latin America will manage the ascendance of its natural leader Brazil, the search for regional unity and convergence, and its interactions with major international powers will determine whether we will have a Latin American decade or yet another failed promise. 2012 will provide interesting indicators on all the three issues.
Gian Luca Gardini is Lecturer in International Relations and Latin American Politics at the University of Bath. He obtained his PhD from the University of Cambridge. His research interests cover the international relations of Latin America, regional integration, foreign policy analysis, and issues of international power, order and institutions. His most recent books are The Origins of Mercosur (Palgrave, 2010), Latin American Foreign Policies (Palgrave, 2011) and Latin America in the 21st Century (Zed Books, 2012).
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About The Author ( Gian Luca Gardini):
Gian Luca Gardini is Senior Lecturer in International Relations and Latin American Politics at the University of Bath. He is the author of Latin America in the 21st Century (Zed Books, 2012), Latin American Foreign Policies (Palgrave, 2011), and The Origins of Mercosur (Palgrave 2010)
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Address: 20 Karlien St, Zinniaville, Rustenburg, 0299, South Africa, North West. See full address and map.
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text:
\section{Introduction}
Graphene stands out amongst other two-dimensional materials due to the presence
of Dirac points in its band structure, with an approximately linear electronic
dispersion and a vanishing band-gap.
One of the most important consequences of this band structure is the possibility
for low energetic electron/hole pairs to interact with plasmons via plasmon
absorption or emission;
This is excluded for in band-gap materials where the energy of plasmons is
typically much smaller than that of single particle excitations
\cite{Giuliani1982,Bostwick2007}.

When graphene is excited with a femtosecond optical pulse, high-energetic
electron/hole pairs are generated far from the Fermi edge.
Carrier-carrier and carrier-plasmon scattering will cause a rapid redistribution
of the energy in the carrier system, on a 10-100fs scale \cite{Song2015},
firstly bringing the plasma into a two-component inverted quasi-equilibrium  and
then into a state of a high-temperature carrier equilibrium
\cite{Dawlaty2008,Li2012,Zhang2013,Meng2016}.
Following the equilibration in the carrier system, the plasma will equilibrate
with the phonon system, albeit on longer timescales, as carrier-phonon
scattering is typically at least one order of magnitude slower than
carrier-carrier scattering \cite{Butscher2007,Rana2009,Wang2010}.
Excitation and relaxation processes are illustrated in Fig.~\ref{concept}.
During this relaxation process, the scattering of carriers and their interaction
with plasmons are mediated by the screened Coulomb potential
$V_\mathbf{q}^\mathrm{eff}(q,\omega) = V_q /\varepsilon(q,\omega)$, where $V_q $ is the
bare Coulomb potential and $\varepsilon(q,\omega)$ the dynamic dielectric
function, which in turn depends on the dynamically evolving nonequilibrium
distribution of carriers.
Therefore, when evaluating the scattering probabilities associated with plasmas
far from equilibrium, it is not sufficient to assume equilibrium dielectric
functions, but instead one needs to evaluate the nonequilibrium dielectric
function of the plasma associated with the momentary carrier distribution.
For graphene this was, to the best of our knowledge, first pointed out in
Ref.~\cite{Tomadin2013}, where the dielectric function is evaluated for a two
component plasma in a quasi-equilibrium to describe the dynamic screening of
Auger processes, using a procedure described by Maldague \cite{Maldague1978,
VanDuppen2016}.
At the same level, the plasmon dispersion, defined by the zeros of the
dielectric function, $\varepsilon(q,\omega)=0$ is also functionally dependent on
the carrier distribution.
A recent experimental study has shown the plasmon response of graphene can be
activated by pulsed optical excitation, due to the increase in Drude weight
associated with generated non-equilibrium carriers \cite{Ni2016}.
It has been shown in Ref.~\cite{Chaves2016} that photoexcitation beyond the
Dirac cone can give support to plasmons with anisotropic dispersion relations.
In a previous work \cite{Page2015}, we have solved the plasmon dispersion for
photo-inverted graphene and found that it differs fundamentally from the
equilibrium dispersion, as the phase space for net stimulated emission grows and
plasmons can become amplified \cite{Hamm2015}.
However, the method used therein to calculate the plasmon dispersion is only
applicable to a two-component plasma in quasi-equilibrium and at temperatures
much smaller than the chemical potentials ($k_B T \ll \mu_e, \mu_h$).
Crucially, it does not accommodate for more general carrier occupations, such as
the presence of high-energetic carriers that are  generated during optical
excitation.

In this work, we show how to efficiently calculate the polarization function
(and hence the dielectric function), and plasmon dispersion of graphene for
arbitrary isotropic nonequilibrium carrier distributions within the
random phase approximation (RPA).
Whilst the formalism can be extended, we further assume the electronic dispersion
of graphene to be a gapless Dirac cone, i.e. as described by the massless Dirac
fermion (MDF) model.
Crucially, the method presented here fully accommodates for the
conditions encountered during pulsed photo-excitation: it is applicable to an
electron/hole plasma at high temperatures ($k_B T \gg \mu_e, \mu_h$) and allows
to incorporate the influence of high-energetic carriers.

This paper is structured as follows: firstly we present the general theory that
allows the polarization function for isotropic nonequilibrium carrier
distributions to be efficiently evaluated on the basis of  zero-temperature
equilibrium polarization functions.
Next, to determine the complex-frequency plasmon dispersion, we present a
contour-integration method that remains valid at high temperatures and when
taking into account the photo-generated high-energetic carriers characteristic
of a plasma far from equilibrium.
To demonstrate the versatility of the method we calculate the energy-loss
function and plasmon dispersion for three different cases: a high-temperature
equilibrium, a high-temperature inverted two-component quasi-equilibrium, and
a nonequilibrium case that combines a hot quasi-equilibrium bath with a
distribution of photo-excited high-energy carriers.

\section{Theory\label{theory}}
\subsection{Non-equilibrium polarization function \label{noneqPol}}
The Coulomb interaction of a pair of electrons in graphene is effectively
screened by the collective of electrons (the MDF plasma) according to
$V^{\mathrm{eff}}_\mathbf{q} = V_q / \varepsilon(\mathbf{q},\omega)$, where
$V_q = e^2/ ( 2\bar{\eps}\varepsilon_0 q )$ is the 2D Fourier transform of the Coulomb
potential, for excitations with frequency, $\omega$, and in-plane
wave-vector, $\mathbf{q}$; and where $\varepsilon(\mathbf{q},\omega)$ is the dynamic dielectric (or
screening) function. It can be expressed, within RPA,
as \cite{giuliani2005quantum}
\begin{equation}\label{epsRPA}
\varepsilon(\mathbf{q}, \omega) = 1 - V_q \Pi(\mathbf{q}, \omega)
\;,
\end{equation}
introducing $\Pi(\mathbf{q}, \omega)$, the polarization function, given by the bare
bubble diagram.
Importantly, the poles of the effective Coulomb potential,
$\varepsilon(\mathbf{q},\omega)=0$, signify collective plasmon excitations, freely
propagating charge density waves that transport energy and momentum
\cite{Giuliani1982}.
It has been shown, that despite an interaction constant larger than one, the MDF
plasma of graphene behaves like a weakly interacting electron gas and
application of RPA is well-justified \cite{Hofmann2014}.

The polarization function which relates to the (non-local) sheet conductivity of
graphene, $\sigma_\mathrm{s}(\mathbf{q}, \omega) = i e^2 \omega / q^2 \, \Pi(\mathbf{q},\omega)$
\cite{giuliani2005quantum}, is calculated using the Lindhard equation
\cite{Wunsch2006,Hwang2007},
\begin{equation}\label{lindhard}
  \Pi[n](\mathbf{q},\omega) = \frac{g}{A} \sum_{\mathbf{k},s,s'}
  M^{ss'}_{\mathbf{k},\mathbf{k}+\mathbf{q}}
  \frac{
      n(\eeps^s_\mathbf{k}) - n(\eeps^{s'}_{\mathbf{k}+\mathbf{q}})
  }{
    \eeps^s_\mathbf{k} - \eeps^{s'}_{\mathbf{k}+\mathbf{q}} + \hbar(\omega + i\eta)
  }
  \;,
\end{equation}
in the limit as $\eta \rightarrow 0^+$.
The equation above introduces the in-plane electron wavevector $\mathbf{k}$, indices
$s, s' = \pm$ labelling the conduction and valence band, electron degeneracy
$g$, and sheet area $A$.
Further, for graphene in the MDF approximation, the matrix element is
given by $M^{ss'}_{\mathbf{k},\mathbf{k}'} = ( 1 + s s' \cos \theta_{\mathbf{k},\mathbf{k}'} )/2$ and
electronic dispersion $\eeps^s_\mathbf{k} = s \hbar v_\mathrm{F} |\mathbf{k}|$, with Fermi velocity
$v_\mathrm{F} \approx c/300$.
In the scope of this work, we assume non-equilibrium carrier distributions
that are functions of energy,
$n(\eeps^s_\mathbf{k})$,
and that the carrier energy is isotropic in $\mathbf{k}$ space, i.e.,
$\eeps^s_\mathbf{k} = \eeps^s_k$.
It follows that the response will thereby be isotropic, so herein we drop the
vector character of $\mathbf{q}$ in favour of $q$.

Fundamentally, the equation for the polarization and dielectric functions remain
valid in non-equilibrium as long as pair-excitations are not too highly damped,
so that it is possible to define non-equilibrium distribution functions,
$n(\eeps^s_k)$ \cite{Kadanoff1994}.
That is, if the relaxation rates of the distributions are long compared to the
oscillation timescales of the excitations, one can neglect non-Markovian
contributions \cite{Vasko2005}.
Experimental measurements of the ultrafast carrier dynamics of optically excited
carriers at 0.8~eV give a fastest relaxation timescale of 10-150~fs, followed by
slower cooling and recombination on a 150~fs-15~ps timescale
\cite{Dawlaty2008,George2008}.
At an oscillation period of 5.2~fs, such carriers would be well described by a
Markovian model, which permits the use of the model presented herein in cases of
equilibrium, two-component quasi-equilibrium, and for hot photoexcited
distributions.

A general expression for the non-equilibrium polarization function can be
constructed based on the equilibrium polarization (Eq.~\ref{lindhard}), as
first demonstrated in Ref.~\cite{Page2015}.
In the following we briefly present this result, before detailing how the result
is applied to both screening and plasmons.
Firstly, we make use of the delta function identity,
\begin{equation}
  \Pi [ n ] = \Pi \left[
    \int_{-\infty}^\infty \mathrm{d}\eeps \:
      n(\eeps)
     \delta(\eeps-\circ)
  \right]
  \;,
\end{equation}
where `$\circ$' is the dummy variable which the outer $\Pi$ functional operates
over.
The $(q, \omega)$ dependence is omitted for clarity.
Then, as Eq.~\ref{lindhard} is a linear functional of the carrier distribution,
we are able to swap the order of the integration and the functional, so that
only the delta function remains inside the functional,
\begin{equation}
  \Pi [ n ] =
  \int_{-\infty}^\infty \mathrm{d}\eeps \:
  \Pi \left[
     \delta(\eeps-\circ)
  \right]
  n(\eeps)
  \;.
\end{equation}
The delta function can be expressed as the derivative of a zero-temperature
Fermi-function, i.e. a Heaviside step function; the polarization function of
which is well known \cite{Wunsch2006,Hwang2007,Pyatkovskiy2009}.
This allows the polarization function to be expressed an integral transform of
the carrier density,
\begin{equation}
  \Pi [ n ] =
  \int_{-\infty}^\infty \mathrm{d}\eeps
  \pd{\Pi_{\mu=\eeps}^{T=0}}{\eeps}
  n(\eeps)
  \;.
\end{equation}
The integral can be split into contributions from electrons and holes by
introducing carrier distribution functions, $n_\elec$ and $n_\hole$, where,
\begin{equation}
n(\eeps) = n_\elec(\eeps) \theta(\eeps) +
  \left[1 - n_\hole(-\eeps) \right] \theta(-\eeps)
\;,
\end{equation}
with the Heaviside step function, $\theta(\eeps)$, separating contributions
from the conduction and valence band.
The polarization function then takes form as a sum of the polarization
function of intrinsic graphene and contributions from electrons and holes,
\begin{equation}\label{elegant0}
  \Pi [ n ] =
  \Pi_{\mu=0}^{T=0} +
  \int_{0}^\infty \mathrm{d}\eeps\,
  \big[
  \Pi'(\eeps) n_\elec(\eeps) + \Pi'(-\eeps) n_\hole(\eeps)
  \big]
\;,
\end{equation}
with $\Pi'(\eeps) = \pd{}{\eeps} \Pi_{\mu=\eeps}^{T=0}$ for brevity.
For systems with particle-hole symmetry, i.e. $\Pi'(\eeps) = \Pi'(-\eeps)$, such
as MDF graphene, Eq.~\ref{elegant0} can be written in terms of the joint
occupation $n_\elec + n_\hole$.
This can be further split into a sum,
$n_\elec + n_\hole = \sum_i n_i$, whose contributions to the polarization function can
be evaluated separately, due to linearity, i.e.,
\begin{equation}\label{elegant}
  \Pi [ n ] =
  \Pi_{\mu=0}^{T=0} +
  \sum_{i}
  \int_{0}^\infty \mathrm{d}\eeps\,
  \Pi'(\eeps) n_i(\eeps)
\;,
\end{equation}
This is a general result that holds for any material with isotropic band
structure and carrier distributions, within the RPA, not just graphene.

For graphene in the MDF approximation, the intrinsic contribution is
\cite{Khveshchenko2006},
\begin{equation}\label{intrisicPol}
  \Pi_{\mu=0}^{T=0} =
  \frac{g}{8 \pi \hbar^2 v_\mathrm{F}^2}
  \frac{
    -i \pi (u-v)^2
  }{
    4 \sqrt{u} \sqrt{v}
  }
  \;,
\end{equation}
where new coordinates
$u = \hbar(\omega + v_\mathrm{F} q)/2$ and $v = \hbar(\omega - v_\mathrm{F} q)/2$,
along the axes parallel and perpendicular to the Dirac cone have been
introduced.
These coordinates are particularly convenient to highlight the position of
branch points in the polarization function.
For graphene in the MDF model, the integral kernel is particle/hole symmetric,
$\Pi'(\eeps) = \Pi'(-\eeps)$, and takes the simple form
\cite{Page2015},
\begin{subequations}
\label{kern}
\begin{align} \label{kern.sum}
  \Pi'(\eeps) &=
  \frac{g}{8 \pi \hbar^2 v_\mathrm{F}^2}
  \left(
    \tilde{\Pi}'_+(\eeps) + \tilde{\Pi}'_-(\eeps)
  \right)
\\ \label{kern.part}
  \tilde{\Pi}'_\pm(\eeps) &=
    \frac{
      2i\sqrt{\pm i(\eeps \mp u)} \sqrt{\pm i(\eeps \mp v)} \pm 2\eeps
    }{
      \sqrt{u} \sqrt{v}
    } - 2
\;,
\end{align}
\end{subequations}

\begin{figure}
 \includegraphics{integrand.pdf}
 \caption{\label{contourReal}
  Complex-$\eeps$ plane diagram for the integration in Eq.~\ref{elegant}.
  The integration path is shown as the green line along the real axis.
  Branch points of the integration kernel (Eq.~\ref{kern}) are shown as blue
  `${\mathord{+\mkern-14mu\times}}$' at $\eeps = \pm u, \pm v$.
  Branch cuts that stem from the branch points and are set vertically such that
  in (a) the function evaluates as if an infinitesimal positive imaginary
  part was added to $u$ and $v$, when $q$ and $\omega$ are real.
  In (b) $\omega$ is complex, so $u$ and $v$ have acquired finite imaginary
  parts, causing the branch points pierce the integration contour.
   }
\end{figure}
There are four branch points in Eq.~\ref{kern} coming from the square root
terms, $\eeps = \pm u$ and $\eeps = \pm v$, shown as blue `${\mathord{+\mkern-14mu\times}}$' in the
complex-$\eeps$ plane in Fig.~\ref{contourReal}a, which lie on the real axis
for real values of $(q,\omega)$.
The branch cuts in  Eq.~\ref{kern} have been positioned vertically (see
Appendix~\ref{apndBranchCuts});
the integration can therefore be carried out directly on the real $\eeps$ axis
(green line), without the infinitesimal shift prescribed by Eq.~\ref{lindhard}.

Equation~\ref{elegant}, along with the definitions in Eqs.~\ref{intrisicPol} and
\ref{kern}, is all that is needed to evaluate the non-equilibrium polarization
function for real-valued arguments $(q,\omega)$.
As such it is sufficient when calculating the screening function and energy-loss
spectrum (see Sec.~\ref{eels}).
However, when tracing the plasmon pole $\eps(q,\omega)=0$ for real-valued
wavevectors $q$, the plasmon frequency becomes complex \cite{Pyatkovskiy2009}
and the branch points in Fig.~\ref{contourReal} are no longer situated on the
real-$\eeps$ axis.
In this circumstance the branch points may cut through the integration contour,
as shown in Fig.~\ref{contourReal}b, and the value of the integral cannot be
uniquely determined.
The following section deals with the proper treatment of the integral in this
situation.

\subsection{Non-equilibrium plasmon dispersion relations \label{noneqPlas}}
Plasmons exist at frequencies where there is a pole in the effective Coulomb
potential, i.e. for the zeros of the dielectric function
(Eq.~\ref{epsRPA}), i.e.
\begin{equation} \label{epsRPAzero}
\eps(q,\omega)=0
\;.
\end{equation}
The dielectric function is complex valued in regions where there is
Landau damping or electron-hole recombination, so it follows that for a real
wavevector, the roots themselves are at complex frequencies,
\begin{equation}
\omega = \omPl - i \gamPl
\;,
\end{equation}
where the real part is proportional to the energy of the plasmon, $\hbar\omPl$,
and the imaginary part is proportional to the net stimulated absorption rate of
the plasmon number density,
$2\gamPl = \gamma_\mathrm{stim} = \gamma_\mathrm{abs} - \gamma_\mathrm{emit}$.
Commonly, an approximation to the root is found by solving,
\begin{equation}
\re \eps(q, \omPl) = 0
\;,
\end{equation}
for a real frequency, $\omPl(q)$, given a real $q$. The imaginary
part is then evaluated by taking a first-order Taylor expansion about $\omPl$,
perturbing by $-i \gamPl$, and solving, which leads to
\cite{Wunsch2006,Wang2007},
\begin{equation}
\gamPl =
\frac{\im \eps(q, \omPl)}{\re \pd{\eps}{\omega}(q, \omPl)}
\;.
\end{equation}
This approximation holds when the solved decay rate small with respect to the
frequency, $\gamPl \ll \omPl$, In this approximation however, as was shown in
\cite{Page2015}, the decay rate grows without bounds for increasing $q$, and
deviations from the exact complex solution are clearly visible in both real and
imaginary parts, as well as a failure to produce correct emission spectra in the
photo-inverted case.
This section shall detail how complex frequency plasmon dispersion relations can
be found for graphene with arbitrary carrier configurations.

Figure~\ref{contourReal} shows the integration of Eq.~\ref{elegant} in the case
of real valued frequencies;
there the integration contour is a straight line along the real axis from zero
to infinity.
When the frequencies become complex, the branch points detach from
the real axis and may cut through the integration contour, in which case the
polarization function evaluates incorrectly as the integral cannot be uniquely
defined.
Evaluation of Eq.~\ref{epsRPAzero} for complex frequencies thus requires a
deformation of the integration contour and the branch cuts such that both vary
continuously with increasing wavevector, $q$, and that neither cross each other
in complex-$\eeps$ space.

In general, an integral is independent of its integration contour;
any two paths with the same endpoints, that can be continuously deformed from
one to the other, will yield the same result, as long as no singularities are
crossed during the deformation.
Applying this principle to the polarization function, the contour of
integration, in Eq.~\ref{elegant}, can be altered by taking it off the real axis
to allow the branch points to move in a region below the real axis.

In addition to branch points, the other singularities that may appear in the
integrand are the poles of the carrier distribution function $n(\eeps)$.
The positions of these poles are fixed in the complex-$\eeps$ plane, independent
of the dynamic variables $(q,\omega)$.
As with branch points, the integration contour must not pass over poles
without being corrected for.
The Fermi function,
$n(\eeps) = (1+\exp((\eeps-\mu)/ T))^{-1}$,
for example, has its poles at the complex fermionic Matsubara frequencies,
$\eeps^{(n)} = \mu + i (2 n + 1) \pi T$ with residue $-T$.
If the integration contour passes over a pole, at $\eeps_{\!\mathord{\times}\!}$,
the integral will gain a contribution due to the residue of the pole,
i.e., the integral of a closed contour encircling that pole,
$\mathcal{C}_{\!\mathord{\times}\!}$.
For a pole in the carrier distribution function this contribution is,
\begin{equation}\label{residue}
\oint_{\mathcal{C}_{\!\mathord{\times}\!}} \mathrm{d} \eeps\: \Pi'(\eeps) n(\eeps) =
2 \pi i \Pi'(\eeps_{\!\mathord{\times}\!}) \res_{\eeps_{\!\mathord{\times}\!}}n(\eeps)
\;.
\end{equation}
Therefore, the integration contour in Eq.~\ref{elegant} can be altered to
pass over poles as long as the contributions of the residues are subtracted.

\begin{figure}
 \includegraphics{integrand2.pdf}
 \caption{\label{contourDeformed}
  Complex contour integration for solving plasmon dispersion relations.
  (a) and (b) show the integration contour (green line) of
  $\int\mathrm{d}\eeps \tilde{\Pi}'_\pm(\eeps) n(\eeps)$.
  Poles are shown as red `${\!\mathord{\times}\!}$', i.e. here for a Fermi distribution.
  Branch points are represented as blue `${\mathord{+\mkern-14mu\times}}$'.
  The black dotted line illustrates how the branch points move as the solutions
  to $\varepsilon(q,\omega) = 0$ evolve with increasing $q$.
  (a) integrates the $\tilde{\Pi}'_-$ kernel, over the positive real line,
  while (b) integrates $\tilde{\Pi}'_+$ over a contour that avoids being crossed
  by the moving branch points.
  One of the contained poles has been circled with a clockwise integration to
  indicate that the contribution of this pole must be subtracted, with the
  contour having passed over it.
  }
\end{figure}

Apart from the poles of the carrier distribution, which are explicitly
accounted for by Eq.~\ref{residue}, the prescription for calculation plasmon
dispersion curves is that the integration contour must not be crossed by
singularities as they move through $\eeps$ space, and that the contour is to be
deformed to enforce this.
It is useful here to break the integral into two parts, and calculate the
$\tilde{\Pi}'_+(\eeps)$ and $\tilde{\Pi}'_-(\eeps)$ contributions to the
integration kernel in Eq.~\ref{kern.sum} separately.
This will allow a different contour to be used in each part, and particularly
will separate the branch point singularities, so that only one pair has to be
considered in each integral.
For such evaluation, the favourable choice of branch cuts changes, as
illustrated in Fig.~\ref{contourDeformed} (see Appendix~\ref{apndBranchCuts}),
and Eq.~\ref{kern.part} is replaced by,
\begin{subequations}
\begin{align}
  \tilde{\Pi}'_+(\eeps) &=
    \frac{
      -2 \sqrt{\eeps - u} \sqrt{\eeps - v} + 2\eeps
    }{
      \sqrt{u} \sqrt{v}
    } - 2 \\
  \tilde{\Pi}'_-(\eeps) &=
    \frac{
      -2 i \sqrt{i(\eeps + u)} \sqrt{i(\eeps + v)} - 2\eeps
    }{
      \sqrt{u} \sqrt{v}
    } - 2
\;,
\end{align}
\end{subequations}
For the first equation, this keeps the branch cut compact, whereas the second
equation orients the cuts at $-u$ and $-v$ vertically upwards (blue zigzag
lines, as shown in Fig.~\ref{contourDeformed}).
The $\tilde{\Pi}'_-$ kernel is integrated along the real line but the
$\tilde{\Pi}'_+$ contribution follows a contour that, starting from zero, cups
the branch points $u$ and $v$ from underneath, and corrects for any poles that
would not have otherwise been included by a contour along the real axis.

Given a form for the polarization function, it still remains to find a solution
to Eq.~\ref{epsRPAzero}.
For the complex frequency plasmon solutions, $\eps(q, \omega)$ takes $q$ as a
real parameter, and a complex-$\omega$ is found for each $q$ which solves
Eq.~\ref{epsRPAzero}.
When treated as described herein, the dielectric function is well behaved as a
function of a complex-$\omega$ in the vicinity of solutions.
As such standard complex root finding methods, e.g. Newton's method,
that are available in numeric software packages, can be used.
The starting value for the root finding method, in the first instance, can be
the solution to an equivalent system with a Drude conductivity, as will be
described in Sec~\ref{sec:Drude} and subsequently a complex
frequency solution for a nearby $q$, that has been previously solved for,
should be used.

Examples of plasmon dispersion relations calculated in this way are presented in
the next section.

\section{High temperature equilibrium and quasi-equilibrium}
\subsection{Energy loss function and plasmon dispersion \label{eels}}

\begin{figure*}
 \includegraphics{finiteTjoint.pdf}
 \caption{\label{finiteT}
  Energy loss function, $S = -\im \eps(q,\omega)^{-1}$, over a range of
  temperatures.
  The solid white diagonal line is the Fermi velocity, whereas the dotted lines
  are the boundaries of regions of Landau damping at zero temperature.
  Complex frequency plasmon dispersion curves are plotted as green lines over
  each plot $(q, \re \omega_\mathrm{pl}(q))$.
  Parts (a-d): equilibrium graphene for temperatures in the range $[0,
  2\mu]$.
  Part (e-h): two-component plasma in quasi-equilibrium, following
  photo-excitation, with temperatures in the range $[0, 4\mu]$.
  The inversion line at $\hbar \omega = 2\mu$ separates regions of
  interband spontaneous emission (gain) and absorption (loss);
  in these regions, the loss function is negative and is shown in red.
  }
\end{figure*}

To demonstrate the contour integration method, we first evaluate the
complex frequency plasmon dispersion, $\omPl - i \gamPl$, together with the
energy loss function, $S = \-\im \eps(q,\omega)^{-1}$ for both equilibrium
and two-component quasi-equilibrium, at finite temperatures, $T$ (given in
units of energy).
In both cases the carrier distribution functions for each band ($n_\elec$, $n_\hole$) are
Fermi functions, $n_{\mathrm{e}/\mathrm{h}} =
\{1+\exp[(\eeps-\mu_{\mathrm{e}/\mathrm{h}})/ T]\}^{-1}$.
The energy loss function is shown in Fig.~\ref{finiteT}, alongside
the corresponding complex frequency plasmon dispersion curves.
Parts (a-d) for equilibrium graphene, where there is a single Fermi-level,
i.e. the chemical potentials of each carrier species are of equal magnitude but
opposite sign ($\mu_\elec = -\mu_\hole = \mu$).
Part (a) is for zero temperature.
Here the Fermi edge is sharp and Landau damping is confined to regions permitted
by energy and momentum conservation.
The energy loss function, $S$, is finite in these regions and zero in the
so-called quasi-loss-free regions, the triangular regions in the plot.
The complex frequency plasmon dispersion (first solved at zero temperature
equilibrium in \cite{Pyatkovskiy2009}) is shown here as a green line.
The energy loss function is resonant about the plasmon dispersion curve, and is
broader where the plasmon loss $\gamPl$ is larger and sharper when it is small.
The solution of the plasmon dispersion curve continues over the Dirac cone
line, $\omega = v_\mathrm{F} q$, into the intraband region. Notably this is not reflected
in the energy loss function as the plasmon curve has moved onto a different
branch of the energy loss function \cite{Kukhtaruk2015}.
Figure~\ref{finiteT}(b-d) show what happens to the plasmon dispersion and the
energy loss function as temperature is increased.
The energy loss function in this equilibrium case is in agreement with
\cite{Ramezanali2009}.
In the band, the Fermi edge stops being sharply defined and has a spread for
finite temperatures.
This results in the energy loss function not being confined to well defined
regions, becoming finite everywhere.
Again the energy loss function is resonant about the plasmon dispersion, and
this is more pronounced in regions of low loss, where the resonance is sharpest.
The plasmon dispersion curves themselves become steeper as $T$ is increased,
remaining in the interband region for a wider range of wavevectors.

Figure~\ref{finiteT}(e-h), accounts for the case of a
two-component quasi-equilibrium of excited intrinsic graphene.
In this case, each band has its own Fermi level, and here
($\mu_\elec = \mu_\hole = \mu$).
Parts (a-d) were scaled by the energy difference between the Dirac point and the
Fermi level, $\mu$, whereas parts (e-h) are scaled to the energy difference
between the two Fermi levels, $2\mu$, as was the convention in \cite{Page2015}.
With a population of electrons in the conduction band sitting above a population
of holes in the valence band, recombination of electron-hole pairs by stimulated
emission of plasmons becomes possible.
At zero temperature, as in Fig.~\ref{finiteT}(e), this manifests as the energy
loss function being negative, indicating gain (red on the plot), in the interband region for
energies less than $2\mu$.
For frequencies above this, stimulated emission is no longer possible, and
plasmons at these frequencies are absorbed by Landau damping (blue areas).
At finite temperatures, where the Fermi edge is not sharply defined,
phase space for stimulated emission at frequencies above $2\mu$ and for
absorption at frequencies below is allowed for.
The net rate of stimulated emission/absorption however remains sharply defined
at $\hbar \omega = 2\mu$, as reflected in parts (f-h).
This is also despite the plasmon curves steepening and growing as they did in
the equilibrium case, and hence dispersion curves for high temperatures having
gain in a comparably narrow wavevector range.

The manner in which the curves for equilibrium and two-component
quasi-equilibrium scale with temperature is similar in both cases.
In fact, for the highest temperature shown of each case,
Fig.~\ref{finiteT}, parts (d) and (h), the dispersion curves and loss functions
are almost identical.
At high temperatures, it is indeed the temperature that is responsible for the
scale of the features of the plasmon dispersion and energy loss function,
whereas at low temperatures, it is the chemical potentials that determine the
scale and features such as inversion.
In the next section, we shall introduce a scaling factor that generalizes for
arbitrary carrier distributions in order to more clearly show how the behaviour
of plasmons change depending on the carrier distribution.

\subsection{Drude weight scaling \label{sec:Drude}}

For regimes where temperature is much greater than the chemical potentials,
the distribution function and hence polarization function is, in relative terms,
insensitive to changes in either chemical potential.
This is the regime of thermoplasma polaritons, where the loss function and
plasmon dispersion scales linearly with temperature \cite{Vafek2006}.
In contrast, for low temperatures, the plasmon dispersion scales primarily
with the chemical potentials.

The Drude weight, defined as,
\begin{equation}
D = \lim_{\omega \rightarrow 0} \omega \im \sigma_\mathrm{s}(0,\omega)
\;,
\end{equation}
can be used as a scaling parameter for the plasmons.
We introduce a derived frequency, $\omega_\mathrm{D}$, which later in this section we
shall plot plasmon dispersion curves in relation to:
\begin{equation}
\omega_\mathrm{D} = \frac{Z_0}{\alpha_0 g} D
\;,
\end{equation}
where $\alpha_0$ is the fine structure constant and $Z_0$ is the impedance of
free space.
To derive an expression for the Drude weight for a non-equilibrium
distribution in RPA, it suffices to use an expression for the intraband surface conductivity
in the local limit ($q \rightarrow 0$), given as \cite{Falkovsky2007b},
\begin{equation}
\sigma_\mathrm{intra}(0, \omega) = -\frac{i e^2 g}{4\pi \hbar^2 \omega}
\sum_{i}\int_0^\infty \mathrm{d} \eeps\, \eeps \pd{n_i}{\eeps} \;,
\end{equation}
or more simply as,
\begin{align}
Z_0\sigma_\mathrm{intra}(0, \omega) = i \alpha_0 g \frac{\omega_\mathrm{D}}{\omega}
\;.
\end{align}
where the integral term is equal to the Drude weight frequency (after
integration by parts),
\begin{equation}\label{epsScale}
\hbar\omScale = \sum_{i}\int_0^\infty \mathrm{d}\eeps\:
n_i(\eeps)
\;,
\end{equation}
Substituting this into the solution for plasmons on a conducting sheet,
$1 + i c q Z_0\sigma_\mathrm{s}(\omega) / (2 \bar{\eps} \omega) = 0$,
returns the familiar square-root plasmon dispersion approximation,
\begin{equation}
\frac{\omega}{\omega_\mathrm{D}} = \sqrt{\frac{g\alpha_0}{2\bar{\eps}}}
\sqrt{\frac{c q}{\omega_\mathrm{D}}}
\;,
\end{equation}
or even more compactly as, $\tilde\omega = \sqrt{2 \alpha_\mathrm{g} \tilde{q}}$, for
$\tilde\omega = \omega/\omega_\mathrm{D}$, $\tilde{q} = v_\mathrm{F} q / \omega_\mathrm{D}$, and
$\alpha_\mathrm{g} = g \alpha_0 c/(4 \bar{\eps} v_\mathrm{F})$.
which is the long wavelength limit for equilibrium distributions at zero
\cite{Wunsch2006,Hwang2007,Abedinpour2011} and finite \cite{Falkovsky2007a,
Ramezanali2009} temperature.
Thus, when the frequency and wavevector are scaled by the Drude weight, the
local limit of the plasmon dispersion is independent of the carrier
distribution.
We shall show later in this section that scaling by the Drude weight is indeed
useful to identify general behaviour when calculating the fully non-local
complex-frequency plasmon dispersion.

The Drude weight in Eq.~\ref{epsScale} is linear in carrier distribution, $n$,
and therefore can be calculated for each contribution independently and summed
for the total Drude weight.
For a Fermi distribution in a single band,
$n = \{1+\exp[(\eeps-\mu)/T]\}^{-1}$,
the contribution to the Drude weight takes the form,
\begin{equation}
\hbar\omScale \rightarrow T \log\left(1 + e^{\mu/T}\right)
\;,
\end{equation}
which in the limit where chemical potential dominates, i.e.
$\mu \gg T$,
is purely
$\hbar\omScale \rightarrow \mu$,
and in the temperature dominated regime,
$T \gg \mu$,
becomes
$\hbar\omScale \rightarrow T \log 2$.

For equilibrium graphene, the electron and hole chemical potentials are equal
and opposite, $\mu_e = -\mu_h = \mu$.
The sum of both contributions to the Drude weight,
$\hbar\omScale = T \log\left[\left(1 + e^{\mu_\mathrm{e}/T}\right)\left(1 +
e^{\mu_\mathrm{h}/T}\right)\right]$,
then becomes equivalent to the form,
$2 T \log \left[2 \cosh(\mu/2T) \right]$,
as appears in Ref.~\cite{Falkovsky2007a}.

\begin{figure}
 \includegraphics{scaleF.pdf}
 \caption{\label{scaleF}
  Plasmon dispersion curves at finite temperature for (a) equilibrium graphene
  and (b) two-component quasi-equilibrium.
  All curves are scaled to the Drude weight, $\omega_\mathrm{D}$, and
  plotted alongside the local Drude limit (dashed line).
  The curves progress from zero temperature (red in (a), blue in (b)), towards
  the case where temperature dominates over chemical potential (black curve,
  which is identical in (a) and (b)).
  For the two component case, (b), the curves continue for negative chemical
  potentials, i.e. for small probabilities of carriers in each band.
  Alongside the dispersion curves, scaled plasmon losses are plotted against
  frequency (right column).}
\end{figure}

Figure~\ref{scaleF} shows complex-frequency RPA non-local plasmon dispersion
curves scaled to the Drude weight $\omega_\mathrm{D}$.
This figure is for graphene in equilibrium in part (a) and two-component
quasi-equilibrium in part (b).
In each case the carrier distribution is a Fermi function in each
band (with both bands adding a contribution to the Drude weight) with a
temperature that is varied to span all ratios with the chemical potential.
Firstly, for equilibrium; part (a) shows that the scaled curves are largely
insensitive to the balance between chemical potential and temperature with all
curves overlapping.
For frequencies on the order of half the Drude weight or smaller ($\omega
\lesssim \omega_\mathrm{D} / 2$), the dispersion curves overlap with the square-root
Drude dispersion predicted by the local model.
Outside this frequency range, the curves each follow the zero-temperature
equilibrium solution (red curve).
Even in the high temperature limit, $T \gg \mu$, ($\mu/T=0$, black curve) the
dispersion is not significantly different from the zero temperature result.
Due to the similarity of these curves, they may all be approximated by a single
function, e.g.,
\begin{equation}
\frac{\omPl(q)}{\omega_\mathrm{D}} = \log\left(\frac{
  e^{
    \kappa(1 + e^{a \kappa})\sqrt{2 \alpha_\mathrm{g} \tilde{q}}
  } +
  e^{
    \kappa \left( a + b \tilde{q} \right)
  }
}{
  1 + e^{a \kappa}
}\right)/\kappa
\;,
\end{equation}
which asymptotically approaches the Drude dispersion for $q \rightarrow 0$
and approaches a straight line at other values,
where $\kappa,a,b$ are parameters to be fit
(with values $\kappa=-2.79$, $a=1.06$, $b=0.45$ for suspended graphene).

The imaginary part of the complex frequency solution, i.e. the loss
curve, does vary with temperature.
This can be seen in the right panel of Fig.~\ref{scaleF}(a) where the red
curve, representing zero temperature, has frequencies where there is no loss,
i.e. where the plasmon dispersion passes through the loss free region.
When the temperature rises, absorption processes, that would be loss free at
zero temperature, are now no longer Pauli blocked and their
rates become finite, resulting in loss in the plasmon dispersion.
The extremal curve of the high temperature limit has a finite loss, and all
intermediate curves are bounded by the high and zero temperature limit.

In part (b) of Fig.~\ref{scaleF}, plasmon dispersion curves are drawn for
two-component quasi-equilibrium.
Here both positive and negative chemical potentials are solved for.
Whereas in the equilibrium case, with a single Fermi level, a positive or
negative chemical potential determines if the Fermi-level is found in the
conduction or valence band;
For a two-component quasi-equilibrium, each band has its own distribution
and Fermi level, and a positive chemical potential places the Fermi level within
it's band, and a negative outside of it.
Particularly a negative chemical potential indicates a low density of carriers
i.e. at the end of the Fermi distribution tail, where the occupation probability
for any state is less than one half.
Such distributions occur early in photo-excitation, when an excitation pulse
thermalizes and there are relatively few carriers but these carriers have a
large energy density.

In contrast to part (a), Fig.~\ref{scaleF}(b) shows that the plasmon dispersion
curves will spread out for different ratios of temperature to chemical
potential.
Each still follows the Drude square root behaviour for
$\omega \lesssim \omega_\mathrm{D} / 2$, but the curves fan out after this.
The black curve that represents temperature dominated behaviour
($\mu/T \rightarrow 0$)
is the same curve as in the equilibrium case, as in both cases the Fermi level
of each band is at the Dirac point.
From here, curves with a positive chemical potential fall underneath,
while curves with a negative chemical potential start to steepen over the
high-temperature curve. For sufficiently negative $\mu$, local extrema appear
in the dispersion, where the group velocity, $\pd{\omega}{q}$, is zero;
these are known as stopped-light points \cite{Tsakmakidis2014,Pickering2014}.
The extremal curve ($\mu / T \rightarrow -\infty$) has an
analytic solution, solved for in Appendix~\ref{lowCarriers}.
The loss curves show that for curves with a positive chemical
potential, there are frequencies ($\hbar \omega < 2\mu$) where the loss is
negative, i.e. there is gain.
As explained in the previous section, this is where the rate of
stimulated emission processes is greater than absorption processes.
In the high temperature limit and for negative chemical potentials, the bands
never have an occupancy probability greater than one half, resulting in greater
rates of plasmon absorption than emission at all frequencies.
For low carrier numbers ($\mu < 0$), the carriers in the system allow
plasmons to be supported, but these plasmons are far more likely to be absorbed
than to stimulate emission, and hence show the highest loss rates in the figure.

This section has shown that the shape of plasmon dispersion curves, once scaled
to the Drude weight, is approximately constant for equilibrium graphene, but
can vary quantitatively and qualitatively for non-equilibrium carrier
distributions such as a two-component quasi-equilibrium, especially for
configurations where there is a low density of high energy carriers in the
system.
Carrier inversion and high temperatures contribute to the gain and loss
channels of the plasmons such that the low-loss approximation,
$|\operatorname{Im} \omega| \ll \operatorname{Re} \omega$,
is no longer valid, and that plasmon dispersion and losses should be calculated
using this non-equilibrium complex-frequency procedure.
In the next section, we show how the non-equilibrium carrier distributions
produced immediately following photoexcitation can further affect the
polarization and plasmons.

\section{Photo-excitation non-equilibrium}

\begin{figure}
 \includegraphics{photoex.pdf}
 \caption{\label{photoex}
  Plasmon dispersion curves of photoexcited graphene with a carrier distribution
  function of a \emph{sech} excitation profile above thermalized Fermi bath,
  with parameters, $\mu_\elec = \mu_\hole$,
  $T=0.4\mathrm{eV}$,
  $\eeps_\mathrm{s} = 1.2\mathrm{eV}$,
  $\gamma_\mathrm{s} = 0.12\mathrm{eV}$.
  In part (a), the bath chemical potential is set at $\mu_\elec = 0.1\mathrm{eV}$ and the
  peak excitation occupation, $n_{\mathrm{s}0}$, varies in the range $[0, 0.5]$ between no
  excitation and transparency.
  In part (b) $\mu_\elec$ varies in the range $[-1.4,1.0]$ with $n_{\mathrm{s}0}$ fixed at 0.5.
  The carrier distribution function is inset in each part, and the plasmon loss
  plotted in the rightmost panels.
  All curves are scaled to the Drude weight, $\omega_\mathrm{D}$.
 }
\end{figure}

Until now, only polarization functions derived from Fermi distributions have
been shown.
The method presented in Sec.~\ref{theory} generalizes to calculations with
arbitrary carrier occupations.
We present a model for photo-excitation which describes a hot-carrier
nonequilibrium model.
That is, a transient population of photo-excited electrons are modelled to be
placed above a bath of thermalized quasiequilibrium carriers.

The photo-excited carriers are modelled with a hyperbolic secant (sech)
profile \cite{Chow2013}, above a background of relaxed quasi-equilibrium
carriers.
Both conduction and valence bands have the distribution,
\begin{equation} \label{photoexMod}
  n_{\elec/\hole}(\eeps) = \frac{1}{1+\exp(\frac{\eeps-\mu}{T})} +
  n_{\mathrm{s}0}
  \sech\left(\frac{\eeps-\eeps_\mathrm{s}}{\gamma_\mathrm{s}}\right)
  \;.
\end{equation}
The extra parameters introduced here are the excitation energy, $\eeps_\mathrm{s}$
(which is half the pump photon energy);
the population width, $\gamma_\mathrm{s}$
(which relates to the pump width and ratio of pump rate to electron
recombination rate);
and the peak occupation, $n_{\mathrm{s}0}$, which ranges from 0, for no pump,
to 0.5 at transparency.
Calculating the complex contour integral of the sech contribution is almost
identical to that of the Fermi distribution;
like the Fermi distribution, the sech distribution has evenly spaced poles at
Matsubara frequencies,
$\eeps^{(n)} = \eeps_\mathrm{s} + i (2 n + 1) \pi \gamma_\mathrm{s} / 2$
with residue $-i (-1)^{n} n_{\mathrm{s}0} \gamma_\mathrm{s} / 2$,
therefore the same algorithm is employed.

Plasmon dispersion curves are shown for a photoexcited carrier distribution
function in Fig.~\ref{photoex}.
In part (a) the height of the sech peak is varied from 0.5 (i.e. transparency)
to 0 (no excitation).
The parameters, given in the figure caption, are such that when the sech peak is
at its maximum height ($n_{\mathrm{s}0} = 0.5$), it contributes approximately twice as
much to the Drude weight as the Fermi bath does.
This contribution decreases linearly to zero as the height of the sech
peak is lowered.
The dispersion curves produced for each height of the sech distribution follow
the Drude dispersion in the local limit and then spread out for higher
wavevectors, being pushed down for as $n_{\mathrm{s}0}$ increases.
This is in a similar manner to when temperature is increased in the
quasi-equilibrium case.
The loss of the plasmon dispersion curves increases with the height of the sech
peak, which is perhaps unexpected since in principal more carrier inversion is
being added to the system, however until these carriers relax within the band,
there are not enough emission channels at any particular energy to compete with
absorption and have a net gain for plasmons.
A more stark change comes in part (b) when $n_{\mathrm{s}0}$ is held
constant at 0.5, and instead, the chemical potential of the thermalized bath is
reduced.
The curves split into two bundles, with those with the smallest chemical
potentials are attracted towards the low density-high energy limit, as in
quasiequilibrium for $\mu/T \rightarrow -\infty$ in Sec.~\ref{sec:Drude}
and Appendix~\ref{lowCarriers}.
In this case points of zero group velocity (turning points) appear in the
dispersion, and the dispersion is significantly different from that solved for
equilibrium graphene.

This section has shown that immediately after photo-excitation, whilst the
carriers relax initially to a two-component quasi-equilibrium, the momentary
plasmon dispersion and screening function can be qualitatively different.
Particularly, excited carriers need to relax within their band before they
contribute to plasmon gain processes.
The sech distribution has been used here to model photo-excitation, however a
sum of one or more sech functions may be used to fit other carrier
distributions, such as the non-thermal carriers observed during relaxation in
Ref.~\cite{Gierz2015}.

\section{Conclusion}
This work has outlined the procedure for efficiently evaluating RPA
polarization functions and plasmon dispersion relations in monolayer graphene
for arbitrary non-equilibrium carrier occupations.
This has allowed for these quantities to be calculated in cases of high
temperature, both in equilibrium and a two-component plasma quasi-equilibrium,
as well as in application to a model for carriers immediately following
photo-excitation, where they are momentarily excited to a ring of high
energy states.
The non-equilibrium carrier occupations explored in this paper are transient and
will evolve through a range of configurations as the system relaxes.
The irreducible polarization function, calculated here, is a key quantity of the
dynamic screening, which influences all carrier-carrier interactions, such as
Auger recombination and others which may play a role in carrier relaxation in
graphene \cite{Gierz2015,Giovanni2015}.
The theory is general and can be modified to describe graphene outside of the
Dirac cone regime \cite{Hill2009,Pyatkovskiy2009,Mihnev2015}, and indeed may be
applied to other two-dimensional materials \cite{Roldan2017} such as transition
metal dichalcogenides.

\section{Acknowledgements}
The authors acknowledge financial support provided by the Engineering and
Physical Sciences Research Council (United Kingdom).

redpajama_set_name: RedPajamaArXiv


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