Early 2016, I was contacted by Thomas Duguet of CEA Saclay to give a one-week lecture series on the topic of Richardson-Gaudin and beyond-integrability. The purpose of these lectures was to get to the core of our beyond-integrability research program, starting from fundamental principles of lie algebras and integrability.
I was immediately won for the idea, so we agreed on a full-week lecturing at CEA in september 2016: 5 days, 6h/day (lunch break included). It was quite a "tour de force", not in the least by my audience, who dauntlessly attended each of my lectures to the very end (impressive)!
It was basically a blackboard performance, so the only support material I had were my hand written notes , of which you can find the scans below. I broke it down into 10 chapters.
Introductory lecture :: what do we understand under quantum integrability?
A blue dahlia, a black tulip
That's where opinions differ
The scholars disagree
Much of this lecture is based on the following (almost philosophical) paper by Jorn Mossel and Jean-Sebastien Caux
History class :: the Coordinate Bethe Ansatz (CBA)
The first one in a series of arXiv papers by Karbach and collaborators are a good reference for this
although I took an additional anisotropy constant Δ into account in my derivation
The fundamentals :: the Algebraic Bethe Ansatz (ABA)
I closely followed the line of reasoning in Jean-Sebastien Caux's (unpublished) Nordita lecture notes for the Heisenberg chain, and the topical review by Jon Links for the so-called classical limit of the R-matrix to get to the Gaudin magnets
Intermezzo :: Lie algebras
The quantum many-body problem can be rephrased entirely in terms of Lie Algebras. A special (important) algebra is the quasi-spin algebra associated with pairing
Richardson-Gaudin :: Conserved charges
Starting from a reasonable proposition for the conserved charges, the (Yang-Baxter-Gaudin) integrability equations are derived. It is shown that the eigenstates of the conserved charges consist of a Bethe Ansatz of creation operators in the Gaudin algebra, provided the rapidities in the Ansatz fulfill the Richardson-Gaudin equations.
Richardson Gaudin :: the rational model (XXX)
The link between the XXX conserved charges and the reduced BCS model is made. The structure of the rapidities is discussed, as well as current methods to solve them numerically. In chronological order
The cluster method
Solving the Richardson equations for fermions
Stefan Rombouts, Dimitri Van Neck, & Jorge Dukelsky
Phys. Rev. C69, 061303R (2004)
[doi:10.1103/PhysRevC.69.061303]
Eigenvalue-based method
Gaudin models solver based on the correspondence between Bethe ansatz and ordinary differential equations
Alexandre Faribault, Omar El Araby, Christoph Sträter, and Vladimir Gritsev
Phys. Rev. B 83, 235124 (2011)
[doi:10.1103/PhysRevB.83.235124]
Pseudo deformation of the quasispin algebra
Richardson-Gaudin integrability in the contraction limit of the quasispin
Stijn De Baerdemacker
Phys. Rev. C86, 044332 (2012)
[doi:10.1103/PhysRevC.86.044332]
Heine-Stieltjes connection
Heine-Stieltjes correspondence and the polynomial approach to the standard pairing problem
Xin Guan, Kristina D. Launey, Mingxia Xie, Lina Bao, Feng Pan, and Jerry P. Draayer
Phys. Rev. C 86, 024313 (2012)
[doi:10.1103/PhysRevC.86.024313]
Richardson-Gaudin :: the hyperbolic model (XXZ)
The XXZ has a much more interesting phase diagram than the XXX Richardson-Gaudin model, with topologically protected Moore-Read and Read-Green states.
The BCS mean-field and exact Richardson-Gaudin discussion of the phase diagram are provided by respectively
Exactly solvable pairing model for superconductors with p(x)+ip(y)-wave symmetry
Miguel Ibañez, Jon Links, Germán Sierra, and Shao-You Zhao
Phys. Rev. B 79, 180501R (2009)
[doi:10.1103/PhysRevB.79.180501]
Quantum phase diagram of the integrable px+ipy fermionic superfluid
Stefan M. A. Rombouts, Jorge Dukelsky, and Gerardo Ortiz
Phys. Rev. B 82, 224510 (2010)
[doi:10.1103/PhysRevB.82.224510]
Overlaps
Key to the beyond-integrability program is the ability to calculate norms, overlaps and correlation functions. I reformulated the Richardson-Gaudin Bethe Ansatz states in the framework of Geminal wave functions, showed the central role of the permanent of a (geminal) matrix, and demonstrated the use of integrability using dual states. Important references here are (note the publication dates!)
Bestimmung der symmetrischen Verbindungen vermittelst ihrer erzeugenden Function.
Carl Wilhelm Borchardt
J. reine angew. Math. 53, 193(1857)
[doi:N.A.]
On the determinant representations of Gaudin models' scalar products and form factors
Alexandre Faribault and Dirk Schuricht
J. Phys. A: Math. Theor. 45 485202 (2012)
[doi:10.1088/1751-8113/45/48/485202]
Eigenvalue-based method and form-factor determinant representations for integrable XXZ Richardson-Gaudin models
Pieter W. Claeys, Stijn De Baerdemacker, Mario Van Raemdonck, and Dimitri Van Neck
Phys. Rev. B91, 155102 (2015)
[doi:10.1103/PhysRevB.91.155102]
Eigenvalue based variables
Arguably the best numerical method on the market to solve the Richardson-Gaudin equations and calculate overlaps and correlation functions is the Eigenvalue-based method, developed by Alexandre Faribault, and extended by Pieter Claeys (same references as above)
Beyond integrability
This constitutes a major part of my current research program, which was basically initiated by the following two papers
A New Mean-Field Method Suitable for Strongly Correlated Electrons: Computationally Facile Antisymmetric Products of Nonorthogonal Geminals
Peter A. Limacher, Paul W. Ayers, Paul A. Johnson, Stijn De Baerdemacker, Dimitri Van Neck, & Patrick Bultinck
J. Chem. Theory Comp. 9, 1394 (2013)
[doi:10.1021/ct300902c]
A size-consistent approach to strongly correlated systems using a generalized antisymmetrized product of nonorthogonal geminals
Paul A. Johnson, Paul W. Ayers, Peter A. Limacher, Stijn De Baerdemacker, Dimitri Van Neck & Patrick Bultinck
[doi:10.1016/j.comptc.2012.09.030]
Please check my research page for more information.
It was an extremely interesting experience to present my research program from the very fundaments, because it allowed me to look back at the past few years, retrace the tracks we have been doing, and find the why and how of it all. I would definitely recommend it to everyone to do this exercise, although I am not sure if you will find a crowd as amazing as the one I had :-).
For those interested in these notes (despite being hand written), please feel free to use them, I hope they can be useful to you. The notes should find an electronic counterpart in due time, so if you feel like sending me some comments, please do so!