Bibliography

Lecture 0 · Why integrability?

M. Toda, Theory of Nonlinear Lattices, §1.1 (the FPU problem, its parameters and the recurrence), §2.7 and §2.10; E. Fermi, J. Pasta and S. Ulam, Los Alamos report LA-1940 (1955); T. Dauxois, Physics Today 61(1) (2008) 55; N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15 (1965) 240; M. Toda, J. Phys. Soc. Japan 22 (1967) 431; G. Benettin, H. Christodoulidi and A. Ponno, J. Stat. Phys. 152 (2013) 195; T. Kinoshita, T. Wenger and D. S. Weiss, Nature 440 (2006) 900; J.-S. Caux and J. Mossel, J. Stat. Mech. (2011) P02023.

Lecture 1 · Phase space and flows

Arnold, Mathematical Methods of Classical Mechanics, Ch. 8 (vector fields and their flows on a manifold); da Silva, Lectures on Symplectic Geometry, §§1–3 (manifolds and vector fields); Babelon, Bernard and Talon, Introduction to Classical Integrable Systems, Ch. 1.

Lecture 2 · Two languages: the Poisson bracket and differential forms

Arnold, Mathematical Methods of Classical Mechanics, Ch. 8 (Hamiltonian vector fields, the Lie algebra of Hamiltonian functions) and §32 (exterior algebra); da Silva, Lectures on Symplectic Geometry, §§1–3 and 18 (Poisson structures) and §6 (forms); Babelon, Bernard and Talon, Introduction to Classical Integrable Systems, Ch. 1. Sign conventions differ across all three sources for precisely the identity of §2.3: check any formula against (2.7) and (2.6) before transcribing it.

Lecture 3 · The exterior derivative, Stokes, and the symplectic form

da Silva, Lectures on Symplectic Geometry, §1 (symplectic forms) and §6 (forms, $\dd$, the Poincaré lemma, Stokes); Arnold, Mathematical Methods of Classical Mechanics, §§35–36 (integration of forms, Stokes), §37 (symplectic structures) and §44 (the action integral this lecture was built to support).

Lecture 4 · One structure: canonical maps and Lagrangian submanifolds

da Silva, Lectures on Symplectic Geometry, §§1–4 (symplectic forms, cotangent bundles and the tautological form, Lagrangian submanifolds, generating functions) and §§7–8 (the Moser trick and Darboux); Arnold, Mathematical Methods of Classical Mechanics, §§37–38 and 40–41 (symplectic structures, integral invariants, the Poisson bracket from $\symp$), §43 (Darboux), §44 (the integral invariant of Poincaré–Cartan) and §§47–48 (Hamilton–Jacobi, generating functions); Bolsinov and Fomenko, Integrable Hamiltonian Systems, §1.3 (Darboux, and its version adapted to commuting functions). Conventions. da Silva’s agree with ours exactly — $\symp=-\dd\alpha$ with $\alpha$ the tautological form, $\iota_{\hvf{H}}\symp=\dd H$, $\pb{f}{g}=\symp(\hvf{f},\hvf{g})$, and the engine’s minus sign. Arnold and Bolsinov–Fomenko write $\symp=\sum\dd p\wedge\dd q$, so their $\symp$ is minus ours: check any formula from them against (3.13) and Theorem 4.17 before transcribing it.

Lecture 5 · The Liouville–Arnold theorem

Arnold, Mathematical Methods of Classical Mechanics, §49 (the theorem and its proof; Lemma 2 of §49 C is our stage (iii), Lemma 3 of §49 D our Lemma 5.10); Babelon, Bernard and Talon, Introduction to Classical Integrable Systems, ch. 2 (the theorem, and the examples). Conventions. Arnold writes $\symp=\sum\dd p\wedge\dd q$, the negative of ours; nothing in this lecture depends on the sign of $\symp$, since every use of it is a statement that something vanishes.

Lecture 6 · Action–angle variables

Arnold, Mathematical Methods of Classical Mechanics, §50 (action–angle variables: the construction and the neighbourhood statement), §51 (conditionally periodic motion) and §8 (central fields, Bertrand’s theorem); Babelon, Bernard and Talon, Introduction to Classical Integrable Systems, ch. 2; DLMF, §§19.2 and 19.12 (complete elliptic integrals and their behaviour as $k\to1$). Conventions. Arnold writes $\symp=\sum\dd p\wedge\dd q$, the negative of ours; his actions come out positive for the same reason ours do (§3.4).

Lecture 7 · When integrability fails

Arnold, Mathematical Methods of Classical Mechanics, App. 8–9; Kozlov, Russian Math. Surveys 38:1 (1983), ch. IV §1; Ott, Chaos in Dynamical Systems, §§4.4, 7.2–7.3; Toda, Theory of Nonlinear Lattices, §§1.2, 2.9; Hénon and Heiles, Astron. J. 69 (1964) 73; Ito (1985) and Morales-Ruiz, Ramis and Simó (2007) for the Hénon–Heiles family.

Lecture 8 · Lax pairs, and the spectral parameter

Babelon, Bernard and Talon, Introduction to Classical Integrable Systems, ch. 3 (Lax pairs) and ch. 2 on Toda; Toda, Theory of Nonlinear Lattices, §§2.10–2.11 (Flaschka’s variables and the periodic chain); H. Flaschka, Phys. Rev. B 9 (1974) 1924; M. Hénon, Phys. Rev. B 9 (1974) 1921; M. Audin, Spinning Tops (the Euler top, in the reading).

Lecture 9 · Why the integrals commute: the Toda ring is integrable

Babelon, Bernard and Talon, Introduction to Classical Integrable Systems, ch. 2–3 (the Sklyanin bracket) and ch. 6 (the periodic chain, its $2\times2$ monodromy matrix and its spectral curve); Faddeev and Takhtajan, Hamiltonian Methods in the Theory of Solitons, Part I ch. 1 (the classical $\rmat$-matrix and the transition to the quantum case); H. Flaschka and D. W. McLaughlin, Prog. Theor. Phys. 55 (1976) 438 (the actions from the gaps). Conventions. Babelon–Bernard–Talon exchange the letters $\lambda$ and $z$, and conventions for the sign of $\rmat$ differ between all three; see the third caveat.

Lecture 10 · The coordinate Bethe ansatz: the Heisenberg chain

H. Bethe, Z. Phys. 71 (1931) 205; Faddeev and Takhtajan, Phys. Lett. A 85 (1981) 375; Essler, Korepin and Schoutens, J. Phys. A 25 (1992) 4115; Korepin, Bogoliubov and Izergin, Quantum Inverse Scattering Method and Correlation Functions, ch. I.

Lecture 11 · The Lieb–Liniger gas

E. H. Lieb and W. Liniger, Phys. Rev. 130 (1963) 1605; C. N. Yang and C. P. Yang, J. Math. Phys. 10 (1969) 1115; M. Olshanii, Phys. Rev. Lett. 81 (1998) 938; T. C. Dorlas, Commun. Math. Phys. 154 (1993) 347; Korepin, Bogoliubov and Izergin, Quantum Inverse Scattering Method and Correlation Functions, §I.2; Sutherland, Beautiful Models, §§2.2, 4.2; Gaudin, The Bethe Wavefunction, §4.3.

Lecture 12 · Scattering without diffraction: the open Toda chain

J. Moser, in Dynamical Systems, Theory and Applications, Lecture Notes in Physics 38 (1975) 467; Toda, Theory of Nonlinear Lattices, §§2.1, 3.10; Babelon, Bernard and Talon, Introduction to Classical Integrable Systems, §4.8; Sutherland, Beautiful Models, §4.2.

Lecture 13 · Three-body consistency: the Yang–Baxter equation

C. N. Yang, Phys. Rev. Lett. 19 (1967) 1312 (labelled particles, and the equation); M. Gaudin, The Bethe Wavefunction, trans. J.-S. Caux, Cambridge University Press (2014), §10.1 (Coxeter’s presentation and the $N$-body ansatz); B. Sutherland, Beautiful Models, World Scientific (2004), Chapter 2 (non-diffraction); R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press (1982) (the equation on lattices); A. B. Zamolodchikov and Al. B. Zamolodchikov, Ann. Phys. 120 (1979) 253; L. D. Faddeev, Sov. Sci. Rev. C1 (1980) 107 (the exchange algebra).

Lecture 14 · The quantum inverse scattering method, and its classical limit

E. K. Sklyanin, L. A. Takhtajan and L. D. Faddeev, Theor. Math. Phys. 40 (1979) 688 (the method); L. D. Faddeev, How algebraic Bethe ansatz works for integrable model, Les Houches lectures 1995, arXiv:hep-th/9605187, §§1–3; V. E. Korepin, N. M. Bogoliubov and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press (1993), §§VI.1–VI.5, §VII.8 (the quantum determinant), §VIII.4 (higher spin); M. C. Gutzwiller, Ann. Phys. 133 (1981) 304 and E. K. Sklyanin, Lect. Notes Phys. 226 (1985) 196 (the quantum Toda chain); A. Molev, Yangians and Classical Lie Algebras, American Mathematical Society (2007), Chapter 1 (the Yangian and its centre); O. Babelon, D. Bernard and M. Talon, Introduction to Classical Integrable Systems, Cambridge University Press (2003), Ch. 2–3 (the classical Yang–Baxter equation) and §4.7 (the open chain’s $\rmat$). The conventions for the ordering of $RTT$, of the monodromy and for the sign in $\Rmat$ differ between these sources; see §14.2.

Lecture 15 · The algebraic Bethe ansatz

L. D. Faddeev, How algebraic Bethe ansatz works for integrable model, Les Houches lectures 1995, arXiv:hep-th/9605187, §§3–4 (the method, in the other ordering); V. E. Korepin, N. M. Bogoliubov and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press (1993), §§VII.1–VII.5 (the algebraic Bethe ansatz), chapter IX (scalar products), §§X.1–X.3 (norms); F. Franchini, An Introduction to Integrable Techniques for One-Dimensional Quantum Systems, Springer (2017), chapters 4 and 5 (the XXZ chain, the algebraic Bethe ansatz, Slavnov’s and Gaudin’s formulas); R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press (1982), chapter 8 (the six-vertex model) and §10.5 (the matrix $Q$), and Ann. Phys. 70 (1972) 193; M. Gaudin, The Bethe Wavefunction, trans. J.-S. Caux, Cambridge University Press (2014);

E. Mukhin, V. Tarasov and A. Varchenko, Commun. Math. Phys. 288 (2009) 1 (completeness); M. P. Grabowski and P. Mathieu, Ann. Phys. 243 (1995) 299 (the local charges). The ordering of $RTT$, the monodromy and the sign in $\Rmat$ differ between these sources; see §14.2 before transcribing an exchange relation.

Lecture 16 · The thermodynamic limit: densities and ground states

M. Takahashi, Thermodynamics of One-Dimensional Solvable Models, Cambridge University Press (1999), §3.3 (Hulthén’s solution); V. E. Korepin, N. M. Bogoliubov and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press (1993), §§I.2–I.3 (the counting, the thermodynamic limit, Lieb’s equation); F. Franchini, An Introduction to Integrable Techniques for One-Dimensional Quantum Systems, Springer (2017), chapter 2 and §§3.4–3.6, §4.4; E. H. Lieb and W. Liniger, Phys. Rev. 130 (1963) 1605; C. N. Yang and C. P. Yang, J. Math. Phys. 10 (1969) 1115; L. Hulthén, Ark. Mat. Astron. Fys. 26A (1938) 1. Finite size: H. W. J. Blöte, J. L. Cardy and M. P. Nightingale, Phys. Rev. Lett. 56 (1986) 742, and I. Affleck, ibid. 746.

Lecture 17 · Dressing, excitations and the thermodynamic Bethe ansatz

C. N. Yang and C. P. Yang, J. Math. Phys. 10 (1969) 1115; E. H. Lieb, Phys. Rev. 130 (1963) 1616 (Types I and II); M. Takahashi, Thermodynamics of One-Dimensional Solvable Models, Cambridge University Press (1999), chapters 3–5, 7 and 8; V. E. Korepin, N. M. Bogoliubov and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press (1993), §§I.4, I.6, I.8; F. Franchini, An Introduction to Integrable Techniques for One-Dimensional Quantum Systems, Springer (2017), §2.9; L. D. Faddeev and L. A. Takhtajan, Phys. Lett. A 85 (1981) 375 (the spinon); F. H. L. Essler, V. E. Korepin and K. Schoutens, J. Phys. A 25 (1992) 4115 (strings); A. H. van Amerongen et al., Phys. Rev. Lett. 100 (2008) 090402 (the experiment).

Lecture 18 · Free fermions: the transverse-field Ising chain

P. Jordan and E. Wigner, Z. Phys. 47 (1928) 631 (the map); E. Lieb, T. Schultz and D. Mattis, Ann. Phys. 16 (1961) 407 (free fermions, the Toeplitz determinant); P. Pfeuty, Ann. Phys. 57 (1970) 79 (the TFIM solved); F. Franchini, An Introduction to Integrable Techniques for One-Dimensional Quantum Systems, Springer (2017), Chapter 1 (the XY chain, in full); G. B. Mbeng, A. Russomanno and G. E. Santoro, The quantum Ising chain for beginners, arXiv:2009.09208 (the two parity sectors, carefully); H. A. Kramers and G. H. Wannier, Phys. Rev. 60 (1941) 252; A. Yu. Kitaev, Phys.-Usp. 44 (2001) 131; M. Girardeau, J. Math. Phys. 1 (1960) 516.

Lecture 19 · Why integrable systems do not thermalise

J. M. Deutsch, Phys. Rev. A 43 (1991) 2046, and M. Srednicki, Phys. Rev. E 50 (1994) 888 (eigenstate thermalisation); M. Rigol, V. Dunjko and M. Olshanii, Nature 452 (2008) 854; G. Biroli, C. Kollath and A. M. Läuchli, Phys. Rev. Lett. 105 (2010) 250401 (the rare states); M. Rigol, V. Dunjko, V. Yurovsky and M. Olshanii, Phys. Rev. Lett. 98 (2007) 050405 (the generalised Gibbs ensemble); E. Barouch, B. M. McCoy and M. Dresden, Phys. Rev. A 2 (1970) 1075; P. Calabrese, F. H. L. Essler and M. Fagotti, Phys. Rev. Lett. 106 (2011) 227203 (the Ising quench); F. H. L. Essler and M. Fagotti, J. Stat. Mech. (2016) 064002 (review); J.-S. Caux and F. H. L. Essler, Phys. Rev. Lett. 110 (2013) 257203; J.-S. Caux and J. Mossel, J. Stat. Mech. (2011) P02023 (counting charges).

Lecture 20 · Generalised hydrodynamics, and the quantum Newton’s cradle

O. A. Castro-Alvaredo, B. Doyon and T. Yoshimura, Phys. Rev. X 6 (2016) 041065, and B. Bertini, M. Collura, J. De Nardis and M. Fagotti, Phys. Rev. Lett. 117 (2016) 207201 (generalised hydrodynamics); B. Doyon, SciPost Phys. Lect. Notes 18 (2020) (hard rods, Example 3.4; the external force, §4.5) and B. Doyon and T. Yoshimura, SciPost Phys. 2 (2017) 014; C. Boldrighini, R. L. Dobrushin and Yu. M. Sukhov, J. Stat. Phys. 31 (1983) 577 (hard rods); E. Ilievski et al., Phys. Rev. Lett. 115 (2015) 157201 (quasi-local charges); J.-S. Caux, B. Doyon, J. Dubail, R. Konik and T. Yoshimura, SciPost Phys. 6 (2019) 070 (the cradle geometry); M. Schemmer, I. Bouchoule, B. Doyon and J. Dubail, Phys. Rev. Lett. 122 (2019) 090601, and N. Malvania et al., Science 373 (2021) 1129 (the experiments); I. E. Mazets, T. Schumm and J. Schmiedmayer, Phys. Rev. Lett. 100 (2008) 210403; B. Bertini, F. H. L. Essler, S. Groha and N. J. Robinson, Phys. Rev. Lett. 115 (2015) 180601 (prethermalisation); G. P. Brandino, J.-S. Caux and R. M. Konik, Phys. Rev. X 5 (2015) 041043.

Optional lecture A · Quantum chaos and level statistics

J.-S. Caux and J. Mossel, J. Stat. Mech. (2011) P02023; M. P. Grabowski and P. Mathieu, Mod. Phys. Lett. A 9 (1994) 2197 and J. Phys. A 28 (1995) 4777; V. Oganesyan and D. A. Huse, Phys. Rev. B 75 (2007) 155111 (the ratio); Y. Y. Atas, E. Bogomolny, O. Giraud and G. Roux, Phys. Rev. Lett. 110 (2013) 084101 ($\langle r\rangle$); M. V. Berry and M. Tabor, Proc. R. Soc. Lond. A 356 (1977) 375; O. Bohigas, M.-J. Giannoni and C. Schmit, Phys. Rev. Lett. 52 (1984) 1; D. Poilblanc, T. Ziman, J. Bellissard, F. Mila and G. Montambaux, Europhys. Lett. 22 (1993) 537; F. Finkel and A. González-López, Phys. Rev. B 72 (2005) 174411; J. C. Talstra and F. D. M. Haldane, J. Phys. A 28 (1995) 2369; D. A. Abanin, E. Altman, I. Bloch and M. Serbyn, Rev. Mod. Phys. 91 (2019) 021001; J. Šuntajs, J. Bonča, T. Prosen and L. Vidmar, Phys. Rev. E 102 (2020) 062144; R. J. Baxter, Exactly Solved Models in Statistical Mechanics, §10.14 (the XYZ chain). The readings’ sources are named in them.

Optional lecture B · Calogero–Moser, classical and quantum

J. Moser, Adv. Math. 16 (1975) 197; F. Calogero, J. Math. Phys. 12 (1971) 419; Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, §§3.1, 3.3; J. Avan and M. Talon, Phys. Lett. B 303 (1993) 33; D. Kazhdan, B. Kostant and S. Sternberg, Comm. Pure Appl. Math. 31 (1978) 481; C. F. Dunkl, Trans. Amer. Math. Soc. 311 (1989) 167; Reed and Simon II, §X.1. The readings name their own sources.

Optional lecture C · Nesting and the Hubbard model

C. N. Yang, Phys. Rev. Lett. 19 (1967) 1312; M. Gaudin, Phys. Lett. A 24 (1967) 55; E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20 (1968) 1445; Essler, Frahm, Göhmann, Klümper and Korepin, The One-Dimensional Hubbard Model (CUP 2005); B. S. Shastry, Phys. Rev. Lett. 56 (1986) 1529. The readings name their other sources.