Audience. This seminar is intended for students of Mathematical Sciences as well as students of Theoretical Physics. Although no prior courses are mandatory to participate, familiarity with algebra (groups, Lie algebras) will be useful. |
Examination. There is no exam. The grade will be composed as follows: hand-in exercises (pass/fail) 20% (you are allowed to drop one failed hand-in exercise, plus a second one in case you pass hand-in exercise 6 and solve the bonus from that exercise correctly), practice presentation 10%, presentation 50%, typed notes of each presentation (pdf and tex-files to hand in within two weeks of the presentation) 20%. |
Format. Lectures (attendance required). |
• Sept 12: BBG 007 | • Oct 03: BBG 007 | • Oct 24: BBG 007 | • Nov 14: MIN 016 | • Dec 05: MIN 025 | • Jan 09: BBG 308 | • Jan 30: BBG 005 |
• Sept 19: BBG 308 | • Oct 10: MIN 016 | • Oct 31: BBG 007 | • Nov 21: MIN 016 | • Dec 12: MIN 022 | • Jan 16: BBG 308 | |
• Sept 26: MIN 022 | • Oct 17: BBG 007 | • Nov 07: BBG 007 | • Nov 28: MIN 025 | • Dec 19: BBG 069 | • Jan 23: BBG 308 |
13-10 | Remarks added to 03-10, based on issues that were discussed during that presentation. |
15-10 | Update in the schedule of Part II. |
24-10 | Major update in the schedule of Part II. |
01-11 | Solution to hand-in exercise 3 added, further update in the schedule of Part II, and an outline of the program of Part III (January). |
12-11 | Venues for the remainder of seminar are known (see above). |
12-12 | Further update in the schedule; rules for the hand-in exercises specified (see above). |
Our main references for Part I are [JM]: Chapters 0–2, [GRS]: Chapter 1, [Bax]: Chapters 8–9. |
Description. The motivation behind the schedule for this part is the following. The goal of the seminar is to understand the algebraic Bethe Ansatz and its mathematical context. The algebraic Bethe Ansatz is quite abstract, so it is nice to obtain some important results and encounter a part of the formalism already before using a different method. In this part we will get acquainted with the relevant spin chain models that we want to solve, with the Bethe-Ansatz equations, and with the transfer matrix method. |
12-09 | Introduction and overview: integrability, and some background in statistical physics (Gleb Arutyunov) |
[GRS]: pp 1–5, [JM]: Chapter 0, [Bax]: Chapter 1 (main ideas) and Section 2.1 | |
19-09 | Magnetism and spin: the XXX model (André Henriques) |
[Fra]: Sections 3.1–3.2, [GRS]: p 10 | |
Description. Heisenberg introduced a family of lattice models for magnetism that are "homogeneous" (= translationally invariant) and only involve "nearest-neighbour" interactions. In one dimension the resulting models are often called "XYZ models". Denote the number of lattice sites by L. We first focus on the easiest Heisenberg model: the "isotropic" (= rotationally invariant) XXX model with local spins equal to 1/2. Since the operator measuring the total spin in any particular direction commutes with the Hamiltonian, we can restrict ourselves to the spectrum in the "M-particle sector" with fixed total spin ħ(L/2 − M) along the z-axis. | |
26-09 | Bethe's Ansatz: coordinate Bethe Ansatz, Bethe-Ansatz equations (Nick Plantz, notes) |
[Fra]: Sections 3.3–3.4, [GRS]: Section 1.2, [Fra]: Section 3.5 (see also [GRS]: Section 3.3.1) | |
Description. We study the spectrum of the XXX model (in the ferromagnetic regime, this is the physical spectrum). The case M = 0 is trivial, and M = 1 is easy. For M > 1 the coordinate Bethe Ansatz allows one to go further. We look at the procedure for M = 2 (which shows the main features of the general case; the computations are already a bit involved), and analyze the resulting spectrum in the two-particle sector. For general M the result is a set of coupled nonlinear equations, called the Bethe-Ansatz equations, which can be interpreted as expressing the remarkable phenomenon of factorized scattering. If time allows, we see that in the "thermodynamic limit" of infinite lattice size, the solutions to the Bethe-Ansatz equations arrange themselves into strings in the complex plane. | |
Remarks. For M = 2 it is not necessary to spell out all computational details on the blackboard (cf. the homework and the hand-in exercise for next week). Sections 3.3–3.4 of [Fra] are based on the exposition in [KM]; however, be careful not to spend too much of the presentation on that analysis! Section 3.5 of [Fra] closely follows Sections 5–6 of [Fad] (up to an irrelevant rescaling of the rapidities by a factor of 1/2). Some working knowledge of complex logarithms may come in handy; see e.g. the first few pages of [unk]. Since we cannot cover everything, unfortunately (at least for physicists) we will skip the analysis of the physical spectrum in the anti-ferromagnetic case. More about this can be found in [Fra]: Sections 3.5–3.8, [KM], [Fad]: Sections 5–7, [GRS]: Section 3.3. | |
Homework. To prepare for this week's lecture, read the short "popular-scientific" article [Bat]. To get some fluency in typical calculations use the su_{2} commutation relations to check that (in physics notation, where |↓…↓〉 stands for the lowest-weight state for the total spin in the z-direction) |k〉 ≔ S^{+}_{k} |↓…↓〉 and |k, l〉 ≔ S^{+}_{k} S^{+}_{l} |↓…↓〉 are orthonormal vectors, and compute the action of the XXX Hamiltonian on these basis vectors. | |
03-10 | Anisotropy: basics of the XXZ model (Erik van der Wurff, slides, notes) |
[JM]: Sections 1.1–1.2, [Fra]: Sections 4.1–4.3 (for simplicity take vanishing external magnetic field: h = 0), [KBI]: p 67 (the part about the Yang-Yang function; see also Theorem 2 and its proof in Section 1.2 to understand the argument completely) | |
Description. To familiarize ourselves further with quantum spin chains we consider the more general XXZ model, which involves an additional parameter describing its "anisotropy". It can again be tackled with the coordinate Bethe Ansatz, leading to Bethe-Ansatz equations, which again provide information about the physical spectrum of the model. Unlike for the XXX model, in the anisotropic case the existence and uniqueness of solutions to the Bethe-Ansatz equations (for fixed M, fixed integers, and solving for real rapidities only) can be proven for a range of values for the anisotropy parameter. | |
Remarks. As we have seen this week, the proof about the Yang-Yang function in [KBI] only establishes uniqueness of real solutions, but not their existence: a function can be convex but with the minimum "running away to infinity", as happens for the function S(λ) = e^{λ} on R. For a proof of the existence of real solutions to the Bethe-Ansatz equations see the original paper [YY]. A proof of the completeness of Bethe solutions for "generic" deformations (consisting of local "inhomogeneities" z_{m}, and of "twisted" boundary conditions μ arising in the presence of an external magnetic field) of the XXZ model can be found in [TV]; for more about the issue of completeness in the context of the XXZ model, see [Ba2]. | |
Hand-in exercise 1 (for 3-10). (i) Although perhaps a bit tedious, it is useful to go through the computational details of the coordinate Bethe Ansatz for the case M = 2 once yourself. (Solution. See e.g. Section 5.1 of [Aru], but be aware of the subtlety with the use of coefficients f(n_{1}, n_{2}) for n_{1} = n_{2} !) (ii) Read the first two pages of [LS] and try to relate that description to what we have seen in class last week. | |
10-10 | Ice: six-vertex model; transfer matrix method (Rob Klabbers, notes) |
[JM]: Chapter 2, [Bax]: Sections 8.1–8.7 (up to equation (8.7.5)), [GRS]: Section 1.3 | |
Description. The formalism that will be used for the algebraic Bethe Ansatz in particular involves transfer matrices. These concepts can be introduced already in a classical setting for the six-vertex (or ice-type) model, which is closely related to the XXZ model. | |
Remark. For an account of Chapters 6–9 of [Bax] that is oriented slightly more towards mathematicians, see Chapters 5–8 of [Tol] (but beware of the typos!). In particular, Sections 7.4 and 8.1–8.7 of [Tol] closely follow Sections 8.1–8.7 of [Bax]. | |
Homework. To prepare for this week's lecture, read [Bax]: Sections 2.1–2.2 (see also [Tol]: Section 5.1). | |
17-10 | Baxter's TQ-construction: an alternative to the Bethe Ansatz (Yassir Awwad, slides) |
[Bax]: Chapter 9, [Fra]: Section 5.3 (except for pp 90–91) | |
Description. This alternative method to obtain the Bethe Ansatz equations was used by Baxter to solve the XYZ (or, equivalently, eight-vertex) model. A discussion of the latter is beyond our scope; we study the method for the six-vertex model. An important ingredient for Baxter's method is the star-triangle relation, which will play a major role in the rest of this seminar under the name of Yang-Baxter equation. | |
Remarks. [Tol]: Sections 7.5–7.11 correspond to [Bax]: Sections 9.6–9.8 (cf. the remark above). The treatment of Baxter's TQ-construction in [Bax] is a bit old-fashioned, but it does not require a lot of prerequisites. Once we know more about quantum groups, in Part II of the seminar, we will come back to a more modern approach to this method. If you are interested in the XYZ model you can take a look at [Bax]: Chapter 10, [GRS]: Chapter 4. | |
Hand-in exercise 2 (for 17-10). Prove equation (2.5) in Section 2.2 of [JM] by fixing a ground state (sector) i ∈ {0, 1} and using perturbation theory in a, b. (Hint. To find all contributing configurations C at fixed d_{i}(C) it is convenient to use some graphical notation for the configurations; particularly nice is the three-colouring of the square lattice described in the introduction of Section 8.13 of [Bax]. Whatever graphical notation you use, do not forget to explain your notation and how you can read off the weight of the corresponding configuration!) |
Our main references for this part are [JM]: Chapter 3, [GRS]: Chapters 2–3 and 6. |
Description. This part revolves around the algebraic Bethe Ansatz and its mathematical foundation: the theory of quantum groups. In line with the historical development of the latter, we will study mathematical concepts such as Hopf algebras motivated by constructions that we encounter through the algebraic Bethe Ansatz. The relation between the more abstract mathematics and the theory of quantum integrability will be stressed throughout. |
24-10 | Integrability: more about the Yang-Baxter equation (Daniel Medina Rincón, slides, notes) |
[JM]: (end of Section 2.4,) Sections 3.1–3.3, [GRS]: Sections 2.1 (except for 2.1.4) and 2.2 | |
Description. In a sense, the algebraic Bethe Ansatz can be used to prove that a model is quantum integrable. Indeed, we have already come accross a remarkable feature of quantum integrable models: M-body scattering factorizes into two-body processes. However, it is not clear from the coordinate Bethe Ansatz why these models exhibit this feature. The algebraic Bethe Ansatz sheds light on this by producing, in addition to the momentum operator and the Hamiltonian, many more conserved quantities; these render the model quantum integrable. At the core of all this lies the famous Yang-Baxter equation. In addition, the formalism of the algebraic Bethe Ansatz also provides a deeper algebraic understanding of the relation between the XXZ and six-vertex models: they share the same R-matrix. | |
Homework. As with the coordinate Bethe Ansatz, the algebraic Bethe Ansatz involves permutations. It is useful to get fluent in the efficient cycle notation. This and other basics of the symmetric group (such as the way to write general permutations as a product of transpositions) can be found e.g in the short [Arm]: Chapter 6. | |
31-10 | Algebraic Bethe Ansatz: deriving all Bethe-Ansatz equations at once (Ben Werkhoven, slides, notes) |
[GRS]: Section 2.1.4, [Fad]: Sections 3–4, [Fra]: Sections 6.1–6.4, [KBI]: Section VII.1 and examples 2–4 of Section VII.3 | |
Description. After all our preliminaries we are finally ready for the main course: the algebraic Bethe Ansatz. These will allow us to derive the Bethe-Ansatz equations for all M-particle sectors, for the XXX spin chain, for deformations thereof involving "inhomogeneities" (= shifts in the spectral parameters of the local R-matrices), and for the XXZ spin chain. | |
Remark. Next to the transfer matrix method from statistical physics, the formalism of the algebraic Bethe Ansatz also owes to the theory of (Liouville) integrable models from classical mechanics. Indeed, there are classical analogues of R-matrices, Lax matrices, spectral parameters (rapidities), and the Yang-Baxter equation. Unfortunately we do not have time to go into any of this. If you are interested you can read about the theory of classical integrable systems e.g. in [Aru]: Sections 1–4, [KBI]: Chapter V, and the book [BBT]. | |
Hand-in excercise 3 (for 31-10). Exercise 2.7 of [GRS], pp 68–70. Along the way, also: spot the typo in (2.158) and fix it, prove (2.163), and show that the unitarity condition for the R-matrix implies that the first two terms on the right-hand side of (2.166) vanish. (Solution. Here is a scan of a solutions of most of the exercise: page 1, page 2.) | |
07-11 | Lie algebras: structure, presentations, and representations (André Henriques, scan of handwritten notes page 1, page 2, page 3) |
The material can be found in the books [Hum] and [FH], and partially also in [Tol]: Sections 1.1, 1.5, 1.6.1, 1.7 and 1.9 | |
Description. We review the basics of the theory of Lie algebras, mainly focussing on sl_{2}, which is the complexification of the spin algebra su_{2}. Although it is a rather ambitious, ideally the following notions (which we will need for the remainder of our seminar) are covered: Cartan subalgebra, root systems, triangular decomposition; presentation via Chevalley generators and Serre relations; Casimir operators, highest weight vectors (cf. pseudovacuum), enveloping algebras, Verma modules. | |
Remark. Although many courses at the UU have an exam this week, our class continues as usual (but without homework). | |
14-11 | Quantum algebra: Hopf algebras, the quantum universal enveloping algebra U_{q}(sl_{2}), and its representations (Alexandros Aerakis, slides, notes) |
[GRS]: Sections 2.3 (only the intro), 6.1 and 6.4.2 (skip quasi-triangularity), [Maj]: Section 1, [Kas]: Sections III.1–3, IV.2, VI.1–4, VII.1 and VII.6–7 | |
Description. The mathematical structure underlying the integrability of our spin chains and the algebraic Bethe Ansatz uses the language of Hopf algebras. Two weeks ago we have already encountered coproducts, so we essentially know what a coalgebra is. Now we have to learn some more abstract algebra: algebra + coalgebra = bialgebra, and bialgebra + extra structure = Hopf algebra. By studying the Yang-Baxter equation in the braid limit we find the Hopf algebra underlying the XXZ model: U_{q}(sl_{2}), the quantum enveloping algebra of sl_{2}. For the generic case (q not a root of unity) the representation theory of U_{q}(sl_{2}) is quite similar to that of sl_{2} from last week. | |
Remarks. In the mathematics literature there are many references about Hopf algebras and quantum groups. The above parts of [Kas] contain many details, and you should try not to get lost in all the details; for instance, you can skip the proofs on a first reading to understand the main line of thought, and then destill those things that you need. For this purpose, [Tol] Sections 2.1–2.5 may come in handy, since that is essentially a shorter account (but beware of the typos!) of the above parts of [Kas]. Further background about representations (= modules) can be found in [Kas]: Sections III.5–6, V.4–5, and VII.2. For more details you can also take a look at [Ma2]: Sections 1.1–1.5 and 3.2 (again skip quasi-triangularity), or at these lectures. | |
Homework. To prepare for this week's lecture try to read this blog entry and the quite chatty introduction [DB+]: Sections 2–6 (you can skip examples involving group algebras, as well as Section 3.2). | |
21-11 | R-matrices and the quantum double: quasi-triangular Hopf algebras, duality, and the quantum double construction (Flore Kunst, notes) |
[GRS]: Sections 2.3.1, 6.2–6.4, [Maj]: Section 5 and 8, [Kas]: V.7, VII.4–5, Sections VIII.1–5 and IX.5–7 | |
Description. The next important ingredient for the algebraic structure of the integrability of the spin chains comes from the Yang-Baxter equation, which in turn crucially involves R-matrices. In the language of Hopf algebras these are encoded in an additional layer, the quasi-triangular structure, that a Hopf algebra may (and for us does) have. We take a closer look at duality, which is an important concept in the theory of Hopf algebras. If time allows, we also discuss a somewhat technical but nice construction by Dinfel'd, which takes a Hopf algebra together with its dual and produces from this a quasi-triangular structure on that Hopf algebra. | |
Remarks. See also the remarks above; [Tol] Section 2.6 correspond to [Kas]: Sections VIII.1–3. For more details you can also take a look at [Ma2]: Sections 2.1–2.2 and 3.2, or at these lectures. | |
Hand-in exercise 4 (for 21-11). | |
28-11 | Yangians: Y(gl_{2}), the quantum determinant, and back to the XXX model (Stefano Lucat , notes) |
[Mol]: Sections 2.1–2.9 (it suffices to consider n = 2), [MNO]: Sections 0–2 (again, you can take N = 2), [Ber]: Section 2, [Mac]: Section 2 (again, you may take g = gl_{2}), [MTV]: Sections 1–3 | |
Description. The symmetry algebra of the XXX spin chain is an infinite-dimensional quasi-triangular Hopf algebra called the Yangian of gl_{2}. | |
Homework. Review what you know about Hopf algebras by going back to this blog entry and [DB+]: Sections 2–6 (again you can skip Section 3.2, but this time try to understand the group algebra for another nice example of a Hopf algebra). | |
05-12 | Rapidities: affine Lie algebras, quantum affine sl_{2}, and back to the XXZ model (Bram van Dijk, notes) |
[Tol]: Sections 1.2–1.4 and 3.1–3.2, [GRS]: Sections 2.3.2, 6.10 (only for G_{l} = sl_{2}), [JM]: Sections 3.4–3.7 | |
Description. Similar to what we saw last week, the algebraic structure underlying the integrability of the XXZ spin chain is an infinite-dimensional version of U_{q}(sl_{2}) which also includes the rapidities: quantum affine sl_{2}. To understand its construction we start by reviewing loop algebras and their central extensions. | |
Hand-in exercise 5 (for 12-12). | |
12-12 | Higher spin: descendant procedure, algebraic Bethe Ansatz (Mpampis Lazaridis-Patsalias, notes) |
[GRS]: Sections 2.4, 3.4 (only the intro), 3.7 (only p 98), [Fad]: Section 8 (up to equation (275)) | |
Description. So far at the lattice sites of our spin chains we have only considered spin-1/2 representations. This week we see how the spin chains with higher local spins can be treated. In particular we see one of the big achievements of the algebraic Bethe Ansatz: the construction of integrable Hamiltonians for e.g. the spin-1 XXX model. | |
19-12 | Baxter's TQ-construction revisited: a more algebraic perspective (Aggelos Tzetzias) |
[FT]: Sections 4–5 (only applied to the six-vertex/XXZ case), [Kor] (see also the other papers of Korff mentioned at the references) | |
Description. Now that we know more about quantum algebra, we come back to Baxter's TQ-construction to understand it in more algebraic terms. The notation from Chapter 9 of [Bax] is quite old fashioned; [FT] contains a more modern account using the formalism of the algebraic Bethe Ansatz, which shows the idea behind Baxter's procedure more clearly. (Although [FT] discuss the eight-vertex/XYZ model, the logic is the same for the six-vertex/XXZ model in which we are interested.) We also take a still more algebraic perspective, relating the method to the representation theory of quantum-affine sl_{2}. | |
Description. In the last part of the seminar we head in a slightly different direction: we will see how integrability can play a role in field theory (in 1+1 dimension), and how such integrable field theories are related to conformal field theory. |
09-01 | Conformal field theory (André Henriques) |
[BPZ] | |
Description. This week we recap conformal field theory: primary fields and conformal weights, stress-energy tensor, OPE, normal ordering, vertex operators; implications of conformal invariance for n-point correlators (n = 2, 3, 4, ...). | |
Hand-in exercise 6 (for 09-01). | |
16-01 | Loop algebras and central extensions (Hamish Forbes, notes) |
[KM+]: Sections 3, 16–17, 24 | |
Description. In part II of the seminar we encountered quantum affine sl_{2} through a construction involving an extension of the Cartan matrix associated with the Lie algebra. There is an alternative construction involving loop algebras and their central extensions. After using this construction to recover quantum affine sl_{2} in a different way and seeing how that is related to the approach via Cartan matrices, we apply this construction to get the Virasoro algebra as a central extension of the Witt algebra. (You can also take a look at Section 7 for the relation to previous week, and at Section 25.2 and the start of Section 26 for examples of the general construction from Sections 16–17.) | |
23-01 | Integrable field theory (Jorgos Papadomanolakis) |
[Dor]: Sections 1–5, [Mac]: Section 3 | |
Description. Integrability in field theory: sine-Gordon model, factorized scattering, Lax formalism, Yangian symmetry. If possible, present the Korteweg-De Vries model as a second example: that one will reappear next week. | |
30-01 | Integrability and CFT (Gleb Arutyunov) |
[BLZ] | |
Description. The integrable structure of CFT. | |
Remark. Although many courses at the UU have an exam this week, our class continues this week (but without homework). (It is possible that the date has to be slightly shifted; in that case we will find a suitable alternative date together.) |