Algebraic Bethe Ansatz for the six-vertex model

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Exactly solvable models for polymers and the IQHE plateau transition

As announced in the previous chapter, both the problems of calculating correlation func-tions of two-dimensional polymers and studying transport observables at the Quantum Hall plateau transitions, as well as many other two-dimensional statistical mechanics or one-dimensional quantum mechanics problems, can be reformulated in the framework of loop or vertex models on a bidimensional lattice.
The rst section of the present chapter aims at introducing the tools and concepts related to the physics of integrable, exactly solvable lattice models. For pedagogical reasons the presentation is mostly based on the well-known six-vertex model, from which we introduce the notions of quantum integrability, Yang{Baxter equation and quantum groups. In section 2.2, we introduce in detail the notion of loop model as well as the related algebraic concepts, and work out the precise relationship between the six-vertex model and the related fully packed loop model on the square lattice. The mappings from the problem of polymer collapse and the computation of transport observables at the IQHE transition to integrable loop and vertex models are then detailed in sections 2.3 and 2.4 respectively.

Introduction to integrability and exactly solvable models

This presentation is very much inspired from the lectures of H. Saleur and J-B. Zuber [66], which we recommend for more details and references on the subject. We also discuss the connection with integrable quantum eld theories and factorizable S-matrices, and refer for this matter to the original work of Zamolodchikov and Zamolodchikov [67] and the more recent review [68].

Vertex models, integrability, and the Yang{Baxter equa-tion

Let us rst give a proper de nition of what a vertex model is: the degrees of freedom are attached to the links of a given lattice, which for us will always be the two-dimensional square lattice, and interactions take place at the vertices as each con guration of four links incident to a vertex is assigned a Boltzmann weight.
A con guration of the system living on a lattice of size L M with some given prescription on the boundary conditions is speci ed by the corresponding values of the degrees of freedom f ig on every edge, and the associated Boltzmann weight is the product of all corresponding vertex contributions. The degrees of freedom i may be of a rather general nature: in the cases we will be dealing with, they will be vectors in some given complex vector space (more precisely, in representations of some q-deformed (super)algebras, a notion which will be introduced further).
The toy-model we will be using throughout this section is Lieb’s six-vertex model (1967), in which the degrees of freedom live in the vector space C2 whose basis fj « i; j #ig can be represented as arrows on the links of the square lattice. The possible con gurations of arrows are restricted by demanding that at each vertex the number of incoming arrows equals the number of outgoing ones (ice rule), restricting to 6 the number of admissible vertex con gurations, Moreover requiring that these weights be invariant under a global reversal of all arrows restricts to 3 the number of independent weights, w1 = w2 a, w3 = w4 b and w5 = w6 c. Such a model was rst introduced for describing crystal lattices with hydrogen bonds such as ice (the position of the hydrogen atom on each bond, closer to one or the other of the lattice atoms, is indicated by the corresponding arrow, and the ice rules indicates that each of the lattice atoms has a valence number equal to two), but is of central importance in the world of spin chains, loop models and quantum integrability.
Going back to the general case, it will turn convenient to temporarily assume that the degrees of freedom on horizontal and vertical edges can be of di erent nature, and we therefore call V and Va the nite-dimensional vector space of degrees of freedom living on each vertical (resp. horizontal) link. A lattice of horizontal width L sites can be viewed in the Hamiltonian picture as a one-dimensional quantum system of Hilbert space H = V L evolving in discrete imaginary time. The discrete time evolution corresponds to the addition of one horizontal row, and the corresponding matrix elements between the state f 1; 2; : : : Lg at time t and the state f 10; 20; : : : L0g at time t + 1 is encoded by the so-called row-to-row transfer matrix where the labels aj denote the states on the horizontal links, and must accordingly be traced over in accordance with the boundary conditions. In the graphical representation we have pictured the space Va by a double line, to insist on its a priori di erent nature than that of V . The horizontal A Va space is refered to as the auxilliary space, whereas H V L is refered to as the quantum space. Choosing for instance periodic boundary conditions in the horizontal direction imposes a0 = aL where the sums are over some basis of Va. The matrix T in (2.3), which acts on Va V L, is called the monodromy matrix, and we can for instance express the partition function of the L M system with periodic boundary conditions in both directions as ZL M = TrV L T M : (2.4)

Integrable vertex models, the R-matrix and the Yang{Baxter equation

We are now ready to turn to the de nition of integrability for such a system. Quite naturally, we associate it with the existence of an in nite set of conserved quantities (although in practice only dim H independent of them are needed; we note however that this de nition actually allows for some loopholes, and alternative de nitions can be proposed, see [69]), or in other terms, of quantities commuting with the transfer matrix. This can be realized quite easily if we actually seek for a one-parameter family of Boltzmann weights w(a; ; 0; a0ju), hence a one-parameter family of transfer matrices T (u), such that [T (u); T (v)] = 0 ; (2.5) for any u and v. Indeed, if this is the case, each coe cient in the Laurent expansion of T (u) furnishes one observable commuting with T (u) for any u, moreover these observables are mutually commuting. The physical interpretation of the parameter u, which is called the spectral parameter, will be discussed shortly. One su cient condition for (2.5) to hold is the existence of a Va Va ! Va Va two-parameter dependent matrix R(u; v) such that the following ‘RTT relation’ holds : (2.6) R(u; v) (T (u) T (v)) = (T (v) T (u)) R(u; v) ; where the tensor product denotes a matrix multiplication over the quantum space, and the dot denotes a matrix multiplication over the product of the two auxilliary spaces.
De ning the braiding matrix R = R, where is the permutation operator of the two auxilliary spaces in Va Va, we have the following graphical representation of (2.6)
It can be shown that the R-matrix satisfying this ‘RTT’ requirement is subject to a consistency condition, which is ful lled if we take R to satisfy the Yang{Baxter equation, R23(v; w)R13(u; w)R12(u; v) = R12(u; v)R13(u; w)R23(v; w) : (2.8)
This equation, which of course holds in a similar form for R, is de ned in the tensor product Va Va Va, and for instance the notation R23 indicates the product of the identity operator acting on the rst space times the braiding of the second and third spaces. It allows for the following simple pictorial representation
In most cases of physical interest, the dependence of R(u; v) in the spectral parameters will be through the di erence u v. It is usual to require the unitarity condition on the R-matrix, R(u)R( u) / 1 ; (2.9) in particular all the solutions we will get to consider are regular solutions, in the sense that they satisfy R(0) = 1, R(0) = P.
We can now go back to the case where the vertical and horizontal spaces are of the same nature, V = Va : assuming that a solution of the Yang Baxter has been found, the RTT relation is readily solved provided the Boltzmann weights
For u = 2 one has a = b, and more generally the weights of the model are in this case isotropic, namely, invariant under discrete rotations of the lattice. The spectral parameter u is thus understood as introducing an anisotropy, which can be thought of as a spatial anisotropy of the lattice and as such is not supposed to a ect the critical behaviour of the model, if any. This statement should however be made more precise, since for instance at a real value of the geometrical interpretation does not hold anymore for complex values of u. It turns out that for the six-vertex model the phase structure can be described in terms of the parameter = a2 +b2 c2 = cos : for1 or < 1, that is for imaginary 2ab the system is in a non-critical, massive phase, whereas for 1< 1, that is for real non zero the system is in a critical, massless phase. In the latter case, the critical behaviour does not depend on the spectral parameter if real, whereas complex values of u perturb the system to a massive phase.
These observations can be related to the more general statement : there are, in general (and forgetting about some higher-genus solutions in some cases), three types of regular solutions of the Yang{Baxter equation. These are said to be rational, trigonometric (or hyperbolic) or elliptic according to their dependence in the spectral parameter, and all the trigonometric solutions found to this date have turned out to be critical. The solution (2.13) for the six-vertex model is trigonometric for generic real , rational for ! 0 and ! , and hyperbolic for imaginary . The eight-vertex model [36] gives in turn an example of elliptic solution of the Yang{Baxter equation. From now on, and unless speci ed, we will therefore restrict to real values of the spectral parameters, keep in mind the geometrical interpretation and use indistinctly the following graphical notations
We close this section with a word on 1+1-dimensional integrable quantum eld theories (IQFTs), and how their study is related to the notions we have just encountered, namely the Yang{Baxter equation, and ultimately, quantum spin chains. As announced in the beginning of the section, this follows the approach of Zamolodchikov and Zamolodchikov in [67].
Consider a 1+1-dimensional quantum theory of relativistic particles of masses ma1 ; : : : maN 1 , whose on-shell light-cone momenta are parametrized by pa = pa0 + pa1 = mae a ;pa = pa0 pa1 = mae a : (2.19)
The real parameters a are the so-called rapidities, and we assume that these actually denote narrow momenta wavepackets such that each particle can be assigned an ap-proximate position at any time. Assuming for simplicity that the theory is massive (in the case of a CFT we consider an in nitesimal massive deformation) and therefore that the interactions are nite-ranged, one can view the N-particle states as a collection of free particles when these do not overlap. In particular, one can de ne the in- and out-asymptotic states jAa1 ( 1) : : : AaN ( N )iin;out ; (2.20) characterized respectively by there being no further interactions at t ! and +1 (the symbol Aai ( i) denotes a particle of type ai with rapidity i). In the former case the particles’ rapidities must be increasing from left to right, whereas in the latter case they must be decreasing, so both states can be represented by thinking of the symbols Aai ( i) as non-commuting operators written in the same order as the corresponding particles.
Such operators are mutually commuting, and we will now see that imposing an in nite set of such conserved charges (the so-called conserved local charges) puts severe constraints on the S-matrix, and how these constraints are so particular in 1+1-dimensional theories.
The rst constraint imposed by the existence of local charges for arbitrarily large values of s is an in nite number of conservation laws of the form q (s) e 1 + : : : q (s) e M = q (s) 0 + : : : q (s) 0 (2.26) a1 aM b1 e 1 bN e N between the in and out momenta of any scattering process, implying the equality between the initial and nal sets of momenta, and in particular, forbidding particle production of annihilation.
The second constraint comes from the fact that for any scattering process we can use the conserved charges to reorder the incoming or outgoing particles at will. Assume indeed, for a sketchy demonstration [68], that the set of charges Ps (P1)s are conserved, where P1 is the spatial part of the energy-momentum. Acting with ei Ps on a particle of momentum pa yields a phase factor ps, and so amounts to shift the position of this particle by s ps 1. In more than one spatial dimensions the particles of any scattering process can therefore be translated so that they simply avoid each other without changing the corresponding scattering amplitude, which forces the S-matrix to be trivial, or, in other terms, the theory to be non interacting (this is a very unrigorous justi cation of the Coleman-Mandula theorem in three spatial dimensions). This however does not apply to 1+1 dimensions where obviously interactions cannot be avoided, yet these are severly constrained: consider for instance the 3 ! 3 scattering process between the in state Aa1 ( 1)Aa2 ( 2)Aa3 ( 3) ( 1 > 2 > 3) and the out state Aa3 ( 3)Aa2 ( 2)Aa1 ( 1). Shifting the in particles as described above shows that the two following processes are equivalent, from which we can deduce that the 3 ! 3 S-matrix, and similarly any N ! N, can be factorized as the product of 2 ! 2 scattering processes. Even once factorizability of the S-matrix is assumed, the equality between the two above diagrams is not trivial as soon as the 2 ! 2 S-matrix is not entirely diagonal in the space of di erent possible kinds of particles (or if these particles have, say, a non zero spin). Assuming (as imposed from Lorentz invariance) that the latter depends on the rapidities only through their di erence, this equality translates into the following matrix equation S23( 3 2)S13( 3 1)S12( 2 1) = S12( 2 1)S13( 3 1)S23( 3 2) ; (2.27) which is nothing but the Yang{Baxter equation (2.8) for the 2 ! 2 S-matrix, where the rapidities i play the role of the spectral parameters. The connection with integrable vertex models is now quite clear, for instance the integrable R matrix found for the six-vertex model (or, equivalently, for the XXZ Hamiltonian) has its equivalent in quantum eld theory as the S-matrix of the deformed SU(2) Nambu-Jona-Lasinio model (also called deformed chiral Gross-Neveu model), which is a 1+1 dimensional relativistic theory of Dirac fermions with action.

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The quantum group Uq(sl2)

The integrable R-matrix found for the six-vertex model belongs to the class of quasi-classical solutions of the Yang{Baxter equation, in the sense that it can be expanded in terms of some ~ (= here) parameter as follows R(~; u) / 1 + ~ r(u) + O(~2) : (2.30)
The matrix r(u) is called the classical limit of R( ; u) and satis es the so-called classical Yang{Baxter equation [r12(u); r13(u + v)] + [r12(u); r23(v)] + [r13(u + v); r23(v)] = 0 ; (2.31) which appears in the context of classical integrable systems, and has for characteristic feature that it involves only commutation relations. As a consequence, if r(u) can be decomposed as r(u) = r (u)X X (2.32) over a basis of generators fX g of some Lie algebra represented in End(V ) (with V = C2 here), we now have the very strong result that any representation of the generators fX g in some vector space V 0 yields a solution of the classical Yang{Baxter equation in V 0 V 0 V 0. For instance, the solution r(u) found for the six-vertex case can be decomposed over the products of spin 12 generators of the algebra sl2, hence any higher spin j representation of sl2 yields a solution of the classical Yang{Baxter equation in C2j+1 C2j+1 C2j+1.
From sl2 to the quantum group Uq(sl2) It is now natural to look for a similar construction for the R( ; u) matrix, or in other terms, to wonder whether the classical R-matrices we have just built may be the classical limit of some new solutions R( ; u) of the Yang{Baxter equation.

Table of contents :

1 Introduction to critical phenomena, polymer collapse and the Integer Quantum Hall Effect 
1.1 Universality in macroscopic systems, and conformal eld theory
1.2 Polymer chains
1.3 The Integer Quantum Hall Eect
1.4 A step towards exact solutions : lattice models
1.5 Non-compactness
1.6 Plan of the rest of the manuscript
2 Exactly solvable models for polymers and the IQHE plateau transition 
2.1 Introduction to integrability and exactly solvable models
2.1.1 Vertex models, integrability, and the Yang{Baxter equation
2.1.2 The quantum group Uq(sl2)
2.1.3 General framework
2.2 Loop models and algebraic aspects
2.2.1 Loop formulation of the six-vertex model and the Temperley{Lieb algebra
2.2.2 Representation theory of the TL algebra
2.2.3 Periodic loop and vertex models
2.2.4 Schur-Weyl duality
2.3 From polymers to the O(n) model, to the a(2) 2 model
2.4 From the IQHE to the b(1) 2 integrable model
2.4.1 The Chalker{Coddington model
2.4.2 A loop model for transport observables
2.4.3 The truncation procedure
2.4.4 Dense and dilute two-colour loop models
2.4.5 The a(2) 3 and b(1) 2 integrable chains
3 From exactly solvable models to non-compact CFTs
3.1 The algebraic Bethe Ansatz
3.1.1 Algebraic Bethe Ansatz for the six-vertex model
3.1.2 Algebraic Bethe ansatz for higher rank models
3.2 Continuum limit of the a(2) 2 model
3.2.1 Continuum limit in regimes I and II
3.2.2 First observations in regime III
3.2.3 The black hole CFT
3.2.4 The density (s) and the discrete states
3.2.5 The parafermions
3.2.6 Conclusion: the continuum limit of the a(2) 2 model
3.3 Continuum limit of the a(2) 3 model
3.3.1 Continuum limit in regimes I and II
3.3.2 Continuum limit in regime III
3.4 General solution of the a(2) n chains
3.4.1 Duality arguments for a non compact regime
3.4.2 The regimes I and II
3.4.3 The regimes III
3.5 The non compact world
3.5.1 A look at the Bethe ansatz kernels
3.5.2 The search for non compact degrees of freedom
3.5.3 Non compact theories in practice
4 Polymer collapse and non compact degrees of freedom 
4.1 The VISAW phase diagram
4.2 Physical properties at the non compact point
4.2.1 Grand-canonical Monte-Carlo simulations
4.2.2 The and exponents
4.2.3 Discrete states and polymer attraction
4.3 Probing the non compact degrees of freedom
4.3.1 Relationship between the black hole and dense phase
4.3.2 Probing degrees of freedom
4.3.3 A model for K KBN
4.4 The collapse transition
5 Application to the Integer Quantum Hall Effect 
5.1 Review of the results by Ikhlef et. al
5.2 Continuum limit of the b(1) n model
5.2.1 Bethe ansatz solution in regime IV
5.2.2 b(1) 2 , a(2) 3 , and Pohlmeyer reduction
5.2.3 Critical exponents of the truncated model
5.3 Continuous exponents at the IQHE transitions
5.3.1 GL(2j2) interpretation of point-contact conductances
5.3.2 Loop interpretation
5.3.3 Pure scaling observables


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