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From the cylinder to the plane
A signicant part of this thesis consists in determining the characteristics of the continuum limit of a model from the hints it leaves in nite size. In particular, upon the hypothesis of conformal invariance, part of the CFT data can be read o from the asymptotic expansion in L of the spectrum of the Hamiltonian or the transfer matrix in size L. In this subsection we remind the reader of the origin of this important link between the discrete quantum or statistical system and its continuum limit, emphasizing the role of the dierent limits.
To that end, we consider a transfer matrix TL on L sites with D degrees of freedom per site that we assume for simplicity to be diagonalizable with orthogonal eigenvectors jii with eigenvalues i;L, and L a given observable in size L. These objects are thus DL DL matrices. On an L M lattice with periodic boundary conditions, the correlation function of L separated by Y rows is denoted G L;M(Y ) and is G L;M(Y ) = tr(TMYL LTY L L) tr(TM L ).
Perturbation by irrelevant operators
In a nite size system, there are clearly always higher-order corrections to equation (2.106). These corrections can be described in an eective way by perturbing the Hamiltonian in the continuum limit by irrelevant operators with a coupling constant g, that depends thus on L. The model in nite size is described in a continuous way by perturbing the limit theory in innite size. Let us consider a Hamiltonian H0 that describes the system in the continuum limit, and we perturb it by a eld V: H = H0 + g 2d Z 2d 0 V(x; t)dx.
A basic application of conformal invariance
For sake of completeness, in this subsection we give a basic example of application of conformal invariance, whose explanation will be useful in the next chapter.
We consider the completely packed O(N) loop model on a square lattice of size (LK) (LK), and ask what is the large-x behaviour of the probability Gp(0; x) that two points separated by X = xL are visited by the same loop in the limit L ! 1;K ! 1 6.
To that end, we use the hypothesis of conformal invariance to map the problem on the geometry of the cylinder, better suited for a transfer matrix approach. On the cylinder, the probability that two strands are propagating without forming a loop on a certain distance Y can be written with a similar equation to (2.100).
The spectrum from the spin chain
In the remainder of this section we provide two alternative means of deriving (3.73), or at least special cases thereof. The rst of these relies on the Bethe-ansatz diagonalization of the spin chain Hamiltonian, and the other on the computation of three-point functions|either directly, or using a trick reminiscent of Wick’s theorem. While both of these methods are of independent relevance, the reader interested mainly in results for other models may chose to skip directly to section 3.3. Let us rst describe the corresponding osp(1j2) spin chain. Like the sigma model which involves as a basic degree of freedom a eld in the vector representation of the algebra, the spin chain involves a tensor product of fundamental representations of osp(1j2). In contrast with the other superalgebras we will encounter in this paper, osp(1j2) has a simple representation theory. Its action on the spin chain is fully reducible, and the Hilbert space decomposes onto a direct sum of ‘spin j’ representations, with dimension 4j + 1. Here j is the eigenvalue of the Jz generator on the highest weight state, and j = 1=2 corresponds to the fundamental.
Relation with 3-point functions
We discuss in this section how the logarithmic corrections can be related to the 3-point functions on the plane. Our calculation parallels the work by Cardy for quasi-primary elds [92], but applies here to the logarithmic case.
The derivation in Section 2.4.3 of the relation between the logarithmic correction and the three point function does not apply to the logarithmic cases discussed above. Indeed, 1 and 2 are not quasi-primary: their correlation involves log that is not a scale covariant function, and the three point function is not constrained as before in Section 2.4.3. It thus needs a more detailed study. The purpose of the subsequent sections is to study the link between the correction to the energy levels for the states j1@1:::@m1i and the three-point function between them and the perturbative potential. Denote 1 m(z) the eld 1(z)@1(z):::@m1(z) and 2 m(z) the eld @m2(z):::@2(z)2(z).
They have scaling dimensions m(m+1)=2. The corresponding state j1 mi is given by the constant coecient in 1 m(z), and is / 1 0 1 1::: 1 m. The state h2 mj is dened as the conjugate of j1 mi, thus / 2 m::: 2 11 0. It is given by the coecient log jzj2zm(m+1) in 2 m(z), denoted.
Table of contents :
1 Introduction
2 Bethe ansatz and related aspects
2.1 Bethe ansatz
2.1.1 Notations and denitions
2.1.2 The ansatz
2.1.3 A word on numerically solving the Bethe equations
2.1.4 Admissible and non-admissible solutions
2.2 An additional TQ relation
2.2.1 Polynomiality of the other solution to the TQ relation
2.2.2 Polynomiality of P() and constructability of the Bethe state
2.2.3 An additional TQ relation
2.3 Geometrical models
2.3.1 Loop models
2.3.2 The Potts model
2.4 Identifying the continuum limit
2.4.1 Conformal eld theory
2.4.2 From the cylinder to the plane
2.4.3 Perturbation by irrelevant operators
2.4.4 A basic application of conformal invariance
3 Logarithms in a non-unitary spin chain
3.1 Introduction
3.1.1 Overview
3.1.2 Denitions
3.2 OSp(1j2)
3.2.1 The spectrum from eld theory
3.2.2 The spectrum from the spin chain
3.2.3 Relation with 3-point functions
3.3 OSp(2j2)
3.3.1 The spectrum from eld theory
3.3.2 The spectrum from the spin chain
3.4 OSp(3j2)
3.4.1 The spectrum from the spin chain
3.5 Physical properties of fully packed trails
3.5.1 A model for loops with crossings
3.5.2 Inclusion of osp spectra
3.5.3 Charges and loop congurations
3.5.4 Transfer matrix eigenvalues and loop congurations
3.5.5 Watermelon 2-point functions for loops with crossings
3.5.6 Away from integrability
4 Excitation spectrum computation
4.1 Introduction
4.1.1 Historical review
4.1.2 Euler-MacLaurin formula
4.1.3 An introductory example: the free fermions
4.2 A glance at the interacting case: revisiting the free fermions
4.3 Finite-size corrections in the interacting case
4.3.1 Presentation
4.3.2 Properties of the distribution S
4.3.3 Computing the shift I
4.3.4 The momentum
4.3.5 The energy
4.4 Strings
4.4.1 Denitions and notations
4.4.2 Riemann sums with logarithmic singularities
4.4.3 Computing the shifts Iq
4.4.4 The energy
4.5 Logarithmic corrections
4.5.1 Presentation
4.5.2 Computing B1
4.5.3 Logarithmic corrections to the energy
4.5.4 Examples and numerical checks
5 Series expansions and magnetic eld inuence
5.1 Introduction
5.2 The XXZ spin chain in a magnetic eld
5.2.1 Historical review
5.2.2 The Bethe equations as a recurrence relation
5.2.3 Example: the Heisenberg spin chain
5.2.4 Example: the XXZ spin chain
5.2.5 Radius of convergence of the series ( p hc h)
5.2.6 An example with complex roots
5.2.7 Finite-size corrections and critical exponents
5.3 Some intermediate results on a non-compact spin chain
5.3.1 Motivations
5.3.2 A dual recurrence relation
5.3.3 Analytic continuation at m = 1
5.3.4 Perspectives
5.4 Fluctuations inside the arctic curve in the interacting six-vertex model
5.4.1 Presentation
5.4.2 The free energy F(x; y)
5.4.3 The free fermion case = 0
5.4.4 Value of K(mx;my) in the interacting case
5.4.5 Numerical checks
A
A.1 Change of grading
A.2 Proof of Lemma 4
A.3 Proof of Lemma 6
A.4 Proof of Lemma 7
A.5 Proof of Lemma 8
