45 research outputs found
Symmetry classes of alternating sign matrices in the nineteen-vertex model
The nineteen-vertex model on a periodic lattice with an anti-diagonal twist
is investigated. Its inhomogeneous transfer matrix is shown to have a simple
eigenvalue, with the corresponding eigenstate displaying intriguing
combinatorial features. Similar results were previously found for the same
model with a diagonal twist. The eigenstate for the anti-diagonal twist is
explicitly constructed using the quantum separation of variables technique. A
number of sum rules and special components are computed and expressed in terms
of Kuperberg's determinants for partition functions of the inhomogeneous
six-vertex model. The computations of some components of the special eigenstate
for the diagonal twist are also presented. In the homogeneous limit, the
special eigenstates become eigenvectors of the Hamiltonians of the integrable
spin-one XXZ chain with twisted boundary conditions. Their sum rules and
special components for both twists are expressed in terms of generating
functions arising in the weighted enumeration of various symmetry classes of
alternating sign matrices (ASMs). These include half-turn symmetric ASMs,
quarter-turn symmetric ASMs, vertically symmetric ASMs, vertically and
horizontally perverse ASMs and double U-turn ASMs. As side results, new
determinant and pfaffian formulas for the weighted enumeration of various
symmetry classes of alternating sign matrices are obtained.Comment: 61 pages, 13 figure
Fusion hierarchies, -systems and -systems for the dilute loop models
The fusion hierarchy, -system and -system of functional equations are
the key to integrability for 2d lattice models. We derive these equations for
the generic dilute loop models. The fused transfer matrices are
associated with nodes of the infinite dominant integral weight lattice of
. For generic values of the crossing parameter , the -
and -systems do not truncate. For the case
rational so that
is a root of unity, we find explicit closure
relations and derive closed finite - and -systems. The TBA diagrams of
the -systems and associated Thermodynamic Bethe Ansatz (TBA) integral
equations are not of simple Dynkin type. They involve nodes if is
even and nodes if is odd and are related to the TBA diagrams of
models at roots of unity by a folding which originates
from the addition of crossing symmetry. In an appropriate regime, the known
central charges are . Prototypical examples of the
loop models, at roots of unity, include critical dense polymers
with central charge , and loop
fugacity and critical site percolation on the triangular lattice
with , and . Solving
the TBA equations for the conformal data will determine whether these models
lie in the same universality classes as their counterparts. More
specifically, it will confirm the extent to which bond and site percolation lie
in the same universality class as logarithmic conformal field theories.Comment: 34 page
Boundary algebras and Kac modules for logarithmic minimal models
Virasoro Kac modules were initially introduced indirectly as representations
whose characters arise in the continuum scaling limits of certain transfer
matrices in logarithmic minimal models, described using Temperley-Lieb
algebras. The lattice transfer operators include seams on the boundary that use
Wenzl-Jones projectors. If the projectors are singular, the original
prescription is to select a subspace of the Temperley-Lieb modules on which the
action of the transfer operators is non-singular. However, this prescription
does not, in general, yield representations of the Temperley-Lieb algebras and
the Virasoro Kac modules have remained largely unidentified. Here, we introduce
the appropriate algebraic framework for the lattice analysis as a quotient of
the one-boundary Temperley-Lieb algebra. The corresponding standard modules are
introduced and examined using invariant bilinear forms and their Gram
determinants. The structures of the Virasoro Kac modules are inferred from
these results and are found to be given by finitely generated submodules of
Feigin-Fuchs modules. Additional evidence for this identification is obtained
by comparing the formalism of lattice fusion with the fusion rules of the
Virasoro Kac modules. These are obtained, at the character level, in complete
generality by applying a Verlinde-like formula and, at the module level, in
many explicit examples by applying the Nahm-Gaberdiel-Kausch fusion algorithm.Comment: 71 pages. v3: version published in Nucl. Phys.
Integrability and conformal data of the dimer model
The central charge of the dimer model on the square lattice is still being
debated in the literature. In this paper, we provide evidence supporting the
consistency of a description. Using Lieb's transfer matrix and its
description in terms of the Temperley-Lieb algebra at , we
provide a new solution of the dimer model in terms of the model of critical
dense polymers on a tilted lattice and offer an understanding of the lattice
integrability of the dimer model. The dimer transfer matrix is analysed in the
scaling limit and the result for is expressed in terms of
fermions. Higher Virasoro modes are likewise constructed as limits of elements
of and are found to yield a realisation of the Virasoro algebra,
familiar from fermionic ghost systems. In this realisation, the dimer Fock
spaces are shown to decompose, as Virasoro modules, into direct sums of
Feigin-Fuchs modules, themselves exhibiting reducible yet indecomposable
structures. In the scaling limit, the eigenvalues of the lattice integrals of
motion are found to agree exactly with those of the conformal integrals
of motion. Consistent with the expression for obtained from
the transfer matrix, we also construct higher Virasoro modes with and
find that the dimer Fock space is completely reducible under their action.
However, the transfer matrix is found not to be a generating function for the
integrals of motion. Although this indicates that Lieb's transfer matrix
description is incompatible with the interpretation, it does not rule out
the existence of an alternative, compatible, transfer matrix description
of the dimer model.Comment: 54 pages. v2: minor correction
Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models
A Temperley-Lieb (TL) loop model is a Yang-Baxter integrable lattice model
with nonlocal degrees of freedom. On a strip of width N, the evolution operator
is the double-row transfer tangle D(u), an element of the TL algebra TL_N(beta)
with loop fugacity beta=2cos(lambda). Similarly on a cylinder, the single-row
transfer tangle T(u) is an element of the enlarged periodic TL algebra. The
logarithmic minimal models LM(p,p') comprise a subfamily of the TL loop models
for which the crossing parameter lambda=(p'-p)pi/p' is parameterised by coprime
integers 0<p<p'. For these special values, additional symmetries allow for
particular degeneracies in the spectra that account for the logarithmic nature
of these theories. For critical dense polymers LM(1,2), D(u) and T(u) are known
to satisfy inversion identities that allow us to obtain exact eigenvalues in
any representation and for all system sizes N. The generalisation for p'>2
takes the form of functional relations for D(u) and T(u) of polynomial degree
p'. These derive from fusion hierarchies of commuting transfer tangles
D^{m,n}(u) and T^{m,n}(u) where D(u)=D^{1,1}(u) and T(u)=T^{1,1}(u). The fused
transfer tangles are constructed from (m,n)-fused face operators involving
Wenzl-Jones projectors P_k on k=m or k=n nodes. Some projectors P_k are
singular for k>p'-1, but we argue that D^{m,n}(u) and T^{m,n}(u) are well
defined for all m,n. For generic lambda, we derive the fusion hierarchies and
the associated T- and Y-systems. For the logarithmic theories, the closure of
the fusion hierarchies at n=p' translates into functional relations of
polynomial degree p' for D^{m,1}(u) and T^{m,1}(u). We also derive the closure
of the Y-systems for the logarithmic theories. The T- and Y-systems are the key
to exact integrability and we observe that the underlying structure of these
functional equations relate to Dynkin diagrams of affine Lie algebras.Comment: 77 page
Fusion hierarchies, -systems and -systems for the models
The family of models on the square lattice includes a dilute loop
model, a -vertex model and, at roots of unity, a family of RSOS models. The
fused transfer matrices of the general loop and vertex models are shown to
satisfy -type fusion hierarchies. We use these to derive explicit
- and -systems of functional equations. At roots of unity, we further
derive closure identities for the functional relations and show that the
universal -system closes finitely. The RSOS models are shown to
satisfy the same functional and closure identities but with finite truncation.Comment: 36 page
La structure de Jordan des matrices de transfert des modèles de boucles et la relation avec les hamiltoniens XXZ
Les modèles sur réseau comme ceux de la percolation, d’Ising et de Potts servent
à décrire les transitions de phase en deux dimensions. La recherche de leur solution
analytique passe par le calcul de la fonction de partition et la diagonalisation de matrices de transfert. Au point critique, ces modèles statistiques bidimensionnels sont
invariants sous les transformations conformes et la construction de théories des
champs conformes rationnelles, limites continues des modèles statistiques, permet
un calcul de la fonction de partition au point critique. Plusieurs chercheurs pensent
cependant que le paradigme des théories des champs conformes rationnelles peut
être élargi pour inclure les modèles statistiques avec des matrices de transfert non diagonalisables. Ces modèles seraient alors décrits, dans la limite d’échelle, par
des théories des champs logarithmiques et les représentations de l’algèbre de Virasoro
intervenant dans la description des observables physiques seraient indécomposables.
La matrice de transfert de boucles D_N(λ, u), un élément de l’algèbre de Temperley-
Lieb, se manifeste dans les théories physiques à l’aide des représentations
de connectivités ρ (link modules). L’espace vectoriel sur lequel agit cette représentation se décompose en secteurs étiquetés par un paramètre physique, le nombre d de défauts. L’action de cette représentation ne peut que diminuer ce nombre ou le laisser constant. La thèse est consacrée à l’identification de la structure de Jordan de D_N(λ, u) dans ces représentations. Le paramètre β = 2 cos λ = −(q + 1/q) fixe la théorie : β = 1 pour la percolation et √2 pour le modèle d’Ising, par exemple.
Sur la géométrie du ruban, nous montrons que D_N(λ, u) possède les mêmes blocs de Jordan que F_N, son plus haut coefficient de Fourier. Nous étudions la non
diagonalisabilité de F_N à l’aide des divergences de certaines composantes de ses
vecteurs propres, qui apparaissent aux valeurs critiques de λ. Nous prouvons dans
ρ(D_N(λ, u)) l’existence de cellules de Jordan intersectorielles, de rang 2 et couplant des secteurs d, d′ lorsque certaines contraintes sur λ, d, d′ et N sont satisfaites.
Pour le modèle de polymères denses critique (β = 0) sur le ruban, les valeurs
propres de ρ(D_N(λ, u)) étaient connues, mais les dégénérescences conjecturées. En
construisant un isomorphisme entre les modules de connectivités et un sous-espace
des modules de spins du modèle XXZ en q = i, nous prouvons cette conjecture.
Nous montrons aussi que la restriction de l’hamiltonien de boucles à un secteur
donné est diagonalisable et trouvons la forme de Jordan exacte de l’hamiltonien
XX, non triviale pour N pair seulement.
Enfin nous étudions la structure de Jordan de la matrice de transfert T_N(λ, ν)
pour des conditions aux frontières périodiques. La matrice T_N(λ, ν) a des blocs de Jordan intrasectoriels et intersectoriels lorsque λ = πa/b, et a, b ∈ Z×. L’approche
par F_N admet une généralisation qui permet de diagnostiquer des cellules intersectorielles dont le rang excède 2 dans certains cas et peut croître indéfiniment avec N. Pour les blocs de Jordan intrasectoriels, nous montrons que les représentations de connectivités sur le cylindre et celles du modèle XXZ sont isomorphes sauf pour certaines valeurs précises de q et du paramètre de torsion v. En utilisant le comportement de la transformation i_N^d dans un voisinage des valeurs critiques (q_c, v_c), nous construisons explicitement des vecteurs généralisés de Jordan de rang 2 et
discutons l’existence de blocs de Jordan intrasectoriels de plus haut rang.Lattice models such as percolation, the Ising model and the Potts model are useful
for the description of phase transitions in two dimensions. Finding analytical solutions is done by calculating the partition function, which in turn requires finding
eigenvalues of transfer matrices. At the critical point, the two dimensional statistical models are invariant under conformal transformations and the construction of rational conformal field theories, as the continuum limit of these lattice models, allows one to compute the partition function at the critical point. Many researchers think however that the paradigm of rational conformal conformal field theories can be extended to include models with non diagonalizable transfer matrices. These models would then be described, in the scaling limit, by logarithmic conformal field theories and the representations of the Virasoro algebra coming into play would be indecomposable.
We recall the construction of the double-row transfer matrix D_N(λ, u) of the
Fortuin-Kasteleyn model, seen as an element of the Temperley-Lieb algebra. This transfer matrix comes into play in physical theories through its representation in link modules (or standard modules). The vector space on which this representation acts decomposes into sectors labelled by a physical parameter d, the number of defects, which remains constant or decreases in the link representations. This thesis is devoted to the identification of the Jordan structure of D_N(λ, u) in the link representations.
The parameter β = 2 cos λ = −(q + 1/q) fixes the theory : for instance β = 1 for percolation and √2 for the Ising model.
On the geometry of the strip with open boundary conditions, we show that D_N(λ, u) has the same Jordan blocks as its highest Fourier coefficient, F_N. We study
the non-diagonalizability of F_N through the divergences of some of the eigenstates of ρ(F_N) that appear at the critical values of λ. The Jordan cells we find in ρ(D_N(λ, u)) have rank 2 and couple sectors d and d′ when specific constraints on λ, d, d′ and N are satisfied.
For the model of critical dense polymers (β = 0) on the strip, the eigenvalues
of ρ(D_N(λ, u)) were known, but their degeneracies only conjectured. By constructing an isomorphism between the link modules on the strip and a subspace of spin
modules of the XXZ model at q = i, we prove this conjecture. We also show that the restriction of the Hamiltonian to any sector d is diagonalizable, and that the XX
Hamiltonian has rank 2 Jordan cells when N is even.
Finally, we study the Jordan structure of the transfer matrix T_N(λ, ν) for periodic
boundary conditions. When λ = πa/b and a, b ∈ Z×, the matrix T_N(λ, ν) has Jordan blocks between sectors, but also within sectors. The approach using F_N admits
a generalization to the present case and allows us to probe the Jordan cells
that tie different sectors. The rank of these cells exceeds 2 in some cases and can
grow indefinitely with N. For the Jordan blocks within a sector, we show that the
link modules on the cylinder and the XXZ spin modules are isomorphic except for
specific curves in the (q, v) plane. By using the behavior of the transformation i_N^d in a neighborhood of the critical values (q_c, v_c), we explicitly build Jordan partners of rank 2 and discuss the existence of Jordan cells with higher rank
The Jordan Structure of Two Dimensional Loop Models
We show how to use the link representation of the transfer matrix of
loop models on the lattice to calculate partition functions, at criticality, of
the Fortuin-Kasteleyn model with various boundary conditions and parameter
and, more specifically,
partition functions of the corresponding -Potts spin models, with
. The braid limit of is shown to be a central element
of the Temperley-Lieb algebra , its eigenvalues are
determined and, for generic , a basis of its eigenvectors is constructed
using the Wenzl-Jones projector. To any element of this basis is associated a
number of defects , , and the basis vectors with the same
span a sector. Because components of these eigenvectors are singular when and , the link representations of
and are shown to have Jordan blocks between sectors and
when and (). When
and do not satisfy the previous constraint, is diagonalizable.Comment: 55 page