arXiv subjects

Yvan Saint-Aubin

Publications and source records attributed to Yvan Saint-Aubin.

At least 19 recordsLinked to original sources

arXiv2022

Fusion hierarchies, $T$-systems and $Y$-systems for the dilute $A_2^{(2)}$ loop models on a strip

We study the dilute $A_2^{(2)}$ loop models on the geometry of a strip of width $N$. Two families of boundary conditions are known to satisfy the boundary Yang-Baxter equation. Fixing the boundary condition on the two ends of the strip leads to four models. We construct the fusion hierarchy of commuting transfer matrices for the model as well as its $T$- and $Y$-systems, for these four boundary conditions and with a generic crossing parameter $λ$. For $λ/π$ rational and thus $q=-e^{4iλ}$ a root of unity, we prove a linear relation satisfied by the fused transfer matrices that closes the fusion hierarchy into a finite system. The fusion relations allow us to compute the two leading terms in the large-$N$ expansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for small $N$. The present work complements a previous study (A. Morin-Duchesne, P.A. Pearce, J. Stat. Mech. (2019)) that investigated the dilute $A_2^{(2)}$ loop models with periodic boundary conditions.

math-ph↗

arXiv2022

Spin chains as modules over the affine Temperley-Lieb algebra

The affine Temperley-Lieb algebra $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(β)$ is an infinite-dimensional algebra parametrized by a number $β\in \mathbb{C}$ and an integer $N\in \mathbb{N}$. It naturally acts on $(\mathbb{C}^2)^{\otimes N}$ to produce a family of representations labeled by an additional parameter $z\in\mathbb C^\times$. The structure of these representations, which were first introduced by Pasquier and Saleur in their study of spin chains, is here made explicit. They share their composition factors with the cellular $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(β)$-modules of Graham and Lehrer, but differ from the latter representations by the direction of about half of the arrows of their Loewy diagrams. The proof of this statement uses a morphism introduced by Morin-Duchesne and Saint-Aubin as well as new maps that intertwine various $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(β)$-actions on the XXZ chain and generalize applications studied by Deguchi $\textit{et al}$ and after by Morin-Duchesne and Saint-Aubin.

math.RT↗

arXiv2021

L'invariance conforme et l'universalité au point critique des modèles bidimensionnels

Résumé. Des quelques articles publiés par Robert P. Langlands en physique mathématique, c'est celui publié dans le {\it Bulletin of the American Mathematical Society} sous le titre {\it Conformal invariance in two-dimensional percolation} qui a eu, à ce jour, le plus d'impact : les idées d'Oded Schramm ayant mené à l'équation de Loewner stochastique et les preuves de l'invariance conforme de modèles de physique statistique par Stanislav Smirnov ont été suscitées, au moins en partie, par cet article. Ce chapitre rappelle sommairement quelques idées de l'article original ainsi que celles issues des travaux de Schramm et Smirnov. Il est aussi l'occasion pour moi de décrire la naissance de ma collaboration avec Robert Langlands et d'exprimer ma profonde gratitude pour cette fantastique expérience scientifique et humaine. Abstract. Of all mathematical physics contributions by Robert P. Langlands, the paper {\it Conformal invariance in two-dimensional percolation} published in the {\it Bulletin of the American Mathematical Society} is the one that has had, up to now, the most significant impact: Oded Schramm's ideas leading to the stocastic Loewner equation and Stanislav Smirnov's proof in two dimensions were at least partially inspired by it. This chapter reviews briefly some ideas of the original paper and some of those by Schramm and Smirnov.

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arXiv2019

The representation theory of seam algebras

The boundary seam algebras $\mathsf{b}_{n,k}(β=q+q^{-1})$ were introduced by Morin-Duchesne, Ridout and Rasmussen to formulate algebraically a large class of boundary conditions for two-dimensional statistical loop models. The representation theory of these algebras $\mathsf{b}_{n,k}(β=q+q^{-1})$ is given: their irreducible, standard (cellular) and principal modules are constructed and their structure explicited in terms of their composition factors and of non-split short exact sequences. The dimensions of the irreducible modules and of the radicals of standard ones are also given. The methods proposed here might be applicable to a large family of algebras, for example to those introduced recently by Flores and Peltola, and Crampé and Poulain d'Andecy.

math-ph↗

arXiv2018

Fusion and monodromy in the Temperley-Lieb category

Graham and Lehrer (1998) introduced a Temperley-Lieb category $\mathsf{\widetilde{TL}}$ whose objects are the non-negative integers and the morphisms in $\mathsf{Hom}(n,m)$ are the link diagrams from $n$ to $m$ nodes. The Temperley-Lieb algebra $\mathsf{TL}_{n}$ is identified with $\mathsf{Hom}(n,n)$. The category $\mathsf{\widetilde{TL}}$ is shown to be monoidal. We show that it is also a braided category by constructing explicitly a commutor. A twist is also defined on $\mathsf{\widetilde{TL}}$. We introduce a module category ${\text{ Mod}_{\mathsf{\widetilde{TL}}}}$ whose objects are functors from $\mathsf{\widetilde{TL}}$ to $\mathsf{Vect}_{\mathbb C}$ and define on it a fusion bifunctor extending the one introduced by Read and Saleur (2007). We use the natural morphisms constructed for $\mathsf{\widetilde{TL}}$ to induce the structure of a ribbon category on ${\text{ Mod}_{\mathsf{\widetilde{TL}}}}(β=-q-q^{-1})$, when $q$ is not a root of unity. We discuss how the braiding on $\mathsf{\widetilde{TL}}$ and integrability of statistical models are related. The extension of these structures to the family of dilute Temperley-Lieb algebras is also discussed.

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arXiv2018

On the computation of fusion over the affine Temperley-Lieb algebra

Fusion product originates in the algebraisation of the operator product expansion in conformal field theory. Read and Saleur (2007) introduced an analogue of fusion for modules over associative algebras, for example those appearing in the description of 2d lattice models. The article extends their definition for modules over the affine Temperley-Lieb algebra $\atl n$. Since the regular Temperley-Lieb algebra $\tl n$ is a subalgebra of the affine $\atl n$, there is a natural pair of adjoint induction-restriction functors $(\Indar{}, \Resar{})$. The existence of an algebra morphism $ϕ:\atl n\to\tl n$ provides a second pair of adjoint functors $(\Indphi{},\Resphi{})$. Two fusion products between $\atl{}$-modules are proposed and studied. They are expressed in terms of these four functors. The action of these functors is computed on the standard, cell and irreducible $\atl n$-modules. As a byproduct, the Peirce decomposition of $\atl n(q+q^{-1})$, when $q$ is not a root of unity, is given as direct sum of the induction $\Indar{\TheS{n,k}}$ of standard $\tl n$-modules to $\atl n$-modules. Examples of fusion products of various pairs of affine modules are given.

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arXiv2017

Restriction and induction of indecomposable modules over the Temperley-Lieb algebras

Both the original Temperley-Lieb algebras $\mathsf{TL}_{n}$ and their dilute counterparts $\mathsf{dTL}_{n}$ form families of filtered algebras: $\mathsf{TL}_{n}\subset \mathsf{TL}_{n+1}$ and $\mathsf{dTL}_{n}\subset\mathsf{dTL}_{n+1}$, for all $n\geq 0$. For each such inclusion, the restriction and induction of every finite-dimensional indecomposable module over $\mathsf{TL}_{n}$ (or $\mathsf{dTL}_{n}$) is computed. To accomplish this, a thorough description of each indecomposable is given, including its projective cover and injective hull, some short exact sequences in which it appears, its socle and head, and its extension groups with irreducible modules. These data are also used to prove the completeness of the list of indecomposable modules, up to isomorphism. In fact, two completeness proofs are given, the first is based on elementary homological methods and the second uses Auslander-Reiten theory. The latter proof offers a detailed example of this algebraic tool that may be of independent interest.

math-ph↗

arXiv2015

On the reality of spectra of $\boldsymbol{U_q(sl_2)}$-invariant XXZ Hamiltonians

A new inner product is constructed on each standard module over the Temperley-Lieb algebra $\mathsf{TL}_n(β)$ for $β\in \mathbb R$ and $n \ge 2$. On these modules, the Hamiltonian $h = -\sum_i e_i$ is shown to be self-adjoint with respect to this inner product. This implies that its action on these modules is diagonalisable with real eigenvalues. A representation theoretic argument shows that the reality of spectra of the Hamiltonian extends to all other Temperley-Lieb representations. In particular, this result applies to the celebrated $U_q(sl_2)$-invariant XXZ Hamiltonian, for all $q+q^{-1}\in \mathbb R$.

math-ph↗

arXiv2014

The principal indecomposable modules of the dilute Temperley-Lieb algebra

The Temperley-Lieb algebra \tln(β) can be defined as the set of rectangular diagrams with n points on each of their vertical sides, with all points joined pairwise by non-intersecting strings. The multiplication is then the concatenation of diagrams. The dilute Temperley-Lieb algebra \dtl n(β) has a similar diagrammatic definition where, now, points on the sides may remain free of strings. Like \tl n, the dilute \dtl n depends on a parameter β\in\mathbb C, often given as β=q+q^{-1} for some q\in\mathbb C^\times. In statistical physics, the algebra plays a central role in the study of dilute loop models. The paper is devoted to the construction of its principal indecomposable modules. Basic definitions and properties are first given: the dimension of \dtl n, its break up into even and odd subalgebras and its filtration through n+1 ideals. The standard modules \U{n,k} are then introduced and their behaviour under restriction and induction is described. A bilinear form, the Gram product, is used to identify their (unique) maximal submodule \dr{n,k} which is then shown to be irreducible or trivial. It is then noted that \dtl n is a cellular algebra. This fact allows for the identification of complete sets of non-isomorphic irreducible modules and projective indecomposable ones. The structure of \dtl n as a left module over itself is then given for all values of the parameter q, that is, for both q generic and a root of unity.

math-ph↗

arXiv2014

Standard Modules, Induction and the Temperley-Lieb Algebra

The basic properties of the Temperley-Lieb algebra $TL_n$ with parameter $β= q + q^{-1}$, for $q$ any non-zero complex number, are reviewed in a pedagogical way. The link and standard (cell) modules that appear in numerous physical applications are defined and a natural bilinear form on the standard modules is used to characterize their maximal submodules. When this bilinear form has a non-trivial radical, some of the standard modules are reducible and $TL_n$ is non-semisimple. This happens only when $q$ is a root of unity. Use of restriction and induction allows for a finer description of the structure of the standard modules. Finally, a particular central element $F_n$ of $TL_n$ is studied; its action is shown to be non-diagonalisable on certain indecomposable modules and this leads to a proof that the radicals of the standard modules are irreducible. Moreover, the space of homomorphisms between standard modules is completely determined. The principal indecomposable modules are then computed concretely in terms of standard modules and their inductions. Examples are provided throughout and the delicate case $β= 0$, that plays an important role in physical models, is studied systematically.

math-ph↗

arXiv2013

Jordan cells of periodic loop models

Jordan cells in transfer matrices of finite lattice models are a signature of the logarithmic character of the conformal field theories that appear in their thermodynamical limit. The transfer matrix of periodic loop models, T_N, is an element of the periodic Temperley-Lieb algebra EPTL_N(β, α), where N is the number of sites on a section of the cylinder, and β= -(q+1/q) = 2 \cos λand αthe weights of contractible and non-contractible loops. The thermodynamic limit of T_N is believed to describe a conformal field theory of central charge c=1-6λ^2/(π(λ-π)). The abstract element T_N acts naturally on (a sum of) spaces V_N^d, similar to those upon which the standard modules of the (classical) Temperley-Lieb algebra act. These spaces known as sectors are labeled by the numbers of defects d and depend on a {\em twist parameter} v that keeps track of the winding of defects around the cylinder. Criteria are given for non-trivial Jordan cells of T_N both between sectors with distinct defect numbers and within a given sector.

math-ph↗

arXiv2013

The idempotents of the TL_n-modules \otimes^nC^2 in terms of elements of U_qsl_2

The vector space \otimes^nC^2 upon which the XXZ Hamilonian with n spins acts bears the structure of a module over both the Temperley-Lieb algebra TL_n(β=q+1/q) and the quantum algebra U_qsl_2. The decomposition of \otimes^nC^2 as a U_qsl_2-module was first described by Rosso [23], Lusztig [15] and Pasquier and Saleur [20] and that as a TL_n-module by Martin [17] (see also Read and Saleur [21] and Gainutdinov and Vasseur [9]). For q generic, i.e. not a root of unity, the TL_n-module \otimes^nC^2 is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of \otimes^nC^2) onto each of these irreducible modules as linear combinations of elements of U_qsl_2. When q=q_c is a root of unity, the TL_n-module \otimes^nC^2 (with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involve some new generators, whose action on \otimes^nC^2 is that of the divided powers (S^\pm)^{(r)}=\lim_{q\to q_c} (S^\pm)^r/[r]!.

math-ph↗

arXiv2013

A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra

We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley-Lieb algebra EPTL_N(β, α) introduced by Martin and Saleur, and Graham and Lehrer. These are labeled by the numbers of sites N and of defects d, and extend the standard modules of the original Temperley-Lieb algebra. Beside the defining parameters β=u^2+u^{-2} with u=e^{iλ/2} (weight of contractible loops) and α(weight of non-contractible loops), this family also depends on a twist parameter v that keeps track of how the defects wind around the cylinder. The transfer matrix T_N(λ, ν) depends on the anisotropy νand the spectral parameter λthat fixes the model. (The thermodynamic limit of T_N is believed to describe a conformal field theory of central charge c=1-6λ^2/(π(λ-π)).) The family of periodic XXZ Hamiltonians is extended to depend on this new parameter v and the relationship between this family and the loop models is established. The Gram determinant for the natural bilinear form on these link modules is shown to factorize in terms of an intertwiner i_N^d between these link representations and the eigenspaces of S^z of the XXZ models. This map is shown to be an isomorphism for generic values of u and v and the critical curves in the plane of these parameters for which i_N^d fails to be an isomorphism are given.

math-ph↗

arXiv2011

Geometric Exponents of Dilute Logarithmic Minimal Models

The fractal dimensions of the hull, the external perimeter and of the red bonds are measured through Monte Carlo simulations for dilute minimal models, and compared with predictions from conformal field theory and SLE methods. The dilute models used are those first introduced by Nienhuis. Their loop fugacity is beta = -2cos(pi/barkappa}) where the parameter barkappa is linked to their description through conformal loop ensembles. It is also linked to conformal field theories through their central charges c = 13 - 6(barkappa + barkappa^{-1}) and, for the minimal models of interest here, barkappa = p/p' where p and p' are two coprime integers. The geometric exponents of the hull and external perimeter are studied for the pairs (p,p') = (1,1), (2,3), (3,4), (4,5), (5,6), (5,7), and that of the red bonds for (p,p') = (3,4). Monte Carlo upgrades are proposed for these models as well as several techniques to improve their speeds. The measured fractal dimensions are obtained by extrapolation on the lattice size H,V -> infinity. The extrapolating curves have large slopes; despite these, the measured dimensions coincide with theoretical predictions up to three or four digits. In some cases, the theoretical values lie slightly outside the confidence intervals; explanations of these small discrepancies are proposed.

cond-mat.stat-mech↗

arXiv2011

The Jordan Structure of Two Dimensional Loop Models

We show how to use the link representation of the transfer matrix $D_N$ of loop models on the lattice to calculate partition functions, at criticality, of the Fortuin-Kasteleyn model with various boundary conditions and parameter $β= 2 \cos(π(1-a/b)), a,b\in \mathbb N$ and, more specifically, partition functions of the corresponding $Q$-Potts spin models, with $Q=β^2$. The braid limit of $D_N$ is shown to be a central element $F_N(β)$ of the Temperley-Lieb algebra $TL_N(β)$, its eigenvalues are determined and, for generic $β$, a basis of its eigenvectors is constructed using the Wenzl-Jones projector. To any element of this basis is associated a number of defects $d$, $0\le d\le N$, and the basis vectors with the same $d$ span a sector. Because components of these eigenvectors are singular when $b \in \mathbb{Z}^*$ and $a \in 2 \mathbb{Z} + 1$, the link representations of $F_N$ and $D_N$ are shown to have Jordan blocks between sectors $d$ and $d'$ when $d-d' < 2b$ and $(d+d')/2 \equiv b-1 \ \textrm{mod} \ 2b$ ($d>d'$). When $a$ and $b$ do not satisfy the previous constraint, $D_N$ is diagonalizable.

math-ph↗

arXiv2009

Behavior of the two-dimensional Ising model at the boundary of a half-infinite cylinder

The two-dimensional Ising model is studied at the boundary of a half-infinite cylinder. The three regular lattices (square, triangular and hexagonal) and the three regimes (sub-, super- and critical) are discussed. The probability of having precisely 2n spinflips at the boundary is computed as a function of the positions k_i's, i=1,..., 2n, of the spinflips. The limit when the mesh goes to zero is obtained. For the square lattice, the probability of having 2n spinflips, independently of their position, is also computed. As a byproduct we recover a result of De Coninck showing that the limiting distribution of the number of spinflips is Gaussian. The results are obtained as consequences of Onsager's solution and are rigorous.

cond-mat.stat-mech↗

arXiv2009

Critical exponents for the homology of Fortuin-Kasteleyn clusters on a torus

A Fortuin-Kasteleyn cluster on a torus is said to be of type $\{a,b\}, a,b\in\mathbb Z$, if it possible to draw a curve belonging to the cluster that winds $a$ times around the first cycle of the torus as it winds $-b$ times around the second. Even though the $Q$-Potts models make sense only for $Q$ integers, they can be included into a family of models parametrized by $β=\sqrt{Q}$ for which the Fortuin-Kasteleyn clusters can be defined for any real $β\in (0,2]$. For this family, we study the probability $π({\{a,b\}})$ of a given type of clusters as a function of the torus modular parameter $τ=τ_r+iτ_i$. We compute the asymptotic behavior of some of these probabilities as the torus becomes infinitely thin. For example, the behavior of $π(\{1,0\})$ is studied along the line $τ_r=0$ and $τ_i\to\infty$. Exponents describing these behaviors are defined and related to weights $h_{r,s}$ of the extended Kac table for $r,s$ integers, but also half-integers. Numerical simulations are also presented. Possible relationship with recent works and conformal loop ensembles is discussed.

cond-mat.stat-mech↗

arXiv2008

Geometric Exponents, SLE and Logarithmic Minimal Models

In statistical mechanics, observables are usually related to local degrees of freedom such as the Q < 4 distinct states of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuum scaling limit, these models are described by rational conformal field theories, namely the minimal models M(p,p') for suitable p, p'. More generally, as in stochastic Loewner evolution (SLE_kappa), one can consider observables related to nonlocal degrees of freedom such as paths or boundaries of clusters. This leads to fractal dimensions or geometric exponents related to values of conformal dimensions not found among the finite sets of values allowed by the rational minimal models. Working in the context of a loop gas with loop fugacity beta = -2 cos(4 pi/kappa), we use Monte Carlo simulations to measure the fractal dimensions of various geometric objects such as paths and the generalizations of cluster mass, cluster hull, external perimeter and red bonds. Specializing to the case where the SLE parameter kappa = 4p'/p is rational with p < p', we argue that the geometric exponents are related to conformal dimensions found in the infinitely extended Kac tables of the logarithmic minimal models LM(p,p'). These theories describe lattice systems with nonlocal degrees of freedom. We present results for critical dense polymers LM(1,2), critical percolation LM(2,3), the logarithmic Ising model LM(3,4), the logarithmic tricritical Ising model LM(4,5) as well as LM(3,5). Our results are compared with rigourous results from SLE_kappa, with predictions from theoretical physics and with other numerical experiments. Throughout, we emphasize the relationships between SLE_kappa, geometric exponents and the conformal dimensions of the underlying CFTs.

cond-mat.stat-mech↗