arXiv subjects

Alexi Morin-Duchesne

Publications and source records attributed to Alexi Morin-Duchesne.

At least 19 recordsLinked to original sources

arXiv2026

Fusion in the periodic Temperley-Lieb algebra: general definition of a bifunctor

The periodic Temperley-Lieb category consists of connectivity diagrams drawn on a ring with $N$ and $N'$ nodes on the outer and inner boundary, respectively. We consider families of modules, namely sequences of modules $\mathsf{M}(N)$ over the enlarged periodic Temperley-Lieb algebra for varying values of $N$, endowed with an action $\mathsf{M}(N') \to \mathsf{M}(N)$ of the diagrams. Examples of modules that can be organised into families are those arising in the RSOS model and in the XXZ spin-$\frac12$ chain, as well as several others constructed from link states. We construct a fusion product which outputs a family of modules from any pair of families. Its definition is inspired from connectivity diagrams drawn on a disc with two holes. It is thus defined in a way to describe intermediate states in lattice correlation functions. We prove that this fusion product is a bifunctor, and that it is distributive, commutative, and associative.

math-ph↗

arXiv2026

Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors

Affine and periodic Temperley-Lieb algebras are families of diagrammatic algebras that find diverse applications in mathematics and physics. These algebras are infinite dimensional, yet most of their interesting modules are finite. In this paper, we introduce finite quotients for these algebras, which we term uncoiled affine Temperley-Lieb algebras and uncoiled periodic Temperley-Lieb algebras. We study some of their properties, including their defining relations, their description with diagrams, their dimensions, and their relations with affine and skew sandwich cellular algebras. The uncoiled algebras all have finitely many one-dimensional modules. We construct a family of Wenzl-Jones idempotents, each of which projects onto one of these one-dimensional modules. Our construction is explicit and uses the similar projectors for the ordinary Temperley--Lieb algebras, as well as the diagrammatic description of the uncoiled algebras in terms of sandwich diagrams. We also discuss the Markov traces for the uncoiled algebras and their evaluations on the newly defined projectors, and find expressions involving Chebyshev polynomials of the first kind.

math.RT↗

arXiv2026

Temperley-Lieb modules and local operators for critical ADE models

We investigate critical restricted solid-on-solid models associated to Dynkin diagrams of type $A$, $D$ and $E$, with fixed, periodic and twisted periodic boundary conditions. These models are endowed with an action of the diagrams of the Temperley-Lieb category. For each model, we obtain the decomposition of the state space as a direct sum of irreducible modules over the Temperley-Lieb algebra $\mathsf{TL}_N(β)$ or its periodic incarnation $\mathsf{\mathcal EPTL}_N(β)$. This allows us to recover the known conformal partition functions for these models in the continuum scaling limit. For each irreducible factor arising in the decompositions, we define an associated local operator on the lattice, which behaves like a connectivity operator. Using knowledge from the Temperley-Lieb representation theory at roots of unity, we show that these operators satisfy certain linear difference relations, which are lattice counterparts of the singular-vector relations in conformal field theory.

math-ph↗

arXiv2025

Modular covariant torus partition functions of dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models

Yang-Baxter integrable dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models are considered on the torus in their simplest physical regimes. A combination of boundary conditions $(h,v)$ is applied in the horizontal and vertical directions with $h,v=0$ and $1$ for periodic and antiperiodic boundary conditions respectively. The fugacities of non-contractible and contractible loops are denoted by $α$ and $β$ respectively where $β$ is simply related to the crossing parameter $λ$. At roots of unity, when $λ/π\in\mathbb Q$, these models are the dense ${\cal LM}(p,p')$ and dilute ${\cal DLM}(p,p')$ logarithmic minimal models with $p,p'$ coprime integers. We conjecture the scaling limits of the transfer matrix traces in the standard modules with $d$ defects and deduce the conformal partition functions ${\cal Z}_{\textrm{dense}}^{(h,v)}(α)$ and ${\cal Z}_{\textrm{dilute}}^{(h,v)}(α)$ using Markov traces. These are expressed in terms of functions ${\cal Z}_{m,m'}(g)$ known from the Coulomb gas arguments of Di Francesco, Saleur and Zuber and subsequently as sesquilinear forms in Verma characters. Crucially, we find that the partition functions are identical for the dense and dilute models. The coincidence of these conformal partition functions provides compelling evidence that, for given $(p,p')$, these dense and dilute theories lie in the same universality class. In root of unity cases with $α=2$, the $(h,v)$ modular covariant partition functions are also expressed as sesquilinear forms in affine $u(1)$ characters involving generalized Bezout conjugates. These also give the modular covariant partition functions for the 6-vertex and Izergin-Korepin 19-vertex models in the corresponding regimes.

math-ph↗

arXiv2024

Fusion of irreducible modules in the periodic Temperley--Lieb algebra

We propose a new family ${\sf Y}_{k,\ell,x,y,[z,w]}$ of modules over the enlarged periodic Temperley--Lieb algebra ${\sf{\cal E}PTL}_N(β)$. These modules are built from link states with two marked points, similarly to the modules ${\sf X}_{k,\ell,x,y,z}$ that we constructed in a previous paper. They however differ in the way that defects connect pairwise. We analyse the decomposition of ${\sf Y}_{k,\ell,x,y,[z,w]}$ over the irreducible standard modules ${\sf W}_{k,x}$ for generic values of the parameters $z$ and $w$, and use it to deduce the fusion rules for the fusion $\sf W \times W$ of standard modules. These turn out to be more symmetric than those obtained previously using the modules ${\sf X}_{k,\ell,x,y,z}$. From the work of Graham and Lehrer, it is known that, for $β=-q-q^{-1}$ where $q$ is not a root of unity, there exists a set of non-generic values of the twist $y$ for which the standard module ${\sf W}_{\ell,y}$ is indecomposable yet reducible with two composition factors: a radical submodule ${\sf R}_{\ell,y}$ and a quotient module ${\sf Q}_{\ell,y}$. Here, we construct the fusion products $\sf W\times R$, $\sf W\times Q$ and $\sf Q\times Q$, and analyse their decomposition over indecomposable modules. For the fusions involving the quotient modules ${\sf Q}$, we find very simple results reminiscent of $\mathfrak{sl}(2)$ fusion rules. This construction with modules ${\sf Y}_{k,\ell,x,y,[z,w]}$ is a good lattice regularization of the operator product expansion in the underlying logarithmic bulk conformal field theory. Indeed, it fits with the correspondence between standard modules and connectivity operators, and is useful for the calculation of their correlation functions. Remarkably, we show that the fusion rules $\sf W\times Q$ and $\sf Q\times Q$ are consistent with the known fusion rules of degenerate primary fields.

math-ph↗

arXiv2023

Critical site percolation on the triangular lattice: From integrability to conformal partition functions

Critical site percolation on the triangular lattice is described by the Yang-Baxter solvable dilute $A_2^{(2)}$ loop model with crossing parameter specialized to $λ=\frac\pi3$, corresponding to the contractible loop fugacity $β=-2\cos4λ=1$. We study the functional relations satisfied by the commuting transfer matrices of this model and the associated Bethe ansatz equations. The single and double row transfer matrices are respectively endowed with strip and periodic boundary conditions, and are elements of the ordinary and periodic dilute Temperley-Lieb algebras. The standard modules for these algebras are labeled by the number of defects $d$ and, in the latter case, also by the twist $e^{iγ}$. Nonlinear integral equation techniques are used to analytically solve the Bethe ansatz functional equations in the scaling limit for the central charge $c=0$ and conformal weights $Δ,\barΔ$. For the groundstates, we find $Δ=Δ_{1,d+1}$ for strip boundary conditions and $(Δ,\barΔ)=(Δ_{γ/π,d/2},Δ_{γ/π,-d/2})$ for periodic boundary conditions, where $Δ_{r,s}=\frac1{24}((3r-2s)^2-1)$. We give explicit conjectures for the scaling limit of the trace of the transfer matrix in each standard module. For $d\le8$, these conjectures are supported by numerical solutions of the logarithmic form of the Bethe ansatz equations for the leading $20$ or more conformal eigenenergies. With these conjectures, we apply the Markov traces to obtain the conformal partition functions on the cylinder and torus. These precisely coincide with our previous results for critical bond percolation on the square lattice described by the dense $A_1^{(1)}$ loop model with $λ=\frac\pi3$. The concurrence of all this conformal data provides compelling evidence supporting a strong form of universality between these two stochastic models as logarithmic CFTs.

math-ph↗

arXiv2022

Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models

In two-dimensional loop models, the scaling properties of critical random curves are encoded in the correlators of connectivity operators. In the dense O($n$) loop model, any such operator is naturally associated to a standard module of the periodic Temperley-Lieb algebra. We introduce a new family of representations of this algebra, with connectivity states that have two marked points, and argue that they define the fusion of two standard modules. We obtain their decomposition on the standard modules for generic values of the parameters, which in turn yields the structure of the operator product expansion of connectivity operators.

math-ph↗

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↗

arXiv2021

Groundstate finite-size corrections and dilogarithm identities for the twisted $A_1^{(1)}$, $A_2^{(1)}$ and $A_2^{(2)}$ models

We consider the $Y$-systems satisfied by the $A_1^{(1)}$, $A_2^{(1)}$, $A_2^{(2)}$ vertex and loop models at roots of unity with twisted boundary conditions on the cylinder. The vertex models are the 6-, 15- and Izergin-Korepin 19-vertex models respectively. The corresponding loop models are the dense, fully packed and dilute Temperley-Lieb loop models respectively. For all three models, our focus is on roots of unity values of $e^{iλ}$ with the crossing parameter $λ$ corresponding to the principal and dual series of these models. Converting the known functional equations to nonlinear integral equations in the form of Thermodynamic Bethe Ansatz (TBA) equations, we solve the $Y$-systems for the finite-size $\frac 1N$ corrections to the groundstate eigenvalue following the methods of Klümper and Pearce. The resulting expressions for $c-24Δ$, where $c$ is the central charge and $Δ$ is the conformal weight associated with the groundstate, are simplified using various dilogarithm identities. Our analytic results are in agreement with previous results obtained by different methods and are new for the dual series of the $A_2^{(1)}$ model.

math-ph↗

arXiv2021

Bipartite fidelity for models with periodic boundary conditions

For a given statistical model, the bipartite fidelity $\mathcal F$ is computed from the overlap between the groundstate of a system of size $N$ and the tensor product of the groundstates of the same model defined on two subsystems $A$ and $B$, of respective sizes $N_A$ and $N_B$ with $N = N_A + N_B$. In this paper, we study $\mathcal F$ for critical lattice models in the case where the full system has periodic boundary conditions. We consider two possible choices of boundary conditions for the subsystems $A$ and $B$, namely periodic and open. For these two cases, we derive the conformal field theory prediction for the leading terms in the $1/N$ expansion of $\mathcal F$, in a most general case that corresponds to the insertion of four and five fields, respectively. We provide lattice calculations of $\mathcal F$, both exact and numerical, for two free-fermionic lattice models: the XX spin chain and the model of critical dense polymers. We study the asymptotic behaviour of the lattice results for these two models and find an agreement with the predictions of conformal field theory.

cond-mat.stat-mech↗

arXiv2021

Finite-size corrections in critical symmetry-resolved entanglement

In the presence of a conserved quantity, symmetry-resolved entanglement entropies are a refinement of the usual notion of entanglement entropy of a subsystem. For critical 1d quantum systems, it was recently shown in various contexts that these quantities generally obey entropy equipartition in the scaling limit, i.e. they become independent of the symmetry sector. In this paper, we examine the finite-size corrections to the entropy equipartition phenomenon, and show that the nature of the symmetry group plays a crucial role. In the case of a discrete symmetry group, the corrections decay algebraically with system size, with exponents related to the operators' scaling dimensions. In contrast, in the case of a U(1) symmetry group, the corrections only decay logarithmically with system size, with model-dependent prefactors. We show that the determination of these prefactors boils down to the computation of twisted overlaps.

quant-ph↗

arXiv2020

Boundary emptiness formation probabilities in the six-vertex model at $Δ= -\frac12$

We define a new family of overlaps $C_{N,m}$ for the XXZ Hamiltonian on a periodic chain of length $N$. These are equal to the linear sums of the groundstate components, in the canonical basis, wherein $m$ consecutive spins are fixed to the state ${\uparrow}$. We define the boundary emptiness formation probabilities as the ratios $C_{N,m}/C_{N,0}$ of these overlaps. In the associated six-vertex model, they correspond to correlation functions on a semi-infinite cylinder of perimeter $N$. At the combinatorial point $Δ= -\frac12$, we obtain closed-form expressions in terms of simple products of ratios of integers.

math-ph↗

arXiv2019

Bipartite fidelity of critical dense polymers

We investigate the bipartite fidelity $\mathcal F_d$ for a lattice model described by a logarithmic CFT: the model of critical dense polymers. We define this observable in terms of a partition function on the pants geometry, where $d$ defects enter at the top of the pants lattice and exit in one of the legs. Using the correspondence with the XX spin chain, we obtain an exact closed-form expression for $\mathcal F_d$ and compute the leading terms in its $1/N$ asymptotic expansion as a function of $x = N_A/N$, where $N$ is the lattice width at the top of the pants and $N_A$ is the width of the leg where the defects exit. We find an agreement with the results of Stéphan and Dubail for rational CFTs, with the central charge and conformal weights specialised to $c=-2$ and $Δ= Δ_{1,d+1} = d(d-2)/8$. We compute a second instance $\mathcal {\tilde F}_2$ of the bipartite fidelity for $d=2$ by imposing a different rule for the connection of the defects. In the conformal setting, this choice corresponds to inserting two boundary condition changing fields of weight $Δ= 0$ that are logarithmic instead of primary. We compute the asymptotic expansion in this case as well and find a simple additive correction compared to $\mathcal F_2$, of the form $-2\log((1+x)/(2\sqrt{x}))$. We confirm this lattice result with a CFT derivation and find that this correction term is identical for all logarithmic theories, independently of $c$ and $Δ$.

cond-mat.stat-mech↗

arXiv2019

Logarithmic correlation functions for critical dense polymers on the cylinder

We compute lattice correlation functions for the model of critical dense polymers on a semi-infinite cylinder of perimeter $n$. In the lattice loop model, contractible loops have a vanishing fugacity whereas non-contractible loops have a fugacity $α\in(0,\infty)$. These correlators are defined as ratios $Z(x)/Z_0$ of partition functions, where $Z_0$ is a reference partition function wherein only simple arcs are attached to the boundary of the cylinder. For $Z(x)$, the boundary is also decorated with simple arcs, but it also has two positions $1$ and $x$ where the boundary condition is different. We investigate two such kinds of boundary conditions: (i) there is a single node at each of these points where a long arc is attached, and (ii) there are pairs of adjacent nodes at these points where two long arcs are attached. We find explicit expressions for these correlators for finite $n$ using the representation of the enlarged periodic Temperley-Lieb algebra in the XX spin chain. The resulting asymptotics as $n\to\infty$ are expressed as simple integrals that depend on the parameter $τ=\frac{x-1}n\in(0,1)$. For small $τ$, the leading behaviours are proportional to $τ^{1/4}$, $τ^{1/4}\log τ$, $\logτ$ and $\log^2τ$. We interpret the lattice results in terms of ratios of conformal correlation functions. We assume that the corresponding boundary changing fields are highest weight states in irreducible, Kac or staggered Virasoro modules, with central charge $c=-2$ and conformal dimensions $Δ= -\frac18$ or $Δ=0$. We obtain differential equations satisfied by the conformal correlators, solve these equations, and find a perfect agreement with the lattice results. We compute structure constants and ratios thereof which appear in the operator product expansions of the boundary condition changing fields. The fusion of these fields is found to be non-abelian.

cond-mat.stat-mech↗

arXiv2019

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

The fusion hierarchy, $T$-system and $Y$-system of functional equations are the key to integrability for 2d lattice models. We derive these equations for the generic dilute $A_2^{(2)}$ loop models. The fused transfer matrices are associated with nodes of the infinite dominant integral weight lattice of $s\ell(3)$. For generic values of the crossing parameter $λ$, the $T$- and $Y$-systems do not truncate. For the case $\fracλπ=\frac{(2p'-p)}{4p'}$ rational so that $x=\mathrm{e}^{\mathrm{i}λ}$ is a root of unity, we find explicit closure relations and derive closed finite $T$- and $Y$-systems. The TBA diagrams of the $Y$-systems and associated Thermodynamic Bethe Ansatz (TBA) integral equations are not of simple Dynkin type. They involve $p'+2$ nodes if $p$ is even and $2p'+2$ nodes if $p$ is odd and are related to the TBA diagrams of $A_2^{(1)}$ models at roots of unity by a ${\Bbb Z}_2$ folding which originates from the addition of crossing symmetry. In an appropriate regime, the known central charges are $c=1-\frac{6(p-p')^2}{pp'}$. Prototypical examples of the $A_2^{(2)}$ loop models, at roots of unity, include critical dense polymers ${\cal DLM}(1,2)$ with central charge $c=-2$, $λ=\frac{3π}{8}$ and loop fugacity $β=0$ and critical site percolation on the triangular lattice ${\cal DLM}(2,3)$ with $c=0$, $λ=\fracπ{3}$ and $β=1$. Solving the TBA equations for the conformal data will determine whether these models lie in the same universality classes as their $A_1^{(1)}$ 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.

math-ph↗

arXiv2018

Two-point boundary correlation functions of dense loop models

We investigate six types of two-point boundary correlation functions in the dense loop model. These are defined as ratios $Z/Z^0$ of partition functions on the $m\times n$ square lattice, with the boundary condition for $Z$ depending on two points $x$ and $y$. We consider: the insertion of an isolated defect (a) and a pair of defects (b) in a Dirichlet boundary condition, the transition (c) between Dirichlet and Neumann boundary conditions, and the connectivity of clusters (d), loops (e) and boundary segments (f) in a Neumann boundary condition. For the model of critical dense polymers, corresponding to a vanishing loop weight ($β= 0$), we find determinant and pfaffian expressions for these correlators. We extract the conformal weights of the underlying conformal fields and find $Δ= -\frac18$, $0$, $-\frac3{32}$, $\frac38$, $1$, $\tfrac θπ(1+ \tfrac{2θ}π)$, where $θ$ encodes the weight of one class of loops for the correlator of type f. These results are obtained by analysing the asymptotics of the exact expressions, and by using the Cardy-Peschel formula in the case where $x$ and $y$ are set to the corners. For type b, we find a $\ln|x-y|$ dependence from the asymptotics, and a $\ln (\ln n)$ term in the corner free energy. This is consistent with the interpretation of the boundary condition of type b as the insertion of a logarithmic field belonging to a rank two Jordan cell. For the other values of $β= 2 \cos λ$, we use the hypothesis of conformal invariance to predict the conformal weights and find $Δ= Δ_{1,2}$, $Δ_{1,3}$, $Δ_{0,\frac12}$, $Δ_{1,0}$, $Δ_{1,-1}$ and $Δ_{\frac{2θ}λ+1,\frac{2θ}λ+1}$, extending the results of critical dense polymers. With the results for type f, we reproduce a Coulomb gas prediction for the valence bond entanglement entropy of Jacobsen and Saleur.

cond-mat.stat-mech↗

arXiv2018

Extended T-systems, Q matrices and T-Q relations for $s\ell(2)$ models at roots of unity

The mutually commuting $1\times n$ fused single and double-row transfer matrices of the critical six-vertex model are considered at roots of unity $q=e^{iλ}$ with crossing parameter $λ=\frac{(p'-p)π}{p'}$ a rational fraction of $π$. The $1\times n$ transfer matrices of the dense loop model analogs, namely the logarithmic minimal models ${\cal LM}(p,p')$, are similarly considered. For these $s\ell(2)$ models, we find explicit closure relations for the $T$-system functional equations and obtain extended sets of bilinear $T$-system identities. We also define extended $Q$ matrices as linear combinations of the fused transfer matrices and obtain extended matrix $T$-$Q$ relations. These results hold for diagonal twisted boundary conditions on the cylinder as well as $U_q(s\ell(2))$ invariant/Kac vacuum and off-diagonal/Robin vacuum boundary conditions on the strip. Using our extended $T$-system and extended $T$-$Q$ relations for eigenvalues, we deduce the usual scalar Baxter $T$-$Q$ relation and the Bazhanov-Mangazeev decomposition of the fused transfer matrices $T^{n}(u+λ)$ and $D^{n}(u+λ)$, at fusion level $n=p'-1$, in terms of the product $Q^+(u)Q^-(u)$ or $Q(u)^2$. It follows that the zeros of $T^{p'-1}(u+λ)$ and $D^{p'-1}(u+λ)$ are comprised of the Bethe roots and complete $p'$ strings. We also clarify the formal observations of Pronko and Yang-Nepomechie-Zhang and establish, under favourable conditions, the existence of an infinite fusion limit $n\to\infty$ in the auxiliary space of the fused transfer matrices. Despite this connection, the infinite-dimensional oscillator representations are not needed at roots of unity due to finite closure of the functional equations.

hep-th↗

arXiv2018

Fusion hierarchies, $T$-systems and $Y$-systems for the $A_2^{(1)}$ models

The family of $A^{(1)}_2$ models on the square lattice includes a dilute loop model, a $15$-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 $s\ell(3)$-type fusion hierarchies. We use these to derive explicit $T$- and $Y$-systems of functional equations. At roots of unity, we further derive closure identities for the functional relations and show that the universal $Y$-system closes finitely. The $A^{(1)}_2$ RSOS models are shown to satisfy the same functional and closure identities but with finite truncation.

math-ph↗