arXiv subjects

Jesper Lykke Jacobsen

Publications and source records attributed to Jesper Lykke Jacobsen.

At least 37 records · Page 2Linked to original sources

arXiv2023

Integrability of rank-two web models

We continue our work on lattice models of webs, which generalise the well-known loop models to allow for various kinds of bifurcations [arXiv:2101.00282, arXiv:2107.10106]. Here we define new web models corresponding to each of the rank-two spiders considered by Kuperberg [arXiv:q-alg/9712003]. These models are based on the $A_2$, $G_2$ and $B_2$ Lie algebras, and their local vertex configurations are intertwiners of the corresponding $q$-deformed quantum algebras. In all three cases we define a corresponding model on the hexagonal lattice, and in the case of $B_2$ also on the square lattice. For specific root-of-unity choices of $q$, we show the equivalence to a number of three- and four-state spin models on the dual lattice. The main result of this paper is to exhibit integrable manifolds in the parameter spaces of each web model. For $q$ on the unit circle, these models are critical and we characterise the corresponding conformal field theories via numerical diagonalisation of the transfer matrix. In the $A_2$ case we find two integrable regimes. The first one contains a dense and a dilute phase, for which we have analytic control via a Coulomb gas construction, while the second one is more elusive and likely conceals non-compact physics. Three particular points correspond to a three-state spin model with plaquette interactions, of which the one in the second regime appears to present a new universality class. In the $G_2$ case we identify four regimes numerically. The $B_2$ case is too unwieldy to be studied numerically in the general case, but it found analytically to contain a simpler sub-model based on generators of the dilute Birman-Murakami-Wenzl algebra.

hep-th↗

arXiv2023

From combinatorial maps to correlation functions in loop models

In two-dimensional statistical physics, correlation functions of the O(N) and Potts models may be written as sums over configurations of non-intersecting loops. We define sums associated to a large class of combinatorial maps (also known as ribbon graphs). We allow disconnected maps, but not maps that include monogons. Given a map with n vertices, we obtain a function of the moduli of the corresponding punctured Riemann surface. Due to the map's combinatorial (rather than topological) nature, that function is single-valued, and we call it an n-point correlation function. We conjecture that in the critical limit, such functions form a basis of solutions of certain conformal bootstrap equations. They include all correlation functions of the O(N) and Potts models, and correlation functions that do not belong to any known model. We test the conjecture by counting solutions of crossing symmetry for four-point functions on the sphere.

hep-th↗

arXiv2023

Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via $U_{\mathfrak{q}}\mathfrak{sl}_2$ Symmetry

We derive by the traditional algebraic Bethe ansatz method the Bethe equations for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [J. Phys. A 37 (2004), 433-440, arXiv:hep-th/0304092]. The technical difficulties due to the breaking of $\mathsf{U}(1)$ symmetry and the absence of a reference state are overcome by an algebraic construction where the two-boundary Temperley-Lieb Hamiltonian is realised in a new $U_{\mathfrak{q}}\mathfrak{sl}_2$-invariant spin chain involving infinite-dimensional Verma modules on the edges [J. High Energy Phys. 2022 (2022), no. 11, 016, 64 pages, arXiv:2207.12772]. The equivalence of the two Hamiltonians is established by proving Schur-Weyl duality between $U_{\mathfrak{q}}\mathfrak{sl}_2$ and the two-boundary Temperley-Lieb algebra. In this framework, the Nepomechie condition turns out to have a simple algebraic interpretation in terms of quantum group fusion rules.

hep-th↗

arXiv2023

Non-invertible symmetries and RG flows in the two-dimensional $O(n)$ loop model

In a recent paper, Gorbenko and Zan [arXiv:2005.07708] observed that $O(n)$ symmetry alone does not protect the well-known renormalization group flow from the dilute to the dense phase of the two-dimensional $O(n)$ model under thermal perturbations. We show in this paper that the required "extra protection" is topological in nature, and is related to the existence of certain non-invertible topological defect lines. We define these defect lines and discuss the ensuing topological protection, both in the context of the $O(n)$ lattice model and in its recently understood continuum limit, which takes the form of a conformal field theory governed by an interchiral algebra.

math-ph↗

arXiv2022

Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model

We define the two-dimensional $O(n)$ conformal field theory as a theory that includes the critical dilute and dense $O(n)$ models as special cases, and depends analytically on the central charge. For generic values of $n\in\mathbb{C}$, we write a conjecture for the decomposition of the spectrum into irreducible representations of $O(n)$. We then explain how to numerically bootstrap arbitrary four-point functions of primary fields in the presence of the global $O(n)$ symmetry. We determine the needed conformal blocks, including logarithmic blocks, including in singular cases. We argue that $O(n)$ representation theory provides upper bounds on the number of solutions of crossing symmetry for any given four-point function. We study some of the simplest correlation functions in detail, and determine a few fusion rules. We count the solutions of crossing symmetry for the $30$ simplest four-point functions. The number of solutions varies from $2$ to $6$, and saturates the bound from $O(n)$ representation theory in $21$ out of $30$ cases.

hep-th↗

arXiv2022

Geometric Algebra and Algebraic Geometry of Loop and Potts Models

We uncover a connection between two seemingly separate subjects in integrable models: the representation theory of the affine Temperley-Lieb algebra, and the algebraic structure of solutions to the Bethe equations of the XXZ spin chain. We study the solution of Bethe equations analytically by computational algebraic geometry, and find that the solution space encodes rich information about the representation theory of Temperley-Lieb algebra. Using these connections, we compute the partition function of the completely-packed loop model and of the closely related random-cluster Potts model, on medium-size lattices with toroidal boundary conditions, by two quite different methods. We consider the partial thermodynamic limit of infinitely long tori and analyze the corresponding condensation curves of the zeros of the partition functions. Two components of these curves are obtained analytically in the full thermodynamic limit.

hep-th↗

arXiv2021

$U_{\mathfrak{q}}(\mathfrak{sl}_3)$ web models: Locality, phase diagram and geometrical defects

We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice H having a U_q(sl_n) quantum group symmetry. We focus here on the n=3 case of cubic webs, based on the Kuperberg A_2 spider, and illustrate its properties by comparisons with the well-known dilute loop model (the n=2 case) throughout. A local vertex-model reformulation is exhibited, analogous to the correspondence between the loop model and a three-state vertex model. The n=3 representation uses seven states per link of H, displays explicitly the geometrical content of the webs and their U_q(sl_3) symmetry, and permits us to study the model on a cylinder via a local transfer matrix. A numerical study of the central charge reveals that for each q $\in \mathbb{C}$ in the critical regime, |q|=1, the web model possesses a dense and a dilute critical point, just like its loop model counterpart. In the dense $q=-e^{i π/4}$ case, the n=3 webs can be identified with spin interfaces of the critical three-state Potts model defined on the triangular lattice dual to H. We also provide another mapping to a $\mathbb{Z}_3$ spin model on H itself, using a high-temperature expansion. We then discuss the sector structure of the transfer matrix, for generic q, and its relation to defect configurations in both the strip and the cylinder geometries. These defects define the finite-size precursors of electromagnetic operators. This discussion paves the road for a Coulomb gas description of the conformal properties of defect webs, which will form the object of a subsequent paper. Finally, we identify the fractal dimension of critical webs in the $q=-e^{i π/3}$ case, which is the n=3 analogue of the polymer limit in the loop model.

cond-mat.stat-mech↗

arXiv2021

The action of the Virasoro algebra in quantum spin chains. I. The non-rational case

We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra ("Koo-Saleur generators"[arXiv:hep-th/9312156]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [arXiv:2007.11539] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules $\mathcal{W}_{j,1}$ for $j\neq 0$ contains pairs of "conjugate states" with conformal weights $(h_{r,s},h_{r,-s})$ and $(h_{r,-s},h_{r,s})$ that give rise to dual structures: Verma or co-Verma modules. The limit of $\mathcal{W}_{0,\mathfrak{q}^{\pm2}}$ contains diagonal fields $(h_{r,1},h_{r,1})$ and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in $\mathfrak{q}^{\pm 2}$. In order to obtain matrix elements of Koo-Saleur generators at large system size $N$ we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.

hep-th↗

arXiv2021

The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic $Q$

The spectrum of conformal weights for the CFT describing the two-dimensional critical $Q$-state Potts model (or its close cousin, the dense loop model) has been known for more than 30 years. However, the exact nature of the corresponding $\hbox{Vir}\otimes\overline{\hbox{Vir}}$ representations has remained unknown up to now. Here, we solve the problem for generic values of $Q$. This is achieved by a mixture of different techniques: a careful study of "Koo--Saleur generators" [arXiv:hep-th/9312156], combined with measurements of four-point amplitudes, on the numerical side, and OPEs and the four-point amplitudes recently determined using the "interchiral conformal bootstrap" in [arXiv:2005.07258] on the analytical side. We find that null-descendants of diagonal fields having weights $(h_{r,1},h_{r,1})$ (with $r\in \mathbb{N}^*$) are truly zero, so these fields come with simple $\hbox{Vir}\otimes\overline{\hbox{Vir}}$ ("Kac") modules. Meanwhile, fields with weights $(h_{r,s},h_{r,-s})$ and $(h_{r,-s},h_{r,s})$ (with $r,s\in\mathbb{N}^*$) come in indecomposable but not fully reducible representations mixing four simple $\hbox{Vir}\otimes\overline{\hbox{Vir}}$ modules with a familiar "diamond" shape. The "top" and "bottom" fields in these diamonds have weights $(h_{r,-s},h_{r,-s})$, and form a two-dimensional Jordan cell for $L_0$ and $\bar{L}_0$. This establishes, among other things, that the Potts-model CFT is logarithmic for $Q$ generic. Unlike the case of non-generic (root of unity) values of $Q$, these indecomposable structures are not present in finite size, but we can nevertheless show from the numerical study of the lattice model how the rank-two Jordan cells build up in the infinite-size limit.

hep-th↗

arXiv2021

Nested Closed Paths in Two-Dimensional Percolation

For two-dimensional percolation on a domain with the topology of a disc, we introduce a nested-path operator (NP) and thus a continuous family of one-point functions $W_k \equiv \langle \mathcal{R} \cdot k^\ell \rangle $, where $\ell$ is the number of independent nested closed paths surrounding the center, $k$ is a path fugacity, and $\mathcal{R}$ projects on configurations having a cluster connecting the center to the boundary. At criticality, we observe a power-law scaling $W_k \sim L^{X_{\rm NP}}$, with $L$ the linear system size, and we determine the exponent $X_{\rm NP}$ as a function of $k$. On the basis of our numerical results, we conjecture an analytical formula, $X_{\rm NP} (k) = \frac{3}{4}ϕ^2 -\frac{5}{48}ϕ^2/ (ϕ^2-\frac{2}{3})$ where $k = 2 \cos(πϕ)$, which reproduces the exact results for $k=0,1$ and agrees with the high-precision estimate of $X_{\rm NP}$ for other $k$ values. In addition, we observe that $W_2(L)=1$ for site percolation on the triangular lattice with any size $L$, and we prove this identity for all self-matching lattices.

cond-mat.stat-mech↗

arXiv2021

$U_q(\mathfrak{sl}_n)$ web models and $\mathbb{Z}_n$ spin interfaces

This is the first in a series of papers devoted to generalisations of statistical loop models. We define a lattice model of $U_q(\mathfrak{sl}_n)$ webs on the honeycomb lattice, for $n \ge 2$. It is a statistical model of closed, cubic graphs with certain non-local Boltzmann weights that can be computed from spider relations. For $n=2$, the model has no branchings and reduces to the well-known O($N$) loop model introduced by Nienhuis. In the general case, we show that the web model possesses a particular point, at $q=e^{i π/(n+1)}$, where the partition function is proportional to that of a $\mathbb{Z}_n$-symmetric chiral spin model on the dual lattice. Moreover, under this equivalence, the graphs given by the configurations of the web model are in bijection with the domain walls of the spin model. For $n=2$, this equivalence reduces to the well-known relation between the Ising and O($1$) models. We define as well an open $U_q(\mathfrak{sl}_n)$ web model on a simply connected domain with a boundary, and discuss in particular the role of defects on the boundary.

cond-mat.stat-mech↗

arXiv2020

Lattice regularisation of a non-compact boundary conformal field theory

Non-compact Conformal Field Theories (CFTs) are central to several aspects of string theory and condensed matter physics. They are characterised, in particular, by the appearance of a continuum of conformal dimensions. Surprisingly, such CFTs have been identified as the continuum limits of lattice models with a finite number of degrees of freedom per site. However, results have so far been restricted to the case of periodic boundary conditions, precluding the exploration via lattice models of aspects of non-compact boundary CFTs and the corresponding D-brane constructions. The present paper follows a series of previous works on a $\mathbb{Z}_2$-staggered XXZ spin chain, whose continuum limit is known to be a non-compact CFT related with the Euclidian black hole sigma model. By using the relationship of this spin chain with an integrable $D_2^2$ vertex model, we here identify integrable boundary conditions that lead to a continuous spectrum of boundary exponents, and thus correspond to non-compact branes. In the context of the Potts model on a square lattice, they correspond to wired boundary conditions at the physical antiferromagnetic critical point. The relations with the boundary parafermion theories are discussed as well. We are also able to identify a boundary renormalisation group flow from the non-compact boundary conditions to the previously studied compact ones.

hep-th↗

arXiv2020

Critical percolation on the kagome hypergraph

We study the percolation critical surface of the kagome lattice in which each triangle is allowed an arbitrary connectivity. Using the method of critical polynomials, we find points along this critical surface to high precision. This kagome hypergraph contains many unsolved problems as special cases, including bond percolation on the kagome and $(3,12^2)$ lattices, and site percolation on the hexagonal, or honeycomb, lattice, as well as a single point for which there is an exact solution. We are able to compute enough points along the critical surface to find a very accurate fit, essentially a Taylor series about the exact point, that allows estimations of the critical point of any system that lies on the surface to precision rivaling Monte Carlo and traditional techniques of similar accuracy. We find also that this system sheds light on some of the surprising aspects of the method of critical polynomials, such as why it is so accurate for certain problems, like the kagome and $(3,12^2)$ lattices. The bond percolation critical points of these lattices can be found to 17 and 18 digits, respectively, because they are in close proximity, in a sense that can be made quantitative, to the exact point on the critical surface. We also discuss in detail a parallel implementation of the method which we use here for a few calculations.

cond-mat.stat-mech↗

arXiv2020

Cylinder partition function of the 6-vertex model from algebraic geometry

We compute the exact partition function of the isotropic 6-vertex model on a cylinder geometry with free boundary conditions, for lattices of intermediate size, using Bethe ansatz and algebraic geometry. We perform the computations in both the open and closed channels. We also consider the partial thermodynamic limits, whereby in the open (closed) channel, the open (closed) direction is kept small while the other direction becomes large. We compute the zeros of the partition function in the two partial thermodynamic limits, and compare with the condensation curves.

hep-th↗

arXiv2020

$N$-cluster correlations in four- and five-dimensional percolation

We study $N$-cluster correlation functions in four- and five-dimensional (4D, 5D) bond percolation by extensive Monte Carlo simulation. We reformulate the transfer Monte Carlo algorithm for percolation [Phys. Rev. E {\bf 72}, 016126 (2005)] using the disjoint-set data structure, and simulate a cylindrical geometry $L^{d-1}\times \infty$, with the linear size up to $L=512$ for 4D and $128$ for 5D. We determine with a high precision all possible $N$-cluster exponents, for $N \! =\!2$ and $3$, and the universal amplitude for a logarithmic correlation function. From the symmetric correlator with $N \! = \!2$, we obtain the correlation-length critical exponent as $1/ν\! =\! 1.4610(12)$ for 4D and $1/ν\! =\! 1.737 (2)$ for 5D, significantly improving over the existing results. Estimates for the other exponents and the universal logarithmic amplitude have not been reported before to our knowledge. Our work demonstrates the validity of logarithmic conformal field theory and adds to the growing knowledge for high-dimensional percolation.

cond-mat.stat-mech↗

arXiv2020

Geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: The interchiral conformal bootstrap

Based on the spectrum identified in our earlier work [arXiv:1809.02191], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the $Q$-state Potts model. Crucial in our approach is the existence of "interchiral conformal blocks", which arise from the degeneracy of fields with conformal weight $h_{r,1}$, with $r\in\mathbb{N}^*$, and are related to the underlying presence of the "interchiral algebra" introduced in [arXiv:1207.6334]. We also find evidence for the existence of "renormalized" recursions, replacing those that follow from the degeneracy of the field $Φ_{12}^D$ in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.

hep-th↗

arXiv2020

Analytic continuation of Bethe energies and application to the thermodynamic limit of the $SL(2,\mathbb{C})$ non-compact spin chains

We consider the problem of analytically continuing energies computed with the Bethe ansatz, as posed by the study of non-compact integrable spin chains. By introducing an imaginary extensive twist in the Bethe equations, we show that one can expand the analytic continuation of energies in the scaling limit around another 'pseudo-vacuum' sitting at a negative number of Bethe roots, in the same way as around the usual pseudo-vacuum. We show that this method can be used to compute the energy levels of some states of the $SL(2,\mathbb{C})$ integrable spin chain in the infinite-volume limit, and as a proof of principle recover the ground-state value previously obtained in [1] (for the case of spins $s=0, \bar{s}=-1$) by extrapolating results in small sizes. These results represent, as far as we know, the first (partial) description of the spectrum of $SL(2,\mathbb{C})$ non-compact spin chains in the thermodynamic limit.

math-ph↗

arXiv2020

Integrable boundary conditions in the antiferromagnetic Potts model

We present an exact mapping between the staggered six-vertex model and an integrable model constructed from the twisted affine $D_2^2$ Lie algebra. Using the known relations between the staggered six-vertex model and the antiferromagnetic Potts model, this mapping allows us to study the latter model using tools from integrability. We show that there is a simple interpretation of one of the known K-matrices of the $D_2^2$ model in terms of Temperley-Lieb algebra generators, and use this to present an integrable Hamiltonian that turns out to be in the same universality class as the antiferromagnetic Potts model with free boundary conditions. The intriguing degeneracies in the spectrum observed in related works (arXiv:nlin/0002050 and arXiv:1707.09260) are discussed.

math-ph↗