The goal of my talk will be to summarise joint work with G. Kotousov and S. Lacroix approaching the quantisation of some integrable non-linear sigma models through their conformal limits. We focus mostly on the example of the Klimcık model, which is a two-parameter deformation of the Principal Chiral Model on a Lie group G. The UV fixed point of this theory is described classically by two...
Integrable spin chains play an important role, both in condensed matter and in high-energy physics. The presence of integrability allows us to apply several advanced techniques to address problems in these fields, especially when the Hamiltonians have nearest-neighbour interactions. When the Hamiltonians have interaction of range higher than two, however, there are several points that remain...
The Haldane--Shastry spin chain has long-range interactions and remarkable properties including Yangian symmetry at finite length and explicit highest-weight wave functions featuring Jack polynomials. This stems from the trigonometric spin-Calogero--Sutherland model, which is intimately related to affine Hecke algebras, already enjoys these properties from affine Schur–Weyl duality and reduces...
In this talk I will consider the energy density of integrable scattering theories, which can
be calculated using a linear integral TBA equation. These include the Gaudin-Yang and Lieb-Liniger models as well as integrable quantum field theories in which a conserved charge is coupled to an external field. I will explain how these TBA equations can be expanded perturbatively and investigate the...
Fermionic basis allows to solve the reflection relations introduced by Fateev, Fradkin, Lukyanov, Zamolodchikov and Zamolodcikov. We explain that the solution is compatible with the singular vectors of CFT. Necessary checks lead to certain integrals which look rather complicated but nevertheless allow explicit computation.
We discuss the hidden fermionic structure of the six vertex model
as well as its applications to the CFT and the sine-Gordon model.
The key role in the above construction is played by the transcendental
function $\omega$. Our special attention will be paid to the discussion
of few recent results on the properties of this function.
I will present an ongoing work on an integrable long-range version of the XXZ spin chain which can also be seen as a quantum deformation of the Haldane-Shastry model. At generic values for the deformation parameter, the model possess quantum affine symmetry, but when q is root of unity we expect extra symmetries to occur. We are studying the case q=i, which in the nearest neighbour case is...
In d>2 dimensions, the bootstrap program for conformal field theory (CFT) explores the constraints on scaling dimensions and OPE coefficients imposed by the crossing symmetry equations of correlation functions. The kinematical constituents of the crossing equations, called conformal blocks, were recently identified as the wave functions of certain many body quantum integrable systems in the...
The XYZ spin chain describes a chain of spin 1/2 quantum particles with a general anisotropic interaction between neighbors. When the anisotropy parameters satisfy J_x J_y + J_x J_z + J_y J_z = 0, the chain has an underlying supersymmetry. It is then possible to obtain exact results even for finite size systems. In the special case of the XXZ chain, this is related to very interesting...
A ’t Hooft loop is the simplest disorder operator in the N=4 super-Yang-Mills theory exactly S-dual to a Wilson loop. Insertion of a ’t Hooft loop defines a defect CFT which is expected to be integrable. I will describe how to compute 1pt functions in this setup by combining S-duality, localization and holography for protected operators and integrability for unprotected.
My talk will be about an application of the snake modules intruduced by Moukhin and Young in 2012. I will give a short summary of the construction regarding a recursion formula for the correlation functions of the rational spin ½ XXX chain that was done by Boos et al. in 2004. Most surprisingly, the behaviour of this recursion is completely determined by the Kirillov Reshetikhin modules....
Fragmentation of the Hilbert space is a specific fenomena that appears in systems with a presence of non-dynamical domain walls which lead to the exponential degeneration of energy levels. Such degeneration was observed in multiple systems. In particular such property was discovered in the folded-XXZ model ArXiv:2105.02252 (see also 2009.04995, 2011.01159). Further generalization of this model...
We introduce orthogonal ring patterns consisting of pairs of concentric circles. They generalize orthogonal circle patterns which can be treated as conformal limit. It is shown that orthogonal ring patterns in euclidean and hyperbolic planes and in a sphere are governed by integrable equations, in particular by the discrete master equation Q4. We deliver variational principles which are used...
We consider relations between discrete surfaces and integrability. More specifically, we consider discretizations of conjugate nets. We introduce a set of projective invariants together with a Poisson structure. This Poisson structure turns out to be invariant under discrete dynamics. In the case of biperiodic nets and circular nets we show the existence of commuting Hamiltonians.
The ODE/IM correspondence is, in a nutshell, the fact that the solutions to the Bethe equations of some integrable quantum field theory can be exactly represented as spectral determinants of some linear differential operators. This discovery goes back to Dorey and Tateo 1998 paper, and there has been tremendous development since then, which, mathematically speaking, amounts to a formidable...
We show that the celebrated six-vertex model of statistical mechanics (along with its multistate generalizations) can be reformulated as an Ising-type model with only a two-spin interaction. Such a reformulation unravels remarkable factorization properties for row to row transfer matrices, allowing one to uniformly derive all functional relations for their eigenvalues and present the...
We introduce a family of geometrical lattice models generalising the well-known loop model on the hexagonal lattice. These models have a $U_q(sl_n)$ quantum group symmetry, the loop model being the $n=2$ case. The general models give rise to branching webs and describe, at a special point, the interfaces in $Z_n$ symmetric spin models. We mainly discuss the $n=3$ case of bipartite cubic webs,...
Although the open XXZ spin chain with non-diagonal boundary terms is known to be integrable, the corresponding Bethe ansatz equations cannot be derived directly because of the lack of a suitable reference state. They also present many unusual features, among which an “inhomogeneous” term which vanishes only if the parameters of the model satisfy a certain quantization condition known as the...
Local operators in interaction round a face models can be expressed in terms of generalized transfer matrices. We use the properties of the local Boltzmann weights to derive discrete functional equations of reduced q-Knizhnik-Zamolodchikov type satisfied by the reduced density matrices for a sequence of consecutive sites in inhomogeneous
generalizations of these models. For the critical...
The periodic staggered six-vertex model with the anisotropy parameter (|q|=1) is critical and exhibits several phases with interesting universal behaviour. In a certain regime its scaling limit possesses a non-compact degree of freedom. In this talk, we discuss the influence of quantum group invariant boundary conditions on the finite-size spectrum of the model in the special case of the...
I will describe a broad class of d-dimensional conformal field theories of SU(N) adjoin scalar fields generalizing the 4d Fishnet CFT (FCFT) discovered by O. Gurdogan and myself, as a special limit of γ-deformed N = 4 SYM theory. In the planar limit the perturbation theory of FCFTs is dominated by the generalized “fishnet” planar Feynman graphs. These graphs are explicitly integrable, as was...
Basso-Dixon integrals evaluate rectangular fishnets – Feynman graphs with massless scalar propagators which form a m × n rectangular grid – which arise in certain one-trace four-point correlators in the ‘fishnet’ limit of N = 4 SYM. Recently, Basso, Dixon, Kosower, Krajenbrink and Zhong explored the thermodynamical limit m → ∞ with fixed aspect ratio n/m. They showed that the thermodynamical...
The Stampedes are a perturbative method for the computation of the leading logarithmic UV divergence of correctors of four or more points in quite general CFT, both in presence of gauge symmetry (e.g.: N=4 SYM) and in its absence (e.g.: Fishnet theories, Loom FCFTs). I will explain its formulation, providing examples about four-point functions and outline some future applications to Feynman...
In this talk, I will report on recent developments in the study of periodic Temperley-Lieb algebras and their applications in physics. I will first discuss ongoing progress done in collaboration with Y. Ikhlef in defining fusion products for representations of these algebras, that reproduce the expected fusion rules in conformal field theory. I will also describe joint work with A....
Evaluating a lattice path integral in terms of the spectral data and matrix elements of a suitably defined qunatum transfer matrix we obtain a `thermal form factor series' for the dynamical two-point functions of local operators in fundamental Yang-Baxter integrable models at finite temperature and, in the same way, in many other physically relevant settings. We shall consider in some detail...
A main goal in the field of nonequilibrium condensed matter physics is the control of electronic orders and the induction of ordered states which do not exist in equilibrium. Striking examples of nonthermal phases have been discovered experimentally, e. g. in fullerides and dichalcogenides, and also in numerical studies of correlated electron systems. Relevant insights have been gained into...
I will discuss a pair of coupled partial differential equations identified as Bogoliubov-de Gennes equations with Dirac operators and show that they appear naturally in the effective dynamics of inhomogeneous quantum many-body systems in one dimension. The equations feature an effective local gap that opens up due to inhomogeneities, coupling right- and left-moving degrees of freedom, leading...
The talk is based on the recent paper with Anton Zabrodin where we discussed an ensemble of particles with logarithmic repulsive interaction on a closed plane contour, a geometric deformation of the Dyson-Selberg integral Z_N(\Gamma)=\oint_\Gamma \prod_{i>j=1}^N|z_i-z_j|^{2\beta} d z_1\dots dz_N. In the limit of a large number of variables, the integral converges to the spectral determinant of...
It was recently recognized that various observables in four-dimensional supersymmetric gauge theories can be computed for an arbitrary 't Hooft coupling as determinants of certain semi-infinite matrices. I will show that these quantities can be expressed as Fredholm determinants of the so-called Bessel kernel and they are closely related to celebrated Tracy-Widom distribution (more precisely,...
I will present new results in the separation of variables (SoV) program for integrable models. The SoV is expected to be very powerful but until recently has been almost undeveloped beyond the simplest gl(2) examples. I will describe how to realize the SoV for any gl(N) spin chain and demonstrate how to solve the longstanding problem of deriving the scalar product measure in SoV. Using these...
We present recent results for the computational treatment of the spectra of the integrable staggered six-vertex model and the $3-\bar 3$ superspin chain.
The staggered six-vertex model has attracted the interest of several groups of authors who derived a wealth of results (e.g.~Ikhlef, Jacobsen, Saleur 08, 12; Frahm, Martins 12; Candu, Ikhlef 13; Frahm, Seel 14; Bazhanov, Kotousov, Koval,...
We prove that the asymptotic behavior of the recoupling coefficients of the symmetric group is characterized by a quantum marginal problem -- namely, by the existence of quantum states of three particles with given eigenvalues for their reduced density operators. This generalizes Wigner's observation that the semiclassical behavior of the 6j-symbols for SU(2) -- fundamental to the quantum...
Affine Gaudin models are field theories built from Kac-Moddy currents. At the classical level, they are known to be integrable, in the sense that they admit an infinite familly of Poisson-commuting charges. However, their quantum integrability is so far still conjectural. A natural starting point for exploring this question is the case of chiral affine Gaudin models, for which the underlying...
