Attendees of the meeting are listed below. For attendees giving a talk, click on their talk title to see the abstract.

Name Talk title (if applicable)
Angus Alexander

In this talk I will discuss Levinson’s theorem from mathematical scattering theory and give a new topological interpretation of Levinson’s theorem as an index pairing.

Murray Batchelor

It is now 35 years since the spin-1 biquadratic model was shown to give a particular representation of the Temperley-Lieb-Jones algebra. This result was used to obtain various physical properties of the biquadratic model from the equivalent spin-1/2 XXZ and 9-state Potts chains. However, relatively little attention has been paid to the ferromagnetic end of the eigenspectrum. This is also true for other models — ferromagnetic ground states have long been overlooked in comparison to seemingly more interesting antiferromagnetic ground states. For the biquadratic model both the physical and mathematical structure of ferromagnetic ground states are seen to be particularly rich. For example, the ground state degeneracies are asymptotically the famous golden spiral. It is also established that the entanglement entropy scales logarithmically, with the prefactor being half the number of type-B Goldstone modes. The latter in turn is identified to be the fractal dimension of the self-similar golden spiral fractal structure in Hilbert space. The appearance of Nambu-Goldstone modes in this context is surprising, as they feature fundamentally in high-energy particle physics, atomic and condensed matter physics. The ground state subspace of the ferromagnetic biquadratic model consists of one-site translation-invariant states, dimerized states, tetramerized states and so on. More generally the observation of such scale-invariant states has implications for the complete classification of quantum states of matter.

Lachlan Bennett

In this talk, we will associate a bosonic network to each complete bipartite graph. A Hamiltonian is defined with hopping terms on the edges of the graph, and a global interaction term depending on the vertex sets. For a generic graph, this Hamiltonian is superintegrable, and we derive a Bethe Ansatz solution for the energy and eigenstates. In certain tunnelling regimes, we explore the quantum dynamics and compute occupancy probabilities, focusing on a specific case we refer to as the ‘diamond models’.

Jean-Emile Bourgine

A new model for the non-Abelian quantum Hall effect is obtain from the diagonalization of a matrix model proposed by Dorey, Tong and Turner. The form of the Hamiltonian is reminiscent of a spin Calogero-Moser model but also involves higher representations of the non-abelian symmetry. I will present the energy spectrum, and ground state wave functions of this model, and discuss the affine Kac-Moody symmetries.

Peter Bouwknegt
Chris Bradly

In order to study long chain polymers many lattice models accommodate a pulling force applied to a particular part of the chain, often a free endpoint. This is in addition to well-studied features such as energetic interaction between the lattice polymer and a surface. However, the critical behaviour of the pulling force alone is less well studied, such as characterising the nature of the phase transition and particularly the values of the associated exponents. We investigate a simple model of lattice polymers subject to forced extension, namely self-avoiding walks (SAWs) on the square and simple cubic lattices with one endpoint attached to an impermeable surface and a force applied to the other endpoint acting perpendicular to the surface. In the thermodynamic limit the system undergoes a transition to a ballistic phase as the force is varied and it is known that this transition occurs whenever the magnitude of the force is positive, i.e. f > fc = 0. Using well established scaling arguments we show that the crossover exponentφfor the finite-size model is identical to the well-known exponentνd, which controls the scaling of the size of the polymer ind-dimensions. With extensive Monte Carlo simulations we test this conjecture and show that the value of φ is indeed consistent with the known values of ν_2 = 3/4 and ν_3 = 0.587597(7). Scaling arguments, in turn, imply the specific heat exponent α is 2/3 in two dimensions and 0.29815(2) in three dimensions.

Nathan Clisby
Jan de Gier

We derive an explicit formula for the transition probability of the asymmetric simple exclusion process on the half line with open boundaries. A new feature and technical complication in this setup is the appearance of a non-trivial empty-to-empty probability given by a Pfaffian formula related to diagonally symmetric alternating sign matrices. We provide new Cauchy identities for polynomial families depending on boundary parameters.

Martijn de Sterke
Harini Desiraju
Serena Dipierro
Peter Drummond
Maciej Dunajski

I discuss the impact of the positive cosmological constant on the interplay between the equivalence principle in general relativity, and the rules of quantum mechanics. There is an ambiguity in the definition of a phase of a wave function measured by inertial and accelerating observes which is a non-relativistic analogue of the Unruh effect. This will be put in the framework of a non-relativistic limit of twistor space.

Justine Fasquel

W-algebras are a large family of vertex algebras associated to nilpotent orbits of simple Lie algebras. For classical Lie algebras, they are parametrized by certain partitions.
Among the W-algebras of type sl(n) those with nilpotent orbits corresponding to hook partitions (m,1,1,…) of n are the most understood ones.
In this talk, we will show that in fact any W-algebras of type sl(n) should be expressed by using several hook-type W-algebras. We will illustrate with examples in small ranks. It’s a work in progress with T. Creutzig, A. Linshaw and N. Nakatsuka.

Zachary Fehily
Laurence Field

Calabi-Yau manifolds, which see interest as internal geometries of string theory compactifications to four dimensions, can be studied from the perspective of arithmetic.

This means that instead of considering the CY over the complex numbers, one instead realises it as a variety over each finite field. A zeta function serves as a generating function for the numbers of solutions in each finite field. On a certain locus in the moduli space, this function will exhibit unusual behaviour such as factorisation.

Remarkably, there is some physical interpretation to this locus. It solves either the rank-two attractor equations that govern the behaviour of scalar fields on the horizon of a 4d N=2 black hole, or the supersymmetric vacuum equations in type IIB flux compactifications.

Tyler Franke
Ethan Fursman

Vertex Operator Algebras (VOAs) provide a rigorous framework for studying conformal field theories. An important class of VOAs, known as W-algebras, are obtained via Quantum Hamiltonian Reduction (QHR). There is a complementary notion of Inverse Quantum Hamiltonian Reduction (IQHR). In this talk, I will discuss the sense in which these are inverses and what IQHR can tell us about W-algebras.

Tim Garoni

Stein’s Method is a powerful tool for studying distances between distributions of random variables. It has been used widely in the context of statistical mechanics to obtain appropriate approximations of various thermodynamic quantities. We will discuss an application of Stein’s Method to the O(N) model, a model of magnetism, and show how recent advances in Stein’s theory allow us to study its limiting behaviour at the phase transition.

Cengiz Gazi
Weiying Guo

Starting from the Izergin–Korepin 19-vertex model in the quadrant, we introduce two families of rational multivariate symmetric functions; these functions are in direct analogy with functions introduced by Borodin in the context of the higher-spin 6-vertex model in the quadrant. Our symmetric functions have many important properties, including a Cauchy identity and symmetrization formula.

This is a joint work with Alexandr Garbali and Michael Wheeler.

Anthony Guttmann
Bolin Han
Phillip Isaac
Mitchell Jones

Integrable systems with supersymmetric properties can be constructed using representations of superalgebras. We construct Gaudin superalgebras using the general linear gl$(m|n)$ Lie superalgebras and examine their properties. These systems are considered to be Yang-Baxter integrable as a result of their commuting transfer matrices. We find certain constructions expressing the associated transfer matrices as being identically zero. This results in a dilemma as to whether these systems can be regarded as Yang-Baxter integrable. In this talk, I will present our findings on how to remedy this issue. In addition, I will also reveal the process on how to identify when a zero transfer matrix will occur.

Nalini Joshi

Elliptic curves possess deep mathematical structures, which have led to cryptographic algorithms in a wide range of applications. They rely on the ease of multiplying points on elliptic curves combined with the difficulty of finding their component factors. An algorithm that was considered for implementing public-key cryptography on quantum computers went one step further; it was based on factoring curves, rather than factoring numbers. Elliptic curves also lie at the foundations of a very different theory in mathematical physics: the construction of integrable maps. In this talk, I describe how such integrable maps fit into the framework of elliptic curve cryptography.

Andrew Kels

The star-star relation is an equation that implies a Yang-Baxter equation for lattice models of statistical mechanics. The quasi-classical expansion of the star-star relation can be interpreted as an equation defined on vertices and edges of an octahedron. I will show how this leads to the new concept of hex systems, which are consistent systems of integrable difference equations that are defined in the honeycomb lattice. The quasi-classical expansion of the interaction-round-a-face form of the Yang-Baxter equation can be used to derive Lax pairs for the resulting difference equations.

Mario Kieburg

The information paradox of black holes is one of the long-standing problems in physics that shows the boundaries of our understanding of quantum physics. How can a quantum system radiate thermally (the Hawking radiation) while on the other hand preserving a unitary evolution? Page proposed the solution that the radiation only looks thermally and has shown that indeed a reduced density matrix originating from a generic pure state in a bipartite system becomes almost maximally mixed in the limit of large system size. Critiques of this idea claimed that there must be long ranged, especially non-local, entanglement between particles in this radiation. We quantified the size of the entries of the correlation matrix describing typical pure bosonic Gaussian states with the help of random matrices and found that with high probability typically two particles are separable in the limit of large system size and hence not entangled, in contrast to the critique. I will report about this study in my presentation. This work has been done in collaboration with Erik Aurell, Lucas Hackl, Pawel Horodecki and Robert Jonnson.

Johanna Knapp
Jonathan Kress
Pengcheng Lu

“The Izergin-Korepin (IK) model is a quantum integrable model related to the Dodd-Bullough-Mikhailov or Jiber-Mikhailov-Shabat model, one of two integrable relativistic models containing one scalar field (the other one is the sine-Gordon model). The R-matrix of the model corresponds to the simplest twisted affine algebra A^(2)_2. The IK model with open boundary conditions is related to the loop models and self-avoiding walks at a boundary. The boundary IK models with U (1)-symmetry, i.e. with periodic boundary condition or diagonal boundaries have been extensively studied. Recently, exact solutions of the model with generic integrable boundary conditions have been obtained by the so-called off-diagonal Bethe ansatz (ODBA). However, physical properties of the model are still missing due to the inhomogeneity of the resulting Bethe ansatz equations (BAEs) and unclear patterns of their Bethe roots.

Very recently, a novel t-W method has been proposed to calculate the physical quantities of quantum integrable systems with or without U(1) symmetry. The key point of the scheme lies in parameterizing the eigenvalues of transfer matrix by the zero roots instead of the Bethe roots. Subsequently, the method is applied successfully to solve certain SU(n) models, such as (anti)periodic XXZ model, (an)isotropic integrable J_1-J_2 models, the supersymmetric t−J model. Despite these progresses, an important open problem is whether the t-W method can be applied to other multi-component integrable models defined beyond the SU(n) algebra.

We study the IK model with generic integrable boundaries by the t-W method.
First, using the properties of the R-matrix and K-matrices, we parameterize the eigenvalues of transfer matrix by the zero roots. Based on them, we obtain the homogeneous zero roots BAEs and the corresponding patterns of the zero roots. We calculate the densities of the zero roots and the surface energies in different regimes of the model parameters. The results indicate that the contributions of the two boundaries to the surface energy are no longer additive and a correlation effect between the two boundary fields appears. The above process shows that the t-W method is also valid for non A-type models. In addition, the results may promote research in related models or fields.”

Ian Marquette

We reexamine different examples of reduction chains g ⊃ g’ of Lie algebras in order to show how the polynomials determining the commutant with respect to the subalgebra g’ leads to polynomial deformations of Lie algebras. These polynomial algebras have already been observed in various contexts, such as in the framework of superintegrable systems. Two relevant chains extensively studied in Nuclear Physics, namely the Elliott chain su(3) ⊃ so(3) and the chain so(5) ⊃ su(2) x u(1) related to the Seniority model, are analyzed in detail from this perspective. We show that these two chains both lead to three-generator cubic polynomial algebras, a result that paves the way for a more systematic investigation of nuclear models in relation to polynomial structures arising from reduction chains. In order to show that the procedure is not restricted to semisimple algebras, we also study the chain hat S(3) ⊃ sl(2,R) x so(2) involving the centrally-extended Schrödinger algebra in (3+1)-dimensional space-time.

Chihiro Matsui

“Understanding the thermalization mechanism of isolated quantum systems is one of the most well-developed studies in recent statistical mechanics. After the eigenstate thermalization hypothesis (ETH) is recasted as the most powerful candidate to explain thermalization phenomena, a plenty of related works have been achieved including the ones which test validity or violation of the ETH. Although generic isolated quantum systems are believed to obey the strong ETH, which requires all the energy eigenstates are macroscopically indistinguishable from the thermal states, it has been found that some energy eigenstates are different from the thermal states by violating the statement of strong ETH. These non-thermal states often show up in systems which do not thermalize, including the systems with integrability or many-body localization, while it has been found that such non-thermal states also show up in the system which does thermalize.

Non-thermal states in the latter case often emerge in association with partial solvability of the system. In the talk, I will show a new example of partially solvable models with the review of known examples.”

Joseph McGovern

Calabi-Yau manifolds, which see interest as internal geometries of string theory compactifications to four dimensions, can be studied from the perspective of arithmetic.

This means that instead of considering the CY over the complex numbers, one instead realises it as a variety over each finite field. A zeta function serves as a generating function for the numbers of solutions in each finite field. On a certain locus in the moduli space, this function will exhibit unusual behaviour such as factorisation.

Remarkably, there is some physical interpretation to this locus. It solves either the rank-two attractor equations that govern the behaviour of scalar fields on the horizon of a 4d N=2 black hole, or the supersymmetric vacuum equations in type IIB flux compactifications.

Damien McLeod
Jock McOrist
William Mead

We consider a family of symmetric functions defined by double-row transfer matrices of the six-vertex model with boundaries. For general open boundary parameters, we will evaluate the limit to the exclusion process and prove a new Cauchy identity.

Catherine Meusburger

Turaev-Viro-Barrett-Westbury state sum models define invariants of 3-manifolds and topological quantum field theories (TQFTs). They are also directly related to models in topological quantum computing and condensed matter physics, in particular to Kitaev’s quantum double models.

A topic of current interest are defects in these models, which involve distinguished submanifolds decorated by higher categorical data and are related to defect TQFTs.

I explain the relation between these models, how to introduce defects and discuss their geometrical and physical interpretation for simple choices of categorical data, namely Dijkgraaf-Witten models.

Madeline Nurcombe

The Temperley-Lieb (TL) algebra has a wide range of applications, from physical models of polymers and quantum spin chains, to knot theory. Its basis elements can be expressed as rectangular diagrams of non-crossing strings, with multiplication based on concatenation of diagrams. There are also one- and two-boundary TL algebras, describing physical systems with boundaries, but the two-boundary TL algebra requires its diagrams to have an even number of strings connected to each boundary. In this talk, I will introduce the ghost algebra, a two-boundary generalisation of the TL algebra that allows diagrams with odd or even numbers of strings at each boundary. Its diagrams contain ghosts: dots on the boundaries that act as bookkeeping devices to ensure associativity of multiplication. I will also introduce the dilute generalisation of the ghost algebra, and discuss the lattice models associated with these algebras. (arXiv: 2308.11966)

Anthony Parr

We extend the method for constructing symmetry operators of higher order for two-dimensional quantum Hamiltonians by Kalnins, Kress and Miller (2010). This expansion method expresses the integral in a finite power series in terms of lower degree integrals so as to exhibit it as a first-order differential operators. One advantage of this approach is that it does not require the a priori knowledge of the explicit eigenfunctions of the Hamiltonian nor the action of their raising and lowering operators as in their recurrence approach (2011). We obtain insight into the two-dimensional Hamiltonians of radial oscillator type with general second-order differential operators for the angular variable. We then re-examine the Hamiltonian of Tremblay, Turbiner and Winternitz (2009) as well as a deformation discovered by Post, Vinet and Zhedanov (2011) which possesses reflection operators. We will extend the analysis to spaces of constant curvature. We present explicit formulas for the integrals and the symmetry algebra, the Casimir invariant and oscillator realisations with finite-dimensional irreps which fill a gap in the literature.

Aram Perez

Stein’s Method is a powerful tool for studying distances between distributions of random variables. It has been used widely in the context of statistical mechanics to obtain appropriate approximations of various thermodynamic quantities. We will discuss an application of Stein’s Method to the O(N) model, a model of magnetism, and show how recent advances in Stein’s theory allow us to study its limiting behaviour at the phase transition.

Robert Pryor

B-twisted N=(2,2) hybrid models represent a class of superconformal field theories of string theoretic relevance. They can be understood as Landau-Ginzburg models fibred over a non-linear sigma model base. The bulk theory of these hybrid models is relatively well understood, but the boundary theory and the associated categories of D-branes are still relatively unexplored. In this talk, I will introduce these hybrid models, as well as some of their key features. I will then present a method of explicitly constructing hybrid B-branes and illustrate some specific examples.

Long Qiang

We have developed the capability in our laboratory to generate solitons with arbitrary dispersion profiles. These solitons satisfy a generalised nonlinear Schrödinger equation. To support this work, we have developed a method to iteratively generate solutions of such equations which take the form of an exponentially converging sum of functions. These functions each have an infinite number of discontinuous derivatives, but these disappear upon summation. Our method has the advantage of being systematic, requiring no initial assumptions or guesses, and can be applied to a range of different systems, to find both stable and unstable solutions.

Thomas Quella

Gaussian unitary transformations are transformations that map bosonic or fermionic Gaussian states into Gaussian states. An important class of examples is provided by the time evolution of Hamiltonians that are quadratic in the modes. We derive a closed-form expression for the expectation value of Gaussian unitaries in Gaussian states by studying the relation between the symplectic and the metaplectic group on the one hand and the orthogonal and the spin group on the other hand. This is achieved in the framework of Kähler geometry following and generalizing ideas by Rawnsley.

Milena Radnovic
Damodar Rajbhandari
Margaret Reid
David Ridout

Quantum hamiltonian reduction is a cohomological construction for chiral algebras of conformal field theory. In particular, it is used to construct certain commonly used W-algebras from affine Wess-Zumino-Witten theories. The catch is that the latter have to involve fractional levels — they are logarithmic CFTs even when the W-algebras are rational.

I will review some aspects of this construction, recalling some successes and failures, before outlining a relatively recent “inverse” approach that is well suited to studying W-algebra representation theory. One hope is that it will shed light on some of the failures of the usual reduction.

Pieter Roffelsen

Recently, B. Yang and J. Yang showed that the Sasa-Satsuma equation admits partial-rogue wave solutions, conditional to certain polynomials not having any real roots. Those polynomials are the generalised Okamoto polynomials, which appear in the theory of Painlevé IV. I will describe how Okamoto-Sakai theory allows for the exact determination of the number of real roots of the generalised Okamoto polynomials. In particular, this tells us which ones have no real roots and, for the corresponding partial-rogue waves, we obtain the exact number of rational solitons into which they split as time grows large.

Konrad Schöbel

Classification results for superintegrable systems are only known in dimensions two and three. We recently developed algebraic geometric methods to classify superintegrable systems in arbitrarily high dimension. In particular, we proved that the classification space for non-degenerate second order superintegrable systems is a quasi-projective variety. In this talk we demonstrate how to use our methods to solve the classification problem in dimension four. Concretely, we show that the corresponding variety is isomorphic to the spinor variety of maximally isotropic subspaces in a 10-dimensional pseudo-Euclidean space of split signature.

This work is an ongoing collaboration with Andreas Vollmer and Hans-Christian Graf von Bothmer (both University of Hamburg).

Alessandro Sfondrini

I will give a pedagogical review of how integrable-models techniques can be used to perform exact computations in string theory, by treating the string worldsheet model as a two-dimensional quantum field theory in finite volume. I will also describe the significance of these results in unraveling the holographic “AdS/CFT” correspondence, one of the major developments in quantum gravity of the last decades.

Alexander Sherman

The Scasimir, or Casimir’s ghost, plays an important role in representation theory of Lie superalgebras, especially that of osp(1|2n). Originally introduced by Pais and Rittenberg in this case, Maria Gorelik made a general study of the so-called ghost centre for the more general class of Kac-Moody superalgebras.

I will discuss a generalization of the ghost centre which in fact makes sense for any supersymmetric space. We will however limit our discussion mostly to the superalgebra sl(1|n), where we obtain a larger algebra, the full ghost centre, which acts nicely on irreducible representations. We hope to mention several open questions at the end as well.

Yang Shi
Gavrilo Šipka

To each classical Lie algebra g, one can associate an infinite-dimensional Hopf algebra (X(g)) Y(g) known as the (extended) Yangian of g. If we further quotient out by generators of “certain orders”, we obtain a sub-algebra known as the truncated (extended) Yangian of order p. In type A, the truncation at the first order simply results in the enveloping algebra (U(glN)) U(slN). Consequently, when one looks at the finite-dimensional irreducible representations of the corresponding (extended) Yangians, they can be described relatively efficiently in terms of the underlying Lie algebra representations. Unsurprisingly, for types B, C and D, this picture becomes more complicated and it is no longer sufficient to consider the truncation at the first order to describe the representation theory of the associated (extended) Yangians. In this talk, we will review the results in type A, and examine what occurs in the orthogonal cases (types B and D)

Yury Stepanyants

We present exact analytical solutions that describe new nonlinear entities governed by the Kadomtsev–Petviashvili (KP) equation with positive dispersion. They have shapes of a compact horseshoe ripplon (two-dimensional wave with an oscillatory structure in space) or horseshoe soliton (a lump-type formation). Both the ripplons and solitons decay in time. These entities can play an important role in the description of turbulence in plasma and other positive dispersion media.

Gabriele Tartaglino Mazzucchelli

Recently, there has been considerable interest in quantum field theories in two space-time dimensions (D=2) deformed by the irrelevant “TTbar” operator defined by the determinant of the energy-momentum tensor. In this talk, I will review how similar “TTbar-like” flows play an interesting role in characterising theories of non-linear electrodynamics in D=3,4, and interacting theories of chiral three-forms in D=6 dimensions.

Roberto Tateo
Enrico Valdinoci
Jaco van Tonder

Central spin models are closely related to Richardson-Gaudin models and have many present and potential physical applications. Recently the XX central spin model was shown to be integrable for both spin-1/2 and spin-1 central spin and arbitrary bath spin, with the eigenstates and eigenvalues found using a Bethe-Ansatz method of solution as well as its supersymmetric structure. We have shown that integrability carries over to the XY model generalisation. The conserved charges were found for both models and through the supersymmetry, the eigenstates for the spin-1/2 central spin Hamiltonian were found, having a similar form as for the XX model. Particular limits of the charges were used to produce several other non-equivalent XY integrable models.

Michael Wheeler

One of the key goals of integrable probability is to obtain exact formulas for the distribution functions of observables, and to show that they belong to the Kardar–Parisi–Zhang (KPZ) universality class. Depending on the model in question, this is often a difficult problem that blends algebraic and analytic techniques.

Occasionally, some of the difficulties may be circumvented by demonstrating a distribution match between the observable in question and a simpler one that is known to admit either determinant or Pfaffian structure. I will describe two instances where such a matching is known, and can be used to carry out asymptotic analysis of observables essentially for free.

Justin Widjaja

Conventional temporal optical solitons—arising from second-order dispersion—are known to emit dispersive waves in the presence of perturbative higher-order dispersion. The same effect appears also in Korteweg-de Vries solitons and has been recently explained mathematically in terms of the Stokes phenomenon. We realised that this formalism is generalisable to any nonlinear wave system supporting solitons which is perturbed by higher-order dispersion.

In a fibre laser, we experimentally generate optical pure-quartic solitons (originating from fourth-order dispersion) and subject them to a sixth-order perturbation. Performing time- and phase-resolved measurements of these pulses shows that the emitted radiation is consistent with the Stokes phenomenon. By showing that it applies to these novel higher-order dispersion systems, our results reinforce the universality of the radiation mechanism across all of wave physics.

Ben Wootton
Junze Zhang

The construction of superintegrable system Hamiltonians based on Lie algebras and their universal enveloping algebras has been widely studied over the past decades. However, most of the constructions rely on explicit differential operator realizations, homogeneous spaces and Marsden-Weinstein reductions.
In this talk, I will talk about an entirely algebraic approach on finding superintegrable systems and their corresponding symmetry algebras based on the subalgebras of a Lie algebra g. As examples, we will present the commutant of one dimensional subalgebras of a 2D conformal algebra and construct the corresponding Hamiltonians with integrals in algebraic forms. This will connect with recent work by Fordy and Huang in which they obtained superintegrable Hamiltonian on Darboux spaces. Moreover, I will briefly discuss the construction of Cartan commutant from A_n,B_n,C_n and D_n-type of semisimple Lie algebras. In this case, the Cartan centralizers contain higher rank polynomial algebras that relates to Racah algebras.

Yao-Zhong Zhang
Zongzheng (Eric) Zhou

The Ising model is one of the most fundamental models in statistical mechanics. Apart from the original spin representation, it is well known there are two geometric representations; one is the 2-state Fortuin-Kasteleyn random-cluster model and the other one is the loop model. In this talk, I will summarize our recent progress on the critical behaviour of the Ising model in the three representations on high-dimensional finite systems.