Integrable Research Articles

Exactly Solvable Inhomogeneous Fermion Systems

Abstract 15 exactly solvable inhomogeneous (spinless) fermion systems on 1D lattices are constructed explicitly based on the discrete orthogonal polynomials of the Askey scheme, e.g. Krawtchouk, Hahn, Racah, Meixner, and q-Racah polynomials. The Schrödinger and Heisenberg equations are solved explicitly, as the entire set of the eigenvalues and eigenstates are known explicitly. The ground-state two-point correlation functions are derived explicitly. The multipoint correlation functions are obtained by Wick’s theorem. 15 corresponding exactly solvable XX spin systems are also displayed. They all have nearest-neighbor interactions. The exact solvability of the Schrödinger equation means that the corresponding Fokker–Planck equation is also exactly solvable. This leads to 15 exactly solvable birth and death fermions and 15 birth and death spin models. These provide plenty of material for calculating interesting quantities, e.g. entanglement entropy, etc.

One loop effective actions in Kerr-(A)dS black holes

We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ordinary differential equations with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasinormal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulas for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover, we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss the exact properties of the thermal version of the BH effective actions by providing a proof of the Denef-Hartnoll-Sachdev formula and explicitly computing it for new relevant cases. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulas can be given for higher spin matter and radiation fields in more general gravitational backgrounds. Published by the American Physical Society 2024

New cases of integrable ninth-order conservative and dissipative dynamical systems

New cases of integrable dynamical systems of the ninth order homogeneous in terms of variables are presented, in which a system on a tangent bundle to a four-dimensional manifold can be distinguished. In this case, the force field is divided into an internal (conservative) and an external one, which has a dissipation of a different sign. The external field is introduced using some unimodular transformation and generalizes the previously considered fields. Complete sets of both first integrals and invariant differential forms are given.

Controlling Energy Storage Crossing Quantum Phase Transitions in an Integrable Spin Quantum Battery.

We investigate the performance of a one-dimensional dimerized XY chain as a spin quantum battery. Such integrable model shows a rich quantum phase diagram that emerges through a mapping of the spins onto auxiliary fermionic degrees of freedom. We consider a charging protocol relying on the double quench of an internal parameter, namely the strength of the dimerization, and address the energy stored in the systems. We observe three distinct regimes, depending on the timescale characterizing the duration of the charging: a short-time regime related to the dynamics of the single dimers, a long-time regime related to the recurrence time of the system at finite size, and a thermodynamic limit time regime. In the latter, the energy stored is almost unaffected by the charging time and the precise values of the charging parameters, provided the quench crosses a quantum phase transition. Finally, we analytically prove that the three-timescale behavior and the strong dependence of the energy stored on the quantum phase diagram also hold in the quantum Ising chain in a transverse field. Our results can play a relevant role in the design of stable solid-state quantum batteries.

Fermionic integrable models and graded Borchers triples

We provide an operator-algebraic construction of integrable models of quantum field theory on 1+1-dimensional Minkowski space with fermionic scattering states. These are obtained by a grading of the wedge-local fields or, alternatively, of the underlying Borchers triple defining the theory. This leads to a net of graded-local field algebras, of which the even part can be considered observable, although it is lacking Haag duality. Importantly, the nuclearity condition implying nontriviality of the local field algebras is independent of the grading, so that existing results on this technical question can be utilized. Application of Haag–Ruelle scattering theory confirms that the asymptotic particles are indeed fermionic. We also discuss connections with the form factor programme.

On Integrability of Certain Classes of Variable Dissipation Systems

We study the non-conservative systems for which the methods for studying, for example, Hamiltonian systems is not applicable in general. Therefore, for such systems, it is necessary, in some sense, to “directly” integrate the main equation of dynamics. Herewith, we offer more universal interpretation of both obtained cases and new ones of complete integrability in transcendental functions in dynamics in a non-conservative force field.

A quadrupole oscillator as an integrable model

A quadrupole oscillator is presented as an integrable model in the Born-Oppenheimer formalism with an electronic Hamiltonian being the quadrupole tensor. The electronic states of present concern are associated with a doubly degenerate positive eigenvalue of the electronic Hamiltonian, and accordingly the nuclear Hamiltonian takes a 2×2 matrix form. While the potential function for nuclear motion is proportional to r2, the kinetic energy operator is rather complicated, containing coupling terms with a Berry connection through adiabatic approximation. The energy eigenvalues, which receive a modification by a Chern number, get closer to those for the 3D isotropic harmonic oscillator if the angular momentum quantum number becomes sufficiently large.

Q-operators are ’t Hooft lines

We study ’t Hooft lines in four-dimensional holomorphic-topological Chern-Simons theory. We relate them to Q-operators in the theory of integrable systems. We give a physical interpretation of the fundamental TQ and QQ relations satisfied by Q-operators and conventional transfer matrices.

Quantum Information Scrambling in Adiabatically Driven Critical Systems.

Quantum information scrambling refers to the spread of the initially stored information over many degrees of freedom of a quantum many-body system. Information scrambling is intimately linked to the thermalization of isolated quantum many-body systems, and has been typically studied in a sudden quench scenario. Here, we extend the notion of quantum information scrambling to critical quantum many-body systems undergoing an adiabatic evolution. In particular, we analyze how the symmetry-breaking information of an initial state is scrambled in adiabatically driven integrable systems, such as the Lipkin-Meshkov-Glick and quantum Rabi models. Following a time-dependent protocol that drives the system from symmetry-breaking to a normal phase, we show how the initial information is scrambled, even for perfect adiabatic evolutions, as indicated by the expectation value of a suitable observable. We detail the underlying mechanism for quantum information scrambling, its relation to ground- and excited-state quantum phase transitions, and quantify the degree of scrambling in terms of the number of eigenstates that participate in the encoding of the initial symmetry-breaking information. While the energy of the final state remains unaltered in an adiabatic protocol, the relative phases among eigenstates are scrambled, and so is the symmetry-breaking information. We show that a potential information retrieval, following a time-reversed protocol, is hindered by small perturbations, as indicated by a vanishingly small Loschmidt echo and out-of-time-ordered correlators. The reported phenomenon is amenable for its experimental verification, and may help in the understanding of information scrambling in critical quantum many-body systems.

Spin(8,C)-Higgs Bundles and the Hitchin Integrable System

Let M(Spin(8,C)) be the moduli space of Spin(8,C)-Higgs bundles over a compact Riemann surface X of genus g≥2. This admits a system called the Hitchin integrable system, induced by the Hitchin map, the fibers of which are Prym varieties. Moreover, the triality automorphism of Spin(8,C) acts on M(Spin(8,C)), and those Higgs bundles that admit a reduction in the structure group to G2 are fixed points of this action. This defines a map of moduli spaces of Higgs bundles M(G2)→M(Spin(8,C)). In this work, the action of triality automorphism is extended to an action on the Hitchin integrable system associated with M(Spin(8,C)). In particular, it is checked that the map M(G2)→M(Spin(8,C)) is restricted to a map at the level of the Prym varieties induced by the Hitchin map. Necessary and sufficient conditions are also provided for the Prym varieties associated with the moduli spaces of G2 and Spin(8,C)-Higgs bundles to be disconnected. Finally, some consequences are drawn from the above results in relation to the geometry of the Prym varieties involved.

Hamiltonian integrable systems in a magnetic field and symplectic-Haantjes geometry

We investigate the geometry of classical Hamiltonian systems immersed in a magnetic field in three-dimensional (3D) Riemannian configuration spaces. We prove that these systems admit non-trivial symplectic-Haantjes manifolds, which are symplectic manifolds endowed with an algebra of Haantjes (1,1)-tensors. These geometric structures allow us to determine separation variables for known systems algorithmically. In addition, the underlying Stäckel geometry is used to construct new families of integrable Hamiltonian models immersed in a magnetic field.

The Construction of Analytical Exact Soliton Waves of Kuralay Equation

Abstract The primary objective of this work is to examine the Kuralay equation, which is a complex integrable coupled system, in order to investigate the integrable motion of induced curves. The soliton solutions derived from the Kuralay equation are thought to be the supremacy study of numerous significant phenomena and extensive applications across a wide range of domains, including optical fibres, nonlinear optics and ferromagnetic materials. The inverse scattering transform is unable to resolve the Cauchy problem for this equation, so the analytical method is used to produce exact travelling wave solutions. The modified auxiliary equation and Sardar sub-equation approaches are used to find solitary wave solutions. As a result, singular, mixed singular, periodic, mixed trigonometric, complex combo, trigonometric, mixed hyperbolic, plane and combined bright–dark soliton solution can be obtained. The derived solutions are graphically displayed in 2-D and 3-D glances to demonstrate how the fitting values of the system parameters can be used to predict the behavioural responses to pulse propagation. This study also provides a rich platform for further investigation.

Supersymmetric Integrable Hamiltonian Systems, Conformal Lie Superalgebras K(1, N = 1, 2, 3), and Their Factorized Semi-Supersymmetric Generalizations

We successively reanalyzed modern Lie-algebraic approaches lying in the background of effective constructions of integrable super-Hamiltonian systems on functional N=1,2,3- supermanifolds, possessing rich supersymmetries and endowed with suitably related compatible Poisson structures. As an application, we describe countable hierarchies of new nonlinear Lax-type integrable N=2,3-semi-supersymmetric dynamical systems and constructed their central extended superconformal Lie superalgebra K(1|3) and its finite-dimensional coadjoint orbits, generated by the related Casimir functionals. Moreover, we generalized these results subject to the suitably factorized super-pseudo-differential Lax-type representations and present the related super-Poisson brackets and compatible suitably factorized Hamiltonian superflows. As an interesting point, we succeeded in the algorithmic construction of integrable super-Hamiltonian factorized systems generated by Casimir invariants of the centrally extended super-pseudo-differential operator Lie superalgebras on the N=1,2,3-supercircle.

Special Joyce structures and hyperkähler metrics

Joyce structures were introduced by T. Bridgeland in the context of the space of stability conditions of a three-dimensional Calabi–Yau category and its associated Donaldson–Thomas invariants. In subsequent work, T. Bridgeland and I. Strachan showed that Joyce structures satisfying a certain non-degeneracy condition encode a complex hyperkähler structure on the tangent bundle of the base of the Joyce structure. In this work we give a definition of an analogous structure over an affine special Kähler (ASK) manifold, which we call a special Joyce structure. Furthermore, we show that it encodes a real hyperkähler (HK) structure on the tangent bundle of the ASK manifold, possibly of indefinite signature. Particular examples include the semi-flat HK metric associated to an ASK manifold (also known as the rigid c-map metric) and the HK metrics associated to certain uncoupled variations of BPS structures over the ASK manifold. Finally, we relate the HK metrics coming from special Joyce structures to HK metrics on the total space of algebraic integrable systems.

Observation of Two-Dimensional Dam Break Flow and a Gaseous Phase of Solitons in a Photon Fluid.

We report the observation of a two-dimensional (2D) dam break flow of a photon fluid in a nonlinear optical crystal. By precisely shaping the amplitude and phase of the input wave, we investigate the transition from one-dimensional (1D) to 2D nonlinear dynamics. We observe wave breaking in both transverse spatial dimensions with characteristic timescales determined by the aspect ratio of the input box-shaped field. The interaction of dispersive shock waves propagating in orthogonal directions gives rise to a 2D ensemble of solitons. Depending on the box size, we report the evidence of a dynamic phase characterized by a constant number of solitons, resembling a 1D soliton gas in integrable systems. We measure the statistical features of this gaslike phase. Our findings pave the way to the investigation of collective solitonic phenomena in two dimensions, demonstrating that the loss of integrability does not disrupt the dominant phenomenology.

CHAOTIC REGIMES GENERATED BY SPECIAL POINTS AND SINGULAR SOLUTIONS

The role of singular solutions and some singular points of ordinary differential equations with respect to the occurrence of chaotic regimes in physical systems is considered. It is shown that in those physical systems in which singular solutions exist, they can cause the occurrence of chaotic regimes. Examples of such systems are given. Of particular interest are the results showing that chaotic regimes can occur even in systems with one degree of freedom and even in completely integrable systems. These results, at first glance, violate known mathematical theorems (Poincare-Bendixson), and also destroy the paradigms of dynamic chaos. It is shown that this contradiction is an apparent contradiction. The reason for the appearance of chaotic dynamics in these cases are special solutions. At the points of this solution, the conditions of the uniqueness theorem are violated. In numerical calculations, this is manifested in the fact that a random component (numerical noise) appears in the calculations. As a result, the system under study becomes a system with 1.5 degrees of freedom. It is shown that in almost all cases, increasing the accuracy of the calculations allows suppressing chaotic dynamics in systems with one degree of freedom.

Higher gauge theory and integrability

In recent years, significant progress has been made in the study of integrable systems from a gauge theoretic perspective. This development originated with the introduction of four-dimensional Chern-Simons theory with defects, which provided a systematic framework for constructing two-dimensional integrable systems. In this article, we propose a novel approach to studying higher-dimensional integrable models employing techniques from higher category theory. Starting with higher Chern-Simons theory on the 4-manifold R×Y, we complexify and compactify the real line to CP1 and introduce the disorder defect ω=z−1dz. This procedure defines a holomorphic five-dimensional variant of higher Chern-Simons theory, which, when endowed with suitable boundary conditions, allows for the localization to a three-dimensional theory on Y. The equations of motion of the resulting model are equivalent to the flatness of a 2-connection (L,H), that we then use to construct the corresponding higher holonomies. We prove that these are invariants of homotopies relative boundary, which enables the construction of conserved quantities. The latter are labeled by both the categorical characters of a Lie crossed module and the infinite number of homotopy classes of surfaces relative boundary in Y. Moreover, we also demonstrate that the three-dimensional theory has left and right acting symmetries whose current algebra is given by an infinite-dimensional centrally extended affine Lie 2-algebra. Both of these conditions are direct higher homotopy analogs of the properties satisfied by the two-dimensional Wess-Zumino-Witten conformal field theory, which we therefore interpret as facets of integrable structures. Published by the American Physical Society 2024

Möbius Invariant Y-systems (Cluster Structures) for Miquel Dynamics

Abstract Miquel dynamics is a discrete time dynamics for circle patterns, which relies on Miquel’s six circle theorem. Previous work shows that the evolution of the circle centers satisfy the dSKP equation on the octahedral lattice $A_{3}$. As a consequence, Miquel dynamics is a discrete integrable system. Moreover, Miquel dynamics give rise to a real-valued cluster structure. The evolution of the cluster variables under Miquel dynamics is also called a Y-system in the discrete integrable systems community. If the Y-system is real positive-valued then the circle pattern is accompanied by an invariant dimer model, an exactly solvable model studied in statistical physics. However, while circle patterns are Möbius invariant, the circle centers and the Y-system are not Möbius invariant, which violates the so called transformation group principle. In this article we show that half the intersection points satisfy the dSKP equation as well, and we introduce two new real-valued Y-systems for Miquel dynamics that involve only the intersection points. Therefore, the new Y-systems are Möbius invariant, and thus satisfy the transformation group principle. We also show that the circle centers and intersection points combined satisfy the dSKP equation on the 4-dimensional octahedral lattice $A_{4}$. In addition, we present two more complex-valued Y-systems for Miquel dynamics, which are real-valued in and only in the case of integrable circle patterns. We also investigate the special cases of harmonic embeddings and s-embeddings, which relate to the spanning tree and Ising model, respectively.

Higher‐order integrable models for oceanic internal wave–current interactions

AbstractIn this paper, we derive a higher‐order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a “flat‐surface” assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long‐wave, and small‐amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations.

On the exploration of new solitary wave solutions for the classical integrable Kuralay-IIA system of equations

In this paper, we derive various new optical soliton solutions for the coupled Kuralay-IIA system of equations using an innovative solution approach known as the ϕ 6 − model expansion technique. This solution methodology employs a traveling wave transformation to reduce the considered problem into an easily solvable higher-order ordinary differential equation. Unlike other existing related methods, this solution approach adopted here allows us to extract a rich list of diverse exact soliton solutions for the considered problem. The obtained solutions incorporate the Jacobi elliptic functions which are shown to degenerate into trigonometric and hyperbolic function solutions. These solutions exhibit distinct wave structures consisting of dark, bright, rational, periodic, singular and mixed optical solitons profiles. In exploring the impact of spatial and temporal variables on the wave patterns of the considered model, physical structures of some of the obtained solitons solutions are characterized through 3D, contour and 2D wave profiles for selected parameter values. This not only ensures the validity of the solutions as well as the constraints arising from the solution technique but also offers researchers a deeper understanding of the properties of the considered problem. The outcomes here demonstrate the applicability, versatility and efficiency of the considered solution approach for deriving diverse new soliton solutions for even more complex systems of nonlinear evolution equations.