Showing papers on "Integrable system published in 2022"

Introduction to the Special Issue on Emergent Hydrodynamics in Integrable Many-Body Systems

Abstract: Alvise Bastianello, Bruno Bertini4,5,∗, Benjamin Doyon and Romain Vasseur 1 Department of Physics, Technical University of Munich, 85748 Garching, Germany 2 Institute for Advanced Study, 85748 Garching, Germany 3 Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München, Germany 4 Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Oxford University, Parks Road, Oxford OX1 3PU, United Kingdom 5 School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom 6 Department of Mathematics, King’s College London, Strand WC2R 2LS, United Kingdom 7 Department of Physics, University of Massachusetts, Amherst, MA 01003, United States of America E-mail: bruno.bertini@nottingham.ac.uk

Fate of Measurement-Induced Phase Transition in Long-Range Interactions

Abstract: We consider quantum many-body dynamics under quantum measurements, where the measurement-induced phase transitions (MIPs) occur when changing the frequency of the measurement. In this work, we consider the robustness of the MIP for long-range interaction that decays as r^{-α} with distance r. The effects of long-range interactions are classified into two regimes: (i) the MIP is observed (α>α_{c}), and (ii) the MIP is absent even for arbitrarily strong measurements (α<α_{c}). Using fermion models, we demonstrate both regimes in integrable and nonintegrable cases. We identify the underlying mechanism and propose sufficient conditions to observe the MIP, that is, α>d/2+1 for general bilinear systems and α>d+1 for general nonintegrable systems (d: spatial dimension). Numerical calculation indicates that these conditions are optimal.

Riemann–Hilbert Problems and Soliton Solutions of Type (λ∗, −λ∗) Reduced Nonlocal Integrable mKdV Hierarchies

Abstract: Reduced nonlocal matrix integrable modified Korteweg–de Vries (mKdV) hierarchies are presented via taking two transpose-type group reductions in the matrix Ablowitz–Kaup–Newell–Segur (AKNS) spectral problems. One reduction is local, which replaces the spectral parameter λ with its complex conjugate λ∗, and the other one is nonlocal, which replaces the spectral parameter λ with its negative complex conjugate −λ∗. Riemann–Hilbert problems and thus inverse scattering transforms are formulated from the reduced matrix spectral problems. In view of the specific distribution of eigenvalues and adjoint eigenvalues, soliton solutions are constructed from the reflectionless Riemann–Hilbert problems.

Lax representation and bi-Hamiltonian structure of nonlinear Qiao model

Abstract: This paper explores accurate, stable, and novel soliton wave solutions of the nonlinear Qiao model. This model, which was derived in 2007, possesses a Lax representation and bi-Hamiltonian structure, where it is a second positive member in the utterly integrable hierarchy. The well-known generalized extended tanh-function method is employed to construct novel soliton wave solutions. The stable property of the obtained solutions is examined along with the Hamiltonian system’s characterizations. Furthermore, the accuracy of the obtained solutions is checked by comparing it with the model’s semi-analytical solutions that have been obtained by employing the variational iteration (VI) method. The obtained analytical and semi-analytical solutions are demonstrated through some distinct graphs to show more physical and dynamical behavior of the investigated model. The used analytical and semi-analytical schemes’ performance is checked to show if it is effective and powerful.

Soliton solutions by means of Hirota bilinear forms

Abstract: The paper aims to provide a brief overview of soliton solutions obtained through the Hirota direct method. A bilinear formulation of soliton solutions in both (1+1)-dimensions and (2+1)-dimensions is discussed, together with applications to various integrable equations. The Hirota conditions for N-soliton solutions are analyzed and a few open questions regarding higher-dimensional cases and generalized bilinear equations are presented.

Investigation into the Explicit Solutions of the Integrable (2+1) - Dimensional Maccari System via the Variational Approach

Abstract: In this paper, the integrable (2+1)-dimensional Maccari system (MS), which can model many complex phenomena in hydrodynamics, plasma physics and nonlinear optics, is investigated by the variational approach (VA). This proposed approach that based on the variational theory and Ritz-like method can construct the explicit solutions via the stationary conditions only taking two steps. Finally, the dynamic behaviors of the solutions are exhibited by choosing the appropriate parameters through the 3-D and density plots. It can be seen that the proposed method is concise and straightforward, and can be adopted to study the travelling wave theory in physics.

Correlation functions and transport coefficients in generalised hydrodynamics

Abstract: We review the recent advances on exact results for dynamical correlation functions at large scales and related transport coefficients in interacting integrable models. We discuss Drude weights, conductivity and diffusion constants, as well as linear and nonlinear response on top of equilibrium and non-equilibrium states. We consider the problems from the complementary perspectives of the general hydrodynamic theory of many-body systems, including hydrodynamic projections, and form-factor expansions in integrable models, and show how they provide a comprehensive and consistent set of exact methods to extract large scale behaviours. Finally, we overview various applications in integrable spin chains and field theories.

Long-lived phantom helix states in Heisenberg quantum magnets

Abstract: Exact solutions for quantum many-body systems are rare but provide valuable insights for the description of universal phenomena such as the non-equilibrium dynamics of strongly interacting systems and the characterization of new forms of quantum matter. Recently, specific solutions of the Bethe ansatz equations for integrable spin models were found. They are dubbed phantom Bethe states and can carry macroscopic momentum yet no energy. Here, we show experimentally that there exist special helical spin patterns in anisotropic Heisenberg chains which are long-lived, relaxing only very slowly in dynamics, as a consequence of such states. We use these phantom spin-helix states to directly measure the interaction anisotropy, which has a major contribution from short-range off-site interactions. We also generalize the theoretical description to higher dimensions and other non-integrable systems and find analogous stable spin helices, which should show non-thermalizing dynamics associated with so-called quantum many-body scars. These results have implications for the quantum simulation of spin physics, as well as many-body dynamics. In generic quantum many-body systems, initial configurations far from equilibrium are expected to undergo general thermalization. An experiment with ultracold atoms now shows evidence of a class of spin-helix states that evade such behaviour.

Reduced nonlocal integrable mKdV equations of type (−λ, λ) and their exact soliton solutions

Abstract: We conduct two group reductions of the Ablowitz–Kaup–Newell–Segur matrix spectral problems to present a class of novel reduced nonlocal reverse-spacetime integrable modified Korteweg–de Vries equations. One reduction is local, replacing the spectral parameter with its negative and the other is nonlocal, replacing the spectral parameter with itself. Then by taking advantage of distribution of eigenvalues, we generate soliton solutions from the reflectionless Riemann–Hilbert problems, where eigenvalues could equal adjoint eigenvalues.

Thermodynamic symmetry resolved entanglement entropies in integrable systems

Abstract: We develop a general approach to compute the symmetry-resolved Rényi and von Neumann entanglement entropies (SREE) of thermodynamic macrostates in interacting integrable systems. Our method is based on a combination of the thermodynamic Bethe ansatz and the Gärtner–Ellis theorem from large deviation theory. We derive an explicit simple formula for the von Neumann SREE, which we show to coincide with the thermodynamic Yang–Yang entropy of an effective macrostate determined by the charge sector. Focusing on the XXZ Heisenberg spin chain, we test our result against iTEBD calculations for thermal states, finding good agreement. As an application, we provide analytic predictions for the asymptotic value of the SREE following a quantum quench.

A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes

Abstract: In recent publications, the construction of explicit symplectic integrators for Schwarzschild- and Kerr-type spacetimes is based on splitting and composition methods for numerical integrations of Hamiltonians or time-transformed Hamiltonians associated with these spacetimes. Such splittings are not unique but have various options. A Hamiltonian describing the motion of charged particles around the Schwarzschild black hole with an external magnetic field can be separated into three, four, and five explicitly integrable parts. It is shown through numerical tests of regular and chaotic orbits that the three-part splitting method is the best of the three Hamiltonian splitting methods in accuracy. In the three-part splitting, optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators exhibit the best accuracies. In fact, they are several orders of magnitude better than the fourth-order Yoshida algorithms for appropriate time steps. The first two algorithms have a small additional computational cost compared with the latter ones. Optimized sixth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators have no dramatic advantages over the optimized fourth-order ones in accuracy during long-term integrations due to roundoff errors. The idea of finding the integrators with the best performance is also suitable for Hamiltonians or time-transformed Hamiltonians of other curved spacetimes including Kerr-type spacetimes. When the numbers of explicitly integrable splitting sub-Hamiltonians are as small as possible, such splitting Hamiltonian methods would bring better accuracies. In this case, the optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström methods are worth recommending.

Fractional Integrable Nonlinear Soliton Equations

Abstract: Nonlinear integrable equations serve as a foundation for nonlinear dynamics, and fractional equations are well known in anomalous diffusion. We connect these two fields by presenting the discovery of a new class of integrable fractional nonlinear evolution equations describing dispersive transport in fractional media. These equations can be constructed from nonlinear integrable equations using a widely generalizable mathematical process utilizing completeness relations, dispersion relations, and inverse scattering transform techniques. As examples, this general method is used to characterize fractional extensions to two physically relevant, pervasive integrable nonlinear equations: the Korteweg-deVries and nonlinear Schrödinger equations. These equations are shown to predict superdispersive transport of nondissipative solitons in fractional media.

On the analytical approximations to the nonplanar damped Kawahara equation: Cnoidal and solitary waves and their energy

Abstract: In this work, the non-integrable nonplanar (cylindrical and spherical) damped Kawahara equation (ndKE) is solved and analyzed analytically. The ansatz method is implemented for analyzing the ndKE in order to derive some high-accurate and more stable analytical approximations. Based on this method, two-different and general formulas for the analytical approximations are derived. The obtained solutions are applied for studying the distinctive features for both cylindrical and spherical dissipative dressed solitons and cnoidal waves in a complex plasma having superthermal ions. Moreover, the accuracy of the obtained approximations is numerically examined by estimating the global maximum residual error. Also, a general formula for the nonplanar dissipative dressed solitons energy is derived in details. This formula can recover the energy of the nonplanar dissipative dressed solitons, the planar dressed solitons, the planar damped dressed solitons, and the nonplanar dresses solitons. Both the suggested method and obtained approximations can help a large sector of authors interested in studying the nonlinear and complicated phenomena in various fields of science such as the propagating of nonlinear phenomena in physics of plasmas, nonlinear optics, communications, oceans and seas.

Matrix integrable fourth-order nonlinear Schrödinger equations and their exact soliton solutions

Abstract: We construct matrix integrable fourth-order nonlinear Schrödinger equations through reducing the Ablowitz-Kaup-Newell-Segur matrix eigenvalue problems. Based on properties of eigenvalue and adjoint eigenvalue problems, we solve the corresponding reflectionless Riemann-Hilbert problems, where eigenvalues could equal adjoint eigenvalues, and formulate their soliton solutions via those reflectionless Riemann-Hilbert problems. Soliton solutions are computed for three illustrative examples of scalar and two-component integrable fourth-order nonlinear Schrödinger equations.

Integrable deformations of sigma models

Abstract: In this pedagogical review we introduce systematic approaches to deforming integrable 2-dimensional sigma models. We use the integrable principal chiral model and the conformal Wess-Zumino-Witten model as our starting points and explore their Yang-Baxter and current-current deformations. There is an intricate web of relations between these models based on underlying algebraic structures and worldsheet dualities, which is highlighted throughout. We finish with a discussion of the generalisation to other symmetric integrable models, including some original results related to Z T cosets and their deformations, and the application to string theory. This review is based on notes written for lectures delivered at the school "Integrability, Dualities and Deformations", which ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.

Data-Driven Soliton Solutions and Model Parameters of Nonlinear Wave Models Via the Conservation-Law Constrained Neural Network Method

Abstract: In the process of the deep learning, we integrate more integrable information of nonlinear wave models, such as the conservation law obtained from the integrable theory, into the neural network structure, and propose a conservation-law constrained neural network method with the flexible learning rate to predict solutions and parameters of nonlinear wave models. As some examples, we study real and complex typical nonlinear wave models, including nonlinear Schrödinger equation, Korteweg-de Vries and modified Korteweg-de Vries equations. Compared with the traditional physics-informed neural network method, this new method can more accurately predict solutions and parameters of some specific nonlinear wave models even when less information is needed, for example, in the absence of the boundary conditions. This provides a reference to further study solutions of nonlinear wave models by combining the deep learning and the integrable theory.

Temporal Entanglement, Quasiparticles, and the Role of Interactions

Abstract: In quantum many-body dynamics admitting a description in terms of noninteracting quasiparticles, the Feynman-Vernon influence matrix (IM), encoding the effect of the system on the evolution of its local subsystems, can be analyzed exactly. For discrete dynamics, the temporal entanglement (TE) of the corresponding IM satisfies an area law, suggesting the possibility of an efficient representation of the IM in terms of matrix-product states. A natural question is whether integrable interactions, preserving stable quasiparticles, affect the behavior of the TE. While a simple semiclassical picture suggests a sublinear growth in time, one can wonder whether interactions may lead to violations of the area law. We address this problem by analyzing quantum quenches in a family of discrete integrable dynamics corresponding to the real-time Trotterization of the interacting XXZ Heisenberg model. By means of an analytical solution at the dual-unitary point and numerical calculations for generic values of the system parameters, we provide evidence that, away from the noninteracting limit, the TE displays a logarithmic growth in time, thus violating the area law. Our findings highlight the nontrivial role of interactions, and raise interesting questions on the possibility to efficiently simulate the local dynamics of interacting integrable systems.

Matrix integrable fifth-order mKdV equations and their soliton solutions

Abstract: We consider matrix integrable fifth-order mKdV equations via a kind of group reductions of the Ablowitz–Kaup–Newell–Segur matrix spectral problems. Based on properties of eigenvalue and adjoint eigenvalue problems, we solve the corresponding Riemann–Hilbert problems, where eigenvalues could equal adjoint eigenvalues, and construct their soliton solutions, when there are zero reflection coefficients. Illustrative examples of scalar and two-component integrable fifth-order mKdV equations are given.