Elliptic three-boson system, “two-level three-mode” JCD-type models and non-skew-symmetric classical r-matrices

Using the technique of the classical r-matrices and quantum Lax operators we construct the most general form of quantum integrable multi-boson and spin-multi-boson models associated with linear Lax algebras and sl(2)⊗sl(2)-valued classical non-dynamical r-matrices with spectral parameters. We consider example of non-skew-symmetric elliptic r-matrix and explicitly obtain one-, two- and three-boson integrable models and the corresponding one-, two- and three-mode two-level Jaynes-Cummings-Dicke-type models. We show that integrable “elliptic” two-level one-mode Jaynes-Cummings-Dicke Hamiltonian is hermitian and contains both rotating and counter-rotating terms.     

Form factors of exponential fields for two-parametric family of integrable models

A two-parametric family of integrable models (the SS model) that contains as particular cases several well known integrable quantum field theories is considered. After the quantum group restriction it describes a wide class of integrable perturbed conformal field theories. Exponential fields in the SS model are closely related to the primary fields in these perturbed theories. We use the bosonization approach to derive an integral representation for the form factors of the exponential fields in the SS model. The same representations for the sausage model and the cosine–cosine model are obtained as limiting cases. The results are tested at the special points, where the theory contains free particles.     

Isospectral Twirling and Quantum Chaos

We show that the most important measures of quantum chaos, such as frame potentials, scrambling, Loschmidt echo and out-of-time-order correlators (OTOCs), can be described by the unified framework of the isospectral twirling, namely the Haar average of a k-fold unitary channel. We show that such measures can then always be cast in the form of an expectation value of the isospectral twirling. In literature, quantum chaos is investigated sometimes through the spectrum and some other times through the eigenvectors of the Hamiltonian generating the dynamics. We show that thanks to this technique, we can interpolate smoothly between integrable Hamiltonians and quantum chaotic Hamiltonians. The isospectral twirling of Hamiltonians with eigenvector stabilizer states does not possess chaotic features, unlike those Hamiltonians whose eigenvectors are taken from the Haar measure. As an example, OTOCs obtained with Clifford resources decay to higher values compared with universal resources. By doping Hamiltonians with non-Clifford resources, we show a crossover in the OTOC behavior between a class of integrable models and quantum chaos. Moreover, exploiting random matrix theory, we show that these measures of quantum chaos clearly distinguish the finite time behavior of probes to quantum chaos corresponding to chaotic spectra given by the Gaussian Unitary Ensemble (GUE) from the integrable spectra given by Poisson distribution and the Gaussian Diagonal Ensemble (GDE).     

Yang-Baxter algebras of monodromy matrices in integrable quantum field theories

We consider a large class of two-dimensional integrable quantum field theories with non-abelian internal symmetry and classical scale invariance. We present a general procedure to determine explicitly the conserved quantum monodromy operator generating infinitely many non-local charges. The main features of our method are a factorization principle and the use of $\text{P}$, $\text{T}$, and internal symmetries. The monodromy operator is shown to satisfy a Yang-Baxter algebra, the structure constants (i.e. the quantum R-matrix) of which are determined by two-particle S-matrix of the theory. We apply the method to the chiral SU(N) and the O(2N) Gross-Neveu models.     

Exact results for the persistent current in an aharonov-bohm ring with a quantum impurity

We consider the problem of a persistent current in a one-dimensional mesoscopic ring with the electrons coupled by a spin exchange to a magnetic impurity. We show that this problem can be mapped onto an integrable model with a quadratic dispersion (with the latter property allowing for an unambiguous definition of the persistent current). We have solved the model exactly by a Bethe ansatz and found that the current is insensitive to the presence of the impurity. We conjecture that this result holds for any integrable quantum impurity model with an electronic dispersionε(k) that is an even function ofk.     

Systematic classical continuum limits of integrable spin chains and emerging novel dualities

We examine certain classical continuum long wave-length limits of prototype integrable quantum spin chains. We define the corresponding construction of classical continuum Lax operators. Our discussion starts with the XXX chain, the anisotropic Heisenberg model and their generalizations and extends to the generic isotropic and anisotropic gln$g{l}_n$ magnets. Certain classical and quantum integrable models emerging from special “dualities” of quantum spin chains, parametrized by c-number matrices, are also presented.     

Quantum nonintegrability and the classical limit for usp(4) systems

We investigate the transition from integrability to chaos in a system built of usp(4) elements, both in the quantum case and in its classical limit, obtained using coherent states. This algebraic Hamiltonian consists in an integrable term plus a nonlinear perturbation, and we see that the level spacing distribution for the quantum system is well approximated by the Berry-Robnik-Brody distribution, and accordingly the classical limit displays mixed dynamics.     

Darboux Transformations of Bispectral Quantum Integrable Systems

We present an approach to higher-dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the algebro-geometric definition of quantum integrability, we utilize the bispectral duality of quantum Hamiltonian systems to construct nontrivial Darboux transformations between completely integrable quantum systems. As an application, we are able to construct new quantum integrable systems as the Darboux transforms of trivial examples (such as symmetric products of one dimensional systems) or by Darboux transformation of well-known bispectral systems such as quantum Calogero–Moser.     

Quantum hydrodynamics from large-<Emphasis Type="Italic">n</Emphasis> supersymmetric gauge theories

We study the connection between periodic finite-difference Intermediate Long Wave (\(\Delta \hbox {ILW}\)) hydrodynamical systems and integrable many-body models of Calogero and Ruijsenaars-type. The former describe quantum cohomology and quantum K-theory of the ADHM moduli space of Abelian instantons, while the latter arise in the instanton counting of four- and five-dimensional supersymmetric gauge theories with eight supercharges in the presence of defects. Using string theory dualities, we provide correspondences between hydrodynamical and many-body integrable systems. In particular, we match the energy spectra on both sides.     

Supersymmetric quantum mechanics

We give a general construction for supersymmetric Hamiltonians in quantum mechanics. We find that N-extended supersymmetry imposes very strong constraints, and for N > 4 the Hamiltonian is integrable. We give a variety of examples, for one-particle and for many-particle systems, in different numbers of dimensions.     

Nekrasov prepotential with fundamental matter from the quantum spin chain

Nekrasov functions were conjectured in Mironov and Morozov (2009) [1] to be related to exact Bohr-Sommerfeld periods of quantum integrable systems. This statement was thoroughly checked for the case of the pure SU(Nc) gauge theory in Mironov and Morozov (2009) [2] and Popolitov (2010) [3]. Here we successfully perform a set of checks in the case of gauge group SU(Nc) with additional Nf fundamental hypermultiplets. We show that the Baxter equation for the spin chain gives the same quantum periods as the one for the Gaudin system in this case.     

Spectral duality in integrable systems from AGT conjecture

We describe relationships between integrable systems with N degrees of freedom arising from the Alday-Gaiotto-Tachikawa conjecture. Namely, we prove the equivalence (spectral duality) between the N-cite Heisenberg spin chain and a reduced gl _N Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the Alday-Gaiotto-Tachikawa relation in the Nekrasov-Shatashvili (and further the Seiberg-Witten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. The quantum duality extends this to the equivalence of the corresponding Baxter-Schrödinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2 × 2 and N × N representations of the Toda chain and the famous Adams-Harnad-Hurtubise duality.     

Quantum corrections to the classical reflection factor in a2(1) Toda field theory

The O(β²) quantum correction to the classical reflection factor is calculated for one of the integrable boundary conditions of a₂⁽¹⁾ affine Toda field theory. This is found to agree with the conjectured exact reflection factor of the quantum theory. We consider the existence of other exact reflection factors consistent with our perturbative answer and examine the question of how duality transformations might relate theories with different boundary conditions.     

An Approach to Loop Quantum Cosmology Through Integrable Discrete Heisenberg Spin Chains

The quantum evolution equation of Loop Quantum Cosmology (LQC)—the quantum Hamiltonian constraint—is a difference equation. We relate the LQC constraint equation in vacuum Bianchi I separable (locally rotationally symmetric) models with an integrable differential-difference nonlinear Schrödinger type equation, which in turn is known to be associated with integrable, discrete Heisenberg spin chain models in condensed matter physics. We illustrate the similarity between both systems with a simple constraint in the linear regime.     

An open-boundary integrable model of three coupled XY spin chains

The integrable open-boundary conditions for the model of three coupled one-dimensional XY spin chains are considered in the framework of the quantum inverse scattering method. The diagonal boundary K-matrices are found and a class of integrable boundary terms is determined. The boundary model Hamiltonian is solved by using the coordinate space Bethe ansatz technique and Bethe ansatz equations are derived.     

Reflection amplitudes of boundary Toda theories and thermodynamic Bethe ansatz

We study the ultraviolet asymptotics in A_n affine Toda theories with integrable boundary actions. The reflection amplitudes of non-affine Toda theories in the presence of conformal boundary actions have been obtained from the quantum mechanical reflections of the wave functional in the Weyl chamber and used for the quantization conditions and ground-state energies. We compare these results with the thermodynamic Bethe ansatz derived from both the bulk and (conjectured) boundary scattering amplitudes. The two independent approaches match very well and provide the non-perturbative checks of the boundary scattering amplitudes for Neumann and (+) boundary conditions.     

Universal R operator with deformed conformal symmetry

We study the general solution of the Yang–Baxter equation with deformed sl(2) symmetry. The universal R operator acting on tensor products of arbitrary representations is obtained in spectral decomposition and in integral forms. The results for eigenvalues, eigenfunctions and integral kernel appear as deformations of the ones in the rational case. They provide a basis for the construction of integrable quantum systems generalizing the XXZ spin models to the case of arbitrary not necessarily finite-dimensional representations on the sites.     

Renormalization character and quantum S-matrix for a classically integrable theory

The quantum theory of a N-component generalization of the sine-Gordon model is investigated. We find at the one-loop order that the model is renormalizable only when the corresponding classical theory is completely integrable: N = 1 (sine-Gordon model) and N = 2 (reduced O(4) σ-model). Moreover the coupling constant does nor renormalize in these two cases. Although the S-matrix for N = 2 is factorizable at the tree level, an anomaly appears at the one-loop order. Its effect is like a local quartic coupling.     

Integrable open boundary conditions for the Bariev model of three coupled XY spin chains

The integrable open-boundary conditions for the Bariev model of three coupled one-dimensional XY spin chains are studied in the framework of the boundary quantum inverse scattering method. Three kinds of diagonal boundary K-matrices leading to nine classes of possible choices of boundary fields are found and the corresponding integrable boundary terms are presented explicitly. The boundary Hamiltonian is solved by using the coordinate Bethe ansatz technique and the Bethe ansatz equations are derived.     

Integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons

A general graded reflection equation algebra is proposed and the corresponding boundary quantum inverse scattering method is formulated. The formalism is applicable to all boundary lattice systems where an invertible R-matrix exists. As an application, the integrable open-boundary conditions for the q-deformed supersymmetric U model of strongly correlated electrons are investigated. The diagonal boundary K-matrices are found and a class of integrable boundary terms are determined. The boundary system is solved by means of the coordinate space Bethe ansatz technique and the Bethe ansatz equations are derived. As a sideline, it is shown that all R-matrices associated with a quantum affine superalgebra enjoy the crossing-unitarity property.