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arXiv:2608.23770

General Construction of Time-Dependent Integrable Chiral Field Theories

Pradip Kattel

hep-thcond-mat.str-elmath-phnlin.SIquant-phSubmitted Aug 24, 2026• Announced Aug 26, 2026• 0 citations
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Abstract

We develop a general procedure for constructing time-dependent integrable chiral field theories from autonomous unitary difference-form $S$-matrices satisfying the Yang-Baxter equation. Spectral parameters are transported along the free right- and left-moving characteristics, defining a map from physical spacetime to spectral space on which the two-body scattering data are evaluated. Requiring spatially homogeneous right-left scattering forces the characteristic map to be affine. The inverse Cayley transform then determines the local contact interaction, while evaluation along the affine spectral trajectory fixes its time dependence. Thus, the nonautonomous interaction is determined by the autonomous scattering data together with chiral kinematics and spatial homogeneity, rather than being introduced independently. The Yang--Baxter equation supplies the factorized many-body transport, and on a spatial circle periodicity leads to quantum Knizhnik--Zamolodchikov (qKZ) equations whose compatibility defines a flat discrete transport in spectral space. Pulling the corresponding spectral-space amplitudes back to physical coordinates gives the time-dependent many-body wavefunctions. Rational $SU(N)$, trigonometric $U_q(\widehat{\mathfrak{sl}}_2)$, and rational $O(N)$ scattering illustrate how the same mechanism generates distinct nonautonomous interactions. The resulting framework gives a geometric route from autonomous factorized scattering data to time-dependent integrable field theories.

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We develop a general procedure for constructing time-dependent integrable chiral field theories from autonomous unitary difference-form SSS-matrices satisfying the Yang-Baxter equation. Spectral parameters are transported along the free right- and left-moving characteristics, defining a map from physical spacetime to spectral space on which the two-body scattering data are evaluated. Requiring spatially homogeneous right-left scattering forces the characteristic map to be affine. The inverse Cayley transform then determines the local contact interaction, while evaluation along the affine spectral trajectory fixes its time dependence. Thus, the nonautonomous interaction is determined by the autonomous scattering data together with chiral kinematics and spatial homogeneity, rather than being introduced independently. The Yang--Baxter equation supplies the factorized many-body transport, and on a spatial circle periodicity leads to quantum Knizhnik--Zamolodchikov (qKZ) equations whose compatibility defines a flat discrete transport in spectral space. Pulling the corresponding spectral-space amplitudes back to physical coordinates gives the time-dependent many-body wavefunctions. Rational SU(N)SU(N)SU(N), trigonometric Uq(sl^2)U_q(\widehat{\mathfrak{sl}}_2)Uq​(sl2​), and rational O(N)O(N)O(N) scattering illustrate how the same mechanism generates distinct nonautonomous interactions. The resulting framework gives a geometric route from autonomous factorized scattering data to time-dependent integrable field theories.
We construct a class of integrable chiral fermionic quantum field theories with interactions that depend on both space and time. The construction starts from a unitary, regular, difference-form two-body scattering matrix satisfying braiding unitarity and the Yang--Baxter equation. Each freely moving right- or left-moving particle carries a characteristic coordinate that remains constant along its trajectory, and the scattering matrix depends on differences of these coordinates. The many-body amplitudes form a discrete connection over the coordinate-ordering sectors. The Yang--Baxter equation gives local path independence, while periodic winding on a spatial circle produces quantum Knizhnik--Zamolodchikov (qKZ) difference equations expressing global flatness. Once a reference amplitude satisfying the exchange and qKZ constraints is specified, these transport equations determine the exact wavefunction in every fixed-particle-number sector. Independent reparametrizations of the right- and left-moving characteristics give couplings that depend on both space and time. For the chiral SU(2)SU(2)SU(2) Gross--Neveu realization, we solve the qKZ equations for this full characteristic family using off-shell Bethe vectors. The resulting amplitudes are written in the characteristic labels, and their pullbacks give exact many-body wavefunctions for the corresponding space-time-dependent couplings. The two-body amplitude is a ratio of Gamma functions that resums the phase shifts from successive scattering processes. Equal-slope affine maps give a spatially homogeneous specialization in which the coupling depends only on time, and the Bethe ansatz with constant coupling is recovered when the qKZ step vanishes.
We present a general method to construct and solve time-dependent integrable chiral quantum field theories on finite intervals from autonomous scattering and reflection data. Spatial homogeneity forces the spectral labels of right- and left-moving particles to be affine with a common slope, so bulk and boundary couplings sample one spectral trajectory. A complete round trip shifts the spectral labels and gives boundary quantum Knizhnik--Zamolodchikov equations. The Yang--Baxter and reflection equations provide local consistency, while boundary-qKZ flatness provides global consistency. We reconstruct the local Hamiltonian by Cayley inversion of the bulk and boundary matching matrices. We apply the construction to rational SU(N)SU(N)SU(N) scattering with fundamental boundary impurities and to an O(3)O(3)O(3) model with a spin-12\tfrac1221​ impurity. For the SU(2)SU(2)SU(2) impurity, we construct explicit boundary-qKZ solutions as lattice sums of off-shell boundary Bethe vectors and recover the static open-chain Bethe equations when the drive rate vanishes
In this paper we consider the problem of solving quantum field theories with time dependent interaction strengths. We show that the recently formulated framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], which is a generalization of the regular Bethe ansatz technique, provides the exact many-body wavefunction. In this framework, the time-dependent Schrodinger equation is reduced to a set of analytic difference equations and matrix difference equations, called the quantum Knizhnik-Zamolodchikov (qKZ) equations. The consistency of the solution gives rise to constraints on the time-dependent interaction strengths. For interaction strengths satisfying these constraints, the system is integrable, and the solution to the qKZ and the analytic difference equations provides the explicit form of the many-body wavefunction that satisfies the time-dependent Schrodinger equation. We provide a concrete example by considering the SU(2)SU(2)SU(2) Gross-Neveu model with time dependent interaction strength. Using this framework we solve the model with the most general time-dependent interaction strength and obtain the explicit form of the wave function.
We develop a generalized coordinate Bethe-ansatz framework for the massive Thirring model with an explicitly time-dependent fermion mass. Unlike the generalized Bethe ansatz for chiral quantum field theories, where particle transport acts in internal spin spaces and leads to quantum Knizhnik-Zamolodchikov equations, the present construction acts directly on particle coordinates and physical rapidities. Its central new feature is a rapidity lattice. Because the time-dependent mass couples right- and left-moving components already in the one-particle sector, a fixed-rapidity block is not closed under spatial winding. Periodicity therefore requires all sectors generated by the finite shift β→β−pβ\toβ-pβ→β−p. Particle exchange relates amplitudes on this lattice through scalar analytic difference equations of Ruijsenaars type, whose solutions determine the complete time-dependent many-body wavefunction. Consistency of the coordinate construction also selects an exponential evolution of the bare mass, providing a new realization of the time dependent integrability-renormalization group correspondence. In the static limit, the rapidity-lattice construction reduces to the conventional massive-Thirring Bethe equations. Through the massive-Thirring/sine Gordon equivalence, the result also connects this exact quantum construction to the generalized space-time dependent Lax formalism.
The generalized Bethe ansatz framework formulated in [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)] provides a unified framework to find exact solutions to quantum many-body systems with time-dependent coupling strengths with periodic boundary conditions. In this work we extend this framework to the case of open boundary conditions where in addition to the time-dependent interactions in the bulk, the boundary conditions are explicitly time-dependent. We show that for integrable time-dependent bulk coupling strengths, the generalized Bethe ansatz framework provides the time-dependent boundary conditions compatible with integrability and reduces the time-dependent Schrodinger equation to a set of matrix difference equations called the boundary quantum Knizhnik-Zamolodchikov (BqKZ) equations. The solution to the BqKZ equations provides the explicit form of the exact wavefunction. We further show that the RG invariants of the corresponding static model identify with the dynamical invariants in the time-dependent model.
Quantum integrability has been applied to a large variety of low dimensional Hamiltonians in Quantum Field Theory, Condensed Matter Physics, and Statistical Mechanics to obtain exact expressions for the spectrum and thermodynamics of these systems. In most of these studies the coupling constants are constant in time. Here we present an exact solution of the nonstationary Schrödinger equation for the Kondo Hamiltonian with a time-dependent spin-exchange coupling J(t) J ( t ) of the form \lambda t + p(t) ± \sqrt{(\lambda t + p(t))^2 + 16/3} λ t + p ( t ) ± ( λ t + p ( t ) ) 2 + 16 / 3 , where p(t) p ( t ) is an arbitrary periodic function, under periodic boundary concditions. Unlike previously studied time-dependent integrable models, which are rooted in the classical Yang–Baxter structure and associated Knizhnik–Zamolodchikov equations, our approach is based on the quantum Knizhnik–Zamolodchikov framework and the quantum Yang–Baxter algebra. Our results broaden the domain of time-dependent integrability to a genuinely quantum class of models and provide new tools for exploring coherent nonequilibrium dynamics in strongly correlated systems.