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arXiv:2609.13377

Integrable Chiral Quantum Field Theories with Spacetime-Dependent Interactions

Pradip Kattel

hep-thcond-mat.str-elmath-phnlin.SIquant-phSubmitted Sep 11, 2026• Announced Sep 15, 2026• 0 citations
v1 · Sep 11, 2026v2 · Oct 12, 2026
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Abstract

We construct a class of integrable chiral fermionic quantum field theories with interactions depending on both space and time. The construction starts with a unitary, regular, difference-form two-body scattering matrix that satisfies 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 coordinate-ordering sectors. The Yang--Baxter equation gives its local flatness, while periodicity on a spatial circle produces quantum Knizhnik--Zamolodchikov difference equations expressing global flatness. In terms of a reference amplitude satisfying the exchange and qKZ constraints, these transports determine the exact fixed-particle-number wavefunction. We derive the characteristic form of the spacetime-dependent interaction and illustrate the construction with a chiral $SU(2)$ Gross--Neveu realization. Independent reparametrizations of the right- and left-moving characteristics yield genuine space-time dependence, whereas equal-slope affine maps recover the spatially homogeneous, time-dependent case.

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We construct a class of integrable chiral fermionic quantum field theories with interactions depending on both space and time. The construction starts with a unitary, regular, difference-form two-body scattering matrix that satisfies 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 coordinate-ordering sectors. The Yang--Baxter equation gives its local flatness, while periodicity on a spatial circle produces quantum Knizhnik--Zamolodchikov difference equations expressing global flatness. In terms of a reference amplitude satisfying the exchange and qKZ constraints, these transports determine the exact fixed-particle-number wavefunction. We derive the characteristic form of the spacetime-dependent interaction and illustrate the construction with a chiral SU(2)SU(2)SU(2) Gross--Neveu realization. Independent reparametrizations of the right- and left-moving characteristics yield genuine space-time dependence, whereas equal-slope affine maps recover the spatially homogeneous, time-dependent case.
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 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
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.
In this paper we consider quantum Hamiltonians with time-dependent interaction strengths, and following the recently formulated generalized Bethe ansatz framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], we show that constraints imposed by integrability take the same form as the renormalization group flow equations corresponding to the respective Hamiltonians with constant interaction strengths. As a concrete example, we consider the anisotropic time-dependent Kondo model characterized by the time-dependent interaction strengths J∥(t)J_{\parallel}(t)J∥​(t) and J⊥(t)J_{\perp}(t)J⊥​(t). We construct an exact solution to the time-dependent Schrodinger equation and by applying appropriate boundary conditions on the fermion fields we obtain a set of matrix difference equations called the quantum Knizhnik-Zamolodchikov (qKZ) equations corresponding to the XXZ R-matrix. The consistency of these equations imposes constraints on the time-dependent interaction strengths J∥(t)J_{\parallel}(t)J∥​(t) and J⊥(t)J_{\perp}(t)J⊥​(t), such that the system is integrable. Remarkably, the resulting temporal trajectories of the couplings are shown to coincide exactly with the RG flow trajectories of the static Kondo model, establishing a direct and universal correspondence between integrability and renormalization-group flow in time-dependent quantum systems.
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.