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Quantum Knizhnik-Zamolodchikov Equations and Integrability of Quantum Field Theories with Time-dependent Interaction Strength

Parameshwar R. Pasnoori

TL;DR

The paper develops an exact framework for solving quantum field theories with time-dependent interaction strengths by extending Bethe-ansatz methods to a qKZ-based formulation. By decomposing the problem into analytic-difference (phase) and qKZ (spin) components, it derives integrability constraints on $g(t)$, reducing the TDSE to a consistent set of equations that yield the full many-body wavefunction; for the SU(2) Gross-Neveu model, explicit constructions are provided and shown to recover the standard Bethe ansatz in the constant-coupling limit. The solution employs the off-shell Bethe ansatz to express the spin amplitudes as Jackson-type integrals over rapidities and uses transfer-matrix/Yang-Baxter machinery to ensure consistency across particle orderings. This work broadens exactly solvable models to time-dependent couplings, with potential applications to circuit QED, cold-atom platforms, and dynamical SPT phenomena, and points toward further generalizations to U(1) symmetry and varied boundary conditions.

Abstract

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 $SU(2)$ 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.

Quantum Knizhnik-Zamolodchikov Equations and Integrability of Quantum Field Theories with Time-dependent Interaction Strength

TL;DR

The paper develops an exact framework for solving quantum field theories with time-dependent interaction strengths by extending Bethe-ansatz methods to a qKZ-based formulation. By decomposing the problem into analytic-difference (phase) and qKZ (spin) components, it derives integrability constraints on , reducing the TDSE to a consistent set of equations that yield the full many-body wavefunction; for the SU(2) Gross-Neveu model, explicit constructions are provided and shown to recover the standard Bethe ansatz in the constant-coupling limit. The solution employs the off-shell Bethe ansatz to express the spin amplitudes as Jackson-type integrals over rapidities and uses transfer-matrix/Yang-Baxter machinery to ensure consistency across particle orderings. This work broadens exactly solvable models to time-dependent couplings, with potential applications to circuit QED, cold-atom platforms, and dynamical SPT phenomena, and points toward further generalizations to U(1) symmetry and varied boundary conditions.

Abstract

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 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.
Paper Structure (9 sections, 71 equations, 1 figure)

This paper contains 9 sections, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The ansatz wavefunction
  3. One particle case
  4. Two particle case and the S-matrix
  5. Three particle case and the Yang-Baxter algebra
  6. N particle case and the quantum Knizhnik-Zamolodchikov equations
  7. The case of constant interaction strength and the regular Bethe ansatz technique
  8. Quantum Knizhnik-Zamolodchikov equations and the exact wavefunction
  9. Discussion

Figures (1)

  • Figure 1: Figure depicts electrons (blue arrows) in a quantum wire which forms a loop. The electrons interact with each other through spin exchange interaction with time dependent strength $g(t)$, which is uniform throughout space.