Non-invertible Kramers-Wannier duality-symmetry in the trotterized critical Ising chain

TL;DR

This work demonstrates that the critical transverse-field Ising model remains integrable under first-order trotterization when formulated via an inhomogeneous transfer-matrix framework from QISM, yielding a sequence of commuting conserved charges that depend on the discrete time-step ${\Omega}$. It reveals a generalized, non-invertible Kramers-Wannier duality for the trotterized evolution, introducing operators ${\frak D}_{\pm}(\Omega)$ that act as half-spatio-temporal translations and commute with the trotterized circuit. The authors extend these ideas to Floquet dynamics, showing integrability persists for certain Floquet protocols and that KW duality can map between trotterizations or between Floquet evolutions, enriching the symmetry structure of discrete-time Ising dynamics. The results offer a rigorous link between transfer-matrix formalisms, non-invertible dualities, and discrete-time quantum simulations, with potential implications for Majorana chains and twisted boundary conditions. In short, the paper provides a concrete, QISM-backed framework for integrable, discrete-time evolutions of critical spin chains and their duality symmetries, applicable to both static trotterized and Floquet settings.

Abstract

Integrable trotterization provides a method to evolve a continuous time integrable many-body system in discrete time, such that it retains its conserved quantities. Here we explicitly show that the first order trotterization of the critical transverse field Ising model is integrable. The discrete time conserved quantities are obtained from an inhomogeneous transfer matrix constructed using the quantum inverse scattering method. The inhomogeneity parameter determines the discrete time step. We then focus on the non-invertible Kramers-Wannier duality-symmetry for the trotterized evolution. We find that the discretization of both space and time leads to a doubling of these duality operators. They account for discrete translations in both space and time. As an interesting application, we find that these operators also provide maps between trotterizations of different orders. This helps us extend our results beyond the trotterization scheme and investigate the Kramers-Wannier duality-symmetry for finite time Floquet evolution of the critical transverse field Ising chain.

Figures (1)

• Figure 1: We can think of ${\frak D}_\pm(\Omega)^2$ as translations along the light-cone coordinates $x\pm t$. We restricted ourselves in the even parity sector ${\sf P}=1$.