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The Floquet Baxterisation

Yuan Miao, Vladimir Gritsev, Denis V. Kurlov

Abstract

Quantum integrability has proven to be a useful tool to study quantum many-body systems out of equilibrium. In this paper we construct a generic framework for integrable quantum circuits through the procedure of Floquet Baxterisation. The integrability is guaranteed by establishing a connection between Floquet evolution operators and inhomogeneous transfer matrices obtained from the Yang-Baxter relations. This allows us to construct integrable Floquet evolution operators with arbitrary depths and various boundary conditions. Furthermore, we focus on the example related to the staggered 6-vertex model. In the scaling limit we establish a connection of this Floquet protocol with a non-rational conformal field theory. Employing the properties of the underlying affine Temperley--Lieb algebraic structure, we demonstrate the dynamical anti-unitary symmetry breaking in the easy-plane regime. We also give an overview of integrability-related quantum circuits, highlighting future research directions.

The Floquet Baxterisation

Abstract

Quantum integrability has proven to be a useful tool to study quantum many-body systems out of equilibrium. In this paper we construct a generic framework for integrable quantum circuits through the procedure of Floquet Baxterisation. The integrability is guaranteed by establishing a connection between Floquet evolution operators and inhomogeneous transfer matrices obtained from the Yang-Baxter relations. This allows us to construct integrable Floquet evolution operators with arbitrary depths and various boundary conditions. Furthermore, we focus on the example related to the staggered 6-vertex model. In the scaling limit we establish a connection of this Floquet protocol with a non-rational conformal field theory. Employing the properties of the underlying affine Temperley--Lieb algebraic structure, we demonstrate the dynamical anti-unitary symmetry breaking in the easy-plane regime. We also give an overview of integrability-related quantum circuits, highlighting future research directions.
Paper Structure (25 sections, 2 theorems, 180 equations, 16 figures)

This paper contains 25 sections, 2 theorems, 180 equations, 16 figures.

Table of Contents

  1. Introduction
  2. The Yang--Baxter integrability
  3. Overview of quantum circuits related to concepts of integrability
  4. Integrable Floquet protocols
  5. Trotterised circuits
  6. Protocols related to set-theoretic solutions of the Yang--Baxter equation
  7. Remarks on the exponential solutions
  8. Some possible future directions
  9. Floquet Baxterisation
  10. Floquet Baxterisation with periodic boundary condition: generic case
  11. Floquet Baxterisation for regular R matrix
  12. Floquet integrability with reflecting ends
  13. Example: Floquet integrability in the staggered 6-vertex model
  14. The affine Temperley--Lieb algebra
  15. The staggered 6-vertex model
  16. ...and 10 more sections

Key Result

Theorem 1

The periodic Floquet evolution operator with depth $n$ and the system size $L$ satisfying $L \, \mathrm{mod} \, n =0$ is integrable, i.e. where the inhomogeneous transfer matrix is defined in Eq. eq:inhomoT. The inhomogeneous transfer matrix $\mathbf{T} ( u , \{ u_j \}_{j=1}^n )$ is regarded as the Baxterised Floquet evolution operator.

Figures (16)

  • Figure 1: Diagrammatic demonstration of the R matrices and permutation operator.
  • Figure 2: Diagrammatic demonstration of the inhomogeneous transfer matrix of period $n=3$ with system size $L=6$ and inhomogeneities $\{ u_j\}_{j=1}^3$.
  • Figure 3: Demonstration of the brick-wall structure of the quantum circuit. Boxes correspond to two-qubit (qudit) gates and are realised by the $\check{\rm R}$-matrices satisfying the Yang-Baxter equation. We denote the layer of quantum gates as the depth of the quantum circuit. For the brick-wall construction, the period of the quantum circuit with respect to the lattice sites equals the depth.
  • Figure 4: Demonstration of the brick-wall circuits with period/depth $n=3$.
  • Figure 5: Diagrammatic demonstration of the Floquet evolution operator with period 2 as a tilted transfer matrix, the so-called "light-cone transfer matrix".
  • ...and 11 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof