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Generalized Kramers-Wanier Duality from Bilinear Phase Map

Han Yan, Linhao Li

Abstract

We present the Bilinear Phase Map (BPM), a concept that extends the Kramers-Wannier (KW) transformation to investigate unconventional gapped phases, their dualities, and phase transitions. Defined by a matrix of $\mathbb{Z}_2$ elements, the BPM not only encapsulates the essence of KW duality but also enables exploration of a broader spectrum of generalized quantum phases and dualities. By analyzing the BPM's linear algebraic properties, we elucidate the loss of unitarity in duality transformations and derive general non-invertible fusion rules. Applying this framework to (1+1)D systems yields the discovery of new dualities, shedding light on the interplay between various Symmetry Protected Topological (SPT) and Spontaneous Symmetry Breaking (SSB) phases. Additionally, we construct a duality web that interconnects these phases and their transitions, offering valuable insights into relations between different quantum phases.

Generalized Kramers-Wanier Duality from Bilinear Phase Map

Abstract

We present the Bilinear Phase Map (BPM), a concept that extends the Kramers-Wannier (KW) transformation to investigate unconventional gapped phases, their dualities, and phase transitions. Defined by a matrix of elements, the BPM not only encapsulates the essence of KW duality but also enables exploration of a broader spectrum of generalized quantum phases and dualities. By analyzing the BPM's linear algebraic properties, we elucidate the loss of unitarity in duality transformations and derive general non-invertible fusion rules. Applying this framework to (1+1)D systems yields the discovery of new dualities, shedding light on the interplay between various Symmetry Protected Topological (SPT) and Spontaneous Symmetry Breaking (SSB) phases. Additionally, we construct a duality web that interconnects these phases and their transitions, offering valuable insights into relations between different quantum phases.
Paper Structure (9 sections, 66 equations, 2 figures, 1 table)

This paper contains 9 sections, 66 equations, 2 figures, 1 table.

Table of Contents

  1. Symmetry-twisting mapping of three-site BPM duality
  2. SPT phase invariant under three-site BPM duality and Kennedy-Tasaki duality
  3. The Hamiltonian of SPT phase
  4. String order parameters, ground state charge under twisted boundary condition and edge modes
  5. The generalized Kennedy-Tasaki transformation
  6. Four-site BPM duality transformation
  7. BPM duality under PBC and fusion rules
  8. Unitarity problem and symmetry-twist transformation
  9. Anomaly of four-site BPM duality symmetry

Figures (2)

  • Figure 1: Three gapped phases with $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry and the dualities between them.
  • Figure 2: Three phase transitions between two different gapped phases with $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry and $c=1$ and the dualities between them.