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A complete classification of 2d symmetry protected states with symmetric entanglers

Alex Bols, Wojciech De Roeck, Michiel De Wilde, Bruno de O. Carvalho

Abstract

We consider symmetry protected topological states of 2d quantum spin systems, with a finite symmetry group $G$. It has been conjectured that such states are classified by the cohomology group $H^3(G,U(1))$, but the completeness of this classfication is an open problem. We restrict ourselves to symmetry protected topological states that can be prepared from a product state by a symmetric entangler. For this class of states, we prove that the classification by $H^3(G,U(1))$ is complete.

A complete classification of 2d symmetry protected states with symmetric entanglers

Abstract

We consider symmetry protected topological states of 2d quantum spin systems, with a finite symmetry group . It has been conjectured that such states are classified by the cohomology group , but the completeness of this classfication is an open problem. We restrict ourselves to symmetry protected topological states that can be prepared from a product state by a symmetric entangler. For this class of states, we prove that the classification by is complete.
Paper Structure (41 sections, 26 theorems, 109 equations, 5 figures)

This paper contains 41 sections, 26 theorems, 109 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Setup and Main Result
  3. Spin systems
  4. Stacking of spin systems
  5. Quantum cellular automata and finite depth quantum circuits
  6. Symmetries
  7. Stable equivalence of LPSs
  8. Symmetric entanglers
  9. Classification of symmetric entanglers
  10. Symmetry protected states
  11. Symmetric short-range entangled states
  12. Stable equivalence of $G$-states
  13. Classification of $G$-states
  14. Classification of 1d symmetric entanglers
  15. An index for 1d symmetric entanglers
  16. ...and 26 more sections

Key Result

Proposition 2.1

For $d = 0, 1, 2$, the monoid $(\mathop{\mathrm{\mathsf{sEnt}}}\nolimits_{G,d} / \sim)$ is a group and there is a group isomorphism

Figures (5)

  • Figure 1: A finite-depth quantum circuit (FDQC) on a spin chain. Each $V_{i,i+1}$ is a unitary gate acting non-trivially on sites $i,i+1$. The dots represent a pure product state $\ket \phi = \otimes_{i \in \mathbb Z} \ket{\phi_i}$, where each $\ket{\phi_i}$ is a unit vector in a finite-dimensional on-site Hilbert space $\mathcal{H}_i$.
  • Figure 2: An illustration of a depth 2 FDQC representation of $\alpha$. The gates comprising $\eta_x$ are shown in blue.
  • Figure 3: The automorphism $\alpha_k$ acts by conjugation with a unitary block. The colored lines represent a non-trivial action. In the bulk (green region), $\alpha_k$ acts as $\alpha$.
  • Figure 4: The block partitioned automorphisms $\alpha_{\text{even}}$ and $\alpha_{\text{odd}}$, each illustrated on a spin chain. Non-trivial actions are represented by colored lines. Analogously as in Figure \ref{['fig:alpha-k-1d']}, they act as $\alpha$ on green regions.
  • Figure 5: An illustration of the the build-up of the example class. In order to better represent the actions of $A_1$ and $A_2$, we take the $y$-axis to point to the right, and the $x$-axis pointing down. Each square lattice represent the spin system $\mathcal{B}$, with different on-site symmetries, given by repsectively $\Gamma, B$ and $\Gamma$. The entanglers between the symmetries are $A_1$ and $A_2$, acting componentwise on the indicated stripes.

Theorems & Definitions (50)

  • Proposition 2.1: Classification of symmetric entanglers
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5: Classification of 2d SPTs with symmetric entanglers
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • ...and 40 more