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Matrix Product Operator Algebras II: Phases of Matter for 1D Mixed States

Alberto Ruiz-de-Alarcón, José Garre-Rubio, András Molnár, David Pérez-García

Abstract

The classification of topological phases of matter is fundamental to understand and characterize the properties of quantum materials. In this paper we study phases of matter in one-dimensional open quantum systems. We define two mixed states to be in the same phase if both states can be transformed into the other by a shallow circuit of local quantum channels. We aim to understand the phase diagram of matrix product density operators that are renormalization fixed points. These states arise, for example, as boundaries of two-dimensional topologically ordered states. We first construct families of such states based on C*-weak Hopf algebras, the algebras whose representations form a fusion category. More concretely, we provide explicit local fine-graining and local coarse-graining quantum channels for the renormalization procedure of these states. Finally, we prove that those arising from C*-Hopf algebras are in the trivial phase.

Matrix Product Operator Algebras II: Phases of Matter for 1D Mixed States

Abstract

The classification of topological phases of matter is fundamental to understand and characterize the properties of quantum materials. In this paper we study phases of matter in one-dimensional open quantum systems. We define two mixed states to be in the same phase if both states can be transformed into the other by a shallow circuit of local quantum channels. We aim to understand the phase diagram of matrix product density operators that are renormalization fixed points. These states arise, for example, as boundaries of two-dimensional topologically ordered states. We first construct families of such states based on C*-weak Hopf algebras, the algebras whose representations form a fusion category. More concretely, we provide explicit local fine-graining and local coarse-graining quantum channels for the renormalization procedure of these states. Finally, we prove that those arising from C*-Hopf algebras are in the trivial phase.
Paper Structure (11 sections, 25 theorems, 141 equations)

This paper contains 11 sections, 25 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Renormalization Fixed Point MPDOs
  4. RFP MPDOs are boundary states of topological 2D PEPS
  5. Classification via shallow circuits of quantum channels
  6. Graphical notation for tensor networks
  7. The Canonical Regular Element
  8. Proof of Lemma 3.6
  9. Proof of Theorem 3.3
  10. Proof of Lemma 5.2
  11. Proof of Lemma 5.6

Key Result

Theorem 2.12

Let ${A}$ be a biconnected C*-WHA. Then, is a cocentral non-degenerate positive idempotent, known as the canonical regular element of $A$. Moreover, there exists a unique linear map $T\in\mathfrak{L}(A)$ such that for all elements $x\in A$, usually referred to as pulling-through identity. In particular, $T$ is an involutive algebra anti-homomorphism.

Theorems & Definitions (64)

  • Definition 2.1: see bohm_1996_coassociativenill_1998_axioms
  • Remark 2.2: see e.g. Subsection 2.1 in bohm_1999_weak
  • Remark 2.3: see Lemma 2.8 and Theorem 2.10 in bohm_1999_weak
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7: see kac_1966_finite
  • Example 2.8: cf. bohm_1996_coassociative
  • Remark 2.9: see e.g. etingof_2005_on_fusionnill_1998_axioms
  • Theorem 2.12: cf. molnar_2022_mpo
  • ...and 54 more