Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States

We propose a tensor network approach known as the locally purified density operator (LPDO) to investigate the classification and characterization of symmetry-protected topological phases in open quantum systems. We extend the concept of injectivity, originally associated with matrix product states and projected entangled pair states, to LPDOs in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mo stretchy="false">(</a:mo><a:mrow><a:mn>1</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:mrow><a:mo stretchy="false">)</a:mo><a:mi mathvariant="normal">D</a:mi></a:mrow></a:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mrow><f:mo stretchy="false">(</f:mo><f:mrow><f:mn>2</f:mn><f:mo>+</f:mo><f:mn>1</f:mn></f:mrow><f:mo stretchy="false">)</f:mo><f:mi mathvariant="normal">D</f:mi></f:mrow></f:math> systems, unveiling two distinct types of injectivity conditions that are inherent for short-range entangled density matrices. Within the LPDO framework, we outline a classification scheme for decohered average symmetry-protected topological (ASPT) phases, consistent with earlier results obtained through spectrum sequence techniques. However, our approach offers an intuitive and explicit construction of ASPT states with the decorated domain-wall picture emerging naturally. We illustrate our framework with ASPT phases protected by a weak global symmetry and strong fermion parity symmetry and then extend it to a general group structure. Moreover, we derive both the classification data and the explicit forms of the obstruction functions using the LPDO formalism, particularly in the case of nontrivial group extension between strong and weak symmetries, where intrinsic ASPT phases may emerge. We demonstrate constructions of fixed-point LPDOs for ASPT phases in both <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:mo stretchy="false">(</k:mo><k:mrow><k:mn>1</k:mn><k:mo>+</k:mo><k:mn>1</k:mn></k:mrow><k:mo stretchy="false">)</k:mo><k:mi mathvariant="normal">D</k:mi></k:mrow></k:math> and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mrow><p:mo stretchy="false">(</p:mo><p:mrow><p:mn>2</p:mn><p:mo>+</p:mo><p:mn>1</p:mn></p:mrow><p:mo stretchy="false">)</p:mo><p:mi mathvariant="normal">D</p:mi></p:mrow></p:math> and discuss their physical realization in decohered or disordered systems. In particular, we construct examples of intrinsic ASPT states in <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mo stretchy="false">(</u:mo><u:mrow><u:mn>1</u:mn><u:mo>+</u:mo><u:mn>1</u:mn></u:mrow><u:mo stretchy="false">)</u:mo><u:mi mathvariant="normal">D</u:mi></u:mrow></u:math> and <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mrow><z:mo stretchy="false">(</z:mo><z:mrow><z:mn>2</z:mn><z:mo>+</z:mo><z:mn>1</z:mn></z:mrow><z:mo stretchy="false">)</z:mo><z:mi mathvariant="normal">D</z:mi></z:mrow></z:math> using the LPDO formalism.