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arXiv:2403.17069

Tensor network formulation of symmetry protected topological phases in mixed states

Hanyu Xue, Jong Yeon Lee, Yimu Bao

Published Mar 25, 2024• 0 citations
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Abstract

We define and classify symmetry-protected topological (SPT) phases in mixed states based on the tensor network formulation of the density matrix. In one dimension, we introduce strong injective matrix product density operators (MPDO), which describe a broad class of short-range correlated mixed states, including the locally decohered SPT states. We map strong injective MPDO to a pure state in the doubled Hilbert space and define the SPT phases according to the cohomology class of the symmetry group in the doubled state. Although the doubled state exhibits an enlarged symmetry, the possible SPT phases are also constrained by the Hermiticity and the semi-positivity of the density matrix. We here obtain a complete classification of SPT phases with a direct product of strong $G$ and weak $K$ unitary symmetry given by the cohomology group $H^2(G, \text{U}(1))\oplus H^1(K, H^1(G, \text{U}(1)))$. The SPT phases in our definition are preserved under symmetric local circuits consisting of non-degenerate channels. This motivates an alternative definition of SPT phases according to the equivalence class of mixed states under a ``one-way" connection using symmetric non-degenerate channels. In locally purifiable MPDO with strong symmetry, we prove that this alternative definition reproduces the cohomology classification. We further extend our results to two-dimensional mixed states described by strong semi-injective tensor network density operators and classify the possible SPT phases.

DOI
10.48550/arxiv.2403.17069

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We define and classify symmetry-protected topological (SPT) phases in mixed states based on the tensor network formulation of the density matrix. In one dimension, we introduce strong injective matrix product density operators (MPDO), which describe a broad class of short-range correlated mixed states, including the locally decohered SPT states. We map strong injective MPDO to a pure state in the doubled Hilbert space and define the SPT phases according to the cohomology class of the symmetry group in the doubled state. Although the doubled state exhibits an enlarged symmetry, the possible SPT phases are also constrained by the Hermiticity and the semi-positivity of the density matrix. We here obtain a complete classification of SPT phases with a direct product of strong GGG and weak KKK unitary symmetry given by the cohomology group H2(G,U(1))⊕H1(K,H1(G,U(1)))H^2(G, \text{U}(1))\oplus H^1(K, H^1(G, \text{U}(1)))H2(G,U(1))⊕H1(K,H1(G,U(1))). The SPT phases in our definition are preserved under symmetric local circuits consisting of non-degenerate channels. This motivates an alternative definition of SPT phases according to the equivalence class of mixed states under a ``one-way" connection using symmetric non-degenerate channels. In locally purifiable MPDO with strong symmetry, we prove that this alternative definition reproduces the cohomology classification. We further extend our results to two-dimensional mixed states described by strong semi-injective tensor network density operators and classify the possible SPT phases.
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We develop a general framework for studying phases of mixed states with strong and weak symmetries, including non-invertible or categorical symmetries. The central idea is to consider a purification of the mixed state density matrix, which lives in a doubled Hilbert space. We propose a systematic classification of phases in this doubled Hilbert space, relying crucially on the Symmetry Topological Field Theory (SymTFT) approach. This framework applies not only to group symmetries but also, importantly, to non-invertible symmetries. We illustrate the approach in 1+1d to classify phases with strong (non-)invertible symmetries, which include strong-to-weak spontaneous symmetry breaking (SWSSB) phases and mixed strong/weak symmetry-protected topological phases (SPTs). We also develop an approach for studying symmetries that involve a combination of strong and weak symmetries. A noteworthy example of this has weak non-invertible Kramers-Wannier duality symmetry and strong Z2\mathbb{Z}_2Z2​ symmetry. The continuum description is complemented by a lattice model analysis informed by the SymTFT framework.
Modulated symmetries are internal symmetries that act in a spatially non-uniform manner. Consequently, when a modulated symmetry GintG_{\text{int}}Gint​ is combined with a spatial symmetry GspG_{\text{sp}}Gsp​, the total symmetry group takes the form of a semidirect product G=Gint⋊GspG=G_{\text{int}}\rtimes G_{\text{sp}}G=Gint​⋊Gsp​. Using matrix product states, we classify topological phases protected by modulated symmetries together with spatial symmetries in one spatial dimension. We show that these modulated symmetry-protected topological (SPT) phases are classified by H2(G,U(1)s)H^{2}(G,U(1)_s)H2(G,U(1)s​), in agreement with the crystalline equivalence principle, which states that SPT phases protected by symmetries involving spatial elements are in one-to-one correspondence with internal SPT phases protected by the same symmetries, viewed as acting internally. Furthermore, we provide a matrix product state derivation of the Lyndon-Hochschild-Serre spectral sequence for the corresponding internal SPT phases, which enables us to construct an explicit correspondence between modulated SPT phases and internal SPT phases. As applications of this classification, we prove a Lieb-Schultz-Mattis (LSM) theorem for modulated symmetries that forbids the existence of symmetric short-ranged entangled ground state, as well as an SPT-LSM constraint that enforces nontrivial entanglement in the SPT ground state. Finally, we use the classification to establish a similar LSM-type constraints for non-invertible Kramers-Wannier reflection symmetries.