[Paper Review] Classification of symmetry protected topological phases in quantum spin chains

Yoshiko Ogata|arXiv (Cornell University)|Oct 10, 2021

Quantum many-body systems4 citations

TL;DR

This paper provides a rigorous operator algebraic classification of symmetry-protected topological (SPT) phases in one- and two-dimensional quantum spin chains, establishing that SPT phases are classified by group cohomology: $H^2(G, U(1))$ in 1D and $H^3(G, U(1))$ in 2D. It constructs explicit invariants using automorphisms and cocycle actions on infinite systems, confirming a conjecture by physicists and extending the Dijkgraaf-Witten model to the infinite setting.

ABSTRACT

We consider the classification problem of symmetry protected topological (SPT) phases on quantum spin systems. SPT phases are gapped short-range-entangled quantum phases with a symmetry $G$. We explain that in one and two-dimensional quantum spin systems, there are $H^{2}(G, U(1))$/ $H^{3}(G, U(1))$-valued invariant, confirming a physicists conjecture.

Motivation & Objective

• To provide a mathematically rigorous classification of symmetry-protected topological (SPT) phases in quantum spin systems using operator algebraic methods.
• To confirm the physicists' conjecture that SPT phases in $\nu$-dimensional systems are classified by $H^{\nu+1}(G, U(1))$ for finite on-site symmetry groups $G$.
• To define and construct $H^2(G, U(1))$- and $H^3(G, U(1))$-valued invariants for SPT phases in 1D and 2D, respectively, using automorphisms and cocycle actions on infinite systems.
• To establish the connection between the Dijkgraaf-Witten topological quantum field theory and the classification of SPT phases in the infinite-volume limit.

Proposed method

• Uses the framework of $C^*$-dynamical systems to describe quantum spin systems and their time evolution as strongly continuous one-parameter groups of automorphisms.
• Applies the split property to ensure the existence of unique gapped ground states in infinite systems, enabling the definition of topological invariants.
• Constructs a matrix product state (MPS) representation for 1D systems and defines an $H^2(G, U(1))$-valued index via projective representations and unitary cocycles.
• For 2D systems, defines a $H^3(G, U(1))$-valued index using a triple of automorphisms $\sigma^{(0)}, \sigma^{(1)}, \sigma^{(2)}$ and their coboundary maps.
• Utilizes the Dijkgraaf-Witten model as a concrete realization of the $H^3(G, U(1))$-valued index, with the 3-cocycle $\Psi^3(\nu)$ arising from the unitary cocycle $V(g,h)$.
• Applies the theory of group cohomology to relate the automorphism actions and their commutators to the cohomology classes, ensuring topological invariance.

Experimental results

Research questions

• RQ1 How can SPT phases in one-dimensional quantum spin chains be classified using group cohomology?
• RQ2 What is the precise mathematical structure underlying the $H^2(G, U(1))$-valued invariant in 1D SPT phases?
• RQ3 How does the $H^3(G, U(1))$-valued index emerge in two-dimensional SPT phases with on-site symmetry $G$?
• RQ4 Can the Dijkgraaf-Witten model be realized as a physical SPT phase in the infinite system limit?
• RQ5 What role does the split property play in defining and stabilizing topological invariants in infinite quantum spin systems?

Key findings

• The paper proves that SPT phases in one-dimensional quantum spin chains with on-site finite group symmetry $G$ are classified by the second group cohomology group $H^2(G, U(1))$.
• In two-dimensional systems, the classification of SPT phases is given by the third group cohomology group $H^3(G, U(1))$, confirming a conjecture by physicists.
• An explicit $H^3(G, U(1))$-valued index is constructed using automorphisms and a 3-cocycle $\Psi^3(\nu)$, which arises from the unitary cocycle $V(g,h)$ in the Dijkgraaf-Witten model.
• The $H^2(G, U(1))$-index is derived from the projective representation of the symmetry group $G$ on the boundary of the system, using matrix product states and unitary operators.
• The classification is stable under smooth deformation of Hamiltonians without closing the gap, provided the symmetry is preserved and the ground state is unique and gapped.
• The construction is valid in the infinite system limit, where the topological invariant is non-trivial only when the Hilbert space is infinite-dimensional, analogous to the Fredholm index.