[Paper Review] Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners

Laurens Lootens, Clement Delcamp+2 more|arXiv (Cornell University)|Dec 16, 2021

Algebraic structures and combinatorial models15 citations

TL;DR

This paper presents a categorical framework for classifying dualities in one-dimensional quantum lattice models by leveraging matrix product operator (MPO) realizations of generalized symmetries. It establishes that dual Hamiltonians arise from distinct module categories over a fusion category, with MPO intertwiners mapping local operators to non-local string-order operators, unifying known dualities like Kramers-Wannier and Jordan-Wigner under a single algebraic structure.

ABSTRACT

We present a systematic recipe for generating and classifying duality transformations in one-dimensional quantum lattice systems. Our construction emphasizes the role of global symmetries, including those described by (non)-abelian groups but also more general categorical symmetries. These symmetries can be realized as matrix product operators which allow the extraction of a fusion category that characterizes the algebra of all symmetric operators commuting with the symmetry. Known as the bond algebra, its explicit realizations are classified by module categories over the fusion category. A duality is then defined by a pair of distinct module categories giving rise to dual realizations of the bond algebra, as well as dual Hamiltonians. Symmetries of dual models are in general distinct but satisfy a categorical Morita equivalence. A key novelty of our categorical approach is the explicit construction of matrix product operators that intertwine dual bond algebra realizations at the level of the Hilbert space, and in general map local order operators to non-local string-order operators. We illustrate this approach for known dualities such as Kramers-Wannier, Jordan-Wigner, Kennedy-Tasaki and the IRF-vertex correspondence, a new duality of the $t$-$J_z$ chain model, and dualities in models with the exotic Haagerup symmetry. Finally, we comment on generalizations to higher dimensions.

Motivation & Objective

• To develop a systematic, symmetry-based classification of dualities in one-dimensional quantum lattice models beyond traditional group symmetries.
• To unify known dualities—such as Kramers-Wannier, Jordan-Wigner, and Kennedy-Tasaki—under a single algebraic framework rooted in categorical symmetries.
• To establish a correspondence between dual Hamiltonians and distinct module categories over a fusion category, generalizing the notion of symmetry beyond unitary group actions.
• To provide explicit constructions of matrix product operator (MPO) intertwiners that map symmetric operators between dual models, including the transformation of local operators into non-local string-order operators.
• To extend the formalism to exotic symmetries like the Haagerup category and to suggest generalizations to higher-dimensional models via higher categorical structures.

Proposed method

• Realize generalized (non-abelian, non-invertible) symmetries as matrix product operators (MPOs) acting on the Hilbert space of 1D quantum lattice systems.
• Characterize the algebra of symmetric operators commuting with MPO symmetries via the bond algebra, which is classified by module categories over a fusion category.
• Define a duality as a pair of distinct module categories that realize the same bond algebra, leading to dual Hamiltonians with categorically Morita-equivalent symmetries.
• Construct explicit MPO intertwiners that map local operators in one model to non-local string-order operators in the dual model, preserving symmetry structure.
• Apply the framework to known dualities (e.g., t-Jz chain, IRF-vertex duality) and exotic symmetries (e.g., Haagerup), using tensor network representations of MPOs.
• Propose a generalization to higher dimensions using spherical fusion 2-categories and module 2-categories, with analogies to bulk-boundary correspondences and topological order.

Experimental results

Research questions

• RQ1 How can dualities in 1D quantum lattice models be systematically classified using generalized symmetries beyond traditional group actions?
• RQ2 What is the role of matrix product operator (MPO) intertwiners in relating dual Hamiltonians that realize the same bond algebra via different module categories?
• RQ3 How do local operators in one dual model transform into non-local string-order operators in the dual model under the categorical duality framework?
• RQ4 Can the framework unify diverse dualities such as Kramers-Wannier, Jordan-Wigner, and Kennedy-Tasaki under a single categorical structure?
• RQ5 What is the higher-dimensional generalization of this duality framework, and how does it relate to topological order and gapped boundaries in (2+1)D?

Key findings

• The paper establishes that dual Hamiltonians arise from distinct module categories over a fusion category, with the bond algebra—encoding symmetric operators—remaining invariant under the duality.
• MPO intertwiners explicitly constructed from the categorical data map local operators in one model to non-local string-order operators in the dual model, providing a concrete realization of duality at the operator level.
• The framework successfully reproduces known dualities, including the Kramers-Wannier duality of the Ising model and the Jordan-Wigner transformation, by identifying their underlying fusion category and module category structure.
• A new duality is identified for the t-Jz chain model, relating a spin chain with non-Abelian anyonic degrees of freedom to a vertex model via the IRF-vertex correspondence.
• The approach generalizes to exotic symmetries such as the Haagerup category, with explicit tensor network constructions of MPOs and intertwiners provided using data from the literature.
• The formalism suggests a higher-dimensional generalization using spherical fusion 2-categories and module 2-categories, with potential connections to bulk-boundary correspondences and (2+1)d topological order.