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

TL;DR

The paper develops a comprehensive categorical framework to generate and classify dualities between one-dimensional quantum lattice Hamiltonians by treating symmetries as fusion-category MPOs. Dualities arise from different module-category realizations over a fixed input fusion category and are implemented by explicit MPO intertwiners, mapping local symmetric operators to dual nonlocal ones while preserving bond-algebra structure. Through numerous examples across Vec, Ising, Rep(S3), and Haagerup categories, it recovers known dualities (Kramers-Wannier, Jordan-Wigner, Kennedy-Tasaki, IRF-Vertex) and reveals new ones (including a t-J_z/XXZ duality) and emergent dualities in constrained subspaces. The work also clarifies non-abelian dualities, provides a Morita-equivalence-based classification of dualities, and outlines extensions to boundaries and higher dimensions, offering a unifying algebraic approach with potential experimental relevance and connections to bulk-boundary physics.

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.

Figures (1)

• Figure 1: Flowchart for generation of dualities. Given a Hamiltonian $\mathbb{H}_0$, we choose the MPO realization of the categorical symmetry $\mathcal{C}_0$, providing a module category ${\cal M}_0$ over $\mathcal{C}_0$. This data in turn stipulates an input category ${\cal D}$ corresponding to a bond algebra. One can then generate all possible dual Hamiltonians $\mathbb{H}_i$ by choosing different module categories $\mathcal{M}_i$ over $\mathcal{D}$. Had one chosen a different MPO symmetry ${\cal C}'_0$ and module category ${\cal M}'_0$, one would have generated a different input category ${\cal D}'$ leading to a different set of dual maps.