Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Intertwiners

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 Abelian and non-Abelian groups but also more general categorical symmetries. These symmetries can be realized as matrix product operators that 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 nonlocal string-order operators. We illustrate this approach for known dualities such as the Kramers-Wannier, Jordan-Wigner, and Kennedy-Tasaki dualities and the interaction-round-the-face–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.

Popular Summary

Dualities are mathematical transformations that relate two seemingly different physical model theories that, at a fundamental level, share similar underlying physical behavior although expressed in different operator languages. For instance, the $t$-$J_z$ chain model of antiferromagnetic Mott insulators is unitarily equivalent (i.e., dual) to interacting spinless fermions in one dimension, effectively unveiling its emergent low-energy physics. In addition, computations difficult to perform in one model may become easier in one of its dual theories, thus allowing analytic exploration of the original model’s thermodynamic phase diagram. Despite its unquestionable analytic power, a constructive and systematic recipe for generating and classifying these duality maps remained elusive.

Here we develop such a constructive framework in a category-theoretical language and extensively illustrate our recipe’s power in several one-dimensional quantum lattice systems. Our systematic method exploits detailed understanding of the symmetries of the original physical model in terms of fusion categories and their associated module categories, and it demonstrates that all possible dualities of a given model are dictated by its symmetries. We illustrate how our implementation not only recovers known dualities but also discovers new ones. Most importantly, one can explicitly construct matrix product operators that conceive the duality map at the level of the Hilbert space, thereby revealing the entanglement structure of these transformations.

We envision extensions of our framework to address non-Abelian and holographic dualities in higher spatial dimensions, of relevance to condensed matter and high-energy physics, and quantum circuit implementations of dualities for simulation purposes.

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 ${\mathcal{M}}_0$ over ${\mathcal{C}}_0$. These data in turn stipulate an input category $\mathcal{D}$ 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 ${\mathcal{C}}_0^{\prime }$ and module category ${\mathcal{M}}_0^{\prime }$, one would have generated a different input category ${\mathcal{D}}^{\prime }$, leading to a different set of dual maps.