Revolutionizing Quantum Lattice Models: New Approach Unlocks Dualities via Categorical Symmetries

Quantum lattice models, one-dimensional models in physics, have been a longstanding challenge due to the difficulty in constructing a general framework that relates the spectra of dual theories. However, progress has been made using matrix product operators to construct explicit symmetry operators that preserve boundary conditions. This has led to a better understanding of quantum theories by interpreting symmetries in terms of topological operators. The paper also discusses the concept of categorical symmetries, which are encoded into abstract higher mathematical structures known as spherical fusion categories. These symmetries are applied in quantum models, including anyonic chains and spin systems with quantum group symmetries.

What is the Problem with Quantum Lattice Models?

Quantum lattice models have been a subject of intense scrutiny in the field of physics. These models, which are one-dimensional, have been a longstanding open problem due to the difficulty in constructing a general framework that relates the spectra of dual theories to each other. This problem has been solved for models with symmetry-twisted boundary conditions. Dualities are defined between categorically symmetric models that only differ in a choice of module category.

The use of matrix product operators has allowed for the construction of explicit symmetry operators that preserve boundary conditions. These operators also serve as intertwiners, mapping topological sectors of dual models onto one another. This construction has been illustrated with a family of examples that are in the duality class of the spin 1/2 Heisenberg XXZ model.

How are Symmetries Interpreted in Quantum Theories?

In recent years, significant progress has been made in understanding quantum theories by interpreting symmetries in terms of topological operators. More specifically, correlation functions of the theories, including symmetry operators, are insensitive to topology-preserving deformations of the submanifolds supporting the operators, unless they pass through charged operators.

This new approach has led to generalizations of the notion of symmetry, whereby operators are not necessarily supported on one-codimensional submanifolds and/or are not necessarily invertible. This paper is concerned with such generalized symmetries in the context of translation invariant one-dimensional quantum lattice models.

What are Categorical Symmetries?

Relaxing the invertibility condition leads to symmetry operators, the properties of which are encoded into abstract higher mathematical structures known as spherical fusion categories. These so-called categorical symmetries have been under intense scrutiny in recent years.

The corresponding operators are typically non-local in the sense that they cannot be written as tensor products of local operators and are realized instead by matrix product operators (MPOs). Although exotic, such categorical symmetries are not uncommon in one-dimensional quantum models and are typically related to rational conformal field theories (CFTs).

How are Categorical Symmetries Applied in Quantum Models?

A large family of lattice models known as anyonic chains that commute with symmetry operators organized into fusion categories can be readily constructed, including spin systems with quantum group symmetries. In virtue of their topological nature, any categorically symmetric model in 1+1D can be lifted to a gapped boundary condition of the Turaev-Viro-Barrett-Westbury topological quantum field theory (TQFT) with as input datum the corresponding spherical fusion category.

Mathematically, gapped boundary conditions admit a classification in terms of module categories over the input category. This holographic viewpoint on symmetries has also garnered a lot of interest. Crucially, the bulk TQFT can be reconstructed from any choice of gapped boundary condition so that bulk topological lines are encoded into the Drinfeld center of the corresponding spherical fusion category of boundary topological lines.

What is the Future of Quantum Lattice Models?

Inspired by these developments, a systematic study of dualities in one-dimensional quantum lattice models from the viewpoint of their categorical symmetries has been initiated. One merit of this approach is to make very concrete the concepts and results alluded to above, which are often formal and abstract otherwise, as well as demonstrate that this approach to dualities agrees and extends traditional ones.

Within this framework, an equivalence class of dual models is given by a choice of input spherical fusion category together with an algebra of local operators. A representative of such a class then corresponds to a specific lattice realization of the underlying theory. Choosing a lattice realization loosely boils down to picking a collection of degrees of freedom which happen to be encoded into a choice of module category over the input fusion category. This means that models that only differ in a choice of module category are dual to one another.