[Paper Review] Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space

Nathan Seiberg, Sahand Seifnashri+1 more|arXiv (Cornell University)|Jan 22, 2024

Quantum many-body systemsPhysics and Astronomy153 references10 citations

TL;DR

The paper analyzes non-invertible Kramers-Wannier-type symmetry on a lattice tensor-product Hilbert space, its relation to continuum symmetries, and derives an LSM-type constraint for systems with this symmetry.

ABSTRACT

We discuss the exact non-invertible Kramers-Wannier symmetry of 1+1d lattice models on a tensor product Hilbert space of qubits. This symmetry is associated with a topological defect and a conserved operator, and the latter can be presented as a matrix product operator. Importantly, unlike its continuum counterpart, the symmetry algebra involves lattice translations. Consequently, it is not described by a fusion category. In the presence of this defect, the symmetry algebra involving parity/time-reversal is realized projectively, which is reminiscent of an anomaly. Different Hamiltonians with the same lattice non-invertible symmetry can flow in their continuum limits to infinitely many different fusion categories (with different Frobenius-Schur indicators), including, as a special case, the Ising CFT. The non-invertible symmetry leads to a constraint similar to that of Lieb-Schultz-Mattis, implying that the system cannot have a unique gapped ground state. It is either in a gapless phase or in a gapped phase with three (or a multiple of three) ground states, associated with the spontaneous breaking of the lattice non-invertible symmetry.

Motivation & Objective

• Motivate and define a simple non-invertible symmetry on a tensor product lattice Hilbert space and contrast it with continuum notions.
• Construct and analyze the non-invertible Kramers-Wannier symmetry on a finite chain.
• Explore how lattice defects and translation interact with the non-invertible symmetry and their anomalies.
• Connect lattice realizations to continuum fusion category descriptions and emergent symmetries in the infrared.

Proposed method

• Introduce the lattice setup with translation and Z2 symmetry and define the non-invertible operator D with D^2 = (1+η) T^{-1}.
• Describe the action of D on lattice operators such as X_j and Z_j Z_{j+1}.
• Provide a matrix product operator (MPO) construction of D via Kramers-Wannier duality or Majorana/bosonization approaches.
• Develop the operator algebra for D with and without defects such as Z2 or duality defects.
• Relate lattice D to its continuum non-invertible symmetry N and compare their algebras and commutation relations.
• Discuss the role of anomalies of non-invertible symmetries, including parity/time-reversal considerations, and the emergence of internal symmetries.

Experimental results

Research questions

• RQ1 How can a non-invertible symmetry be realized on a lattice with a tensor product Hilbert space and how does it relate to lattice translation?
• RQ2 What is the operator and defect algebra generated by the non-invertible symmetry on the lattice and in the continuum?
• RQ3 How do lattice defects and gauging procedures produce non-invertible defects and operators on the lattice?
• RQ4 What is the interplay between non-invertible lattice symmetries, anomalies, and emergent continuum symmetries?
• RQ5 What Lieb-Schultz-Mattis-type constraints arise from the presence of the non-invertible lattice translation symmetry?

Key findings

• The non-invertible lattice translation operator D satisfies D^2 = (1+η) T^{-1} and commutes with η, projecting onto Z2-even states.
• The action of D on lattice operators maps X_j to Z_{j-1} Z_j times D, and maps Z_{j-1} Z_j to X_{j-1} times D, illustrating a Kramers-Wannier duality on the lattice.
• The lattice symmetry algebra differs from the continuum one and depends on the chain length L, becoming infinite-dimensional on an infinite chain.
• A lattice-to-continuum discussion shows an emanant non-invertible symmetry in the continuum, with TY fusion category structure relevant for the Ising-like systems.
• An LSM-type constraint is derived: a finite-range Hamiltonian preserving D must be gapless or break the symmetry, with a three-sector condition in the symmetry-broken case.
• The tricritical Ising model serves as a concrete example illustrating the phase diagram and the action of D and η at criticality.