[Paper Review] Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries

Arkya Chatterjee, Ömer M. Aksoy+1 more|arXiv (Cornell University)|May 8, 2024

Quantum many-body systems8 citations

TL;DR

The paper constructs and analyzes two one-dimensional spin-chain models with invertible S3 and non-invertible Rep(S3) symmetry, explores their phase diagrams, self-dual lines, and SymTO descriptions, and identifies all spontaneous symmetry breaking patterns and transitions.

ABSTRACT

Generalized symmetries often appear in the form of emergent symmetries in low energy effective descriptions of quantum many-body systems. Non-invertible symmetries are a particularly exotic class of generalized symmetries, in that they are implemented by transformations that do not form a group. Such symmetries appear in large families of gapless states of quantum matter and constrain their low-energy dynamics. To provide a UV-complete description of such symmetries, it is useful to construct lattice models that respect these symmetries exactly. In this paper, we discuss two families of one-dimensional lattice Hamiltonians with finite on-site Hilbert spaces: one with (invertible) $S^{\,}_3$ symmetry and the other with non-invertible $\mathsf{Rep}(S^{\,}_3)$ symmetry. Our models are largely analytically tractable and demonstrate all possible spontaneous symmetry breaking patterns of these symmetries. Moreover, we use numerical techniques to study the nature of continuous phase transitions between the different symmetry-breaking gapped phases associated with both symmetries. Both models have self-dual lines, where the models are enriched by so-called intrinsically non-invertible symmetries generated by Kramers-Wannier-like duality transformations. We provide explicit lattice operators that generate these non-invertible self-duality symmetries. We show that the enhanced symmetry at the self-dual lines is described by a 2+1D symmetry-topological-order (SymTO) of type $\mathrm{JK}^{\,}_4\boxtimes \overline{\mathrm{JK}}^{\,}_4$. The condensable algebras of the SymTO determine the allowed gapped and gapless states of the self-dual $S^{\,}_3$-symmetric and $\mathsf{Rep}(S^{\,}_3)$-symmetric models.

Motivation & Objective

• Motivate the study of generalized, non-invertible symmetries in lattice quantum systems.
• Provide UV-complete lattice models that realize S3 and Rep(S3) symmetries exactly.
• Characterize all spontaneous symmetry breaking patterns for these symmetries.
• Investigate continuous phase transitions between symmetry-broken gapped phases.
• Link lattice results to symmetry-topological-order (SymTO) descriptions.

Proposed method

• Construct two 1D lattice Hamiltonians with finite local Hilbert spaces: one with S3 symmetry and one with Rep(S3) symmetry.
• Gauging subgroups (Z3 or Z2) to obtain non-invertible self-duality symmetries and dual bond algebras.
• Identify and analyze four SSB patterns via analytical arguments and tensor-network numerics.
• Implement non-invertible self-duality operators explicitly as lattice sequential circuits.
• Describe the enhanced symmetry on self-dual lines via 2+1D SymTO of type JK4 x overline{JK4} and its condensable algebras.
• Discuss dualities, order/disorder operators, and an incommensurate gapless region.

Experimental results

Research questions

• RQ1 What SSB patterns are realized for S3 and Rep(S3) symmetries in the examined spin chains?
• RQ2 How do non-invertible self-duality symmetries emerge and what are their lattice implementations?
• RQ3 What is the SymTO/PSA description of the self-dual points and their allowed gapped/gapless states?
• RQ4 How do phase transitions occur between different SSB phases, including continuous transitions?
• RQ5 Is there an incommensurate gapless region, and how is it characterized?

Key findings

• The S3-symmetric spin chain is dual to the Rep(S3)-symmetric chain under gauging a Z2 subgroup, mapping their phase diagrams one-to-one.
• All four SSB patterns are realized for both S3 and Rep(S3) symmetries, corresponding to four inequivalent module categories over the fusion categories.
• On self-dual lines, the models possess intrinsic non-invertible self-duality symmetries with explicitly constructed lattice circuits.
• The multi-critical point on each self-dual line is symmetric under the non-invertible self-duality, with three relevant perturbations identified.
• An extended gapless region consistent with an incommensurate phase is found, with an analogy to a Z2 KW self-duality spin chain illustrating related features.
• The SymTO description corresponds to 2+1D JK4 ⊗ overline{JK4} topological order, constraining allowed gapped and gapless states.