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Exact quantization of topological order parameter in SU($N$) spin models, $N$-ality transformation and ingappabilities

Hang Su, Yuan Yao, Akira Furusaki

TL;DR

This work provides a rigorously quantized nonlocal order parameter for one-dimensional SPT phases protected by $[\mathrm{U}(1)]^{N-1}\rtimes\mathbb{Z}_N$ symmetry, realized as the ground-state expectation value of a Lieb–Schultz–Mattis twisting operator. For SU$(N)$ spins, translation symmetry implements an $N$-ality transformation that links $N$ distinct SPT phases, with the order parameter taking values among the $N$-th roots of unity. The authors prove a fundamental LSM ingappability result and show how magnetic translations extend these constraints in the SU$(2)$ case, while providing a framework to generate multi-critical phase transitions from a single gapped Hamiltonian. These results offer a sharp, numerically accessible diagnostic for phase identification and have potential implications for cold-atom realizations and experimental probes of SU$(N)$ spin systems.

Abstract

We show that the ground-state expectation value of twisting operator is a topological order parameter for $\text{U}(1)$- and $\mathbb{Z}_{N}$-symmetric symmetry-protected topological (SPT) phases in one-dimensional "spin" systems -- it is quantized in the thermodynamic limit and can be used to identify different SPT phases and to diagnose phase transitions among them. We prove that this (non-local) order parameter must take values in $N$-th roots of unity, and its value can be changed by a generalized lattice translation acting as an $N$-ality transformation connecting distinct phases. This result also implies the Lieb-Schultz-Mattis ingappability for SU($N$) spins if we further impose a general translation symmetry. Furthermore, our exact result for the order parameter of SPT phases can predict a large number of LSM ingappabilities by the general lattice translation. We also apply the $N$-ality property to provide an efficient way to construct possible multi-critical phase transitions starting from a single Hamiltonian with a unique gapped ground state.

Exact quantization of topological order parameter in SU($N$) spin models, $N$-ality transformation and ingappabilities

TL;DR

This work provides a rigorously quantized nonlocal order parameter for one-dimensional SPT phases protected by symmetry, realized as the ground-state expectation value of a Lieb–Schultz–Mattis twisting operator. For SU spins, translation symmetry implements an -ality transformation that links distinct SPT phases, with the order parameter taking values among the -th roots of unity. The authors prove a fundamental LSM ingappability result and show how magnetic translations extend these constraints in the SU case, while providing a framework to generate multi-critical phase transitions from a single gapped Hamiltonian. These results offer a sharp, numerically accessible diagnostic for phase identification and have potential implications for cold-atom realizations and experimental probes of SU spin systems.

Abstract

We show that the ground-state expectation value of twisting operator is a topological order parameter for - and -symmetric symmetry-protected topological (SPT) phases in one-dimensional "spin" systems -- it is quantized in the thermodynamic limit and can be used to identify different SPT phases and to diagnose phase transitions among them. We prove that this (non-local) order parameter must take values in -th roots of unity, and its value can be changed by a generalized lattice translation acting as an -ality transformation connecting distinct phases. This result also implies the Lieb-Schultz-Mattis ingappability for SU() spins if we further impose a general translation symmetry. Furthermore, our exact result for the order parameter of SPT phases can predict a large number of LSM ingappabilities by the general lattice translation. We also apply the -ality property to provide an efficient way to construct possible multi-critical phase transitions starting from a single Hamiltonian with a unique gapped ground state.
Paper Structure (13 sections, 31 equations)

This paper contains 13 sections, 31 equations.

Table of Contents

  1. Introduction.---
  2. Setting and Results.---
  3. Theorem 1:
  4. Proof of Theorem 1.---
  5. SU(N) case:---
  6. Theorem 2:
  7. Applications to SPT phases.---
  8. Corollary 3:
  9. Applications to LSM ingappabilities and multi-phase transitions.---
  10. Corollary 4:
  11. Conclusions and discussions.---
  12. Acknowledgements.---
  13. Supplemental Materials: Symmetrization of local Hamiltonian terms