Validity of the Lieb-Schultz-Mattis Theorem in Long-Range Interacting Systems

TL;DR

This paper extends the Lieb-Schultz-Mattis theorem to one-dimensional systems with long-range interactions, showing it holds for decay exponents greater than 2 and exploring implications for experimental platforms with tunable long-range couplings.

Contribution

The authors generalize the LSM theorem to long-range interacting systems, identifying the decay exponent threshold for the theorem's applicability and supporting findings with numerical simulations.

Findings

LSM theorem holds for decay exponent $ ext{} extgreater 2$

Constraints do not apply for $ extless 2$ decay exponent

Numerical simulations confirm theoretical predictions

Abstract

The Lieb-Schultz-Mattis (LSM) theorem asserts that microscopic details of the system can impose non-trivial constraints on the system's low-energy properties. While traditionally applied to short-range interaction systems, where locality ensures a vanishing spectral gap in large system size limit, the impact of long-range interactions on the LSM theorem remains an open question. Long-range interactions are prevalent in experimental platforms such as Rydberg atoms, dipolar quantum gases, polar molecules, optical cavities, and trapped ions, where the interaction decay exponent can be experimentally tuned. We extend the LSM theorem in one dimension to long-range interacting systems and find that the LSM theorem holds for exponentially or power-law two-body interactions with a decay exponent $\alpha > 2$. However, for power-law interactions with $\alpha < 2$, the constraints of the LSM theorem on the ground state do not apply. Numerical simulations of long-range versions of the Heisenberg and Majumdar-Ghosh models, both satisfying the LSM symmetry requirements, are also provided. Our results suggest promising directions for experimental validation of the LSM theorem in systems with tunable long-range interactions.