• Corpus ID: 246822895

The Lieb-Schultz-Mattis Theorem: A Topological Point of View

@inproceedings{Tasaki2022TheLT,
  title={The Lieb-Schultz-Mattis Theorem: A Topological Point of View},
  author={Hal Tasaki},
  year={2022}
}
  • H. Tasaki
  • Published 13 February 2022
  • Mathematics
We review the Lieb-Schultz-Mattis theorem and its variants, which are no-go theorems that state that a quantum many-body system with certain conditions cannot have a locally-unique gapped ground state. We restrict ourselves to one-dimensional quantum spin systems and discuss both the generalized Lieb-Schultz-Mattis theorem for models with U(1) symmetry and the extended Lieb-Schultz-Mattis theorem for models with discrete symmetry. We also discuss the implication of the same arguments to systems… 

Figures from this paper

From Lieb-Robinson Bounds to Automorphic Equivalence
. I review the role of Lieb-Robinson bounds in characterizing and utilizing the locality properties of the Heisenberg dynamics of quantum lattice systems. In particular, I discuss two definitions of

References

SHOWING 1-10 OF 60 REFERENCES
Lieb-Schultz-Mattis in higher dimensions
A generalization of the Lieb-Schultz-Mattis theorem to higher dimensional spin systems is shown. The physical motivation for the result is that such spin systems typically either have long-range
Twisted Boundary Condition and Lieb-Schultz-Mattis Ingappability for Discrete Symmetries.
TLDR
It is shown that, if the system is gapped, the ground-state degeneracy under the twisted boundary condition also implies a ground- state (quasi-degeneracy) under the periodic boundary conditions, giving a compelling evidence for the recently proposed Lieb-Schultz-Mattis-type ingappability due to the on-site discrete symmetry in two and higher dimensions.
Lieb–Schultz–Mattis Theorem with a Local Twist for General One-Dimensional Quantum Systems
We formulate and prove the local twist version of the Yamanaka–Oshikawa–Affleck theorem, an extension of the Lieb–Schultz–Mattis theorem, for one-dimensional systems of quantum particles or spins. We
General Lieb–Schultz–Mattis Type Theorems for Quantum Spin Chains
We develop a general operator algebraic method which focuses on projective representations of symmetry group for proving Lieb-Schultz-Mattis type theorems, i.e., no-go theorems that rule out the
Generalized Boundary Condition Applied to Lieb-Schultz-Mattis-Type Ingappabilities and Many-Body Chern Numbers
We introduce a new boundary condition which renders the flux-insertion argument for the Lieb-Schultz-Mattis type theorems in two or higher dimensions free from the specific choice of system sizes. It
Lattice Homotopy Constraints on Phases of Quantum Magnets.
TLDR
It is conjecture that all LSM-like theorems for quantum magnets can be understood from lattice homotopy, which automatically incorporates the full spatial symmetry group of the lattice, including all its point-group symmetries.
A Multi-Dimensional Lieb-Schultz-Mattis Theorem
For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear
A Proof of the Bloch Theorem for Lattice Models
The Bloch theorem is a powerful theorem stating that the expectation value of the U(1) current operator averaged over the entire space vanishes in large quantum systems. The theorem applies to the
Lieb–Schultz–Mattis Type Theorems for Quantum Spin Chains Without Continuous Symmetry
We prove that a quantum spin chain with half-odd-integral spin cannot have a unique ground state with a gap, provided that the interaction is short ranged, translation invariant, and possesses
An elementary proof of 1D LSM theorems
The Lieb-Schultz-Mattis (LSM) theorem and its generalizations forbids the existence of a unique gapped ground state in the presence of certain lattice and internal symmetries and thus imposes
...
...