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Gappability Index for Quantum Many-Body Systems.

Yuan Yao1, Masaki Oshikawa2,3,4, Akira Furusaki1,5

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|July 16, 2022
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Summary
This summary is machine-generated.

We introduce an index I_{G} to quantify the difficulty of achieving a unique gapped ground state under symmetry G. This index helps classify gapped theories and analyze magnetic systems.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Field Theory
  • Statistical Mechanics

Background:

  • Understanding the conditions for unique gapped ground states in quantum systems is crucial for classifying phases of matter.
  • The Lieb-Schultz-Mattis theorem provides a fundamental constraint on gapped phases with certain symmetries.
  • Characterizing the 'theory space' of gapped systems requires robust theoretical tools.

Purpose of the Study:

  • To propose a new index, I_{G}, that quantifies the degree of gappability in the presence of a symmetry G.
  • To establish I_{G} as a measure of the difficulty in inducing a unique ground state with a non-vanishing excitation gap.
  • To demonstrate the utility of I_{G} in analyzing the phase diagrams of magnetic systems.

Main Methods:

  • Developing a theoretical framework to define and calculate the index I_{G}.
  • Interpreting I_{G} as the dimension of a subspace of uniquely gapped theories within the G-invariant theory space.
  • Relating the Lieb-Schultz-Mattis theorem to the case where I_{G}=0, indicating complete ingappability.

Main Results:

  • The proposed index I_{G} successfully characterizes the gappability of theories with symmetry G.
  • The index provides a quantitative measure for the difficulty of achieving a unique gapped ground state.
  • The Lieb-Schultz-Mattis theorem is reformulated within this framework as a condition for complete ingappability (I_{G}=0).

Conclusions:

  • The index I_{G} offers a powerful new tool for classifying gapped quantum phases and understanding symmetry constraints.
  • The framework is applicable to systems with various symmetries, including those beyond lattice translation symmetry.
  • The usefulness of I_{G} is demonstrated by its application to the phase diagrams of spin-1/2 antiferromagnets.