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A one-spin path can close its spectral gap

Statement

Let one site of $Z^{d}$ have spin dimension $2$ , let $Z = d i a g (1, - 1)$ , and define the finite-volume Hamiltonian path $H (s) = (1 - 2 s) Z$ on $0 \leq s \leq 1$ , with all other interaction terms zero. This is a $C^{1}$ finite-range path with a uniform finite weighted-interaction norm and gapped endpoint Hamiltonians, but its gap closes at $s = \frac{1}{2}$ . Thus endpoint gaps alone do not imply a uniform path gap.

Statement references

Weighted interactions and gapped finite-volume paths

Facts & Assumptions

Given: A single spin space $C^{2}$ , the path in the Statement, and the weighted-interaction convention. Energies are dimensionless in the chosen positive energy unit, $ℏ = 1$ , and $s$ is dimensionless.

[F1]

Weighted interactions and gapped finite-volume paths: a one-site self-adjoint matrix path with a bounded $C^{1}$ coefficient is a finite-range path with finite $G$ -interaction and derivative norms.

Proof

technique · direct

Given: $H (s) = (1 - 2 s) Z$ and $Z = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ .

1.1givenF1algebra

The interaction has only one nonzero term, on a singleton support, with coefficient $1 - 2 s$ ; its derivative is the constant $- 2$ . The two weighted norms are finite because the support and coefficient are bounded and every admissible weight is positive at distance zero.

2.1givenstep 1.1algebra∎

The eigenvalues of $H (s)$ are $1 - 2 s$ and $- (1 - 2 s)$ . The ground eigenvalue is $- ∣ 1 - 2 s ∣$ , the excited eigenvalue is $+ ∣ 1 - 2 s ∣$ , so the gap is $2 ∣ 1 - 2 s ∣$ . It is $2$ at each endpoint and $0$ at $s = \frac{1}{2}$ . The path therefore has gapped endpoints but no uniform positive gap on $[0, 1]$ .

Depends on

Weighted interactions and gapped finite-volume paths

Dependency tree · 2 levels

Sources

S. Bachmann, S. Michalakis, B. Nachtergaele, and R. Sims, Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems, §2 and discussion after Theorem 5.5