Let
$$
H_\pm=\pm J\sum_jh_{j,j+1},
\qquad
h_{j,j+1}=\mathbf S_j\mathbin\cdot\mathbf S_{j+1},
$$
where the two signs describe the <Heisenberg antiferromagnet> and <Heisenberg ferromagnet>. A standard <Lieb-Robinson bound> is obtained by iterating the <Heisenberg picture> equation and bounding nested <commutator>[commutators] by <operator norm>[operator norms]. Its constants depend on the interaction only through quantities such as
$$
\sup_x\sum_{X\ni x}|X|\,\|\Phi(X)\|e^{\mu\operatorname{diam}X}.
$$
Changing $J$ to $-J$ changes neither the supports nor the norms $\|\Phi(X)\|$. Every term in the nested-commutator estimate acquires at most an irrelevant sign before its absolute value is taken. Therefore both chains obey exactly the same estimate
$$
\|[A(t),B]\|
\leq C\|A\|\|B\|e^{-\mu(d(A,B)-v_{\rm LR}|t|)}
$$
with the same $C,\mu$, and Lieb-Robinson velocity $v_{\rm LR}$. No unitary equivalence of the two Hamiltonians is required.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 343 / 3 / a / Solution

{c}

A Heisenberg ferromagnet has negative exchange coupling in $H=J\sum_{\langle ij\rangle}\mathbf S_i\mathbin\cdot\mathbf S_j$, so neighboring spins energetically prefer parallel alignment.


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 Incoming links: Heisenberg ferromagnet