Fig. 4: Energy levels of the model are fragmented by the resonance condition Eq. (8). | Communications Physics

Fig. 4: Energy levels of the model are fragmented by the resonance condition Eq. (8).

From: Hilbert space fragmentation at the origin of disorder-free localization in the lattice Schwinger model

Fig. 4: Energy levels of the model are fragmented by the resonance condition Eq. (8).The alternative text for this image may have been generated using AI.

a The parity \({\sum }_{j}{\hat{\sigma }}_{j}^{z}\) over some regions gives the total charge in the original gauge theory; this must sum to zero over the whole chain. b Exchanging \(| 10\left.\right\rangle \leftrightarrow | 01\left.\right\rangle\) at sites 2ℓ and 2ℓ + 1 creates an e−e+ pair. This is resonant if the combination of gauge and background fields L2ℓ + θ/2π = ± 1/2 both before and after this process. c For a particular charge sector and spin configuration, only certain bonds satisfy the previous condition and thus permit dynamics in the large-J limit. These “active regions” are independent and so the effective Hamiltonian is a sum of local commuting terms. d Each energy level \({{{{\mathcal{K}}}}}_{a}\) therefore fragments into Krylov subspaces \(\{{{{{\mathcal{K}}}}}_{a}^{b}\}\) which are disconnected under the action of \({\hat{H}}_{\pm }\). e For finite J, degenerate perturbation theory (see Methods) allows us to obtain an effective Hamiltonian within \({{{{\mathcal{K}}}}}_{0}\) as an expansion in powers of w/J. For example, a series of three spin exchanges via a nearby energy level contributes a term to the third-order effective Hamiltonian H[3] of magnitude \({{{\mathcal{O}}}}({w}^{3}/{J}^{2})\).

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