Gapped ground states of quantum spin systems have been referred to in the physics\n literature as being `in the same phase' if there exists a family of Hamiltonians H(s), with\n finite range interactions depending continuously on
sin[0,1], such that for each
s,\n H(s) has a non-vanishing gap above its ground state and with the two initial states being\n the ground states of H(0) and H(1), respectively. In this work, we give precise conditions\n under which any two gapped ground states of a given quantum spin system that 'belong to the\n same phase' are automorphically equivalent and show that this equivalence can be\n implemented as a flow generated by an
s-dependent interaction which decays faster than\n any power law (in fact, almost exponentially). The flow is constructed using Hastings'\n 'quasi-adiabatic evolution' technique, of which we give a proof extended to\n infinite-dimensional Hilbert spaces. In addition, we derive a general result about the\n locality properties of the effect of perturbations of the dynamics for quantum systems with\n a quasi-local structure and prove that the flow, which we call the {\\em spectral flow},\n connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As\n a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a\n co-cycle of automorphisms of the algebra of quasi-local observables of the infinite spin\n system. This proves that the ground state phase structure is preserved along the curve of\n models
H(s),0leqsleq1.
Abstract We prove a Lieb–Robinson bound for lattice fermion models with polynomially decaying interactions, which can be used to show the locality of the quasi-local inverse Liouvillian. This allows us to prove automorphic equivalence and the local perturbations perturb locally (LPPL) principle for these systems. The proof of the Lieb–Robinson bound is based on the work of Else et al. [5], and our results also apply to spin systems. We explain why some newer Lieb–Robinson bounds for long-range spin systems cannot be used to prove the locality of the quasi-local inverse Liouvillian, and in some cases may not even hold for fermionic systems.
We apply the technique of quasi-adiabatic continuation to study systems with continuous symmetries. We first derive a general form of Goldstone's theorem applicable to gapped nonrelativistic systems with continuous symmetries. We then show that, for a fermionic system with a spin gap, it is possible to insert π flux into a cylinder with only exponentially small change in the energy of the system, a scenario which covers several physically interesting cases such as an s-wave superconductor or a resonating valence bond state.
We bound the error in ground-state expectations of local observables caused by truncating the spatial tails of a gapped quantum lattice Hamiltonian. We show that the error is controlled by the interaction strength discarded near each site, rather than by the extensive norm of the omitted Hamiltonian. If the gap remains open along a path that removes the interaction tail, the resulting bounds are uniform in system size and extend, under suitable assumptions, to thermodynamic-limit ground states. The convergence rate reflects the decay of the interaction tail: it is algebraic for power-law interactions, superpolynomial for superpolynomial interactions, and exponential at any strictly smaller rate for exponentially decaying interactions. For superpolynomial interactions, we also derive a direct infinite-volume estimate using automorphic equivalence. The results extend to isolated low-energy sectors and parity-even fermionic systems. For two-body interactions decaying as
r−p in
d dimensions, we prove an error bound
O(R−(p−d)) for
p>2d, where
R is the truncation range. We construct a gapped non-translation-invariant example that saturates this scaling, showing that the bound is optimal for the general class considered. Our results quantify when finite-range truncations faithfully reproduce the local physics of gapped long-range systems.
We prove that weak long-range many-body interactions preserve the bulk gap of a free fermion system and that a quasi-local spectral flow transports the free ground state to ground states of the interacting system. Let
Φh be a free fermionic interaction, possibly including pairing terms, whose infinite-volume Majorana coefficient operator
h has no spectrum in
(−g,g), and let
V be an even self-adjoint many-body interaction. For sufficiently small
∣λ∣, we construct pure ground states
ωλ of the interacting bulk system
Φλ=Φh+λV and quasi-local automorphisms
β0,λ such that
ωλ=ω0∘β0,λ, where
ω0 is the free ground state of
Φh. Each
ωλ has a GNS Hamiltonian with gap at least
g−C∗∣λ∣∥V∥∗, where
∥⋅∥∗ is the relevant interaction norm.
Our result allows polynomial decay of both the free coefficients and the many-body interaction. For two-site interactions on a uniformly discrete point set
Γ⊂Rd, a sufficient condition is decay as
(1+∣x−y∣)−ϖ with
ϖ>2d, for both the hopping and pairing coefficients and the interaction terms. The spectral-flow generator preserves the spatial exponent of the polynomial interaction norm, and its automorphisms satisfy locality bounds at that exponent. If
h and
V have stretched-exponential decay, so does the spectral-flow generator.
The proof constructs the spectral flow and establishes the gap through an excitation-number comparison, using polynomial propagation bounds for evolved interactions proved in a companion paper.