We consider bulk-gapped fermion systems whose infinite-volume bulk is obtained by adding sufficiently weak even many-body interactions to a free Hamiltonian with spectral gap $g>0$, allowing polynomial spatial decay of both parts. We prove quantitative local determination of low-energy states: whenever another interaction agrees with the bulk interaction inside a finite or infinite region $U$, with bulk terms crossing its boundary unchanged or removed, all its states with excitation energies in $[0,3g/4]$ have local expectations close to those of the bulk ground state deep inside $U$. The error decays with the distance to the complement of $U$, independently of the interaction terms supported outside $U$ and of the chosen low-energy state.
Quantitative local determination has strong consequences. (i) The bulk interaction has a unique algebraic ground state. (ii) A strong "local perturbations perturb locally" (LPPL) principle holds: arbitrary distant defects change local ground-state expectations only by a spatially decaying error. (iii) Low-energy excitations in finite samples remain localized near the boundary for all times, up to spatially decaying tails, uniformly over arbitrary even boundary terms. (iv) Correlations from deep inside $U$ to outside $U$ are small, with quantitative bounds in terms of spatial separation and distance to $U^c$.
To prove local determination of low-energy states, we combine quasi-adiabatic transport of the free ground state to the interacting bulk ground state with a new condition, local excitation detectability (LED). A companion paper establishes gap stability and constructs the transport for weak interactions, while we prove LED at sufficiently weak coupling. The abstract theorem is nonperturbative and applies whenever the transport and LED hypotheses hold.
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A bulk gap does not by itself establish that boundaries or strong distant defects leave interior low-energy states unchanged. This paper proves quantitative local determination for fermionic systems obtained by sufficiently weak, even many-body perturbations of a gapped quadratic Hamiltonian, including pairing and polynomially decaying interactions, without translation invariance.
Suppose a modified interaction agrees with the bulk inside $U$, while each bulk term crossing its boundary is retained or removed. In any positive-energy representation, with $P=\mathbf{1}_{[0,3g/4]}(L)$ and free gap $g$, the main bound is $\|P(A-\omega_\lambda(A))P\|\le C(1+|Y|)^q\|A\|G(\operatorname{dist}(Y,U^c))$ for observables supported on $Y$. Constants are independent of exterior interaction strength, geometry, and the chosen low-energy state; the bound also holds under matrix amplification. Consequences include uniqueness among all algebraic bulk ground states, a bulk GNS gap of at least $3g/4$, strong local-perturbations-perturb-locally estimates, and boundary localization of low-energy states uniformly in time. Analogous determination follows from ground-state inequalities imposed only inside $U$.
The proof combines quasi-local ground-state transport, constructed in a companion paper, with energy-filtered annihilation operators and a local excitation detectability inequality proved here at weak coupling. A contraction iteration controls excitation densities. Polynomial decay gives $G(D)=(1+D)^{-\min\{p_h,p_{\mathcal V}\}}$ at weak coupling, requiring weighted exponents above the volume-growth dimension, a geometric partition condition, and an exponential support-size weight. Stretched-exponential input yields every strictly smaller decay order. Beyond weak coupling, transport and detectability remain assumptions; uniqueness and a gap alone do not imply the chosen detectability condition.
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We consider bulk-gapped fermion systems whose infinite-volume bulk is obtained by adding sufficiently weak even many-body interactions to a free Hamiltonian with spectral gap g>0, allowing polynomial spatial decay of both parts. We prove quantitative local determination of low-energy states: whenever another interaction agrees with the bulk interaction inside a finite or infinite region U, with bulk terms crossing its boundary unchanged or removed, all its states with excitation energies in [0,3g/4] have local expectations close to those of the bulk ground state deep inside U. The error decays with the distance to the complement of U, independently of the interaction terms supported outside U and of the chosen low-energy state.
Quantitative local determination has strong consequences. (i) The bulk interaction has a unique algebraic ground state. (ii) A strong "local perturbations perturb locally" (LPPL) principle holds: arbitrary distant defects change local ground-state expectations only by a spatially decaying error. (iii) Low-energy excitations in finite samples remain localized near the boundary for all times, up to spatially decaying tails, uniformly over arbitrary even boundary terms. (iv) Correlations from deep inside U to outside U are small, with quantitative bounds in terms of spatial separation and distance to Uc.
To prove local determination of low-energy states, we combine quasi-adiabatic transport of the free ground state to the interacting bulk ground state with a new condition, local excitation detectability (LED). A companion paper establishes gap stability and constructs the transport for weak interactions, while we prove LED at sufficiently weak coupling. The abstract theorem is nonperturbative and applies whenever the transport and LED hypotheses hold.
A bulk gap does not by itself establish that boundaries or strong distant defects leave interior low-energy states unchanged. This paper proves quantitative local determination for fermionic systems obtained by sufficiently weak, even many-body perturbations of a gapped quadratic Hamiltonian, including pairing and polynomially decaying interactions, without translation invariance.
Suppose a modified interaction agrees with the bulk inside U, while each bulk term crossing its boundary is retained or removed. In any positive-energy representation, with P=1[0,3g/4](L) and free gap g, the main bound is ∥P(A−ωλ(A))P∥≤C(1+∣Y∣)q∥A∥G(dist(Y,Uc)) for observables supported on Y. Constants are independent of exterior interaction strength, geometry, and the chosen low-energy state; the bound also holds under matrix amplification. Consequences include uniqueness among all algebraic bulk ground states, a bulk GNS gap of at least 3g/4, strong local-perturbations-perturb-locally estimates, and boundary localization of low-energy states uniformly in time. Analogous determination follows from ground-state inequalities imposed only inside U.
The proof combines quasi-local ground-state transport, constructed in a companion paper, with energy-filtered annihilation operators and a local excitation detectability inequality proved here at weak coupling. A contraction iteration controls excitation densities. Polynomial decay gives G(D)=(1+D)−min{ph,pV} at weak coupling, requiring weighted exponents above the volume-growth dimension, a geometric partition condition, and an exponential support-size weight. Stretched-exponential input yields every strictly smaller decay order. Beyond weak coupling, transport and detectability remain assumptions; uniqueness and a gap alone do not imply the chosen detectability condition.
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.
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.
Abstract Based on a result by Yarotsky (J Stat Phys 118, 2005), we prove that localized but otherwise arbitrary perturbations of weakly interacting quantum spin systems with uniformly gapped on-site terms change the ground state of such a system only locally, even if they close the spectral gap. We call this astrong versionof thelocal perturbations perturb locally(LPPL) principle which is known to hold for much more general gapped systems, but only for perturbations that do not close the spectral gap of the Hamiltonian. We also extend this strong LPPL-principle to Hamiltonians that have the appropriate structure of gapped on-site terms and weak interactions only locally in some region of space. While our results are technically corollaries to a theorem of Yarotsky, we expect that the paradigm of systems with a locally gapped ground state that is completely insensitive to the form of the Hamiltonian elsewhere extends to other situations and has important physical consequences.
This thesis analyses several locality properties of extended quantum lattice systems, considering both spins and fermions. We begin by assuming locality of the Hamiltonian and then discuss how this translates into locality of the time evolution and the thermal states. Starting from a local Hamiltonian, it is well known that the Heisenberg time evolution satisfies Lieb-Robinson bounds, which quantify its locality. We strengthen these bounds for long-range interacting fermions and for commuting interactions. Furthermore, we establish the first Lieb-Robinson bounds for Hamiltonians whose local terms can grow linearly away from the origin. We also prove stability of Lieb-Robinson bounds with respect to local perturbations in the Hamiltonian. For the thermal states we focus on three types of locality known as decay of correlations, the local perturbations perturb locally (LPPL) principle and local indistinguishability. As a key tool we use spectral filters, which were introduced by Hastings in the 2000s for various applications. It enable us to use Lieb-Robinson bounds to obtain information about locality properties of states. In particular, we use quantum belief propagation to prove an equivalence of decay of correlations, LPPL and local indistinguishability for Gibbs states at any positive temperature. Using the improved Lieb-Robinson bounds mentioned before, we obtain improvements for decay of correlations in the ground state, the locality of the spectral flow and LPPL for long-range interacting systems and commuting Hamiltonians. Furthermore, we adjust the filter functions to achieve exponential decay for certain applications that were previously limited to stretched exponential decay. Spectral filters are also used to rigorously justify response theory, which usually requires a gapped system to begin with. To relax this assumption, we introduce various notions of local gaps, discuss their relations and use one to justify response theory and Kubo’s formula in the absence of a global gap. In another result we use local indistinguishability as an input to prove bulk-edge correspondence at finite temperature, in the sense that the bulk magnetisation is equal to the edge current. This result has mostly been studied for non-interacting models or at zero temperature before. Finally, we present a class of weakly interacting spin systems with an on-site gap and prove that the ground state satisfies a strong form of LPPL. Moreover, we show that these systems satisfy decay of correlations and local indistinguishability uniformly in temperature.
We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding GNS-Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite volume Hamiltonians need not have a spectral gap.
Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. When a fixed boundary condition is imposed, this problem is undecidable: no algorithm, even with unlimited resources, can decide for every Hamiltonian whether it is gapped. Yet the thermodynamic limit is meant to capture the intrinsic properties of very large systems, which should not depend on how their boundaries are chosen. We therefore consider the bulk spectral gap, which concerns only the intrinsic excitations of the infinite system, and ask what can and cannot be computed about it. First, we give a general algorithm producing a hierarchy of certified upper bounds on the bulk gap, arbitrarily tight at the cost of more computation. Second, we prove that the spectral gap problem remains undecidable in the bulk, already for translationally invariant nearest-neighbour chains of fixed local dimension. Hence no general algorithm produces convergent lower bounds, and no convergent upper-bounding algorithm, ours included, admits a computable convergence rate. The bulk spectral gap is thus certifiable from above but uncomputable from below; more precisely, the bulk-gaplessness promise problem is RE-complete, exactly as hard as the halting problem. The same construction shows that no general algorithm can decide whether a given product state is a ground state, and that convergent bounds on ground-state expectation values admit no computable convergence rate in general. Finally, we obtain the first nontrivial certified upper bounds on the bulk gap of the spin-1/2 kagome Heisenberg antiferromagnet.