arXiv:2410.10809 · Journal of Statistical Physics

Response Theory for Locally Gapped Systems

Joscha Henheik, Tom Wessel

Published Oct 12, 2026• 0 citations

Abstract

Abstract We introduce a notion of a local gap for interacting many-body quantum lattice systems and prove the validity of response theory and Kubo’s formula for localized perturbations in such settings. On a high level, our result shows that the usual spectral gap condition, which concerns the system as a whole, is not a necessary condition for understanding local properties of the system. More precisely, we say that an equilibrium state $$\rho _0$$ ρ 0 of a Hamiltonian $$H_0$$ H 0 is locally gapped in $${\varLambda ^{\textrm{gap}}}\subset \varLambda $$ Λ gap ⊂ Λ , whenever the Liouvillian $$- \textrm{i}\, [H_0, \, \cdot \, ]$$ - i [ H 0 , · ] is almost invertible on local observables supported in $${\varLambda ^{\textrm{gap}}}$$ Λ gap when tested in $$\rho _0$$ ρ 0 . To put this into context, we provide other alternative notions of a local gap and discuss their relations. The validity of response theory is based on the construction of non-equilibrium almost stationary states (NEASSs). By controlling locality properties of the NEASS construction, we show that response theory holds to any order, whenever the perturbation $$\varepsilon V$$ ε V acts in a region which is further than $$|\log \varepsilon |$$ | log ε | away from the non-gapped region $$\varLambda \backslash \varLambda \textrm{gap}$$ Λ \ Λ gap .

DOI: 10.1007/s10955-026-03703-1

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arXiv:2410.10809 · Journal of Statistical Physics

Response Theory for Locally Gapped Systems

Joscha Henheik, Tom Wessel

Published Oct 12, 2026• 0 citations

arXiv HTML PDF

Abstract

\rho _0

ρ 0 of a Hamiltonian

H_0

H 0 is locally gapped in

{\varLambda ^{\textrm{gap}}}\subset \varLambda

Λ gap ⊂ Λ , whenever the Liouvillian

- \textrm{i}\, [H_0, \, \cdot \, ]

- i [ H 0 , · ] is almost invertible on local observables supported in

{\varLambda ^{\textrm{gap}}}

Λ gap when tested in

\rho _0

ρ 0 . To put this into context, we provide other alternative notions of a local gap and discuss their relations. The validity of response theory is based on the construction of non-equilibrium almost stationary states (NEASSs). By controlling locality properties of the NEASS construction, we show that response theory holds to any order, whenever the perturbation

\varepsilon V

ε V acts in a region which is further than

|\log \varepsilon |

| log ε | away from the non-gapped region

\varLambda \backslash \varLambda \textrm{gap}

Λ \ Λ gap .

DOI: 10.1007/s10955-026-03703-1

Related papers

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arXiv:2002.08669Reviews in Mathematical PhysicsFeb 21, 202018 citations

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Tom Wessel

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arXiv:2012.15238Journal of Mathematical PhysicsJan 1, 20228 citations

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Sven Bachmann, Wojciech De Roeck, Martin Fraas

The first proof of the quantum adiabatic theorem was given as early as 1928. Today, this theorem is increasingly applied in a many-body context, e.g., in quantum annealing and in studies of topological properties of matter. In this setup, the rate of variation ϵ of local terms is indeed small compared to the gap, but the rate of variation of the total, extensive Hamiltonian, is not. Therefore, applications to many-body systems are not covered by the proofs and arguments in the literature. In this Letter, we prove a version of the adiabatic theorem for gapped ground states of interacting quantum spin systems, under assumptions that remain valid in the thermodynamic limit. As an application, we give a mathematical proof of Kubo's linear response formula for a broad class of gapped interacting systems. We predict that the density of nonadiabatic excitations is exponentially small in the driving rate and the scaling of the exponent depends on the dimension.The first proof of the quantum adiabatic theorem was given as early as 1928. Today, this theorem is increasingly applied in a many-body context, e.g., in quantum annealing and in studies of topological properties of matter. In this setup, the rate of variation ϵ of local terms is indeed small compared to the gap, but the rate of variation of the total, extensive Hamiltonian, is not. Therefore, applications to many-body systems are not covered by the proofs and arguments in the literature. In this Letter, we prove a version of the adiabatic theorem for gapped ground states of interacting quantum spin systems, under assumptions that remain valid in the thermodynamic limit. As an application, we give a mathematical proof of Kubo's linear response formula for a broad class of gapped interacting systems. We predict that the density of nonadiabatic excitations is exponentially small in the driving rate and the scaling of the exponent depends on the dimension.

arXiv:1612.01505Physical Review LettersAug 11, 201769 citations

Local determination of low-energy states in gapped fermion systems with weak long-range interactions

Stefan Teufel

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.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.

math-phCross-listedarXiv:2610.40337Oct 30, 20260 citations

\rho _0

ρ 0 of a Hamiltonian

H_0

H 0 is locally gapped in

{\varLambda ^{\textrm{gap}}}\subset \varLambda

Λ gap ⊂ Λ , whenever the Liouvillian

- \textrm{i}\, [H_0, \, \cdot \, ]

- i [ H 0 , · ] is almost invertible on local observables supported in

{\varLambda ^{\textrm{gap}}}

Λ gap when tested in

\rho _0

\varepsilon V

ε V acts in a region which is further than

|\log \varepsilon |

| log ε | away from the non-gapped region

\varLambda \backslash \varLambda \textrm{gap}

Λ \ Λ gap .

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.