TY - JOUR AB - We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalized super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians, which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem also holds for certain perturbations of gapped ground states that close the spectral gap (so it is also an adiabatic theorem for resonances and, in this sense, “generalized”), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called “super-adiabatic”). In addition to the existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations for infinite systems. While we consider the result and its proof as new and interesting in itself, we also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article. AU - Henheik, Sven Joscha AU - Teufel, Stefan ID - 10600 IS - 1 JF - Journal of Mathematical Physics KW - mathematical physics KW - statistical and nonlinear physics SN - 0022-2488 TI - Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap VL - 63 ER - TY - JOUR AB - 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 a strong version of the local 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. AU - Henheik, Sven Joscha AU - Teufel, Stefan AU - Wessel, Tom ID - 10642 IS - 1 JF - Letters in Mathematical Physics KW - mathematical physics KW - statistical and nonlinear physics SN - 0377-9017 TI - Local stability of ground states in locally gapped and weakly interacting quantum spin systems VL - 112 ER - TY - JOUR AB - 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 Gelfand–Naimark–Segal 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. AU - Henheik, Sven Joscha AU - Teufel, Stefan ID - 10643 JF - Forum of Mathematics, Sigma KW - computational mathematics KW - discrete mathematics and combinatorics KW - geometry and topology KW - mathematical physics KW - statistics and probability KW - algebra and number theory KW - theoretical computer science KW - analysis TI - Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk VL - 10 ER - TY - JOUR AB - We investigate the BCS critical temperature Tc in the high-density limit and derive an asymptotic formula, which strongly depends on the behavior of the interaction potential V on the Fermi-surface. Our results include a rigorous confirmation for the behavior of Tc at high densities proposed by Langmann et al. (Phys Rev Lett 122:157001, 2019) and identify precise conditions under which superconducting domes arise in BCS theory. AU - Henheik, Sven Joscha ID - 10623 IS - 1 JF - Mathematical Physics, Analysis and Geometry KW - geometry and topology KW - mathematical physics SN - 1385-0172 TI - The BCS critical temperature at high density VL - 25 ER - TY - JOUR AB - We compute the deterministic approximation of products of Sobolev functions of large Wigner matrices W and provide an optimal error bound on their fluctuation with very high probability. This generalizes Voiculescu's seminal theorem from polynomials to general Sobolev functions, as well as from tracial quantities to individual matrix elements. Applying the result to eitW for large t, we obtain a precise decay rate for the overlaps of several deterministic matrices with temporally well separated Heisenberg time evolutions; thus we demonstrate the thermalisation effect of the unitary group generated by Wigner matrices. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 10732 IS - 8 JF - Journal of Functional Analysis SN - 0022-1236 TI - Thermalisation for Wigner matrices VL - 282 ER -