The adiabatic theorem and linear response theory for extended quantum systems

Abstract

The adiabatic theorem refers to a setup where an evolution equation contains a time-dependent parameter whose change is very slow, measured by a vanishing parameter $\epsilon$. Under suitable assumptions the solution of the time-inhomogenous equation stays close to an instantaneous fixpoint. In the present paper, we prove an adiabatic theorem with an error bound that is independent of the number of degrees of freedom. Our setup is that of quantum spin systems where the manifold of ground states is separated from the rest of the spectrum by a spectral gap. One important application is the proof of the validity of linear response theory for such extended, genuinely interacting systems. In general, this is a long-standing mathematical problem, which can be solved in the present particular case of a gapped system, relevant e.g.~for the integer quantum Hall effect.

25 pages; v1-->v2 minor typos, change in abstract, references; v2-->v3 remark added after main theorem on p.6; v3-->v4 Lemma 4.8 added, minor changes, one additional reference

References in corpus (7)

• Propagation of Correlations in Quantum Lattice Systems
• Linear response theory for magnetic Schroedinger operators in disordered media
• Persistence of exponential decay and spectral gaps for interacting fermions
• Adiabatic currents for interacting electrons on a lattice
• $C^1$-classification of gapped parent Hamiltonians of quantum spin chains
• Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory
• Quasi-particles in weak perturbations of non-interacting quantum lattice systems

Cited by in corpus (35)

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• Shortcuts to Adiabaticity in Krylov Space
• A ${\mathbb{Z}}_2$-index of symmetry protected topological phases with time reversal symmetry for quantum spin chains
• Local Commuting Projector Hamiltonians and the Quantum Hall Effect
• Adiabatic currents for interacting electrons on a lattice
• Time-dependent topological systems: A study of the Bott index
• Justifying Kubo's formula for gapped systems at zero temperature: a brief review and some new results
• A new approach to transport coefficients in the quantum spin Hall effect
• Lieb-Robinson bounds, the spectral flow, and stability of the spectral gap for lattice fermion systems
• Local Density Approximation for Almost-Bosonic Anyons
• Quantization of the interacting Hall conductivity in the critical regime
• Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap
• Exactness of linear response in the quantum Hall effect
• Local stability of ground states in locally gapped and weakly interacting quantum spin systems
• On ${\mathbb{Z}}_2$-indices for ground states of fermionic chains
• Modification of quantum many-body relaxation by perturbations exhibiting a banded matrix structure
• Adiabatic Lindbladian Evolution with Small Dissipators
• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• Improved energy estimates for a class of time-dependent perturbed Hamiltonians
• Coherent electronic transport in periodic crystals
• Quantum adiabatic theorem for unbounded Hamiltonians with a cutoff and its application to superconducting circuits
• Stability of the spectral gap and ground state indistinguishability for a decorated AKLT model
• Fighting Exponentially Small Gaps by Counterdiabatic Driving
• Adiabatic Evolution of Low-Temperature Many-Body Systems
• Non-equilibrium almost-stationary states and linear response for gapped quantum systems
• A ${\mathbb{Z}}_2$-Topological Index for Quasi-Free Fermions
• Quantum adiabaticity in many-body systems and almost-orthogonality in complementary subspace
• On adiabatic theory for extended fermionic lattice systems
• Purely linear response of the quantum Hall current to space-adiabatic perturbations
• The index of a pair of pure states and the interacting integer quantum Hall effect
• Spin transport and lack of quantisation for time-reversal symmetric insulators on the honeycomb structure
• Automorphic equivalence within gapped phases in the bulk