Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory

1. Front Matter

   Pages i-vii

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2. Introduction

   • J.-B. Bru, W. de Siqueira Pedra

   Pages 1-4

3. Algebraic Quantum Mechanics

   • J.-B. Bru, W. de Siqueira Pedra

   Pages 5-15

4. Algebraic Setting for Interacting Fermions on the Lattice

   • J.-B. Bru, W. de Siqueira Pedra

   Pages 17-30

5. Lieb–Robinson Bounds for Multi–commutators

   • J.-B. Bru, W. de Siqueira Pedra

   Pages 31-61

6. Lieb–Robinson Bounds for Non-autonomous Dynamics

   • J.-B. Bru, W. de Siqueira Pedra

   Pages 63-87

7. Applications to Conductivity Measures

   • J.-B. Bru, W. de Siqueira Pedra

   Pages 89-101

8. Back Matter

   Pages 103-109

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Lieb-Robinson bounds for multi-commutators are effective mathematical tools to handle analytic aspects of infinite volume dynamics of non-relativistic quantum particles with short-range, possibly time-dependent interactions.
In particular, the existence of fundamental solutions is shown for those (non-autonomous) C*-dynamical systems for which the usual conditions found in standard theories of (parabolic or hyperbolic) non-autonomous evolution equations are not given. In mathematical physics, bounds on multi-commutators of an order higher than two can be used to study linear and non-linear responses of interacting particles to external perturbations. These bounds are derived for lattice fermions, in view of applications to microscopic quantum theory of electrical conduction discussed in this book. All results also apply to quantum spin systems, with obvious modifications. In order to make the results accessible to a wide audience, in particular to students in mathematicswith little Physics background, basics of Quantum Mechanics are presented, keeping in mind its algebraic formulation. The C*-algebraic setting for lattice fermions, as well as the celebrated Lieb-Robinson bounds for commutators, are explained in detail, for completeness.

• Constructive Quantum Field Theory
• Constructive QFT
• Minimally connected graph
• Interacting fermions
• Quantum spins on lattices
• Perturbed autonomous dynamics
• AC-conductivity measure
• Paramagnetic conductivity
• Representation theory
• Fermion Fock spaces
• Lattice Fermi
• Quantum spin systems

“This volume deals with Lieb-Robinson bounds for multi-commutators of fermionic systems. … This book is reasonably self-contained … and has complete proofs in the parts describing more recent developments of the theory (Chapters 4– 6). It is recommended for researchers wanting to learn about generalization of Lieb-Robinson bounds and some of its important applications in the description of quantum matter.” (Fernando Lledó, zbMATH 1379.81047, 2018)

• Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Bilbao, Spain

  J.-B. Bru

• Department of Mathematical Physics, Institute of Physics, University of São Paulo , São Paulo, Brazil

  W. de Siqueira Pedra

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