[Paper Review] Linear Response Theory: A Modern Analytic-Algebraic Approach

Giuseppe De Nittis, Max Lein|arXiv (Cornell University)|Dec 6, 2016

Advanced Operator Algebra Research36 references3 citations

TL;DR

This paper presents a rigorous, modern analytic-algebraic framework for Linear Response Theory (LRT) in quantum systems using non-commutative integration and von Neumann algebra techniques. It establishes the Kubo formula and its adiabatic limit in a general setting, enabling a topological interpretation of the quantum Hall effect and extending LRT to random and aperiodic systems beyond standard periodic models.

ABSTRACT

Linear response theory is a tool with which one can study systems that are driven out of equilibrium by external perturbations. This monograph presents a thoroughly modern framework to make linear response theory rigorous for a wide array of systems, that is suitable for novel applications such as periodic and random light conductors not yet covered in the literature. Our analytic-algebraic approach, based on von Neumann algebras and associated non-commutative $L^p$-spaces, can deal with discrete and continuous models alike, and include effects of disorder. First, we explain the mathematical setting, give a complete list of our hypotheses and state the main results, which include Kubo and Kubo-Streda formulas. To make our book accessible to a wide audience, we spend Chapters 3 and 4 explaining the mathematical underpinnings such as non-commutative $L^p$- and Sobolev spaces, and generalized commutators. Furthermore, we show how to construct a von Neumann algebras from a topological dynamical system and a 2-cocycle, a procedure which applies to discrete and continuous quantum systems. We dedicate Chapters 5 and 6 to the proofs of our main results. We close the book by sketching a novel application, linear response theory for periodic and random light conductors. This monograph is aimed at advanced students in mathematical physics and researchers.

Motivation & Objective

• To develop a mathematically rigorous foundation for Linear Response Theory applicable to a broad class of quantum systems, including random and aperiodic media.
• To unify the treatment of linear response in systems with ergodic dynamics, extending beyond periodic or homogeneous settings.
• To provide a framework that justifies the Kubo formula and its adiabatic limit in a general von Neumann algebra setting.
• To enable a topological interpretation of the quantum Hall effect via the Kubo-Strěda formula in non-periodic and disordered systems.
• To generalize the concept of conductivity and current response in systems with unbounded operators and non-trivial spectral properties.

Proposed method

• Formulates the dynamics using a von Neumann algebra of observables equipped with a normal, semifinite, faithful trace (tracial state).
• Introduces non-commutative Lp-spaces and measurable operators via the trace per unit volume for ergodic systems.
• Applies generalized commutators and non-commutative Sobolev spaces to handle unbounded perturbations and spatial derivatives.
• Uses isospectral transformations and T-compatible derivations to define non-commutative gradients and regularity conditions.
• Derives the Kubo formula via comparison of time-evolved equilibrium states under adiabatic perturbations.
• Establishes the adiabatic limit using convergence in measure and spectral theory, leading to the Kubo-Strěda formula.

Experimental results

Research questions

• RQ1 How can Linear Response Theory be rigorously formulated in non-periodic and random quantum systems?
• RQ2 What is the role of the trace per unit volume in defining macroscopic response functions like conductivity?
• RQ3 How does the adiabatic limit of the conductivity tensor relate to topological invariants in disordered systems?
• RQ4 Can the Kubo formula be derived in a general von Neumann algebra setting with unbounded observables?
• RQ5 What is the connection between the linear response of current and the spectral properties of the Hamiltonian in aperiodic media?

Key findings

• The paper establishes a general framework for linear response in systems with ergodic dynamics, using direct integral decompositions and covariant random operators.
• It proves the existence of the unitary propagator for adiabatic perturbations and derives the time evolution of observables in the interaction picture.
• The Kubo formula for conductivity is rigorously derived in the general setting of non-commutative Lp-spaces and measurable operators.
• The adiabatic limit of the conductivity tensor is shown to converge to the Kubo-Strěda formula, linking it to the Chern number in topological insulators.
• The zero-temperature limit yields a topological interpretation of the Hall conductivity as a spectral invariant, consistent with the TKNN formula.
• The framework applies to both continuum and discrete models, including Schrödinger operators with random potentials and tight-binding Hamiltonians.