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Linear Response Theory — a Modern Analytic-Algebraic Approach

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Linear Response Theory — a Modern Analytic-Algebraic Approach Linear Response Theory A Modern Analytic-Algebraic Approach Giuseppe De Nittis & Max Lein arXiv:1612.01710v1 [math-ph] 6 Dec 2016 For Raffa & Tenko Acknowledgements We are indebted to François Germinet who introduced G. D. to [BGK+05; DG08], the works which have inspired this whole endeavor. In addition to François, many other colleagues were not just kind enough to share their insights with us, but also encouraged us to see this book project to an end. We would particularly like to express our gratitude to Jean Bellissard, Hermann Schulz-Baldes, Daniel Lenz, Georgi Raikov and Massimo Moscolari. Thanks to their help we were able to overcome many of the obstacles more quickly and more elegantly. Lastly, G. D. thanks FONDECYT which supported this book project through the grant “Iniciación en Investigación 2015 -No 11150143”. i Contents 1 Introduction 1 2 Setting, Hypotheses and Main Results 7 2.1 Descriptionoftheabstractsetting . ...... 7 2.2 The perturbed dynamics: bridge to analysis . ...... 12 2.3 Linear response and the Kubo formula . 19 2.4 The adiabatic limit and the Kubo-Stredaformula..............ˇ 22 2.5 Zero temperature limit and topological interpretation ........... 24 2.6 Thetight-bindingtypesimplification . ...... 26 3 Mathematical Framework 29 3.1 Algebraofobservables ............................ 29 3.1.1 The von Neumann algebra of observables . 29 3.1.2 The algebra of affiliated operators . 31 3.1.3 Finite vs. semi-finite von Neumann algebras . 32 3.2 Non-commutative Lp-spaces .......................... 33 3.2.1 F.n.s.tracestate ............................. 33 3.2.2 Convergence in measure and measurable operators . 35 3.2.3 Integration and Lp-spaces ....................... 38 3.2.4 Isospectral transformations and induced isometries ........ 41 3.3 Generalized commutators . 44 3.3.1 Commutators between -measurable operators . 44 T 3.3.2 Commutators between -measurable and affiliated operators . 45 T 3.3.3 Commutators between unbounded operators . 50 3.4 Non-commutativeSobolevspaces . 51 3.4.1 -compatible spatial derivations . 51 T 3.4.2 Non-commutative gradient and Sobolev spaces . 55 4 A Unified Framework for Common Physical Systems 57 4.1 Von Neumann algebra associated to ergodic topological dynamical sys- tems ......................................... 57 4.1.1 Projective representations of G .................... 58 4.1.2 Randomly weighted Hilbert spaces . 59 iii Contents 4.1.3 Direct integral of Hilbert spaces . 60 4.1.4 The algebra of covariant random operators . 62 4.2 Thetraceperunitvolume........................... 65 4.3 Generators compatible with the trace per unit volume . ....... 69 4.4 Reductiontothenon-randomcase . 70 5 Studying the Dynamics 73 5.1 Unperturbeddynamics............................. 73 5.1.1 The generator of the unperturbed dynamics . 74 5.1.2 A formula for the projection in Theorem 2.4.1 . 76 5.2 Perturbeddynamics............................... 79 5.2.1 Adiabatic isospectral perturbations . ..... 80 5.2.2 Additive vs. multiplicative perturbations . ...... 81 5.2.3 Existenceoftheunitarypropagator . 84 5.2.4 Evolutionofobservables . 90 5.2.5 Interaction evolution of observables . 94 5.3 Comparison of perturbed and unperturbed dynamics . ...... 97 6 The Kubo Formula and its Adiabatic Limit 101 6.1 Comparing the evolutions of equilibrium states . ........ 102 6.1.1 Initial equilibrium states . 102 6.1.2 Existence of ρfull(t) and its expansion in Φ ............. 107 6.2 TheKuboformulafortheconductivity . 112 6.2.1 The macroscopic net current and the conductivity tensor . 112 6.2.2 ProofoftheKuboformula .. .. .. .. .. .. .. .. .. .. .. 118 6.3 The adiabatic limit of the conductivity tensor . ........ 120 7 Applications 125 7.1 Linear response theory for periodic and random light conductors . 125 7.1.1 Schrödinger formalism of electromagnetism . 126 7.1.2 Randommedia .............................. 128 7.1.3 Openquestions.............................. 130 7.2 Quantum Hall effect in solid state physics . 132 7.2.1 Continuummodels............................ 132 7.2.2 Discretemodels.............................. 135 iv Chapter 1 1 Introduction Linear response theory is a tool with which one can study systems that are driven out of equilibrium by external perturbations. The prototypical example is a first-principles d justification of Ohm’s empirical law J = j=1 σj Ej [Ohm10], which states that the current is linearly proportional to the applied external electric field: These ideas have P been pioneered by Green [Gre54] and Kubo [Kub57] in the context of statistical me- chanics, and later used by Stredaˇ [Str82] to link the transverse conductivity in a 2d electron gas to the number of Landau levels below the Fermi energy. The aim of this book is to provide a modern tool for mathematical physicists, allowing them to make linear response theory (LRT) rigorous for a wide range of systems — including some that are beyond the scope of existing theory. We will explain all the moving pieces of this analytic-algebraic framework below and put it into context with the literature. But first let us give a flavor of the physics. Initially, the unperturbed system, governed by a selfadjoint operator H, is at equi- librium, meaning that it is an a state described by a density operator ρ commuting with H. Then, in the distant past we adiabatically switch on a perturbation which drives the system out of equilibrium. Here we distinguish between a set of pertur- bation parameters Φ = Φ1,..., Φd (e. g. components of the electric field) and the adiabatic parameter ǫ which quantifies how quickly the perturbation is ramped up. Thus, both enter as parameters in the perturbed Hamiltonian HΦ,ǫ(t)= HΦ,ǫ(t)∗. The perturbation is switched on at t < 0 (which in principle could be ) and at t = 0 0 −∞ the Hamiltonian has reached the perturbed state. The adiabatic switching allows us to start with the same initial state ρ as in the unperturbed case, and evolve it according to ρ(t)= UΦ,ǫ(t, t0) ρ UΦ,ǫ(t, t0)∗, (1.0.1) 1 1 Introduction where UΦ,ǫ(t, t0) is the unitary propagator associated to the time-dependent Hamil- tonian HΦ,ǫ(t). In the simplest case we want to see the effects of the perturbation by studying the net current Φ,ǫ(t)= J (t) ρ(t) J ρ (1.0.2) J T Φ,ǫ −T which is the difference of the expectation values of the current operators JΦ,ǫ(t) and J with respect to ρ(t) and ρ, computed with the trace-per-volume . Typically, T J (t)= i X , H (t) and J = i [X , H] are given in terms of commutators with Φ,ǫ − Φ,ǫ − the appropriate Hamiltonians, thereby explaining why the two current operators are different and one of them depends on Φ, ǫ and t. The crucial step in making LRT rigorous is to justify the “Taylor expansion” of the net current to first order in ΦΦ= 0, d Φ,ǫ(t)= Φ,ǫ(t) + Φ ∂ Φ,ǫ(t) + o(ΦΦΦ) J J ΦΦ=0 j Φj J ΦΦ=0 j=1 X d ǫ = Φj σj (t)+ o(ΦΦΦ), (1.0.3) j=1 X where the 0th order terms vanish — no perturbation, no net current — and the con- ǫ ductivity coefficients σj (t) quantify the linear response. To wash out some of the de- tails of the interpolation between the perturbed and the unperturbed system, typically one takes the adiabatic limit ǫ 0 of the conductivity coefficients. → ǫ Kubo’s contribution [Kub57] was the derivation of an explicit formula for the σj (t) (cf. equation (2.3.3)). Stredaˇ has a second expression in case ρ is a spectral projec- tion; This Kubo-Stredaˇ formula (cf. equation (2.4.5)) has helped give a topological interpretation to the Quantum Hall Effect [TKN+82; Hat93], giving birth to the field of Topological Insulators in the process. That is why a significant share of the mathematically rigorous literature concerns LRT for various models of the Quantum Hall Effect (e. g. [BES94; BS98; BGK+05; DG08; ES04]). Roughly speaking, there are two approaches, those that attack LRT from the functional analytic side (e. g. [BGK+05; KLM07]) and those which formulate the problem in algebraic terms (e. g. [BES94; BS98; JP02; JOP06]). Typically, the main advantage of algebraic approaches is that they give a scheme for how to make LRT rigorous, which applies to a whole class of systems, at the expense of rather strong assumptions on H, ρ and . Very often these approaches require H to lie T in a C∗- or von Neumann algebra , and therefore H is necessarily bounded, or A that is finite. This excludes many physically interesting and relevant models, most T 2 notably continuum (as opposed to discrete) models. Conversely, analytic approaches typically focus on one particular system, including those described by unbounded Hamiltonians. However, the details are usually specific to the Hamiltonian of interest, and these techniques do not readily transfer from one physical system to another. Therefore we have developed a unified and thoroughly modern framework which combines the advantages of both approaches: we give an explicit scheme for LRT, based on von Neumann algebras and associated non-commutative Lp-spaces, that applies to discrete and continuous models alike, that can deal with disorder and is not tailored to one specific model. It not only subsumes many previous results, no- tably [BS98; SB98; BGK+05; DG08], but also applies to systems that have not yet been considered in the literature. We will detail the precise setting, all hypotheses and our main results in Chapter 2. Nevertheless, let us anticipate the most important aspects in order to contrast and compare with the literature. Our book is inspired by the works of Bouclet, Germinet, Klein and Schenker [BGK+05] as well as Dom- browski and Germinet [DG08], who make LRT rigorous for Schrödinger operators H =( i A )2 + V for a non-relativistic particle on the continuum subjected to ω − ∇− ω ω a random electric and magnetic field (cf.
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