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arXiv:1911.09620 · Journal of Statistical Mechanics Theory and Experiment

About the foundation of the Kubo generalized cumulants theory: a revisited and corrected approach

Marco Bianucci, Mauro Bologna

Published Apr 1, 2020• 12 citations
arXivHTMLPDF

Abstract

Abstract More than fifty years ago, in a couple of seminal works (Kubo 1962 J. Phys. Soc. Japan 17 1100–20; Kubo 1963 J. Math. Phys . 4 174–83) Kubo introduced the important idea of generalized cumulants, extending to stochastic operators this concept implicitly introduced by Laplace in 1810. Kubo’s idea has been applied in several branches of physics, where the result of the average process is a Lioville operator or an effective time evolution operator for the density matrix of spin systems or the reduced density matrix for boson-fermions etc. Despite this success, the theoretical developments in these Kubo works pose problems that were highlighted many years ago by Fox and van Kampen and never solved. These weaknesses and errors, in particular concerning the factorization property of exponentials of cumulants and the explicit expressions that give generalized cumulants in terms of generalized moments and vice-versa, caused some perplexity (and confusion) about the possible application of this procedure, limiting its use, in practice. In the present paper, we give a sound ground to the approach to cumulant operators, working in a general framework that shows the potentiality of the old Kubo idea, today not yet fully exploited. It results that, for the same moment operators, different definitions of generalized cumulants can be adopted. A general Kubo–Meeron closed-form formula giving cumulant operators in terms of moment operators cannot be obtained, but the reverse one, cumulants in terms of operators, is given and, noticeably, formally it does not depend on the specific nature of the moments, but just on the definition of the generalized cumulants.

DOI
10.1088/1742-5468/ab7755

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Abstract More than fifty years ago, in a couple of seminal works (Kubo 1962 J. Phys. Soc. Japan 17 1100–20; Kubo 1963 J. Math. Phys . 4 174–83) Kubo introduced the important idea of generalized cumulants, extending to stochastic operators this concept implicitly introduced by Laplace in 1810. Kubo’s idea has been applied in several branches of physics, where the result of the average process is a Lioville operator or an effective time evolution operator for the density matrix of spin systems or the reduced density matrix for boson-fermions etc. Despite this success, the theoretical developments in these Kubo works pose problems that were highlighted many years ago by Fox and van Kampen and never solved. These weaknesses and errors, in particular concerning the factorization property of exponentials of cumulants and the explicit expressions that give generalized cumulants in terms of generalized moments and vice-versa, caused some perplexity (and confusion) about the possible application of this procedure, limiting its use, in practice. In the present paper, we give a sound ground to the approach to cumulant operators, working in a general framework that shows the potentiality of the old Kubo idea, today not yet fully exploited. It results that, for the same moment operators, different definitions of generalized cumulants can be adopted. A general Kubo–Meeron closed-form formula giving cumulant operators in terms of moment operators cannot be obtained, but the reverse one, cumulants in terms of operators, is given and, noticeably, formally it does not depend on the specific nature of the moments, but just on the definition of the generalized cumulants.
A detailed account of the density matrix formalism and the resolvent operator theory is given. The statistical mixture of quantum states is described in terms of the density matrix operator, a concept which in an elegant manner allows one to obtain the ensemble average of an observable [a Hermitian operator, which eigenspectrum is complete]. On the basis of an integral equation for the evolution operator, described in the Interaction Representation, the Liouville equation, describing the dynamical evolution of the density matrix operator, is established. A comprehensive analysis of thermal equilibrium in electron systems is presented. For a description of the dynamical time evolution of a system in contact with a heat bath, a reservoir in thermal equilibrium, the den- sity matrix operator approach is shown to be extremely convenient. The Kubo formalism for the time-dependent density matrix is discussed in its most general form. The so-called resolvent propagator formalism is set up starting from a discussion of the analytical part of a Dirac delta function. Having established integral equations for the retarded and advanced scalar propagator operators, the resolvent propagator concept is introduced and analyzed. The resolvent of the Hamiltonian is of particular importance in physics, and the singularities of this resolvent (poles and branch cuts) are identified. We finish by a study of correlated systems in terms of the reduced density matrix operator concept.
The study of electrical currents in a regime where a description as a continuous flow fails and the particle nature becomes important, but where notions such as particle density or currents continue to be meaningful, will be at the heart of the present dissertation. Given such a mesoscopic system through which electrons flow, either actively through quantum pumping or passively due to a potential difference, we investigate quantities such as transferred charge, noise, current fluctuations, and relations between them. More generally we shall examine the statistics of particle transport. All discussions will concentrate on compact devices transferring charge between two one-dimensional reservoirs (the leads) of independent electrons. Firstly, we consider the generating function of the moments of transferred charge and we prove that it is given by the determinant of a single-particle operator. It is formally equivalent to a formula of Lesovik and Levitov, but is shown to be well-defined in the situations under consideration, whereas the former is not, in general. The mechanism leading to regularization is typical for mesoscopic systems: we consider a limiting case that reveals the fundamental structure of the problem. In fact, as suited for an open system, the leads contain an infinite number of charges. It follows mathematically that the second quantization ought to be carried out upon the Fermi sea and not upon the vacuum, a procedure known as the GNS construction. We shall rely heavily on operator algebraic techniques. The set of observables, the C*-algebra of canonical anticommutation relations, admits different representations for systems in the thermodynamic limit, which are described by their reduced one-particle density matrix. An important question will be the existence of operators such as the charge or the time evolution in a given representation, which induce so-called Bogoliubov transformations. These prerequisites at hand, we shall pose the problem of counting statistics clearly, prove that the regularized determinant emerges naturally, and show that it describes the transport of integer charges with non-negative probabilities. Secondly, we introduce a model of instantaneous scattering in which various examples can be computed explicitely. In particular, we show that the above theorem applies for scattering through a smoothly operating pump over a finite time interval. We also exhibit situations where the propagator does exist but the generating function is ambiguous. In a more computational approach, we then turn to the relation of individual moments of transferred charge to current correlators. It turns out that the former can indeed be expressed as integrals of the latter over time, provided the current operators are ordered according to Matthews’ T∗ ordering. It is given by first arranging the charge operators according to their time indices, and only then taking the time derivatives, which induce contact terms whose singular support lies along time coincidences. We shall give two derivations of this result, because the first one is transparent, and the second one is related to contradictory computations in the literature where the system is considered to be measured by an external spin. In the model of constant instantaneous scattering, we then argue from the Lesovik-Levitov determinant that the transport can be interpreted in terms of a binomial process in the long time limit. In a parallel computation, we concentrate on the third cumulant as defined by integration of current correlators and show how the contact terms of the T∗ ordering play an essential role in providing the expected binomial result, too. To close this thesis, we suggest an algebraic framework that lays the foundations for a rigorous discussion of the Casimir effect. The problem of infinites arising in the comparison of two physical situations for a bosonic field is interpreted as the non-equivalence of states on the algebra of canonical commutation relations. We exhibit sufficient conditions on the one-particle Hamiltonians which ensure that the Casimir energy is finite and give an expression for it. We discuss its formal equivalence with the result obtained from the standard mode summing method.
Recently it has been shown that cumulants significantly simplify the analysis of multipartite weak measurements. Here we consider the mathematical structure that underlies this and find that it can be formulated in terms of what we call the moment algebra. Apart from resulting in simpler proofs, the flexibility of this structure allows generalizations of the original results to a number of weak measurement scenarios, including one where the weakly interacting pointers reach thermal equilibrium with the probed system.
First, we consider generalized wave and scattering operators and derive modifications of commutation relations (between scattering operators and unperturbed operators) when the corresponding deviation factors behave as exp⁡{itC±}\exp\{i t {\mathcal C}_{\pm}\}exp{itC±​} for t→±∞t\to \pm \inftyt→±∞. Then, we construct so called secondary generalized scattering operators for the related case of linear divergence in QED, which gives a positive answer (in that case) to the well-known problem of J. R. Oppenheimer regarding scattering operators in QED: "Can the procedure be freed of the expansion in ε\varepsilonε and carried out rigorously?"
In a seminal work, Aharonov, Albert, and Vaidman showed that by having a weak interaction between a system and a detecting apparatus, the average output of the latter could be much larger than the maximum eigenvalue of the observed quantity (times the amplification factor). This does not always happen, however: the observed system must subsequently undergo a second measurement, on the output of which the result of the first one is conditioned. This procedure is known as postselection. On the other hand, linear response theory describes how the observables of a quan- tum system change upon perturbation by a weak classical external force. In a measurement, the measured system applies a generalized force to the measuring apparatus, leading to an observable change in the latter. It appears natural, then, to unify the treatment of weak measurements with an extended version of linear response theory that accounts for a force introduced by an external quantum system. Here, we show how the postselection introduces a complex force term and we provide a modified Kubo formula working in the non-linear regime.