This article is the first of a trilogy that addresses various aspects of the perturbative response of general quantum systems, with possibly nontrivial ground state geometry, beyond linear order. Here, we use group theoretical considerations to investigate the structure of retarded correlators, and demonstrate how they decompose according to irreducible representations of a `time-reversal group' and relevant permutation groups, with the former probing dissipative and time-reversal properties, and the latter discerning configurational properties -- longitudinal, transverse and their generalizations. We establish second order fluctuation-dissipation and fluctuation-reaction theorems, and connect them to well-known second order transport effects such as the shift and injection currents. Exploiting the Schur-Weyl duality between irreducible representations of general linear groups and irreducible representations of permutation groups, we show how to decide which terms in the decomposition based on the latter can be supported by different crystals described via the 32 point groups, and perform the full point group classification for rank 3 and 4 polar and axial tensors. Our results provide a formal basis for the extraction of uniquely differing physical effects from retarded correlators. Applications to second order charge current responses in selected cases are given in part III.
This article is the first of a trilogy that addresses various aspects of the perturbative response of general quantum systems, with possibly nontrivial ground state geometry, beyond linear order. Here, we use group theoretical considerations to investigate the structure of retarded correlators, and demonstrate how they decompose according to irreducible representations of a `time-reversal group' and relevant permutation groups, with the former probing dissipative and time-reversal properties, and the latter discerning configurational properties -- longitudinal, transverse and their generalizations. We establish second order fluctuation-dissipation and fluctuation-reaction theorems, and connect them to well-known second order transport effects such as the shift and injection currents. Exploiting the Schur-Weyl duality between irreducible representations of general linear groups and irreducible representations of permutation groups, we show how to decide which terms in the decomposition based on the latter can be supported by different crystals described via the 32 point groups, and perform the full point group classification for rank 3 and 4 polar and axial tensors. Our results provide a formal basis for the extraction of uniquely differing physical effects from retarded correlators. Applications to second order charge current responses in selected cases are given in part III.
This article is the second of a trilogy that addresses the perturbative response of general quantum systems, with possibly nontrivial ground state geometry, beyond linear order. Here, we establish concise, general formulae for second order response to a spatially uniform, time-varying electric field in the velocity gauge that are manifestly free of static limit spurious divergences. We first discuss general quantum evolution in a curved space, then detail how such a situation is a natural byproduct of Hilbert space truncation, and point out crucial subtleties associated with the resulting finite curvatures. We then present a geometric perspective of the two popular gauges often used in quantum transport theories, the velocity gauge and the length gauge, and discuss how they, taking truncation-induced curvature effects into account, naturally lead to the same results in spite of the truncation. We highlight subtle formal discrepancies in the literature. Finally, we provide a general scheme for removing static limit spurious divergences in the velocity gauge without frequency expansions and present concise and comprehensive Green's function formulae for responses up to second order. As an application of specific aspects of our theory, second order charge current responses in selected cases are analyzed in parts I and III.
This thesis is a collection of three exact results on correlation and response functions in integrable systems. In the first part we study transport (two-point functions) in the 1D Hubbard model. First, we analyze the limit of large on-site repulsion and characterize spin transport as a function of temperature. Then, we consider the case where the model displays non-abelian symmetries, e.g., spin SU(2), and argue that in this case transport is anomalous with dynamical scaling exponent z=3/2 and follows \acrshort{kpz} scaling. In the next two parts we focus instead on (perturbative) nonlinear response functions, which can be expressed as n-point correlators with n>2. Given the lack of a systematic understanding of the information which can be extracted from these, we compute them in integrable 1D systems. We therefore focus on two distinct scenarios. In the first scenario, we consider a finite temperature system perturbed by some inhomogeneous field coupling to local charges --- thus the perturbations can only accelerate quasiparticles, but not create/annihilate them. In this context, we develop a diagrammatic framework based on generalized hydrodynamics that allows a systematic calculation of nonlinear response functions. We show that, in the hydrodynamic limit, nonlinear response functions qualitatively distinguish between non-interacting and interacting systems. In the second scenario, we instead consider a zero temperature system whose ground state coincides with the quasiparticle vacuum. In this simpler setting we consider more general perturbations that can also create and annihilate quasiparticles. Focusing on the Ising chain as a paradigmatic model we show that the long-time limit of four-point functions (third-order response) grows linearly in time. We interpret these divergences in terms of semiclassical processes where quasiparticles propagate ballistically and scatter when their trajectories intersect.
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.
While the theory of linear response functions in many body theory has developed into a substantial lore over the past decades, rather little is known about the properties and behavior of higher order (quadratic, etc.) response functions of many body systems. In this poster we first review essential elements of quadratic response theory such as the quadratic fluctuation dissipation theorems and sum rules for the quadratic density response function. Next we show how the correlational local field G(k,ω) in the linear response is fundamentally linked to the quadratic response function which opens a way to a non-perturbational analysis of the former. We proceed to present the results of the explicit calculation of the quadratic density response of 2D and 3D electron gas. Both, real and imaginary part of the latter reveal an extremely rich behavior. Of particular interest is the comparison between the two- and the three-dimensional case.
Quantum systems of indistinguishable particles are commonly described using the formalism of second quantization, which relies on the assumption that any admissible quantum state must be either symmetric or antisymmetric under particle permutations. Coherence-induced many-body effects such as superradiance, however, can arise even in systems whose constituents are not fundamentally indistinguishable as long as all relevant dynamical observables are permutation-invariant. Such systems are not confined to symmetric or antisymmetric states and therefore require a different theoretical approach. Focusing on noninteracting systems, here we combine tools from representation theory and thermodynamically consistent master equations to develop such a framework. We characterize the structure and properties of the steady states emerging in permutation-invariant ensembles of arbitrary multilevel systems that are collectively weakly coupled to a thermal environment. As an application of our general theory, we further explore how these states can in principle be used to enhance the performance of quantum thermal machines. Our group-theoretical framework thereby makes it possible to analyze various limiting cases that would not be accessible otherwise. In addition, it allows us to show that the properties of multilevel ensembles differ qualitatively from those of spin ensembles, which have been investigated earlier using the standard Clebsch-Gordan theory. Our results have a large scope for future generalizations and pave the way for systematic investigations of collective effects arising from permutation invariance in quantum thermodynamics.
Given an integrable (solvable) quantum system with many equivalent degrees of freedom it is inevitable that consideration of the permutation group will arise. For example the least degenerate stationary states of the system must fall into irreducible representations of the permutation group. The ground state, as well as a class of low‐lying excited states, must fall into a particular irreducible representation. It will be demonstrated that a superposition of states within this representation leads to marginal probability distributions (partial traces), which are time‐independent. These marginal probability distributions are constant in time, but not in space. They provide a resource that can be exploited in reversible quasi‐static (equilibrium) changes of state, i.e. they are a quantum contribution to the entropy. They are necessarily an essential part of any quantum statement of the second law.