Translational invariance of response functions

Malte Henkel

A generalised form of time-translation-invariance permits to re-derive the known generic phenomenology of ageing, which arises in classical many-body systems after a quench from an initially disordered system to a temperature T ≤ T c , at or below the critical temperature T c . Generalised time-translation-invariance is obtained, out of equilibrium, from a change of representation of the Lie algebra generators of the dynamical symmetries of scale-invariance and time-translation-invariance. Observable consequences include the algebraic form of the scaling functions for large arguments of the two-time auto-correlators and auto-responses, the equality of the auto-correlation and the auto-response exponents λ C = λ R , the cross-over scaling form for an initially magnetised critical system and the explanation of a novel finite-size scaling if the auto-correlator or auto-response converge for large arguments y = t / s ≫ 1 to a plateau. For global two-time correlators, the time-dependence involving the initial critical slip exponent Θ is confirmed and is generalised to all temperatures below criticality and to the global two-time response function, and their finite-size scaling is derived as well. This also includes the time-dependence of the squared global order-parameter. The celebrate Janssen-Schaub-Schmittmann scaling relation with the auto-correlation exponent is thereby extended to all temperatures below the critical temperature. A simple criterion on the relevance of non-linear terms in the stochastic equation of motion is derived, taking the dimensionality of couplings into account. Its applicability in a wide class of models is confirmed, for temperatures T ≤ T c . Relevance to experiments is also discussed.

J. Ding, Hudson A. Loughlin, Vivishek Sudhir

Systems theory provides a general framework to analyze and synthesize complex systems. Central to this discipline is the linear time-translation invariant (LTI) abstraction. We extend LTI systems theory into the quantum regime without relying on additional assumptions such as lumped-ness, Markovianity, or knowledge of an internal description. We develop a framework for the quantization of any classical LTI system based solely on its input-output response, bypassing the need to quantize its internal constituents. It describes the minimum additional quantum noise accompanying a given classical input-output response and characterizes the resulting class of quantum LTI systems via the conjugate symplectic Lie group. This algebraic structure enables the synthesis of a quantum LTI system with a prescribed response, the characterization of the invariant structure of its quantum noise, the classification of the specific quantum states it can emit, and the design of measurements optimal for it.

John W. Negele, Henri Orland

This chapter describes the salient properties of zero-temperature, real-time finite temperature, and thermal Green's functions in turn. It considers the effect of the energy and momentum dependence of the self-energy on quasiparticle propagation in an interacting Fermi system. The chapter illustrates the major points for the case of a nucleon in translationally invariant nuclear matter. Although Green's functions were introduced because of their physical connection to response functions, they contain sufficient information to evaluate all ground state observables or thermal averages. The real-time Green's function, which describes the physical response of a system, may thereby, be obtained from the thermal Green's function. The essential point in relating the optical potential to the self-energy is the observation that the wave function can be written in terms of the one-particle Green's function. A good approximation to the response may be obtained using the response function for a non-interacting Fermi gas corrected for the medium dependence of the nucleon self-energy.

Wilhelm Brenig