Correlation Functions In Quantum Systems Research Articles

Time correlation functions for quantum systems: Validating Bayesian approaches for harmonic oscillators and beyond

The quantum harmonic oscillator is the fundamental building block to compute thermal properties of virtually any dielectric crystal at low temperatures in terms of phonons, extended further to cases with anharmonic couplings, or even disordered solids. In general, Path Integral Monte Carlo or Path Integral Molecular Dynamics methods are powerful tools to determine stochastically thermodynamic quantities without systematic bias, not relying on perturbative schemes. Addressing transport properties, for instance calculating thermal conductivity from PIMC, however, is substantially more difficult. Although correlation functions of current operators can be determined by PIMC from analytic continuation on the imaginary time axis, Bayesian methods are usually employed for the numerical inversion back to real time response functions. This task not only strongly relies on the accuracy of the PIMC data but also introduces noticeable dependence on the model used for the inversion. Here, we address both difficulties with care. In particular, we first devise improved estimators for current correlations, which substantially reduce the variance of the PIMC data. Next, we provide a neat statistical approach to the inversion problem, blending into a fresh workflow the classical stochastic maximum entropy method together with recent notions borrowed from statistical learning theory. We test our ideas on a single harmonic oscillator and a collection of oscillators with a continuous distribution of frequencies and provide indications of the performance of our method in the case of a particle in a double well potential. This work establishes solid grounds for an unbiased, fully quantum mechanical calculation of transport properties in solids.

Semiclassical calculation of spectral correlation functions of chaotic systems

We present a semiclassical approach to n-point spectral correlation functions of quantum systems whose classical dynamics is chaotic, for arbitrary n. The basic ingredients are sets of periodic orbits that have nearly the same action and therefore provide constructive interference. We calculate explicitly the first correlation functions, to leading orders in their energy arguments, for both unitary and orthogonal symmetry classes. The results agree with corresponding predictions from random matrix theory, thereby giving solid support to the conjecture of universality.

Full perturbative calculation of spectral correlation functions for chaotic systems in the unitary symmetry class

Starting from a semiclassical approach recently developed for spectral correlation functions of quantum systems whose classical dynamics is chaotic, we focus on the case of broken time-reversal symmetry, the so-called unitary class. We obtain to all orders in perturbation theory the non-oscillatory parts of all correlation functions, showing that the off-diagonal contributions to these correlation functions cancel and the conjectured universailty holds. The innovation that allows this calculation to be performed is the introduction of an auxiliary matrix model which is governed by the same diagrammatic rules as the semiclassical approach and which can be exactly solved.

Thermal out-of-time-order correlators, KMS relations, and spectral functions

We describe general features of thermal correlation functions in quantum systems, with specific focus on the fluctuation-dissipation type relations implied by the KMS condition. These end up relating correlation functions with different time ordering and thus should naturally be viewed in the larger context of out-of-time-ordered (OTO) observables. In particular, eschewing the standard formulation of KMS relations where thermal periodicity is combined with time-reversal to stay within the purview of Schwinger-Keldysh functional integrals, we show that there is a natural way to phrase them directly in terms of OTO correlators. We use these observations to construct a natural causal basis for thermal n-point functions in terms of fully nested commutators. We provide several general results which can be inferred from cyclic orbits of permutations, and exemplify the abstract results using a quantum oscillator as an explicit example.

Noninvasive measurement of dynamic correlation functions

The measurement of dynamic correlation functions of quantum systems is complicated by measurement backaction. To facilitate such measurements we introduce a protocol, based on weak ancilla--system couplings, that is applicable to arbitrary (pseudo)spin systems and arbitrary equilibrium or nonequilibrium initial states. Different choices of the coupling operator give access to the real and imaginary parts of the dynamic correlation function. This protocol reduces disturbances due to the early time measurements to a minimum, and we quantify the deviation of the measured correlation functions from the theoretical, unitarily-evolved ones. Implementations of the protocol in trapped ions and other experimental platforms are discussed. For spin-$1/2$ models and single-site observables we prove that measurement backaction can be avoided altogether, allowing for the use of ancilla-free protocols.

Quantum Monte Carlo estimation of complex-time correlations for the study of the ground-state dynamic structure function.

We present a method based on the path integral Monte Carlo formalism for the calculation of ground-state time correlation functions in quantum systems. The key point of the method is the consideration of time as a complex variable whose phase δ acts as an adjustable parameter. By using high-order approximations for the quantum propagator, it is possible to obtain Monte Carlo data all the way from purely imaginary time to δ values near the limit of real time. As a consequence, it is possible to infer accurately the spectral functions using simple inversion algorithms. We test this approach in the calculation of the dynamic structure function S(q, ω) of two one-dimensional model systems, harmonic and quartic oscillators, for which S(q, ω) can be exactly calculated. We notice a clear improvement in the calculation of the dynamic response with respect to the common approach based on the inverse Laplace transform of the imaginary-time correlation function.

Green-Kubo Formulas with Symmetrized Correlation Functions for Quantum Systems in Steady States: The Shear Viscosity of a Fluid in a Steady Shear Flow

For a quantum system in a steady state with a constant current of heat or particles driven by a temperature or chemical potential difference between two reservoirs attached to the system, the fluctuation theorem for the current was previously shown to lead to the Green-Kubo formula for the linear response coefficient for the current expressed in terms of the symmetrized correlation function of the current density operator. In this article, we show that for a quantum system in a steady state with a constant rate of work done on the system, the fluctuation theorem for a quantity induced in the system also leads to the Green-Kubo formula expressed in terms of the symmetrized correlation function of the induced quantity. As an example, we consider a fluid in a steady shear flow driven by a constant velocity of a solid plate moving above the fluid.

DENSITY-MATRIX RENORMALISATION GROUP FOR DYNAMIC CORRELATION FUNCTIONS

The calculation of dynamic correlation functions in quantum systems is formulated as a variational problem. For low-dimensional many-body systems this variational problem can be solved numerically using the density-matrix renormalisation group (DMRG). This dynamic DMRG method is demonstrated on the linear optical conductivity in the Mott insulating phase of the one-dimensional extended Hubbard model at half filling. The full optical spectrum of this model can be calculated almost exactly for chains with more than 100 sites, which is large enough to investigate the spectral properties in the thermodynamic limit. The accuracy of the method is illustrated by comparison with analytical results in the field-theoretical regime and in the strong-coupling limit.

On the Bayesian approach to calculating time correlation functions in quantum systems; reaction dynamics and spectroscopy

We assess the current status, advantages and limitations of the numerical analytic continuation approach to computing time correlation functions in large many-body quantum systems characteristic of condensed phase chemical processes. We determine the quantum correlation function as a function of complex time, and use its analytic properties to select a suitable contour in the complex time plane along which the function can be evaluated efficiently by stochastic simulation methods. The simulation data are then used to obtain the values of the correlation function along the real-time axis through a maximum entropy numerical analytic continuation procedure. This approach is used to compute the dynamical properties of several condensed phase processes including vibrational relaxation lineshapes and canonical reaction rates. We discuss how to improve the accuracy of the numerical analytic continuation methods.

Statistics and quantum chaos

We use multi-time correlation functions of quantum systems to construct random variables with statistical properties that reflect the degree of complexity of the underlying quantum dynamics.

Effective potential, Mori product and quantum dynamics

We present a method that permits the calculation of the dynamical correlation functions for quantum systems. These are obtained by evaluating the generating functionals of the static moments of the relaxation functions in a self-consistent approximation that can be obtained by an appropriate extension of the effective potential theory.

Methods for simulating time correlation functions in quantum systems

A review of the basic strategies that we have developed for the simulation of dynamic correlation functions in quantum systems is provided. Three methods are considered: (a) the analytic continuation of imaginary (thermal) time correlation functions to real times; (b) direct evaluation of thermally symmetrized version of the time correlation functions; (c) direct simulation of the power spectrum. All of these methods are based on the path integral representation of time corellation functions. The generalization of these methods for the computation of electronic spectrum, which involves the consideration of two potential energy surfaces, is also presented. We also discuss the approach based on numerical integration of an effective time-dependent Schrödinger equation for the simulation of electronic absorption and emission spectra for mixed quantum classical systems.

Time correlation functions in quantum systems

First Page

Dynamic correlation functions in quantum systems: A Monte Carlo algorithm

We propose a new algorithm to compute dynamic correlation functions in quantum systems at zero temperature using Monte Carlo techniques. The method, which avoids inverse-Laplace transforms, is based on a representation of the Dirac $\ensuremath{\delta}$ function as the limit of a Gaussian. We illustrate the method on the one-dimensional harmonic oscillator.

Quasiprobabilities and correlation functions in quantum systems

A general survey is given of the different methods that exist for the description of time correlations in quantum systems. A complete equivalence between all these methods is established on the basis of a consequent rule for the transcription of operator products into phase space. Particular attention is given to possible ambiguities in the definition of quantum mechanical Markov processes. The classical limit is discussed in detail.