Correlation functions for Hermitian many-body systems: Necessary conditions

1994 ◽  
Vol 49 (6) ◽  
pp. 4305-4306 ◽  
Author(s):  
Edward B. Brown
Keyword(s):  
Correlation Functions ◽  
Necessary Conditions ◽  
Many Body ◽  
Many Body Systems

Coarse-Graining, Scaling, and Hierarchies

2004 ◽  
Author(s):  
Juan Pérez-Mercade
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Correlation Functions ◽  
Nonlinear Partial Differential Equations ◽  
Coarse Graining ◽  
Power Laws ◽  
Effective Parameters ◽  
Many Body ◽  
Anomalous Dimensions ◽  
Many Body Systems

We present a scenario that is useful for describing hierarchies within classes of many-component systems. Although this scenario may be quite general, it will be illustrated in the case of many-body systems whose space-time evolution can be described by a class of stochastic parabolic nonlinear partial differential equations. The stochastic component we will consider is in the form of additive noise, but other forms of noise such as multiplicative noise may also be incorporated. It will turn out that hierarchical behavior is only one of a class of asymptotic behaviors that can emerge when an out-of-equilibrium system is coarse grained. This phenomenology can be analyzed and described using the renormalization group (RG) [6, 15]. It corresponds to the existence of complex fixed points for the parameters characterizing the system. As is well known (see, for example, Hochberg and Perez-Mercader [8] and Onuki [12] and the references cited there), parameters such as viscosities, noise couplings, and masses evolve with scale. In other words, their values depend on the scale of resolution at which the system is observed (examined). These scaledependent parameters are called effective parameters. The evolutionary changes due to coarse graining or, equivalently, changes in system size, are analyzed using the RG and translate into differential equations for the probability distribution function [8] of the many-body system, or the n-point correlation functions and the effective parameters. Under certain conditions and for systems away from equilibrium, some of the fixed points of the equations describing the scale dependence of the effective parameters can be complex; this translates into complex anomalous dimensions for the stochastic fields and, therefore, the correlation functions of the field develop a complex piece. We will see that basic requirements such as reality of probabilities and maximal correlation lead, in the case of complex fixed points, to hierarchical behavior. This is a first step for the generalization of extensive behavior as described by real power laws to the case of complex exponents and the study of hierarchical behavior.

scholarly journals Time Evolution of Correlation Functions in Quantum Many-Body Systems

2020 ◽  
Vol 124 (11) ◽  
Author(s):  
Álvaro M. Alhambra ◽  
Jonathon Riddell ◽  
Luis Pedro García-Pintos
Keyword(s):  
Time Evolution ◽  
Correlation Functions ◽  
Many Body ◽  
Many Body Systems

scholarly journals Statistics of correlation functions in the random Heisenberg chain

2019 ◽  
Vol 7 (5) ◽  
Author(s):  
Luis A. Colmenarez ◽  
Paul A. McClarty ◽  
Masud Haque ◽  
David J. Luitz
Keyword(s):  
Correlation Functions ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Many Body ◽  
Heisenberg Spin ◽  
Heisenberg Spin Chain ◽  
Spin Flip ◽  
Local Operators ◽  
Many Body Systems ◽  
Degenerate Perturbation Theory

Ergodic quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH). However, strong disorder can destroy ergodicity through many-body localization (MBL) – at least in one dimensional systems – leading to a clear signal of the MBL transition in the probability distributions of energy eigenstate expectation values of local operators. For a paradigmatic model of MBL, namely the random-field Heisenberg spin chain, we consider the full probability distribution of eigenstate correlation functions across the entire phase diagram. We find gaussian distributions at weak disorder, as predicted by pure ETH. At intermediate disorder – in the thermal phase – we find further evidence for anomalous thermalization in the form of heavy tails of the distributions. In the MBL phase, we observe peculiar features of the correlator distributions: a strong asymmetry in S_i^z S_{i+r}^zSizSi+rz correlators skewed towards negative values; and a multimodal distribution for spin-flip correlators. A quantitative quasi-degenerate perturbation theory calculation of these correlators yields a surprising agreement of the full distribution with the exact results, revealing, in particular, the origin of the multiple peaks in the spin-flip correlator distribution as arising from the resonant and off-resonant admixture of spin configurations. The distribution of the S_i^zS_{i+r}^zSizSi+rz correlator exhibits striking differences between the MBL and Anderson insulator cases.

Hydrodynamics

2021 ◽  
pp. 49-66
Author(s):  
Robert W. Batterman
Keyword(s):  
Correlation Functions ◽  
Linear Response ◽  
Order Parameters ◽  
Equilibrium States ◽  
Conserved Quantities ◽  
Many Body ◽  
Hydrodynamic Description ◽  
Relative Autonomy ◽  
Many Body Systems ◽  
The Continuum

This chapter begins the argument that the best way to understand the relations of relative autonomy between theories at different scales is through a mesoscale hydrodynamic description of many-body systems. It focuses on the evolution of conserved quantities of those systems in near, but out of equilibrium states. A relatively simple example is presented of a system of spins where the magnetization is the conserved quantity of interest. The chapter introduces the concepts of order parameters, of local quantities, and explains why we should be focused on the gradients of densities that inhabit the mesoscale between the scale of the continuum and that of the atomic. It introduces the importance of correlation functions and linear response.

scholarly journals Correlation functions and transport coefficients in generalised hydrodynamics

2022 ◽  
Vol 2022 (1) ◽  
pp. 014002 ◽  
Author(s):  
Jacopo De Nardis ◽  
Benjamin Doyon ◽  
Marko Medenjak ◽  
Miłosz Panfil
Keyword(s):  
Correlation Functions ◽  
Large Scale ◽  
Transport Coefficients ◽  
Integrable Models ◽  
Hydrodynamic Theory ◽  
Many Body ◽  
Exact Methods ◽  
Integrable Spin ◽  
Many Body Systems ◽  
Integrable Spin Chains

Abstract We review the recent advances on exact results for dynamical correlation functions at large scales and related transport coefficients in interacting integrable models. We discuss Drude weights, conductivity and diffusion constants, as well as linear and nonlinear response on top of equilibrium and non-equilibrium states. We consider the problems from the complementary perspectives of the general hydrodynamic theory of many-body systems, including hydrodynamic projections, and form-factor expansions in integrable models, and show how they provide a comprehensive and consistent set of exact methods to extract large scale behaviours. Finally, we overview various applications in integrable spin chains and field theories.

DENSITY-MATRIX RENORMALISATION GROUP FOR DYNAMIC CORRELATION FUNCTIONS

2003 ◽  
Vol 17 (28) ◽  
pp. 5453-5457
Author(s):  
E. JECKELMANN
Keyword(s):  
Density Matrix ◽  
Variational Problem ◽  
Correlation Functions ◽  
Renormalisation Group ◽  
Many Body ◽  
Dynamic Correlation ◽  
Many Body Systems ◽  
The One ◽  
Low Dimensional ◽  
Insulating Phase

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.

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