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arXiv:cond-mat/0108511

Scaling Limit and Renormalisation Group in General (Quantum) Many Body Theory

Manfred Requardt

Published Aug 30, 2001• 1 citation
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Abstract

Using the machinery of smooth scaling and coarse-graining of observables, developed recently in the context of so-called fluctuation operators (originally developed by Verbeure et al), we extend this approach to a rigorous renormalisation group analysis of the critical regime. The approach is completely general, encompassing classical, quantum, discrete and continuous systems. Our central theme is the analysis of the famous `scaling hypothesis', that is, we make a general investigation under what cluster conditions of the l-point correlation functions a scale invariant (non-trivial) limit theory can be actually attained.

DOI
10.48550/arxiv.cond-mat/0108511

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Using the machinery of smooth scaling and coarse-graining of observables, developed recently in the context of so-called fluctuation operators (originally developed by Verbeure et al), we extend this approach to a rigorous renormalisation group analysis of the critical regime. The approach is completely general, encompassing classical, quantum, discrete and continuous systems. Our central theme is the analysis of the famous `scaling hypothesis', that is, we make a general investigation under what cluster conditions of the l-point correlation functions a scale invariant (non-trivial) limit theory can be actually attained.
We employ the machinery of smooth scaling and coarse-graining of observables, developed recently by us in the context of so-called fluctuation operators (inspired by prior work of Verbeure et al) to make a rigorous renormalisation group analysis of the critical regime. The approach appears to be quite general, encompassing classical, quantum, discrete and continuous systems. One of our central topics is the analysis of the famous `scaling hypothesis', that is, we make a general investigation under what conditions on the l-point correlation functions a scale invariant (non-trivial) limit theory can be actually attained. Furthermore, we study in a rigorous manner questions like the quantum character of the system in the scaling limit, the phenomenon of critical slowing down etc.
With the help of a smooth scaling and coarse-graining approach of observables, developed recently by us in the context of so-called fluctuation operators (inspired by prior work of Verbeure et al) we perform a rigorous renormalisation group analysis of the critical regime. The approach is quite general, encompassing classical, quantum, discrete and continuous systems, the main thrust going to quantum many body systems. Our central topic is the analysis of the emergent properties of critical systems on the intermediate scales and in the scaling limit. To mention some particularly interesting points, we show that systems typically loose part of their quantum character in the scaling limit (vanishing of commutators) and we rigorously prove, with the help of the KMS-condition, the emergence of the phenomenon of critical slowing down together with the necessity of renormalising the time variable. These general features are then illustrated with the help of an instructive class of models and are related to the singular structure of quasi particle excitation modes for vanishing energy-momentum.
The basic concepts and formulation of the renormalization group are explained beginning at an elementary level. Discussion is in the framework of classical statistical mechanics with emphasis on applications to the theory of critical phenomena. The details are worked out in the large n limit for 2 < d < 4, where n is the number of components of the fluctuating field of interest and d is the dimension of the thermodynamical system. In the large n limit, the infinite sum of ``tree graphs'' offers an exact and analytically tractable description of the renormalization group. It illustrates many concepts including the fixed point, the critical surface in the space of coupling parameters, and critical exponents. Most important, it illustrates the origin and the limitation of the scaling hypothesis. The critical behavior of various correlation functions and the free energy is examined. Attention is paid to terms often ignored in qualitative scaling arguments. We have attempted to make this paper self-contained and of pedagogical value to a wide audience.
Abstract We consider the most general scale invariant radial Hamiltonian allowing for anisotropic scaling between space and time. We formulate a renormalisation group analysis of this system and demonstrate numerically the existence of a universal quantum phase transition from a continuous scale invariant phase to a discrete scale invariant phase. Close to the critical point, the discrete scale invariant phase is characterised by an isolated, closed, attracting trajectory in renomalisation group space (a limit cycle). Moving in appropriate directions in the parameter space of couplings this picture is altered to one controlled by a quasi periodic attracting trajectory (a limit torus) or fixed points. We identify a direct relation between the critical point, the renormalisation group picture and the power laws characterising the zero energy wave functions.
This chapter explains the static scaling hypothesis, and shows how it leads to scaling laws. It discusses the topic of dynamical scaling, in which time-dependent correlations in equilibrium near the critical point exhibit scaling. The chapter introduces the topic of scaling in non-equilibrium systems. Scaling and scaling laws near the critical point of equilibrium systems follow from the renormalisation group theory, which is capable of calculating the values of the critical exponents and the form of scaling functions when used in combination with approximate methods such as perturbation theory. The static scaling hypothesis, written in the form appropriate for a magnetic system, is an attempt to encode two experimental results in one equation. Equilibrium statistical mechanics is primarily concerned with static quantities. Nevertheless, time-dependent fluctuations of a system in equilibrium fall within the scope of equilibrium statistical mechanics, through the use of the fluctuation dissipation theorem.
This chapter explains how the scaling hypothesis follows from the presence of a diverging correlation length. It shows the relationship between coupling constants defined at different length scales is more complicated than assumed. K. G. Wilson elaborated and completed L. P. Kadanoff’s argument, showing how the relationship between coupling constants at different length scales could be explicitly computed, at least approximately. Wilson’s theory — the renormalisation group (RG) — is capable of estimating the critical exponents. The RG also provides a natural framework in which to understand universality. The chapter provides the argument in two parts, the first being the thermodynamic scaling laws, and the second being the scaling laws for the two-point correlation function. Kadanoff’s block spin argument successfully motivates the functional form of the scaling relations. The conceptual importance of the argument is that it suggests how fruitful it might be to get away from the conventional statistical mechanical approach of treating all the degrees of freedom at once.