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arXiv:quant-ph/0209119

A Renormalisation Approach to Effective Interactions in Hilbert Space

J. Richert

Published Sep 23, 2002• 0 citations
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Abstract

The low-lying bound states of a microscopic quantum many-body system of $n$ particles and the related physical observables can be worked out in a truncated $n$--particle Hilbert space. We present here a non-perturbative analysis of this problem which relies on a renormalisation concept and work out the link with perturbative approaches.

DOI
10.48550/arxiv.quant-ph/0209119

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The low-lying bound states of a microscopic quantum many-body system of nnn particles and the related physical observables can be worked out in a truncated nnn--particle Hilbert space. We present here a non-perturbative analysis of this problem which relies on a renormalisation concept and work out the link with perturbative approaches.
In this chapter, we introduce the analytical machinery to investigate the properties of many-body systems perturbatively. Specifically, employing the “ø 4 -theory” as an example, we learn how to describe systems that are not too far from a known reference state by perturbative means. Diagrammatic methods are introduced as a tool to efficiently implement perturbation theory at large orders. The new concepts are then applied to the analysis of various properties of the weakly interacting electron gas. In previous chapters we have emphasized repeatedly that the majority of many-particle problems cannot be solved in closed form. Therefore, in general, one is compelled to think about approximation strategies. One promising ansatz leans on the fact that, when approaching the low-temperature physics of a many-particle system, we often have some idea, however vague, of its preferred states and/or its low-energy excitations. One may then set out to explore the system by using these prospective ground state configurations as a working platform. For example, one might expand the Hamiltonian in the vicinity of the reference state and check that, indeed, the residual “perturbations” acting in the low-energy sector of the Hilbert space are weak and can be dealt with by some kind of approximate expansion. Consider, for example, the quantum Heisenberg magnet. In dimensions higher than one, an exact solution of this system is out of the question. However, we know (or, more conservatively, “expect”) that, at zero temperature, the spins will be frozen into configurations aligned along some (domain-wise) constant magnetisation axes.
In this chapter, we introduce the analytical machinery to investigate the properties of many-body systems perturbatively. Specifically, employing the “φ 4 -theory” as an example, we learn how to describe systems that are not too far from a known reference state by perturbative means. Diagrammatic methods are introduced as a tool to efficiently implement perturbation theory at large orders. The new concepts are then applied to the analysis of various properties of the weakly interacting electron gas. In previous chapters we have emphasized repeatedly that the majority of many-particle problems cannot be solved in closed form. Therefore, in general, one is compelled to think about approximation strategies. One promising ansatz leans on the fact that, when approaching the low-temperature physics of a many-particle system, we often have some idea, however vague, of its preferred states and/or its low-energy excitations. One may then set out to explore the system by using these prospective ground state configurations as a working platform. For example, one might expand the Hamiltonian in the vicinity of the reference state and check that, indeed, the residual “perturbations” acting in the low-energy sector of the Hilbert space are weak and can be dealt with by some kind of approximate expansion. Consider, for example, the quantum Heisenberg magnet. In dimensions higher than one, an exact solution of this system is out of the question. However, we know (or, more conservatively, “expect”) that, at zero temperature, the spins will be frozen into configurations aligned along some (domain-wise) constant magnetisation axes.
In close analogy to the Bloch-Feshbach formalism known from the theory of nuclear dynamics, I develop a mathematical framework that allows one to understand renormalization in terms of purely algebraic operations (projections, dilatations) in Hilbert space. This algebraic approach is put to the test in the study of the low-energy dynamics of interacting quantum gases, and proves to be efficient in deriving such diverse results as the renormalization group equation for an interacting Bose gas, the βββ function of ϕ4ϕ^4ϕ4 theory, the screening of fermion-fermion interactions or the BCS instability.