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arXiv:cond-mat/9607198

Algebraic approach to renormalization

Jochen Rau

Published Jul 27, 1996• 1 citation
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Abstract

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 $β$ function of $ϕ^4$ theory, the screening of fermion-fermion interactions or the BCS instability.

DOI
10.48550/arxiv.cond-mat/9607198

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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.
The shell model solves the nuclear many-body problem in a restricted model space and takes into account the restricted nature of the space by using effective interactions and operators. In this paper two different methods for generating the effective interactions are considered. One is based on a partial solution of the Schrödinger equation (Bloch-Horowitz or the Feshbach projection formalism) and other on linear algebra (Lee-Suzuki). The two methods are derived in a parallel manner so that the difference and similarities become apparent. The connections with the renormalization group are also pointed out.
The shell model solve the nuclear many-body problem in a restricted model space and takes into account the restricted nature of the space by using effective interactions and operators. In this paper two different methods for generating the effective interactions are considered. One is based on a partial solution of the Schrodinger equation (Bloch-Horowitz or the Feshbach projection formalism) and other on linear algebra (Lee-Suzuki). The two methods are derived in a parallel manner so that the difference and similarities become apparent. The connections with the renormalization group are also pointed out.
Spin systems are known for their purely interacting nature, causing them to lack a canonical perturbative limit, which usually helps to understand the fundamental physics of a system. A method that has proven effective for the description of strongly correlated systems is Functional Renormalization Group (FRG), which can be thought of as an alternative to the path integral formalism as an approach to many-body quantum mechanics. The basic concept is to cure infrared divergences by introducing a cutoff to correlation functions. Continuously lowering the infrared cutoff and hence allowing the system to occupy lower-lying energy states is called the FRG flow and is formulated in terms of differential equations. The solutions to these equations are functions which describe effective interactions between particles and hence contain information about particle correlations and thermodynamic quantities. Formulating FRG in terms of spins turns out to be a subtle endeavor, though, as the spin algebra relations are relatively complicated compared to, for example, the canonical anti-commutation relations of fermions. In order to use the advantages of FRG for spin systems, one can map spins onto fermionic operators, which leads to pseudo-fermion FRG (PFFRG), developed by Reuther and Wölfle.[198] Alter- natively, one can express spins in terms of Majorana operators, leading to pseudo- Majorana FRG (PMFRG), which has been developed by Niggemann, Sbierski, and Reuther.[163] PMFRG is particularly powerful at finite temperatures, where useful properties of Majorana operators render the method more accurate than PFFRG. In contrast to many other methods, PFFRG and PMFRG are applicable to any spin system, even frustrated ones. In this work, PMFRG is being generalized and applied to highly frustrated spin systems. In particular, spin representations in terms of Majorana operators for spins with arbitrary large spin magnitude S are being investigated and classified thoroughly, which closes a gap in the literature about the second quantization of spin operators. Moreover, it is presented how to generalize PMFRG for Heisenberg models with full SU(2) symmetry to XXZ models exhibiting only a U(1) symmetry. This opens up a wide range of interesting applications for PMFRG. An example is the XXZ model on the pyrochlore lattice, which is known to exhibit conventional long-range order but also exotic spin liquid ground states in the so-called spin ice phase. PMFRG is applied to this model, and the results are being compared to experiments. Furthermore, the entire phase diagram of the XXZ model is being mapped out. Also, the capability of PMFRG to reproduce low-energy field theory predictions and to determine critical exponents is being tested.