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Physical review. B, Condensed matter

Renormalization-group theory of the λ transition

J. C. Lee

Published Aug 1, 1979• 11 citations
Publisher

Abstract

The renormalization-group transformation is performed on the microscopic quantum Hamiltonian of a boson system. The transformation involves a step in which the occupation number of each plane-wave state is rescaled. This is equivalent to a transformation of the quantum amplitude and leads to the same fixed point that Singh found previously. The quantum effect again disappears, but however, for a different reason.

DOI
10.1103/physrevb.20.1277

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The renormalization-group transformation is performed on the microscopic quantum Hamiltonian of a boson system. The transformation involves a step in which the occupation number of each plane-wave state is rescaled. This is equivalent to a transformation of the quantum amplitude and leads to the same fixed point that Singh found previously. The quantum effect again disappears, but however, for a different reason.
We saw in Chapter 3 that the hamiltonian for a System at a critical point flows under the renormalization group into a critical fixed point. Under a renormalization group transformation, the microscopic length scale is rescaled by a constant factor b , and so the coordinates of a given point, as measured in units of this length scale, transform according to r → b −1 r. This is called a scale transformation . Once the flows reach such a fixed point, the parameters of the hamiltonian no longer change, and it is said to be scale invariant . As well as being scale invariant, the fixed point hamiltonian usually possesses other spatial symmetries. For example, if the underlying model is defined on a lattice, so that its hamiltonian is invariant under lattice translations, the corresponding critical fixed point hamiltonian is generally invariant under arbitrary uniform translations. This is because terms which might be added to the hamiltonian which break the symmetry under continuous translations down to its subgroup of lattice translations are irrelevant at such a fixed point. Similarly, if the lattice model is invariant under a sufficiently large subgroup of the rotation group (for example, if the interactions in the x and y directions on a Square lattice are equal), then the fixed point hamiltonian enjoys full rotational invariance. As discussed on p.58, even if the interactions are anisotropic, rotational invariance may often be recovered by a suitable finite relative rescaling of the coordinates.
We propose inverse renormalization group transformations within the context of quantum field theory that produce the appropriate critical fixed point structure, give rise to inverse flows in parameter space, and evade the critical slowing down effect in calculations pertinent to criticality. Given configurations of the two-dimensional ϕ^{4} scalar field theory on sizes as small as V=8^{2}, we apply the inverse transformations to produce rescaled systems of size up to V^{'}=512^{2} which we utilize to extract two critical exponents. We conclude by discussing how the approach is generally applicable to any method that successfully produces configurations from a statistical ensemble and how it can give novel insights into the structure of the renormalization group.
The renormalization group transformation near four space-dimensions is extended to irreversible processes of a quantum Bose system in terms of a coherent-state formalism. Recursion relations for a kinetic equation are obtained explicitly up to O(ε2), where ε≡4-d and d denotes the space dimensionality. It is shown that terms representing the quantum effects become “irrelevant” by renormalization group scale transformation which can be carried out in the case of quantum fields due to coherent-state formalism. The complex parameter S introduced by Abrahams and Tsuneto phenomenologically is obtained from the microscopic point of view by eliminating microscopic (non-critical) fluctuations. It is also shown that S is driven to the TDGL type at the fixed point, so that the dynamical critical exponent z agrees with that obtained by Halperin, Hohenberg and Ma in the TDGL model.
The ability of entanglement renormalization (ER) to generate a proper real-space renormalization group (RG) flow in extended quantum systems is analyzed in the setting of harmonic lattice systems in D =1 and 2 spatial dimensions. A conceptual overview of the steps involved in momentum-space RG is provided and contrasted against the equivalent steps in the real-space setting. The real-space RG flow, as generated by ER, is compared against the exact results from momentum-space RG, including an investigation of a critical fixed point and the effect of relevant and irrelevant perturbations.
Abstract Chapter 5 first recalls the importance of the concept of scale decoupling in physics. It then emphasizes that quantum field theory and the theory of critical phenomena have provided two examples where this concepts fails. To deal with such a situation, a new tool has been invented: the renormalization group. In the framework of effective quantum field theory, a perturbative renormalization group has been formulated. Its implementation has led to the discovery of fixed points as zeros of beta functions, and calculations of critical exponents of a class of macroscopic phase transitions in the form of Wilson–Fisher epsilon or fixed dimension expansions. These expansions being divergent, they could summed by methods based on the Borel transformation and the determination of the large order behaviour of perturbation theory.