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arXiv:cond-mat/9909418

O(N)-invariant Hierarchical Renormalization Group Fixed Points by Algebraic Numerical Computation and ε-Expansion

J. Goettker-Schnetmann

Published Sep 29, 1999• 0 citations
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Abstract

Generalizing methods developed by Pinn, Pordt and Wieczerkowski for the hierarchical model with one component (N=1) and dimensions d between 2 and 4 we compute O(N)-symmetric fixed points of the hierarchical renormalization group equation for some N and d with 0 < d < 4 and -2 <= N <= 20. The spectra of the linearized RG equation at the fixed points are calculated and the critical exponents νare extracted from the spectrum and compared to Borel-Pade-resummed ε-expansion.

DOI
10.48550/arxiv.cond-mat/9909418

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Generalizing methods developed by Pinn, Pordt and Wieczerkowski for the hierarchical model with one component (N=1) and dimensions d between 2 and 4 we compute O(N)-symmetric fixed points of the hierarchical renormalization group equation for some N and d with 0 < d < 4 and -2 <= N <= 20. The spectra of the linearized RG equation at the fixed points are calculated and the critical exponents νare extracted from the spectrum and compared to Borel-Pade-resummed ε-expansion.
Critical exponents for the three-dimensional O(n)-symmetric model with n>3 are estimated on the basis of six-loop renormalization-group (RG) expansions. A simple Pad\'e-Borel technique is used for the resummation of the RG series and the Pad\'e approximants [L/1] are shown to give rather good numerical results for all calculated quantities. For large n, the fixed point location gc{\mathit{g}}_{\mathit{c}}gc​ and the critical exponents are also determined directly from six-loop expansions without addressing the resummation procedure. An analysis of the numbers obtained shows that resummation becomes unnecessary when n exceeds 28 provided an accuracy of about 0.01 is adopted as satisfactory for gc{\mathit{g}}_{\mathit{c}}gc​ and the critical exponents. Further, results of the calculations performed are used to estimate the numerical accuracy of the 1/n expansion. The same value n=28 is shown to play the role of the lower boundary of the domain where this approximation provides high-precision estimates for the critical exponents.
Algebraic Computation of the Hierarchical Renormalization Group Fixed Points and their εεε-Expansions
Nontrivial fixed points of the hierarchical renormalization group are computed by numerically solving a system of quadratic equations for the coupling constants. This approach avoids a fine tuning of relevant parameters. We study the eigenvalues of the renormalization group transformation, linearized around the non-trivial fixed points. The numerical results are compared with εεε-expansion.
For pt.II see ibid., vol.11, p.375 (1978). Exact renormalisation group equations (RGE) are obtained for the d-dimensional n-vector and spin-S Heisenberg versions of Dyson's hierarchical model. In the quantum mechanical case the fixed point and neighbourhood properties of the RGE are shown to be independent of S indicating universality of critical behaviour in spin magnitude. Expansions for critical exponents, analogous to the usual epsilon =4-d expansions, are obtained and the RGE solved numerically to obtain relevant scaling indices. The n to infinity or spherical limit is also discussed and 1/n corrections to critical exponents are calculated to first order in 1/n.
The numerical renormalization group method is used to investigate zero-temperature phase transitions in quantum impurity systems, in particular in the particle–hole symmetric soft-gap Anderson model. The model displays two stable phases whose fixed points can be built up of non-interacting single-particle states. In contrast, the quantum phase transitions turn out to be described by interacting fixed points, and their excitations cannot be described in terms of free particles. We show that the structure of the many-body spectrum of these critical fixed points can be understood using renormalized perturbation theory close to certain values of the bath exponents which play the role of critical dimensions. Contact is made with perturbative renormalization group calculations for the soft-gap Anderson and Kondo models. A complete description of the quantum critical many-particle spectra is achieved using suitable marginal operators; technically this can be understood as epsilon-expansion for full many-body spectra.
The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the non linear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order ε and also for the field anomalous dimension to order ε 2. An exact marginal operator for the full RG equations is also constructed.
With a view to study the convergence properties of the derivative expansion of the exact renormalization group (RG) equation, I explicitly study the leading and next-to-leading orders of this expansion applied to the Wilson–Polchinski equation in the case of the N -vector model with the symmetry O ( N ). As a test, the critical exponents η and ν as well as the subcritical exponent ω (and higher ones) are estimated in three dimensions for values of N ranging from 1 to 20. I compare the results with the corresponding estimates obtained in preceding studies or treatments of other O ( N ) exact RG equations at second order. The possibility of varying N allows the derivative expansion method to be better valued. The values obtained from the resummation of high orders of perturbative field theory are used as standards to illustrate the eventual convergence in each case. Particular attention is drawn to the preservation (or not) of the reparameterization invariance.