Derivative Expansion of the Exact Renormalization Group

The functional flow equations for the Legendre effective action, with respect to changes in a smooth cutoff, are approximated by a derivative expansion; no other approximation is made. This results in a set of coupled non-linear differential equations. The corresponding differential equations for a fixed point action have at most a countable number of solutions that are well defined for all values of the field. We apply the technique to the fixed points of one-component real scalar field theory in three dimensions. Only two non-singular solutions are found: the gaussian fixed point and an approximation to the Wilson fixed point. The latter is used to compute critical exponents, by carrying the approximation to second order. The results appear to converge rapidly.Comment: 14 pages (with figures), Plain TeX, uses psfig, 4 postscript figures appended as uuencoded compressed tar file, SHEP 93/94-16, CERN-TH.7203/94. (Added small details and minor improvements in rigour : the version to be published in Phys.Lett.B

On Truncations of the Exact Renormalization Group

We investigate the Exact Renormalization Group (ERG) description of ($Z_2$ invariant) one-component scalar field theory, in the approximation in which all momentum dependence is discarded in the effective vertices. In this context we show how one can perform a systematic search for non-perturbative continuum limits without making any assumption about the form of the lagrangian. Concentrating on the non-perturbative three dimensional Wilson fixed point, we then show that the sequence of truncations $n=2,3,\dots$, obtained by expanding about the field $\varphi=0$ and discarding all powers $\varphi^{2n+2}$ and higher, yields solutions that at first converge to the answer obtained without truncation, but then cease to further converge beyond a certain point. No completely reliable method exists to reject the many spurious solutions that are also found. These properties are explained in terms of the analytic behaviour of the untruncated solutions -- which we describe in some detail.Comment: 15 pages (with figures), Plain TeX, uses psfig, 5 postscript figures appended as uuencoded compressed tar file, SHEP 93/94-23, CERN-TH.7281/94. (Corrections of typos, and small additions to improve readability: version to be published in Phys. Lett. B

Monitoring and Control of Temperature in Networks-on-Chip

Increasing integration densities and the emergence of nanotechnology cause issues related to reliability and power consumption to become dominant factors for the design of modern multi-core systems. Since the arising problems are enforced by high circuit temperatures, monitoring and control of on-chip temperature profiles need to be considered during design phase as well as during system operation. Hence, in this paper different approaches for the realization and integration of a monitoring system for temperature in multi-core systems based on Networks-on-Chip (NoCs) in combination with Dynamic Frequency Scaling (DFS) are investigated. Results show that both combinations using event-driven and time-driven forwarding more than double overall execution time and considerably reduce throughput of application data. Regarding performance of notification and reaction to temperature development event-driven forwarding clearly outperforms time-driven forwarding

Derivative expansion of the renormalization group in O(N) scalar field theory

We apply a derivative expansion to the Legendre effective action flow equations of O(N) symmetric scalar field theory, making no other approximation. We calculate the critical exponents eta, nu, and omega at the both the leading and second order of the expansion, associated to the three dimensional Wilson-Fisher fixed points, at various values of N. In addition, we show how the derivative expansion reproduces exactly known results, at special values N=infinity,-2,-4, ... .Comment: 29 pages including 4 eps figures, uses LaTeX, epsfig, and latexsy

Momentum Scale Expansion of Sharp Cutoff Flow Equations

We show how the exact renormalization group for the effective action with a sharp momentum cutoff, may be organised by expanding one-particle irreducible parts in terms of homogeneous functions of momenta of integer degree (Taylor expansions not being possible). A systematic series of approximations -- the $O(p^M)$ approximations -- result from discarding from these parts, all terms of higher than the $M^{\rm th}$ degree. These approximations preserve a field reparametrization invariance, ensuring that the field's anomalous dimension is unambiguously determined. The lowest order approximation coincides with the local potential approximation to the Wegner-Houghton equations. We discuss the practical difficulties with extending the approximation beyond $O(p^0)$.Comment: 31 pages including 5 eps figures, uses harvmac and epsf. Minor additions -- not worth the bandwidth if you already have a cop

Monitoring and Control of Temperature in Networks-on-Chip

Abstract Increasing integration densities and the emergence of nanotechnology cause issues related to reliability and power consumption to become dominant factors for the design of modern multi-core systems. Since the arising problems are enforced by high circuit temperatures, monitoring and control of on-chip temperature profiles need to be considered during design phase as well as during system operation. Hence, in this paper different approaches for the realization and integration of a monitoring system for temperature in multi-core systems based on Networks-on-Chip (NoCs) in combination with Dynamic Frequency Scaling (DFS) are investigated. Results show that both combinations using event-driven and time-driven forwarding more than double overall execution time and considerably reduce throughput of application data. Regarding performance of notification and reaction to temperature development event-driven forwarding clearly outperforms time-driven forwarding

Three dimensional massive scalar field theory and the derivative expansion of the renormalization group

We show that non-perturbative fixed points of the exact renormalization group, their perturbations and corresponding massive field theories can all be determined directly in the continuum -- without using bare actions or any tuning procedure. As an example, we estimate the universal couplings of the non-perturbative three-dimensional one-component massive scalar field theory in the Ising model universality class, by using a derivative expansion (and no other approximation). These are compared to the recent results from other methods. At order derivative-squared approximation, the four-point coupling at zero momentum is better determined by other methods, but factoring this out appropriately, all our other results are in very close agreement with the most powerful of these methods. In addition we provide for the first time, estimates of the n-point couplings at zero momentum, with n=12,14, and the order momentum-squared parts with n=2 ... 10.Comment: 33 pages, 1 eps figure, 7 tables; TeX + harvmac; version to appear in Nucl. Phys.

Large N and the renormalization group

In the large N limit, we show that the Local Potential Approximation to the flow equation for the Legendre effective action, is in effect no longer an approximation, but exact - in a sense, and under conditions, that we determine precisely. We explain why the same is not true for the Polchinski or Wilson flow equations and, by deriving an exact relation between the Polchinski and Legendre effective potentials (that holds for all N), we find the correct large N limit of these flow equations. We also show that all forms (and all parts) of the renormalization group are exactly soluble in the large N limit, choosing as an example, D dimensional O(N) invariant N-component scalar field theory.Comment: 13 pages, uses harvmac; Added: one page with further clarification of the main results, discussion of earlier work, and new references. To be published in Phys. Lett.

Non-Compact Pure Gauge QED in 3D is Free

For all Poincar\'e invariant Lagrangians of the form ${\cal L}\equiv f(F_{\mu\nu})$, in three Euclidean dimensions, where $f$ is any invariant function of a non-compact $U(1)$ field strength $F_{\mu\nu}$, we find that the only continuum limit (described by just such a gauge field) is that of free field theory: First we approximate a gauge invariant version of Wilson's renormalization group by neglecting all higher derivative terms $\sim \partial^nF$ in ${\cal L}$, but allowing for a general non-vanishing anomalous dimension. Then we prove analytically that the resulting flow equation has only one acceptable fixed point: the Gaussian fixed point. The possible relevance to high-$T_c$ superconductivity is briefly discussed.Comment: 11 pages, plain tex, uses harvmac. Minor additions - version to be published in Physics Letters

A gauge invariant exact renormalization group I

A manifestly gauge invariant continuous renormalization group flow equation is constructed for pure SU(N) gauge theory. The formulation makes sense without gauge fixing and manifestly gauge invariant calculations may thus be carried out. The flow equation is naturally expressed in terms of fluctuating Wilson loops, with the effective action appearing as an integral over a `gas' of Wilson loops. At infinite N, the effective action collapses to a path integral over the trajectory of a single particle describing one Wilson loop. We show that further regularization of these flow equations is needed. (This is introduced in part II.)Comment: TeX, harvmac, epsf; 35 pages, 15 figs; a few typos correcte