arXiv subjects
Publications and source records attributed to F. Niedermayer.
We argue that there is no essential violation of universality in the continuum limit of mixed $\RPn$ and $\On$ lattice sigma models in 2 dimensions, contrary to opposite claims in the literature.
In order to support the formal renormalization group arguments that the fixed point action of an asymptotically free model gives cut--off independent physical predictions in 1--loop perturbation theory, we calculate the finite volume mass--gap $m(L)$ in the non--linear $σ$--model. No cut--off effect of the type $g^4\left(a/L\right)^n$ is seen for any $n$. The results are compared with those of the standard and tree level improved Symanzik actions.
In this paper (the first of a series) we describe the construction of fixed point actions for lattice $SU(3)$ pure gauge theory. Fixed point actions have scale invariant instanton solutions and the spectrum of their quadratic part is exact (they are classical perfect actions). We argue that the fixed point action is even 1--loop quantum perfect, i.e. in its physical predictions there are no $g^2 a^n$ cut--off effects for any $n$. We discuss the construction of fixed point operators and present examples. The lowest order $q {\bar q}$ potential $V(\vec{r})$ obtained from the fixed point Polyakov loop correlator is free of any cut--off effects which go to zero as an inverse power of the distance $r$.
In this paper (the second of a series) we extend our calculation of a classical fixed point action for lattice $SU(3)$ pure gauge theory to include gauge configurations with large fluctuations. The action is parameterized in terms of closed loops of link variables. We construct a few-parameter approximation to the classical FP action which is valid for short correlation lengths. We perform a scaling test of the action by computing the quantity $G = L \sqrt{σ(L)}$ where the string tension $σ(L)$ is measured from the torelon mass $μ= L σ(L)$. We measure $G$ on lattices of fixed physical volume and varying lattice spacing $a$ (which we define through the deconfinement temperature). While the Wilson action shows scaling violations of about ten per cent, the approximate fixed point action scales within the statistical errors for $ 1/2 \ge aT_c \ge 1/6$. Similar behaviour is found for the potential measured in a fixed physical volume.
We present an overview of the construction and testing of actions for SU(3) gauge theory which are approximate fixed points of renormalization group equations (at $β\rightarrow \infty$). Such actions are candidates for use in numerical simulations on coarse lattices.
The classical equations of motion of the perfect lattice action in asymptotically free $d=2$ spin and $d=4$ gauge models possess scale invariant instanton solutions. This property allows the definition of a topological charge on the lattice which is perfect in the sense that no topological defects exist. The basic construction is illustrated in the $d=2$ O(3) non--linear $σ$--model and the topological susceptibility is measured to high precision in the range of correlation lengths $ξ\in (2 - 60)$. Our results strongly suggest that the topological susceptibility is not a physical quantity in this model.
There exist lattice actions which give cut--off independent physical predictions even on coarse grained lattices. Rotation symmetry is restored, the spectrum becomes exact and, in addition, the classical equations have scale invariant instanton solutions. This perfect action can be made short ranged. It can be determined by combining analytical calculations with numerical simulations on small lattices. We illustrate the method and the benefits on the $d=2$ non--linear $σ$--model.
The low temperature and large volume effects in the d=2+1 antiferromagnetic quantum Heisenberg model are dominated by magnon excitations. The leading and next-to-leading corrections are fully controlled by three physical constants, the spin stiffness, the spin wave velocity and the staggered magnetization. Among others, the free energy, the ground state energy, the low lying excitations, staggered magnetization, staggered and uniform susceptibilities are studied here. The special limits of very low temperature and infinite volume are considered also.