ar X iv : h ep - l at / 0 11 01 21 v 1 1 5 O ct 2 00 1 1 Physical Observables from Lattice QCD at Fixed Topology ∗

Because present Monte Carol algorithms for lattice QCD may become trapped in a given topological charge sector when one approaches the continuum limit, it is important to understand the effect of calculating at fixed topology. In this work, we show that although the restriction to a fixed topological sector becomes irrelevant in the infinite volume limit, it gives rise to characteristic finite size effects due to contributions from all θ-vacua. We calculate these effects and show how to extract physical results from numerical data obtained at fixed topology. 1. TOPOLOGICAL CHARGE SECTORS AND θ-VACUA In principle, one should calculate observables in QCD with finite quark masses in a fixed θ-vacuum. However in practical lattice calculations , algorithms that change the gauge field configuration in small steps tend to become trapped in a fixed topological charge sector because they cannot overcome the action barriers between sectors. One sees concrete evidence of this problem , for example, from the large equilibration times for the topological charge required in hybrid Monte Carlo calculations, which increase as the quark mass is decreased [1]. In addition, improved pure gauge actions, such as DBW2, reduce the low eigenmodes that are undesirable for overlap and domain wall fermions by suppressing small instantons and thereby reduce tunneling between sectors. Since we know that the restriction to fixed topology becomes irrelevant in the infinite volume limit, we have calculated the the finite volume dependence of this restriction and thereby show how to extract physical results from practical calculations at fixed topology. The partition function of the QCD Hamilto-nian H with eigenstates |n, θ in a given sector may be written in tems of the lowest eigenvalue E 0 (θ) = V e 0 (θ) in the large volume, low temperature limit as follows: