Microcanonical determination of the order parameter crit - ical exponent

– In this Letter we investigate the density of states ΩN (E,M) of twoand threedimensional Ising models with N spins as a function of energy E and magnetization M . For a fixed energy lower than a critical value Ec,N the density of states exhibits two sharp maxima at M = ±Msp(E) which define the spontaneous magnetization. An analysis of the form Msp(E) ∝ (Ec,∞ − E) βε yields very good results for the critical exponent βε. Corrections to the asymptotic behaviour are discussed and compared to the corrections observed in a canonical analysis. Furthermore, a new and highly efficient Monte Carlo method for the calculation of ΩN (E,M) is presented. Substantial progress has been achieved in the understanding of discontinuous phase transitions since the problem has been formulated in microcanonical terms [1, 2]. Till now the microcanonical analysis of continuous phase transitions has only been partly successful. On the one hand it is a great accomplishment that the typical features of symmetry breaking, i.e. the abrupt inset of the order parameter as the critical point is approached from above and the diverging susceptibilities, turn up already for finite systems [3]. This is in contrast to the canonical ensemble where singularities appear exclusively in the thermodynamic limit [4]. On the other hand it is an intrigueing and disappointing fact that mean field values have been found for all critical exponents and for all finite system sizes [3]. In this Letter we do not contest the earlier analysis where values of the critical exponents of an Ising system were determined from the expansion of the entropy SN (E,M) in the vicinity of the critical point of the finite system, situated at E = EN,c and M = 0. Here it is our aim to show that the non classical infinite lattice exponents can be obtained directly from the density of states if the expansion of the finite system entropy SN (E,M) is carried out at the critical point E = Ec,∞ and M = 0 of the infinite system. (The critical point of the finite or infinite system is defined as the value of the energy E = Ec,N , where the second derivative [∂SN/∂M ]E,M=0 of the entropy changes its sign.) For the purpose of determining the density of states ΩN (E,M) = expSN (E,M) (in units where kB = 1) we have developed a highly efficient new algorithm which is based on the