Effective long range order and phase transitions in finite, macroscopic one and two dimensional systems

Abstract

The concept of long range order is discussed for finite macroscopic systems. It is shown that if the interaction range or strength, in units of kT, is of O(log N), long range order is obtained in one dimensional systems. Short range and long range interactions are able to combine to cause long range order even when each of the two interactions by itself is much too weak to do so. A physical picture of the effects of the long range interactions is presented and supported by detailed calculations. We also show that the theorem stating that the two dimensional Heisenberg Model does not exhibit long range order is inapplicable when the interaction strength in units kT is of O(log N), or when the range of the interaction is of O( log N) 1 4 . The two dimensional Bose gas will condense at temperatures of O( T 0 log N ) , where T 0 is the condensation temperature in three dimensions. The recent proof that superfluid long range order is impossible in two dimensions, is shown to be based, for finite N, on the analogous fact in the free gas, which is invalid according to the above. Consideration of thin Bose films shows that the transition from three dimensional (condensation at T 0) to two dimensional (condensation below T 0) behavior occurs at film thickness of order log N. This result can be of practical importance.