Reptons bonds

The motion of unentangled loops between the cells along the confining tube is represented in the discretized model by the hopping of reptons from site to site. In the repton bond representation foi, this motion corresponds to a random interchange of nearest neighbors... 

For example, the jump of a repton 6 in Fig. 1b from site 4 to site 3, corresponding to the motion of loop DDi through the gate between cells 4 and 3 in Fig. la, is described by the interchange of the last pair in the repton bond representation (100110) >( 100101). If oj = oj+i, their interchange does not lead to a new state and in the repton language implies that this transition is forbidden (e.g.. Jump of repton 3 in Fig. 1b in either direction). 

It is common to find various physical models related to each other. These relations often provide a deeper insight into the behavior of the systems in question. The repton bond representation oj (Fig. 5a) is identical to the one-dimensional Ising spin representation si of a state... 

Axiom 1 we conclude that ai can only be equal to 0 or 1. It is very easy to prove that for each connected cluster of N reptons there is a unique set of A -1 zeros and ones oj, and vice versa. Ones correspond to the spacings between occupied lattice points (forbidden faces between the cells along the tube), while zeroes correspond to multiple reptons on a site. For example, the cluster of reptons in Fig. lb can be represented by the set (100110). Note that the mode corresponding to the translation of the cluster as a whole (the center of mass coordinate) is not included in the bond representation. 

Thus, the axioms of repton dynamics are equivalent to very simple local rules 3 in the bond representation oj, with ai-O or 1. 

In the bond representation each state of the cluster of N reptons is represented by iV-1 values of o. The translation of the cluster as a whole... 

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