Entity tracking is essential for complex reasoning. To perform in-context entity tracking, language models (LMs) must bind an entity to its attribute (e.g., bind a container to its content) to recall attribute for a given entity. For example, given a context mentioning ``The coffee is in Box Z, the stone is in Box M, the map is in Box H'', to infer ``Box Z contains the coffee'' later, LMs must bind ``Box Z'' to ``coffee''. To explain the binding behaviour of LMs, existing research introduces a Binding ID mechanism and states that LMs use a abstract concept called Binding ID (BI) to internally mark entity-attribute pairs. However, they have not captured the Ordering ID (OI) from entity activations that directly determines the binding behaviour. In this work, we provide a novel view of the BI mechanism by localizing OI and proving the causality between OI and binding behaviour. Specifically, by leveraging dimension reduction methods (e.g., PCA), we discover that there exists a low-rank subspace in the activations of LMs, that primarily encodes the order (i.e., OI) of entity and attribute. Moreover, we also discover the causal effect of OI on binding that when editing representations along the OI encoding direction, LMs tend to bind a given entity to other attributes accordingly. For example, by patching activations along the OI encoding direction we can make the LM to infer ``Box Z contains the stone'' and ``Box Z contains the map''.
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