I would like to understand the motivation for studying quantum statistical mechanics, such as spin models, on an infinite lattice, or in other word, in the operator algebraic framework. I learned that treating questions, such as classification of phases of matter, need to be done on an infinite lattice to be rigorous. For instance, certain index (such as fredholm index) can only be defined on an infinite lattice. It turns out that topological entanglement entropy should also be defined in the operator algebraic framework to be mathematically consistent. On the other hand, it seems that most mathematical physicists do not mind working on a finite size lattice.
- The first point I would like to address is that systems, such as spin chains, are never infinite, and so it doesn't seem trivial to me that a quantum phase classification in an infinite dimensional lattice is relevant (for instance coming up with an index classification that can not be defined on a finite sized lattice). Maybe it is easier to mathematically classify or study models on an infinite lattice. But then how can we verify the validity of our results in the case of a finite size system. Shouldn't the classification be robust to lattice size ?
- Secondly, how rigorous would be a classification approach on a finite size system, where bonds of validity on the result are found for each sizes of the lattice. It sounds to me even more rigorous. Could you please discuss the pros and cons, and provide some illustrative papers.
- Could you discuss the validity of classification of lattice realization of topological quantum field theories (TQFTs), since they are mostly defined on a finite size manifold. How can we translate from TQFTs result to operator algebraic results. Can infinite systems be defined on a non trivial topological manifold (such as an infinite cylinder, or a lattice with defects), in order to obtain comparable result to TQFTs. Do lattice realizations of TQFTs exist on an infinite lattice ?
- The last point I would like to address is the methods for taking a lattice in increasingly bigger size. This is discussed in introduction of several papers of X. G. Wen and his book https://arxiv.org/abs/1508.02595 but I can not find a proof, or quantitative elements, such as bonds of validity, limits, etc. Other strategies might be possible. Please provide information on this matter.
