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When dealing with topological phases of matter (topological insulators, quantum hall effect, etc...) one says that the system remains in the same phase as long as any continuous transformation of the Hamiltonian does not close the gap (or break symmetries in the case of symmetry-protected topological phases).

This is clear when we limit the ways a hamiltonian can change only to a few scalar or vector parameters (hopping parameters in tight-binding models, magnetic and electric field magnitudes or directions). Then one could say that the the transformation is continuous if for instance in a given basis the matrix elements of the hamiltonian are continuous functions of the parameters (which has the usual definition for complex/real valued functions).

In general one says that a function is continuous on some topological space if the pre-images of open sets are open sets. This requires a topology on the space. In this case the space of linear operators, possibly limited to hermitian operators or an even smaller space. This space has a lot of structure and there are for instance various ways to define limit of operators (I only know the limits in the weak and strong sense, but I know there are other definitions, never seen used in physics).

Does then "continuous" mean that we choose some topology on our space of hamiltonians (whatever this may be - perhaps not even hermitian is needed in general) and the definition is as usual? If yes, is there a "usual" choice for the topology, derived from some other property of the space? If not, what does continuous mean in this case?

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The topology can be induced by some kind of "local norm" on the vector space of Hamiltonians. Roughly, the local norm should be defined such that Hamiltonians that are sufficiently "local" have finite norm in the thermodynamic limit. One way to define it would be as follows. Consider a Hamiltonian that acts on a lattice system, with a set $\Lambda$ of lattice sites, and suppose there is a metric $d(\cdot,\cdot)$ on $\Lambda$.

Suppose the Hamiltonian can be written in the form $$H = \sum_{X \subseteq \Lambda} h_X,$$

where the term $h_X$ acts only on the degrees of freedom at the lattice sites in $X$. Then we choose some $\kappa > 0$ and define

$$ \| H \|_\kappa := \sup_{i \in \Lambda} \sum_{X : i \in X \subseteq \Lambda} \| h_X \| e^{\kappa \, \mathrm{diam}(X)}$$,

where we defined \begin{equation} \mathrm{diam}(X) := \sup_{i,j \in X} d(i,j) \end{equation}

The reason for defining the norm like this, or something similar, is (among other things) that it ensures that under the induced topology, a continuous deformation of gapped Hamiltonians induces a quasi-local unitary transformation between the respective ground states; see, for example:

https://arxiv.org/abs/1008.5137

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"Continuous" in this context usually means to the author of the paper that if the Hamiltonian has some parameters, then you vary the parameters continuously. This fine for finite-dimensional matrix Hamiltonians for which all the natural operator topologies (norm, weak, strong) are equivalent. Condensed matter physicists very rarely bother to define their topology carefully --- the exception being people like Emil Prodan, who work explicitly with C$^*$ algebras.

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