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I would like to understand the relation between a change of the topological invariant (e.g. when the Chern number changes from $1$ to $2$) and the closure of the gap of a condensed matter system.

I know the relation is almost clarified in the case of the quantum Hall state, when one changes from one plateau to an other (which characterise insulating phases) by passing through a metallic phase, and the conductance is proportional to a the Chern number, but how can one generalises the argument to any other model ?

I have the feeling this statement is at the heart of several articles published since a few years, but I have deep difficulties to understand whether this is a postulate / hypothesis / claim / theorem / conjecture or anything else.

Any reference, advise and comment to improve the question is welcome, both along mathematical or physics direction.

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The basic principle is that, if there is a nonzero gap, then the ground state varies continuously with the parameters of the Hamiltonian. Therefore, any quantity (such as the Chern number) which is a topological invariant -- that is, it does not change under continuous deformations -- can only change at a point in the phase diagram in which the gap closes. (Which is not to say, of course, that it must change at such a point.)

Let us first consider the easy case of a system of non-interacting particles on a lattice, with momentum conservation. Then first-order perturbation theory allows us to evaluate the derivative of any single-particle eigenstate with respect to some parameter $s$ of the Hamiltonian: $$ \partial_s |i\rangle = \sum_{j \neq i} \frac{|j\rangle\langle j | \partial_s H | i \rangle}{E_j - E_i}.$$ Momentum conservation ensures that there is only a finite number of terms in the sum and those correspond to states in different bands, so if there is a gap, so that $|E_j - E_i| > 0$ for states in different bands at the same momentum, we see that $\partial_s |i\rangle$ is finite; which means that $|i\rangle$ varies continuously. Therefore, the Chern number of a band separated from the rest of the spectrum by a gap cannot change as the Hamiltonian is varied continuously.

Much more generally, we can consider an arbitrary many-body interacting system, where the ground state is separated from the rest of the spectrum by a gap. Then one must be a little bit more careful about what one means by the "ground state varying continuously". However, one can prove a theorem (see http://arxiv.org/abs/1004.3835 and references therein) that for any continuous path of Hamiltonians $H(s)$ indexed by a parameter $s$, such that the gap does not close for any $0 \leq s \leq 1$, then the ground states $|\Psi_0\rangle$ and $|\Psi_1\rangle$ of $H(0)$ and $H(1)$ are related by a local unitary $U$, such that $|\Psi_1 \rangle = U |\Psi_0 \rangle$. (More generally, if there is a space of degenerate states separated from the rest of the spectrum by a gap for $0 \leq s \leq 1$, then the degenerate ground states at $s=0$ and $s=1$ are related by a local unitary.) See the linked paper for the definition of local unitary, but basically it means that $|\Psi_1\rangle$ can be constructed from $|\Psi_0\rangle$ by local operations in finite time (not scaling with the system size.) Thus, if we can find a property of a quantum state that is invariant under local unitaries, then its value in the ground state can only change as a function of the parameters of the Hamiltonian if the gap closes.

To take just one example: when defined on a torus, the Kitaev toric code (https://en.wikipedia.org/wiki/Toric_code) has 4 degenerate ground states which are locally indistinguishable; that is, they differ only globally, and cannot be distinguished by the expectation value of any local observable: $\langle \psi | \hat{o} | \psi \rangle = \langle \psi^{\prime} | \hat{o} | \psi^{\prime} \rangle$ for any two ground states $|\psi\rangle$ and $|\psi^{\prime}\rangle$ and any local observable $\hat{o} $. This property is invariant under local unitaries, because if there exists a local unitary $U$ such that $|\psi_1\rangle = U |\psi_0\rangle$ and $|\psi_1^{\prime}\rangle = U |\psi_0^{\prime}\rangle$, then $$\langle \psi_1 | \hat{o} | \psi_1 \rangle - \langle \psi^{\prime}_1 | \hat{o} | \psi^{\prime}_1 \rangle = \langle \psi_0 | U^{\dagger} \hat{o} U | \psi_0 \rangle - \langle \psi_0^{\prime} | U^{\dagger} \hat{o} U | \psi_0^{\prime} \rangle.$$ One can show that if $U$ is a local unitary then $U^{\dagger} \hat{o} U$ is a local observable if and only if $\hat{o}$ is, so it followsj that $|\psi^{\prime}_1\rangle$ and $|\psi_1\rangle$ are locally indistinguishable if and only if $|\psi_0\rangle$ and $|\psi^{\prime}_0\rangle$ are locally indistinguishable. Therefore, the four-fold "topological degeneracy" of the toric code cannot be removed as the parameters of the Hamiltonian are varied without closing the gap.

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