quantum mechanics - Periodic vs Open boundary conditions

In condensed matter, people often use periodic boundary conditions to perform calculations about bulk properties of a material. It's generally argued that in the $N\rightarrow\infty$ limit the boundary conditions don't affect the bulk properties, so you can use periodic boundary conditions to calculate bulk properties of systems with open boundaries.

Are there any formal mathematical proofs of this fact? I'm thinking about statements like:

• As $N\rightarrow\infty$, any observable that only looks at the bulk is not affected by the boundary conditions.


• As $N\rightarrow\infty$, the overlap of the ground state of the periodic system and the ground state of the open system goes to 1.


• As $N\rightarrow\infty$, the reduced density matrix in the bulk does not depend on the boundary conditions.

Or anything similar. I'm not looking for intuitive arguments, but rather for proofs in the literature, if they exist.

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One thought I had after thinking about this for awhile: I believe it must be true that the Hamiltonian with periodic boundary conditions and the Hamiltonian with open boundary conditions must be adiabatically connected.

One simple example is the effect of boundary magnetic fields in the Ising model. If I modify the Hamiltonian of the Ising model at the boundary, I can change the physics in the bulk. Consider

$$H=-\sum_{n=1}^{N-1} J \sigma^z_n\sigma^z_{n+1} +h\sigma_1$$ If $h$ is positive, the ground state is all spin down; if $h$ is negative, it's all spin up. Changing the Hamiltonian at the boundary changed the physics in the bulk. Why did this happen? It's because there's a level crossing when tuning $h$ from positive to negative; an excited state becomes the ground state, and vice versa. In general, if changing the boundary conditions induces a level crossing, we should expect bulk physics to change.

So I think any theorem that says the bulk physics between the two systems is identical must only be true if the open and periodic systems are adiabatically connected.