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Suppose I have a sequence of gapped, spin-$1/2$, translationally invariant quantum spin chains $\{H_1, H_2,H_3\cdots\}$ with interactions of range $\leq 2$ (i.e. no further than nearest-neighbors).

Furthermore, these chains are such that $H_N$ is the Hamiltonian for a chain of length $N$. However, we are not just trivially extending a fixed chain: the interactions in the Hamiltonians are also changing as $N\to\infty$.

Now suppose that the gaps $\{\Delta_1,\Delta_2,\Delta_3,\cdots\}$ are converging to a finite value, i.e. $$\lim_{N\to\infty}\Delta_N~~\text{exists.}$$ Furthermore, the groundstate in the many-body Hilbert space is converging as well to a thermodynamic groundstate. My question is then: does a MPS representation for the thermodynamic groundstate exist? If so, how would I go about calculating it efficiently?

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  • $\begingroup$ Furthermore, the groundstate in the many-body Hilbert space is converging as well to a thermodynamic groundstate. --- What does that even mean (given that these states live in different Hilbert spaces)? Could you give a formal definition? $\endgroup$ – Norbert Schuch Dec 1 '16 at 7:21
  • $\begingroup$ I'll think about it. I'll get back to you in a few hours $\endgroup$ – David Roberts Dec 1 '16 at 23:08

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