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In the former Phys.SE post Can gapped state and gapless state be adiabatically connected to each other?, I saw different answers from Norbert Schuch and Xiaogang Wen.

I am confused by the question: Can we define if a STATE is gapped or gapless without mentioning any Hamiltonian?

In Norbert Schuch's example on the Toric code, he gave an example that a state can be the ground state of a gapped Hamiltonian and of another gapless Hamiltonian as well. So it seems we have to specify the Hamiltonian when we talk about gapped or gapless state.

In Wen's answer talking about the definition of gapped and gapless state (here), he mentioned

I wonder, if some one had consider the definition of gapped many-body system very carefully, he/she might discovered the notion on topological order mathematically.

Here it seems that Prof. Wen suggest, that the gapped or gapless property is intrinsic to the state itself (as in his papers to explain phases of states as equivalent sets w.r.t. finite depth quantum circuits).

So my question:

If Schuch is right, then there is no absolute definition of the phase of a state since it's Hamiltonian dependent.

If Wen is right, then the phase of a state will not depend on the Hamiltonian, or there exists a kind of mapping between the state and the Hamiltonian so that there is no ambiguity on the gapped or gapless property of a state.

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Although it's true, as Norbert Schuch pointed out, that the same state can be the ground state of Hamiltonians with and without gap, in general this behavior seems rather fine-tuned. I would expect that, for a generic perturbation to the gapless Hamiltonians he discussed, a gap would open up. For this reason, physicists tend to ignore this subtlety and treat gapped/gapless as a property of the state. In particular:

  • A ground state of gapped Hamiltonian must have correlations which decay exponentially with distance (this has been proved rigorously by Hastings and Koma).

  • Empirically, it appears that the ground state of a gapless Hamiltonian generically has correlations with decay as a power-law with distance. (Though this may fail at some fine-tuned points, as Norbert Schuch's examples demonstrate).

However, if one wants to make the notion of a "gapped state" totally rigorous without reference to its parent Hamiltonian, one could make the following definition:

A state $|\psi\rangle$ is gapped if there exists a gapped Hamiltonian $H$ such that $|\psi\rangle$ is its ground state.

This obviously does not rule out the existence of another Hamiltonian $H'$ which is gapless and has the same ground state. However, the existence of a gapped parent Hamiltonian already ensures that $|\psi\rangle$ has sufficiently nice properties (e.g. exponentially decaying correlations, area law for entanglement entropy) that its topological order can be defined (you don't ever need to refer to the Hamiltonian $H$ -- its only role is to ensure that the state has these nice properties).

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    $\begingroup$ I fully agree that these are fine-tuned -- I never claimed otherwise (and it was never my point). On a different note, I don't entirely understand the obsession with gaps -- my feeling is that in the end, it is more about stability of the phase. $\endgroup$ – Norbert Schuch Nov 16 '17 at 11:32
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    $\begingroup$ @NorbertSchuch The question of stability is a rather orthogonal one to gapfulness. There are gapped Hamiltonians that are unstable and gapless ones that are stable. But as I said in my answer, I think the important thing is not so much the gap itself as its consequences for the ground state: short-range correlations, area-law entanglement entropy (well, this one is not actually proven), and so forth. $\endgroup$ – Dominic Else Nov 16 '17 at 17:15
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    $\begingroup$ @Dominic Else Then it seems there is some kind of mapping between the state space and the Hamiltonian space. When we are discussing the phase of a state, we usually combine both the property of the state itself (as the entanglement or correlation pattern) and its stability under a certain Hamiltonian. For me this kind of 'stereo vision' is not very convenient if we regard phase as the intrinsic property of a state. Can we only use one eye by finding the mapping between states and Hamiltonians? $\endgroup$ – XXDD Nov 17 '17 at 2:20
  • $\begingroup$ @DominicElse What's an example of an unstable gapped Hamiltonian? $\endgroup$ – tparker Nov 11 '18 at 1:52
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    $\begingroup$ @tparker On a 1-D chain of spin 1/2's, let $H = -\sum_i \sigma_i^z \sigma_{i+1}^z - \sigma^z_1$. Then the unique gapped ground state is the state with all spins polarized in the $+z$ direction. But as soon as you add a uniform magnetic field, $h \sum_j \sigma_j^z$, then the ground state flips to being the state with spins polarized in the $-z$ direction for any $h > 0$. $\endgroup$ – Dominic Else Nov 21 '18 at 20:24

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