In many-body physics, Hilbert spaces are usually equipped with a tensor structure (ie: $\mathcal{H}=\mathcal{V}^{\otimes N}$).

If the dimension of local degrees of freedom is set to be $dim(\mathcal{V})=G$, it is obvious that the dimension of total space $\mathcal{H}$ exhibit an exponential dependence of system size: $dim(\mathcal{H})=G^N$.

But could there be a Hilbert space whose dimension is order $O(N!)$ ? Surely it can never be found in a 1-D many-body system, but it seems proper to be a subspace of a 2-D system. Since $G^{N^2}>N!>G^N$, it looks like something lies between 1-D and 2-D, and unlike those systems live on fractals.

And if there is such a Hilbert space, what are the base vectors? Shall they be some combinatorial objects like elements of $S_N$ or chord diagrams?

Any example would be welcomed, and please feel free to edit my question if it is needed.

  • $\begingroup$ One simple example would be a chain of eg spins where the first site has only one state, the second one two etc. The many-body eigenstates then follow from the combinatorics of combining the single-site spectra. But of course, in this system the sites are not equivalent $\endgroup$ – Wouter Sep 9 at 15:05
  • $\begingroup$ @Wouter I think the problem is: interactions in your system are non-local; At the thermodynamic limit, you have to combine infinite many spins together, which I dont think is reasonable. Also, to make total dimension $N!$, you have assumed system size to be $1+2+......N=\frac{N(N+1)}{2}$, which is not typically a 1-D chain, whose size is linear in N. Thanks anyway :-) $\endgroup$ – SSSSiwei Sep 9 at 15:31

The way to construct such a system would be to take a classical system that can be in $N!$ states, and then consider the corresponding quantum state space.

In this case, a very natural example would be that of $N$ copies of a system with $G = N$, i.e. one that can be in $N$ different classical states, such that the different subsystems cannot be in the same state. This is a subspace of the full tensor product space.

It is e.g. the state space of $N$ distinguishable systems that can each be in $N$ different states, no two in the same state but otherwise non-interacting, like "quantum disks" stacked on top of each other.

  • $\begingroup$ Sounds like something in 3-D. Since the original space is larger ($G^{N^3}$), it might be easier to find such a subspace. But I think there`s no reason that you cannot find such a thing in 2-D......Maybe the trick is hidden in the 'stack operation' you described. $\endgroup$ – SSSSiwei Sep 9 at 15:42
  • $\begingroup$ How can the local dimension depend on the system size? $\endgroup$ – Ryan Thorngren Sep 9 at 15:48

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