Dear all,
Are there any ways to calculate excited states using VUMPS?
In addition, how does VUMPS choose the state if the ground state is degenerate (not necessarily spontaneously symmetry broken)?
Thanks,
Tingyu
You can see: SciPost: SciPost Phys. Lect. Notes 7 (2019) - Tangent-space methods for uniform matrix product states for a discussion of one way to compute excited states on top of the ground state found by VUMPS based on a quasiparticle ansatz.
It chooses it arbitrarily based on the results of the local eigensolver, biases of the initial state, etc. just like DMRG does.
Hi Tingyu,
To give a little more details about how methods like VUMPS and DMRG choose ground states in the presence of a degeneracy, there is now an understanding that these methods choose linear combinations of ground states which have the lowest entanglement entropy. These ground states have been called “minimum entropy states” or MES. There is work by Tarun Grover characterizing these MES for systems with topological order which you might find helpful.
I would say to be cautious with the above statement though, because in the case of eg regular symmetry breaking DMRG and VUMPS might initially break symmetries for lower bond dimensions then gradually restore them for higher bond dimensions, and VUMPS can behave differently from finite DMRG too. So it takes a bit of reasoning about each method adapted to the particular physics you are studying.
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Thanks for the clarification Miles. I didn’t realize there was a deeper understanding now of which states these methods choose. It was definitely an oversimplification when I said they choose the states “arbitrarily”, and in practice as your answer implies the methods definitely bias towards linear combinations that give lower-entangled states. Which work is that in?
Sure thing. So it’s not a definitive statement, and of course the exact behavior of DMRG can depend on the implementation and system details (e.g. for a very small system and very high bond dimension DMRG might find a “Schrodinger cat” state).
But the main work I was thinking of is Tarun Grover’s work on “minimum entropy states” which he proposed in an earlier paper then wrote this article proposing that DMRG finds them:
https://arxiv.org/abs/1112.2215
Hongchen Jiang and Leon Balents researched this some more in DMRG studies:
https://arxiv.org/pdf/1309.7438.pdf
https://arxiv.org/pdf/1205.4289.pdf
Also, a work with a similar spirit is Cincio and Vidal’s work on finding topological order in infinite cylinders (so this relates more to VUMPS):
http://dx.doi.org/10.1103/PhysRevLett.110.067208
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