Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

Please welcome our new moderators!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

122 submissions , 103 unreviewed
3,497 questions , 1,172 unanswered
4,545 answers , 19,342 comments
1,470 users with positive rep
408 active unimported users
More ...

State of Matrix Product States

+ 16 like - 0 dislike
24 views

What is a good summary of the results about the correspondence between matrix product states (MPS) or projected entangled pair states (PEPS) and the ground states of local Hamiltonians? Specifically, what "if and only if" type of equality/approximation hold?

I know about this review-like paper by Verstraete, Cirac, and Murg but I feel some of its results are superseded by new ones...

This post has been migrated from (A51.SE)
asked Nov 1, 2011 in Theoretical Physics by Kaveh_kh (120 points) [ no revision ]
retagged Mar 7, 2014 by dimension10

1 Answer

+ 10 like - 0 dislike

A short summary of the relation between MPS/PEPS and ground states local Hamiltonians:

First the direction from MPS/PEPS to Hamiltonians:

  • Every MPS/PEPS naturally appears as the exact ground state of a frustration free local Hamiltonian. ("parent Hamiltonian")

  • For generic MPS/PEPS, this ground state will be unique.

  • There is a number of cases beyond the generic one where one can make statements about the ground state degeneracy. In particular, for translational invariant MPS the ground state degeneracy is always constant.

  • For translational invariant MPS, there is always a spectral gap above the ground state manifold; for PEPS, this holds only in certain cases.

Conversely, from Hamiltonians to MPS/PEPS:

  • Given a gapped local 1D Hamiltonian, its ground state is well approximated by an MPS. (cf. [Hastings '07] for the scaling)

  • Given a local 2D Hamiltonian where the density of states does not grow too quickly, its ground state (as well as thermal states) is well described by a PEPS (cf. [Hastings '05],[Hastings '07] for the scaling)

This concerns only the analytical relations between MPS/PEPS and Hamiltonians. In practice, better bounds concerning approximability etc. will typically hold.

Concerning "if and only if" relations, I think all these results only hold rigorously in one direction (though typically they might be "if and only if", e.g., I would think that typical gapless Hamiltonians will not have MPS ground states).

If you are looking for something more concrete, please let us know.

This post has been migrated from (A51.SE)
answered Nov 1, 2011 by Norbert Schuch (290 points) [ no revision ]
Thanks for the summary and the links. I was thinking along similar lines: that is typically (but perhaps not an "almost for all" type) the correspondence is solid but degeneracies affect the Hamiltonians and the MPSs in slightly different ways. I am actually not looking at anything more concrete and this came up as a discussion with my friend but it is a favorite subject of mine. I will happily get a rain check for when concrete things come up (as they do!).

This post has been migrated from (A51.SE)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$y$\varnothing$icsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...