Thank you very much for the reply ! Now I have another question about DMRG of finite version.
By using infinite DMRG method, first I increase the size of the system to 2L. Then I use the finite DMRG method with 2L fixed. After finite-size sweeping for 6-10 times, the energy of the system converges very well. However, when paying attention to the correlation function, I am confused by the following results.
In the infinite-method procedure, I have obtained the size-dependence of the correlation function of the two sites in the middle of the chain. which we label as G(i), i=1,2,...,2L. While in the finite procedure, after several sweep, I also obtain a position-dependence of the same two-site correlation function, which we label as P(i), i=1,2,..., 2L. P(i) is converged and the curve (or shape) does not change any more with further finite-size sweeps, just like the energy.
Sometimes I find the shape of G(i) is very similar to the converged shape of P(i), which means that the ground-state wave-function can be obtained accurately by the infinite method and the finite-size sweep is not needed at all. Please see "good.png".
However, sometimes the shape of G(i) is quite different from the shape of P(i), which seems that the ground-state wave-function obtained with the infinite-method is quite inaccurate. Please see "bad.png".
Now my question is: The infinite-method result is the starting point of the finite-size sweep. Suppose the starting point (the results from the infinite-method ) is not reliable. Then the following finite-size sweep is still credible or not incredible at all ? I'd like to mention that the energy of the system converges very well in the finite-size sweep.
Any comments are valuable to me.
2011/4/21 comp-phys-alps-users-request@lists.phys.ethz.ch:
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1. About superposition state and DMRG (???) 2. Re: About superposition state and DMRG (Matthias Troyer)
Message: 1 Date: Wed, 20 Apr 2011 19:30:39 +0800 From: ??? sunzhaoyu2000@gmail.com Subject: [ALPS-users] About superposition state and DMRG To: comp-phys-alps-users@lists.phys.ethz.ch Message-ID: BANLkTindbVsV1z8Fn5rT96x9iXSQUwyD=w@mail.gmail.com Content-Type: text/plain; charset=ISO-8859-1
Dear ALPS users and developers:
Suppose the ground-state of a quantum chain is twofold degenerate in the thermodynamic limit(L is large enough), while in a finite-size system(such as L=14,16,18), the ground state is just a superposition of these two states, and no degeneration would occur.
Now my question is: Is DMRG applicable to a small size system whose ground-state is a superposition state ? ( I know that by imposing some special boundary conditions, one of the two states could be projected out. Here I need to use a standard open boundary condition. )
Any suggestion or comment would be valuable to me.
sunzhaoyu2000@gmail.com
Message: 2 Date: Wed, 20 Apr 2011 13:34:17 +0200 From: Matthias Troyer troyer@phys.ethz.ch Subject: Re: [ALPS-users] About superposition state and DMRG To: comp-phys-alps-users@lists.phys.ethz.ch Message-ID: F267D0BA-C0EA-421B-8CEB-4743AE7C5A12@phys.ethz.ch Content-Type: text/plain; charset=utf-8
DMRG certainly can be applied. If the finite size gap between the two degenerate states is large enough then you should even be able to get both states individually.
Matthias
On Apr 20, 2011, at 1:30 PM, ??? wrote:
Dear ALPS users and developers:
Suppose the ground-state of a quantum chain is twofold degenerate in the thermodynamic limit(L is large enough), while in a finite-size system(such as L=14,16,18), the ground state is just a superposition of these two states, and no degeneration would occur.
Now my question is: Is DMRG applicable to a small size system whose ground-state is a superposition state ? ( I know that by imposing some special boundary conditions, one of the two states could be projected out. Here I need to use a standard open boundary condition. )
Any suggestion or comment would be valuable to me.
sunzhaoyu2000@gmail.com
End of Comp-phys-alps-users Digest, Vol 61, Issue 10
On Apr 21, 2011, at 3:07 PM, 孙照宇 wrote:
Thank you very much for the reply ! Now I have another question about DMRG of finite version.
By using infinite DMRG method, first I increase the size of the system to 2L. Then I use the finite DMRG method with 2L fixed. After finite-size sweeping for 6-10 times, the energy of the system converges very well. However, when paying attention to the correlation function, I am confused by the following results.
In the infinite-method procedure, I have obtained the size-dependence of the correlation function of the two sites in the middle of the chain. which we label as G(i), i=1,2,...,2L. While in the finite procedure, after several sweep, I also obtain a position-dependence of the same two-site correlation function, which we label as P(i), i=1,2,..., 2L. P(i) is converged and the curve (or shape) does not change any more with further finite-size sweeps, just like the energy.
Sometimes I find the shape of G(i) is very similar to the converged shape of P(i), which means that the ground-state wave-function can be obtained accurately by the infinite method and the finite-size sweep is not needed at all. Please see "good.png".
However, sometimes the shape of G(i) is quite different from the shape of P(i), which seems that the ground-state wave-function obtained with the infinite-method is quite inaccurate. Please see "bad.png".
Now my question is: The infinite-method result is the starting point of the finite-size sweep. Suppose the starting point (the results from the infinite-method ) is not reliable. Then the following finite-size sweep is still credible or not incredible at all ? I'd like to mention that the energy of the system converges very well in the finite-size sweep.
Any comments are valuable to me.
The case that the infinite size method gives you a "bad" result is rather common and the sweeps typically improve it and converge to the true ground state. However, you can always be trapped and a careful analysis of the data is always needed.
Matthias
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