+2 votes
asked by (140 points)

I am trying to calculate a dynamical spin structure factor S(k, w) of a spin chain (e.g., 1d Heisenberg Model, in order to check if I can see the spin wave dispersion).

An implementation that I am thinking of is;

(Step 1) get the groundstate |psi_0> by DMRG for a given Hamiltonian (I can already do this easily)

(Step 2) calculate the time-dependent correlation function,
where the superscript a={x, y, z} is a spin component, and the subscript r is a site index, and t is time.

Simplifying the above expression by using the interaction picture,
C^a(r, t)=<psi0| S^zr(0) exp(-i(H-E0)t) S^z0(0) |psi0> ,
where E
0 is the groundstate energy, which I already calculated in step(1)

(Step 3) then Fourier transforming,
C^a(r,t) ---> S^a(k, w)

I want to know an efficient way of doing Step 2. It would be great if you can provide a sample code (for me, looking at a sample working code is the best way to learn).
Also if you have any practical tips (e.g., how to choose the interval of time-evolutions to get the most accurate S(k,w) etc ), please make some comments as well.



(The most naive implementation I can think of for Step2 is;
for each r & t,
(Step2.1) define three MPOs,
A=S^zr(0), B=exp(-i(HE0) t), C=S^z0(0)
(Step2.2) calculate the total MPO by multiplying the three MPOs,
(Step2.3) calculate the overlap,
I am not sure if this implementation works, and even if it does, I believe it is computationally super inefficient. )

commented by (140 points)
Can u tell me how u find ground state energy for 1d Heisenberg hamiltonian.
I am confused.

1 Answer

+2 votes
answered by (70.1k points)

Hi, so it's a good question about the best way to do these things. However, it's a very complicated question to answer with many different aspects that are too detailed to get into here. So I think the best thing I can do is to point you to the relevant literature, which is pretty extensive. A good paper about obtaining exactly the quantity you are asking about is this one:

Another paper with some helpful information is the review article by Schollwock (see near page 156):

Typically for step (2), if the Hamiltonian is sufficiently short ranged, the nicest approach for doing the time evolution is just to use a Trotter decomposition of exp(-i t H) and apply the resulting unitary 'gates' to an MPS (after acting on the MPS with the Sz operator, say).

Based on your question, I just added a "Code Formula" about how to set up Trotter gate time evolution with ITensor:
I hope you find it useful. Please also see the closely related one about time evolution with an MPO, which is an alternative techniques with some advantages and disadvantages.

Best regards,

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