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# Choosing DMRG Parameters

Miles Stoudenmire— Jan 4, 2017

When using the density matrix renormalization group (DMRG) algorithm, a common question is how to choose the best algorithm parameters, such as the number of sweeps or truncation error cutoff for each sweep. There are no perfect choices for these parameters, but this tutorial attempts to provide a basic starting point for understanding. Fortunately DMRG is quite a robust algorithm, so you do not have to worry excessively about getting the wrong answer; the more common concern is about minimizing the calculation time. When in doubt, perform your own experiments: do additional sweeps or vary the cutoff or bond dimension parameters to see if your results change significantly.

This tutorial only discusses the basic, ground-state DMRG algorithm. ITensor also supports more advanced algorithms such as infinite DMRG which may have additional parameters. For more information on these algorithms see the in-depth documentation.

# Basic DMRG Parameters

The most basic parameters controlling a DMRG calculation are

• Number of sweeps
• Maximum MPS bond dimension (maxm)
• Truncation error cutoff (cutoff)

For slightly more advanced parameters see further below.

These parameters are set by creating a Sweeps object. See the Sweeps class documentation for more information on creating a Sweeps object and for very precise definitions of what each parameter does.

## Number of Sweeps

For an easy-to-converge system such as a one-dimensional (1D) spin chain, DMRG can give excellent results in as few as 4-5 sweeps.

For more challenging lattice models, such as quasi-two-dimensional (2D) systems or systems with widely varying energy scales such as the 2D Hubbard model, you may find as many as 10 sweeps are needed. One reason is simply because DMRG can take longer to converge properties related to small energy scales or longer-ranged interactions. Another reason is that you may want to extrapolate your results as a function of the truncation error; in this case more sweeps are needed to get a reliable extrapolation.

In short, the number of sweeps can in principle be small, but it never hurts to do one more sweep to check. Try to find a good number by doing a less accurate calculation for your system before trying more accurate and expensive runs.

## Maximum Bond Dimension

The maxm parameter sets the maximum bond dimension "m" the MPS is allowed to have on each sweep, and can be different for each sweep. Below some sample sweeping schedules are provided with different strategies for choosing maxm.

The most important thing about choosing maxm is to make it small in the first 1-3 sweeps. MPS wavefunctions with bond dimensions as small as m=10 can be surprisingly good at capturing the essential physics, while being extremely cheap for DMRG, so the first sweep should have a maxm in the 10-50 range, followed by slightly larger maxm values until going to a high, or very high maxm in the last few sweeps.

For 1D systems, bond dimensions in the hundreds are often sufficient for high accuracy. For ladder or quasi-2D systems, the bond dimension must be raised exponentially as a function of the transverse system size, and can reach many thousands for large 2D calculations.

## Truncation Error Cutoff

The cutoff parameter is very useful because it controls the bond dimension of the MPS in a "smart" and adaptive way. Setting the cutoff to a modestly small value such as 1E-8 guarantees accuracy, assuming maxm is sufficiently large. But in regions where the bond dimension could be smaller, setting a cutoff will let the bond dimension shrink as much as possible without sacrificing accuracy.

Very roughly speaking, a cutoff of 1E-5 gives sensible accuracy; a cutoff of 1E-8 is high accuracy; and a cutoff of 1E-12 is near exact accuracy.

# Basic Sample DMRG Parameter Schedules

Below are some sample parameter schedules which use different strategies to converge a DMRG calculation. Which strategy to use depends on your resource constraints and your research goals.

In the schedule tables below, each row is a different sweep.

## Maxm Dominated Schedule

If your goal is to reach an accurate ground state while ensuring an efficient calculation, then controlling the accuracy primarily through the maxm parameter is a good approach. Often you may know in advance that a certain final maxm will give sufficient accuracy, for example m=200 is quite good for the S=1 Heisenberg spin chain.

nsweeps = 5
maxm  minm  cutoff  niter  noise
10    1     1E-5    2      0
20    1     1E-8    2      0
80    1     1E-12   2      0
200   1     1E-12   2      0
200   1     1E-12   2      0


## Cutoff Dominated Schedule

If your main priority is finding an accurate ground state, and you are willing to spend the resources necessary to do this or do not have a good idea up-front of what final maxm to choose, then you can quickly increase maxm to a very high value and let the truncation error cutoff set the actual bond dimension DMRG will choose, which could be much less than the maxm specified. It is still smart to keep maxm low initially, though, so as not to waste time during the initial few sweeps.

nsweeps = 6
maxm  minm  cutoff  niter  noise
20    1     1E-5    2      0
80    1     1E-6    2      0
200   1     1E-7    2      0
400   1     1E-8    2      0
800   1     1E-8    2      0
800   1     1E-8    2      0


• Number of Davidson algorithm interations (niter)
• Noise term strength (noise)
• Minimum MPS bond dimension (minm)

## Number of Davidson Iterations

The core of DMRG is the Davidson algorithm, which is type of iterative exact diagonalization algorithm, somewhat similar to Lanczos. The parameter setting the maximum number of Davidson iterations at each step of DMRG is niter. Due to the way the ITensor Davidson code is defined, the minimum value of niter you should use is 2 (two vectors in the basis built by the algorithm).

Often just keeping niter equal to 2 is sufficient and fast for most systems. But for tough systems, such as Hubbard models or long-range models, increasing niter can help.

It is essentially never a good idea to fully converge the inner Davidson loop of a DMRG calculation, since the MPS environment defining the projected Hamiltonian used in the Davidson calculation is only approximate anyway. DMRG can still perfectly converge with the minimum number of Davidson steps since it does multiple sweeps over the system.

## Noise Term

The noise term is a technique originally developed for the single-site DMRG method, but which is also useful for two-site DMRG (the algorithm provided with ITensor). It can be especially useful for ensuring convergence of calculations which conserve quantum numbers or calculations of quasi-2D systems.

To read about the definition of the noise term, see the original paper by White.

The noise term is basically an ad-hoc perturbation that is added to the density matrix at each step before diagonalizing it to get the new MPS basis. It can improve MPS which are deficient in some way (e.g. lacking certain "quantum fluctuations" which are present in the true ground state). But taking the noise term too large can prevent DMRG from finding an optimal MPS, so it should be reduced to a small value or turned off in the last few sweeps.

Roughly speaking, 1E-5 is a lot of noise and 1E-12 is a minimal amount of noise that can still be considered non-zero.

# Advanced Sample DMRG Parameter Schedules

## Schedule with Noise Term

The schedule below could be for a Hubbard chain or ladder, or some other model where we want to conserve quantum numbers and can find the system difficult to converge.

nsweeps = 7
maxm  minm  cutoff  niter  noise
10    1     1E-5    4      1E-5
20    1     1E-6    3      1E-5
80    1     1E-7    3      1E-8
200   1     1E-8    2      1E-9
300   1     1E-8    2      1E-10
400   1     1E-8    2      1E-10
400   1     1E-8    2      1E-10


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