Choosing Between Dynamical DMRG and ITensor's TDVP for Time Evolution of a Doped Mott Insulator on Multi-Core Clusters

DMRG and Numerical Methods
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Hello ITensor community,

I’m currently working on calculating the time evolution of a doped Mott insulator. I’m trying to decide between two methods: the dynamical DMRG proposed by Jeckelmann [Phys. Rev. B 66, 045114] and the Time-Dependent Variational Principle (TDVP) as implemented in ITensor.

From what I understand:

  • Dynamical DMRG computes quantities directly in the frequency space. It’s well-suited for parallel computations, which is advantageous. However, it faces challenges when calculating at low frequencies (ω), potentially leading to broadening effects.
  • TDVP operates in the time domain and is efficient for time evolution problems. The implementation seems robust and user-friendly, but I’m unsure how it compares to dynamical DMRG in terms of parallelization on multi-core clusters.

Given that I have access to a multi-core cluster, I’m curious about the following:

  1. What are the advantages and disadvantages of using TDVP over dynamical DMRG for simulating the time evolution and spectral properties of a doped Mott insulator?
  2. Considering my computational resources (a multi-core cluster), which method would be more suitable for accurately and efficiently calculating the time evolution of a doped Mott insulator?

I would greatly appreciate any insights, experiences, or recommendations you might have regarding these two approaches.

Thank you in advance!

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This is a good question, and I think unfortunately it’s somewhat of an open question still. There have been high hopes in the past for methods like DDMRG and related ideas but from what I understand there is evidence that time-dependent DMRG methods (e.g. TDVP etc.) are overall more efficient, especially if you want results over a wide range of frequencies. It may also be a problem-dependent issue: do you need high accuracy results at low frequencies (long times)? Or are you more looking for good accuracy across many frequencies?

In a nutshell, my belief is that TDVP is currently the “safest” option (in the sense of being easier to use and diagnose and giving you more data in fewer steps).

I think the best thing to do is to read some relevant literature, especially if you find one doing similar applications to yours.

  1. a nice paper directly contrasting DDMRG versus time-dependent methods is Time-step targeting time-dependent and dynamical density matrix renormalization group algorithms with ab initio Hamiltonians by Ronca et al. It has a very well-written introduction and the later parts of the paper contrast DDMRG and TDVP in terms of overall computation needed to obtain similar results.
  2. very recently, there have been some improvements to the methodology for using TDVP to obtain time-dependent correlation functions. The method in the first paper Complex Time Evolution in Tensor Networks is particularly easy to set up and use. And there is a related paper Dynamical correlation functions from complex time evolution you may also find interesting. Similar methods were used quite successfully in a recent paper Spectroscopy and complex-time correlations using minimally entangled typical thermal states.
  3. finally, there has been a certain amount of ongoing research about methods that directly target frequencies, similar in spirit to DDMRG but using different mathematics. In particular there is the Chebyshev MPS approach as described in this paper Chebyshev Matrix Product State Impurity Solver for the Dynamical Mean-Field Theory.

I would basically recommend trying TDVP plus the “parallel contour” method described in Complex Time Evolution in Tensor Networks.

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Thank you so much for your comprehensive reply!

I’m not entirely sure yet which frequency regime I should focus on. However, am I correct in understanding that dynamical DMRG (DDMRG) is more accurate for calculating low-frequency properties, while TDVP is more accurate at higher frequencies? If that’s the case, would it be advantageous to combine both methods to achieve better accuracy across all frequencies?

Thank you again for your insights!

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If you look at some of the results in the “Complex Time Evolution…” papers linked above, you’ll see that they get excellent accuracy at very low frequencies. So I would suggest starting with that method then only switching to DDMRG if that first method does not turn out to be good enough for your needs.

I’m not sure if DDMRG is more accurate than time evolution – I think that’s a good question but a subtle one becaues one would have to compare the resources needed to reach each result, as each method can be made essentially arbitrarily accurate but it’s a question of the computational cost needed to do so.

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