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I know that iDMRG (infinite-density matrix renormalization group) calculations are usually treated as capturing all the thermodynamic-limit behavior of a system, and I have also heard that there are add-on techniques for calculating quantities like correlation lengths. I know this all can't be perfect or no one would do finite-size DMRG calculations.

Here is my question: what are the limitations of iDMRG, and how does it go wrong?

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The iDMRG algorithm is tailored for translationally invariant 1D systems with area law ground states (gapped states). If that's what you're interested in, iDMRG is hard to beat for accuracy, assuming it does converge to the ground state. (In some cases, analytical solutions by e.g. the Bethe ansatz are possible. They may offer an improvement, but it can be very hard to calculate interesting properties from such solutions.) In practice, iDMRG can also be used to approximate other types of states. For example, iDMRG is often applied also to 2D systems, which are treated as quasi-1D ones by wrapping the lattice on a cylinder of infinite length but finite circumference. Of course, this procedure introduces finite-size effects.

There's also a more subtle way finite-size effects can enter. It is sometimes said that iDMRG trades finite-size scaling for finite-entanglement scaling. This is controlled by the (finite) so-called bond dimension parameter $\chi$ of the matrix product state (MPS). It essentially encodes how large ground state entanglement can be captured numerically. Too low $\chi$ can give inaccurate results. For area law states in 1D systems you can, at least in principle, always find a $\chi$ large enough. However, for critical systems the entanglement entropy $S\propto \ln L$ where $L$ is the system length, more care is needed, see F. Pollmann, S. Mukerjee, A.M. Turner and J.E. Moore, Phys. Rev. Lett. 102, 255701 (2009) (arXiv link). In quasi-1D systems the entanglement cut involves more sites (higher "area"), and so one needs to choose large enough $\chi$ for the given circumference, which practically has to be balanced against the computational cost of increasing either parameter. Similarly, excited states are typically not area law states. Another area of interest is in studies of thermalization of quantum systems, or other time evolution problems. In such cases, the entanglement entropy tends to grow with time, and $\chi$ puts a limit on how long one can evolve for. (Of course, all of these aspects also apply to finite MPS studies.)

The main technical limitation of iDMRG relative to finite-size DMRG is that it uses a translationally invariant ansatz. That assumption certainly doesn't always hold. Recently, disordered systems have attracted a lot of attention due to e.g. many-body localization. Further, I want to note that people only really realized how capable iDMRG is around 2008 and thereafter, through works such as I. P. McCulloch's (arXiv:0804.2509), so inertia and availability of existing well-developed general codes probably still favors finite-size DMRG calculations a bit. I'm not sure how iDMRG compares to finite-size DMRG in terms of ease-of-use or performance, which may be more significant considerations than numerical accuracy.

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