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Periodic vs Open or Infinite Boundary Conditions for DMRG, Which Should You Choose?

Miles Stoudenmire—April 28, 2014

One of the weaknesses of the density matrix renormalization group (DMRG) [1] is that it works poorly with periodic boundary conditions. This stems from the fact that conventional DMRG optimizes over open-boundary matrix product state (MPS) wavefunctions whether or not the Hamiltonian includes periodic interactions.

But this begs the question, when are periodic boundary conditions (PBC) really needed? DMRG offers some compelling alternatives to PBC:

However, there are cases where PBC remains preferable despite the extra overhead. A few such cases are:

(Note that in the remaining discussion, by PBC I mean fully periodic boundary conditions in all directions. For the case of DMRG applied to quasi-two-dimensional systems, it remains a good practice to use periodic boundaries in the shorter direction, while still using open (or infinite) boundaries in the longer direction along the DMRG/MPS path.)

Below I discuss more about the problems with using PBC, as well as some misconceptions about when PBC seems necessary even though there are better alternatives.

Drawbacks of Periodic Boundary Conditions

Periodic boundary conditions are straightforward to implement in conventional DMRG. The simplest approach is to include a "long bond" directly connecting site 1 to site N in the Hamiltonian. However this naive approach has a major drawback: if open-boundary DMRG achieves a given accuracy when keeping m states, then to reach the same accuracy with PBC one must keep m2 states! The reason is that now every bond of the MPS not only carries local entanglement as with OBC, but also the entanglement between the first and last sites. (There is an alternative DMRG algorithm for periodic systems which may have better scaling than the above approach but has not been widely applied and tested, as far as I am aware, especially for 2D or critical systems [3].)

The change in scaling from m to m2 is a severe problem. For example, many gapped one-dimensional systems only require about m=100 to reach good accuracy (truncation errors of less than 1E-9 or so). To reach the same accuracy with naive PBC would then require using 10,000 states, which can easily fill the RAM of a typical desktop computer for a large enough system, not to mention the extra time needed to work with larger matrices.

But poor scaling is not the only drawback of PBC. Systems that exhibit spontaneous symmetry breaking are simple to work with under OBC, where one has the additional freedom of applying edge pinning terms to drive the bulk into a specific symmetry sector. Using edge pinning reduces the bulk entanglement and makes measuring order parameters straightforward. Similarly one can use infinite DMRG to directly observe symmetry breaking effects.

But under PBC, order parameters remain equal to zero and can only be accessed through correlation functions. Though using correlation functions is often presented as the "standard" or "correct" approach, such reasoning pre-supposes that PBC is the best choice. Recent work in the quantum Monte Carlo community demonstrates that open boundaries with pinning fields can actually be a superior approach [4].

Cases Where Periodic BC Seems Necessary, But Open/Infinite BC Can be Better

Below are some cases where periodic boundary conditions seem to be necessary at a first glance. But in many of these cases, not only can open or infinite boundaries be just as successful, they can even be the better choice.

In conclusion, consider carefully whether you really need to use periodic boundary conditions, as they impose a steep computational cost within DMRG. Periodic BC can actually be worse for the very types of measurements where they are often presented as the best or "standard" choice. Many of the issues periodic boundaries circumvent can be avoided more elegantly by using infinite DMRG, or when that is not applicable, by using open boundary conditions with sufficient care.


[1] By DMRG, I also mean DMRG-like algorithms such as time-dependent DMRG (tDMRG a.k.a. time-evolving block decimation TEBD) or really any other algorithm that works with open-boundary MPS.

[2] References on smooth boundary conditions:

[3] Efficient matrix-product state method for periodic boundary conditions, P. Pippan, Steven R. White, and H.G. Evertz, Phys. Rev. B 81, 081103

[4] Pinning the Order: The Nature of Quantum Criticality in the Hubbard Model on Honeycomb Lattice, Fakher F. Assaad and Igor F. Herbut, Phys. Rev. X 3, 031010

[5] Luttinger liquid physics from the infinite-system density matrix renormalization group, C. Karrasch and J.E. Moore, Phys. Rev. B 86, 155156

[6] Quasiparticle statistics and braiding from ground-state entanglement, Yi Zhang, Tarun Grover, Ari Turner, Masaki Oshkawa, and Ashvin Vishwanath, Phys. Rev. B 85, 235151

[7] Characterizing Topological Order by Studying the Ground States on an Infinite Cylinder, L. Cincio and G. Vidal, Phys. Rev. Lett. 110, 067208

[7] Infinite boundary conditions for matrix product state calculations, Ho N. Phien, G. Vidal, and Ian P. McCulloch Phys. Rev. B 86, 245107


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