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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)  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:
Use open boundary conditions (OBC). Though this introduces edge effects, the number of states needed to reach a given accuracy is drastically lower than with PBC (see next section below). For gapped systems DMRG scales linearly with system size, meaning often one can study systems with many hundreds or even thousands of sites. Last but not least, open boundaries are often more natural. For studying systems which spontaneously break symmetry, adding "pinning" fields on the edge is often a very nice way to tip the balance toward a certain symmetry broken state while leaving the bulk unmodified.
Use smooth boundary conditions. The basic idea is to use OBC but send the Hamiltonian parameters smoothly to zero at the boundary so that the system can not "feel" the boundary. For certain systems this can significantly reduce edge effects .
Use "infinite boundary conditions," that is, use infinite DMRG. This has a cost that can be even less than with OBC yet is completely free of finite-size effects.
However, there are cases where PBC remains preferable despite the extra overhead. A few such cases are:
Benchmarking DMRG against another code that uses PBC, such as a Monte Carlo or exact diagonalization code.
Extracting the central charge of a critical one-dimensional system described by a CFT. In practice, using PBC can give an accurate central charge even for quite small systems by fitting the subsystem entanglement entropy to the CFT scaling form.
Checking for the presence or absence of topological effects. These could be edge effects (the Haldane phase has a four-fold ground state degeneracy with OBC, but not with PBC), or could be related to some global topological sector that is ill-defined with PBC (e.g. periodic vs. antiperiodic boundary conditions for the transverse field Ising model).
(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 .)
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 .
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.
Measuring asymptotic properties of correlation functions: much of our understanding of gapless one-dimensional systems comes from field-theoretic approaches which make specific predictions about asymptotic decays of various correlators. To test these predictions numerically, one must work with large, translationally invariant systems with minimal edge effects. Using fully periodic boundary conditions satisfies these criteria. However, a superior choice is to use infinite DMRG, which combines the much better scaling of open-boundary DMRG with the ability to measure correlators at arbitrarily long distances by repeating the unit cell of the MPS wavefunction. Although truncating to a finite number of states imposes an effective correlation length on the system, this correlation length can reach many thousands of sites for quite moderate MPS bond dimensions. Karrasch and Moore took advantage of this fact to convincingly check the predictions of Luttinger liquid theory for one-dimensional systems of gapless fermions .
Studying two-dimensional topological order: a hallmark of intrinsic topological order is the presence of a robust ground state degeneracy when the system is put on a torus. Also many topological phases have gapless edge states which can cause problems for numerical calculations. Thus one might think that fully periodic BC are the best choice for studying topological phases. However, topological phases have the same ground-state degeneracy on an infinite cylinder as they do on a torus . Cincio and Vidal exploited this fact to use infinite DMRG to study a variety of topological phases . One part of their calculation did actually require obtaining ground states on a torus, but they accomplished this by taking a finite segment of an infinite MPS and connecting its ends. This approach does not give the true ground state of the torus but was sufficient for their calculation and was arguably closer to the true two-dimensional physics.
Obtaining bulk gaps: DMRG has the ability to "target" low-lying excited states or to obtain such states by constraining them to be orthogonal to the ground state. However, with OBC, localized excitations can get stuck to the edges and not reveal the true bulk gap behavior. Thus one may conclude that PBC is necessary. But using open or infinite boundaries remains the better choice because they allow much higher accuracy.
To deal with the presence of edges in OBC, one can use "restricted sweeping". Here one sweeps across the full system to obtain the ground state. Then, to obtain the first excited state one only sweeps through the bulk of the system, allowing the wavefunction to change arbitrarily there but remain fixed in the ground state near the edges. This traps the particle in a "soft box" which still lets its wavefunction mix with the basis that describes the ground state outside the restricted sweeping region.
Within infinite DMRG, boundary effects are rigorously absent if the calculation has converged. To compute bulk gaps one again uses a type of restricted sweeping known in the literature as "infinite boundary conditions". For more see the work by Phien, Vidal, and McCulloch .
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.
Smooth boundary conditions for quantum lattice systems, M. Vekic and Steven R. White, Phys. Rev. Lett. 71, 4283 (1993) cond-mat/9310053
Grand canonical finite-size numerical approaches: A route to measuring bulk properties in an applied field, Chisa Hotta and Naokazu Shibata, Phys. Rev. B 86, 041108 (2012)
 Efficient matrix-product state method for periodic boundary conditions, P. Pippan, Steven R. White, and H.G. Evertz, Phys. Rev. B 81, 081103
 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
 Luttinger liquid physics from the infinite-system density matrix renormalization group, C. Karrasch and J.E. Moore, Phys. Rev. B 86, 155156
 Quasiparticle statistics and braiding from ground-state entanglement, Yi Zhang, Tarun Grover, Ari Turner, Masaki Oshkawa, and Ashvin Vishwanath, Phys. Rev. B 85, 235151
 Characterizing Topological Order by Studying the Ground States on an Infinite Cylinder, L. Cincio and G. Vidal, Phys. Rev. Lett. 110, 067208
 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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