# Definition of open boundary conditions

1. Apr 29, 2015

### king vitamin

I have a question I'm a little embarrassed to be asking: what is meant in condensed matter when someone describes a system with "open boundary conditions," say in one-dimension for simplicity? I am comfortable with the statement of fixed (Dirichlet) or free (von Neumann) boundary conditions, as well as periodic of course. But I also see the terms open/closed, which I want to be identical with free/fixed respectively, but I can't find a simple source which just states a definition.

If it helps, I'm asking because I would like to set up an analytic calculation to compare with a numerical DMRG paper which simply states that they take open boundaries. Is there a simple definition somewhere to tell me what this means? Maybe a better understanding of DMRG would help me. I would be fine with a definition in terms of lattice models, though I'm interested in taking the continuum limit quickly.

I saw one source which mentioned imagining the system on an infinite lattice but simply "turning off" couplings at the edges of the system. This seems weird to me; the eigenfunctions of (say) the Laplacian with these boundary conditions are Dirichlet (sine) in the continuum limit, but I expect "open" to be used in a thermodynamic sense, where the system can exchange energy through its boundaries.

Last edited: Apr 29, 2015
2. Apr 30, 2015

### Omega0

I am not that expert on "open boundary conditions" but from a first glance on the matter: If you have a source term in your field and if you don't have external conditions (outside of your numerical space the world continues without influence) I would exactly expect "open boundaries". Whatever your conditions are, I would expect from your system a signal transport. This signal will transport energy. If I assume to be far away the transport trough a surface depends from the physics. Example: In electromagnetics it is useful to use for open boundaries "absorbing boundary conditions (ABC)".

3. May 7, 2015

Are you talking about a 1D chain? It just means you are talking about a chain without putting it on a ring. If you put it on a ring you can have either periodic or antiperiodic BCs

4. May 7, 2015

### Physics Monkey

Usually it just means deleting the degrees of freedom outside some region and removing all the terms in the Hamiltonian that coupled to them.

However, sometimes one adds additional boundary-only terms as part of the general notion of open boundary conditions.

In the context of DMRG to the best of my knowledge this phrase means you solve the 1d model on some finite line segment. Alternatively, one solves a 2d model rolled up into a finite cylinder. However, one does permit to add forcing terms at the boundary to test for symmetry breaking, sensitivity to boundary conditions, etc.