Overview ======== In the past, we discussed a new format for Kwant systems [1] that will allow more general symmetries. Joe and Daniel have implemented a prototype [2] that supports general symmetries. Since general symmetries turn out to be tricky, the current consensus is to concentrate on translational symmetries at a first time. More general symmetries can then be added later. This staged approach does not have to complicate the final format since it seems anyway a good idea to have separate layers for translational and other spatial symmetries. This way the non-translational symmetries can remain hidden from solvers who are not interested in them.
There is another important aspect of the "new system" effort: we need a way to couple systems of different symmetry into one composite system. Such composite systems exist in today's Kwant but are not a separate concept from finite systems (this is easy to overcome). However, they are limited to attaching 1d leads to 0d scattering regions. Overcoming this limitation is tricky and I'd like to start a discussion on how to best do it. [1] https://www.mail-archive.com/kwant-devel@kwant-project.org/msg00063.html [2] https://gitlab.kwant-project.org/kwant/system2 Idea 1: unique sites ==================== Our experience has shown that it is useful to associate high-level sites with low-level systems. Users who build systems in terms of such high-level sites expect to find them again in the low-level systems. One attractive way of defining composite systems is to give additional importance to high-level sites: to require that they have a unique identity *across* the multiple systems that constitute a composite system. Observe that this is not the case in current Kwant: it is possible to attach a lead to the center of a ring even if both share the same lattice. The sites where the lead "hovers" over a sector of the ring exist twice as physical degrees of freedom. This "unique site" approach allows to elegantly specify complex composite systems. For example, to embed a 0d island in a 2d bulk it is enough to describe both constituents separately and to specify that sites From the lower-symmetry subsystem "overwrite" sites from the higher-symmetry subsystem. The "unique site" approach has the problem of not defining any canonical dense numbering of the present degrees of freedom. However any solver that performs useful computations for a composite system will have to establish such a numbering. Since the different parts of the composite system are connected using sites the usage of hash-tables to map sites to integer site indices (that necessarily appear at some point) seems inevitable. An additional problem is that the coupling between different parts is implicit and must be computed. For example a composite system that consists of a 0d central region with an attached 1d lead would require a precalculation that is equivalent to today's Builder.attach_lead() function. This preprocessing is also tricky: consider the simple configuration of a central region with two parallel leads (both belonging to the same lattice). Nothing remarkable happens if one lead is rotated slightly *away* from the other, but rotating it *towards* the other suddenly changes the very topology of the composite system! This simple example shows that correctly interpreting a composite system that uses the "unique site" approach is rather tricky. If this preprocessing can be done in a general way, it should obviously be cached but then shouldn't the cached results *become* the low-level composite system description instead? Idea 2: Systems as numbering schemes for degrees of freedom =========================================================== The following is an attempt to reconcile various ideas that came up recently (e.g. putting all operators on equal footing, reunifying symmetries of lattices with system symmetries) with a low-level format that is easy and efficient to use by solvers. I propose to redefine "system" within Kwant to mean a numbering scheme for degrees of freedom together with associated symmetries. This corresponds to what we understand today as system minus the Hamiltonian operator. An example of a very basic system is a single site (numbered 0) with two orthogonal translational symmetries. Note that this can be interpreted as a lattice! Thanks to the numbering, operators that act on a system can be defined easily. An operator is a mapping from group elements of the system's symmetry to (sparse) matrices of some form. We can introduce systems that correspond to a parent system but with a reduced symmetry. For example, a 1d wire could be derived from the above 2d lattice by picking 3 sites of the lattice (e.g. at the coordinates (0, 0), (0, 1), (0, 2)), numbering them starting with 0, and specifying the remaining symmetry (which must be compatible with that of the parent system). Note that the choice of the sites fixes the unit cell of the wire. Operators on the wire are specified by a mapping from the remaining symmetry to 3x3 matrices. A particular kind of derived system that seems useful are interrupted systems. That would make it explicit that a system with n-dimensional symmetry has been interrupted (in some way) to turn it into a system with n-1-dimensional symmetry. For example, a 1d wire could be turned into a semi-inifinite lead. A lead could be also represented as a reduced-symmetry system derived from a wire that adds a barrier but this will make it difficult to distinguish the specific case of a 1d lead From the general case of a 1d wire with some perturbation. Interrupted systems would provide an explicit way of creating semi-inifinite systems. Note that in the same spirit, a 2d plane could be "interrupted" to become a half-plane. Finally, in order to be able to represent general composite systems, another kind of derived system will be necessary: one that combines multiple systems with the same symmetry into one. This can be seen as addition of systems. For simplicity, we could require that operator matrices may only couple a first "central" subsystem to any of the others. I will now discuss some concrete examples. Example a: A 0d central region with two 1d leads ------------------------------------------------ This would be represented as follows: - A 2d square lattice is made. A 1d wire of some width is derived from that. The lead "interrupted system" is derived from the wire (by having it stop at some unit cell) - A 0d central system is derived from the same lattice as the wire. - A composite system is defined that couples the central system twice (at different placed) to the lead. Note that this corresponds to two physically separate leads, even if they have the same Hamiltonian. I have deliberately chosen this example to highlight a difference with the "unique site" approach. In practice it is preferable to have a builder that follows the "unique site" approach and creates two separate leads that live at different regions of space. But I want to emphasize that the low-level format is agnostic to this. It describes the system in a bottom-up, simple and explicit way. Example b: a wire with a local perturbation ------------------------------------------- - The wire is defined like in the previous example. - A 0d reduced-symmetry system is derived from the wire that modifies some of its elements locally. Note that this could be also realized (already in today's Kwant) as a central region with two attached leads. But this description makes it explicit that the left and right lead are the same wire and that there is only a (possibly small) perturbation. Remark 1 -------- The above says nothing about how the matrices that define operators are to be represented. That could be vanilla sparse matrices, but this is probably not the right approach if we want to be able to evaluate them in a vectorized way. Probably something along [2] is preferable. But, you see, this is a discussion that is independent from the present one, that concerns only numbering of degrees of freedom. Remark 2 -------- Another kind of derived systems could be useful: one that zeros-out all sites of the parent system that are not explicitly listed. That would make it possible to "cut-out" a wire (or a dot) from a 2d bulk. Remark 3 -------- Multiplication of systems could be naturally defined as tensor product, e.g. the product of the following - 1d chain in x direction - 1d chain in y direction - spin 1/2 would give a 2d plane with spin. The product of 2 2d planes would be a 2-particle system, etc. Rempark 4 --------- I think we should adopt numbering the real degrees of freedom instead of sites that may contain multiple orbitals. I.e. even if a system consists of 3 sites of 2 orbitals each, it seems preferable to number them 0, 2, 4. This way, the same scheme is used for numbering sites and degrees of freedom.
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