Extreme finite size effects

 The standard approach for describing and analysing macroscopic properties of physical systems is to consider a family of many-body quantum Hamiltonians defined on an increasing sequence of finite lattices, and then look for properties, like the groundstate degeneracy or the energy gap between the groundstate and the first excited states, that are satisfied uniformly for all system sizes large enough. This idea is at the core of the mathematical definition of thermodynamic limit, and is how most experiments (either numerical or in a lab) are carried over. This approach has been highly successful in predicting properties of physical systems, but on the other hand recently has been shown that determining whether a 2D model will be gapped or gapless in the limit is an undecidable problem.

Therefore, in order to correctly extrapolate the thermodynamic properties of a physical model, it is important to distinguish and recognise features that are consequence of finite-size effects, i.e. properties of the model which are not present in the thermodynamic limit but appear as a by-product of conditions which only hold for systems sizes smaller than some threshold.

While intuition would say that such finite-size effects can only produce small perturbations of the real properties of the model, they can instead be dominant to the point of completely obscuring the physics of the thermodynamic limit. I will show the existence of such extreme finite size effect by presenting two examples of finite-range, translational invariant Hamiltonian on a regular spin lattice, with the following properties: if the system size is lower than some threshold, the groundstate and first excited states will be classical (product) states, while above such threshold they will be the groundstate and first excitation of the Planar Code, a topological model similar to the famous Toric code.

The construction of these examples is based, in one case, on periodic tiling of the plane, in the other case, on embedding of a Turing machine into a classical Hamiltonian, and therefore might be of independent interest. 

This is based on joint work with Johannes Bausch,Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf.