Garrity, D: Fill & Download for Free

This isn’t a perfect match to your question, but the US Military dealt out a serious punishment to Major John Riley. Major Riley was the leader of the San Patricio Battalion during the Mexican-American War. The battalion was made up of mostly deserters from the US Army. When Riley was captured, he was forced to watch as the remainder of his soldiers were hanged. He was then branded on both cheek with a letter D, for deserter. The fact that he deserted before the outbreak of war saved his life. Desertion in a time of war could have gotten him hanged along with his men.

Density matrix renormalization group is an algorithm used primarily for finding numerically exact ground state solutions of one-dimensional quantum systems on a lattice, like the Hubbard model or the Heisenberg model or generalizations. The basic problem with solving such systems directly is that for a chain of sites of length L, the size of the Hilbert space where the wavefunction lives grows exponentially, like d^L, where d is related to the degrees of freedom at each site, so any direct numerical algorithm will fail.Roughly speaking, in DMRG, a solution is reached starting with a small problem you can solve. Then, you split the problem into two small pieces and then add additional sites at the boundary between the pieces and solve the larger problem. The “trick” is that at each step in the problem, after adding sites, you have to throw away a bunch of degrees of freedom that are unimportant, or else the exponential growth will kick in. The algorithm uses a density matrix formulation to identify the “best” way to truncate the Hilbert space every time you add sites.After a certain point, instead of adding sites, you keep a constant number of sites, but keep doing the trick of splitting the problem into subsystems plus additional sites, solving the new problem, and the choosing the “best” truncated solution-space to use in the next step. You sweep which sites you treat back and forth until the energy converges.You can show that this algorithm will work very well for local quantum systems in 1D, because it is equivalent finding the best matrix product state that describes the solution, which is an efficient but general way to write the wavefunction in 1D. You can also understand it using renormalization group principles, where you systematically “flow” towards simpler low-energy descriptions of a difficult problem by transforming it many times.In higher dimensions, splitting the problem into sub-pieces doesn’t work nearly as well, because the pieces will be correlated all along the boundary between them. In 1D, the boundary between pieces is constant as the problem size grows, but in higher dimensions, the boundary grows as the problem size grows.

In principle, a perfect clean and ordered weak topological insulator (TI) will have side surfaces states on the appropriate sides according to its weak topological indices. The question is what will happen in the case of either weak or strong disorder.The conventional expectation is that in 3D weak TIs, the surface states are not robust. This is because you can understand a 3D weak TI as a stack of 2D TI’s (quantum spin Hall systems) that are weakly coupled to each other. If there are an even number of layers, the layers can form pairs that will couple the surface states and open a gap, destroying the conductivity. Therefore, a weak TI is dependent on boundary conditions and the weak topological indices are not robust to perturbations that reduce the translation symmetry. (See Sec. II C-D in https://arxiv.org/pdf/cond-mat/0611341.pdf).This idea may be too simple however, as other researchers have found that conductivity in weak TI models is actually robust to weak and strong random disorder, at least in some cases (see https://arxiv.org/pdf/1109.3201.pdf and https://arxiv.org/pdf/1105.4351.pdf). A basic explanation is that the odd number of layers behavior, where the layers cannot pair off perfectly, is actually more generic. The even behavior that gaps the surface states requires a coherent pairing interaction that does not generally happen from random disorder.The situation begins to clarify if you think about what happens if you allow for spontaneously broken surface translational symmetry, like with a charge density wave (CDW) that doubles the unit cell. This will eliminate surface states and gap the system, as per the first argument. However, in this case, there will be domain walls between different CDW regions, and at these walls there will be edge states that look like 2D TIs, which have robust conduction. So in the presence of disorder, you could get CDWs plus domain walls that are pinned by disorder. These domain walls would be the source of the conduction (See https://arxiv.org/pdf/1110.3420.pdf).So basically, the simple answer is yes, there are surface states. The real story is quite complicated and depends on the assumptions you make about translational symmetry, disorder, and interactions between electrons, but some form of surface state often survives.