Part of the background for the project to be proposed here is a recently published article (B. Hetényi, Phys. Rev. Research, 2 023277 (2020)). In this work a simple correlated model was investigated. The model is a one-dimensional lattice model, including a nearest and next-neatest neighbor (V, V', respectively) interaction. The phase diagram of the model has been known for a long time. For a finite V, it is possible to encounter four phase, if the variable V' is scanned: large-V charge density wave (CDW-V), Luttinger liquid, bond order phase, and large-V' charge density wav (CDW-V'). The unusual feature reported in the article was that the polarization undergoes a discrete jump inside the Luttinger liquid phase. Such behavior is reminiscent of the Su-Schrieffer-Heeger model, in which a discrete jump in polarization characterizes a quantum phase transition (gap closure), and separates two topologically distinct phases. In other aspects,
however, the state of affairs in the t-V-V' model parts from the SSH model. Inside the Luttinger liquid phase, the variance of the polarization diverges with system size, regardless of the value of the polarization. In the other three (insulating) phases, of course, the polarization takes definite values and the variance is finite. To summarize this situation: in the t-V-V' model phases of distinct polarization are not separated by a single gap closure point (as is the case in the Su-Schrieffer-Heeger model), but by an entire gapless phase, the Luttinger liquid phase.
The study leaves some important questions unanswered. Topological insulators descibed by band theory exhibit zero energy edge states (bulk-boundary correspondence principle). How does the bulk-boundary correspondence principle manifest in the t-V-V' model? More generally, in correlated topological insulators? Are there edge states present, and if yes what are their characteristics? In band topological insulators, edge states are generic, but in correlated topological insulators there are other possible behaviors. Edge state are still one possibility, but it is also possible that symmetry breaking occurs at the edges. One study (2) cast the topological invariant via the one-particle Green's function and found that topologically non-trivial systems can exhibit poles or zeros of the Green's function, and in addition, there are also scenarios in which the one-particle Green's function will not indicate the edge effects, a higher-order Green's function is needed. Our aim is to clarify these issues in the t-V-V' model, via a number of other possible methods, such as mapping to spin models(3), and study the entanglement characteristics of the model via the matrix product state method (2,4).
While the above proposal aims at characterizing a specific model, the calculations carry the potential to extend knowledge about the properties of correlated topological systems.
(1) B. Hetényi, Phys. Rev. Research, 2 023277 (2020).
(2) S. R. Manmana, A. M. Essin, R. M. Noack, and V. Gurarie,
Phys. Rev. B, 86 205119 (2012).
(3) I Affleck, J. Phys.: Condens. Mat. 1 3047 (1989).
(4) F. Pollmann, E. Berg, A. M. Turner, M. Oshikawa, Phys. Rev. B 81 064439 (2010).
-Great knowledge of quantum mechanics, solid state physics, statistical physics, and mathematical methods in physics
-Knowledge of field theoretical methods (Haldane mapping)
-Knowledge of entanglement
-Numerical experience (programming in either fortran, C++, matlab, or python)