Effect of the boundary on thermodynamic quantities such as magnetization

Abstract

In this paper, we investigate in general how thermodynamic quantities such as the polarization, magnetization and the magneto-electric tensor are affected by the boundaries. We show that when the calculation with periodic boundary conditions does not involve a Berry's phase, the quantity in question is determined unambiguously by the bulk, even in the presence of gapless surface states. When the calculation involves a Berry's phase, the bulk can only determine the quantity up to some quantized value, given that (i) there are no gapless surface states, (ii) the surfaces do not break the symmetries preserved by the bulk, and (iii) the system is kept at charge neutrality. If any of the above conditions is violated, the quantity is then determined entirely by the details at the boundary. Due to the strong dependence on the boundary, this kind of thermodynamic quantity, such as the isotropic magneto-electric coefficient, cannot be measured in the bulk without careful control at the boundary. One thus cannot distinguish between a topological insulator and a trivial insulator in three dimensions by any local measurement in the bulk.

Equilibrium surface current and role of U(1) symmetry: Sum rule and surface perturbations

“…However, we find that, both for the random potential and constant potential, the left surface current I from I 0 y = (1/N y ) i∈surf j 0 yi , which is confirmed for several different system sizes. Conservation of the surface currents were also seen in the previous study for a constant surface potential in an insulating state 16 . In the case of disorder potential, distribution of I L y for different configurations of the potential is well localized around its mean value I 0 y as shown in Fig.…”

“…In this view point, the conservation of the U(1) surface current in the presence of surface perturbations is understood as a result of non-trivial cancellations of changes in j i at each site. This sum rule argues that surface currents and corresponding OM are bulk quantities which are independent of surface details as suggested in the modern theories on OM [7][8][9][10][11][12][13][14][15][16] , and this supports existence of the bulk-surface correspondence for these quantities. We note that, the sum rule is useful not only for developments in understanding fundamental aspects of surface currents, but also for practical calculations of them.…”

Kinetic magnetoelectric effect in topological insulators

“…Magnetoelectric tensors may require careful consideration of boundaries. While orbital magnetization is independent on the boundary [34][35][36][37][38][39] , the orbital magnetization when an electric field is applied may not have such properties. The general orbital magnetoelectric response [30][31][32][33] depends on the boundary 37 .…”

Current response of a topological insulator to a static Zeeman field

“…Nevertheless, Bianco and Resta [24] provided a real-space expression of orbital magnetization for any two-dimensional insulator with finite Chern number, and demonstrated that the formula is independent of boundary conditions for a large sample. On the other hand, Chen and Lee [25] argued that, for a system composed of two insulators with opposite Chern numbers, the orbital magnetization should be independent of the details at the boundaries. They also demonstrated, by numerical calculations, that the orbital magnetization is insensitive to local perturbed potentials near the edge in a Chern insulator, and it is also unchanged with magnetic moments at the edges, which gap out the edge states, in spin Hall insulator.…”