The problem of exponentiation of connected-graph contributionsC, when one carries out only a partial trace of the density matrix of an assembly of bosons in order to construct an effective, low-momentum Hamiltonian, is examined. It is found that besides accounting for the exponentiation of connected graphs, disconnected graphs contribute certain termsD to connected-graph contributions. TheD-terms diminish as the number of iterations increases in the Singh’s renormalization-group theory for the present system. Therefore, these terms play no role in determining critical behaviour of the system.

We calculate the electronic band dispersion of graphene monolayer on a two-dimensional transition metal dichalcogenide substrate (GrTMD) around $\bf{K}$ and $\bf{K'}$ points by taking into account the interplay of the ferromagnetic impurities and the substrate-induced interactions. The latter are (strongly enhanced) intrinsic spin–orbit interaction (SOI), the extrinsic Rashba spin–orbit interaction (RSOI) and the one related to the transfer of the electronic charge from graphene to substrate. We introduce exchange field $(M)$ in the Hamiltonian to take into account the deposition of magnetic impurities on the graphene surface. The cavalcade of the perturbations yield particle–hole symmetric band dispersion with an effective Zeeman field due to the interplay of the substrate-induced interactions with RSOI as the prime player. Our graphical analysis with extremely low-lying states strongly suggests the following: The GrTMDs, such as graphene on $\rm{WY_2}$, exhibit (direct) band-gap narrowing/widening (Moss–Burstein (MB) gap shift) including the increase in spin polarisation $(P)$ at low temperature due to the increase in the exchange field $(M)$ at the Dirac points. The polarisation is found to be electric field tunable as well. Finally, there is anticrossing of non-parabolic bands with opposite spins, the gap closing with same spins, etc. around the Dirac points. A direct electric field control of magnetism at the nanoscale is needed here. The magnetic multiferroics, like $\rm{BiFeO_{3}}$ (BFO), are useful for this purpose due to the coupling between the magnetic and electric order parameters.

We studied 2D surface state in the periodic Anderson model of a bulk topological Kondo insulator (TKI) system, involving strongly correlated f orbital electrons screened by d orbital electrons, performing a mean-field theoretic (MFT) calculation within the framework of slave-boson protocol. We found that bulk insulation together with the surface metallicity is possible for the system under infinite-U condition. The possibility of the intraband andthe interband unconventional plasmons with linear dispersion for the surface states without substitutional impurities exists in our report. In fact, our analysis demonstrates that, both type of solutions for the group velocity (v$_g$) andthe phase velocity (v$_p$), viz. (v$_g$, v$_p$) > 0 and (v$_g$ > 0, v$_p$ ≤; 0), could be possible for surface plasmons depending on parameters. The inducement of ‘helical liquids’ in the presence of impurities on the TKI surface is the importantoutcome of this paper. The accessibility of the gapless Dirac spectrum leads to the conventional q$^{1/2}$ spin plasmons. The Kondo screening remains unaffected by Rashba coupling.