Edge States in Doped Antiferromagnetic Nanostructures

Abstract

We study competition between different phases in a strongly correlated nano-structure with an edge. Making use of the self-consistent Green's function and density matrix renormalization group methods, we study a system described by the t-Jz and t-J models on a strip of a square lattice with a linear hole density n || . At intermediate interaction strength J/t we find edge stripe-like states, reminiscent of the bulk stripes that occur at smaller J/t. We find that stripes attach to edges more readily than hole pairs, and that the edge stripes can exhibit a peculiar phase separation.PACS numbers: 71.10. Fd, 71.10.Pm It is well established by now that in the ground state of the two-dimensional (2D) strongly correlated cuprates, much studied in the context of the high-T c problem, 1,2 charge carriers tend to be distributed inhomogeneously. Such inhomogeneities occur as an attempt to reduce the frustration between the exchange and kinetic energies arising from the antiferromagnetic (AF) background. 1 It is conceivable that such frustration can be further reduced at a boundary. This would induce an effective attraction of the bulk inhomogeneity to the edge, thus leading to a novel, many-body type edge state. A bulk doped Mott insulator might have various types of striped, paired, or unpaired ground states. Ordinarily one is not concerned with surface states. However, for cuprate nanoscale structures, which are of interest due to recent progress in nano-synthesis, 3 one may potentially have a system with a finite doping in which the holes are bound to the edges. In this case, understanding the possible edge states is crucial.Here, we consider the simplified case of a half-plane, or similarly, one end of a long open cylinder, with a vanishingly small doping sufficient to form a finite linear concentration n = N h /L of holes near the edge. N h is the number of holes and L is the linear size of the system. Given this configuration, several questions arise. For example, if the holes form a single stripe in the bulk, does this stripe attach itself to the edge? Do single pairs attach to the edge? Can one have a one-dimensional (1D) "edge phase separation" with the holes clustered together in edge droplets?In this paper we address some of these questions in the context of the t-J z and t-J models, using the selfconsistent Green's function and the density matrix renormalization group (DMRG) methods. We obtain an approximate edge phase diagram in which the linear hole density n is a control parameter. We find analytically, for the t-J z model, that the stripe edge states do exist in a substantial region of the phase diagram. This is verified for both the t-J z and t-J models using DMRG for systems up to 11×8 sites with periodic/open and 20×5 with open boundary conditions. We also find that the Ising anisotropy enhances tendency to the edge states formation. A remarkable feature of these states is that they are reminiscent of and, as we show in some detail, are directly related to the bulk stripe states observed in the studie...

The density-matrix renormalization group in the age of matrix product states

“…Indeed this is possible in one dimension: if we consider the mixed-canonical representation, we see that for the exponentially decaying eigenvalue spectra of reduced density operators (hence exponentially decaying singular values s a ) it is possible to cut the spectrum following Eq. (27) at the D largest singular values (in the sense of an optimal approximation in the 2-norm) without appreciable loss of precision. This argument can be generalized from the approximation incurred by a single truncation to that incurred by L À 1 truncations, one at each bond, to reveal that the error is at worst [96] kjwi À jw trunc ik 2 2 6 2…”

“…In fact, both questions are intimately related: as was realized quite soon, DMRG is only moderately successful when applied to two-dimensional lattices: while relatively small systems can be studied with high accuracy [21,22,23,24,25,26,27,28], the amount of numerical resources needed essentially increases exponentially with system size, making large lattices inaccessible. The totally different behaviour of DMRG in one and two dimensions is, as it turned out, closely related [29,30] to the different scaling of quantum entanglement in many-body states in one and two dimensions, dictated by the so-called area laws (for a recent review, see [31]).…”

Studying Two-Dimensional Systems with the Density Matrix Renormalization Group

“…Working with a modified t − J model where J is replaced by an Ising J z interaction, the two methods show remarkable correspondence in their predictions of energies and hole distributions, especially for the case of a single hole. 56,57 Further DMRG investigations of the t − J model have examined the existence of checkerboard order, 58 edge states of holes in nanosystems 59 and the competition between stripes and pairing. 60 DMRG has also been used to demonstrate a striped phase in the Hubbard model on systems up to width six.…”

New trends in density matrix renormalization

“…A very interesting application of the DMRG to study single wall carbon nanotubes, where the tube is mapped onto a 1D chain with longer range interactions (depending on the quirality) can be found in Ref. [220] and the study of edge states in doped nanostructures in [221]. Inspired in recent experiments of trapped bosonic atoms [222], a one dimensional Bose gas was studied using ab-initio stochastic simulations at finite temperatures covering the whole range from weak to strong interactions [223].…”