Density matrix renormalization group analysis of the Nagaoka polaron in the two-dimensional $t-J$ model

It is widely believed that a single hole in the two- (or three-)dimensional $t-J$ model, for sufficiently small exchange coupling J, creates a ferromagnetic bubble around itself, a finite J remnant of the ferromagnetic ground state at $J=0\mathrm{}$ (the infinite U Hubbard model), first established by Nagaoka [Phys. Rev. 147, 392 (1966)]. We investigate this phenomenon in two dimensions using the density matrix renormalization group, for system sizes up to $9\times 9.$ We find that the polaron forms for $J/t<\mathrm{}0.02\mathrm{}-\mathrm{0.03.}$ Although finite-size effects appear large, our data seems consistent with the expected ${1.1(J/t)\mathrm{}}^{-1/4}$ variation of polarion radius. We also test the Brinkman-Rice model of nonretracing paths in a Néel background, showing that it is quite accurate, at larger J. Results are also presented in the case where the Heisenberg interaction is dropped (the $t-J^z$ model). Finally we discuss a “dressed polaron” picture in which the hole propagates freely inside a finite region but makes only self-retracing excursions outside this region.