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arXiv:1204.3824 · Physical Review B

Effective approach to the Nagaoka regime of the two-dimensional t - J model

Maciej M. Maśka, Marcin Mierzejewski, E. A. Kochetov, Lev Vidmar, J. Bonča, Oleg P. Sushkov

Published Jun 14, 2012• 16 citations
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Abstract

We argue that the $t$-$J$ model and the recently proposed Ising version of this model give the same physical picture of the Nagaoka regime for $J/t\ensuremath{\ll}1$. In particular, both models are shown to give compatible results for a single Nagaoka polaron as well as for a Nagaoka bipolaron. When compared to the standard $t$-$J$ or $t$-${J}_{z}$ models, the Ising version allows for a numerical analysis on much larger clusters by means of classical Monte Carlo simulations. Taking advantage of this fact, we study the low doping regime of the $t$-$J$ model for $J/t\ensuremath{\ll}1$ and show that the ground state exhibits phase separation into hole-rich ferromagnetic and hole-depleted antiferromagnetic regions. This picture holds true up to a threshold concentration of holes $\ensuremath{\delta}\ensuremath{\le}{\ensuremath{\delta}}_{t}\ensuremath{\simeq}0.44\sqrt{J/t}$. Analytical calculations show that ${\ensuremath{\delta}}_{t}=\sqrt{J/2\ensuremath{\pi}t}$.

DOI
10.1103/physrevb.85.245113

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•Nov 20, 2012•2 citations
We argue that the ttt-JJJ model and the recently proposed Ising version of this model give the same physical picture of the Nagaoka regime for J/t\ensuremath≪1J/t\ensuremath{\ll}1J/t\ensuremath≪1. In particular, both models are shown to give compatible results for a single Nagaoka polaron as well as for a Nagaoka bipolaron. When compared to the standard ttt-JJJ or ttt-Jz{J}_{z}Jz​ models, the Ising version allows for a numerical analysis on much larger clusters by means of classical Monte Carlo simulations. Taking advantage of this fact, we study the low doping regime of the ttt-JJJ model for J/t\ensuremath≪1J/t\ensuremath{\ll}1J/t\ensuremath≪1 and show that the ground state exhibits phase separation into hole-rich ferromagnetic and hole-depleted antiferromagnetic regions. This picture holds true up to a threshold concentration of holes \ensuremathδ\ensuremath≤\ensuremathδt\ensuremath≃0.44J/t\ensuremath{\delta}\ensuremath{\le}{\ensuremath{\delta}}_{t}\ensuremath{\simeq}0.44\sqrt{J/t}\ensuremathδ\ensuremath≤\ensuremathδt​\ensuremath≃0.44J/t​. Analytical calculations show that \ensuremathδt=J/2\ensuremathπt{\ensuremath{\delta}}_{t}=\sqrt{J/2\ensuremath{\pi}t}\ensuremathδt​=J/2\ensuremathπt​.
In the limit of weak exchange J at low hole concentration $\ensuremath{\delta}$ the ground state of the two-dimensional $t\ensuremath{-}J$ model is believed to be ferromagnetic. We study the leading instability of this Nagaoka state, which emerges with increasing J. Both exact diagonalization of small clusters, and a semiclassical analytical calculation of larger systems show that above a certain critical value of the exchange, ${J}_{\mathrm{cr}}\ensuremath{\sim}t{\ensuremath{\delta}}^{2},$ Nagaoka's state is unstable to phase separation. In a finite-size system a bubble of antiferromagnetic Mott insulator appears in the ground state above this threshold. The size of this bubble depends on $\ensuremath{\delta}$ and scales as a power of the system size N.
We present analytic and numerical results for the ground state of a dynamical hole in a quantum Ising antiferromagnet (the t-Jz{J}_{z}Jz​ model) on a two-dimensional square lattice. We find that the ground state has a localized region of reduced N\'eel order associated with the hole, with an energy that is well described by ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{h}}$/t\ensuremath{\approxeq}-3.63+2.93(Jz{\mathit{J}}_{\mathit{z}}Jz​/t)23{)}^{23})23 for 6\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$\ensuremath{\lesssim}Jz{\mathit{J}}_{\mathit{z}}Jz​/t\ensuremath{\lesssim}1. This resolves a disagreement in the literature regarding the coefficient of (Jz{J}_{z}Jz​/t)2/3{)}^{2/3})2/3. A simple string picture gives a reasonable description of the extent of the ground-state polaron in this intermediate-coupling regime. Disruption of long-range antiferromagnetic order in the bulk limit is estimated to occur at Jz{J}_{z}Jz​/t\ensuremath{\approxeq}0.3 for a hole-doping fraction of x=(1/16.
A theoretical investigation of boson versions of the $t\ensuremath{-}J$ and $t\ensuremath{-}{J}_{z}$ models on the square lattice is carried out, by means of Green's function Monte Carlo simulations. Accurate ground-state energy estimates as a function of hole doping are obtained, allowing one to investigate the stability of the uniform phase against separation of the system into hole-rich and hole-free phases. In the boson $t\ensuremath{-}{J}_{z}$ model, such a separation is found to occur for arbitrarily small values of Jz,{J}_{z},Jz​, at sufficiently low hole doping. Phase separation is suppressed in the boson $t\ensuremath{-}J$ model, which features a uniform ground state at any doping, for $J/t\ensuremath{\lesssim}1.5.$ Relevance of this study to the corresponding fermion models is discussed. Fermi statistics enhances the tendency toward phase separation; in particular, phase separation at low doping is predicted in the fermion $t\ensuremath{-}{J}_{z}$ model at any Jz>0.{J}_{z}>0.Jz​>0. The possible formation of stripes of holes is investigated for systems featuring both periodic and cylindrical boundary conditions. No evidence of a striped ground state is found in either the $t\ensuremath{-}J$ or $t\ensuremath{-}{J}_{z}$ boson models.
It is widely believed that a single hole in the two- (or three-)dimensional $t\ensuremath{-}J$ model, for sufficiently small exchange coupling J, creates a ferromagnetic bubble around itself, a finite J remnant of the ferromagnetic ground state at J=0J=0J=0 (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 $9\ifmmode\times\else\texttimes\fi{}9.$ We find that the polaron forms for $J/t<0.02\ensuremath{-}0.03.$ Although finite-size effects appear large, our data seems consistent with the expected ${1.1(J/t)}^{\ensuremath{-}1/4}$ variation of polarion radius. We also test the Brinkman-Rice model of nonretracing paths in a N\'eel background, showing that it is quite accurate, at larger J. Results are also presented in the case where the Heisenberg interaction is dropped (the $t\ensuremath{-}{J}^{z}$ 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.
Within the recently proposed doped-carrier representation of the projected lattice electron operators we derive a full Ising version of the t-J model. This model possesses the global discrete Z(2) symmetry as a maximal spin symmetry of the Hamiltonian at any values of the coupling constants, t and J. In contrast, in the spin anisotropic limit of the t-J model, usually referred to as the t-J(z) model, the global SU(2) invariance is fully restored at J(z) = 0, so that only the spin-spin interaction has in this model the true Ising form. We discuss a relationship between these two models and the standard isotropic t-J model. We show that the low-energy quasiparticles in all three models share qualitatively similar properties at low doping and small values of J/t. The main advantage of the proposed Ising t-J model over the t-J(z) one is that the former allows for the unbiased Monte Carlo calculations on large clusters of up to 10(3) sites. Within this model we discuss in detail the destruction of the antiferromagnetic (AF) order by doping as well as the interplay between the AF order and hole mobility. We also discuss the effect of the exchange interaction and that of the next-nearest-neighbour hoppings on the destruction of the AF order at finite doping. We show that the short-range AF order is observed in a wide range of temperatures and dopings, much beyond the boundaries of the AF phase. We explicitly demonstrate that the local no-double-occupancy constraint plays the dominant role in destroying the magnetic order at finite doping. Finally, a role of inhomogeneities is discussed.
Abstract. Within the recently proposed doped-carrier representation of the projected lattice electron operators we derive a full Ising version of the t–J model. This model possesses the global discrete Z2 symmetry as a maximal spin symmetry of the Hamiltonian at any values of the coupling constants, t and J. In contrast, in the spin anisotropic limit of the t–J model, usually referred to as the t–Jz model, the global SU(2) invariance is fully restored at Jz = 0, so that only the spin-spin interaction has in that model the true Ising form. We discuss a relationship between those two models and the standard isotropic t–J model. We show that the low-energy quasiparticles in all three models share the qualitatively similar properties at low doping and small values of J/t. The main advantage of the proposed Ising t–J model over the t–Jz one is that the former allows for the unbiased Monte Carlo calculations on large clusters of up to 10 3 sites. Within this model we discuss in detail the destruction of the antiferromagnetic (AF) order by doping as well as the interplay between the AF order and hole mobility. We also discuss the effect of the exchange interaction and that of the next nearest neighbour hoppings on the destruction of the AF order at finite doping. We show that the short-range AF order is observed in a wide range of temperatures and dopings, much beyond the boundaries of the AF phase. We explicitly demonstrate that the local no double occupancy constraint plays the dominant role in destroying the magnetic order at finite doping. Finally, a role of inhomogeneities is discussed. PACS numbers: 71.10.Fd, 71.10.Pm, 74.25.JbIsing t–J model close to half filling: A Monte Carlo study 2 1.