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arXiv:cond-mat/0108523 · Physical review. B, Condensed matter

Breakdown of the Nagaoka phase in the two-dimensional t−J model

Eli Eisenberg, Richard Berkovits, David A. Huse, B. L. Altshuler

Published Mar 26, 2002• 27 citations
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Abstract

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.

DOI
10.1103/physrevb.65.134437

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In the limit of weak exchange J at low hole concentration \ensuremathδ\ensuremath{\delta}\ensuremathδ the ground state of the two-dimensional t\ensuremath−Jt\ensuremath{-}Jt\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, Jcr\ensuremath∼t\ensuremathδ2,{J}_{\mathrm{cr}}\ensuremath{\sim}t{\ensuremath{\delta}}^{2},Jcr​\ensuremath∼t\ensuremathδ2, 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 \ensuremathδ\ensuremath{\delta}\ensuremathδ and scales as a power of the system size N.
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 $J/t\ensuremath{\ll}1$. 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 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 $J/t\ensuremath{\ll}1$ 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 ${\ensuremath{\delta}}_{t}=\sqrt{J/2\ensuremath{\pi}t}$.
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.
The phase diagram of the planar t-J model at small hole doping is investigated by finite-size scaling of exact diagonalization data of \ensuremath{\surd}N \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}N clusters (N\ensuremath{\le}26). Hole-droplet binding energies, compressibility, and static spin and charge correlations are calculated. Short-range antiferromagnetic correlations can produce attractive forces between holes leading to a very rich phase diagram including a liquid of d-wave hole pairs (for J/t\ensuremath{\gtrsim}0.2), a liquid of hole droplets (quartets) for larger J/t ratios (J/t\ensuremath{\gtrsim}0.5), and, at even larger coupling J/t, an instability towards phase separation.
The solution of a three-body Schr\"odinger equation (for two holes and a spin flip) shows that in the U=\ensuremath{\infty} Hubbard model the saturated ferromagnetic state with two holes is unstable (in a finite volume and for periodic boundary conditions). The ground state carries a finite momentum, and becomes degenerate with the Nagaoka state in the thermodynamic limit. This is in agreement with exact diagonalization results for small lattices where, in addition, we find that the true two-hole ground state is a singlet. As a result, our notions of ferromagnetism in the Hubbard model might need revision.
Ferromagnetic domains in the large-UUU Hubbard model with a few holes: an FCIQMC study
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of UUU in the large-UUU limit. In the so-called Nagaoka limit (one-hole system at infinite UUU), it is known that the Hubbard model exhibits a ferromagnetic ground-state. Here, by means of exact FCIQMC simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing UUU. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or 12\frac{1}{2}21​, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than one) number of holes.
We present different numerical calculations based on variational quantum Monte Carlo simulations supporting a ferromagnetic ground state for finite and small hole densities in the two-dimensional infinite-U Hubbard model. Moreover, by studying the energies of different total spin sectors, these calculations strongly suggest that the paramagnetic phase is unstable against a phase with a partial polarization for large hole densities delta approximately 0.40 with evidence for a second-order transition to the paramagnetic large doping phase.