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arXiv:1510.06881 · Journal of the Physical Society of Japan

Ferromagnetic Clouds Caused by Hole Motion in a One-Dimensional t–J Model

Kazuhiro Sano, Ken’ichi Takano

Published Aug 26, 2016• 2 citations
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Abstract

The one-dimensional $t_1$-$t_2$-$J_1$-$J_2$ model is examined in the one-hole case, in which the total number of electrons is one less than the number of the lattice sites. The ground-state phase diagram includes a series of partial ferromagnetic phases, which are stacked in a regime of positive and small $J_1$. We find that the ground state in each of these partial ferromagnetic phases includes a ferromagnetic cloud, which is a multiple-spin bound state together with the hole. The ferromagnetic cloud is a large magnetic polaron with a heavy mass in a single-band electronic system and is supposedly formed as a result of Nagaoka ferromagnetism which locally works around the hole.

DOI
10.7566/jpsj.85.094714

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•Apr 1, 2026•0 citations
The one-dimensional t1t_1t1​-t2t_2t2​-J1J_1J1​-J2J_2J2​ model is examined in the one-hole case, in which the total number of electrons is one less than the number of the lattice sites. The ground-state phase diagram includes a series of partial ferromagnetic phases, which are stacked in a regime of positive and small J1J_1J1​. We find that the ground state in each of these partial ferromagnetic phases includes a ferromagnetic cloud, which is a multiple-spin bound state together with the hole. The ferromagnetic cloud is a large magnetic polaron with a heavy mass in a single-band electronic system and is supposedly formed as a result of Nagaoka ferromagnetism which locally works around the hole.
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.
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 full configuration interaction quantum Monte Carlo 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, centered on individual holes. The overall total spin of the wave functions remains 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 1) number of holes.
In this paper we present results of finite-lattice studies of the t-J and the t-t'-J model for 18 (20) sites with up to 4 (2) holes. A modified Lanczos algorithm allowed for the classification of the ground state according to total spin S. In the one-hole sector we find evidence for the existence of hole pockets. We calculate the effective quasiparticle band and show qualitative differences between the t-J and t-t'-J model for J\ensuremath{\gtrsim}0.2. As U increases we notice a substantial band narrowing, which is maximal at the Nagaoka transition, accompanied by S-level crossings rendering a quasiparticle description difficult. In the high-doping regime we present hole binding energies for up to six holes for the t-J model. Evidence for phase separation is only found for J>t. We also studied the magnetic structure of the ground state on the 18-site lattice in the limit J=0. We present a complete ground-state classification for all numbers of holes Nh{\mathit{N}}_{\mathit{h}}Nh​ and find S=Smax{\mathit{S}}_{\mathrm{max}}Smax​ at quarter filling in addition to the Nagaoka case.
In the spin-wave approximation the energy spectrum of one-hole states of the two-dimensional t-J model is considered in the parameter range 0.01\ensuremath{\le}J/t\ensuremath{\le}1, where J is the exchange constant and t the hopping-matrix element. Attention is focused on single-particle states (in contrast to hole-magnon continua) that form energy bands characterized by different values of the z component of the total spin Sz{\mathit{S}}_{\mathit{z}}Sz​, corresponding to ferromagnetically ordered regions of different size around the hole. For an infinite lattice, the shapes and relative positions of the lowest bands are computed by means of the Lanczos method. The formation process of the ferromagnetically ordered region in the one-hole ground state is discussed. It is connected with a characteristic dependence of the band energy on J/t for different Sz{\mathit{S}}_{\mathit{z}}Sz​. The shapes of the magnon cloud around the hole and the point symmetries of the eigenfunctions are determined for various points in the Brillouin zone and for different bands of interest.
In a t-J model on a chain, an additional next-nearest-neighbor hopping term destroys the complete spin-charge separation at J/t\ensuremath{\ll}1 and induces the formation of stable spin-polarized S>0 states. The properties of a single ferromagnetic polaron are investigated by the exact diagonalization of small systems and by the approximative treatment of an effective model, describing a spinless fermion coupled to the spin background. Both methods give consistent results on the critical Jc{\mathit{J}}_{\mathit{c}}Jc​ for the polaron formation on the increase of the spin S and the polaron size with decreasing J. A very large enhancement of the polaron mass with increasing S is observed. Systems with a finite concentration of holes are also studied within the same framework. Analytical considerations indicate on the bound many-hole states and consequently on phase-separated regions in the phase diagram.
Solid state physics, a very broad field in natural science, aims to explain physical properties of materials modeled as a many-body ensemble of the order of ∼ 10^23 interacting particles. For interacting metallic systems, one of the most powerful tools for unraveling the consequences of the collective interaction is the Fermi liquid theory. While still being an approximation, it excellently models many materials especially in the low-temperature and low-energy regime by treating themany-body problem as being equivalent to effectively non-interacting ”particles”, which is done by introducing the concept of quasiparticles. During the last few decades, several exotic phases of matter have been discovered, which originate from strong Coulomb interactions between the electrons, whose description is beyond the Fermi liquid theory. This thesis addresses the question of the validity of Fermi liquid theory once a single hole is introduced into an antiferromagnet with completely localized spins due to the strong interactions. Here, the most-studied case concerns the so-called 2D t − J model, which hosts a well-defined quasiparticle, the spin polaron. Focusing on a more tractable case, the t − Jz model without quantum spin fluctuations, we study the stability of the spin polaron solution once the nextnearest neighbor hopping t′ is introduced into the system. We focus on observations at the Γ-point and the X-point of systems with fixed Jz = 0.4t and varying t′ ∈ [−0.5t, 0.5t]. The model is numerically solved using exact diagonalization on finite-size lattices of up to 32 sites. A study of the quasiparticle weight under finite-size scaling reveals an “anomalous” regime at the Γ-point fort′ ≤ −0.3t, indicated by a vanishing quasiparticle weight. In order to characterize the low-energy dynamics within this “anomalous” regime, we investigated the spin correlation centered around the hole, as well as the magnon number distribution of a single-doped 20-site system. Based on the results, we present a preliminary picture of the antiferromagnetic ground state doped with one hole within that “anomalous” regime (t′ ≤ −0.3t) that deviates from the standard spin polaron as the cloud of magnetic fluctuations, typically localized around the hole, delocalizes over the whole finite-size lattice. Consequently, such systems show a weakly suppressed antiferromagnetic configuration under single-hole doping.