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Physical review. B, Condensed matter

Hole motion in quantum antiferromagnets: String and hopping results

William G. Macready, Allan E. Jacobs

Published Sep 1, 1991• 10 citations
Publisher

Abstract

We have investigated the t-J model and its variants using the string basis (of Shraiman, Siggia, Emery, and Lin) and the hopping basis (of Trugman); both bases are generated by the hopping term in the Hamiltonian applied repeatedly to the N\'eel state with a single hole. For the string basis, states to order ${\mathit{t}}^{10}$ are treated exactly, so that the Green function is obtained to order ${\mathit{t}}^{20}$; the tree approximation (of Brinkman and Rice) is used for the remainder. The hopping basis, which relaxes constraints imposed in the string basis, contains all states generated by up to eight hops. Properties studied include the wave function, the ground-state energy, the effective mass, the bandwidth, the spectral function, the self-energy, and the density of states. For the U=\ensuremath{\infty} model, there are no quasiparticles; the ferromagnetic polaron is missed. For the t-${\mathit{J}}_{\mathit{z}}$ model, both the string and hopping bases provide excellent results for ${\mathit{J}}_{\mathit{z}}$>0.1t. The Ising limit of the t-J model is treated well by the string basis, but the Heisenberg limit ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=${\mathit{J}}_{\mathit{z}}$ requires the hopping basis, which gives apparently good results for all J>0.1t. The mass is highly anisotropic; for example, at J=0.4t the masses parallel and perpendicular to the magnetic zone face are ${\mathit{m}}_{@}$nN=20.4m and ${\mathit{m}}_{\mathrm{\ensuremath{\perp}}}$=2.1m.

DOI
10.1103/physrevb.44.5166

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We have investigated the t-J model and its variants using the string basis (of Shraiman, Siggia, Emery, and Lin) and the hopping basis (of Trugman); both bases are generated by the hopping term in the Hamiltonian applied repeatedly to the N'eel state with a single hole. For the string basis, states to order t10{\mathit{t}}^{10}t10 are treated exactly, so that the Green function is obtained to order t20{\mathit{t}}^{20}t20; the tree approximation (of Brinkman and Rice) is used for the remainder. The hopping basis, which relaxes constraints imposed in the string basis, contains all states generated by up to eight hops. Properties studied include the wave function, the ground-state energy, the effective mass, the bandwidth, the spectral function, the self-energy, and the density of states. For the U=\ensuremath{\infty} model, there are no quasiparticles; the ferromagnetic polaron is missed. For the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model, both the string and hopping bases provide excellent results for Jz{\mathit{J}}_{\mathit{z}}Jz​>0.1t. The Ising limit of the t-J model is treated well by the string basis, but the Heisenberg limit J\ensuremath⊥{\mathit{J}}_{\mathrm{\ensuremath{\perp}}}J\ensuremath⊥​=Jz{\mathit{J}}_{\mathit{z}}Jz​ requires the hopping basis, which gives apparently good results for all J>0.1t. The mass is highly anisotropic; for example, at J=0.4t the masses parallel and perpendicular to the magnetic zone face are m@{\mathit{m}}_{@}m@​nN=20.4m and m\ensuremath⊥{\mathit{m}}_{\mathrm{\ensuremath{\perp}}}m\ensuremath⊥​=2.1m.
Exact diagonalizations of small clusters up to 26 sites are used to extrapolate the ground-state energy of a single hole in the N\'eel state of the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model or in the quantum antiferromagnetic (t-J model). A surprisingly rapid convergence with system size is observed for a wide range of parameters and for the largest clusters considered. In the Ising limit the energy dependence with Jz{\mathit{J}}_{\mathit{z}}Jz​ is remarkably close to a Jz2/3{\mathit{J}}_{\mathit{z}}^{2/3}Jz2/3​ law for intermediate coupling in agreement with Brinkman-Rice type of calculations. Nevertheless a small but significant bandwidth indicates that the hole can propagate coherently in this limit. Clearly the polaronic behavior \ensuremath{\sim}Jz1/2{\mathit{J}}_{\mathit{z}}^{1/2}Jz1/2​, for very small Jz{\mathit{J}}_{\mathit{z}}Jz​, is out of the range of the method. When quantum fluctuations are introduced (isotropic case) the ground-state energy behavior still remains close to that of the Ising limit although an overall increase of the energy by a factor 0.3t is observed. This suggests that the ``string scenario'' might have some relevance there.
The t-J model in the large-t/J and low-doping limit is approximated by an effective Hamiltonian that treats spin degrees of freedom within the linear-spin-wave approximation and the hole-hopping term with a hole-spin interaction vertex that is linear in the spin-wave operators. The Dyson's equation for the single-hole Green's function is solved numerically on finite- but sufficiently large-size square lattices. Our calculation is based on an approximate equation for the hole motion that has been obtained using a self-consistent perturbation approach where crossing diagrams are neglected. We have studied the contribution of vertex corrections and found that the leading two-loop crossing diagram is exactly zero. Thus, the leading nonzero contribution to the proper self-energy from crossing diagrams is a three-loop correction. In two dimensions, it is possible to solve Dyson's equation numerically including these three-loop corrections. The results demonstrate that the contribution of such vertex corrections is indeed small. Our solution for the hole spectral function, obtained by neglecting such small contributions from vertex corrections, shows that most features describing the hole motion are in close agreement with the results of the exact diagonalization on the 42{4}^{2}42 lattice in the region of J/t\ensuremath{\le}0.2. The results obtained on sufficiently large-size lattices suggest that certain important features of the spectral function remain while others change due to finite-size effects. In particular, we find well-defined peaks above the lowest-energy peak that survive in the thermodynamic limit. We analyze these higher-energy peaks of the spectral functions and find that they can be understood as ``string'' excitations, which are created by the hole moving in a nearly linear potential.
We report numerical results for the single-hole properties in the t-J model and the strong-coupling approximation to the Hubbard model in two dimensions. Using the hopping basis with over 106{10}^{6}106 states we discuss (for an infinite system) the bandwidth, the leading Fourier coefficients in the dispersion, the band masses, and the spin-spin correlations near the hole. We compare our results with those obtained by other methods. The band minimum is found to be at (\ensuremath{\pi}/2,\ensuremath{\pi}/2) for the t-J model for 0.1\ensuremath{\le}t/J\ensuremath{\le}10, and for the strong-coupling model for 1\ensuremath{\le}t/J\ensuremath{\le}10. The bandwidth in both models is approximately 2J at large t/J, in rough agreement with loop-expansion results but in disagreement with other results. The strong-coupling bandwidth for t/J\ensuremath{\gtrsim}6 can be obtained from the t-J model by treating the three-site terms in first-order perturbation theory. The dispersion along the magnetic zone face is flat, giving a large parallel/perpendicular band mass ratio.
The motion of a single hole in an arbitrary magnetic background is investigated for the two-dimensional $t\ensuremath{-}J$ model. The wave function of the hole is described within a generalized string picture which leads to a modified concept of spin polarons. We calculate the one-hole spectral function using a large-string basis for the limits of a N\'eel ordered and a completely disordered background. In addition we use a simple approximation to interpolate between these cases. For the antiferromagnetic background we reproduce the well-known quasiparticle band. In the disordered case the shape of the spectral function is found to be strongly momentum dependent, the quasiparticle weight vanishes for all hole momenta. Finally, we discuss the relevance of results for the lowest energy eigenvalue and its dispersion obtained from calculations using a polaron of minimal size as found in the literature.
We formulate a quasiparticle theory for a single hole in a quantum antiferromagnet in the limit that the Heisenberg exchange energy is much less than the hopping matrix element, J\ensuremath{\ll}t. We consider the ground state of the spins to be either a quantum N\'eel state or a d-wave resonating-valence-bond (RVB) state. We show in a self-consistent perturbation theory that the hole spectrum is strongly renormalized by the interactions with spin excitations. The hole can be described by a narrow quasiparticle band located at an energy of order -t with a quasiparticle residue of order J/t and a bandwidth of order J. Above the quasiparticle band is an incoherent band of width of order t. Our results indicate that the energy scale for any coherent phenomenon involving the holes is \ensuremath{\delta}J, where \ensuremath{\delta} is the doping concentration. In the N\'eel state we perform a spin-wave expansion on an anisotropic Heisenberg model. In the Ising limit we reproduce previously known results and then expand perturbatively about that limit. In this expansion we find that the holes have a quasiparticle residue of Jz{J}_{z}Jz​/t and a bandwidth of ${J}_{\ensuremath{\perp}}$. In the Heisenberg limit we employ a ``dominant pole'' approximation in which we ignore contributions to the self-energy from the incoherent part of the hole spectrum. A similar technique is used to study the d-wave RVB state. The relevance of our results to recent optical experiments is discussed.