We present in detail a Green's-function approach for studying charged-spin systems which preserves the local constraints prohibiting double occupancy. This approach satisfies Wick's theorem, uses a fermionic expansion around a singly occupied N\'eel state, and treats charge and spin degrees of freedom on an equal footing. For the antiferromagnetic Heisenberg model we recover gapless spin excitations (renormalized spin waves) in a straightforward real-space random-phase-approximation approach. This expansion is strictly controlled by a geometrical factor, 1/z, where z is the coordination number. We describe the incoherent motion of charges (holes) in the t-J model by a self-retracing-path approximation and consider two competing contributions to the coherent hole propagation. These approximations are made conserving in a constructive fashion by mapping Feynman diagrams to an equivalent tight-binding model. To study the accuracy of this procedure, we have made a detailed numerical check against the results obtained by exact diagonalization of a 4\ifmmode\times\else\texttimes\fi{}4 system with one hole, finding excellent agreement both in and near the Ising limit.
An approximation method is presented for the calculation of the Green's function for a single hole moving in a correlated spin background. General formulas are obtained for calculating the moments of the Green's function, generalizing the Brinkman-Rice method to a general spin state. Numerical results are presented for a one-dimensional Gutzwiller resonating-valence-bond (RVB) state. In the two-dimensional case, the bandwidth is derived. The kinetic energy of a single hole in a correlated antiferromagnet or Gutzwiller RVB state is higher than in a N\'eel antiferromagnet.
We investigate the t-J model including local disorder for a hole in an antiferromagnetic spin background on a hypercubic lattice in d dimensions. We show that the well-known approximation of a linear potential, caused by the Ising term of the spin-spin interaction, is correct up to order 1/
d2 and hence becomes exact in d=\ensuremath{\infty}. We also investigate the influence of disorder and compute exactly the Green function and the density of states for a hole in an antiferromagnetic spin background for arbitrary disorder and J in the limit d\ensuremath{\rightarrow}\ensuremath{\infty}. Moreover, for finite J and disorder we determine the dynamical conductivity in high dimensions exactly. Remarkable structure both in the density of states and in the dynamical conductivity is found. Finally, we systematically derive the retraceable path approximation by Brinkman and Rice via a self-consistent 1/d expansion, which is correct up to order 1/
d4.
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 are treated exactly, so that the Green function is obtained to order
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 model, both the string and hopping bases provide excellent results for
Jz>0.1t. The Ising limit of the t-J model is treated well by the string basis, but the Heisenberg limit ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=
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@nN=20.4m and ${\mathit{m}}_{\mathrm{\ensuremath{\perp}}}$=2.1m.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
The motion of a single hole in a two-dimensional Heisenberg antiferromagnet (hAF) is studied in a representation where holes are described as spinless fermions (holons) and spins as normal bosons. Assuming long-range AF order the spin dynamics is treated in linear spin-wave theory. The formulation highlights the close relation with the conventional polaron problem. The holon Green's function is calculated self-consistently within the Born approximation using finite-cluster geometries for the numerical solution. As a remarkable result we find close agreement with the spectral function A(k,\ensuremath{\omega}) of a hole calculated by exact diagonalization methods. A(k,\ensuremath{\omega}) is characterized by a narrow quasiparticle (QP) peak at the low-energy side of the spectrum, which is well separated from the incoherent part for large enough J values. A complete characterization of our solution is given, including the spectral weight, the dispersion relation, and effective masses of the QP state. A finite-size-scaling study gives a nonvanishing spectral weight of the QP in the thermodynamic limit for values J/t typical for copper oxide superconductors. Our calculations indicate that the self-consistent Born approximation is a valuable scheme for characterizing the dynamics of a hole in the t-J model, even in the strong-coupling regime.