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Physical review. B, Condensed matter

Single-hole effective masses in thet-Jmodel

X. Zotos, P. Prelovek, Igor Sega

Published Nov 1, 1990• 53 citations
Publisher

Abstract

The ground state of a single hole in the one- and two-dimensional t-J model is studied for finite systems as a function of the phase in the kinetic energy term. The coherent effective hopping ${\mathit{t}}_{\mathit{c}}^{\mathrm{*}}$ and the incoherent hopping ${\mathit{t}}_{\mathit{i}}^{\mathrm{*}}$, evaluated in this way, are connected via the conductivity sum rule. It is shown that for even and odd chains ${\mathit{t}}_{\mathit{c}}^{\mathrm{*}}$\ensuremath{\sim}${\mathit{t}}_{\mathit{i}}^{\mathrm{*}}$, while their reduction from the free value t remains small even for large J/t. The Bethe ansatz solution, obtained for J=2t, is consistent with the latter result, yielding ${\mathit{t}}_{\mathit{c}}^{\mathrm{*}}$/t\ensuremath{\sim}0.938. In the 4\ifmmode\times\else\texttimes\fi{}4 system, the phase variation yields a nontrivial energy manifold. The minimum appears to be near the (\ensuremath{\pi}/2,\ensuremath{\pi}/2)-(\ensuremath{\pi},0) line in the Brillouin zone for J/t1, while moving towards the (\ensuremath{\pi}/2,\ensuremath{\pi}/2)-(\ensuremath{\pi},\ensuremath{\pi}) line for J/t\ensuremath{\gg}1.

DOI
10.1103/physrevb.42.8445

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Physical review. B, Condensed matter•Feb 1, 1993•15 citations
The ground state of a single hole in the one- and two-dimensional t-J model is studied for finite systems as a function of the phase in the kinetic energy term. The coherent effective hopping tc∗{\mathit{t}}_{\mathit{c}}^{\mathrm{*}}tc∗​ and the incoherent hopping ti∗{\mathit{t}}_{\mathit{i}}^{\mathrm{*}}ti∗​, evaluated in this way, are connected via the conductivity sum rule. It is shown that for even and odd chains tc∗{\mathit{t}}_{\mathit{c}}^{\mathrm{*}}tc∗​\ensuremath{\sim}ti∗{\mathit{t}}_{\mathit{i}}^{\mathrm{*}}ti∗​, while their reduction from the free value t remains small even for large J/t. The Bethe ansatz solution, obtained for J=2t, is consistent with the latter result, yielding tc∗{\mathit{t}}_{\mathit{c}}^{\mathrm{*}}tc∗​/t\ensuremath{\sim}0.938. In the 4\ifmmode\times\else\texttimes\fi{}4 system, the phase variation yields a nontrivial energy manifold. The minimum appears to be near the (\ensuremath{\pi}/2,\ensuremath{\pi}/2)-(\ensuremath{\pi},0) line in the Brillouin zone for J/t1, while moving towards the (\ensuremath{\pi}/2,\ensuremath{\pi}/2)-(\ensuremath{\pi},\ensuremath{\pi}) line for J/t\ensuremath{\gg}1.
We have studied hole states on square lattices with 8, 10, 16, 18, and 20 sites by exactly diagonalizing the single-hole t-J model with twisted boundary conditions in the whole Brillouin zone. The bottom of the hole band is found to be extremely flat and anisotropic. The hole effective mass is highly anisotropic and enhanced in a quantum antiferromagnet. The momentum of a hole in the ground state is not located at high-symmetry points such as (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2), (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi},0), or (0,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}). They are strongly lattice-size dependent and weakly parameter J/t (t is the hopping constant and J is the exchange constant) dependent. As the lattice size increases, the momentum of the ground state approaches the zone face center (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2).
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.
Using exact diagonalization techniques, we study one dynamical hole in the two-dimensional t-J and Hubbard models on a square lattice including a next-nearest-neighbor hopping t'. We present the phase diagram in the parameter space (J/t,t'/t), discussing the ground-state properties of the hole. At J=0, a crossing of levels exists at some value of t' separating a ferromagnetic from an antiferromagnetic ground state. For nonzero J, at least four different regions appear where the system behaves like an antiferromagnet or a (not fully saturated) ferromagnet. We study the quasiparticle behavior of the hole, showing that for small values of \ensuremath{\Vert}t'\ensuremath{\Vert} the previously presented string picture is still valid. We also find that, for a realistic set of parameters derived from the Cu-O Hamiltonian, the hole has momentum (\ensuremath{\pi}/2,\ensuremath{\pi}/2), suggesting an enhancement of the p-wave superconducting mode due to the second-neighbor interactions in the spin-bag picture. Results for the t-t'-U model are also discussed with conclusions similar to those of the t-t'-J model. In general we found that t'=0 is not a singular point of these models.
We consider a single hole in the t-J model where the Heisenberg and the hole-hopping terms are expanded in powers of the spin-deviation operators. The lowest order corresponds to the linear spin-wave approximation and leads to an effective Hamiltonian studied earlier by means of self-consistent perturbation theory. In this work we shall include all terms generating up to two-loop diagrams in the perturbative expansion of the single-hole Green's function. The hard-core constraints are partially respected and the overall effect gives a renormalization factor of 1.158 to the hopping parameter t. The vertex that describes the process where a spin wave is absorbed and emitted at the same time by the hole is far more important than others. We find that the numerical solutions of Dyson's equation including all two-loop diagrams lower considerably the single-hole ground-state energies over the previous one-loop results and our results on the 4\ifmmode\times\else\texttimes\fi{}4 lattice are in better agreement with those obtained by exact diagonalization in the physical regime (small J/t). Our method can provide numerical solutions to the hole Green's function on very large lattices. We find that with the inclusion of the two-loop diagrams the spectral function retains all its features found in the one-loop calculation and the bandwidth, the quasiparticle residues, and the ``string'' excitations feel very small corrections.
We calculate the energy of flux states described by Slater determinants of single-particle levels in a uniform magnetic field with p/q flux quanta per plaquette and with no double occupancy. We find the exchange energy of flux states is lower than that of the Gutzwiller wave function by 10%--20%. The minimum kinetic energy is obtained by introducing phases associated with the holes. The phases are calculated by mapping onto a classical frustrated XY model. At 10% doping, the 1/2 flux state (\ensuremath{\pi} phase) is stabilized for the parameter regime J/t\ensuremath{\gtrsim}0.2. At about 20% doping, the time-reversal-breaking (1/3 flux state has the lowest energy for J/t\ensuremath{\gtrsim}1.0.