Numerical approach to the low-doping regime of the $t\text{−}J$ model

We develop an efficient numerical method for the description of a single-hole motion in the antiferromagnetic background. The method is free of finite-size effects and allows the calculation of physical properties at an arbitrary wave vector. A methodical increase of the functional space leads to results that are valid in the thermodynamic limit. We found good agreement with cumulant expansion, exact-diagonalization approaches on finite lattices as well as self-consistent Born approximations. The method allows a straightforward addition of other inelastic degrees of freedom, such as lattice effects. Our results confirm the existence of a finite quasiparticle weight near the band minimum for a single hole and the existence of stringlike peaks in the single-hole spectral function.

Figure 1

(Color online) (a) Single-hole energy $E_{\mathbf{k}=(\pi ∕2,\pi ∕2)}$ vs $J∕t$. The full line represents EDLFS results of the present work that were obtained using $N_{st}=37402972$ states. Dotted lines represent CE results from Ref. 10. The dashed line, given by $E_{\mathbf{k}=(\pi ∕2,\pi ∕2)}=J∕t-3.24+2.65{(J∕t)}^{0.72}$ is an interpolation based on ED results obtained on 32 sites.[13] The double-dot-dashed line represents interpolation $E_{\mathbf{k}=(\pi ∕2,\pi ∕2)}=-3.36+3.50{(J∕t)}^{2∕3}$ based on the WMC method from Ref. 15. The dot-dashed line, $E_{\mathbf{k}=(\pi ∕2,\pi ∕2)}=-3.17+2.83{(J∕t)}^{0.73}$ is a result of the SCBA calculation (Refs. 4, 26). Squares represent QMC calculations from Ref. 17. (b) Quasiparticle weight $Z_{\mathbf{k}=(\pi ∕2,\pi ∕2)}$ vs $J∕t$. The full line represents the EDLFS calculation using $N_{st}=5213618$ states. The dot-dashed line represents ED results as summarized in the extrapolation given by $Z_{\mathbf{k}=(\pi ∕2,\pi ∕2)}=-0.136+0.664{(J∕t)}^{0.33313}$, while the dashed line represents the SCBA calculation $Z_{\mathbf{k}=(\pi ∕2,\pi ∕2)}=0.63{(J∕t)}^{0.667}$ from Ref. 4 (similar extrapolation was also obtained using SCPA in Ref. 5). Diamonds represent the WMC calculation from Ref. 15, and squares are QMC results from Ref. 17. Note that WMC and QMC data were multiplied by a factor of 2 due to a different definition of $Z_{\mathbf{k}}$ used in Refs. 15, 17. (c) The bandwidth $W∕t$ vs $J∕t$ calculated with EDLFS (full line), ED (Ref. 13) (dot-dashed line), SCBA (Ref. 4) (dashed lines), QMC calculations from Ref. 17 (squares), and $W∕t=2t^2∕J$ (dotted line), proposed in Ref. 4. (d) $Z_{\mathbf{k}}$ vs $J∕t$ (ten nearly overlapping curves) obtained by using ten different sizes of LFS as explained in the text.

Figure 2

(Color online) (a) Single-hole energy $E_{\mathbf{k}}$ and (b) quasiparticle weight $Z_{\mathbf{k}}$. Both quantities were calculated at fixed $J∕t=0.3$ and 0.4, as noted in the figure. EDLFS results of the present work were obtained with $N_{st}=37402972$states (full lines); ED results (circles), calculated on a 32 site system, are from Ref. 13, QMC results (squares) are from Ref. 17, while SCBA results (diamonds) in (b) are from Ref. 4.

Figure 3

(Color online) (a) Surface plot of $Z_{\mathbf{k}}$ at $J∕t=0.3$, computed on a mesh of 400 $\mathbf{k}$ points. Variational space with $N_{st}=5213618$ was used. Only $1∕4$ of the AFM BZ is shown. (b) Contour plot of $Z_{\mathbf{k}}$ with equidistant contours while wave vectors are spanning the whole AFM BZ. The rest is the same as in (a).

Figure 4

(Color online) [(a)–(c)] Spectral functions $A_{\mathbf{k}}\left(\omega \right)$ for three typical values of $\mathbf{k}$. [(d)–(f)] $A_{\mathbf{k}}\left(\omega \right)$ at $\mathbf{k}=(\pi ∕2,\pi ∕2)$ calculated using different sizes of LFS’s, as indicated in the insets. In all cases, we have used $J∕t=0.3$ and an artificial damping of $ϵ=0.05$.

Figure 5

Spectral functions $A_{\mathbf{k}}\left(\omega \right)$ for wave vectors $\mathbf{k}=(\pi ∕2,\pi ∕2)$ through (0, 0). In all cases, we have used $J∕t=0.3$, $N_{st}=3109626$, and artificial damping $ϵ=0.1$. Dark-shaded areas are proportional to the quasiparticle weight $Z_{\mathbf{k}}$.

Figure 6

Spectral functions $A_{\mathbf{k}=(\pi ∕2,\pi ∕2)}\left(\omega \right)$ for various values of $J∕t$ are shown. We have used $N_{st}=3109626$ and artificial damping $ϵ=0.1$. Dark-gray areas are proportional to the quasiparticle weight $Z_{\mathbf{k}=(\pi ∕2,\pi ∕2)}$.