Single-hole wave function in two dimensions: A case study of the doped Mott insulator

We study a ground-state ansatz for the single-hole-doped $t\text{−}J$ model in two dimensions via a variational Monte Carlo method. Such a single-hole wave function possesses finite angular momenta generated by hidden spin currents, which give rise to a novel ground-state degeneracy in agreement with recent exact diagonalization (ED) and density matrix renormalization group (DMRG) results. We further show that the wave function can be decomposed into a quasiparticle component and an incoherent momentum distribution in excellent agreement with the DMRG results up to an $8\times 8$ lattice. Such a two-component structure indicates the breakdown of Landau's one-to-one correspondence principle, and in particular, the quasiparticle spectral weight vanishes by a power law in the large sample size limit. By contrast, turning off the phase string induced by the hole hopping in the so-called $\sigma ·t\text{−}J$ model, a conventional Bloch-wave wave function with a finite quasiparticle spectral weight can be recovered, also in agreement with the ED and DMRG results. The present study shows that a singular effect already takes place in the single-hole-doped Mott insulator, by which the bare hole is turned into a non-Landau quasiparticle with translational-symmetry breaking. Generalizations to pairing and finite doping are briefly discussed.

Figure 1

(a) Schematic illustration of the single-hole wave function ansatz [Eq. (3)], in which the mutual entanglement between a hole and surrounding spins is explicitly characterized by the phase string operator [Eqs. (4) and (5)]. Such a single-hole ground state can be labeled by the following quantum numbers: hole number $N_{\text{h}}=1$, spin $S=1/2,S_z=\pm 1/2$, and for a lattice with a discrete $C_4$ rotational symmetry, an angular momentum $L_z=\pm 1$ with nontrivial spin and hole currents, $J_s$ and $J_h$, in agreement with the ED and DMRG results [29]. (b) The quasiparticle spectral weight of the ground state (3) shows four sharp peaks at momenta $(\pm \pi /2,\pm \pi /2)$ at a finite-size system $\left(N=16\times 16\right)$.

Figure 2

Momentum distribution of the hole, $n^h\left(\mathbf{k}\right)$, and the quasiparticle weight $Z_{\mathbf{k}}$, as calculated with VMC in the ground-state ansatz of Eq. (3): (a) and (b) show excellent agreement with the DMRG on an $8\times 8$ lattice; (c) and (d) show a larger size at $16\times 16$. The finite-size scaling results obtained with VMC are presented in Fig. 3.

Figure 3

The scaling analysis of the momentum distribution $n^h\left(\mathbf{k}\right)$ and the quasiparticle weight $Z_{{\mathbf{K}}^0}$. (a) The broad background part of $n^h\left(\mathbf{k}\right)$ scales with the inverse of $L^2$ in the square sample $N=L\times L$. (b) The quasiparticle weight $Z_{{\mathbf{K}}^0}$ at the peak momentum ${\mathbf{K}}^0=(\pm \frac{\pi }{2},\pm \frac{\pi }{2})$ vanishes in a power-law fashion by $L^{-1.31}$.

Figure 4

(a) The kinetic energies of the Bloch-like states $|{\mathrm{\Psi }}_{\mathrm{Bloch}}{\left(\mathbf{k}\right)\rangle }_{1\mathrm{h}}$ [Eq. (7)] and $|{\tilde{\mathrm{\Psi }}}_{\mathrm{Bloch}}{\left(\mathbf{k}\right)\rangle }_{1\mathrm{h}}$ [Eq. (17)] as a function of momentum vs that of the variational ground state $|{\mathrm{\Psi }}_{\mathrm{G}}{\rangle }_{1\mathrm{h}}$ in Eq. (3) (dashed line). Note that the hole wave function ${\phi }_{\mathrm{h}}\left(i\right)$ determined variationally in Eq. (3) is not translationally invariant as shown in (b). (b) The distribution of the absolute value of ${\phi }_{\mathrm{h}}\left(\mathbf{k}\right)$ as the Fourier transformation of ${\phi }_{\mathrm{h}}\left(i\right)$ in the momentum space.

Figure 5

The neutral spin current $\left(J_{ij}^s\right)$ pattern surrounding the hole in one of the degenerate ground states with $L_z=1$ in a lattice of $N=8\times 8$ with OBC. (The red vertical and horizontal dashed lines mark the bonds with vanishing spin currents.) The dashed blue closed loops and numbers indicate the Wilson loops that count the Berry phase defined in Eq. (23).

Figure 6

The charge (hole) current $\left(J_{ij}^h\right)$ pattern surrounding an $\uparrow$ spin projected onto a given lattice site in a degenerate ground state with $L_z=1$ in a lattice of $N=8\times 8$ with OBC. The dashed red closed loops and numbers indicate the Wilson loops counting the Berry phase defined in Eq. (24).

Figure 7

The propagation amplitude of a bare hole, ${\mathcal{C}}_{ij}$, is much reduced as compared to that of the “twisted” hole, ${\mathcal{D}}_{ij}$, in the ground-state ansatz (3). The latter is comparable to the hole propagation ${\mathcal{C}}_{ij}^0$ for a Bloch-wave state defined in Eq. (7) (with momentum ${\mathbf{K}}^0)$. Here the spatial distance $r_{ij}=|x_i-x_j|+|y_i-y_j|$, and ${\mathcal{C}}_{ij}$ and ${\mathcal{D}}_{ij}$ are calculated by averaging over all the lattice sites with each given distance $r_{ij}$ based on the definitions given in Eq. (29) and Eq. (31), respectively. The lattice size is $20\times 20$.

Figure 8

Hole momentum distribution $n^h\left(\mathbf{k}\right)$ and quasiparticle weight $Z_{\mathbf{k}}$ for the $\sigma ·t\text{−}J$ model. Differently from the $t\text{−}J$ model in Fig. 2, $n^h\left(\mathbf{k}\right)$ only has single peak located at the symmetry point ${\mathbf{K}}_0=(\pi ,\pi )$ with null spin currents.