Hole in the two-dimensional Ising antiferromagnet: Origin of the incoherent spectrum

We develop a “self-avoiding walks” approximation and use it to calculate the spectral function of a single hole introduced into a two-dimensional square lattice Ising antiferromagnet. The local spectral function obtained qualitatively agrees with the exact diagonalization result and is largely incoherent. Such a result stays in contrast with the spectrum obtained on a Bethe lattice, which consists of the well-separated quasiparticle-like peaks and stems from the motion of a hole in an effective linear potential. We determine that this onset of the incoherent spectrum on a square lattice (i) is not triggered by the so-called Trugman loops but (ii) originates in the warping of the linear potential by the interactions between magnons created along the tangential paths of the moving hole.

Figure 1

Cartoon picture of the motion of a hole (due to tunneling of electrons) in the ground state of an Ising antiferromagnet on a square lattice. Two antiferromagnetic sublattices are colored in light blue and gray; electrons may jump over blue and green bonds; red arrows depict spins with unsatisfied bonds and misaligned with respect to the ground state as a result of the hole motion. Top panels show the creation of one misaligned spin at each hole hop. Middle and bottom panels show a partial or full reconstruction of the antiferromagnetic order either by moving along a path tangential to itself (subfigures $4A\to 5A\to 6A\to 7A$; as discussed in this paper) $\mathit{or}$ by going around a loop (panels $4B\to 5B\to 6B\to 7B$; as discussed by Trugman in Ref. [5]).

Figure 2

Cartoon picture of the hole motion on a square lattice in terms of holes $\left(h_i^{†}{h}_i\right)$ and magnons $\left(a_i^{†}{a}_i\right)$. Blue squares represent the hole, red squares represent magnons created by the hole, gray squares represent empty sites. Within the self-avoiding walks approximation, propagation via the dashed bond is forbidden. In addition, panels (d) and (e) show a satellite hole-magnon proximity interaction (blue dashed bond) and a satellite magnon-magnon interaction (red dashed bond), both are not possible on the Bethe lattice (cf. Fig. 3). See main text for further details.

Figure 3

Cartoon picture of the hole motion on a Bethe lattice with the coordinate number $z=4$ in terms of holes $\left(h_i^{†}{h}_i\right)$ and magnons $\left(a_i^{†}{a}_i\right)$. Symbols are the same as in Fig. 2. Note the lack of the satellite bonds (no dashed bonds) in contrast with Fig. 2. See main text for further details.

Figure 4

Graph of the reachable states on the Bethe lattice case belonging to the corresponding sets $S_n$. The initial state ${\psi }_0^{\left(0\right)}$ is shown in Fig. 3. The red arrows correspond to the three hopping processes shown in Figs. 3–3. The potential energy of the group of states corresponding to a certain walk (and its symmetries) is denoted next to each node. States with the same number of magnons are indistinguishable. Therefore, a tree-like structure connecting reachable states on the Bethe lattice can be mapped onto a chain.

Figure 5

Graph of the reachable states on the square lattice belonging to the corresponding sets $S_n$ within the self-avoiding walks approximation. Initial state ${\psi }_0^{\left(0\right)}$ is shown in Fig. 2. Red arrows correspond to the three hopping processes shown in Figs. 2–2. The potential energy of the group of states corresponding to a certain walk (and its symmetries) is denoted next to each node. States with the same number of magnons are distinguishable, i.e., they may have different energy. A tree-like structure connecting reachable states cannot easily be mapped onto a chain.

Figure 6

Spectral function $A\left(\omega \right)$ of a single hole in the $t-J_z$ model on the 2D square lattice calculated using self-avoiding walks approximation (blue) and ED (red) for $J=0.4t$ (top left), $J=0.6t$ (top middle), $J=0.8t$ (top right), $J=1.2t$ (bottom left), $J=1.6t$ (bottom middle), and $J=2.0t$ (bottom right). The self-avoiding walks approximation includes up to 20 magnons and ED calculations (red) are performed on a 26-site lattice with periodic boundary conditions. Broadening $\delta =0.05t$.

Figure 7

Correlation $\xi \left(J\right)$ between the spectral functions $A\left(\omega \right)$ calculated on a square and the Bethe lattice within the self-avoiding walks approximation with the ED result on a 26-site square lattice. Note that the self-avoiding walks approximation is an exact method on the Bethe lattice.

Figure 8

Spectral function $A\left(\omega \right)$ of a single hole in the $t-J^z$ model on the Bethe lattice with the coordinate number $z=4$ (left) and the square lattice (right). Results are obtained using the self-avoiding walks approximation with $J=0.4t$ and broadening $\delta =0.05t$.

Figure 9

Spectral function $A\left(\omega \right)$ of a single hole in the $t-J^z$ model on the square lattice. Left (right) panel shows results without (with) magnon-magnon interactions correctly included in the model Hamiltonian. Darker (lighter) lines depict results with (without) the hole-magnon proximity interaction properly included. All results are obtained using self-avoiding walks approximation with $J=0.4t$ and broadening $\delta =0.05t$.

Figure 10

Spectral function $A(k,\omega )$ of a single hole in the $t-J^z$ model on the square lattice. Results obtained using exact diagonalization (ED) on (a) a 26-site and (b) a 20-site square lattice with $J=0.4t$ and broadening $\delta =0.05t$.