Anomalous Eigenstates of a Doped Hole in the Ising Antiferromagnet

Abstract

The problem of a mobile hole doped into an antiferromagnet Mott insulator is believed to underly the rich physics of several paradigmatic strongly correlated electron systems, ranging from heavy fermions to high-Tc superconductivity. Arguably the simplest incarnation of this problem corresponds to a doped Ising antiferromagnet, a problem widely considered essentially solved since almost 60 years by a popular yet approximate mapping to a single-particle problem on the Bethe lattice. Here we show that, despite its deceptive simplicity, the local spectrum of a single hole in a classical Ising-Néel state contains a series of anomalous, long-lived states that go beyond the well-known ladder-like spectrum with excited energies spaced as $J^{2/3} t^{1/3}$. The anomalous states we find through exact diagonalization and within the self-avoiding path approximation have excitation energies scaling approximately linear with $J$ and lead to a series of avoided crossings with the more pronounced ladder spectrum. By also computing different local, rotational spectra we explain the origin of the anomalous states as rooted in an approximate emergent local $C_3$ symmetry of the problem. From their direct spectral signatures we further conclude that these states lead to anomalously slow thermalization behavior -- hence representing a new type of quantum many-body scar state, potentially related to many-body scars predicted in lattice gauge theories.

Figures (25)

• Figure 1: We compute the local spectral function $A(\omega)$ of a single hole introduced to the $t$-$J^z$ model for different strengths of the spin interaction $J/t \in [0, 1]$ of the $t$-$J^z$ model. Energies $\omega$ are measured relative to the ground state (lowest-energy peak). Calculations were performed using the self-avoiding walks approximation ( cf. Wrzosek2021 and text for further details). (a) On the Bethe lattice, where an exact analytical solution is possible, $A(\omega)$ consists only of the ladder spectrum. The corresponding peaks ($nS$) can be labeled by their vibrational quantum number $n=1,2,...$ and have no angular momentum ($S$-wave states). On the square lattice, but neglecting magnon-magnon interactions, (b), a new set of anomalous states is found in this work, which lead to avoided level crossings with the more pronounced ladder states. We show in this article that this new set of states $nSx$ can be labeled by an approximate $C_3$ angular momentum $x$. Importantly, both the ladder states and the new anomalous peaks remain clearly visible, on top of an emerging incoherent background, when magnon-magnon interactions on the square lattice are included (c). Dashed lines on top of each spectrum represent the locations of the most pronounced low lying vibrational and rotational modes of the hole identified throughout the paper, using rotational spectral functions which can also resolve states with non-zero $C_4$ lattice angular momentum (e.g. the series $n\tilde{x}$ with $\tilde{x}=P,D,F$). Under idealized conditions in a), rotational excitations have zero spectral weight and dashed lines only serve as a reference. In b), c), dashed lines help identify avoided crossings between different ro-vibrational states.
• Figure 2: Cartoon picture of an initial state with (a) $l = 0$, (b) first-generation $l = 1$, and (c) second generation $l = 2$ rotational excitations. Gray circle represents a hole while red square (with rounded corners) stands for a magnon (string). The hole delocalized on multiple sites is denoted with dashed edges -- it also means a sum over multiple configurations of the hole and magnons on the lattice: (b) 4 configurations and (c) 12 configurations. Yellow area covers the corresponding sites in panels (b) and (c). In panel (c) only one of four rotations around the origin is shown. Varying $m^{(l)}$ yields different rotational states.
• Figure 3: Cartoon picture of a square lattice with denoted direction dependent phase factors $\varphi$ of the Fourier transform used in the definition of the operator $\hat{R}_{\sigma,M_l}(\bm{i})$. The hole is introduced at site $\bm{i}$. We consider cases where the hole can be initially propagated up to $l=2$ times: $\mathrm{(a)}$ for first-generation excitations $l=1$ the hole acquires a phase $\varphi_{\bm{j_1}-\bm{i}} \in \{0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\}$; the values wind anticlockwise, where $\varphi_{\bm{j_1}-\bm{i}} = 0$ corresponds to the positive $x$ direction from site $\bm{i}$; $\mathrm{(b)}$ for second-generation excitations $l=2$ the hole acquires a phase $\varphi_{\bm{j_1}-\bm{i}} \in \{0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\}$ and then phase $\varphi_{\bm{j_1}-\bm{i}, \bm{j_2}-\bm{j_1}} \in \{0, \frac{2\pi}{3}, \frac{4\pi}{3}\}$. Since we consider forward propagation of the hole (no returns), there are exactly three possible directions for the propagation of the hole after the first hop. Note that the choice of phases $\varphi_{\bm{j_1}-\bm{i}, \bm{j_2}-\bm{j_1}}$ preserves the $C_4$ symmetry around site $\bm{i}$.
• Figure 4: Different types of hole paths in the AF square lattice. Panels (a) and (b) show examples of non-crossing paths -- these type of paths are included in the SAW approximation Wrzosek2021. Note that this also includes a tangential path, where the bond at which the path becomes tangential is highlighted in yellow. Panels (c) and (d) show examples of paths with loops Tru88 -- those are excluded from the considerations in our calculations.
• Figure 5: Dependence of the spectral function $A(\omega)$ (zeroth generation) of a single hole in the Ising antiferromagnet on the coupling constant $J/t$. Results obtained without the magnon-magnon interactions and: (a) on the square lattice and (b) on the Bethe lattice.