Two-channel physics in a lightly doped antiferromagnetic Mott insulator revealed by two-hole spectroscopy

Abstract

Understanding pairing in the strong-coupling regime of doped Mott insulators remains an open problem in the context of cuprate superconductors. We perform ultra-high resolution numerical simulations of spectral functions in the highly underdoped $t-J$ model and discover two coupled branches of hole pairs emerging at low energies in the largely unexplored two-particle spectrum. As spin anisotropy is tuned from the Ising limit to the $SU(2)$-symmetric Heisenberg regime, the lowest $d$-wave pair evolves from a single bipolaronic branch into two hybridized branches separated by an avoided crossing. We explain this behaviour using an effective two-channel model involving a tightly bound bipolaronic state and a second channel associated with two magnetic polarons. The model reproduces the qualitative low-energy spectra and implies near-resonant $d$-wave interactions in the $SU(2)$-symmetric $t-J$ model, consistent with proximity to an emergent Feshbach-type resonance. To probe these predictions experimentally, we propose a Raman spectroscopy scheme for the attractive Hubbard model that can be directly implemented using ultracold atoms in optical lattices. Our work establishes two-particle spectroscopy, beyond single-particle Green's functions, as a powerful tool for revealing the microscopic origins of unconventional superconductivity.

Figures (9)

• Figure 1: Signatures of two-channel physics in the two-hole rotational spectrum with $d$-wave symmetry, $A^{(2)}(\mathbf{k},\omega)$, starting from an initially undoped Heisenberg , computed at fixed $k_y=\pi/2$ using MPS Bohrdt2023. Dashed lines are guides to the eye tracking local maxima in the low-energy part of the spectrum to highlight the two hybridized branches we find. The inset illustrates the effective two-channel model Homeier2025 and its ingredients -- magnetic polaron (sc) and bipolaron (cc) channels -- which we use to explain the observed level splitting. These numerical simulations were performed in Ref. Bohrdt2023 on a $40 \times 4$-cylinder, at $t/J=3$, using time-dependent Zaletel2015.
• Figure 2: Emergence of two paired branches in the two-hole spectrum $A^{(2)}(\mathbf{k}, \omega)$ of the $t-J$ model. In a), b) we consider an easy-axis , $J_\perp = 0.1 J$ ($J_z=J$). The full numerical spectrum Bohrdt2023 shown in b) is in excellent qualitative agreement with the effective single-channel model a), based on two holes connected by a string of displaced spins. In particular, all observed branches -- labeled ① - ③ -- are found to be in one-to-one correspondence. The effective model in a) describes only the tightly-bound bipolaronic pairing channel, without coupling to the magnetic polarons; it has been artificially broadened for better comparison. In c) we increase $J_\perp = J_z =J$ and show spectra in the $SU(2)$ invariant $t-J$ model Bohrdt2023, which we compare to ultra-high resolution spectra obtained by the CTKSCTKS approach described in the methods (inset). The lowest pair resonance, ①, is found to split into two branches, ①a and ①b -- indicating the presence of an additional paired eigenstate hybridizing with the bipolaronic hole pair. We work at fixed $k_y=\pi/2$ and in the d-wave channel, $m_4 = 2$, at $t/J=3$.
• Figure 3: Emergent scattering resonance between low-energy two-hole states. a) We perform high-resolution simulations of the two-hole spectrum $A^{(2)}(\mathbf{k}, \omega)$ on a $40 \times 4$ cylinder using the method. We fix $\mathbf{k}=(0,\pi/2)$, $t/J=3$ and tune $J_\perp / J_z$, indicated by baseline offsets of the curves along the $y$-axis. Around $J_\perp/J_z \approx 0.8$, the shoulder of the lowest-energy peak develops into an avoided level crossing of two paired states, highlighted by small arrows. Energies $\Delta \omega$ are plotted with respect to the lowest quasi-particle peak, and spectra are normalized to their maximal value. The blue dots mark some of the peaks to highlight the kink of the higher peak positions. b) The avoided level crossing is captured by an effective two-channel model, showing only the lowest-lying branches. The single fit parameter, $\Delta E$, in the two-channel model is obtained by comparison to the simulations in a). These results can be interpreted as an emergent Feshbach resonance between two branches of paired states: In panel c), we show the ratio of the coupling, $V$, between the two channels, and $\Delta E$, the fitted bare energy offset of the effective two-channel model. $-|V|^2/\Delta E \propto g_d$ is proportional to the effective Feshbach interaction between two magnetic polarons, mediated by the coupling to the tightly-bound bipolaronic pair. The shaded region marks the strong coupling regime, i.e., the location of the emergent Feshbach resonance, where the hybridization $V>2\Delta E$ exceeds the offset.
• Figure 4: Comparison of the d-wave ($m_4 = 2$) two-hole spectrum $A^{(2)}(\mathbf{k},\omega)$ computed numerically, a), to the effective two-channel model, b). We study the $SU(2)$ invariant $t-J$ model ($J_\perp = J_z=J$) at $k_y = \pi/2$ as a function of $k_x$. In a) we performed high-resolution - simulations on a $40 \times 4$ cylinder. For the two-channel model calculations in b) we used the fit parameter $\Delta E = 0.15 J$. We indicate the lower edge of the two-hole scattering continuum (dashed) and the dispersion of the tightly-bound (cc) pair (dotted).
• Figure 5: Two-hole spectral functions $A^{(2)}(\mathbf{k},\omega)$ evaluated at $\mathbf{k}=(0,\frac{\pi}{2})$ and d-wave symmetry $m_4=2$ as a function of $\omega$ and $J_\perp$. Using a complex-time Krylov space augmentation, we increased the frequency resolution by an order of magnitude compared to standard methods employing a real-time evolution with subsequent Fourier transformation into frequency space Paeckel2024.