\section*{}} 

\begin{document}
\normalem

\title{Dichotomy of heavy and light pairs of holes in the $t-J$ model}


\author{A. Bohrdt}
\email[Corresponding author email: ]{bohrdt@fas.harvard.edu}
\address{ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA}
\affiliation{Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA}

\author{E. Demler}
\affiliation{Institut f\"{u}r Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland}

\author{F. Grusdt}
\affiliation{Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC), Ludwig-Maximilians-Universit\"at M\"unchen, Theresienstr. 37, M\"unchen D-80333, Germany}
\affiliation{Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M\"unchen, Germany}




\date{\today}

\begin{abstract}

 A key step in unraveling the mysteries of materials exhibiting unconventional superconductivity is to understand the underlying pairing mechanism. While it is widely agreed upon that the pairing glue in many of these systems originates from  antiferromagnetic spin correlations \cite{Scalapino1999,Moriya2003,OMahony2022}, a microscopic description of pairs of charge carriers remains lacking.
 Here we use state-of-the art numerical methods to probe the internal structure and dynamical properties of pairs of charge carriers in  quantum antiferromagnets in four-legged cylinders. Exploiting the full momentum resolution in our simulations, we are able to distinguish two qualitatively different types of bound states: a highly mobile, meta-stable pair, which has a dispersion proportional to the hole hopping $t$, and a heavy pair, which can only move due to spin exchange processes and turns into a flat band in the Ising limit of the model. 
Understanding the pairing mechanism can on the one hand pave the way to boosting binding energies in related models \cite{Bohrdt2022_bilayer}, and on the other hand enable insights into the intricate competition of various phases of matter in strongly correlated electron systems \cite{Scalapino2012,Qin2019}. 


\end{abstract}

\maketitle


\textbf{Introduction.--}
Following the discovery of high temperature superconductivity in the cuprates, understanding the mechanism by which pairs of charge carriers can form in a system with repulsive interactions has been a key question in the field. 
Motivated by experimental results on the cuprate materials, a lot of theoretical and numerical work has focused on identifying the potential pairing symmetry \cite{Scalapino1995,Tsuei2000} as well as the binding energies in these microscopic models \cite{Chernyshev1998,Leung2002}. 
Despite a vast research effort over several decades, the existence of a superconducting phase in the simplest model describing interacting electrons, the Fermi-Hubbard model, remains debated \cite{Qin2019}. 
Competing orders, such as charge density waves and stripes, contribute to the difficulty in realizing as well as understanding superconductivity \cite{Scalapino2012}. 
In order to unravel the competition between different orders, and thus the conditions for the existence of a superconducting phase, it is essential to gain a deeper understanding of the nature of pairs of charge carriers.
The existence of pairs close to half-filling does not imply that for a finite density of holes, the system necessarily realizes a $d$-wave paired state. Instead, a finite number of charge carriers can for example self-organize into a charge or pair density wave state \cite{Fradkin2015}. However, understanding whether and how pairs form in the two-hole problem is crucial to the subsequent understanding of self-organization of many holes.


Here we approach the question of the underlying binding mechanism from a new perspective: through novel spectroscopic tools, we search for bound states of charge carriers in a quantum antiferromagnet and directly probe their internal structure.
In particular, we numerically simulate rotational two-hole spectra, where different angular momenta can be imparted on the system, using time-dependent matrix product states. Crucially, these rotational spectra go beyond the standard pairing correlations through the momentum resolution they provide. The momentum dependence of the peaks in the spectral function enables direct insights into the effective mass of the pairs, which is an essential property for understanding their ability to condense at finite doping and temperature.

We study pairing between holes doped into the two-dimensional $t-J$ model, which corresponds to the enigmatic Fermi-Hubbard model to second order in $t/U$ (up to next-nearest neighbor hopping terms, where $U$ is the on-site interaction) and describes electrons in cuprates \cite{Zhang1988}:
    \begin{multline}
\H_{t-J} = - t ~ \hat{\mathcal{P}} \sum_{\ij} \sum_\sigma \l  \hat{c}^\dagger_{\vec{i},\sigma} \c_{\vec{j},\sigma} + \text{h.c.} \r \hat{\mathcal{P}} + \\
+ J \sum_{\ij} \hat{\vec{S}}_{\vec{i}} \cdot \hat{\vec{S}}_{\vec{j}} - \frac{J}{4} \sum_{\ij} \hat{n}_{\vec{i}} \hat{n}_{\vec{j}},
\label{eqtJModel}
\end{multline}
where $\hat{\mathcal{P}}$ projects to the subspace with maximum single occupancy per site; $\hat{\vec{S}}_{\vec{j}}$ and $\hat{n}_{\vec{j}}$ denote the on-site spin and density operators, respectively. In our numerical simulations, we consider a 40 site long, four-legged cylinder.

In order to probe a possible bound state of two charges, we consider an extension of conventional angle-resolved photoemission spectroscopy (ARPES). In particular, we excite the system by creating not one, but two charges while simultaneously imparting angular momentum on the system. The resulting spectra thus directly contain information about the existence of possible bound states, their ground state energy, as well as their dispersion relation. 
In our numerical matrix product state calculations, we find well-defined peaks in the rotational spectral function for all angular momenta, for spin singlet as well as triplet pairs, and throughout an extended frequency range. 

In order to gain a deeper understanding of the rotational two-hole spectra, we also consider the conceptually simpler $t-J_z$ model, where the $SU(2)$ invariant spin interactions are replaced by Ising type interactions. Without additional spin dynamics, a direct comparison of our numerical results to an effective theory describing pairs of charge carriers bound by strings is possible, yielding excellent agreement in terms of the existence as well as the dispersion of the various bound states we observe. 
In particular, we discover a strongly dispersive low-energy peak, with a dispersion scaling with the hole hopping $t$, as well as completely flat bands at competitive energies. We attribute the flat bands to destructive interference of pairs with $d$-wave symmetry \cite{supp}.

Upon introducing spin dynamics, the flat bands develop into weakly dispersive bands, whereas the $t$-dependent feature remains largely unchanged. We thus discover two qualitatively different kinds of bound states: highly dispersive peaks, including a high energy feature with strong spectral weight in the $s$-wave spectra; and a weakly dispersive band, which has a high amount of spectral weight in the $d$-wave spectra. The dispersion of the latter is determined by the spin coupling $J$. The emergence of a slow time-scale set by $J$ is intuitive in the case of a single hole, which forms a spinon-holon bound state and can thus only move as fast as the spin excitation \cite{Bohrdt2020_dynamics}. In contrast, it is surprising to find a bound state of two holes, which in principle can individually move with a hopping amplitude $t$, but when bound into a pair have a dispersion given by the spin exchange. 

The remainder of this paper is organized as follows. We start by introducing the rotational two-hole spectra. We then discuss results for the $t-J_z$ model, where the SU(2) invariant spin interactions are replaced by Ising-type interactions. We discuss the features found in the numerically obtained spectra in detail and compare to a semi-analytical theoretical description of pairs of charge carriers \cite{Grusdt2022_arxiv}. Finally, we consider the full $t-J$ model.
\\






\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{spectrum_tJ_m0.pdf}
\caption{
\textbf{Rotational spectroscopy of two holes} in a singlet state in the $t-J$ model with $t/J=3$, on a $40\times 4$ cylinder, based on a time evolution up to $T_{max}/J=3$ and bond dimension $\chi=1200$. (a) Sketch of the response probed by the rotational spectrum. (b) The upper, middle, and lower plots correspond to $k_y=0,\pi/2,\pi$ and data is shown as a function of momentum $k_x$ and frequency $\omega/J$ for $m_4=0$; $m_4=2$ exhibits similar features, see \cite{supp}. Gray dashed lines correspond to a cosine dispersion $-2 J \alpha \cos (k_x) + b_J$, black line corresponds to a cosine dispersion $-2 t \alpha \cos(k_x) + b_t $, where $\alpha = 0.33 $ in both cases, $b_J=-11$, and $b_t= -9$. 
}
\label{fig:fig1}
\end{figure}






\textbf{Rotational Spectra.--}
In order to probe the internal structure of pairs of charge carriers, we study rotational spectra.
We define an operator $\hat{\Delta}_{m_4}(\vec{j},\sigma,\sigma')$ that creates a pair of holes on the bonds adjacent to site $\vec{j}$ with discrete $C_4$ angular momentum $m_4=0,1,2,3$ as
\begin{equation}
    \hat{\Delta}_{m_4}(\vec{j},\sigma,\sigma') = \sum_{\vec{i}: \ij} e^{i \frac{\pi}{2} m_4 \varphi_{\vec{i} - \vec{j}}}    \c_{\vec{i},\sigma'} \c_{\vec{j},\sigma},
\end{equation}
with $\varphi_{\vec{r}} = {\rm arg}(\vec{r})$ the polar angle of $\vec{r}$.
In order to annihilate a spin-singlet, we define the singlet pair operator (and similar for triplets) as
\begin{equation}
    \hat{\Delta}_{m_4}^{(s)}(\vec{j}) =  \hat{\Delta}_{m_4}(\vec{j},\uparrow,\downarrow) - \hat{\Delta}_{m_4}(\vec{j},\downarrow,\uparrow).
    \label{eq:singlet}
\end{equation}

The simplest term creating a spin-singlet excitation with discrete angular momentum $m_4$, charge two, and total momentum $\vec{k}$ is directly given by the spatial Fourier transform of the singlet pair operator as
\begin{equation}
\hat{\Delta}_{m_4}^{(s)}(\vec{k}) =  \sum_{\vec{j}} \frac{e^{- i \vec{k} \cdot \vec{j}}}{\sqrt{V}}\hat{\Delta}_{m_4}^{(s)}(\vec{j}) 
\label{eqDefRmkSinglet}
\end{equation}
with volume $V$. The discrete angular momentum $m_4$ is a good quantum number at $C_4$ invariant momenta $\vec{k} = (0,0), (\pi,\pi)$ only. 
Based on this operator, we now consider the rotational Green's function
\begin{equation}
\mathcal{G}_{\rm rot}^{(m_4)}(\vec{k},t) = \theta(t)  \bra{\Psi_0} \hat{\Delta}_{m_4}^{(s) \dagger}(\vec{k},t) \hat{\Delta}_{m_4}^{(s)}(\vec{k},0) \ket{\Psi_0},
\label{eq:rotationalGF}
\end{equation}
which we calculate using time-dependent matrix product states \cite{Kjall2013,Zaletel2015,Paeckel2019}. The corresponding two-hole rotational spectrum, $- \pi^{-1} {\rm Im} \mathcal{G}_{\rm rot}^{(m_4)}(\vec{k},\omega)$, in Lehmann representation is
\begin{equation}
 A_{\rm rot}^{(m_4)}(\vec{k},\omega) =  \sum_{n} \delta \l \omega-E_n+E_0^0 \r  | \bra{\Psi_n} \hat{\Delta}_{m_4}^{(s)}(\vec{k})  \ket{\Psi_0^0} |^2,
 \label{eqDefArot}
\end{equation}
where $\ket{\Psi_0^0}$ ($E_0^0$) is the ground state (energy) of the un-doped system and $\ket{\Psi_n}$ ($E_n$)  are the eigenstates (eigenenergies) with two holes. 

The two-hole rotational spectral function defined above is closely related to the dynamical pairing correlations frequently considered in the literature \cite{Dagotto1990pair,Poilblanc1994,Qin2019},
\begin{equation}
    P(\omega) = \int dt ~e^{i\omega t} \left< \hat{\Delta}_{m_4}^{(s) \dagger}(t) \hat{\Delta}_{m_4}^{(s)}(0) \right>,
\end{equation}
where $    \hat{\Delta}_{m_4}^{(s)} = \sum_{\vec{j}} \hat{\Delta}_{m_4}^{(s)}(\vec{j})$. We, however, consider the full momentum dependence of the pairing correlations, which enables direct insights into the center-of-mass dispersion of pairs of charge carriers. 

The resulting rotational spectra thus directly probe the existence of bound states and their internal structure: If a bound state of two holes with long-lived rotational excitations exists, the rotational spectra should exhibit well-defined coherent peaks. If on the other hand such bound states do not exist, the excitation with the rotational operator $\hat{\Delta}_{m_4}(\vec{k})$ will lead to a broad continuum in the corresponding spectral function.

In Fig.~\ref{fig:fig1}, we show the two-hole spectral function without additional angular momentum, i.e. $m_4=0$ ($s$-wave), for the $t-J$ model for momenta $k_y=0,\pi/2$, and $\pi$ and $0\leq k_x \leq \pi$. We find a well-defined coherent peak at low energies for all momenta, indicating the existence of a bound state. The spectrum furthermore reveals a plethora of different features, including a highly dispersive band (black line) as well as bands with a dispersion proportional to the spin-exchange $J$ (gray dashed lines). At momentum $\mathbf{k}=(\pi,\pi)$, the spectral weight vanishes for all energies since $\hat{\Delta}_{0}^{(s)}(\vec{k}=(\pi,\pi))=0$.

In order to gain a deeper understanding of these intriguing results, we take a step back and analyze the conceptually simpler $t-J_z$ model in the following section.
\\

\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{spectrum_tJz_m0.pdf}
\caption{
\textbf{Two hole rotational spectra in the $t-\rm{XXZ}$ model}
for $t/J_z = 3$ and $J_\perp/J_z=0.1$ and $m_4=0$ ($s$-wave) on a $40 \times 4$ cylinder, based on time evolution up to $T_{max}/J_z = 10$ and bond dimension $\chi=600$. The colormap corresponds to numerical matrix product state simulations of the singlet two-hole rotational spectrum, blue lines are geometric string theory predictions for the position of states (all shifted by $-0.35J$), and the black line is a cosine fit. The upper, middle, and lower plots correspond to $k_y=0,\pi/2,\pi$ and data is shown as a function of momentum $k_x$ and frequency $\omega/J_z$. 
The overall ground state energy for two holes is marked by orange circles for $k_y=0$, and the green dashed line in the top panel corresponds to twice the energy of a single hole.
}
\label{fig:tJz}
\end{figure}
\textbf{The $t-\rm{XXZ}$ model.--}
We now consider a modification of the $t-J$ model, where the SU(2) invariant spin interactions are replaced by in-plane and Ising-type spin interactions with coupling constants $J_\perp$ and $J_z$:
\begin{multline}
\H_{t-\rm{XXZ}} = \sum_{\ij} \left( J_\perp \left(\hat{S}^x_{\vec{i}} \hat{S}^x_{\vec{j}}+\hat{S}^y_{\vec{i}} \hat{S}^y_{\vec{j}}\right)  +J_z\hat{S}^z_{\vec{i}} \hat{S}^z_{\vec{j}} \right) -
\\
- t ~ \hat{\mathcal{P}} \sum_{\ij} \sum_\sigma \l  \hat{c}^\dagger_{\vec{i},\sigma} \c_{\vec{j},\sigma} + \text{h.c.} \r  \hat{\mathcal{P}} - \frac{J_z}{4} \sum_{\ij} \hat{n}_{\vec{i}} \hat{n}_{\vec{j}}  .
\label{eqtJzModel}
\end{multline}
In the limit of $J_\perp \ll J_z$, also called the $t-J_z$ model, the lack of spin dynamics facilitates our theoretical understanding. Experimentally, the anisotropic interactions can for example be realized by employing Rydberg interactions \cite{GuardadoSanchez2020Ry}.





\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{peakPos_full.pdf}
\caption{
\textbf{Strongly dispersive pair state} in the $t-\rm{XXZ}$ model
for $t/J_z=1,2,3$ and $J_\perp/J_z=0.1,1.0$ and $m_4=0$ ($s$-wave). The symbols correspond to the position of the lowest energy peak at $k_y=\pi$ extracted from numerical matrix product state simulations of the singlet two-hole rotational spectrum. Yellow crosses correspond to the isotropic case, $J_z=J_\perp=J$ with $t/J=3$. All data points are shifted vertically to collapse at $k_x=0$. 
Blue dashed lines are geometric string theory predictions for the position of the lowest states. For $t/J_z=2,3$, the lowest two states predicted by the geometric string theory are shown. 
Black line is a cosine fit, $0.62 \cos(k_x) + 0.72$, to the extracted peak positions for $t/J_z=3$, $J_\perp/J_z=0.1$. 
}
\label{fig:peak_pos}
\end{figure}



Remarkably, the two-hole spectral function, Fig.~\ref{fig:tJz}, exhibits a highly dispersive peak with a mass proportional to $1/t$; I.e., we find a long-lived, tightly bound state of two holes, which can move as fast as the hole hopping $t$. This is in stark contrast to the case of a single hole in the same model, which has a very high effective mass $\gg 1/t$ and thus an almost flat dispersion \cite{Grusdt2018tJz}, since it can only move due to Trugman loops \cite{Trugman1988}, which are higher order processes.

In Fig.~\ref{fig:peak_pos}, we further analyze the scaling of the mass of the pair by analyzing the position $\omega_{\rm max}$ of the lowest energy peak at $k_y=\pi$ as a function of $k_x$ for different values of $t/J_z=1,2,3$ and $J_\perp/J_z=0.1$. Note that here, the frequency $\omega_{max}$ is shown in units of the hole hopping $t$. We added a constant to the curves for different values of $t/J_z$ to show  the remarkable agreement in the overall shape of the dispersive peak for different values of $t/J_z$, thus highlighting the scaling with the hole hopping $t$. 

The lowest lying peak for $k_y=0$ and $k_y=\pi/2$ is -- within our numerical resolution -- completely flat. Note that the situation of two unbound, and thus approximately immobile, holes should have a very small matrix element in the spectral function considered here, and therefore cannot account for the pronounced flat band peaks we find in the two-hole spectra. This is further corroborated by the direct comparison of the energy for two holes ($E_{2h}-E_{0h} = -12.08 J_z$, orange circles in top panel Fig.~\ref{fig:tJz}) and twice the energy of a single hole ($2 \cdot (E_{1h}-E_{0h}) = -11.01 J_z$, green dashed line in top panel Fig.~\ref{fig:tJz}). 

In a companion paper \cite{Grusdt2022_arxiv}, we extend the geometric string theory developed for a single hole \cite{Grusdt2018tJz,Grusdt2019} to the case of two charge carriers. In particular, this geometric string theory approach describes the properties of two holes bound together by a string of displaced spins, and thus provides estimates of the energy and dispersion relation of such pair states, see blue lines in Figs.~\ref{fig:tJz} and \ref{fig:peak_pos}. Note that the existence of a state at a given energy does not imply that said state is visible in the spectral function, since the spectral weight, i.e. the overlap with the excitation we consider, can still be zero. 

The geometric string theory correctly predicts the highly dispersive peak, as well as the existence of completely flat bands. The latter have also been predicted by a related theoretical study \cite{Shraiman1988}.
\\







\textbf{Results for the $t-J$ model.--}
In the isotropic case, $J_\perp = J_z$, the strongly dispersive peak remains visible, see Fig.~\ref{fig:fig1} bottom. In Fig.~\ref{fig:peak_pos}, we compare the peak position for the isotropic $t-J$ model with the $t-\rm{XXZ}$ model with $J_\perp/J_z=0.1$ and find that the momentum dependence along the $x$-direction of the lowest lying peak for $k_y=\pi$ is qualitatively very similar between the two cases. This indicates in particular that also in the $t-J$ model, a highly mobile, tightly bound pair state exists. 

\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{peakPos_JperpDep.pdf}
\caption{
\textbf{Weakly dispersive pair} in the $t-\rm{XXZ}$ model
for $t/J_z=3$, $J_\perp/J_z=0.1,0.5,1.0$, and $m_4=2$ ($d$-wave). The symbols corresponds to the position of the lowest energy peak at $k_y=0$ extracted from matrix product state simulations of the singlet two-hole rotational spectrum. 
Dashed lines are a cosine fit to the extracted peak positions with prefactor as indicated in the legend, and an additional offset (not indicated). 
}
\label{fig:peak_pos2}
\end{figure}

The flat bands, particularly visible in the $J_\perp/J_z=0.1$ case at momenta $k_y = 0,\pi/2$, acquire a dispersion approximately proportional to $J_\perp^2/J_z$, as can be seen in the top and middle plot in Fig.~\ref{fig:fig1}. We analyze this behavior in more detail by explicitly extracting the peak position $\omega_{\rm max}$ at $k_y=0$ for $t/J_z=3$ and different values of $J_\perp/J_z$ in Fig.~\ref{fig:peak_pos2}. 

Again, the geometric string theory \cite{Grusdt2022_arxiv} correctly captures the highly dispersive peak with mass proportional to $1/t$ (black lines) discussed in Fig.~\ref{fig:peak_pos}. However, since this theoretical description does not account for spin dynamics, it does not predict the dispersion proportional to $J_\perp^2/J_z$ (gray dashed lines). The corresponding bands in the $t-J_z$ model are flat, as predicted by the geometric string theory. 

\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{spectrum_ky0_m0vs2.pdf}
\caption{
\textbf{Angular momentum dependence} of the rotational two-hole spectra in the $t-J$ model with $t/J=3$ for $m_4=0,2$ (left, right column) and $k_y=0$, obtained from time-dependent matrix product state simulations of the singlet two-hole rotational spectrum.
}
\label{fig:m_dep_tJ}
\end{figure}


\textbf{Different angular momenta.--}
Thus far, we considered the case of $s$-wave pairs, i.e. $m_4=0$ and found a well-defined coherent peak at low energies. 
In Fig.~\ref{fig:m_dep_tJ}, we show the spectral function for the $t-J$ model at $t/J=3$ at momentum $k_y=0$ for $m_4=0,2$, corresponding to $s$- and $d$-wave excitations. 
Note, however, that the $40 \times 4$ cylinder geometry used in our numerical simulations breaks the $C_4$ symmetry, and therefore $m_4=0$ and $m_4=2$ excitations can hybridize.

The spectral weight exhibits a strong dependence on the excitation, see Fig.~\ref{fig:m_dep_tJ}. Before, we have identified two qualitatively different types of bound states: a strongly dispersive band, see Fig.~\ref{fig:peak_pos}, and a weakly dispersive band, see Fig.~\ref{fig:peak_pos2}. By considering different rotational excitations, we find that the weakly dispersive pair, realizing the overall ground state, has most spectral weight for the $d$-wave ($m_4=2$) excitation, whereas for the $s$-wave ($m_4=0$) excitation, a large fraction of the spectral weight appears in a strongly dispersive high energy feature. This high-energy feature is also predicted by the geometric string theory \cite{Grusdt2022_arxiv}. 

The hybridization of different $m_4$ excitations in the cylinder geometry leads to the appearance of the weakly dispersive feature, found with strong spectral weight for $m_4=2$, also at $m_4=0$, albeit with a smaller spectral weight. 
The overall lowest energy peak appears at momentum $\mathbf{k}=(0,0)$ and $m_4=0,2$.

We note that in early exact diagonalization studies on small systems \cite{Dagotto1990pair,Poilblanc1994}, the integration over all momenta, or the consideration of only momentum zero, leads to a much sharper low energy peak in the $d$-wave than in the $s$-wave case. We attribute this to the widely different dispersions of the $s$- and $d$-wave peaks revealed here, and the exact $C_4$ symmetry in small systems.

The geometric string theory correctly predicts the $d$-wave character of the weakly dispersive band, as well as the accumulation of spectral weight at $\vec{k}=(0,0)$ \cite{Grusdt2022_arxiv}. A more detailed comparison of spectral weights for different values of $m_4$ shows good qualitative agreement at low energies (see supplementary material). 




So far, we considered a singlet excitation. In order to annihilate a triplet instead, we define the triplet pair operator as
\begin{equation}
    \hat{\Delta}_{m_4}^{(t)}(\vec{j}) =  \hat{\Delta}_{m_4}(\vec{j},\uparrow,\downarrow) + \hat{\Delta}_{m_4}(\vec{j},\downarrow,\uparrow).
    \label{eq:triplet}
\end{equation}
Upon considering the corresponding triplet spectral function, we  find that the lowest lying peaks are at higher energies than in the case of the singlet spectral function. This finding furthermore suggests that the lowest energy peak in the singlet spectral function cannot be attributed to unbound states of two holes. 
\\

\textbf{Summary and Outlook.--}
In this work, we extensively studied the properties of pairs of charge carriers in the $t-J$ and $t-\rm{XXZ}$ model through rotational spectra. We find well defined coherent peaks at low energies for all momenta and angular momenta $m_4$. Our work provides the first extensive numerical study of the mass of pairs of charge carriers in extended systems. We have revealed two qualitatively different types of bound states: First, a weakly dispersive peak with a dispersion approximately proportional to $J_\perp^2/J_z$, which has most spectral weight for a $d$-wave excitation. And second, a highly dispersive peak, which corresponds to tightly bound pairs with a mass proportional to $1/t$. 
We find the same signatures of these light pairs of charge carriers in the $t-J_z$ and the SU(2) invariant $t-J$ models. The band corresponding to these light pairs, as well as the bands corresponding to heavy pairs, are qualitatively captured by a semi-analytic geometric string theory approach \cite{Grusdt2022_arxiv}.


An intriguing direction for future research is the direct experimental probe of the two-hole rotational states. Understanding the pairing mechanism in the Fermi-Hubbard and related models has been one of the key motivations in the development of quantum simulators, and in particular cold atoms in optical lattices \cite{Hofstetter2002}. In the past two decades, remarkable progress has been made in the field \cite{Tarruell2018,Bohrdt2021_review}, and several proposals to probe the pairing symmetry have been put forward \cite{Rey2009,Kitagawa2011}. More recently, the single-hole spectral function has been measured experimentally \cite{Brown2019,Bohrdt2018}. Using additional lattice modulations, the two-hole rotational spectral function considered here could be accessed experimentally. In solid state experiments, the $s$-wave two-hole spectral function can be accessed through coincidence angle-resolved photo-emission spectroscopy \cite{Su2020}, which relies on simultaneous measurements of two photo-electrons and provides direct insights into the pair Green’s function.
A different approach is based on Anderson-Goldman pair tunneling in a tunnel junction setup \cite{Anderson1967a,Anderson1972,Scalapino1970}: to study the structure of individual pairs in a strongly underdoped quasi-2D material as considered here, we propose to tunnel-couple the latter to a probe-superconductor along $z$-direction. Momentum resolution can in this case be obtained through an in-plane magnetic field.


The observation of light as well as heavy pairs in the spectra shown here furthermore suggests a real-space and -time experiment. 
Upon slowly releasing two holes next to each other in a cold atom experiment, a low energy state of the pair can be prepared. In the ensuing time evolution, we predict the pairs to spread through the system in two distinct wave-fronts, corresponding to the light and the heavy pair, respectively. 

Finally, the existence of flat or weakly dispersive bands opens a new avenue to understand the many competing orders found experimentally in cuprate materials as well as numerically in Fermi-Hubbard and $t-J$ models at finite doping \cite{Corboz2014,Qin2019,Arovas2022}. In a next step, we will investigate the dichotomy between two types of light and heavy pairs in the Fermi-Hubbard model.



\textbf{Acknowledgements.--}
We thank Immanuel Bloch, Antoine Georges, Markus Greiner, Mohammad Hafezi, Lukas Homeier, and Ulrich Schollw\"ock for fruitful discussions. This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -- EXC-2111 -- 390814868, by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programm (Grant Agreement no 948141) — ERC Starting Grant SimUcQuam, by the ARO (grant number W911NF-20-1-0163), and by the NSF through a grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University and the Smithsonian Astrophysical Observatory.



\section*{References}
