## Spin and charge go their separate ways

Strongly interacting chains of fermions are predicted to exhibit two types of collective excitations: spinons, which carry only spin, and holons, which carry only charge. These excitations move at different velocities. Signatures of this so-called spin-charge separation have been observed in solid-state systems, but obtaining direct dynamical evidence is tricky. With this goal in mind, Vijayan *et al.* perturbed a chain of ultracold interacting fermions housed in a one-dimensional optical lattice by removing one of the atoms. This gave rise to two independent excitations, which the researchers identified as spinons and holons using a quantum gas microscope.

*Science*, this issue p. 186

## Abstract

Elementary particles carry several quantum numbers, such as charge and spin. However, in an ensemble of strongly interacting particles, the emerging degrees of freedom can fundamentally differ from those of the individual constituents. For example, one-dimensional systems are described by independent quasiparticles carrying either spin (spinon) or charge (holon). Here, we report on the dynamical deconfinement of spin and charge excitations in real space after the removal of a particle in Fermi-Hubbard chains of ultracold atoms. Using space- and time-resolved quantum gas microscopy, we tracked the evolution of the excitations through their signatures in spin and charge correlations. By evaluating multipoint correlators, we quantified the spatial separation of the excitations in the context of fractionalization into single spinons and holons at finite temperatures.

Strongly correlated quantum many-body systems often exhibit behaviors that cannot be attributed to the properties of the individual particles. Instead, the collective nature of the excitations can lead to the emergence of quasiparticles that are fundamentally distinct from free electrons. For example, in one-dimensional (1D) quantum systems, electron-like excitations do not exist but are replaced by decoupled collective spin and charge modes (*1*). These two independent excitation branches feature different propagation velocities (*2*) and have previously been studied in the Luttinger liquid regime (*3*) of quasi-1D solids using spectroscopic techniques, such as angle-resolved photoemission spectroscopy (*4*–*6*) and conductance measurements in metallic quantum wires (*7*–*9*). Cold-atom experiments have been used extensively to study attractive 1D bosonic and fermionic gases (*10*–*15*), but the investigation of repulsive 1D fermionic gases has been more recent (*16*–*18*). Trapping fermionic spin mixtures in optical lattices has enabled a clean and well-controlled realization of the 1D Fermi-Hubbard model. Being an exactly solvable paradigmatic model (*19*–*21*) for strongly correlated electrons, this has allowed for quantitative comparisons between theory and experiments. Such experiments can probe the regime that lies in between the low-energy Luttinger liquid and the spin-incoherent Luttinger liquid; for the latter, the temperature is on the order of or exceeds the magnetic energy (*22*). Recent equilibrium signatures of spin-charge separation have been observed in ultracold lattice gases using quantum gas microscopy (*23*, *24*). However, real-space tracking of the dynamics of the individual excitations signaling their deconfinement has been more challenging to accomplish.

Here, we demonstrate dynamical spin-charge separation directly by performing a local quench in a 1D gas of ultracold fermionic atoms and subsequently monitoring the evolution of the system with spin- and density-resolved quantum gas microscopy (*18*) (see Fig. 1). The local quench is realized by the high-fidelity removal of one atom from a single site of a 1D optical lattice; the lattice is initially filled with nearly one atom per site, and the system exhibits short-range antiferromagnetic spin correlations (*18*, *25*–*27*). In the subsequent dynamics, we observed the emergence of two apparently independent excitations propagating at different velocities (*28*–*30*), which we assigned to spinons and holons on the basis of their characteristic signatures in the spin and charge (density) sectors.

Our experiment (*24*) began by loading a 2D balanced spin mixture of ^{6}Li atoms in the lowest two hyperfine states into several 1D tubes using an optical lattice of spacing *a _{y}* = 2.3 μm along the

*y*direction. Next, a lattice of spacing

*a*= 1.15 μm was ramped up along the

_{x}*x*direction. By varying the hopping strength along the

*x*direction from

*t*/

*h*= 190 Hz to

*t*/

*h*= 410 Hz, we realized Fermi-Hubbard chains with

*U*/

*t*~ 8 to 20, where

*U*is the onsite interaction energy,

*t*is the tunneling energy, and

*h*is Planck’s constant. We fixed the total atom number in the gas to around 75 through the choice of the evaporative cooling parameters, such that the resulting Hubbard chains were prepared close to half-filling in the center of the harmonically confined cloud. This produced at least three 1D chains of mean length 13 atoms, each with a unity-filled region of about nine sites. To perform a local quench in which a single atom is simultaneously removed from each chain, we used an elliptically shaped near-resonant laser beam at 671 nm focused to a waist of ~0.5 μm along its narrow direction. This pushout beam was pulsed on for 20 μs, addressing the central sites. The power and alignment of the pushout beam was adjusted such that the probability of spin-independent removal of an atom from the addressed site was ~78%, with ~14% chance of affecting the nearest neighboring sites (

*31*). After the quench, we let the system evolve for a variable hold-time before imaging the spin and density distributions. To collect statistics, the experiment was repeated several thousand times for a given evolution time.

We first investigated the difference in the dynamics of holons and spinons by preparing 1D Hubbard chains with *t* = *h* × 250 Hz and *U*/*t* = 15, corresponding to an exchange interaction of *J* = *h* × 65 Hz, where *J* is the spin-exchange energy, and then performing the local quench. A natural observable to characterize the subsequent dynamics of holons is the spatially resolved hole density distribution *i* labels the lattice sites. The observed distribution broadens as a function of time with a light-cone-like ballistic propagation of the wavefront (see Fig. 2A). It starts from the addressed site and reaches the edge of the unity-filled region of the chain in

To study the time evolution of the spin excitation, we measured nearest-neighbor spin correlations *23*, *24*). For strong interactions *32*–*34*) to which we compared our results. We observed a strong reduction of the antiferromagnetic correlations in the direct vicinity of the quenched site immediately after the quench, demonstrating an enhanced probability of finding parallel spins on neighboring sites. Such a suppression was expected from the creation of spinons by the local quench (see Fig. 1A). The region with reduced antiferromagnetic correlations spread with time, with a light-cone-like propagation of the wavefront (see Fig. 2C). It reached the edge of the unity-filled region in *k*_{B}*T*/*J* ~ 0.75, where *k*_{B} is the Boltzmann constant and *T* is temperature, in our system prevented us from observing any interference effects in the spin dynamics. However, the observed ballistic wavefront was still expected from the Heisenberg model at our temperatures (*31*, *35*, *36*).

Next, we extracted the velocities of the spin and charge excitations emerging from the quench. We monitored the spatial width of the squeezed space correlator *U*/*t* = 15, we found a ratio of 5.31 ± 0.43 between the two propagation velocities, indicating a large difference in the velocities of the two excitation channels. Despite the finite nonzero temperature in our system, the extracted velocities are in excellent agreement with both exact diagonalization results of an extended *t* − *J* model (*31*) as well as a single-particle quantum walk for a hole and a Heisenberg model prediction at our temperature for the spin excitations.

To investigate the scaling of the velocities with the tunneling and spin-exchange energies *t* and *J*, we repeated the experiment for different *U*/*t* by tuning the lattice depth. Within our experimental uncertainties, we found the extracted velocities to be in good agreement with the maximum expected group velocities

An essential feature of spin-charge deconfinement is the existence of unbound states of spin and charge excitations, allowing them to spatially separate over arbitrary large distances. To quantify the dynamical deconfinement, we studied the spin correlations across the propagating hole as a function of time, through the spin-hole-spin (SHS) correlator *i* + 1 (*23*, *24*) (Fig. 4A). Immediately after the quench, the hole is likely to be surrounded by parallel spins, and *C*_{SHS} retains a positive value. The measured spin correlations are consistent with the next-nearest-neighbor correlations *C*_{SHS} becomes negative and, by *31*). The absence of binding between the spin and charge excitations beyond the immediate vicinity of the hole is shown by calculating the normalized deviation from the mean nearest-neighbor correlations *d* is the distance of the hole from the closest of sites *i* and *i* + 1, ● indicates occupied sites, and ○ indicates the position of the hole. δ*C*_{1} shows no dependence on *d*, indicating the lack of influence of the holon on the spin excitation.

To locate the excess spin excitation in a fluctuating spinon background, we introduced an operator quantifying the local spin fluctuations in squeezed space *37*). A single spinon located at site

To study the spatial separation of the spin and charge excitations, we considered the chains at time *k*_{B}*T* = 0.75*J*, taking into account our quench efficiency (*31*).

An interesting extension of this work would be to study spin-charge confinement dynamics in the dimensional crossover from 1D to 2D, where polaronic signatures were recently observed (*24*, *38*). The protocol used here could be directly implemented to extract the effective mass of a polaron. Our work opens avenues to dynamically probe the doped Fermi-Hubbard model in higher dimensions and explore fractionalization in topological phases of matter.

## Supplementary Materials

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We thank G. Baskaran, E. Demler, T. Giamarchi, R. Moessner, and R. Shankar for useful discussions. P.S. acknowledges support from the Development and Promotion of Science and Technology Talents Project (DPST) of Thailand.

**Funding:**We acknowledge funding by the Max Planck Society (MPG), the European Union (UQUAM grant no. 319278 and PASQuanS grant no. 817482), and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany`s Excellence Strategy (EXC-2111-390814868). J.K. acknowledges funding from the Hector Fellow Academy, and G.S. acknowledges funding from the Max Planck Harvard Research Center for Quantum Optics. F.G. and A.B. acknowledge support from the Technical University of Munich–Institute for Advanced Study, funded by the German Excellence Initiative and the European Union FP7 under grant agreement 291763, from DFG grant no. KN 1254/1-1 and DFG TRR80 (Project F8). A.B. also acknowledges support from the Studienstiftung des deutschen Volkes.

**Author contributions:**J.V. and P.S. acquired the data underlying this study and, together with G.S., analyzed them. J.V., P.S., G.S., J.K., and S.H. maintained and improved the experimental setup. A.B. and F.G. did the theoretical studies. I.B. and C.G. supervised the study. All authors worked on the interpretation of the data and contributed to the final manuscript.

**Competing interests:**The authors declare no competing interests.

**Data and materials availability:**The data that support the plots presented in this paper are publicly available from the Open Access Data Repository of the Max Planck Society (

*39*).