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Time-resolved observation of spin-charge deconfinement in fermionic Hubbard chains

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Science  10 Jan 2020:
Vol. 367, Issue 6474, pp. 186-189
DOI: 10.1126/science.aay2354

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 (46) and conductance measurements in metallic quantum wires (79). Cold-atom experiments have been used extensively to study attractive 1D bosonic and fermionic gases (1015), but the investigation of repulsive 1D fermionic gases has been more recent (1618). 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 (1921) 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, 2527). In the subsequent dynamics, we observed the emergence of two apparently independent excitations propagating at different velocities (2830), which we assigned to spinons and holons on the basis of their characteristic signatures in the spin and charge (density) sectors.

Fig. 1

Probing spin-charge deconfinement with cold atoms. (A) Cartoon depicting the fractionalization of a fermionic excitation into quasiparticles. The dynamics are initiated by removing a fermion from the Hubbard chain. This quench creates a spin (spinon) and a charge (holon) excitation, which propagate along the chain at different velocities vJ and vt. (B) Using quantum gas microscopy, we simultaneously detected the spin and density on every site of the chain after a variable time after the quench. (C) Average number of holes in the chain as a function of time (top). Error bars of 1 SEM are smaller than the symbol size. The quench, performed at 0 ms creates a hole with a probability of ~78% in the central site of the chain (bottom).

Our experiment (24) began by loading a 2D balanced spin mixture of 6Li atoms in the lowest two hyperfine states into several 1D tubes using an optical lattice of spacing ay = 2.3 μm along the y direction. Next, a lattice of spacing ax = 1.15 μm was ramped up along the 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 n^ih in each chain, where 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 5 τt, where τt=h×(4πt)1=0.32 ms is the time it takes for a hole propagating at the theoretically expected maximum group velocity vmaxt=ax/τt to move by one site. The coherent evolution of the hole can be seen in the evolving interference pattern of n^ih over time. This dynamic is found to be in excellent agreement with a single-particle quantum walk (see Fig. 2B), as expected for a spin-charge separated system.

Fig. 2 Time evolution of spin and charge excitations.

(A) Hole density distribution n^ih as a function of time after the quench. The wavefront of the distribution starts at the center of the chain and expands outward linearly with time. Interference peaks and dips are visible throughout the dynamics, indicating the coherent evolution of the charge excitation. (B) 1D cuts of the experimental hole density distributions at times 0 τt, 1.88 τt, and 3.77 τt (blue circles) are compared with simulations of a single-particle quantum walk (gray squares). (C) Nearest-neighbor squeezed-space spin correlation C(x˜=1) distribution as a function of time after the quench. (D) 1D cuts of the experimental C(x˜=1) distributions at times 0 τJ, 1.54 τJ, and 3.08 τJ (red circles) along with exact diagonalization simulations of the Heisenberg model (gray squares) (31). Error bars denote 1 SEM.

To study the time evolution of the spin excitation, we measured nearest-neighbor spin correlations Ci˜ (x˜=1)=4(S^i˜zS^i˜+1zS^i˜zS^i˜+1z) in squeezed space (denoted by ~), obtained by removing holes and doublons from the chain in the analysis (23, 24). For strong interactions U/t  8, the spin dynamics in squeezed space is expected to be well captured by an antiferromagnetic Heisenberg model (3234) 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 4 τJ, where τJ=h×(π2J)1=1.56  ms is the time it takes for a spinon propagating at the theoretically expected maximum group velocity vmaxJ=ax/τJ to move by one site. In contrast to the highly coherent evolution of the hole, the finite temperature kBT/J ~ 0.75, where kB 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 Ci˜ (x˜=1) and hole distributions as a function of time (Fig. 3A, inset). We then used a linear fit to determine their respective velocities (Fig. 3A). For the data shown in Fig. 3A, taken at 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 tJ 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.

Fig. 3 Quasiparticle velocities of spinons and holons.

(A) Time evolution of the widths of the hole density distributions (blue circles) and nearest-neighbor spin correlation distributions (red circles) after the quench. The measured widths are defined as the full width at 30% of maxima of the distributions (see inset). Density and spin excitations reach the edge of the unity-filled region of the chain (central nine sites) after different evolution times. Their dynamics are in quantitative agreement with both a single-particle quantum walk for the hole and exact diagonalization calculations of the Heisenberg model for the spin (gray squares). They are also found to reproduce the predictions of the extended tJ model at our temperature (gray dashed lines) (31). The velocities of the spin (0.58 ± 0.04 sites/ms) and the charge (3.08 ± 0.09 sites/ms) excitations are obtained as half the slope of a linear fit to the data (solid blue and red lines), ignoring the width immediately after the quench. (B) Holon velocities as a function of t/h. The velocities of the holon (blue circles) increase linearly with the tunneling rate in the chain, consistent with vmaxt=4πtax/h sites/ms (blue dashed line). (C) Spin-excitation velocities as a function of J/h. The velocities of the spin excitation (red circles) increase linearly with the spin-exchange coupling in the chain, consistent with vmaxJ=π2Jax/h sites/ms (red dashed line). Error bars denote 1 SEM.

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 vmaxt and vmaxJ for the two excitation channels (see Fig. 3, B and C). These correspond to the velocities of a free holon and spinon at the maximum group velocity allowed by their dispersion. Unlike the Luttinger liquid regime, which only describes low-energy excitations, our local quench excited all momentum modes, the fastest of which is tracked here.

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 CSHS (2)=4  S^iz n^i+1h S^i+2z , a spin correlator conditioned on having a hole at site i + 1 (23, 24) (Fig. 4A). Immediately after the quench, the hole is likely to be surrounded by parallel spins, and CSHS retains a positive value. The measured spin correlations are consistent with the next-nearest-neighbor correlations C(2)=4(S^izS^i+2zS^izS^i+2z) in the absence of the quench. As the hole propagates, the sign of CSHS becomes negative and, by 4 τt, approaches the nearest-neighbor correlations C(1)=4(S^izS^i+1zS^izS^i+1z) without the quench. These observations indicate the decoupling of the two excitations. At longer evolution times, the antiferromagnetic correlations across the hole are reduced. We attribute this to the holon oscillating in the chain, owing to the harmonic confinement present in our system and hence to the changing overlap of the spin and charge distributions (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 δC1(d)=S^izS^i+1zS^izS^i+1z1ii+1i+1+d  id (see Fig. 4A, inset), where 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. δC1 shows no dependence on d, indicating the lack of influence of the holon on the spin excitation.

Fig. 4 Spatial deconfinement of spin and charge excitations.

(A) Spin-hole-spin correlations (CSHS) averaged over the entire chain as a function of time after the quench. The correlator starts with a positive value consistent with the next-nearest-neighbor spin correlations C(2) in the absence of the quench (top gray-shaded region) and turns negative, approaching the nearest-neighbor spin correlations C(1) without the quench (bottom gray-shaded region) by 4 τt. At longer evolution times, the correlator shows reduced antiferromagnetic correlations, owing to the oscillating dynamics of the hole in our finite size system. The inset shows the lack of dependence of the normalized deviation from the mean nearest-neighbor correlations δC1 on the distance d from the hole at times 4τt (purple) and 19 τt (yellow). Error bars denote 1 SEM. (B) Spatially resolved magnetization fluctuations Σ^j˜2 in subregions of the chain with (red) and without (gray) the quench at 3.77 τt. The background fluctuations Σ^j˜2BG are caused by quantum and thermal fluctuations in our system. The peak in the difference signal Σ^j˜2Σ^j˜2BG indicates the location of the spin excitation. Gray- and red-shaded areas denote 1 SEM without and after the quench, respectively. (C) Efficiency of initially creating, at the central site, a single local spinon ηspin=4(Σ^j˜=02Σ^j˜=02BG) with σ = 1.5 sites (orange) and holon ηhole=1(n^i=01)2 (blue) after an ideal quench as a function of temperature, as predicted from exact diagonalization of the Heisenberg model (for the spinon) and the Hubbard model (for the holon). With increasing temperature, ηspin (ηhole, inset) decreases because of the increase of thermal spin (density) excitations, preventing the creation of a localized spinon (holon) by the quench. Taking into account our quench efficiency, the measured amplitude is consistent with the prediction at a temperature of kBT/J = 0.75 (gray-shaded region).

To locate the excess spin excitation in a fluctuating spinon background, we introduced an operator quantifying the local spin fluctuations in squeezed space Σ^j˜2=(i˜S^i˜zfj˜σ(i˜))2, where fj˜σ(i˜)=exp[(i˜j˜)22σ2] is a smooth window function centered at lattice site j˜ with a characteristic size of σ. At zero temperature, this operator is expected to capture local fractional quantum numbers (37). A single spinon located at site j˜, carrying a spin 1/2, would increase Σ^j˜2 by 1/4, provided that the mean distance between thermal spin fluctuations is larger than σ.

To study the spatial separation of the spin and charge excitations, we considered the chains at time 3.77 τt, where the highest probability of detecting the hole is at sites ±2. We postselected on chains with a single hole outside the central three sites of the unity-filled region and computed Σ^j˜2 for a window size σ = 1.5 (see Fig. 4B). Comparing Σ^j˜2 with the quench to its value Σ^j˜2BG without the quench, where BG indicates the background value in the absence of the quench, we observed a well-localized signal extending over the central three sites, distinct from the position of the holon. The maximum deviation Σ^j˜2Σ^j˜2BG reached 0.13 ± 0.01, about half the value expected at zero temperature. We attribute this difference mainly to the finite temperature of our system leading to a background density of thermal spin excitations. In this case, even an ideal quench would not create an initially localized spinon with unity probability, and the fractionalization scenario, where a single removed particle breaks up into precisely one holon and one spinon, holds only asymptotically at zero temperature (see Fig. 4C). A reduction in the deviation Σ^j˜2Σ^j˜2BG from 1/4 is thus expected in our system, and the measured value is in good agreement with exact diagonalization calculations of the Heisenberg model at kBT = 0.75J, 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

science.sciencemag.org/content/367/6474/186/suppl/DC1

Supplementary Text

Figs. S1 to S8

Reference (40)

References and Notes

  1. See supplementary materials.
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).
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