Revealing Hidden Antiferromagnetic Correlations in Doped Hubbard Chains via String Correlators

TL;DR

This paper demonstrates the direct measurement of hidden antiferromagnetic correlations in doped Hubbard chains using quantum gas microscopy, revealing topological order through non-local string correlators and advancing understanding of spin-charge separation.

Contribution

It introduces a novel experimental approach to observe hidden magnetic correlations in doped Hubbard chains via non-local string correlators.

Findings

Detection of finite-range antiferromagnetic order in doped chains

Observation of hidden magnetic correlations due to spin-charge separation

Method extension potential to higher-dimensional systems

Abstract

Topological phases, like the celebrated Haldane phase in spin-1 chains, defy characterization through local order parameters. Instead, non-local string order parameters can be employed to reveal their hidden order. Similar diluted magnetic correlations appear in doped one-dimensional lattice systems due to the phenomenon of spin-charge separation. Here we report on the direct observation of such hidden magnetic correlations via quantum gas microscopy of hole-doped ultracold Fermi-Hubbard chains. The measurement of non-local spin-density correlation functions reveals a hidden finite-range antiferromagnetic order, a direct consequence of spin-charge separation. Our technique demonstrates how topological order can directly be measured in experiments and it can be extended to higher dimensions to study the complex interplay between magnetic order and density fluctuations.

Equations

$$\hat{H}=-t\sum _{i,\sigma }({\hat{c}}_{i,\sigma }^{†}{\hat{c}}_{i+1,\sigma }+\text{h.c.})+U\sum _i{\hat{n}}_{i,\uparrow }{\hat{n}}_{i,\downarrow }+\sum _{i,\sigma }{ϵ}_i{\hat{n}}_{i,\sigma }$$

$$C(d)=4\left(⟨{\hat{S}}_i^z{\hat{S}}_{i+d}^z{⟩}_{{\text{newmoon}}_i{\text{newmoon}}_{i+d}}-⟨{\hat{S}}_i^z{⟩}_{{\text{newmoon}}_i}⟨{\hat{S}}_{i+d}^z{⟩}_{{\text{newmoon}}_{i+d}}\right)$$

$$C^{\text{str}}(d)=4{⟨{\hat{S}}_i^z\left(\prod _{j=1}^{d-1}(-1{)}^{(1-{\hat{n}}_{i+j})}\right){\hat{S}}_{i+d}^z⟩}_{{\text{newmoon}}_i{\text{newmoon}}_{i+d}}$$

$$C_{SH}^{\text{str}}(d,s)=4{⟨{\hat{S}}_i^z\left(\prod _{\begin{array}{c}j=1,j\ne s\end{array}}^{d-1}(-1{)}^{(1-{\hat{n}}_{i+j})}\right){\hat{S}}_{i+d}^z⟩}_{{\text{newmoon}}_i{\text{fullmoon}}_{i+s}{\text{newmoon}}_{i+d}}$$

$$C(d)$$

$$C_{SH}(d,s)$$

$$C^{\text{str}}(d)$$

$$C_{SH}^{\text{str}}(d,s)$$

$$4{\hat{M}}^2=\hat{N}+2\sum _{d=1}^{N-1}(N-d)\hat{C}(d)$$

$$C^{\text{offset}}(N,M)=$$

$$-\underset{A}{\underbrace{\sum _{d=1}^{N-1}\frac{2(N-d)}{N(N-1)}{C}^{\text{corr}}(d)}}$$

$$\hat{\rho }={\hat{\rho }}_{\mathrm{s}}\otimes {\hat{\rho }}_{\mathrm{ch}}.$$

$${\hat{\mathcal{H}}}_{t-J\ast }={\hat{\mathcal{P}}}_s$$

$$+J\sum _{⟨i,j⟩}\left({\hat{\vec{S}}}_i\cdot {\hat{\vec{S}}}_j-\frac{{\hat{n}}_i{\hat{n}}_j}{4}\right)+{\hat{\mathcal{H}}}_{\mathrm{NNN}}]{\hat{\mathcal{P}}}_s,$$

$$\begin{array}{rl}{\hat{\mathcal{H}}}_{\mathrm{NNN}}=& -\frac{J}{8}\sum _{⟨i,j,k⟩,\sigma }^{i\ne k}({\hat{c}}_{i,\sigma }^{†}{\hat{c}}_{k,\sigma }{\hat{n}}_j\\ & -\sum _{{\sigma }^{\prime },\tau ,{\tau }^{\prime }}{\hat{c}}_{i,\sigma }^{†}{\vec{\sigma }}_{\sigma ,{\sigma }^{\prime }}{\hat{c}}_{k,{\sigma }^{\prime }}\cdot {\hat{c}}_{j,\tau }^{†}{\vec{\sigma }}_{\tau ,{\tau }^{\prime }}{\hat{c}}_{j,{\tau }^{\prime }}),\end{array}$$

$$\begin{array}{rl}{\hat{\mathcal{H}}}_{t-J\ast }=& t\sum _{⟨i,j⟩}\left({\hat{h}}_i^{†}{\hat{h}}_j{\hat{\mathcal{F}}}_{ij}^{†}+\text{h.c.}\right)\\ & +J\sum _{⟨i,j⟩}\left(1-{\hat{n}}_i^h\right){\hat{\vec{S}}}_i\cdot {\hat{\vec{S}}}_j\left(1-{\hat{n}}_j^h\right)\\ & -\frac{J}{4}\sum _{⟨i,j⟩}{\hat{n}}_i^h{\hat{n}}_j^h+\frac{J}{4}\sum _{⟨i,j,k⟩}^{i\ne k}{\hat{h}}_k^{†}\left(1-{\hat{n}}_j^h\right){\hat{h}}_i{\hat{\mathcal{A}}}_{ij}^{†}{\hat{\mathcal{A}}}_{kj}.\end{array}$$

$$\frac{J}{4}{\hat{\mathcal{A}}}_{ij}^{†}{\hat{\mathcal{A}}}_{kj}=\frac{J}{2}\left({\hat{\vec{S}}}_{\tilde{j}}\cdot {\hat{\vec{S}}}_{\tilde{k}}-\frac{1}{4}\right),$$

$${\hat{\mathcal{H}}}_{\mathrm{ch}}=t\sum _{⟨i,j⟩}\left({\hat{h}}_i^{†}{\hat{h}}_j+\text{h.c.}\right)=2t\sum _k\cos (k){\hat{h}}_k^{†}{\hat{h}}_k,$$

$${\hat{\mathcal{H}}}_{\mathrm{s}}=J_{\mathrm{eff}}(n)\sum _{⟨\tilde{i},\tilde{j}⟩}{\hat{\vec{S}}}_{\tilde{i}}\cdot {\hat{\vec{S}}}_{\tilde{j}}.$$

$$J_{\mathrm{eff}}^{(1)}=J\frac{⟨\left(1-{\hat{n}}_i^h\right)\left(1-{\hat{n}}_{i+1}^h\right)⟩}{⟨\left(1-{\hat{n}}_i^h\right)⟩}.$$

$$J_{\mathrm{eff}}^{(2)}=\frac{J}{2}\frac{⟨\left(1-{\hat{n}}_i^h\right)\left({\hat{h}}_{i-1}^{†}{\hat{h}}_i+{\hat{h}}_i^{†}{\hat{h}}_{i-1}\right)⟩}{⟨\left(1-{\hat{n}}_i^h\right)⟩}.$$

$$J_{\mathrm{eff}}(n,T)=Jn\left[1+\frac{1}{\pi n}{\int }_0^{\pi }dk\cos (2k)n_k^F(n,T)\right].$$

$$J_{\mathrm{eff}}(n,T=0)=Jn\left[1-\frac{\sin (2\pi n)}{2\pi n}\right].$$

$$ϵ(p)=\frac{\pi }{2}Jn\left[1-\frac{\sin (2\pi n)}{2\pi n}\right]\left|\sin \left(p/n\right)\right|,$$

$$f(M)=\mathcal{N}{e}^{-\frac{4M^2}{2\left(N/2\right)}},$$

Full text

The Fermi-Hubbard model, describing systems of strongly correlated fermions on a lattice, lies at the heart of our understanding of the Mott insulator-metal transitions and quantum magnetism Auerbach (1994). The complexity of the interplay between hole doping and magnetic ordering in this model is believed to give rise to a rich phase diagram, including a High-Tc superconducting phase as, for example, observed in cuprate compounds Lee et al. (2006). In one dimension however, the competition between the spin and density sectors is largely absent due to the separation of the spin and density modes at low energy. This phenomenon of spin-charge separation, generally appearing in Luttinger liquids, is well understood theoretically Giamarchi (2004), but there are only limited experimental observations. All experimental evidences of this foundational phenomenon are based so far on spectroscopic Kim et al. (1996); Segovia et al. (1999); Kim et al. (2006) or transport measurements Auslaender et al. (2005); Jompol et al. (2009) in condensed matter systems. Nevertheless, quasi long-range antiferromagnetic order at zero temperature, as conventionally measured by two-point spin correlation functions, gets suppressed by a finite hole density in the system. However, due to the independence of the spin and charge sectors the order is not truly reduced, but rather hidden den Nijs and Rommelse (1989); Kennedy and Tasaki (1992); Kruis et al. (2004a). It can be revealed by measurements over an extensive part of the system allowing to construct string correlation functions. In analogy to the spin-1 Haldane phase Haldane (1983a, b); Affleck et al. (1987) this requires measuring all spins in the chain. A closely related way to unveil the hidden order is to work directly in "squeezed space", where empty sites are completely removed from the system Ogata and Shiba (1990); Ren and Anderson (1993); Zaanen et al. (2001); Kruis et al. (2004b). In traditional condensed matter systems neither string order can be measured, nor is squeezed space accessible to experiments. Fermionic quantum gas microscopes Haller et al. (2015); Cheuk et al. (2016); Parsons et al. (2016); Omran et al. (2015); Edge et al. (2015); Brown et al. (2016), in contrast, give access to snapshots of the full spin and density distribution with single site resolution Boll et al. (2016), such that non-local correlation functions can be extracted Endres et al. (2011). Here we report on the direct measurement of string correlations in ultracold Fermi-Hubbard chains. The ability to locally detect holes, doublons and the spin state allows for an analysis of the system directly in squeezed space, in which Heisenberg spin correlations are restoredKruis et al. (2004b). Our observations provide a microscopic picture of spin-charge separation independent of the more frequently discussed spectral properties or the excitation dynamics.

We probed the physics of the doped one-dimensional Fermi-Hubbard model using a balanced spin mixture of 6Li trapped in a single plane of a two-dimensional optical lattice. A versatile quantum gas microscope allowed for the simultaneous local detection of both spin states Boll et al. (2016) (see Fig. 1A). By controlling the lattices depths in the different spatial directions som we created independent one-dimensional systems described by the single band Hubbard Hamiltonian

[TABLE]

The fermion creation (annihilation) operator is denoted by $\hat{c}^{\dagger}_{i,\sigma}$ ($\hat{c}_{i,\sigma}$) at site $i$ for each of the two spin states $\sigma=\ \uparrow,\downarrow$ and the operator $\hat{n}_{i,\sigma}=\hat{c}^{\dagger}_{i,\sigma}\hat{c}_{i,\sigma}$ counts the number of atoms with spin $\sigma$ at site $i$. The energy offsets $\epsilon_{i}$ result from an additional confinement due to the lattice beams, which leads to a smoothly changing local density ${n_{i}=\langle\hat{n}_{i}\rangle=\langle\hat{n}_{i,\uparrow}+\hat{n}_{i,\downarrow}\rangle}$. At half filling in the strong coupling limit ($U/t\gg 1$) the Fermi-Hubbard model reduces to a Heisenberg spin chain with $J=4t^{2}/U$ and supports quasi long-range antiferromagnetic (AFM) order at zero temperature Auerbach (1994). The doped system is described at long wavelength by Luttinger liquid theory, which predicts at zero temperature an algebraic decay of the spin correlations with distance that is faster than the one of the Heisenberg model Giamarchi (2004). This decay can be understood from spin-charge separation, allowing holes to freely move in the AFM spin-chain. Consequently, the spins around the hole are anti-aligned and the sign of the staggered magnetization $(-1)^{i}S_{i}^{z}$, called AFM parity, changes. This implies that a hole acts as a domain wall of the AFM parity, which reduces the spin correlations. The spin order however, is still present and can be revealed either in squeezed space by effectively removing the holes in the analysis or by evaluating string correlators, which take the AFM parity domain walls into account by flipping the sign of the correlator (Fig. 1C). Analytic and numerical studiesKruis et al. (2004b) have shown that at zero temperature, the two-point spin correlations in squeezed space are comparable to the ones of a pure Heisenberg chain, for any doping and any repulsive interaction $U$. This is readily understood in the $U/t\rightarrow\infty$ limit, where the many-body wave function $\Psi(\{x_{j,\sigma}\})=\Psi_{\rm ch}(\{x_{j}\})\,\Psi_{\rm s}(\{\tilde{x}_{j,\sigma}\})$ factorizes exactly into a density $\Psi_{\rm ch}$ and a spin $\Psi_{\rm s}$ part Woynarovich (1982); Ogata and Shiba (1990). The spin degree of freedom is described by a Heisenberg model in squeezed space with the spins "living" on a lattice defined by the positions of spinless, non-interacting fermions Ren and Anderson (1993). Distances in squeezed space are rescaled by the spinless fermion density $\tilde{x}\sim nx$. Also at non-zero temperature and finite interactions, the spin correlations in squeezed space are governed by a Heisenberg model with a renormalized exchange coupling $J_{\textrm{eff}}(n)$ that depends on the original density $n$ som .

The experiment started with a two-dimensional degenerate two-component Fermi gas. Using the large spacing component of an optical superlattice ($a_{sl}=2.3\,\mu$m), the system was divided into about ten independent one-dimensional tubes. The Fermi Hubbard chains were then realized using a lattice of $1.15\,\mu$m spacing along the tubes. The atom number was set such that the maximum density in the chains was typically just below unity. At the final lattices depths the tunneling amplitude reached $t=h\times 400\,$Hz, and the confinement due to the lattice beams fixed the length of the central tubes to about $15$ sites. The onsite repulsion $U$ was tuned to $h\times 2.9\,$kHz using the broad Feshbach resonance between the hyperfine states $\left|\downarrow\right\rangle=\left|F,m_{F}\right\rangle=\left|1/2,-1/2\right\rangle$ and $\left|\uparrow\right\rangle=\left|1/2,1/2\right\rangle$ to set a scattering length of $2000$ Bohr radii at the end of the lattice ramps. These parameters and the lattice ramps have been optimized to produce cold, strongly interacting doped Hubbard chains som . For the detection of the spin and density degrees of freedom the lattice depth along the tubes was rapidly increased, followed by a local Stern-Gerlach like detection using a magnetic field gradient and the short scale component of a superlattice transverse to the tubes Boll et al. (2016). Applying Raman sideband cooling for $500\,$ ms, we collected fluorescence photons on an EMCCD camera to form a high contrast and site resolved image of the atomic distribution Omran et al. (2015) as shown in Fig. 1A. From comparison of the measured spin-correlations at half filling to Quantum Monte-Carlo results Boll et al. (2016), we estimated the temperature in the central chains to be $0.51(2)\,t$ or $0.90(3)\,J$, which corresponds to an entropy per particle of $0.63(2)k_{B}$.

To investigate the magnetic environment around a hole, we calculate the conditional three-point spin-hole correlation function $C_{SH}(2)=4\,\langle\hat{S}^{z}_{i}\hat{S}^{z}_{i+2}\rangle_{\tiny\newmoon_{i}\fullmoon_{i+1}\newmoon_{i+2}}$, where the symbols describe the condition that the correlator is only evaluated on configurations with the sites $i$ and $i+2$ singly occupied and the middle site empty som . The correlator indeed reveals anti-alignment of the spins around individual holes ($C_{SH}(2)<0$) and Fig. 2A highlights the hole induced sign change by comparison to the standard two-point correlator

[TABLE]

To obtain unity filling, the latter was evaluated on a hole-free subset of the data. The additional condition indicated by the symbols is important in this paper as it removes the trivial $n^{2}$ density dependence of this correlator, but has no effect here som . The measured modulus of the correlation around a hole is $|C_{SH}(2)|=0.184(4)$, considerably larger than $C(2)=0.057(3)$ and about half of the next-neighbor value of $|C(1)|=0.316(2)$. At zero temperature for $U/t\rightarrow\infty$ one expects $|C_{SH}(2)|=|C(1)|$, since the hole has no effect on the magnetic alignment of its surrounding spins. For our interaction strength, the measured difference agrees with exact diagonalization results at a temperature $0.94(5)\,J$. These calculations take the experimental fluctuations of the magnetization per chain into account. Due to finite size effects the correlation function shows a small offset at large distances, for which we correct in the subsequent analysis throughout this paper som .

The influence of larger doping on the spin order is revealed by studying $C_{SH}(d)$ as a function of the number of holes between the two spins, that is by evaluating $C_{SH,N_{h}}(d)=4\,\langle\hat{S}^{z}_{i}\hat{S}^{z}_{i+d}\rangle_{\tiny\newmoon_{i}\{\fullmoon\}_{N_{h}}\newmoon_{i+d}}$ with exactly $N_{h}$ holes on the otherwise singly occupied string of length $d+1$. The results of this analysis shown in Fig. 2B reveal a sign change of $C_{SH,N_{h}}$ at fixed distance $d$ for each newly introduced hole and antiferromagnetic correlations versus distance for fixed hole number $N_{h}$. Thus, each hole indeed corresponds to a flip of the antiferromagnetic parity. In a thermodynamic ensemble, the hole number between the two measured spins fluctuates, resulting in a weighted averaging over the alternating correlations for different hole numbers. This directly explains the suppression of magnetic correlations with hole doping (cf. Fig. 3A).

The strong reduction of spin correlations due to hole fluctuations does not imply the absence of magnetic order in the system, but rather suggests that it is hidden by the fluctuations in the position of the atoms. This situation is similar to the Haldane phase of spin-1 chains Haldane (1983a, b); Affleck et al. (1987); den Nijs and Rommelse (1989), where fluctuating $\left|0\right\rangle$ spins hide correlations between the $\left|\pm 1\right\rangle$ components leading to exponentially decaying local correlators. The intrinsic AFM order is unveiled by considering a non-local correlation function. By identifying double occupancies and holes with spin $\left|0\right\rangle$ states, one can use the same procedure to construct a string correlator that probes the underlying spin order in the doped Hubbard chain Kruis et al. (2004b):

[TABLE]

This string correlator takes the antiferromagnetic parity flips into account by a corresponding sign flip for each hole (cf. Fig. 1C). The unique ability to detect the spin and density locally on single images Boll et al. (2016) enables the direct measurement of the string correlator $C^{\textrm{{str}}}(d)$ for different densities. The dependence of the string correlator on distance reported on Fig. 3A is in stark constrast with the standard two-point spin correlation function $C(d)$. While $C(d)$ quickly vanishes when the system is doped away from half filling, staggered correlations at distances up to four sites are detected with the string correlator $C^{\textrm{str}}(d)$ (Fig. 3A). When analyzing the data in regions of fixed density, we additionally observe an increasing periodicity of the AFM correlations with decreasing density Giamarchi (2004). The amplitude of the string correlations, on the other hand, even slightly increase in magnitude at a given real space distance $d$ which we attribute to the decreasing distance $\tilde{d}\sim nd$ in squeezed space som .

An analysis of the correlations directly in squeezed space is also possible with the quantum gas microscope by removing the empty and doubly occupied sites in the analysis before evaluating the standard two-point correlator $C(d)$. This corresponds to a weighted summation along the diagonals of Fig. 2B, and thus mixes events that had different distances in real space. Similar to the string correlator, the squeezed space analysis (Fig. 3B) reveals the finite-range hidden antiferromagnetic order. A quantitative comparison to a Heisenberg model with renormalized coupling $J_{\textrm{eff}}(n)$, that decreases with doping, agrees well at a temperature of $T=0.87(2)~{}J$, which demonstrates that the concept of squeezed space can be successfully applied even away from the $U/t\rightarrow\infty$ limit Imambekov et al. (2012). Here, $J_{\textrm{eff}}$ was determined independently from the microscopic parameters of the Hubbard model som . The discrepancy between theory and experiments at densities below $0.45$ might arise from adiabatic cooling when decreasing the density during the preparation of the chains.

In order to further confirm the independence of the spin and density sectors, we define a tailored string correlator

[TABLE]

which isolates the effect of a single hole at distance $s$ from the first spin independent of the density. Here, the effect of extra charge fluctuations is taken care of by inserting string correlators around the hole. For a system with a single hole this correlator is identical to the three-point correlation function introduced before $C_{SH}^{\textrm{str}}$ = $C_{SH}$. The dependence of $C_{SH}^{\textrm{str}}$ on the spin separation $d$ and the position of the hole in the string $s$ is shown in Fig. 4A. For $s=0$ and $s=d$ the hole crosses one of the two spins, which causes the previously discussed AFM parity flip, while the correlation signal is almost independent of the position of the hole between the two spins. This observation emphasizes spin-charge separation by the absence of polaron-like effects, which would result in a local change of the spin correlations around the hole. The rectified correlator $(-1)^{d}C_{SH}^{\textrm{str}}(d,s)$ in Fig. 4B highlights the two domains of opposite AFM parity, demonstrating that the hole acts as a domain wall for the magnetic order Zaanen (1998). To emphasize the symmetries of the three-point correlator, the position of the hole is measured here relative to the center of mass of the two spins.

Through the analysis of various local and non-local correlation functions our measurements revealed striking equilibrium signatures of spin-charge separation in one-dimensional Hubbard chains. An interesting extension of this work would be the detection of dynamic signatures of spin-charge separation in quench experiments through the measurements of different spin and charge velocities Recati et al. (2003); Kollath et al. (2005). In higher dimensions the experimental evaluation of non-local correlations in synthetic hole doped antiferromagnetic materials is also of prime interest for the investigation of exotic many-body phases relevant to high temperature superconductors Carlson et al. (2004); Zhang et al. (2002). The extension to two-dimensional frustrated quantum magnets would, for example, enable the detection of deconfined criticality through Wilson loops Senthil et al. (2004). Hence, our experiments mark a first step towards experimental studies of emergent gauge structures and topological order Wen (2004).

Acknowledgements We acknowledge J. Koepsell and J. Vijayan for a critical reading of the manuscript, A. Sterdyniak and M. Zvonarev for useful discussions, and financial support by MPG and EU (UQUAM). FG and ED acknowledge support from Harvard-MIT CUA, NSF Grant No. DMR-1308435, the Moore Foundation, AFOSR Quantum Simulation MURI, AFOSR MURI Photonic Quantum Matter.

Supplementary material

I Preparation of the ultracold lattice gas

The preparation of degenerate Hubbard chains closely followed the protocol detailed in Ref. Boll et al. (2016). We started from a quasi two-dimensional degenerate mixture of the two lowest energy Zeeman states of 6Li in a single plane of an optical lattice with $3.1\,\mu$m spacing in the vertical direction. The vertical lattice depth was $110\,E_{r}^{z}$ and the scattering length was set to $530\,a_{B}$. Here $E_{r}^{i}=h^{2}/8ma_{i}^{2}$ is the recoil energy for a lattice of period $a_{i}$ in the $i$-direction, $m$ the atomic mass and $a_{B}$ the Bohr radius. Next, the preparation of about $10$ Fermi-Hubbard chains started with ramping up the large scale component of an optical superlattice ($a_{sl}=2.3\,\mu$m) in the $y$-direction in $15\,$ms to a depth of $18\,E_{r}^{y}$. A lattice of period $a_{l}=1.15\,\mu$m along $x$ was then ramped up in $15\,$ms to $3\,E_{r}^{x}$ and finally to $5\,E_{r}^{x}$ in $80\,$ms, while the lattice depth in $z$-direction was linearly decreased to $17\,E_{r}^{z}$ in $50\,$ms. Simultaneously the lattice in $y$-direction was increased to $27\,E_{r}^{y}$ in $60\,$ms. Using a magnetic offset field of $714\,$G near the broad Feshbach resonance located at $834.1\,$G Zürn et al. (2013) the scattering length was linearly increased during these ramps to $2000\,a_{B}$. The lattice and the low peak densities of about one atom per site ensured collisional stability by suppressing three-body recombination losses. At the end of the ramps, the onsite interaction energy was $U=h\times 2.9\,$kHz, as estimated from Wannier function calculations without taking into account finite band gap corrections Büchler (2010). The tunneling amplitude was $t=h\times 400\,$Hz and the exchange energy $J=4t^{2}/U=h\times 220\,$Hz. A local Stern-Gerlach detection technique operating at a transverse magnetic field gradient of $95\,$G/cm detailed in Boll et al. (2016) was used to detect both the spin and the density on each lattice site with a fidelity larger than $98\%$.

II Properties of the atomic cloud

The prepared clouds contained in total $131\pm 5.5$ atoms with a global magnetization of $\frac{1}{2}(N_{\uparrow}+N_{\downarrow})=+1.2\pm 2.9$ compatible with zero, where the uncertainties are the standard deviations of the distributions, i.e. not the standard error of the mean. The sub-shot-noise fluctuations are attributed to our magnetic gradient assisted evaporation in a stiff optical trap, which cuts into the Fermi sea Serwane et al. (2011). The gaussian intensity profiles of the lattices beams introduced an additional confining potential, which lead to an inhomogeneous density distribution (Fig. S1 A). The atom number was chosen to obtain $N\lesssim N_{0}=2\left(\frac{4t}{\frac{1}{2}m\bar{\omega}^{2}a_{l}^{2}}\right)^{1/2}\simeq 13$ in most chains, corresponding to a filling lower than one atom per site Rigol et al. (2003). Here, the frequency corresponding to the harmonically approximated confinement along the chains is $\bar{\omega}\approx 2\pi\times 300\,$Hz. The atom number distribution in the tubes is shown in Fig. S1 B.

Equipped with the measurement of the full counting statistics, we binned the $38000$ chains from all shots of the dataset by their number of atoms $N=\langle\sum_{i}(\hat{n}_{i\uparrow}+\hat{n}_{i\downarrow})\rangle$, where $i$ is running over all sites of the chain. For each bin of fixed $N$, we then analyzed the magnetization $\hat{M}=\sum_{i}\hat{S}^{z}_{i}$ and magnetization fluctuations $\langle\hat{M}^{2}\rangle$. For typical atom numbers of $N=7-15$, we observed less than half a spin of net magnetization $\langle\hat{M}\rangle=0.011(1)\,N$ and sub-shot-noise fluctuations $\langle 4\hat{M}^{2}/N\rangle=0.519(6)$ per chain, which is about half the value expected for uncorrelated spins (see Fig. S1 C). These fluctuations were taken into account when comparing to numerical predictions.

III Data post selection

The fidelity of the spin resolved imaging depends on the phase fluctuations of the superlattice in $y$-direction. In $3\%$ of the experimental runs we detected unusually large global spin imbalances, which we attribute to environmentally induced lattice phase fluctuations. Thus, runs with more than $16$ excess spins in the entire cloud have been discarded. The presence of short scale imperfections in the trapping potential, visible in Fig. S1A, resulted in a consistent failure of spin detection on some sites Boll et al. (2016). While the precise superlattice phase control ensured a local mean spin of $S^{z}_{i}=0.004(3)$ for typical "good" sites, we filtered out sites with detected mean spin of more than $S^{z}_{i}=0.025$ on a $3.5\sigma$ level. This concerns $15$ lattice sites out of about $300$ (Fig. S1A). In about $1\%$ of the chains we detected one or more site occupied by two atoms of the same spin. These events might be due to a small chance of having atoms in the second band or due to hopping during the detection. We removed all of those lines from the dataset. We ensured that none of the filters discussed above critically affect the results reported in this manuscript.

IV Correlation functions

An empty or doubly occupied site has spin zero and thus these sites reduce the magnitude of spin correlations trivially. In order to compare spin correlations at different densities, we evaluate the spin operators on singly occupied sites only. The conventional connected spin correlation function including this condition can be written as

[TABLE]

where we defined the "singlon" operator $\hat{n}_{i}^{s}=\hat{n}_{i,\uparrow}+\hat{n}_{i,\downarrow}-2\hat{n}_{i,\uparrow}\hat{n}_{i,\downarrow}$. The first relation is the definition of the conditional correlation function $C(d)$, which we directly measured experimentally. The doublon fraction as well as density correlations beyond one site were negligible, which justifies the approximation of the "singlon-singlon" correlations in the denominator by the measured densities resulting in a simple normalized spin-spin correlator. These last expressions are just given for clarity, they have not been used in the analysis.

In the same spirit, the spin-hole correlation function, which selects on two occupied and one empty site, can be rewritten as a normalized three-point correlator:

[TABLE]

where the hole operator $\hat{n}_{i}^{h}=(1-\hat{n}_{i,\uparrow})(1-\hat{n}_{i,\downarrow})\approx(1-n_{i})$ detects the presence of a hole at site $i$.

The measured non-local correlation functions can equivalently be expressed in terms of unconditional spin and density correlation functions,

[TABLE]

When studying both local and non-local spin correlations at large distances, the presence of the trap complicates the definition of density. For the data presented in the main text in Fig. 3 we defined the density $n$ as the mean density over all the sites connecting the two operators evaluated at sites $i$ and $i+d$: $n=\frac{1}{d+1}\sum_{k=i}^{i+d}\langle\hat{n}_{k}\rangle$.

V Correction for finite size effects

In each chain, the atom number and magnetization weakly fluctuate from shot-to-shot, but for a single realization the atom number $N=\langle\sum_{i}\hat{n}_{i}\rangle=N_{\uparrow}+N_{\downarrow}$ and magnetization $M=\langle\sum_{i}\hat{S}^{z}_{i}\rangle=\frac{1}{2}(N_{\uparrow}-N_{\downarrow})$ are fixed. A fixed magnetization $M$ in a single spin-$1/2$ chain without holes or doublons (length $L=N$) acts as a constraint on the spin-spin correlations because of the following exact sum rule:

[TABLE]

Here, the line average of the spin correlation operator $\hat{C}(d)=4/(N-d)\sum_{i=1}^{N-d}\hat{S}_{i}^{z}\hat{S}_{i+d}^{z}$ has been introduced.

Even for completely uncorrelated spins in the chain the correlations are thus non-zero and their value $C(d)=4M^{2}/(N^{2}-N)-1/(N-1)$ is necessarily constant with distance $d$. This effect only vanishes in the case of Poisson (shot-noise) magnetization fluctuations $4M^{2}=N$, or in the infinite system size limit assuming non-extensive fluctuations. For sub-shot-noise fluctuations of the magnetization this offset is negative, in agreement with our experimental and numerical observations.

For systems with non-trivial correlations $C^{\textrm{corr}}$ we assume a constant additive offset $C(d)=C^{\text{corr}}(d)+C^{\text{offset}}(N,M)$ due to this finite size effect. Solving Eq. S1, reveals an offset $C^{\textrm{offset}}(N,M)$ depending on the correlations:

[TABLE]

Experimentally, we find that our data is well described by $C^{\textrm{offset}}(N,M)$ with constant $A=-0.045(5)$ obtained from the correlations for $d>6$, where we do not observe any staggered correlations in the conventional spin correlator (Fig. S2 B,C). In order to reveal the non-trivial staggered spin correlations, we measured the atom number and magnetization for each chain, calculated $C^{\textrm{offset}}(N,M)$ using the experimental value for $A$, and finally subtracted it from each single outcome contributing to $\langle\hat{S}^{z}_{i}\hat{S}^{z}_{i+d}\rangle$ before averaging spatially and over different experimental runs. This procedure was followed for all different versions of correlation functions, even though the string correlators are largely insensitive to the offset (Fig. S2A).

VI Comparison of correlation functions

We analyzed three types of spin correlation functions, the conventional connected two-point correlation function $C(d)$, the string correlation function $C^{\textrm{str}}(d)$ and the squeezed space correlation function $C^{sq}(\tilde{d})$. Without any holes all of these correlators give the same results. To emphasize the difference between them at finite doping, Fig. S3 shows an exemplary direct comparison of the $C^{\rm x}(2)$ values as a function of density. The conventional correlation function $C(d)$ is very sensitive to doping due to the discussed AFM parity flips and $C(2)$ even changes sign at $n=0.70(3)$, when the contribution from spin-hole-spin events $C_{SH,N_{h}=1}(d=2)$ dominates over events without a hole $C_{SH,N_{h}=0}(d=2)$. The spin string correlation function, on the other hand, is insensitive to the hole induced AFM parity flips, thus, $C^{\textrm{str}}(2)$ stays positive and even increases slightly in magnitude with decreasing density due to the effectively shorter spin-spin distance in squeezed space. The squeezed space correlation function $C^{sq}(\tilde{d}=2)$ is also insensitive to the AFM parity flips and stays positive, but decreases with doping at finite temperature due to the decreasing effective coupling strength between the spins (see Sec. VII).

VII Effective Heisenberg model in squeezed space

In the limit of zero temperature and infinite repulsion $U/t\to\infty$, Ogata and Shiba have shown that the ground state wavefunction of the doped Fermi-Hubbard model factorizes Ogata and Shiba (1990), $\Psi(\{x_{j,\sigma}\})=\Psi_{\rm ch}(\{x_{j}\})\,\Psi_{\rm s}(\{\tilde{x}_{j,\sigma}\})$. The first part is a wavefunction of free, spinless fermions which describes the charge sector (holes). The second part corresponds to a pure spin wavefunction given by the ground state of an anti-ferromagnetic Heisenberg chain. As described in the main text of our paper, this factorization can be illustrated in the squeezed space picture Zaanen et al. (2001); Kruis et al. (2004b) where holes are removed from the lattice.

For sufficiently large $U$, we expect that a similar factorization remains valid even for finite temperatures $T\geq 0$, i.e. spin and charge degrees of freedom remain uncorrelated. This is a direct consequence of the separation of the spin-exchange and hole-hopping energy scales, $J\ll t$. In analogy with Ogata and Shiba’s wavefunction, we will make the following ansatz for describing the thermal density matrix $\hat{\rho}$ in the large-$U$ limit,

[TABLE]

For the ground state at $T=0$, the presence of the holes has no effect on the wavefunction $\Psi_{\rm s}$ of spins on the squeezed lattice. Nevertheless the typical exchange energy scales are suppressed. Simply speaking, the spins are less likely to be located next to each other when the hole doping increases. At finite temperature $T>0$, in contrast to the ground state case, this reduction of the spin-exchange energies $J\to J_{\rm eff}$ modifies the spin-correlations. To describe this effect quantitatively we will now derive an expression for $J_{\rm eff}$, as a function of density $n$ and temperature $T$, see Eq. (S8) below. Our results are shown in Fig. S4. Using this data we calculated the density dependence of spin-correlators presented in Fig. 3b of our paper.

VII.1 The $t-J^{*}$ model

Our starting point is the Fermi-Hubbard model at fillings less than or equal to one half. We consider the limit when $U\gg t$, where double occupancy of sites by two fermions is strongly suppressed and the Fermi-Hubbard Hamiltonian reduces to the $t-J^{*}$ model. To leading order in $J=4t^{2}/U$ one obtains the effective Hamiltonian

[TABLE]

see for example Ref. Auerbach (1994). Here $\hat{c}_{j,\sigma}$ annihilates a fermion with spin $\sigma$ on site $j$ and the spin operators are defined as $\hat{\vec{S}}_{j}=\frac{1}{2}\sum_{\sigma,\sigma^{\prime}}\hat{c}^{\dagger}_{j,\sigma}\vec{\sigma}_{\sigma,\sigma^{\prime}}\hat{c}_{j,\sigma^{\prime}}$. We defined the fermion density as $\hat{n}_{j}=\sum_{\sigma}\hat{c}^{\dagger}_{\sigma,j}\hat{c}_{\sigma,j}$, and $\vec{\sigma}$ denotes a vector of Pauli matrices. Note that the Hamiltonian in Eq. (S4) must be projected onto the subspace of single-occupied sites, which is ensured by the projection operators $\hat{\mathcal{P}}_{s}$. The first two terms in Eq. (S4) correspond to the commonly studied $t-J$ Hamiltonian, where the additional next-nearest neighbor hopping processes described by $\hat{\mathcal{H}}_{\rm NNN}$ are neglected. In our case such terms are crucial, however, and we will discuss them in more detail below.

The first term in Eq. (S4) describes hopping processes of a fermion to an unoccupied neighboring site. The second term in Eq. (S4) corresponds to the usual spin-exchange interaction between two fermions on neighboring sites. Its energy scale is set by $J=4t^{2}/U$ because it derives from a second order tunneling processes where a state with two fermions on the same site is virtually occupied. The exchange interaction between two neighboring spins leads to a zero-point energy of $J/4$, which manifests itself by the nearest neighbor density-density interaction $J\hat{n}_{i}\hat{n}_{j}/4$ in Eq. (S4).

In addition to the nearest neighbor spin-exchange interaction, virtual processes where a site is occupied by two fermions also leads to a next-nearest neighbor tunneling process of a hole. As illustrated in Fig. S5, a hole can tunnel from site $i$ to $k=i+2$ in the following way. First a fermion hops from site $k=i+2$ onto another fermion on site $j=i+1$, where a virtual two-fermion state with energy $U$ is formed. Then one of the two fermions hops from site $j=i+1$ to site $i$. No double-occupancies are remaining now, and the hole has effectively moved to site $k=i+2$. The energy scale for this process is also set by $J$, and it can be formally described by the following term in the $t-J^{*}$ Hamiltonian Auerbach (1994),

[TABLE]

where $\langle i,j,k\rangle$ denotes a triple of neighboring sites.

VII.2 The $t-J^{*}$ model in squeezed space

To understand the physics of the one-dimensional $t-J^{*}$ model (S4), we would like to introduce spin-less hole operators describing the charge degrees of freedom as well as spin operators defined in squeezed space. First the hole operators $\hat{h}_{j}$ can be introduced by describing the spins with Schwinger bosons. This leads us to the following Hamiltonian (up to a constant energy offset), see Ref. Auerbach (1994),

[TABLE]

Here $\hat{h}^{\dagger}_{i}$ creates a hole on site $i$, $\hat{n}_{i}^{h}=\hat{h}^{\dagger}_{i}\hat{h}_{i}$ is the hole density, and $\hat{\vec{S}}_{i}$ denotes the spin operator at site $i$. The operators $\hat{\mathcal{F}}_{ij}$ and $\hat{\mathcal{A}}_{ij}$ can be expressed in terms of Schwinger bosons, see Ref. Auerbach (1994) for details, but we will explain their meaning below by mapping the Hamiltonian to squeezed space.

The first term in Eq. (S5) describes the hopping of the holes, where the operators $\hat{\mathcal{F}}_{ij}$ ensure that the spins are physically moved in real space. This has no effect in squeezed space however, where the ordering of the spins is unchanged and we can effectively set $\hat{\mathcal{F}}_{ij}=1$. The second term describes the anti-ferromagnetic exchange interactions. Note that two neighboring spins in squeezed space can only interact if they are not separated by holes in real space. Thus, on average we expected a reduction of the exchange energy scale in squeezed space with increasing hole density.

The third term in Eq. (S5) describes an attractive interaction between the holes, which, however, is small compared to the kinetic energy term and can be neglected in the limit when $t\gg J$.

The last term in Eq. (S5) describes the next-nearest neighbor tunneling of the hole from site $i$ to $k$, across a site $j$ which is occupied by a spin. This process is accompanied by a spin-flip interaction, described by the operators $\hat{\mathcal{A}}_{ij}$. The term can be simplified considerably by mapping it to a squeezed space representation: Consider a situation with two spins on sites $k=-1$ and $j=0$ and a hole on site $i=1$. In squeezed space we can label the two spins by $\tilde{j}$ and $\tilde{k}$, and their labels do not change when the hole hops from $i$ to $k$, see Fig. S5. In the new basis, this next-nearest neighbor hopping process becomes

[TABLE]

and the hole operators $\hat{h}^{\dagger}_{k}\left(1-\hat{n}^{h}_{j}\right)\hat{h}_{i}$ are unmodified.

Like before, in the limit $t\gg J$ the next-nearest neighbor hopping term $\sim J$ has no effect on the hole dynamics which is dominated by the nearest neighbor hopping term of order $\sim t$. However, the spin-dynamics can be substantially modified by this term, because it is of comparable strength $\sim J$ as the exchange interactions.

VII.3 Effective Hamiltonians: spin-charge separation

Now we make use of the separation of energy scales, $t\gg J$ valid in the large-$U$ limit. This justifies neglecting spin-charge correlations and thus the use of our ansatz Eq. (S3). As a result we obtain an effective Hamiltonian for the holes,

[TABLE]

which is derived from the $t-J^{*}$ model Eq. (S5) by discarding terms of order $\mathcal{O}(J/t)$. As expected, this corresponds to free fermions hopping on a one-dimensional lattice.

From the correlations of the holes, determined by $\hat{\rho}_{\rm ch}$, we can now derive an effective Hamiltonian for the spin sector. Formally it is obtained by tracing out the charge sector, i.e. $\hat{\mathcal{H}}_{\rm s}={\rm tr}_{\rm ch}(\hat{\rho}_{\rm ch}\hat{\mathcal{H}}_{t-J*})$. As a result we obtain a pure spin Hamiltonian,

[TABLE]

Note that in contrast to $\hat{\vec{S}}_{j}$ in Eq. (S5), $\hat{\vec{S}}_{\tilde{j}}$ denotes operators for the spin chain in squeezed space now.

VII.4 Effective spin-exchange interaction

To obtain the effective exchange interaction $J_{\rm eff}$ in squeezed space, we need to calculate conditional probabilities in real space. To understand this, let us start by considering the second term in Eq. (S5). If there is a spin on a given site $i$ in real space, corresponding to spin $r$ in squeezed space, it interacts with its neighboring spin $s=r+1$ in squeezed space only if there is a spin on site $i+1$ in real space. This leads to a contribution $J_{\rm eff}^{(1)}$ to the exchange interactions in squeezed space, given by

[TABLE]

Similarly we obtain a contribution $J_{\rm eff}^{(2)}$ to the spin-exchange interactions from the next-nearest neighbor hopping processes from Eq. (S6). In this case we need to calculate the following conditional correlator,

[TABLE]

Assuming that the holes at density $n_{h}=1-n$ have a temperature $T$, we can easily calculate all correlation functions assuming a Fermi-Dirac distribution function $n^{F}_{k}(n,T)$ of the holes (Eq. (S7)) and making use of Wick’s theorem. As a result we obtain for $J_{\rm eff}=J_{\rm eff}^{(1)}+J_{\rm eff}^{(2)}$,

[TABLE]

At zero temperature the integral on the right-hand side of Eq. (S8) can be evaluated exactly and we obtain

[TABLE]

This expression is in agreement with results from exact Bethe-ansatz calculations for the 1D Fermi-Hubbard model. Using this analytical technique the spin-wave dispersion $\epsilon(p)$ was derived in Ref. Coll (1974) in the large-$U$ limit considered here,

[TABLE]

where $p$ is the momentum of the spin wave. As expected for a spin-wave in squeezed space, the dispersion relation is given by $\epsilon(p)=\pi/2J_{\rm eff}|\sin p_{\rm sq}|$, where the momentum $p_{\rm sq}$ in squeezed space is related to real space by rescaling all length scales with the density, $p_{\rm sq}=p/n$.

VIII Comparison to theory: exact diagonalization

In the main part of the paper, we compared our experimental data to theoretical results from the $t-J^{*}$ Hamiltonian described in Sec.VII. To this end we performed exact numerical calculations of thermal states within the $t-J^{*}$ model. We implemented the Schwinger-boson representation of our model, see Eq. (S5), and considered small systems with periodic boundary conditions. In our calculations, the number of holes and the number of spins were fixed, as well as the total magnetization $M$ in $z$-direction.

We changed the density by changing the number of holes, and calculated the observables $\mathcal{O}(M)$ of interest separately for all possible values of the magnetization. To compare our results directly with the experimental data, we assumed the following distribution function for the total magnetization,

[TABLE]

which approximates the experimental magnetization fluctuations (Fig. S1 C). Here $\mathcal{N}$ is a normalization which ensures that $\sum_{M}f(M)=1$ and $N$ is the number of fermions in the chain. The final observables are given by $\mathcal{O}=\sum_{M}f(M)\mathcal{O}(M)$.

To calculate the ground state of our Hamiltonian, we used a standard numerical Lanczos technique. At finite temperatures, $T>0$, the extension of the Lanczos method introduced in Ref. Aichhorn et al. (2003) was implemented. We sampled over a few thousand random vectors and worked with a dimension between $50$ and $200$ of the Krylof basis, taking system sizes up to $L=14$ sites.