Competition of fermion pairing, magnetism, and charge order in the spin-doped attractive Hubbard gas

TL;DR

The study addresses how fermion pairing, magnetism, and charge order compete in the spin-doped attractive Hubbard model and explores the possibility of magnetized superfluidity. It combines a two-dimensional optical-lattice realization with single-atom, spin- and charge-resolved microscopy and state-of-the-art theory (AFQMC and NLCE) to map pairing and ordering across densities $n$ and magnetizations $m$, at $T/t \approx 0.3$. The key findings reveal a progression from a mixture of nonlocal pairs and excess fermions at low $m$ and moderate $U/t$ to a hard-core Bose-Fermi mixture at higher coupling, with a CDW-to-polarons crossover and indications of spin- and pair-density wave tendencies. The work highlights partial pairing and potential FFLO-like correlations as precursors to magnetized superfluidity, offering insights into spin-charge ordering relevant to cuprates and related platforms.

Abstract

The tension between fermion pairing and magnetism affects numerous strongly correlated electron systems, from high-temperature cuprates to twisted bilayer graphene. Exotic forms of fermion pairing and superfluidity are predicted when attraction between fermions competes with spin doping. Here, we follow the evolution of fermion pairing and charge and spin order in a spin-imbalanced attractive Hubbard gas of fermionic $^{40}$K atoms, covering a wide range of densities, magnetizations, and interactions with single-atom resolution. At low spin imbalance and weak interactions, we find a mixture of nonlocal fermion pairs coexisting with itinerant excess fermions. For stronger interactions an effective hard-core Bose-Fermi mixture emerges. Spin doping drives a crossover from charge-density wave correlations to a Fermi liquid of polarons. Beyond a certain spin imbalance and interaction strength, we find evidence for the onset of combined spin- and pair-density wave order, a possible precursor for the existence of magnetized superfluidity in the attractive Hubbard system.

Figures (6)

• Figure 1: Realizing the spin-doped attractive Hubbard model.(a) Schematic phase diagram of the spin-doped attractive Hubbard model away from half filling (density $n<1$) at magnetization $m$. The spin-balanced gas features a BEC-BCS crossover below a critical temperature $T_C$ and preformed pairs above $T_C$ that display charge-density wave (CDW) and superfluid (SF) correlations. With increasing magnetization, exotic forms of superfluidity (FFLO) are predicted, including strange spin metal behavior at elevated temperatures and a Fermi liquid of polarons at high polarization. (b) Spin- and atom-resolved images of a trapped spin-doped Hubbard gas with central magnetization $m = 0$, $m = 0.1$, and $m = 0.2$ at interaction strength $U/t=5.8(3)$, detected via quantum gas microscopy. (c) Upper row: radially averaged densities of spin-up ($n_\uparrow$) atoms, spin-down ($n_\downarrow$) atoms, and doubly occupied sites (doublons, $d$) as a function of radial position in the trap. Lower row: singly occupied site (singlon) densities, where $s_\uparrow = n_\uparrow - d$, $s_\downarrow = n_\downarrow - d$, and magnetization $m=n_\uparrow - n_\downarrow$. The flat dependence of magnetization on position in the trap is indicative of a Bose-Fermi mixture where excess fermions are repelled by bosonic pairs that compensate the trapping potential. Solid lines are from AFQMC calculations (see main text).
• Figure 2: Evolution from fermion pairs to a Fermi liquid of polarons.(a) The doublon density $d$, measuring the strength of short-range pair correlations, evolves smoothly from the regime of paired fermions at spin balance to the expected polaronic Fermi liquid at large magnetization. Dotted lines give the doublon density for a gas of $n_\downarrow = (n-m)/2$ Fermi polarons at $T=0$ from the variational Ansatz Chevy2006Pascual2024SI. (b) The fraction of singlons among minority atoms $s_\downarrow/n_\downarrow$ quantifies nonlocal pair correlations, and gradually decreases with $m$ to the limit expected for Fermi polarons Pascual2024SI (arrows). (c) The vanishing of total magnetization fluctuations $\sum_\delta \langle \hat{m}_i \hat{m}_{i+\delta}\rangle_c = \chi_m T$ indicates full fermion pairing at $m = 0$ beyond $U/t \gtrsim 5.76$. Fluctuations increase with magnetization but remain sub-poissonian (dotted line is $\chi_m T = m$), implying a degenerate Fermi gas of excess fermions at small $m$, and a degenerate Fermi mixture at large $m$. The gray shaded region is the noise floor for detecting fluctuations, given by the standard deviation at $m=0$ and $U/t=11.93$. Lines are AFQMC results at $T/t=0.33$ (solid) and $T/t=0.50$ (dashed).
• Figure 3: Spatial structure of pair correlations. (a) Inter-spin connected singlon correlations $\langle s_{\uparrow,i} s_{\downarrow, i+\delta} \rangle_c$ at $U/t=5.76$ and $n\approx 0.7$. Bunching is observed between spin-up and spin-down singlons at all magnetizations, whether due to pairing or polaronic attraction. (b) Excess density of spin-up singlons surrounding an isolated spin down ($\Delta s_\uparrow = \sum_{\delta \neq 0}\langle s_{\uparrow,i} s_{\downarrow, i+\delta} \rangle_c/s_\downarrow$) versus magnetization, at various densities and interaction strengths. For two-body pairing each isolated spin is located near exactly one opposite spin, thus $\Delta s_\uparrow =1$. An excess density less than one in a gas with full pairing (at $U/t = 8.44$ and $5.76$ and $m \lesssim 0.05$) implies that each spin attracts only a portion of an opposite spin, a many-body form of pairing similar to a BCS gas. (c) Intra-spin connected correlations $\langle s_{\uparrow,i} s_{\uparrow, i+\delta} \rangle_c$ between isolated alike fermions at $U/t=5.76$ and $n\approx 0.7$. At large magnetization, negative correlations develop: the Fermi hole forms due to Pauli exclusion. (d) Nonlocal correlations ($\sum_{\delta \neq 0} \langle s_{\uparrow,i} s_{\uparrow, i+\delta} \rangle_c$) of majority singlons ($s_\uparrow$ for $m > 0$) converge at large $U/t$ to the expectation for a degenerate Fermi gas of excess spins (gray dotted line, right panel: free Fermi gas of dopants at $T/t = 0.33$). Minority singlons ($s_\uparrow$ for $m < 0$) become nearly uncorrelated. (b, d) Lines are AFQMC results at $T/t = 0.33$ (solid) and $T/t = 0.50$ (dashed); ☆: BCS expectation value at $T=0$, $\times$: from AFQMC at $T/t = 0.1$, both at $n = 0.6$.
• Figure 4: Effective pair-pair and pair-dopant interactions.(a) Measured doublon-doublon correlations ($g^{(2)}_{d,d} = \langle \hat{d}_i \hat{d}_{i+1}\rangle/d^2$) at three different magnetizations and $n\approx0.7$ showing pair-pair repulsion. (b) Pair-dopant repulsion, detected through doublon-singlon up correlations ($g^{(2)}_{d,s_\uparrow} = \langle \hat{d}_i \hat{s}_{\uparrow,i+1}\rangle/d s_\uparrow$) at $n\approx0.7$. (c) Doublon-doublon repulsion is dependent only on density, revealed by $g^{(2)}_{d,d}$ at displacement $(1,0)$. (d-e) In contrast, at high magnetization and high density, doublon-singlon repulsion is overcome by many-body effects, producing positive correlations in $g^{(2)}_{d,s_\uparrow}$ at nearest-neighbor (d) and next-nearest-neighbor (e) displacements. Data is shown for $U/t=5.76$. Lines in (c-e) are AFQMC results at $T/t = 0.33$ (solid) and $T/t = 0.50$ (dashed).
• Figure 5: Classifying emergent behavior of a spin-imbalanced Hubbard gas with microscopic observables.(a) Many-body order is revealed by the sign of the diagonal density-density correlation ($g^{(2)}_{n,n}(1,1) = \langle \hat{n}_i \hat{n}_{i+(1,1)}\rangle/n^2$). At low density fermion pairs become larger and repel, leading to negative diagonal density correlations. At high density, pairs order into a charge-density wave (CDW) having positive diagonal density correlations. (b) Two-dimensional map of $g^{(2)}_{n,n}(1,1)$ measured at various densities and magnetizations with $U/t=8.44$. Markers show experimental data, and background color shows a second order two dimensional polynomial interpolation. The pink band is the crossover region where $g^{(2)}_{n,n}(1,1)\approx 1$ within the statistical uncertainty of the fit (1$\sigma$ confidence level from bootstrapping). Data in (a) shows a cut of the interpolation in (b) at $m=0.1$, including statistical uncertainty (red shading). (c-d) The dominant behavior of isolated fermions is revealed by the sign of the total nonlocal singlon-singlon (moment-moment) correlations. Isolated fermions include both excess majority fermions and the nonlocal fluctuations of bosonic pairs. Total nonlocal singlon correlations cross over from pairing dominated (positive) at low magnetization to Fermi dominated (negative) at high magnetization. Data in (c-d) is analogous to (a-b), and (d) shows a cut through (c) at $n=0.6$. (e) Resulting experimental classification of the emergent order of the spin-imbalanced mixture for multiple attraction strengths, ranging from strongly attractive ($U/t=8.44$) to below the threshold for full pairing ($U/t=3.97$). Each map is obtained using the the observables in (a-d), with similar statistical certainty.