$d$-wave FFLO state and charge-2e supersolidity in the $t$-$t'$-$J$ model under Zeeman fields

TL;DR

This work investigates the $t$-$t'$-$J$ model under Zeeman fields to address FFLO superconductivity beyond the Pauli limit. Using state-of-the-art tensor-network methods (zero-temperature DMRG and finite-temperature tanTRG), it maps the temperature–field phase diagram on ladders and wider cylinders, identifying a robust zero-momentum $d$-wave SC that persists until the spin gap closes and coexists with CDW (a 2e-SS1 phase). At higher Zeeman fields, a $d$-wave FFLO state emerges with finite pairing momentum, accompanied by spin- and charge-density waves (2e-SS2), and the FFLO pairing momentum is found to lock to the underlying Fermi surface. The study reveals intertwined density orders and superconductivity, offering microscopic insight into field-induced unconventional pairing and suggesting feasible routes to realize the FFLO state and charge-2e supersolidity in ultracold-atom optical lattices, while noting the large Pauli limit might challenge experimental cuprate observation.

Abstract

Unconventional superconductivity under strong Zeeman fields--particularly beyond the Pauli paramagnetic limit--remains a central challenge in condensed matter physics. The exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in particular, remains in need of definitive study within fundamental electronic models. Here we employ state-of-the-art finite-temperature and ground-state tensor network approaches to systematically explore the superconducting (SC) phase diagram of the $t$-$t'$-$J$ model subjected to Zeeman fields. We find that zero-momentum $d$-wave superconductivity persists until the spin gap closes, coexisting with charge density waves. A novel $d$-wave FFLO phase emerges under a higher Zeeman field even above the Pauli limit, concomitant with a field-enhanced spin density waves. We identify these phases, characterized by the simultaneous presence of pairing condensate and density wave orders, as charge-2e supersolids. Analysis of Matsubara Green's function reveals that the FFLO pairing momentum is locked to the underlying Fermi surface. Our results provide microscopic insights into field-induced unconventional pairing mechanisms and reveal the long-sought FFLO state in a fundamental correlated electron model, offering a promising route for its realization in ultracold atom optical lattice.

Figures (8)

• Figure 1: (a) Magnetization $m \equiv (N_\uparrow - N_\downarrow) / N_{\mathrm{site}}$ as a function of Zeeman field $h$ at temperature $T = 0$. Here $h_0 = 0.360J$ is the Zeeman field where spin gap closes, $h_\mathrm{P} = 0.744J$ denotes the Pauli limit and $h_c \approx 1.4 J$ is the superconducting critical field. (b) Ground-state pairing correlations $|\Phi_{yy}(r)|$ under different Zeeman fields. The red dashed line decaying as $r^{-2}$ is also plotted as a reference. $h = 0.5J$ and $h = 0.9J$ locate in the FFLO phase. (c) Calculated FFLO pairing pattern at $h=0.9J$, where the size of circles is proportional to the local hole density, and the thickness of bonds represents the pairing amplitude. Doping $\delta = 0.1$ is fixed in all the calculations.
• Figure 2: (a) Finite-temperature and ground-state phase diagram of $t$-$t'$-$J$ ladder ($t'/t > 0$) in the presence of a Zeeman field $h$. For Zeeman fields below $h_0$, we identify a zero-momentum $d$-wave superconducting phase ($d$-SC) emerging at $T_c^*$ that coexists with a charge density wave, which constitutes a charge-2e supersolid (2e-SS1). The magnitude of $\Phi_{yy}(\mathbf{k=0})$ (indicated by the blue colorbar) reflects the strength of the zero-momentum pairing correlations. For $h > h_0$, the $d$-SC persists at elevated temperatures. The finite-momentum superconducting phase emerges with intertwined spin density wave and charge density wave, which constitutes another charge-2e supersolid phase (2e-SS2). Here $T_c^*$ denotes the onset temperatures of $d$-wave FFLO superconductivity. The difference $\Phi_{yy}(\mathbf{q}_{\mathrm{pair}}) - \Phi_{yy}(\mathbf{k=0})$ (depicted by the orange colorbar) quantifies the relative magnitude of the finite-momentum pairing correlations, also capturing the finite-momentum SC fluctuations at high temperatures. (b) Magnetic susceptibility $\chi_m = \partial m / \partial h$ in the FFLO (fluctuating) phase. $T^*$ denotes the magnetic characteristic temperature at which spin density wave emerges. (c) Pairing susceptibility $\chi_{\mathrm{SC}} = \partial \langle \Delta_{\mathrm{tot}} \rangle / \partial h_{\mathrm{pair}}$ in the FFLO (fluctuating) phase, which diverges algebraically as $\chi_{\mathrm{SC}} \sim 1/T^\gamma$ below the SC onset temperature $T_c^*$. The red line corresponds to the fitted power-law curve.
• Figure 3: (a,c) Rung pairing correlation $\Phi_{yy}(k_x, k_y=0)$ at $T = 0.02 J$ and $T = 0.05 J$. (b, d) $\beta G_\sigma (\mathbf{k}, \beta/2)$ at low temperature $\beta = 50 J$ ($T = 0.02 J$). In (a) and (b) we take $h = 0.5 J$, exhibiting zero-momentum $d$-wave superconductivity at high temperature while FFLO state appearing at low temperature. (c, d) show the FFLO state with finite-momentum pairing ($h = 0.9 J$).
• Figure 4: Contour plots of the spin structure factor $S(k_x, k_y = \pi)$ and charge structure factor $S_c (k_x, k_y = 0)$. In (a, c), the temperature is fixed at $T = 0.02 J$. Both the spin gapless point $h_0$ and the superconducting critical field $h_{c}$ are identified. In (b, d) the Zeeman field is set at $h = 0.9 J$ (2e-SS2 phase). The magnetic characteristic temperatures $T^*$ is indicated.
• Figure S1: (a) Convergence check of ground-state DMRG calculations. We show the rung pairing correlations at $D = 3000$ and $4000$ DMRG states, obtained from $L_x = 80$, $L_y = 2$ ladder with Zeeman field $h = 0.5 J$. (b) Convergence check of finite-temperature tanTRG calculations. We show the magnetization $m$ at different temperatures, obtained from $L_x = 60$, $L_y = 2$ ladder with $D = 3000$ and $4000$ states.