Charge-2e Supersolids in Correlated Systems

• Charge-2e supersolids are quantum phases characterized by simultaneous charge-2e pairing and spatial density modulation arising from density wave orders in strongly correlated systems.
• They are realized in various models including t-t'-J ladders under Zeeman fields, mixed triplet-singlet d-density-wave systems, and coupled Luther-Emery chains, each demonstrating unique pairing and modulation mechanisms.
• Experimental and theoretical studies reveal rich phase diagrams, topological transport phenomena, and distinctive collective excitations that bridge superconducting and density orders.

A charge-2e supersolid is a quantum phase where electronic pair condensate (superfluidity/superconductivity with charge 2e) and charge or spin density wave order (solid-like, spatial modulation) coexist. Distinct from conventional BCS superconductors or classical supersolids, these phases are realized in strongly correlated electronic systems via mechanisms that intertwine off-diagonal (pair) and diagonal (density) long-range order. They involve either the coexistence of pair condensate and density modulation at the microscopic level or condensation of topological defects (such as charge-2e Skyrmions) in a broken symmetry background. Recent theoretical advances have established explicit models in which charge-2e supersolidity emerges and clarified its experimental signatures, phase diagrams, and microscopic mechanisms (Qu et al., 23 Jan 2026, Hsu et al., 2012, Xie et al., 4 Jan 2025).

1. Microscopic Realizations and Model Hamiltonians

Charge-2e supersolidity appears in various correlated electron models. Three paradigmatic cases have been rigorously analyzed:

1. $t$-$t'$-$J$ Model Under Zeeman Fields: The no-double-occupancy Hamiltonian describes electrons with nearest- and next-nearest-neighbor hopping ($t$, $t'$), antiferromagnetic exchange $J>0$, and Zeeman energy $h$. The full Hamiltonian is

$H_{t\!-\!t'\!-\!J} = -\!\!\!\sum_{\<i,j\>,\sigma} t_{ij} (c_{i\sigma}^\dagger c_{j\sigma} + \mathrm{h.c.}) + \!\!\!\sum_{\<i,j\>} J_{ij} \left(\mathbf{S}_i \cdot \mathbf{S}_j - \frac14 n_i n_j\right) - h \sum_i S_i^z,$

where $t_{ij} = t, t'$, $c_{i\sigma}$ is the annihilation operator, and $S_i^z$ is the spin (Qu et al., 23 Jan 2026).

2. Mixed Triplet-Singlet $d$-Density-Wave (DDW) Model: Here the electron system develops both triplet and singlet $d$-density-wave order with form factor $f_2(\mathbf{k})$, supporting topologically nontrivial Skyrmion configurations in the spin sector. The order parameter is

$$\Delta_{\ell=2}(\mathbf{r}) \equiv \langle c^\dagger(\mathbf{r}+\mathbf{Q})\,\sigma^\mu\,c(\mathbf{r})\rangle\,f_2(\mathbf{k}),$$

with $\mathbf{Q} = (\pi, \pi)$ (Hsu et al., 2012).

3. Coupled Luther-Emery Chains: These are quasi-one-dimensional systems of spin-gapped chains with bosonized charge sector, interchain Josephson (pair hopping, $J_\perp$) and density-density (CDW, $V_\perp$) interactions in the bosonized framework:

$$H_{\rm chain} = \frac{v_c}{2\pi} \int dx\, [K_c(\partial_x \theta_c)^2 + K_c^{-1} (\partial_x \phi_c)^2],$$

where $K_c$ is the Luttinger parameter, plus interchain couplings promoting either CDW or pair coherence (Xie et al., 4 Jan 2025).

2. Order Parameters and Supersolid Criterion

The essential features of charge-2e supersolidity are captured by two classes of order parameters:

• Superconducting Pair Condensate:
  • Zero-momentum $d$-wave gap:

  $$\Delta_0 = \frac{1}{N} \sum_{\mathbf{k}} f_d(\mathbf{k}) \langle c_{-\mathbf{k},\downarrow}\,c_{\mathbf{k},\uparrow}\rangle$$ - Finite-momentum FFLO gap:

  $$\Delta_Q = \frac{1}{N} \sum_{\mathbf{k}} f_d(\mathbf{k}) \langle c_{-\mathbf{k}+\mathbf{Q},\downarrow}\,c_{\mathbf{k},\uparrow}\rangle$$

(Qu et al., 23 Jan 2026).

• Density-Wave Orders:

  • Charge density wave (CDW): $$\rho_Q = \frac{1}{N} \sum_i e^{i \mathbf{Q}\cdot\mathbf{r}_i} \langle n_i \rangle$$
  • Spin density wave (SDW): $$S_Q = \frac{1}{N} \sum_i e^{i \mathbf{Q}\cdot\mathbf{r}_i} \langle S^z_i \rangle$$
• Supersolid Criterion: Simultaneity $(\Delta_0 \neq 0\,\,\text{or}\,\,\Delta_Q \neq 0)$ and $(\rho_Q \neq 0\,\,\text{or}\,\,S_Q \neq 0)$ defines a charge-2e supersolid (Qu et al., 23 Jan 2026). In the Skyrmionic context, condensation of bosonic charge-2e Skyrmions means both $|\Psi| \neq 0$ (superfluidity of pairs) and $|\Delta| \neq 0$ (density order) (Hsu et al., 2012).

3. Microscopic Mechanisms and Topological Aspects

Distinct mechanisms underlie charge-2e supersolidity in different models:

• FFLO-Induced Supersolidity: For the Zeeman-split $t$-$t'$-$J$ ladder, strong Zeeman splitting shifts spin-up and spin-down Fermi surfaces. Pairing at finite center-of-mass momentum $\mathbf{Q}^* = k_{F\uparrow} - k_{F\downarrow}$ becomes energetically favorable (FFLO state). This develops a pair density wave, which by coupling to the charge sector, induces CDW order; concomitant oscillations of local magnetization (SDW) emerge at commensurate wavevector (Qu et al., 23 Jan 2026). The FFLO momentum is determined by the difference in Fermi momenta, as established via maxima of the single-particle spectral function $$A_\sigma(\mathbf{k},0) \simeq \beta G_\sigma(\mathbf{k},\beta/2)$$.
• Skyrmion Condensation: In a mixed triplet-singlet $d$-density-wave, spin textures (Skyrmions) in the triplet sector are topologically nontrivial, each carrying charge 2e as dictated by a Chern-Simons term in the effective action ($$S_{CS} = \frac{C}{4\pi}\int \epsilon_{\mu\nu\lambda}A_\mu^c \partial_\nu A_\lambda^s,\; C=-2$$). Bose condensation of Skyrmions yields a charge-2e superconductor within the density wave phase, leading to supersolidity via coexistence and topological intertwining (Hsu et al., 2012).
• Quasi-1D Luther-Emery Supersolids: Coupled finite-length chains support both nonzero $\Delta_{SC}$ (pair gap) and $\Delta_{CDW}$ (density gap) when interchain Josephson and CDW couplings have similar strength and chain size is finite. Power-law scaling $\Delta\sim L^{-1/(4\pi)}$ indicates finite but size-suppressed coexistence, unattainable in strictly infinite 1D due to the competing nature of $\phi_c$ and $\theta_c$ (Xie et al., 4 Jan 2025).

4. Phase Diagrams and Stability Regimes

Detailed numerical and analytical studies have mapped the occurrence domains for charge-2e supersolidity:

• $t$-$t'$-$J$ Ladder with Zeeman Field: Tensor-network calculations at doping $\delta=0.1$, $t/J=3$, $t'/t=0.17$ reveal:
  • For $h < h_0 \approx 0.36J$ and $T < T_c^*(h)$: zero-momentum $d$-wave superconductivity coexists with CDW ($2e$-SS1).
  • For $h_0 < h < h_P \approx 0.74J$ and $T < T_c^*(h)$: FFLO (finite momentum) pairing with SDW and weakened CDW ($2e$-SS2).
  • For $T > T_c^*(h)$ or $h > h_c \approx 1.4J$: normal (Luttinger-liquid) regime.
  • $T_c^*(h)$ is deduced from divergence of the pairing susceptibility in the FFLO region (Qu et al., 23 Jan 2026).
• Ginzburg-Landau Framework for Skyrmion Supersolidity: Phases are categorized by $(r_\Delta, r_\Psi)$, the mass parameters for the density wave and Skyrmion condensate, respectively:
  • $r_\Delta<0, r_\Psi>0$: DDW (“hidden-order”)
  • $r_\Delta>0, r_\Psi<0$: charge-2e superconductivity
  • $r_\Delta<0, r_\Psi<0$: charge-2e supersolid
  • $r_\Delta>0, r_\Psi>0$: semimetal
  • The DDW-to-SC transition may be continuous, described by a noncompact $CP^1$ (NCCP$^1$) theory for deconfined quantum criticality (Hsu et al., 2012).
• Quasi-1D Supersolid Stability: For coupled chains,
  • $V_\perp \gg J_\perp$: pure CDW
  • $J_\perp \gg V_\perp$: pure superconductivity
  • $V_\perp \sim J_\perp$, $K_c \approx 1$, small $\lambda \equiv (1/2\pi)\ln(\Lambda/\epsilon)$ (finite $L$): supersolid phase with both gaps finite. Analytic phase boundaries are accessible from the coupled gap equations (Xie et al., 4 Jan 2025).

5. Collective Excitations and Dynamical Signatures

Supersolid phases support unique collective modes arising from the coexistence of diagonal and off-diagonal order:

• Plasmon–Phason Coupling and Quasi-Goldstone Modes: In quasi-1D supersolids, the effective Gaussian action for phase fields predicts gapped quasi-Goldstone (plasmon–phason) modes with hybridized charge and superfluid character:

$$\omega_q^2 \approx \frac{4\pi e^2}{m q^2}(\rho_{s\parallel} q_x^2 + \rho_{s\perp} q_\perp^2),$$

along with Josephson plasma resonances ($\omega_{q_\perp} = v_J\sqrt{|q_\perp|}$) in the transverse directions (Xie et al., 4 Jan 2025).

• Topological Transport Phenomena: In Skyrmion condensates, time-dependent textures induce quantized charge pumping (per cycle, $\Delta Q = 2e$) and a quantized spin Hall response ($\sigma_{xy}^s = e/2\pi$), characteristic of phases that entwine topological and symmetry-breaking order (Hsu et al., 2012).

6. Experimental Signatures and Realizations

Charge-2e supersolidity is amenable to detection in several platforms and via multiple observables:

• Ultracold Atom Optical Lattices: Engineering $t$-$t'$-$J$-like Hamiltonians with spin imbalance realizes the Zeeman field. Key signatures:
  • Time-of-flight and momentum-resolved radio-frequency spectroscopy mapping spin-split Fermi surfaces.
  • Noise correlations revealing finite-momentum pairing at $\mathbf{Q}^*$.
  • Quantum gas microscopes imaging CDW oscillations at $\mathbf{Q}^*$.
  • Joint measurement of superfluid response and static structure factor confirming coexistence (Qu et al., 23 Jan 2026).
• Strongly Correlated Solids (e.g., URu$_2$Si$_2$): The superconducting phase within the hidden-order state suggests possible Skyrmion-driven supersolidity. Quantized charge pumping, nontrivial spin Hall conductivity, and in-gap s-wave features (not captured by BCS theory) would provide evidence (Hsu et al., 2012).
• Narrow Superconducting Ribbons and Stripe-Ordered Cuprates: Suppression of CDW amplitude with system size $(\Delta \sim L^{-1/(4\pi)})$, coexisting Bragg peaks and superfluid stiffness, and a plasma–phason hybrid mode in optical/THz spectroscopy provide direct dynamical probes (Xie et al., 4 Jan 2025).

7. Theoretical Significance and Connections

Charge-2e supersolids provide a unifying framework for novel intertwined orders in correlated systems. Their theoretical analysis elucidates the nontrivial interplay of Fermi surface topology, strong correlations, and symmetry, and enables exploration of deconfined quantum criticality and non-BCS mechanisms. These phases generalize classical supersolid concepts to electronic systems with topological, density, and superconducting order, offering concrete microscopic models and experimentally accessible predictions (Qu et al., 23 Jan 2026, Hsu et al., 2012, Xie et al., 4 Jan 2025).