Exotic topological density waves in cold atomic Rydberg-dressed fermions

Versatile controllability of interactions in ultracold atomic and molecular gases has now reached an era where quantum correlations and unconventional many-body phases can be studied with no corresponding analogues in solid-state systems. Recent experiments in Rydberg atomic gases have achieved exquisite control over non-local interactions, allowing novel quantum phases unreachable with the usual local interactions in atomic systems. Here we study Rydberg-dressed atomic fermions in a three-dimensional optical lattice predicting the existence of hitherto unheard-of exotic mixed topological density wave phases. By varying the spatial range of the non-local interaction, we find various chiral density waves with spontaneous time-reversal symmetry breaking, whose quasiparticles form three-dimensional quantum Hall and Weyl semimetal states. Remarkably, certain density waves even exhibit mixed topologies beyond the existing topological classification. Our results suggest gapless fermionic states could exhibit far richer topology than previously expected.


Supplementary Figures
is rescaled by a factor 4 for illustration purposes.

Supplementary Tables
1, cos kx cos ky, (cos kx + cos ky) cos kz i(cos kx + cos ky), i cos kz (cos kx + cos ky) sin kz i sin kz Here a is the lattice constant, α index the three directions x, y and z, and c r the fermionic annihilation operator positioned at r. The atomic interaction potential, assuming the Rydberg state is s-wave, takes an isotropic form, To describe a density wave ground state, we take a variational state Here |vac is the vacuum, and we have a normalization condition |u k | 2 + |v k | 2 = 1. In k , k runs over one half of the Brillouin zone, say k z ≥ 0. We can define u k and v k in the whole Brillouin zone by the relation Half filling is already enforced within this variational ansatz. The variational energy cost E = Ψ|H 0 + H int |Ψ is obtained to be . Minimizing the energy, we get the Bogoliubov-de Gennes (BdG) equation where the self-energy is and the off-diagonal term is the density wave order, The chemical potential has been shifted by 1 2Ṽ (0) to compensate Hartree corrections to the self-energy. By definition, the density wave order satisfies ∆ k = ∆ * k+Q , which is an important difference of particle-hole pairing from particleparticle paring in superconducting states. Diagonalizing the BdG Hamiltonian, quasiparticle energies are given as ε k = ± ( k + Σ k ) 2 + |∆ k | 2 . In variational calculations, we choose the negative branch, and the self-consistent equations are then obtained to be The effective potential for ∆ k , defined by With weak interaction, the density wave order ∆ k is restricted to the region near the Fermi surface. We thus rewrite the momentum in terms of Fermi momentum and the distance from the Fermi surface as where e ⊥ is a unit vector perpendicular to the Fermi surface. The effective potential now reads

where Λ is a high-momentum cutoff, and the coupling matrix [g] is the matrix [Ṽ Q ] projected onto the Fermi surface.
To see the momentum dependence of the ∆ k , we expand U [∆ k ] to the second order as, where an infrared cutoff τ is introduced to regularize the perturbative expansion. This scale τ can be physically identified to be temperature. Minimizing the effective potential, with the minimal eigenvalue, or equivalently the eigenvector of γ k f k f with the maximal eigenvalue λ max . The corresponding transition temperature is estimated to be It is worth noting that the couplings g k f k f actually depend on the cutoff Λ. The effective potential U [∆] is, on the other hand, independent of Λ [1], i.e., from which the renormalization group β-function follows This renormalization equation can also be derived by momentum shell scheme [2]. Analyzing the renormalization group flow, the momentum dependence of the density wave order can be determined from leading divergent channels in the renormalization group flow, analogous to the procedure of extracting dominant channels in unconventional superconductivity [3,4]. This approach is equivalent to minimizing the effective potential at quadratic order.
In this section we give density wave phases supported by p-wave dressed Rydberg atoms [5], whose interaction takes an anisotropic form V (r = 0) = V 6 1 + (|r|/r c ) 6 sin 4 (θ), with θ the polar angle. The symmetry group to classify density waves in this system is then D 4h × T . The symmetry classification is shown in Supplementary Table 1 .
For simplicity, we truncate the long-range part of the interaction by taking r max = √ 2 × a. The interaction is then modeled to be All other terms are taken to vanish. With tunability of the lattice geometries, the ratio V z nnn /V nn and V h nnn /V nn is largely tunable between 0 and 1. In this model, the approximate criterion (equation 10 in the main text) for unconventional density waves implies We then numerically solve self-consistent equations for p-wave Rydberg dressed atoms. Both symmetric ( t z = t x = t y ) and asymmetric (t z = 4t x = 4t y ) tunneling cases are studied. For the symmetric case (see Supplementary Figure 2a), we find A + 1g , A + 1g + B − 1g , B − 1g and B − 1g + B + 2g density waves. The Fermi surface is fully gapped out for both A + 1g and A + 1g + B − 1g states. In the B − 1g state the Fermi surface is only partially gapped out, and we have gapless lines (protected by T symmetry), from which the density of states at low energy has linear behavior, i.e., D(ε) ∝ ε. In the B − 1g + B + 2g state, the Fermi surface for this state is fully gapped out, and the complex order ∆ k has a vortex line (with vorticity 2) elongated along the k z axis. This B − 1g + B + 2g state is topologically equivalent to the discussed E − g + T + 2g state in the main text. For the asymmetric case (see Supplementary Figure 2b), we find A + 1g , A − 1g + B − 1g , A − 1g + B − 1g + E + u density waves. In the A − 1g + B − 1g state, T is not broken and we have gapless lines. In the A − 1g + B − 1g + E + u state, the density wave order takes a form, (cos k x + cos k y ) + i∆ B − 1g (cos k x − cos k y ) + ∆ E + u sin k x cos k y , which has two straight vortex lines along the k z direction, given by (k x = π/2, k y = ±π/2). These vortex lines cross the Fermi surface, and we have four Weyl nodes. Topological numbers for this state are (2, 0). For both cases, the transition from trivial A + 1g state to other density waves roughly occurs at V h nnn /V nn = 0.35, with V z nnn /V nn fixed at 0.2, which agrees with the estimate from Supplementary Equation 4.