Bogoliubov corner excitations in a conventional s-wave superfluid

Higher-order topological superconductors and superfluids have triggered a great deal of interest in recent years. While Majorana zero-energy corner or hinge states have been studied intensively, whether superconductors and superfluids host higher-order topological Bogoliubov excitations with finite energies remain elusive. In this work, we propose that Bogoliubov corner excitations with finite energies can be induced through only mirror-symmetric local potentials from a trivial conventional s-wave superfluid. The topological Bogoliubov excited modes originate from the nontrivial Bogoliubov excitation bands. These modes are protected by the mirror symmetry and are robust against mirror-symmetric perturbations as long as the Bogoliubov energy gap remains open. Our work provides a new insight into higher-order topological excitation states in superfluids and superconductors.


Introduction and motivation
Topological phases have ignited intensive research interests in the past two decades.Intrinsic topological states with n-th order in d dimension exhibit d − n dimensional gapless boundary states.Due to the bulk-boundary correspondence, the nontrivial bulk topology for the higher-order topological states (n > 1) is different from the conventional (n = 1) topological states [1][2][3][4].The celebrated tenfold way can characterize the first-order topological insulators and superconductors in a unified way in terms of three nonspatial symmetries, i.e. time-reversal, particle-hole, and chiral symmetries [5][6][7].However, higher-order topological states are usually related to crystalline symmetries, and comprehensive topological classifications have been made recently with point group symmetries [8][9][10][11][12].
In this paper, we show that Bogoliubov corner excited modes could emerge in a conventional s-wave superfluid on a honeycomb lattice with the mirror-symmetric onsite potential.To gain more intuitive insight, we first showcase an s-wave superconductor on a one-dimensional (1D) lattice with inversion-symmetric potential hosting topologically nontrivial edge excitation modes, despite the ground state for the superconductor is in a trivial phase.These topological modes could extend to a defect chain with non-zero mirror symmetric onsite potentials on a two-dimensional (2D) square lattice.Furthermore, the higher-order topological Bogoliubov corner excitation modes are present in an s-wave superfluid on a 2D honeycomb lattice.These Bogoliubov excitation modes are protected by the nontrivial topology for the Bogoliubov excitation bands, and are robust against mirror-symmetric perturbations.
The remainder of this paper is organized as follows.In section 2, we take a simple 1D s-wave superconductor as an example to show that the inversion-symmetric onsite potential could induce localized edge excitation modes.The topological origin of edge modes is explored and demonstrated.In section 3, we consider a defect chain on a 2D square lattice exhibiting robust edge modes.In section 4, we consider an s-wave superfluid on a honeycomb lattice with the mirror-symmetric potential, and show the Bogoliubov corner excitation modes.In section 5, we discuss relevant topics including experiment realizations, and draw a conclusion.

Bogoliubov edge excitations in 1D s-wave superconductors
We first consider a simple model, i.e. 1D s-wave superconductor, to present topologically protected Bogoliubov edge excitations induced by the inversion-symmetric potentials, as shown in figure 1(a).Its physics is described by the Hamiltonian Ĥ = Ĥ0 + ĤV .The first part reads where t denotes the tunneling strength between nearest neighbor sites, ⟨. ..⟩ represents the summation over all nearest-neighbor sites, σ = (↑, ↓) is the spin index, and ∆ 0 is an s-wave superconductor order parameter.
The second term ĤV = ∑ i V i ĉ † i,σ ĉi,σ describes onsite potentials with inversion symmetry.In the following, we consider each unit cell consisting of four sublattices with the potentials V i = V a if mod (i, 4) = 0 or 1 and V i = V b if mod (i, 4) = 2 or 3.For simplicity, we set V a = −V b = V throughout the paper if not otherwise specified.We set nearest-neighbor hopping as the unit of energy.
Through the Fourier transformation, the Hamiltonian Ĥ for the system with periodic boundary conditions can be written as 2 (1 ± cos k), and ξ 0 = t 2 sin k.The Pauli matrices s, σ, τ act on sublattice space, spin space and particle-hole space, respectively, while the nought subscripts represent identity matrices.
The system preserves time-reversal (T ), particle-hole (P) and chiral symmetries (C).The energy spectra for the system are given by Each energy level is four-fold degenerate.The energy gap for the two excitation bands E +,+ and The system preserves inversion symmetry with I = s x s x σ 0 τ 0 .In the absence of inversion-symmetric potentials, namely V = 0, the two Bogoliubov excitation bands are degenerate at inversion-symmetric k = π with ∆E = 0, as illustrated in figure 1(b).When V ̸ = 0, an energy gap ∆E ̸ = 0 is opened, as shown in figure 1(c).Therefore, the introduction of inversion-symmetric potentials opens the gap for excitation bands, which implies a topological phase transition as discussed in the following.
To demonstrate the topological properties of the system, the eigenenergies for a chain with open boundaries are computed and plotted in figure 2(a).We observe four degenerate states emerge at the gap between the excitation bands.Two states localize at the left end and the other two localize at the right end of the chain, as illustrated in figure 2(c).So far, we have focused on the special cases V a = −V b = V for simplicity.We would like to remark that if V a ̸ = −V b , the system also preserve the inversion symmetry, and the Bogoliubov edge states could also be driven from a the conventional s-wave superconductor.In  We also note that the energy gap ∆E between the two excitation bands above zero energy is determined by ∆ 0 and the onsite mirror-symmetric potential.The topological corner modes remain isolated from the bulk excitations in the energy gap even if ∆ 0 decreases an infinitesimal quantity when We would use the Wilson loop approach to characterize the bulk topology of the system with inversion symmetry.The base momentum point is set to be k .The corresponding Bloch wave functions are denoted by |u m (k)⟩ with m representing the band index.We construct a matrix where n stands for the number for occupied bands.The Wilson loop operator then is defined as where ∆k = 2π /N, and N is the number of unit cells.The effective Hamiltonian is defined by H = −i ln W/π .The eigenvalues for H are denoted by v s with s = 1, 2, . .., n.The bulk topological invariant is then given by ξ = ∑ n s=1 v s .Through numerical calculations, the topological invariant is given by ξ = ±2 in superconductor phase if V ̸ = 0, suggesting two localized Bogoliubov edge excitations would appear at each one edge of the 1D lattice, as demonstrated in figure 2. The localized Bogoliubov edge excitations are topologically protected and robust against inversion-symmetric perturbations as long as the bulk energy gap remains open.See appendix for detailed discussions.
The above simple toy model exhibits interesting topological properties induced by inversion-symmetric potentials.However, the long-range superconductor order in a 1D realistic system is forbidden due to the strong quantum fluctuations.In the following, we would propose a realistic platform to manifest intrinsic first-order and higher-order topology, whose topology can be explicitly understood from the above 1D model.

Bogoliubov corner excitations in an s-wave superfluid on a square lattice
Here we consider ultracold Fermionic atoms with pseudo spin loaded in a 2D square lattice.The physics for the system is described by a tight-binding Hamiltonian as where U is the strength of an onsite attractive SU(2)-invariant interaction, and µ denotes the chemical potential.Given a local dip potential with mirror symmetry applied to a one-dimensional line as shown in figure 3(a), the one-dimensional defect chain also enjoys the mirror symmetry along x.The total Hamiltonian then becomes where 3 on sites on the defect chain "Def".
As the interaction U becomes stronger, the fermions would be paired and enter a superfluid phase when U exceeds a critical value.The superfluid order at the lattice site i is assumed as Through the Bogoliubov-Valatin transformation, the creation operators ĉ † i,↑ and ĉ † i,↓ are written as where N u is the number of unit-cells, and ψ † n and ψn are creation and annihilation operators for Bogoliubov quasi-particles such that the Hamiltonian Ĥsqu can be diagonalized.The coefficients u n i,σ and υ n i,σ can be derived from the following equations where Ĥ0,ij,σ denotes the element of the Hamiltonian matrix Ĥsqu with U = 0 under the basis Ψ = ( Ĉ1 , . .., Ĉm , . .., Ĉ2Nu ) T with Ĉm = (ĉ m,↑ ,ĉ m,↓ ).
Through the numeric calculations, we compute the superfluid order parameter at each lattice site on a square optical lattice under open boundary conditions, as shown in figure 3(c).The superfluid order on the defect chain is weaker than that in other regions due to the non-zero mirror symmetric potentials.In addition, we observe that the superfluid order on the boundary is stronger than that in the bulk, and the superfluid order in the bulk is nearly uniform.This is consistent to the intuition that the lattice sites in the bulk are less affected by the boundary.The eigenenergy distributions versus potential V have been shown in figure 3(b), indicating isolated states (denoted by red lines) emerge in the energy gap for Bogoliubov quasiparticles.The particle density distributions for the isolated states, as plotted in figure 3(d), showcase these in-gap states are localized at the end of the defect chain.
We would like to remark that the above defect chain can be considered as a one-dimensional s-wave superfluid imprinted on the 2D lattice.While the defect chain couples with other chains, it also exhibits topological nontrivial properties as long as the energy gap remains open.

Bogoliubov corner excitations in an s-wave superfluid on a honeycomb lattice
Consider a two-component Fermi gas loaded in a 2D honeycomb optical lattice with a uniform chemical potential.Turning on the onsite attractive interaction for fermionic atoms, the fermions would be paired and enter the s-wave superfluid phase from the semimetal phase when the interaction exceeds a critical value [41].Its Bogoliubov excitations are gapless and show trivial properties.Hereafter, we would consider there exists onsite potential with mirror symmetry as shown in figure 4(a), and showcase Bogoliubov corner excitations could be induced from the excitation bands.
At the mean-field level, the physics of a system with a mirror-symmetric potential is described by the Hamiltonian as under the basis vector 2,3,4 denotes the superfluid order parameter on the sublattice site m as indexed in figure 4(a).The Hamiltonian preserves mirror symmetries, M x = (s x s x .D) σ 0 τ z and M y = (s 0 s 0 .D) σ 0 τ 0 , where D is a diagonal unitary matrix [42].The self-consistent equations for the superfluid order and particle filling ratio are given by Through numerical self-consistent calculations for equations ( 11) and ( 12), we find ∆ m ≡ ∆ and n m ≡ n for all m at zero temperature.The rich phase diagram, which has been shown in figure 4(b), exhibits a range of interesting and physically distinctive phases including semimetal (SM), second-order topological insulator (STI), and superfluid phases with a non-zero superfluid order (normal superfluid and superfluid with Bogoliubov corner excitations).Here the chemical potential µ is set to be zero, so the filling ratio is not fixed.Figure 4(b) showcases that at fixed interaction, for example U = 4, as V becomes larger, the system would enter STI since it is hard to form pairing when V exceeds a critical value.We compute the energy spectra for the superfluid under periodic boundary conditions and the numerical results are presented in figure 5. We observe the s-wave superfluid order opens the energy gap for the Dirac semimetal, as shown in figures 5(a) and (b).However, the Bogoliubov excitation band remains gapless.After turning on the onsite potential with mirror symmetry, a direct energy gap emerges, as depicted in figure 5(c).As the potential strength increases, the energy gap becomes larger and a full gap exists when the potential exceeds a critical value, as illustrated in figure 5(d).
To explore the nontrivial properties of Bogoliubov excitation bands, we calculate the eigenenergies for the superfluid versus V with fixed interaction U under open boundary conditions.Four degenerate states emerge in the energy gap for Bogoliubov excitations, as shown by the red lines with four-fold degenerates in figure 6(a).They are localized at two corners of the sample as shown in figure 6(c).Through the numeric calculations, we compute the superfluid order parameters at each lattice site on a honeycomb optical lattice under open boundary conditions, as shown in figure 6(b), we can observe that the bulk superfluid order is uniform.Through comparing figure 6(c) with (d), it is also clear that with increasing potential V, the Bogoliubov corner modes become more localized.In summary, there are two different Bogoliubov excitations in the superfluid phase, one with gapless Bogoliubov excitation bands dubbed NSF, and the other with gapped Bogoliubov excitation bands called CSF, as shown in figure 4(b).We would like to emphasize that figure 4(b) combines the quantum phase diagram (consisting of semimetal, second topological insulator and superfluid) and the state diagram (consisting of supefluid with and without Bogoliubov corner excitations).The transition between STI and CSF is a second-order phase transition from a band insulator to an s-wave superfluid.For example, at a fixed interaction U = 6, the superfluid phase gradually evolves as V decreases.The energy gaps for the Bogoliubov excitation bands in CSF and single-particle excitation bands in The emergence of Bogoliubov corner excitations is the exhibition of the bulk topology, characterized by the topological invariant protected by mirror symmetry.Taking a similar procedure as in the one-dimensional case above, the topological invariant at each k y is defined by where the Wilson loop operator reads W x,k = F x,k+Nx∆kx . ..F x,k+∆kx F x,k , ∆k x = 2π /N x , and N x is the number of unit cells in the x direction.The entry of matrix with N y the number of unit cells in the y direction, and ξ ′ y takes similar form as ξ ′ x .Through numeric calculations, we obtain the topological invariant (ξ ′ x , ξ ′ y ) = (2, 0) in the CSF regime and (0, 0) in NSF regime, as shown in figure 4(b).In summary, the s-wave superfluid phase have two different types of Bogoliubov excitations: trivial Bogoliubov excitations in NSF regime, and higher-order Bogoliubov corner excitations in CSF regime.We emphasize that the ground state of the s-wave superfluid in both regimes is topologically trivial.The topological property of excited corner modes originates from the Bogoliubov excitation bands.

Discussions and conclusions
The s-wave superfluid with a uniform chemical potential exhibits trivial Bogoliubov excitation on a 1D lattice, and 2D square or honeycomb lattice.Intriguingly, we find the onsite potential with mirror symmetry could open the energy gap in Bogoliubov excitation spectrums.The Bogoliubov excitation bands exhibit topological nontrivial properties and the edge modes manifest themselves as zero-dimensional (0D) Bogliubov excitations localized at the end of a 1D lattice and the corners of a 2D honeycomb lattice, although the ground states for the systems remains in a trivial phase.Since the systems preserve inversion or mirror symmetry, the winding number can characterize the nontrivial excitation band.
We would like to remark that our model in this work can be implemented in ultracold atoms.The 2D square and honeycomb optical lattices have been implemented in experiments [43][44][45][46][47].The mirror symmetric potential could be realized by tuning the laser beams.For example, the mirror-symmetric potential on 2D square lattice can be achieved through a pair of coherent counterpropagating laser beams with wave length 2a and 8a along x [48].The onsite attractive interaction could be finely tuned through Feshbach Resonance technique [49][50][51][52].The Bogoliubov excitation band could be detected through the momentum-resolved spectroscopy based on two-photon process that transfers energy and momentum to the New J. Phys.26 (2024) 033050 W Tu et al ensemble of atoms [53].The localization character of topological corner modes can be determined by looking at their localization length detected by a spectroscopy setup [54].
In summary, we propose that topological Bogoliubov excitations can be induced solely by onsite potentials in a topologically trivial conventional s-wave superfluid.The edge excitations manifest themselves as 0D modes localized at edges or corners of the system.These modes are robust against inversion or mirror symmetric perturbations as it preserves the degeneracy.Our work provides new insights for understanding higher-order topological excited states in conventional superconductors and superfluids, and also provides realistic platforms for engineering nontrivial Bogoliubov corner excitations in real experiments.

Appendix. Robustness of edge and corner modes against perturbations
We consider two cases to show the robustness of edge and corner modes against mirror-symmetric random perturbations on onsite potentials.We first impose the perturbations on a 1D lattice with edge modes.It takes the form Ĥper = ∑ i,σ Γ i ĉ † i,σ ĉi,σ , where the random potential Γ i = κγ i , κ represents the amplitude, γ i = γ N−i+1 ∈ [0, 1] is a random quantity with i ⩽ N/2 and N the number of lattice sites.We take a 1D lattice with N = 40 and compute the energies of the system for different onsite perturbations, as shown figure 7(a).Figure 7(b) showcases the particle density distribution of the corner modes for κ = 0.1.The above results indicate that four degenerate in-gap corner modes are robust against mirror-symmetric perturbations while they may acquire a finite energy shift.Next we consider the mirror-symmetric perturbations on the honeycomb lattice with corner modes.Similar results are obtained as shown in figures 7(c) and (d).In summary, the degenerate edge and corner modes are robust against the mirror-symmetric perturbations on onsite potentials as long as the energy gap remains open.We would like to remark that the degeneracy of edge and corner excitations is protected by the inversion or mirror symmetry.If the inversion or mirror symmetry is broken by the random local disorder, the degeneracy would be lifted and the edge and corner modes may disappear.

Figure 1 .
Figure 1.(a) Illustration of a 1D lattice with inversion-symmetric onsite potentials Va and V b .Each unit cell consists of four sublattice sites indexed by 1-4 (from left to right).(b) and (c) Energy spectra for 1D superconductor with onsite potentials Va = −V b = 0 and Va = −V b = 0.2, respectively.The superconductor order parameter is set to be ∆0 = 0.6 in (b) and (c).Common parameter is set to be t = 1.

Figure 2 .
Figure 2. (a) and (b) Eigenspectrum versus inversion-symmetric onsite potential V for a 1D lattice with 40 sites.The in-gap red lines denote four-degenerate Bogoliubov edge excitation modes, where each edge of 1D lattice hosts two localized modes.In panel (a), we set Va = −V b = V while in (b), we choose Va = V, V b = −0.8V.(c) and (d) The particle density distribution versus the site index, corresponding to the colored dish in panels (a) and (b) respectively.To be specific, in panel (c) Va = −V b = V = 1 and in panel (d) Va = 1, V b = −0.8.Common parameters are set to be ∆0 = 2, t = 1.

Figure 3 .
Figure 3. (a) Illustration of a square lattice with a defect chain respecting the mirror symmetry.(b) Eigenspectrum for the s-wave superfluid versus mirror-symmetric onsite potential V on the 20 × 19 square lattice.The in-gap red lines indicate four-degenerate Bogoliubov corner excitation modes.(c) Distributions of s-wave superfluid order parameters on the square lattice with a defect chain.(d) Particle density distributions of the in-gap states.The blue dashed line denotes the defect chain.The Bogoliubov excited states shown in (c) and (d) have been indicated by the blue dot in (b) with the mirror-symmetric onsite potential V = 5.Common parameters in (b)-(d) are set to be t = 1, U = 10, µ = 0.05.

Figure 4 .
Figure 4. (a) Illustration of a honeycomb lattice with a mirror-symmetric onsite potential.Each unit-cell consists of four sublattice sites indexed by 1-4 with onsite potential configuration (V1, V2, V3, V4) = (Va, V b , V b , Va).(b) A rich global phase (state) diagram plotted against the potential and interaction strength including second-order topological insulators (STI), semimetal (SM), normal superfluid (NSF) and superfluid with Bogoliubov corner excitations (CSF).The parameters are set to be t = 1 and µ = 0.

Figure 6 .
Figure 6.(a) Eigenspectrum versus mirror-symmetric onsite potential V for the honeycomb lattice.The in-gap red lines denote four-degenerate Bogoliubov corner excitation modes.(b) Distributions of s-wave superfluid order parameters on the honeycomb lattice.(c) and (d) Particle density distributions of the in-gap states.These chosen parameters in sub-figures also have been indicated by colored dots in (a).In (b) and (c), Va = −V b = V = 2.2.In (d), Va = −V b = V = 4.5.Common parameters are set to be t = 1, U = 10.

Figure 7 .
Figure 7. (a) Eigenenergies E vs the state index in the presence of mirror-symmetric perturbations with different amplitude k = κ on a 1D lattice.The parameters are ∆0 = 2, V = 1 (b) The particle distribution of the in-gap modes denoted by stars in (a).(c) and (d) Similar to (a) and (b) but with U = 6 and V = 2 on a honeycomb lattice.Common parameter is set to be t = 1.
⟨u m,k+∆kx |u m,k ⟩ with |u m,k ⟩ being the Bloch wave function of the energy bands E m (k), i.e. h h (k) u m,k = E m (k) while ξ x