Detecting chiral pairing and topological superfluidity using circular dichroism

Realising and probing topological superfluids is a key goal for fundamental science, with exciting technological promises. Here, we show that chiral px + ipy pairing in a two-dimensional topological superfluid can be detected through circular dichroism, namely, as a difference in the excitation rates induced by a clockwise and counter-clockwise circular drive. For weak pairing, this difference is to a very good approximation determined by the Chern number of the superfluid, whereas there is a non-topological contribution scaling as the superfluid gap squared that becomes significant for stronger pairing. This gives rise to a competition between the experimentally driven goal to maximise the critical temperature of the superfluid, and observing a signal given by the underlying topology. Using a combination of strong coupling Eliashberg and Berezinskii-Kosterlitz-Thouless theory, we analyse this tension for an atomic Bose-Fermi gas, which represents a promising platform for realising a chiral superfluid. We identify a wide range of system parameters where both the critical temperature is high and the topological contribution to the dichroic signal is dominant.

Introduction The realisation and manipulation of topological superfluids and superconductors is presently one of the most actively pursued goals in physics.In addition to being interesting from a fundamental science point of view, their Majorana edge modes promise applications for quantum computing [1].Zero-energy states at the ends of one-dimensional (1D) nanowires have been observed, consistent with the presence of Majorana modes [2,3].So far, there has however been no observation of topological superfluidity in 2D.The most promising solid-state candidate for a 2D topological superconductor is Sr 2 RuO 4 , but the precise symmetry of the order parameter in this crystal remains subject to intense debate [4,5].It has recently been shown that an atomic 2D Fermi gas immersed in a BEC offers a promising platform for realising a topological superfluid [6][7][8].The fermions form Cooper pairs with chiral symmetry by exchanging sound modes in the BEC, and the system offers sufficient flexibility so that one can tune the superfluid critical temperature to be within experimental reach.Experimentally, such a Bose-Fermi mixture has been realized using 173 Yb- 7 Li atoms, which constitutes an important step towards the first unequivocal realisation of a topological p x +ip y superfluid [9].
A key question concerns the detection of topological superfluidity in atomic gases.Their topological properties are not easily extracted from thermodynamic measurements nor using common probes such as radiofrequency spectroscopy [10].Contrary to the chiral edge modes of single-particle band structures, which have been detected in experiments [11], the observation of Majorana states [12,13] is complicated by their small number and their particle-hole nature.
It was recently proposed [14,15] and experimentally demonstrated [16] that the topologically invariant Chern number can be detected in atomic gases through circular dichroism, namely, by analyzing excitation rates upon applying a circular drive.This topological probe was first introduced for non-interacting Chern insulators [14], and later applied to interacting many-body systems [17][18][19].Inspired by this approach, we hereby demonstrate that the chirality of the p x +ip y pairing is revealed in the circular dichroism of the superfluid.In particular, the difference of excitation rates obtained from opposite drive orientations, integrated over the drive frequency [14], is shown to be determined by the Chern number of the topological superfluid for weak pairing, whereas a non-topological contribution scaling as the superfluid gap squared becomes significant for strong pairing.The resulting competition between maximising the superfluid critical temperature while detecting a genuine topological signature is analysed for a concrete atomic Bose-Fermi mixture.Using the strong-coupling Eliasberg equations combined with Berezinskii-Kosterlitz-Thouless (BKT) theory, we identify a wide and accessible parameters regime where the superfluid critical temperature is high and the dichroic signal dominated by the topological Chern number.Our results demonstrate that the dichroic probe offers an experimentally promising pathway to detect topological superfluidity.

Topological responses in superfluids
We first establish a connection between circular dichroism, the Hall conductivity, and the Chern number of the superfluid.Consider a 2D system of spin-polarised fermions described by arXiv:2003.11610v1[cond-mat.quant-gas]25 Mar 2020 the Hamiltonian where ψ(r) is the fermion field and V (r−r ) is an interaction giving rise to p x + ip y pairing.The latter may result from a p-wave Feshbach resonance or, as we will consider later, from the induced interaction in a Bose-Fermi mixture.Within BCS theory, this p-wave superfluid can be described by the Hamiltonian where T and where τ = (τ 1 , τ 2 , τ 3 ) T with τ i the Pauli matrices.Here where µ is the chemical potential, and ∆ k is the gap parameter ( = 1 throughout).The latter has a chiral p-wave symmetry, i.e. ∆ k = ∆ k e iφ , where φ is the polar angle of the momentum k.This results in a topological phase characterised by a Chern number C = −1 for µ > 0 whereas C = 0 for µ < 0 [20].The Chern number reads where is the BCS quasiparticle energy.The second term in the integrand scales as ∆ 2 k /µ 2 so that the Chern number can be approximated by the first term in the regime ∆ k µ.We now show that the Chern number in Eq. ( 3) can be extracted from circular dichroism, namely, by monitoring excitation rates upon a circular drive [14,16].We consider a circular drive of the form and we will set q → 0 at the end of calculations, corresponding to a uniform circular shaking [16].This reads in second quantization, where n(r) = ψ † (r)ψ(r) is the density operator, n(q) is its Fourier transform and ñ(q) = [n(q) − n(−q)]/2.Within linear response, the excitation rate out of the ground state of H 0 induced by V ± (q) can be calculated using Fermi's golden rule as where |g and |f denote the ground and excited states of H 0 with energy E g and E f , respectively.The observable of interest is provided by the differential integrated rate (DIR), which is defined as [14] Substituting Eq. ( 6) into Eq.( 7), we find where we have used momentum conservation to eliminate terms [21].We now use the continuity equation to write ∆Γ in terms of the density-current correlation function.
From ∂ t n(r, t) + ∇ • j(r, t) = 0, we find where the Fourier transform of the current reads Using Eqs. ( 8)-( 9) and noting that lim we find the relation which connects the DIR to the static Hall conductivity Here A is the system's area, and χ A,B (q, ω) is the Fourier transform of the retarded correlation function with θ(x) the Heaviside function.
The Hall conductivity of a superfluid was previously shown to be related to a Chern number; see Refs.[23,24] in the context of chiral 3 He superfluids.Using BCS theory, and upon taking the limits (q, Ω) → 0 in the proper order [22][23][24][25][26], one obtains an expression for the Hall conductivity [21] σ xy ≡ lim q→0 χ jx,n (q, 0) where the last equality is obtained by comparing with the Chern number in Eq. ( 3).In contrast with the case of Chern insulators, where the Hall conductivity is genuinely topological in the thermodynamic limit [29], the Hall response of the superfluid [Eq.( 15)] contains a correction scaling as O(∆ 2 /µ 2 ).This result was previously related to the fact that the edge current of a chiral p-wave superconductor is not strictly topological, as opposed to the presence of edge (Majorana) states [30][31][32].
Finally, combining Eqs. ( 12) and ( 15) yields the central result of this work, which shows that the DIR related to the dichroic probe is closely related to the Chern number of the superfluid phase: this observable exhibits a jump proportional to the Chern number to order O(∆ 2 /µ 2 ) whenever the superfluid enters the topological phase with C = −1.
Dichroic probe for a topological Bose-Fermi mixture We now explore the dichroic probe for a concrete system consisting of a 2D gas of fermionic atoms immersed in a 3D BEC.The fermions interact by exchanging sound modes in the BEC, which leads to an induced attractive interaction and Cooper pairing [6].Since both the range and strength of the induced interaction can be varied, one can tune the mixture in order to reach a high critical temperature.This makes such a mixture a strong candidate for observing a chiral pairing.Recently, progress towards realising this goal was reported with the experimental realisation of a 173 Yb-7 Li mixture [9].We now analyse how the dichroic probe can be used to detect topological pairing in this specific system.
Due to the finite speed of sound in the BEC, the interaction between the fermions mediated by the bosons is not instantaneous, thus giving rise to retardation effects.The latter are included in the frequency-dependent Eliashberg equations [21].It has been shown that retardation effects are small when the bosons in the BEC are light compared to the fermions such as for the 173 Yb- 7 Li mixture [33].The induced interaction is then close to the static Yukawa form Here, n B and m B is the density and mass of the bosons, ξ B = 1/ √ 8πn B a B is the BEC healing length with a B the boson-boson scattering length, and a eff is the mixed dimensional Bose-Fermi scattering length.
According to Eq. ( 7), one should measure the differential rate Γ + −Γ − integrated over all frequencies.However, any real measurement necessarily introduces an upper cut-off frequency Ω c above which there is no signal [16].Using Eqs. ( 12) and ( 15), the resulting signal reads The cut-off θ(Ω c − 2E k ) reflects that the probe breaks pairs with energy 2E k in the long wave length limit.We note that ∆Γ = lim Ωc→∞ ∆Γ trunc (Ω c ).
In Fig. 1, we plot ∆Γ trunc (Ω c ) for a 7 Li-173 Yb mixture with a Bose-Fermi coupling n  18).This quantity is zero for cut-off frequencies be- low twice the gap, i.e. for Ω c < ∼ 0.1E F where E F is the 2D Fermi energy, reflecting that there is not enough energy in the probe to break pairs.Above this threshold, the DIR quickly converges towards to the Chern number for Ω c > ∼ E F .Since ∆ 0.05E F E F for this set of parameters, the deviation of 2∆Γ/AE 2 away from the Chern number is small.We also plot in Fig. 1 the differential rate at a given frequency Ω, This difference is large for frequencies just above the threshold given by twice the gap, where the density of states of the superfluid is highest, and Fig. 1 shows that that one only needs to measure the difference up to a few times the pairing gap to resolve the Chern number.One of the appealing features of the Bose-Fermi mixture is that the critical temperature for the 2D superfluid can be tuned to be close to the maximum value T c /T F = 1/16 allowed by BKT theory.Maximising T c will however also increase the gap and thereby increase corrections to the DIR away from the Chern number as seen from Eq. ( 16).To analyse this tension, we plot in Fig. 2 the DIR ∆Γ and the critical temperature T c as a function of the gas parameter n and two different Bose-Fermi interaction strengths.The critical temperature is calculated by combining strong coupling Eliashberg and BKT theory, which includes the frequency dependence of the gap; see Refs.[6,21] for details.We see that the critical temperature increases with decreasing gas parameter reflecting that the range of interaction in Eq. ( 17), given by the BEC coherence length, increases.The gap consequently also increases leading to a larger correction term for the DIR away from ∆Γ = AE 2 C/2.Nevertheless, Fig. 2 shows that there is a significant region where both the DIR is close to the topological value and the critical temperature is close to its maximum value T c /T F = 1/16.In fact, any non-zero value indicates chiral pairing, since the DIR is zero in a phase with time-reversal symmetry.Note that we expect our calculation to give a lower bound on the DIR, since BCS theory likely overestimates the gap.To further illustrate the competition between maximising the critical temperature and measuring a value of ∆Γ determined by the underlying topology, we plot ∆Γ as a function of T c in Fig. 3 for the same parameters as in Fig. 2.This demonstrates that in order for the dichroic probe to yield a value close to that given by the Chern number, one should cool to around T ∼ 0.06E F , which is within present day technology.This makes the dichroic probe promising for detecting topological superfluidity in 2D.It also shows that a stronger Bose-Fermi interaction strength is slightly more favorable although the difference between the two interaction strengths is small.

Conclusion
We showed that chiral p x + ip y pairing in a 2D superfluid can be detected through circular dichroism.Contrary to the case of topological insulators [14], the DIR is not purely dictated by the Chern number due to a correction term scaling as ∆ 2 /E 2 F , giving rise to a competition between maximising the critical temperature of the superfluid and observing the Chern number from such a dichroic probe.As a concrete example, we considered an atomic Bose-Fermi mixture.Using a combination of Eliashberg and BKT theory, it was demonstrated that there is in fact a wide range of values for the system parameters where both the critical temperature is high and the dichroic signal close to the value given by the Chern number.This combined with the fact that a similar scheme was recently successfully applied to detect topological order in a Chern Bloch band [16], leads to the conclusion that the dichroic probe is a strong candidate for detecting topological p x + ip y pairing in an atomic system.Here, we discuss the role of momentum conservation in the derivation of Eq. ( 12) in the main text.When deriving Eq. ( 8), we use that A(q), B(q ) ∝ δ q,−q for a strictly infinite translationally invariant system.From this, it follows that lim q→0 n(q)n(q) = 0 and such terms can be discarded.However, for a finite physical system of size L, momentum is only defined with a resolution ∼ 1/L.This means that n(q)n(q) starts to become non-zero for q < ∼ 1/L and in particular lim q→0 n(q)n(q) = n(0)n(0) for a finite system, which leads to the extra factor of 2 on the right hand side of Eq. ( 12).Physically, it means that a finite system cannot distinguish between a force with wave length much greater than the system size from a uniform force.We note that uniform circular shaking (q = 0) can be realized in ultracold-atom experiments.

THE HALL CONDUCTIVITY FOR THE SUPERFLUID
In this section, we derive the Hall conductivity of the superfluid resulting in Eq. (15) in the main text.The current-density correlation function is in terms of the Green's function of the superfluid given by [34] χ jx,n (q) = − k k x 2m tr [G 0 (k − q/2)G 0 (k + q/2)τ 3 ] (S. 1) where in shorthand notation k = (k, ω n ) and q = (q, ω m ) with ω n,m being fermionic Matsubara frequencies.We can calculate χ jx,n to see how this encodes the Chern number.One way of doing it is by inserting the Green's function on the spectral form, where Inserting this in Eq. (S.1) and performing the Matsubara sum, we get the limit lim ω→0 χ jx,n (q, ω) = χ jx,n (q, 0) = − 1 To first order we have The first of these terms vanish, since the summand is odd in k y .This can be seen if we fix the phase of the gap function and look at, for instance, the simple example ∆ k = k x + ik y .With this, it is clear that lim q→0 lim ω→0 χ jx,n (q, ω) In the main text, it is shown how this relates to the Chern number.

1 / 3 B 1 / 3 B 2 F /n 1 / 3 B
a eff = 0.12, BEC gas parameter n a B = 0.1, and density ratio n 1/= 0.5, where n F is the 2D Fermi density.These results are obtained by first solving the BCS equations self-consistently at zero temperature and then evaluating the DIR from Eq. (

FIG. 1 .
FIG. 1. (Color online).The differential integrated rate ∆Γtrunc(Ω) between a clockwise and counterclockwise perturbation as a function of the cut-off frequency.The insert shows the difference in the heating rates between a clockwise and counterclockwise perturbation as a function of frequency.

FIG. 2 . 3 B
FIG. 2. (Color online).The critical temperature (red) and the differential integrated rate ∆Γ (black) as a function of the BEC gas parameter n 1/3 B aB for two different Bose-Fermi interaction strengths.

FIG. 3 . 3 B 3 B
FIG. 3. The DIR as a function of the critical temperature for the same parameters as in Fig. 2. The dashed line corresponds to n 1/3 B a eff = 0.1 and the solid line to n 1/3 B a eff = 0.15.
Acknowledgements.-J.M.M. and G.M.B. wish to acknowledge the support of the Danish Council of Independent Research -Natural Sciences via Grant No. DFF -4002-00336.Z.W. acknowledges the support by the National Science Foundation of China (Grant No.11904417) and the Key-Area Research and Development Program of GuangDong Province (Grant No.2019B030330001).N.G. is supported by the ERC Starting Grant TopoCold, and the Fonds De La Recherche Scientifique (FRS-FNRS, Belgium).