-wave chiral superfluidity from an -wave interacting atomic Fermi gas
Chiral -wave superfluids are fascinating topological quantum states of matter that have been found in the liquid He-A phase and arguably in the electronic SrRuO superconductor. They are shown fundamentally related to the fractional quantum Hall state which supports fractional exotic excitations. A common understanding is that such states require spin-triplet pairing of fermions due to -wave interaction. Here we report by controlled theoretical approximation that a center-of-mass Wannier -wave chiral superfluid state can arise from spin-singlet pairing for an -wave interacting atomic Fermi gas in an optical lattice. Despite a conceptually different origin, it shows topological properties similar to the conventional chiral -wave state. These include a non-zero Chern number and the appearance of chiral fermionic zero modes bounded to domain walls. Several signature quantities are calculated for the cold atom experimental condition.
Introduction.— Topological superconductors, like the type of -wave pairing studied in the liquid He Volovik (2003) and strontium ruthenates Kallin (2012), are among the most desirable unconventional many-body states in condensed matter physics Nayak et al. (2008). In two dimensions, their topological properties are fundamentally linked to a class of fractional quantum Hall states of non-Abelian statistics Read and Green (2000). Studies of vortices in such materials point to fascinating braiding statistics and applications in topological quantum computing. The fate of topological superconductivity in two-dimensional electronic matter remains however debatable. In the field of ultracold atoms, this phase was predicted to appear near the -wave Feshbach Resonance in Fermi gases Gurarie and Radzihovsky (2007). However, the life time of such systems is severely limited by the three-body collisions, and achieving superfluidity in the resonance regime was found experimentally challenging Regal et al. (2003). Other strategies like spin-orbit coupling or dipolar interaction also meet new difficulties such as heating or ultracold chemical reactions Galitski and Spielman (2013); Baranov et al. (2012). There is a separate approach being proposed to get around—hybridizing materials of separate topological and superconducting properties, which also encounters some engineering difficulties Qi and Zhang (2011). After all, the search for the homogeneous chiral -wave superconductivity in two dimensions has stood largely open for both electronic and atomic matter systems.
Here we report the discovery of a new mechanism to achieve chiral topological superfluidity. We shall demonstrate this with cold fermionic atoms in optical lattices with the model to be introduced below. The key concept dramatically departing from the conventional wisdom that relies on the or higher partial wave pairing in relative motion is to keep the fermion interaction within the usual -wave channel by pairing fermions from different Wannier orbitals, and the center-of-mass orbital motion of condensed pairs is examined for possible nontrivial topology. Recently the research of higher orbital bands in optical lattices has evolved rapidly 201 (), where the orbital degrees of freedom are found to play a crucial role as in solid state materials. From the early experimental attempt Müller et al. (2007) to the breakthrough observation Wirth et al. (2011) of long-lived -band bosonic atoms in a checkerboard lattice, a growing evidence points to an exotic orbital Bose-Einstein condensate 201 (). For fermions with attractive interaction, superfluid states similar to the type of Fulde-Ferrell-Larkin-Ovchinnikov were found in the theoretical studies of pairing in the -bands Cai et al. (2011) and that between the -band and a single -band Zhang et al. (2010). As we shall show with the model below, pairing fermions from the orbitals of different angular momenta can lead to other unexpected results.
Let us consider an attractive -wave interacting Fermi gas composed of two hyperfine states, to be referred to as spin and , loaded in a spin-dependent 2D optical lattice shown in Fig. 1(a). The spin dependence of the lattice is motivated by various theoretical designs Liu et al. (2004); Daley (2011) and most importantly the recent experimental demonstration of it with bosons McKay and DeMarco (2010); Soltan-Panahi et al. (2012). Further let the gas be tuned with a population imbalance between the two spin species by the techniques developed in the recent experimental advances Zwierlein et al. (2006); Partridge et al. (2006); Nascimbne et al. (2010). A key condition that we propose here is to tune the population imbalance (or equivalently the chemical potential difference) sufficiently large such that the spin and Fermi levels reside in the and orbital bands, respectively. The rotation symmetry () of the lattice dictates that the two orbital bands, and , are degenerated at the high symmetry points in the momentum space. Later on we shall see that this symmetry and hence degeneracy are necessary for the -wave paired superfluidity. Technically speaking, the Bravais lattices for the spin up and down fermions are rotated from each other.
Effective model.— A system of fermionic atoms, say Li, loaded into an optical lattice (Fig. 1) in the tight binding regime is described by a multi-orbital Fermi Hubbard model
where describes tunneling pictorially represented in Fig.1(b) and (c) (the expression for is standard and is given in Supplementary Materials) and is the Hubbard interaction,
Here and are fermionic annihilation operators for the localized (, or ) orbitals on and sites, respectively. The interactions between and orbitals originate from interactions between two hyperfine states, which are tunable by the -wave Feshbach Resonance in ultracold atomic gases. We focus on the case with attractive interaction where superconducting pairing is energetically favorable.
The system, as described by the Hamiltonian in Eq. (1) exhibits lattice rotation and reflection symmetries. For the reflection in the and direction, the fermionic operators transform as and , respectively. Under the lattice rotation, These symmetries, reflection symmetries in particular, play an essential role in the following theory.
Two-Flavor Ginzburg-Landau theory.— From the analysis of Cooper’s problem (see Supplementary Materials), we conclude that condensation of Cooper pairs at is energetically favorable for the ground state, where is the lattice constant. Then, it is convenient to introduce two slowly varying bosonic fields and , which represent Cooper pairs and , respectively. That gives a two-flavor Ginzburg-Landau free energy respecting all the symmetries of the microscopic model as follows
with , and .
This free energy generalizes the theory of two-gap superconductivity as proposed in the context of transition metalsBabaev (2002). We have neglected temporal fluctuations of Cooper pair fields and such a treatment is valid at finite temperature away from quantum critical regime. In this theory, we want to emphasize two key points due to the reflection symmetries: first, and are decoupled at quadratic level; second, linear derivatives such as are prohibited. The absence of linear derivatives makes the fluctuations of suppressed, and condensation of Cooper pairs at is expected to be stable at least when is infinitesimal. For finite the stability (i.e., in Eq. (3)) is confirmed in our numerics (see Supplementary Materials).
With and obtained from integrating out fermions, we find a phase diagram shown in Fig. 2. With moderate attraction , a first order phase transition from the to phase occurs when is above some critical value. Surprisingly, when the attraction is strong enough , we find that even infinitesimal makes the favorable, opening a wide window for this non-trivial state. When , the system has symmetry, which means no phase coherence between the two components and . We also study the finite temperature phase transitions (see Supplementary Materials) and find that the Kosterlitz-Thouless transition temperature can reach about nk, being accessible in the current experimentsSchneider et al. (2008); Jordens et al. (2008), when the lattice strengths are and for and orbitals, respectively.
Gapless chiral fermions.— We now show that the superfluid state possesses important measurable signatures due to the broken time reversal symmetry which belongs to the Ising universality class. Following the standard procedure, our calculation finds that the state is topologically nontrivial by a non-zero Chern number, which is and for the and state, respectively. The topological properties are manifested in the existence of gapless chiral fermions, emergent on a domain wall connecting topologically distinct regions. In experiments, Ising domains of and are expected to spontaneously form as have been observed in the recent cold atom experiment studying ferromagnetic transitions Parker et al. (2013). In the following, we show that a domain wall defect carrying gapless fermions as bounded surface states is experimentally accessible.
Considering a lattice geometry in the presence of a domain wall decorated superconducting background as in Fig. 3 (a), the mean-field Hamiltonian is given by
The energy spectrum of fermionic excitations is obtained by diagonalizing Eq. (4). With the periodical boundary condition chosen in the direction (Fig. 3(a)), the momentum is a good quantum number and the energy spectra in Fig. 3(b) is thus labeled by . For the same reason as in quantum Hall insulators, the number of gapless chiral modes moving along the interface is topologically determined by the difference of the Chern numbers in regions on either side of the interface Haldane and Raghu (2008); in this case . This conclusion is confirmed in our numerics. As shown in Fig. 3(b), we find four gapless chiral modes, with two localized on the domain wall (purple color) and the other two on the outer edges of the lattice (red color). From their spectra , the two chiral modes on the domain wall have positive group velocities, which lead to anomalous mass flow along the domain wall. To characterize the localization of chiral fermions, we calculate the local density of states (LDOS) where is the eigenvector corresponding to the eigenenergy of Hamiltonian Eq. (4) and runs over all the Wannier orbitals (, or ) on and sites. The peak of LDOS located at the position of the domain wall, as shown in Fig. 3(c), (d) and (e), illustrates the existence of localized gapless surface states, reminiscent of the quantum Hall edge states. Taking a laser wavelength of nm typical for the current optical lattices, the width of the LDOS peak is estimated about . This is greater than the reported spatial resolution (about ) in the radio frequency spectroscopy measurement Shin et al. (2007), which makes the detection of this signal experimentally accessible.
In summary, when studying a spin imbalanced atomic Fermi gas with an -wave interaction, we find surprisingly a topological -wave superfluid state whose pairing symmetry and topological origin differ from the previous known superconducting or superfluid phases. To emphasize a remarkable difference, this phase does not require an interaction beyond the usual attractive -wave component. Hence a short-ranged contact interaction as has been widely realized in cold gases should satisfy well. A key concept is the fermionic Cooper pairing between the orbitals of different angular momenta in an optical lattice. For the example presented here, they are the parity even and odd orbitals. The -wave symmetry refers to the center-of-mass motion, not to the relative motion of each fermion pair as in the well-known He superfluid. For free or repulsively interacting systems, previous studies found that mixing orbitals of opposite parities leads to topological semimetal and insulator phases Li et al. (2013); Sun et al. (2012). Whether or how the two phenomena from either sign of the interaction are topologically related is an intriguing question for the future research.
Experimentally, one may consider the existing proposals for realizing spin-dependent optical lattices Liu et al. (2004); Daley (2011). Alternatively, the recent progress in group-II (alkaline-earth-metal) atoms points to the possibility of having even greater spin-dependence tunability if to load two-species fermionic atoms from the ground level and the long-lived metastable levels like and to take the advantage of the atomic orbit dependent AC Stark effect. This should in principle be able to make the lattices for different components being completely independent (so maximally spin-dependent lattice) by selection of the appropriate wavelengths Daley (2011). The appearance of chiral fermionic zero modes bounded to domain walls associated with the orbital Ising order is predicted to be a fascinating and concrete experimental signature for this novel state. Both zero and finite temperature phase diagram are also established, providing the estimates for potential experiments.
Acknowledgements.— The authors want to thank Randy Hulet and Andrew Daley for helpful discussions. This work is supported by AFOSR (FA9550-12-1-0079), ARO (W911NF-11-1-0230), DARPA OLE Program through ARO and the Charles E. Kaufman Foundation of The Pittsburgh Foundation (B.L., X.L. and W.V.L.), the National Basic Research Program of China (Grant No. 2013CB921903, 2012CB921300) and NSF of China (11274024, 11334001) (B.W.), and Overseas Collaboration Program of NSF of China (11128407) (W.V.L., B.W.). X.L. acknowledges support by JQI-NSF-PFC, ARO-Atomtronics-MURI, and AFOSRJQI- MURI.
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Appendix S-1 Hopping term
The hopping term in Eq. (1) can be written as
where the matrices and are given as
Here is the hopping amplitude between s orbital fermions; and are the longitudinal -bond and transverse -bond hopping amplitude for p orbitals, respectively; is the hopping amplitude between and orbitals and is the fermion annihilation operator located at .
Appendix S-2 Cooper’s Problem
From hopping term in Eq. (1), we find that there are five Bloch bands. The corresponding operators and for and bands are introduced, respectively. Because the width of p band is much larger than that of s band, intuitively we know that the condensation of Cooper pairs between these two bands at center-of-mass momentum , which is the energy minimal of p band, will be energetically favorable. Besides this intuitive picture, systematically, this conclusion is borne out by solving energy spectra of Cooper’s bound states, which are defined as where is the vacuum state and is the two-particle wavefunction. The summation here is over modes above the fermi level. Due to translational symmetry, the center-of-mass momentum is a good quantum number, which is used to label the energy spectra obtained from the eigenvalue problem, Resulting from lattice rotation symmetry , there are two branches of Cooper’s bound states, which are related to each other by rotation. These two branches are most clear in the limit of , i.e., without coupling between and orbitals. In this case, particle numbers of and orbitals are separately conserved. One type of bound state is formed by and orbital fermions leading to an energy dispersion ; while the other formed by and orbitals leads to a dispersion .
As shown in Fig. S1, we find that the bound state energy varies as a function of center-of-mass momentum and the energy minimal is located at . Due to symmetry, the energy minimal of also locates at . Condensation of Cooper pairs at is energetically favorable. The effect of finite coupling between and orbitals has been discussed in the frame work of effective field theory.
Appendix S-3 Path Integral approach
To calculate free energy from the path integral method, we introduce the Grassman fields and and express the grand partition function of the system as
with . The quartic term in the interaction term of action S can be decoupled with the Hubbard-Stranovich transformations,
Then the partition function can be written as
The action in Eq. (S4) is
where . After doing an unitary transformation of fermionic fields, we replace and in Eq. (LABEL:eq:action) by two slowly varying and time-independent bosonic fields and , respectively. Integrating the fermionic fields, we get an effective action
where is the inverse Green’s function and . By calculating free energy from Eq. (S6), we obtain coefficients , , and in Eq. (3). As shown in Fig. S2(a), since the low temperature limit is much smaller than the Fermi energy, the coefficient changes sign from positive to negative with increasing , which implies a second order phase transition from normal to a superfluid state with at mean field level (Fig. 2). Our numerics also find that and . Minimizing the free energy gives a field configuration with . The relative phase between and is fixed by as shown in Fig. S2(b) and (d). The coupling makes the relative phase locked at and leads to a superfluid state where the ‘’ sign is spontaneously chosen; while favors a state (Fig. 2).
Appendix S-4 Finite temperature phase transition
It is well known that in 2D the transition from the normal to superfluid state is of the Kosterlitz-Thouless type. To obtain the KT transition temperature, we should rewrite the complex order parameters and with the phase fluctuations and . Introducing new variables and , from the Gaussian fluctuation part of free energy in Eq.(3), we derive the well-known XY model in terms of as . Here, the relative phase is determined by the sign of in Eq. (3) at finite temperature. Specifically, is locked at [or ] for [or ]. The KT transition temperature is determined by the formula . Solving this equation self-consistently, we get the KT transition temperature and plot it in Fig. S3. We find that in the weak-coupling regime approaches the mean-field transition temperature as determined by in Eq. (3) at finite temperature. With stronger interaction, there is a large derivation of the two as expected Dupuis (2004), for the reason that mean field analysis underestimates fluctuation effects. Our numerics also find that can reach accompanying with increasing of interaction strength when the lattice strengths are and for and orbitals, respectively. As shown in the inset plot of Fig. S3, we also find that decreasing of lattice strength will increase .
Appendix S-5 Anisotropic superconducting gap
In this section, we discuss the superconducting gap for fermions resulting from this unconventional paring. The superconducting gap is calculated by solving the Mean field Hamiltonian (Eq. (4)) without a domain wall defect. The anisotropy of the gap which is a remarkable property being absent in the conventional s-wave superconductors is characterized by the structure functions and , where is the total site.
We find near the Fermi surface is highly anisotropic, that is it strongly depends on the polar angle of , , as shown in Fig. S4. In Fig. S4(a), when and , the Fermi surface of and orbital bands fermions are matched very well when , so the gap are almost at the same maximum value when in that region. However, when , the gap decreases by increasing due to the mismatch of Fermi surface. The situation is different for and , where the Fermi surfaces are mismatched. The gap is non-monotonic when varies from to , and it is maximal at (Fig. S4(b)). This peculiar non-monotonic behavior is related to van-Hove singularities which lead to large density of states nearby.