From semimetal to chiral Fulde-Ferrell superfluids
The recent realization of two-dimensional (2D) synthetic spin-orbit (SO) coupling opens a broad avenue to study novel topological states for ultracold atoms. Here, we propose a new scheme to realize exotic chiral Fulde-Ferrell superfluid for ultracold fermions, with a generic theory being shown that the topology of superfluid pairing phases can be determined from the normal states. The main findings are two fold. First, a semimetal is driven by a new type of 2D SO coupling whose realization is even simpler than the recent experiment, and can be tuned into massive Dirac fermion phases with or without inversion symmetry. Without inversion symmetry the superfluid phase with nonzero pairing momentum is favored under an attractive interaction. Furthermore, we show a fundamental theorem that the topology of a 2D chiral superfluid can be uniquely determined from the unpaired normal states, with which the topological chiral Fulde-Ferrell superfluid with a broad topological region is predicted for the present system. This generic theorem is also useful for condensed matter physics and material science in search for new topological superconductors.
pacs:03.75.Mn, 03.75.Lm, 05.90.+m, 05.70.Ln
Introduction.–The recent experimental realization of two-dimensional (2D) spin-orbit (SO) coupling for ultracold atoms Zhan2016 (); Zhang2016a (); Zhang2016b (), which corresponds to synthetic non-Abelian gauge potentials Ruseckas2005 (); Zoller2005 (); Liu2009 (), advances an essential step to explore novel topological quantum phases beyond natural conditions. Ultracold fermions with SO coupling can favor the realization of topological superfluids (TSFs) (similar as topological superconductors in solids Read2000 (); Kitaev2001 (); Kitaev2003 (); Fu2008 (); Sau2010 (); Lutchyn2010 (); Oreg2010 ()) based on an -wave Feshbach resonance Zhang2008 (); Sato2009 (), which are highly-sought-after quantum phases for their ability to host non-Abelian Majorana zero modes and implement topological quantum computation Wilczek2009 (); Alicea2012 (); Franz2013 (); Elliott2015 (). Note that a superfluid phase has to exist in 2D or 3D regime, so having a 2D or 3D SO coupling is the basic requirement for such realization of gapped TSFs. While experimental studies of TSFs are yet to be available, different proposals have been introduced for Rashba and Dirac type SO coupled systems Zhu2011 (); Chunlei2013 (); Zhang2013 (); Hu2014 (); Liu2014 (); Chan2016 (), with BCS or FFLO pairing orders FF1964 (); LO1964 (). When the Fermi energy crosses only a single (or odd number of) Fermi surface (FS), the SO coupling forces Cooper pairs into effective -wave type, rendering a TSF phase Alicea2012 (). However, in the generic case it is not known so far whether there is a universal way to precisely determine the topology, e.g. Chern numbers, of a superfluid phase induced in normal bands. On the other hand, to achieve a TSF phase integrates several essential ingredients, which may bring challenges for the experiment. The minimal schemes of realization are therefore desired to ensure high feasibility of real studies.
In this letter we propose an experimental scheme to realize chiral Fulde-Ferrell (FF) superfluids, and show a generic theory to compute the Chern number of TSF phases through the normal states. The findings are of significance in both experiment and fundamental theory. First, we propose that a new type of 2D SO coupled Dirac semimetal can be realized based on a scheme adopting only a single Raman transition and simpler than the recent experiments Zhan2016 (); Zhang2016a (); Zhang2016b (). By readily breaking the inversion symmetry of the Dirac system, we find that the FF superfluid phase can be favored under an attractive interaction. Moreover, we show a generic formalism for the Chern number of a 2D superfluid induced in the normal states, with which the topological chiral FF superfluid with a broad topological region is predicted.
2D Dirac Semimetal.–We start with the effective tight-binding Hamiltonian on a square lattice, whose realization shall be presented below:
where () is annihilation (creation) operator with spin , is the hoping constant along direction, is the strength of spin-flip hopping, and denote the effective Zeeman couplings. As described in Fig. 1(a), the above Hamiltonian describes a Dirac semimetal if and , with two Dirac points at . The term breaks the inversion symmetry and leads to an energy difference between the two Dirac points. Finally, a nonzero opens a local gap at the two Dirac points. Without loss of generality, we take that and to facilitate the further discussion. As we show below, the above Hamiltonian can be readily realized in experiment.
The realization is sketched in Fig. 1(b,c). The ingredients of realization include a blue-detuned spin-independent square lattice and a Raman lattice generated via only a single Raman transition (details can be found in the Supplementary Material SI ()). In particular, the two standing-wave lights (red and black lines in Fig. 1(b)) of frequency form the electric field components and , and generate the square lattice potentials and , respectively, with and . For our purpose we take that is large compared with , namely, the field is mainly polarized in the direction. The -phase difference between and polarized components is easily achieved by putting a -wave plate before mirror , as shown in Fig. 1(b). All the initial phases of the lights are irrelevant and have been neglected. To generate the Raman lattice another running light of frequency is applied along a tilted direction so that its wave vector (blue line in Fig. 1 (b)). together with the light induces the Raman transition between spin-up and spin-down , as illustrated in Fig. 1(c). The generated Raman potential takes the form , with and . The former -term in leads to a spin-flip hopping along direction , and the latter -term gives an onsite Zeeman term , with SI (). Note that a small -term can induce a relatively large compared with hopping couplings, since is induced by on-site coupling. Through a gauge transformation Liu2014 (), we finally get the effective Hamiltonian (1), with the parameters being and . It can be seen that the inversion symmetry is controlled by the tilt angle , and the gap opening at Dirac points is controlled by the , the -component of the field.
The above realization is clearly simpler than the recent experiments on 2D SO coupling, for it involves only one Raman transition. We note that a 2D Dirac semimetal driven by SO interaction, being different from graphene whose Dirac points are protected by symmetry only when there is no SO coupling, has not been discovered in solid state materials Kane2015 (). The high feasibility and controllability ensure that the 2D Dirac semimetal proposed here can be well observed based on the current experiment.
Anomalous Hall effect.–When both the inversion and time-reversal symmetries are broken, the Hamiltonian (1) leads to anomalous Hall effect, which reflects the nontriviality of 2D SO interaction AHE (). Fig. 2 shows the hall conductance versus chemical potential and the spectra in different regimes. The hall conductance at zero temperature is calculated by ( is the step function)
It can be seen that is always zero at or when is in the band gap [Fig. 2(b,d)]. However, the nonzero for finite crossing bulk bands shows the nontrivial 2D SO effect, implying that the superfluid phase of nontrivial topology may be obtained.
States of superfluidity.– The superfluid phase can be induced by considering an attractive Hubbard interaction. The total Hamiltonian . Due to the existence of multiple FSs corresponding to different Dirac cones, in general we can have intra-cone pairing (FFLO) and inter-cone (BCS) pairing orders, defined by , with or Chan2016 (); Yao2015 (); Burkov2015 (); Wang2016 (). On the other hand, since the inversion symmetry is broken, the BCS pairing would be typically suppressed. The topology of the superfluid phase can be characterized by the Chern number.
However, to compute the Chern number of the present system is highly nontrivial. The reason is because the FFLO order breaks translational symmetry can fold up the original Brillouin zone into many sub-Brillouin zones. In particular, if , with and being mutual prime integers, the system includes sub-Brillouin zones. Moreover, if is incommensurate with original lattice, the system folds up into infinite number of sub-Brillouiin zones. In such generic way, it is not realistic to compute numerically the Chern number of the system.
Generic theory for Chern number.–To resolve the difficulty, we show a generic theorem to determine the Chern number of a 2D system after opening a gap through having superfluid/superconductivity. For convenience we classify the normal bands of the system without pairing into three groups: upper (lower) bands which are above (below) the Fermi energy, and the middle bands which are crossed by Fermi energy. Each middle band may have multiple FSs (loops), and we denote by the -th FS loop of the -th middle band. Let the total Chern number of the upper (lower) bands be (). We can show that the Chern number the superfluid pairing phase induced in the system is given by
Here is the Chern number of the -th middle band and is the phase of the pairing order projected onto the -th FS loop note1 (). The integral direction is specified by arrows along FS lines in Fig. 3 (a,b), which defines the boundary of the vector area in space. The quantities are then determined by “right-hand rule” specified below. The quantity (or ) if the energy of normal states within the area is positive (or negative), while (or ) if the regions () unenclosed by any FS loop have positive (or negative) energy (Fig. 3). With this theorem the Chern number of the superfluid phase can be simply determined once we know the properties of lower and upper bands, and the normal states at the Fermi energy, which govern the phase of .
To facilitate the presentation, here we show the above theorem for a multiband system, but with only a single FS. The generic proof can be found in the Supplementary Material SI (). We write down the BdG Hamiltonian by
where is the normal Hamiltonian, is the pairing order matrix, and is the local momentum measured from FS center. Note that if FS is not symmetric with respect to its center, one can continuously deform it to be symmetric without closing the gap. Finally we can always write down in the above form to study the topology. Denote by the eigenvector of the normal band crossing Fermi energy. Focusing on the pairing on FS, the eiegenstates of has the form . The Chern number of the superfluid phase: , with
Let be the phase of order parameter on FS. Taking that with (without closing bulk gap), and with some algebra we can reach
with being the Berry connection for the normal band states. If we choose a gauge so that is smooth on , the two terms regarding in the right hand side of Eq. (6) cancels. We then reach the formula (3) for the case with a single FS.
The proof can be generalized to the case with generic multiple FSs which may be closed or open [Fig. 3(b)], with multiple bands crossed by Fermi energy, and with the pairing within each FS or between two different FSs, given that the pairing fully gaps out the bulk SI (). Since the result is not restricted by pairing types, the generic theorem shown here is powerful to quantitatively determine the topology of the superfluid phases. On the other hand, it should be noted that this theorem is applied to judge the topology of the phase gapped out by small pairing orders. Starting from the phase governed by Eq. (3), when the magnitude of superfluid order increases, the system may undergo further topological phase transition and enter a new phase with different topology. Monitoring such phase transitions can determine the topological region of the phase diagram.
Phase Diagram.– With the above theorem we can easily determine the topology of the superfluid phase. Note that for the present Dirac system, a BCS pairing cannot fully gap out the bulk but leads to nodal phases, while the FFLO or FF order can. When there is only one FS, the system with an FF order is topological since and , giving the Chern number . In contrast, when there are two FSs, from the same or different bands, one can readily find that the contributions from both FSs cancels out and the Chern number is zero, rendering a trivial phase. This result implies that the system can be topological when it is in an FF phase.
From the mean field results shown in Fig. 4 (a,b), we can see that the BCS pairing is greatly suppressed, and dominates over , with for positive . A rich phase diagram is given in Fig. 4 (c,d), where the topological and trivial FF phases, and FFLO phase are obtained. It is particularly interesting that in Fig. 4 (d) a broad topological region is predicted when is away from . The broad topological region implies that the upper critical value , characterizing the transition from topological FF state to other phases, is largely enhanced compared with the case for . This is a novel effect explained below. Note that the superfluid order also couples the particle-hole states at . Increasing to closes the bulk gap at momentum, with the critical value being solved from BdG Hamiltonian as
where . For , we have and a small critical value . This is because the gap closes at the right hand Dirac point , where the original bulk gap less than before having superfluid pairing [Fig. 4(e)]. Importantly, for , we find that , which is of the order of band width. In this regime is away from the right hand Dirac point, and corresponds to a relatively large bulk gap before adding [Fig. 4(f)]. As a result, a large is necessary to drive the phase transition, giving a broad topological region, as shown in Fig. 4 (d).
Conclusion.–In conclusion, we have proposed an experimental scheme to realize chiral Fulde-Ferrell (FF) superfluids, and showed a generic theorem to determine the topology of TSF phases through the normal states. We start with a tunable Dirac semimetal driven by 2D SO coupling, which has not been discovered in condensed matter materials but can be readily realized with the current ultracold atom experiments, and show how a superfluid phase with nonzero center of mass momentum is typically favored in the system. Moreover, we have shown a generic formalism for the Chern number of a 2D superfluid induced in the normal states, with which the topological chiral FF superfluid with a broad topological region is predicted. Our findings are of significance for both ultracold atoms and condensed matter physics, and also can be useful for material science in search for new topological superconducting states.
This work is supported by MOST (Grant No. 2016YFA0301604), NSFC (No. 11574008), and Thousand-Young-Talent Program of China.
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From semimetal to chiral Fulde-Ferrell superfluids
In this Supplementary Material we provide the details of the model realization, and the proof of the generic theorem to determine the topology, i.e. the Chern number of a chiral superfluid/superconductor.
Appendix S-1 Effective Hamiltonian
S-1.1 Atomic States
The spin-1/2 pseudospin is defined by the two atomic states and of the atom . They are then coupled to the excited states in the manifold and through various two-photon processes depicted in Fig. S1(b-d).
S-1.2 Light fields
The electric field of standing-wave lights for the present realization can be written as:
where , . In the last line of the above equation we have made the change of parameters and and a multiplication of an overall phase factor .
The Rabi-frequencies of the transitions described in Fig. S1(b-d) can be derived. The light denoted by black line has the Rabi-frequency . Those lights depicted as red lines and connecting existing states with have the Rabi-frequency , and the one depicted as blue line is . The phase are irrelevant and so are omitted here. The quantities is the dipole matrix element , where and is the corresponding ground state and excited state, and is the spherical component of vector .
S-1.3 Lattice and Raman potentials
The lattice potential is generated by the two-photon processes which intermediate states are in the manifold (described in the diagram Fig. S1(b,c) by the process generated by black and red lines). Each process gives a contribution to the lattice potential, where is the corresponding Rabi-frequency, runs through all possible atomic states and or if the intermediate state considered is in the manifold or . According to experimental data, for reference, see TGK40 ()) all different or corresponding to intermediate states of different are of the same order of magnitude (order of THz) and the differences among each group are negligible (order of 100MHz to 1GHz).
The lattice potential of both spin is
where we have omitted the constant term, and () is the reduced matrix element between the total angular momentum and . (The coefficients in can be calculated with the help of Table 1 in a straightforward way.) Notice that the coupling between spin-up and spin-down state through these processes are negligible since , where is the energy of the spin-up or spin-down state.
The Raman lattice is generated via only one Raman process (described in the diagram Fig. S1(d)). The Raman potential generated is given by the formula . In this case, the generated potential is
where . The Zeeman term is generated by the a small off-resonant in the Raman process, where .
S-1.4 Tight-binding Model
We derive the tight-binding model by considering the hopping contributed from lattice and Raman potentials, respectively. The lattice potential contributes the spin-conserved hopping terms as
where , and , and is the wavefunction at centered at . On the other hand, for the spin-flip term induced by Raman coupling, we have that the first term of (S3) provides spin-flip hopping along the -direction of strength
where . The same term gives no contribution to the hopping along -direction since is antisymmetric in the -direction at the local minimum of lattice potential. Likewise, the second term of (S3) gives rise to an onsite Zeeman term
where . This term gives negligible contribution to the hopping term since . Therefore, in the tight-binding model, the Raman potential contributes
The total tight-binding Hamiltonian reads , which can be simplified by applying the gauge transformation
With the Fourier transformation , we can finally get the Bloch Hamiltonian of the tight-binding model
where and . All the parameters are independently tunable except that and are related by .
It can be seen that the inversion symmetry is controlled by the tilt angle , and the gap opening at Dirac points is controlled by the , the -component of the field. Note that is induced by onsite spin-flip transition. Thus a small -term in Eq. (S3) can generate a relatively large . For our purpose, we shall consider a small -term compared with -term. Thus the field is mainly polarized in the direction.
Appendix S-2 BdG Hamiltonian
An attractive Hubbard interaction can be described effectively as . We introduce three order parameters and when considering superconducting pairing, where and or . If only one of them is non-zero, the BdG Hamiltonian can be written as , where
and the basis of is