Higher first Chern numbers in one dimensional Bose-Fermi mixtures

We propose to use a one-dimensional system consisting of identical fermions in a periodically driven lattice immersed in a Bose gas, to realise topological superfluid phases with Chern numbers larger than 1. The bosons mediate an attractive induced interaction between the fermions, and we derive a simple formula to analyse the topological properties of the resulting pairing. When the coherence length of the bosons is large compared to the lattice spacing and there is a significant next-nearest neighbour hopping for the fermions, the system can realise a superfluid with Chern number +/- 2. We show that this phase is stable in a large region of the phase diagram as a function of the filling fraction of the fermions and the coherence length of the bosons. Cold atomic gases offer the possibility to realise the proposed system using well-known experimental techniques.

We propose to use a one-dimensional system consisting of identical fermions in a periodically driven lattice immersed in a Bose gas, to realise topological superfluid phases with Chern numbers larger than 1. The bosons mediate an attractive induced interaction between the fermions, and we derive a simple formula to analyse the topological properties of the resulting pairing. When the coherence length of the bosons is large compared to the lattice spacing and there is a significant next-nearest neighbour hopping for the fermions, the system can realise a superfluid with Chern number ±2. We show that this phase is stable in a large region of the phase diagram as a function of the filling fraction of the fermions and the coherence length of the bosons. Cold atomic gases offer the possibility to realise the proposed system using well-known experimental techniques.

I. INTRODUCTION
Until fairly recently, only a few topological phases were known to exist in nature [1][2][3][4]. We now know that these phases are surprisingly ubiquitous in nature, and their study has consequently witnessed an explosion of activities [5][6][7]. Topological superfluids [8] are particularly interesting, since they support gapless edge states, the so-called Majorana modes, which have possible applications for quantum computation [9,10]. In a seminal paper, Kitaev introduced a one dimensional (1D) model of spin-less fermions forming p-wave pairs [11] which has attracted enormous attention since it is one of the simplest playgrounds to study superfluids with non-trivial topological properties. There is an intense effort to observe these modes experimentally, and promising evidences for their existence have been reported in systems where a 1D wire is placed in proximity to a conventional superconductor [12][13][14][15][16][17][18]. The Kitaev wire has, in addition to a particle-hole symmetry, also a time-reversal symmetry which squares to be one. It follows that it belongs to class BDI in the classification scheme of topogical insulators and superfluids. This class has a Z topological invariant in 1D [19][20][21], and it therefore allows the existence of multiple orthogonal Majorana edge states. In order to bring out this Z character of the topological invariant and the associated multiplicity of Majorana states, one however needs to generalise the Kitaev model to include long range pairing and hopping. This has been demonstrated in several models where the long range nature has been introduced by hand [22][23][24].
The aim of this paper is to show how the Z character of the topological invariant of a generalised Kitaev model can be realised in an experimentally realistic system consisting of identical fermions in a 1D lattice immersed in a 1D Bose gas. We calculate the induced interaction between the fermions caused by the exchange of density oscillations in the Bose gas. The topological properties of the resulting pairing between the fermions is then analysed by deriving a simple formula for the Chern number. We demonstrate that the system can realise a phase with Chern number ±2, when the coherence length of the bosons is large and the fermions has a significant nextnearest neighbour hopping. Furthermore, we show that this phase is stable in large regions of the phase diagram as a function of the filling fraction of the lattice and the coherence length of the bosons. Finally, we briefly discuss the experimental realisation of the proposed system using cold atomic mixtures in a periodically driven lattice.

II. SYSTEM
We consider identical (spin-polarised) non-interacting fermions of mass m F mixed with bosons of mass m B in a cigar-shaped trap. The transverse trapping frequency is much larger than any relevant energy scale, so that the system can be treated as one-dimensional (1D). The fermions are subjected to an additional periodic potential forming a 1D lattice whereas the bosons move freely in 1D as illustrated in Fig. 1. We further assume that the 1D lattice is sufficiently deep so that the fermions stay in the lowest energy band. We take the x-axis to 1. (Color online). In the 1D system, the fermions (blue spheres) are confined to one dimension along the x-axis and further trapped in an optical lattice (grey), with lattice constant d. We include nearest and next-nearest neighbour hopping, t1 and t2 respectively. The fermions interact with the surrounding Bose gas (red background) which is also confined to one dimension along the x-axis. The Bose-Fermi interaction results in induced interactions between the fermions (black wiggly lines).
be along the 1D direction and densities of the fermions and bosons are n F and n B respectively. The Bose-Fermi interaction is modelled by a contact interaction g BF δ(r) with a coupling strength g BF . The Bose-Bose interaction is also of the form g B δ(r), where g B is the Bose-Bose interaction strength, whereas the fermions do not interact with themselves. We describe the bosons using Bogoliubov theory, which has been shown to yield the correct excitation spectrum despite the fact that there is no true Bose-Einstein condensation in a 1D homogeneous system [25,26]. The Hamiltonian is where c † j creates a fermion in lattice site j with the nearest and next-nearest neighbour hopping t 1 and t 2 respectively, and γ † k creates a Bogoliubov mode with momentum k and energy We have definedg B = g B /2πl 2 ⊥ as the effective boson-boson coupling in 1D, where l ⊥ is the harmonic oscillator length for the transverse trap. The Bose-Fermi interaction is expressed in terms of Bose and Fermi field operators ψ B (r) and ψ F (r) and will be simplified later. In a typical static optical lattice employed in current experiments, the next-nearest neighbour hopping t 2 is at least one order of magnitude smaller than t 1 . However, by shaking the optical lattice one can increase the effective next-nearest neighbour hopping such that the ratio t 2 /t 1 can be tuned. This technique has in fact been implemented by several experimental groups [27][28][29][30][31][32]. In momentum space, we can write the tight-binding part of the Hamiltonian for the fermions, i.e. the first line in (1), as π/d k=−π/2d ε k c † k c k with ε k = −2t 1 cos kd − 2t 2 cos 2kd. The appearance of the cos(2kd) term in the fermion dispersion will turn out to be one of the crucial ingredients to realise a phase with a Chern number larger than one.
To treat the Bose-Fermi interaction, we expand the fermion field operator as ψ F (r) = j φ ⊥ (y, z)w(x−x j )c j where φ ⊥ (y, z) is the ground state of the transverse harmonic trap and w(x − x j ) is the lowest band Wannier function located at x j in the 1D optical lattice. Likewise, the boson field operator is expanded as where L is the 1D length of the system and b k creates a boson with momentum k. Using these expansions, we can write the boson-fermion interaction, i.e. the third term in (1), as where we have approximated the Wannier function by the ground state of the local harmonic potential and used dx w * (x − x i )w(x − x j )e iqx ≃ 0 with i = j for well localized Wannier functions. To relate the b k operator to the γ k operator, we note that the Bogoliubov theory gives the usual relation

FIG. 2. (Color online).
Feynman diagram for the induced interaction between the fermions. Two fermions with momentum/frequencyp1 andp2 scatter into states with momentum/frequencyp1 +q andp2 −q. The wiggly lines denote the boson-fermion interaction gBF, the dashed lines are bosons in the condensate, and the double blue line is the Bogoliubov mode propagator.

III. INDUCED INTERACTION
The fermions will interact with each other via the exchange of Bogoliubov modes in the BEC. For a weak Bose-Fermi interaction n F g BF /2πl 2 ⊥ ≪ 1, the leading order diagram is the one shown in Fig. 2.
The resulting induced interaction in real space is where is the density-density correlation function for the bosons within Bogoliubov theory. It has been shown that this approximation for the density-density correlation function is accurate for a weakly interacting 1D Bose gas [33]. This justifies our use of Bogoliubov theory, since the bosons enter the theory only through their densitydensity correlation function. The frequency dependence of the induced interaction (3) reflects that the interaction is not instantaneous due to the finite speed of the Bogoliubov modes, which gives rise to retardation effects. It can be shown that these effects are small when the Fermi velocity v F of the fermions is much less than the speed of sound in the Bose gas, while for larger v F they suppress the magnitude of the pairing without changing the qualitative behavior [34]. We therefore neglect retardation effects in the following and restrict ourselves to the ω = 0 component of (3). Doing the q-integral in (3) yields the real space induced interaction between fermions located at sites j and l where is the coherence length in the 1D Bose gas and is the interaction strength. It follows from (5) that one can tune both the strength and the range of the induced interaction by changing the boson-fermion and bosonboson interaction strength, which is a very attractive feature of this setup. For a spin polarised 2D Fermi gas interacting via density oscillations in a 3D BEC, this flexibility enables one to tune the critical temperature of a topological superfluid to be close to the maximum value allowed by Kosterlitz-Thouless theory [34]. Note that whereas the induced interaction mediated by a 3D BEC gas is of the Yukawa form ∝ e − √ 2|xj−x l |/ξB /|x| [34], the interaction (5) mediated by a 1D Bose gas falls of as ∼ e − √ 2|xj−x l |/ξB . We shall see that, in addtion to the next-nearest neighbour hopping, the missing 1/|x| factor in the induced interaction is another crucial ingredient for producing states with a Chern number larger than one. In the following, we take the temperature to be zero.

IV. SUPERFLUID PAIRING
The induced interaction between the fermions is attractive, which means that they can form Cooper pairs. We shall investigate this using mean-field BCS theory. The validity of this approach is not obvious, since there is no true long range order corresponding to a broken continuous symmetry for our 1D system due to the Mermin-Wagner-Hohenberg theorem [35,36]. However, meanfield theory in general works better when the interaction is long range since each particle interacts with a cloud of particles in distance. Because the induced interaction (5) is long range for large coherence lengths, ξ B /d ≫ 1, we expect that BCS theory will be qualitatively correct for the present system. This expectation is in part supported by recent work concerning an exactly solvable 1D model with a 1/r interaction, which was shown to support a bulk energy gap indicating order on a macroscopic scale [37]. Moreover, since the interaction in the present case can be made effectively constant over several lattice sites, we expect our system to be ordered on a macroscopic scale compared to the system size, thereby making BCS theory qualitatively correct.
The BCS Hamiltonian describing pairing of identical fermions with opposite momentum is with ξ k = ε k −µ and µ being the chemical potential. The gap is given by where N is the number of lattice sites, and E k = (ξ 2 k + ∆ 2 k ) 1/2 is the usual energy dispersion of the quasiparticle eigenstates. The form of the effective interaction entering the gap equation is given by W ind (k, q) = 1/2 j [cos((k − q)x j ) − cos((k + q)x j )]Ṽ ind (0, j) where the cosines arise from the Fermi antisymmetry c q c −q = − c −q c q . We assume without loss of generality that the gap is real. The filling fraction 0 ≤ n ≤ 1 of the lattice is and the BCS ground state energy is For a vanishing next-nearest neighbour hopping, t 2 = 0, the system has a filling fraction symmetry between n and 1 − n around n = 1/2. Specifically, any solution with n, µ and ∆ k has a corresponding solution 1 − n, −µ and ∆ k+π/d . For nonzero t 2 this symmetry is broken. We will see this explicitly below in the phase diagrams.

V. SYMMETRIES AND TOPOLOGY
As mentioned in Sec. I, the Kitaev chain belongs to the BDI Cartan class due to the presence of an intrinsic particle-hole symmetry and a time-reversal symmetry, both squaring to +1. As a result the topological index is given by the Chern number where |e − k is the lowest eigenstate of V † H(k)V , with We apply the unitarity transformation matrix V to H(k) in order to make the eigenvector |e − k smooth for all k [38]. In (10), θ k = arctan(∆ k /ξ k ) is the polar angle of the vector h(k) = (∆ k , 0, ξ k ) with the z-axis, where H(k) = h(k) · (σ x , σ y , σ z ). Equation (10) therefore explicitly demonstrates the well-known result that the topological invariant given by the Chern number is simply the same as the winding number of h for a 1D system.
We can obtain the Chern number by simply examining the behaviour of ∆ k and ξ k as a function of k, without having to actually evaluate the integral in (10). The vector h(k) = (∆ k , 0, ξ k ) is confined to the xzplane and it starts and ends parallel to the z-axis since ∆ −π/d = ∆ π/d = 0. Using the symmetries, ∆ k = −∆ −k and ξ k = ξ −k , we can then obtain the winding number simply by counting how many times h(k) crosses the x-axis for k > 0. Now, h(k) crosses the x-axis when ξ k = 0. Assuming ∆ k > 0, the crossing is in the clockwise direction with increasing k when ∂ k ξ k < 0 and is in the counterclockwise direction when ∂ k ξ k > 0. On the other hand, if ∆ k < 0, the crossing is in the clockwise/counterclockwise direction with increasing k for ∂ k ξ k > 0 and ∂ k ξ k < 0 respectively. A clockwise crossing increases the winding number by one, whereas a counterclockwise crossing decreases it by one. This finally gives the formula sgn(∆ kn ∂ k ξ k=kn ) (11) for the Chern number. Here k n are the solutions to ξ k = 0 and sgn(x) is the sign function.
We can now see why it is important to have nextnearest neighbour hopping in order to get a Chern number larger than 1. For nearest-neighbour hopping only, ξ k = 0 has at most one solution for k > 0 and the winding number is therefore C = 0, ±1 from (11). With nextnearest neighbour hopping t 2 = 0 included, we can have up to two zero points of ξ k for k > 0. Since ∂ k ξ k has opposite signs at the two zero points, we can obtain a winding number C = ±2 only if the pairing ∆ k changes sign between the two points. Such a pairing indeed occurs and is functionally similar to ∆ k ∼ sin 2kd as we will see in the numerical analysis below. To summarize, we need non-zero next-nearest neighbour hopping t 2 and a high filling fraction such that there are two pairs of zeroes of ξ k and a pairing ∆ k ∼ sin 2kd. For a given interaction strength G, we will therefore search for values of (ξ B , n) where the pairing ∆ k ∼ sin 2kd is favourable.

VI. NUMERICAL RESULTS
We now discuss our results obtained by solving (8) and (9) numerically. We look for solutions with Chern number |C| = 1 or |C| = 2 by using the pairing functions ∆ k ∝ sin kd and ∆ k ∝ sin 2kd as initial guesses respectively.
In Fig. 3, we show the resulting zero temperature phase diagrams obtained by varying the BEC coherence length ξ B and the filling fraction n, for a fixed coupling strength G/t 1 = 4 and a fixed next-nearest neighbour hopping t 2 /t 1 = 1. Physically, this corresponds to varying the boson-boson interaction strength g B and the filling fraction n, while keeping the Bose-Fermi coupling strength g BF and the BEC density n B fixed. Consider first the left phase diagram, which is obtained by using ∆ k ∝ sin kd as an initial guess. First, note that the phase diagram is highly asymmetrical around n = 1/2, consistent with the observation that a nonzero next-nearest neighbour hopping breaks the n ↔ 1 − n symmetry. Second, we see that there is a topologically non-trivial region with Chern number |C| = 1 for intermediate filling fractions. To understand this in more detail and to prepare ourselves for the ensuing analysis of the phase with Chern number 2, we plot in Fig. 4 ∆ k and ξ k for the two specific points indicated by the symbols × and * in the phase diagram Fig. 3 (left). For the point ×, corresponding to (n, ξ B /d) = (0.2, 5), Fig. 4 (left) shows that there is only one solution to ξ k = 0 for k > 0, and we get from (11) a Chern number C = −1 since ∆ k > 0 and ∂ k ξ k > 0 at that point. The point * corresponds to a larger filling fraction (n, ξ B /d) = (0.5, 5), and in this case ξ k = 0 has two solutions as can be seen from Fig. 4 (right). Since ∆ k > 0 for both solutions whereas ∂ k ξ k has opposite signs, the Chern number is zero. Consider next the phase diagram shown Fig. 3 (right), which is obtained using the initial guess ∆ k ∝ sin 2kd. We see that there is now a large region with a Chern number |C| = 2. To analyse this, we plot in Fig. 4 (right) ∆ k and ξ k for the point marked by * in the phase diagram Fig. 3 (right) corresponding to (n, ξ B /d) = (0.5, 5). Again, ξ k = 0 has two solutions with opposite sign of the slopes ∂ k ξ k . Contrary to the self-consistent solution above obtained from using ∆ k ∝ sin kd as an initial guess, ∆ k now has opposite signs at the two solutions, and it follows from (11) that the Chern number is C = −2. Using the bulk-boundary principle, we conclude that this phase supports two mutually orthogonal Majorana modes at its ends. Fig. 3 (right) shows that the phase with Chern number |C| = 2 is stable only for coherence lengths ξ B /d > ∼ 2. This can be understood from the fact that ∆ k ∼ sin kd corresponds to predominantly nearest neighbour pairing, whereas ∆ k ∼ sin 2kd corresponds to predominantly next-nearest neighbour pairing. Thus, in order to obtain a Chern number number larger than 1, we need a significant next-nearest neighbour pairing, which in turn  Fig. 3 (left). The dotted blue line gives ∆ k for the same parameters (n, ξB/d) = (0.5, 5), but now in the phase diagram in Fig. 3 (right).
requires that the range of the induced interaction ξ B be on the order of twice the lattice spacing or larger. Comparing Fig. 3 (left) and (right), we see that there is a large region of the phase diagram in which two possible self-consistent solutions exist corresponding to phases with Chern numbers C = 0 and |C| = 2. Interestingly, the energy of the two phases turn out to differ only by a few percent, with the C = 0 being the lower one. Such a small energy difference means that the two phases are practically degenerate, and there is therefore a large probability that the system ends up in the |C| = 2 phase when cooled down.
To illustrate the significance of the next-nearest neighbour hopping, we plot in Fig. 5 the phase diagram obtained for G/t 1 = 4 and t 2 /t 1 = 0.63. The smaller nextnearest neighbour hopping has two effects on the system. The dominant one is that |C| = 2 phase only exists for even larger coherence lengths, ξ B /d > ∼ 5. The reason is that for a smaller t 2 , the zeros of ξ k are closer in kspace. In order to a get a Chern number |C| = 2, the gap then has to change sign faster, which requires higher harmonics of ∆ k . Such higher harmonics are generated by pairing between fermions with large spacial separation beyond next-nearest neighbour, which in turn requires a longer range interaction and thus a larger ξ B /d. Secondly, the |C| = 2 phase appears at higher filling frac- tions, n ≈ 0.6. This is because the filling fraction now needs to be higher to create two zeroes of ξ k .

VII. DISCUSSION
All the ingredients necessary for realising the proposed system have been realised experimentally with cold atom systems. Periodically modulated optical lattices as well as Bose-Fermi mixtures have been created by several groups, and this includes, in particular, multicomponent systems in 1D [39] with a species selective potential [40]. This makes the experimental realisation of the system promising, which is a strong feature of our proposal. There are however two caveats about our proposal: one should be able to cool the system sufficiently, and fluctuations away from mean-field can in principle modify some of the results we found above. Nevertheless, we speculate that the topological features should be fairly robust since they are determined by invariants which take on integer values. In addition, fluctuation effects are suppressed for a long range interaction.
We end by briefly discussing the case where the 1D lattice for the fermions is immersed in a 3D BEC. Our numerical calculations show that it is much more difficult to find phases with Chern numbers larger than 1 compared to the pure 1D system discussed above. Their energy is furthermore much higher than phases with Chern number 0 or 1, again in constrast to the pure 1D case. The reason is that induced interaction between the fermions is of the Yukawa form ∝ exp(− √ 2|x|/ξ B )/|x| when the BEC is 3D; the extra factor 1/|x| compared to the pure 1D interaction (5) suppresses the next-nearest neighbour pairing compared to nearest neighbour pairing irrespective of the value of ξ B .

VIII. CONCLUSIONS
In conclusion, we considered a spin-polarised Fermi gas in a 1D lattice immersed in a 1D homogeneous Bose gas. The fermions interact attractively via the exchange of density fluctuations in the Bose gas, which gives rise to pairing. We derived a simple formula for the Chern number of the superfluid phase, which was used to demonstrate that the fermions can realise a topological super-fluid with Chern number ±2, provided that the nextnearest neighbour hopping for the fermions is significant and the induced interaction is sufficiently long range. This phase was shown to be stable in a large region of the phase diagram as a function of the filling fraction of the fermions and the coherence length of the bosons. Atomic gases provide a promising system to realise our proposed setup using well established experimental techniques.