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Methods for detecting charge fractionalization and winding numbers in an interacting fermionic ladder

We consider a spin-1/2 fermionic ladder with spin-orbit coupling and a perpendicular magnetic field, which shares important similarities with topological superconducting wires. We fully characterize the symmetry-protected topological phase of this ladder through the identification of fractionalized edge modes and non-trivial spin winding numbers. We propose an experimental scheme to engineer such a ladder system with cold atoms in optical lattices, and we present two protocols that can be used to extract the topological signatures from density and momentum-distribution measurements. We then consider the presence of interactions and discuss the effects of a contact on-site repulsion on the topological phase. We find that such interactions could enhance the extension of the topological phase in certain parameters regimes.


Introduction
The experimental engineering of topological phases of matter in ultracold atomic gases [1][2][3][4][5][6] lays the foundations for a deeper understanding of phase transitions that transcend the Landau paradigm of symmetry breaking. In these experiments, models displaying non-local forms of order are realized in highly controllable environments, where the parameters driving the system in and out the topological phases can be tuned with wide freedom, and where observables complementary to those of a typical solid-state experiment can be measured. These experiments have already revealed interesting properties associated with two-dimensional (2D) topological Bloch bands: the anomalous (Hall-like) velocity, which was detected in response to an external force [3,4], the topologically invariant Chern number [4], and chiral edge currents [2,5,6].
Following these advances, an important objective would be to probe the edge modes of one-dimensional (1D) topological systems, which typically appear at zero energy and exhibit charge fractionalization. In particular, identifying an observable that unambiguously signals their presence in experiments constitutes a remarkable challenge. Detecting the properties of zero-energy edge modes would strongly complement the Zakphase measurement, recently demonstrated with bosonic atoms in a 1D optical superlattice [1].

The model
We consider a spinful fermionic ladder in the presence of external gauge potentials as depicted in figure 1.
Here the two legs of the ladder are associated with a pseudo-spin z t and the lattice sites along the main axis (x direction) are labelled by r L 1, , .

= ¼
We introduce the four-component fermionic operator a , r defined on Figure 1. Schematic representation of the non-interacting model. The system is engineered in such a way that the two chains in the ladder present opposite kinetic energies. This is obtained through the introduction of a π flux in each ladder plaquette. Here, the pseudo-spin τ refers to the two legs of the ladder, while the spin σ is associated with the internal states ('spin') of the atoms, see equation (1).
the lattice site r, which acts on both the pseudo-spin z t and the spin related to the two internal states of the fermion σ z (two commuting sets of Pauli matrices t a and σ α are used to describe these degrees of freedom and is schematically represented in figure 1. The first term in equation (1) describes the intra-chain tunneling along the x direction, with hopping amplitude t and spin-dependent Peierls phase-factor B exp i 2 , z ( ) s which represents a 'spin-orbit coupling' analogous to those already realized in fermionic [5] and bosonic [6] chains. Note that, because of the z t factor, the two chains have opposite kinetic energy; consequently, the motion around each plaquette of the ladder acquires a π-phase independently of the σ z -spin component. The second term describes an on-site spin-flip term with amplitude Ω, the inter-chain tunneling with amplitude J, the potential difference between the two chains μ, and the overall chemical potential μ 0 .
In the following we will elaborate on the fact that this system effectively reproduces some of the physical features of TSCs without recurring to any physical pairing mechanism. This analogy is based on the idea that atoms in the first chain ( 1 z t = + ) can be identified with conduction electrons of a generic 1D superconducting model, a c r , whereas those in the second chain ( 1 z t = -) can be identified with its holes, In this picture any tunneling from one chain to the other constitutes an effective pairing interaction, i.e. a a c c c c , where the last equality is justified by the Bogoliubov-De Gennes treatment of the superconductor. Within this parallelism we interpret the four-band Hamiltonian (1) as a Bogoliubov-de Gennes Hamiltonian in the superconducting picture. Specifically, the particle-hole symmetry, which plays a key role in the physics of TSCs, is here represented by a swap of the two chains, C , y y t s = which have opposite kinetic energy in the same way as particles and holes do. Such mapping, though, must be seen only as an analogy, since the number of degrees of freedoms in the system (1) is doubled with respect to the superconducting wire and this has important physical consequences, as will be discussed in the following. Finally, note that Hamiltonian (1) is unitarily related to those considered in the four-wire setup of [29,30] and may have a relevance also for the study of electronic gases.
The model in equation (1) can be realized using cold atoms trapped in an optical lattice. We present here an overview of the experimental proposal and refer the interested reader to section 5 for a detailed analysis of the implementation of the model. Let's start considering a 2D setup. The realization of a spin-dependent intra-chain tunneling, described by the first term in equation (1), is particularly challenging, as it requires a subtle control over the hopping amplitudes. This effect could be engineered by exploiting the laser-induced-tunneling methods implemented in recent experiments [2,4,49,50]. Specifically, we propose to achieve this task by combining a spin-dependent staggered potential with large energy offset Δ between neighboring sites, inhibiting the bare hopping along the x direction, together with an onsite energy modulation set at the resonant frequency . w = D The spin-dependent staggered potential is chosen to be opposite for the two internal states, i.e. V r 1 2 , r z stag ( ) ( ) ( )s = -D which can be realized by considering an appropriate anti-magic wave-length [51]; this choice is motivated by the fact that the resonant modulation will then generate effective tunneling matrix elements of the desired form t r t r exp i , [49,52,53]. In order to make the Peierls phasefactors constant over the whole lattice, i.e. r B 2, ( ) f = we propose to modulate the lattice with two pairs of lasers; such a configuration allows to address individual links independently [4], hence realizing the desired Peierls phase factors on all links, (see section 5). Using additional fields resonant with the energy difference between the two spin-states a tunable onsite spin-flip term Ωσ x can be realized. Finally, the two-leg ladder can be isolated using an additional superlattice, or a light-intensity mask [54,55].

Topological phases in the non-interacting system
Hamiltonian (1) is characterized by four energy bands; as shown in figure 2, by varying the filling of the ladder, and thus the chemical potential μ 0 , the system is driven through a series of metallic and insulating quantum phases (we consider in this article only the case of zero temperature). For half filling, corresponding to the case where μ 0 = 0 and the particle density is N L 2 r º = (N is the number of fermions), the single-particle Hamiltonian shows both the particle-hole symmetry we sought for, defined by the operator C y y t s = and an additional time-reversal symmetry, T , x s = which bring the system into the topological class BDI (see appendix for more details). This class includes, for example, the SSH model and, according to the periodic table of TIs and superconductors [23,24], it may present topological phases with zero-energy modes. Specifically, our model displays a non-trivial topological insulating phase for , We find that the topological phase is surrounded by two topologically trivial phases. For μ 0 = 0, trivial and topological phases are distinguished by a topological order parameter  which takes the respective values of 1 + and 1 -(see the appendix and in particular equation (A.7) for the definition which is based on the technique developed in [56]).

Fractionalized edge modes
In the topological phase, two zero-energy fermionic modes appear in ladders with open boundary conditions, as showed in figure 2. These modes are exponentially localized at the ends of the system and have important consequences on the density distribution characterizing the topological insulating phase when N = 2L + 1 fermions are introduced in the system with hard-wall boundary conditions, as displayed in figure 3. We observe that such modes are described by Dirac operators and they are not Majorana modes as it would be expected in the superconducting analog wire. The figure shows that the density in the bulk of the system indeed corresponds to the expected value ρ = 2. Moreover, analogously to the SSH model, a charge 1/2 is exponentially localized at each boundary. This important signature of charge fractionalization can be suitably identified through the  Such a signature can be observed even in the presence of a harmonic confinement, described by the following contribution to the Hamiltonian: H wa a, r r r r The effect of an external potential can be understood in a Thomas-Fermi approach as a space-dependent chemical potential r . 0 ( ) m Due to the four energy bands, the system has three insulating phases for intermediate fillings, and, in the presence of a harmonic trap, this yields a typical wedding-cake structure with integer density plateaus (see figure 4, first column). Remarkably, even in the presence of the trapping it is possible to identify the fractionalized modes, as we see next. These zero-energy modes extend in the intermediate metallic region between one trivial plateau 1, 3 ( ) r = and the topological one 2 , ( ) r = up to exponential corrections. In figure 4, second column, we show the expectation value of n n 1 , which is particularly suited for the detection of fractionalized edge modes in cases where the density in the center of the trap is ρ = 2. Moving from the center of the trap to the next plateau (ρ = 1), this operator measures the excess density with respect to the ρ = 1 value. One can obtain either an integer (no fractional modes) or an half-integer value (presence of one fractional mode). This is an unambiguous signature of the non-trivial topological phase (third row): indeed, in this case, the quantity n ĵ ** á ñ becomes half-integer for values of j corresponding to the distance of the ρ = 1 plateau from the center of the trap. As a final remark, let us stress that the problem of detecting fractionalized edge modes through a density measurement was first addressed in [57,58], where this detection method relies on the optical measurement of reflected light. The recent experimental advances, however, allow for the challenging method presented above, since the feasibility of a single-atom detection for ultracold fermions in optical lattices has indeed been demonstrated [59][60][61]. In particular, a combination of laser cooling and fluorescence detection enables an unambiguous measurement of the occupancy of single sites for both 40 K [59, 60] and 6 Li [61] gases. This is of particular importance for the detection scheme that we are proposing, because it could suffer from the experimental inability to fix the total number of atoms which are used in the many experimental realizations necessary to reconstruct the signal n . ĵ á ñ The novel single-atom microscopes will also allow a post-selection based on the global number of particles of the system, necessary to obtain an accurate measurement.  shows the winding number relative to the second band S k 2 ( ) computed for a system with trap (black) and without a trap (blue).

Spin winding number
Another interesting signature of the topological phase is offered by the expectation value of the spin operator: where a k is the four-component annihilation operator in momentum space. This approach is inspired by the techniques presented for 2D systems in [43][44][45][46][47] and for ladders in [62], where it was shown that the expectation value of the spin k ( ) S á ñ provides a good observable to identify the topological invariant (winding number) of certain TIs. In the following, we generalize this procedure to our quasi-1D ladder model and show that also here the topological invariant ,  which clearly identifies the non-trivial topological regime, can be extracted from k .
ñ This is thus another example of the interesting concept that a time-of-flight measurement can detect topological order.
The Hamiltonian (1) can be written in a real form thanks to its symmetries; therefore k 0 ŷ ( ) áS ñ = for each eigenstate of the system and k ( ) S á ñ always lies in the x ẑ-plane. The 'spin winding number'  is defined as the number of times the vector k ( ) S á ñ encircles the origin for k going from 0 to 2p (the lattice spacing is set to 1). Let us denote with S k 2 2 ( ) the expectation value of the spin operator (3) for a state of non-interacting fermions filling completely and solely the second energy band. Remarkably, the parity of , 2  which is the winding number of S k , 2 ( ) coincides with the topological order parameter of the model: (see appendix for a demonstration and for details on the analytical calculation of this topological index for this specific model). Equation (4) is analogous to those derived for several other 2D models [43][44][45][46][47]: it relates a topological invariant to a quantity, , 2  to be extracted via time-of-flight imaging. A similar behavior was discussed in [62] for a two-band generalizations of the SSH model. In our case, we stress that the second band of the model is the lowest-energy band with non-trivial topological order; this is also related to the fact that the fractionalized edge modes appear in the second bulk gap.
We now describe how to measure , 2  for realistic systems, even in the presence of a harmonic trap. The main problem is that the spin winding number has to be probed for the second band of the Hamiltonian only: in a physical realization of the topological phase, both the first and the second band are filled, and the acquired signal includes information of both. The right column of figure 3 shows the expectation value of the spin k ( ) S á ñ obtained when the system with hard-wall boundaries is filled with 2L fermions (top panel) and with only L fermions (middle panel). Both signals are not particularly interesting. If we consider the artificial situation where atoms populate the second band only (bottom panel), the spin expectation value is characterized by a winding number that reproduces the behavior of  and encircles the origin in the topological phase. In the ideal case of a hard-wall confining potential, the required value of the second band can be extracted by repeating the experiment twice, at densities ρ = 1 and ρ = 2: the difference of the measured distributions returns the sought information S k k k.
In the presence of a harmonic trap, the wedding cake density profile suggests that the many-body wavefunction can be roughly thought as a state where each energy band α is uniformly populated by N α atoms In this case the measurement of the observable k ( ) S á ñ returns: is the expectation value of the spin calculated in the thermodynamic limit for the single particle eigenstate of the th a energy band (see appendix).  is the discretization adopted for the Brillouin zone in the time-of-flight imaging (see, for example, [43,46]).
If we consider the case in which the density profile shows only two plateaus, the value of S k 2 ( ) can be estimated by comparing the observed k 2 ( ) S á ñ r= with that of a realization with a single plateau only, k : where N 1 and N 2 are the occupations of the two bands for the state with two plateaus, and N 1 ¢ is the total number of atoms in the reference state with a single plateau. All the quantities N 1 , N 2 and N 1 ¢ can be experimentally accessed and we report in the right column of figure 4 the comparison of the data obtained for hard wall and harmonic potentials. Our numerical simulations confirm that even in the presence of the trap 2  is equal to ±1 in the topological phase, whereas in the trivial phases, it is either 0 or ±2 (see figure 4).
Let us conclude with some information on how to measure k ( ) S á ñ through spin-resolved time-of-flight imaging [43][44][45][46][47] in our setup. Special care is required in time-modulated systems with spin-dependent features [63], as considered in the specific proposal detailed in section 5 because spin-dependent observables can potentially undergo large and complicated micro-motion (rapid motion with a time-scale of the order of the driving period 2p w), which typically alters the accuracy of measurements. In such schemes, stroboscopic measurements performed at specific times, n 2p w´, where n is integer, are generally required to extract relevant information relative to the spin-dependent quantities [63]. For the scheme detailed in section 5, the micro-motion can be estimated from the unitary operators K t ( ) and R(t) defined in that section, through the method of [53]. We find that k ẑ ( ) áS ñis unaffected by the micro-motion; in contrast, an accurate analysis of x áS ñ does require a stroboscopic measurement. Moreover, we note that measuring the expectation value of x S also necessitates a 2 p pulse, which has to be short compared to the driving period in order to probe the system stroboscopically.

Interacting system
Let us now consider the role of interactions, with a special emphasis on the robustness of symmetry-protected topological order. It is experimentally relevant to consider an on-site Hubbard interaction in each leg: The transition between TI and MI can be located via the analysis of the charge gap at N = 2L. In particular, the single-particle gap is defined as: where E(N) is the ground-state energy of the system with N fermions. Clearly, N 0 1 ( ) D > for the MI because the system has a thermodynamic gap. On the other hand, the TI has zero-energy modes which ensure that = Unfortunately, the mere calculation of Δ 1 does not permit to discriminate the TI from a generic gapless phase, for which N 0 1 ( ) D = too. We thus consider also the two-particle gap: Whereas for a gapless phase is also equal to zero, for a TI it is larger than zero, signaling the gap which is protecting the phase.
Based on this discussion, we now consider a systematic study of the Hamiltonian, focusing on the competition of the two terms which are responsible for a gap opening, namely, the interaction term proportional to U responsible for the MI, and the interchain tunneling proportional to J. Roughly speaking, we identify the pairing term as the one inducing the TI, since at U = 0 the system is in a topological phase for J J B 0 2 cos 2 ( ( ) ) m < < º W --(see appendix for more detail) and the two chains decouple at J = 0.  numerical possibilities; additionally, the 2D space is studied with a grid of 0.2 along both axis. Despite these limitations, the qualitative nature of the phase diagram emerges quite clearly. Indeed, through the study of Δ 1 and Δ 2 we are able to identify the TI, the MI and the critical regions which separate them, resulting in the schematic phase diagram presented in figure 6. The topological region is identified with the large region where Δ 1 = 0 but Δ 2 > 0 whereas for the MI both Δ 1 and Δ 2 are larger than zero. Critical regions with Δ 1 = Δ 2 = 0 separate the two insulators. The schematic phase diagram discriminates the MI, whose appearance is driven by the on-site repulsion, from the trivial band insulator (BI), which appears instead also at U = 0. Our investigation did not identify a phase transition between these two trivial insulating phases, which are adiabatically connected.
To better analyze the transition between TI and MI, in figure 7 we focus on the line at J t 1, = which entails a phase transition for U t u 1.9. For U t u cr < the energy cost and gain for adding and removing one particle to/from the state with N = 2L are both equal to U/2. We interpret this as a signature of the fractionalization of the zero-energy modes of the TI: the charge excess n 1 2 on top of the density plateau ρ = 2 (one particle per site) does cost a repulsive energy U n. Since the zero-energy modes have fermionic nature, they cannot accommodate more than one particle: the energy cost for adding one additional particle to the state with N L 2 1 = + becomes significantly larger than U/2 (vice versa for removing one particle from the state with N = 2L − 1). Thus, In this regime, the term proportional to Ω dominates and the ground state of H pert is close to a trivial product state in which all the spins are oriented in the x direction. We expect that such state, characterizing the MI phase, might be adiabatically connected to the trivial BI at U = 0 and J J . The phase diagram in figure 5 shows that the topological region appears clearly as a thermodynamic region, within a well defined parameter regime. We emphasize that, for J 1. 8 2.2,   the system is in a topologically trivial phase for U = 0, and enters the symmetry-protected topological phase when the interaction parameter U is increased. Therefore the interaction is not necessarily obnoxious to the purpose of experimentally obtaining the topological phase but, on the contrary, it can also drive the system into it by shifting the position of the critical point. This has been verified also in the corresponding TSC systems [35][36][37], where the addition of repulsive interactions is proven to expand the topological phase for certain ranges of the physical parameters (see also [39,40] for related models in terms of Majorana modes). This means that, for some particular value of J J c,1 > the presence of a repulsive interaction allows the formation of edge modes otherwise absent. A similar behavior is also observed in 2D systems with time-reversal invariance [66]. Let us stress, however, that this has nothing to do with the physics of fractional Chern insulators, where interactions drive the system into distinct (strongly correlated) topological phases. As the phase diagram clearly shows, there is only one TI phase, which is strictly equivalent to that of the non-interacting system. Importantly, the phase diagram in figure 5 shows that interactions have a non-trivial role in tuning the system in and out the TI phase. In order to further clarify this last point, we now investigate in more detail the properties of the topological phase in the interacting system. Numerical investigations reported in figure 8 show that the signatures of the non-interacting TI persist in the presence of interactions. First, the density profile of the gas allows for a clear identification of the presence of fractionalized edge modes located at the boundaries of the ladder via the computation of n ĵ á ñ and n . ĵ * á ñ Indeed, for U t u , cr  figure 8 shows that n ĵ * á ñ saturates to 0.5 within few sites, which is strongly different from the behavior for U t u .


It is interesting to observe that within the topological region the localization length of the edge modes has a weak dependence on U t. Second, the system displays also within the interacting region a non-zero winding number associated with the second band of the system. As in an interacting system bands are not well defined, the winding number is computed by subtraction of the data relative to 1 r = to those relative to 2 r = (see similar discussion in section 3). This robustness of the spin winding number against local interaction is consistent with similar results in 2D systems [48]. Finally, on a more abstract side, the analysis of the Schmidt spectrum presents the robust two-fold degeneracy of symmetryprotected topological phases [67].

Physical realization of the model
The physical realization of the ladder system in equation (1) can be obtained by extending the 2D setup elaborated and realized in [4]. The present proposal builds on a 2D optical superlattice subjected to a welldesigned time-modulation as displayed in figure 9. Along the y direction a superlattice potential is used to partition the lattice into a 1D array of isolated ladders. Hopping between the two legs of the ladder corresponds to transitions 1 1, ẑ t = -« + see figure 1. The main challenge in realizing the Hamiltonian in equation (1) consists in engineering the spin-dependent complex matrix elements for tunneling processes taking place along the legs of the ladder. In the following, we will show that this can be achieved by combining a spin-dependent superlattice potential x (figure 9), which introduces a spin-dependent energy offset Δ between neighboring sites, together with the space-dependent time-modulation of the lattice discussed in [4]. We start by considering the time-independent part of the system, which can be described by the 2D tightbinding Hamiltonian Figure 9. Schematic drawing of the proposed experimental setup. The 2D lattice configuration consists of a spin-independent superlattice along y to isolate individual ladders from each other and a superlattice potential along x, which creates a spin-dependent energy offset z s D between neighboring sites in order to inhibit tunneling. Tunneling is then restored resonantly with two pairs of beams denoted as r and b following the scheme introduced in [4]. Each of the pairs consists of a standing wave along x and a runningwave along y. For r b with spin-dependent complex tunneling-matrix elements, here a is the lattice constant of the potential along x.  where m and n label the horizontal and vertical integer coordinates. The spin-dependent staggered potential could be realized, for instance, by considering an appropriate anti-magic wavelength, for which the polarizability is opposite for the two spin species [51]. In order to keep the bare tunneling processes of strength J x, y spinindependent the remaining lattice potentials need to be created using a magic wavelength, for which the polarizability is the same for the two spin-species. The bare tunneling is suppressed along the legs due to the offset J , x y , D  which allows for a complete control over induced-tunneling-matrix elements, such as those realized by modulating the lattice resonantly [53]. Following [4], the modulation is taken to be produced by two pairs of laser beams with frequency difference r b w = D (figure 9) in order to restore resonant tunneling. The corresponding time-dependent potential defined by these four lasers is then of the form This requires a stabilization of the phase of the modulation relative to the static lattice potential, which is challenging and was not yet demonstrated in previous realizations [2,4,49,50]. This specific choice of the potential v m n , ( )is made in order to independently address successive hopping terms along the x direction, which is generally required when engineering Peierls phase-factors in superlattice structures, see [4,53] and below.
The time-evolution of the system is ruled by the Schrödinger equation is defined by equations (11) and (12). The long-time dynamics of the system can be suitably described by an effective-Hamiltonian approach [53], which is valid in the high-frequency regime . w  ¥ Since the static Hamiltonian H 0 contains a staggered-potential term that explicitly diverges linearly with Δ = ω, we first apply the unitary transformation [53]  which removes the diverging term. The effective Hamiltonian can then be derived in this moving frame, using the method of [53,63] (see also [68]). For the sake of simplicity, let us first consider the dynamics associated with the species σ z = +1. For these atoms, the transformed Hamiltonian reads: where we assumed that J x k  and we defined v m n arg , .

Conclusions
In this work we presented a ladder setup for ultracold fermions subject to both the presence of an artificial π-flux magnetic potential and a spin-orbit coupling. Such a model may be seen as the synthesis of two accessible experimental techniques to realize synthetic gauge fields in optical lattices: on one side, the realization of complex tunneling matrix elements using time-modulated optical lattice [2-4, 49, 50], and, on the other, the implementation of spin-orbit terms through the control over internal atomic degrees of freedom [5,6]. Analogously to models already discussed in the context of nanowires [29,30], the combination of these two elements gives rise to a particle-hole symmetry which protects non-trivial topological phases within the Bogoliubov-de Gennes and chiral classes of TIs and superconductors. As the model conserves the number of particles and can be realized with state-of-the-art experimental techniques, our results might give a substantial advance towards the observation of topologically protected zero-energy modes in fermionic systems with and without interactions. In particular the physical realization that we present does not require the engineering of any pairing mechanism, neither the coupling to external molecular gases or superfluids, as, for example in [9,11,12,14], nor an interchain pair-hopping, as exploited in the ladder model presented in [13].
No interaction is indeed necessary for the appearance of the symmetry-protected topological phase. Consequently, the realization of the particle-hole symmetry through the ladder geometry is not as robust as its counterpart in TIs due to the absence of a true superconducting gap, thus it should be considered an extrinsic feature. However, due to the absence of disorder and to the high degree of isolation in ultracold atom setups, we expect the topological features of the system to be experimentally detectable. For example, the introduction of a trapping potential does not spoil the observation of edge modes, despite breaking the particle-hole symmetry.
We analyzed two experimentally relevant signatures of the appearance of topological phases: the presence of fractionalized edge modes, detectable through site-resolved density measurements (as recently reported in [59][60][61]), and the winding behavior of the spin degree of freedom, which can be observed through spin-resolved time-of-flight imaging. In particular, we have shown how to detect these observables even in the presence of a trapping potential, which induces soft boundaries, often believed to be particularly disruptive for the detection of topological signatures.
Our study has also considered the effect of a contact repulsive interaction; this is possible thanks to the presence of the spin degree of freedom that differentiate the main features of our model from its spinless counterparts as the SSH and its interacting generalizations (see for example [41,42]). Apart from mapping out the phase-diagram of the model, which entails two gapped phases, with and without topological properties, we have found that a Hubbard interaction can enhance the extension of the topological phase, instead of being detrimental. Furthermore, the spin winding number introduced in the article provides a good topological parameter also in the interacting case, where the usual order parameters based on single-particle wavefunctions fail.
Concluding, we mention that the model under scrutiny may be an interesting platform for the study of further fractionalization effects, based on a particular fine tuning of the parameters, which can be reminiscent of the physics of parafermionic zero-energy modes (see [69] for a recent review) in the spirit of [30].
Analogously we consider only t 0, > because of the symmetry between the two chains k t k t , , , , .
Hereafter we rescale all the energies in units of t in such a way that, below, we will always consider t = 1. Finally we observe that the expectation value of y s is always null because the Hamiltonian (A.1) is real. The system is characterized by an anti-unitary time-reversal-like symmetry T , and, for 0 0 m = (the system is exactly at half filling), we obtain the particle-hole symmetry C : These non-unitary symmetries characterize the topological symmetry class BDI (see, for example, [23,24]), which is also characterized by the unitary chiral symmetry P TC .
y z t s = = Additional terms in the Hamiltonian may break the C symmetry which is indeed fragile, due to the lack of a physical pairing interaction; we emphasize however that such perturbations become significant only if their magnitude is comparable with the energy gap. To this concern, in a cold atom gas, the presence of noise and defects are negligible and the main effect we must consider is the trapping potential bringing to a space dependent chemical potential . 0 m As discussed in section 3, however, the local shift in energy provided by the trapping allows in general to isolate topological regions of the chain with the effect of binding fractionalized modes at the interface between these regions and the trivial ones.
An additional Zeeman term proportional to σ z , which breaks only the time reversal symmetry, brings instead the system in the symmetry class D, which is still topologically non-trivial in one dimensions. Therefore, this sort of term does not alter in a fundamental way the properties of the system.
1D systems in the BDI class possess topological phases labelled by a topological invariant in  [23]. This topological invariant, can be evaluated by exploiting the chiral basis defined by the symmetry P [56]. In this basis the Hamiltonian assumes the simple form: A k det 0 ( ( )) ¹ for all k then the system is gapped. In this case, constitutes the topological invariant which distinguish topological and nontopological phases [56]. In this model, this winding number may assume only the values 0 or ±1 characterizing trivial and topological phases respectively. In more detail, one obtains: cannot change its value from 0 to π unless A det crosses zero and the gap closes; thus the case B = 0 is either trivial or gapless. In all the other cases, for B 0 2 , p < < the topological invariant  is evaluated by considering the behavior of ξ between k = 0 and k = π. In particular, the tangent of the phase ξ is given by: In order to evaluate the spin expectation value in the plane x z,-it is useful to adopt the Hamiltonian form in equation (A.4) which allows a simple diagonalization. In this basis the components of the physical spin, S x and S z , become: where σ i and i t are Pauli matrices in this basis. To obtain the eigenstates of the Hamiltonian we consider:  The expectation value of S z in the second band does not depend on the phase α and it is easily evaluated.