Topological Devil's staircase in atomic two-leg ladders

We show that a hierarchy of topological phases in one dimension - a topological Devil's staircase - can emerge at fractional filling fractions in interacting systems, whose single-particle band structure describes a topological or a crystalline topological insulator. Focusing on a specific example in the BDI class, we present a field-theoretical argument based on bosonization that indicates how the system phase diagram, as a function of the filling fraction, hosts a series of density waves. Subsequently, based on a numerical investigation of spectral properties, Wilczek-Zee phases, and entanglement spectra, we show that these phases can support symmetry-protected topological order. In sharp contrast to the non-interacting limit, these topological density waves do not follow the boundary-edge correspondence, as their edge modes are gapped. We then discuss how these results are immediately applicable to models in the AIII class, and to crystalline topological insulators protected by inversion symmetry. Our findings are immediately relevant to cold atom experiments with alkaline-earth atoms in optical lattices, where the band structure properties we exploit have been recently realized.


Introduction
In the last few years, a series of remarkable experiments has demonstrated how cold atomic gases in optical lattices can realize topological band structures [1][2][3][4][5][6][7] with a high degree of accuracy and tunability [8][9][10][11][12]. In the context of one-dimensional (1D) systems, ladders pierced by synthetic gauge fields [13][14][15][16][17][18][19][20][21][22][23] have been experimentally shown to display a plethora of phenomena, including chiral currents [24] and edge modes akin to the two-dimensional Hall effect [7], accompanied with the long-predicted-but hard to directly observeskipping orbits [25,26]. While such phenomena have required relatively simple microscopic Hamiltonians apt to describe electrons in a magnetic field [27], the flexibility demonstrated in very recent settings utilizing alkaline-earth-like atoms [28][29][30][31][32][33] has shown how a new class of model Hamiltonians-where nearest neighbor couplings on multi-leg ladders can be engineered almost independently one from the other-is well within experimental reach. Remarkably, these works have not only demonstrated the capability of realizing spin-orbit couplings utilizing clock transitions [29,30], but also the observation of band structures where topology is tied to inversion symmetry [31,34], a playground for crystalline topological insulators [35][36][37]. A natural question along these lines is whether these new recently developed setups offer novel opportunities for the observation of intrinsically interacting topological phases-e.g. symmetry-protected topological phases which appear at fractional filling fractions.
In this work, we show how, starting from experimentally realized microscopic Hamiltonians [31], interactions can generically stabilize novel topological phases in regimes where single-particle Hamiltonians cannot host any. We consider a 1D ladder with two internal spin states, supporting a topological phase at integer filling, and we show that, when the particle filling is reduced to a fractional value, repulsive interactions can stabilize a hierarchy of unconventional topological gapped phases, namely a topological Devil's staircase [38,39]. Such topological density-wave phases are characterized by a well-defined topological number, the Wilczek-Zee phase [40,43], thus signaling that the topological properties of the non-interacting bands are inherited at fractional fillings in the presence of interactions. These gapped states present a degenerate entanglement spectrum [44][45][46] and, in some regimes, an unconventional edge physics without zero-energy modes.
The appearance of these fractional topological phases is reminiscent of the quantum Hall physics, where a similar transition from the integer to the fractional regime is observed when interactions are considered. Owing to the 1D context, here the main difference is that all the phases are symmetry-protected topological phases, as true topological order cannot take place. We note that, for specific filling fractions, our results are closely related to other topological density waves found in single-band models [47,48]. The mere existence of a full class of topological density waves is surprising in view of the fact that, typically, non-interacting topological phases at integer filling appearing in the context of 1D systems with two internal spin states such as the Su-Schrieffer-Hegger model [49] or Creutz ladders [50] are robust against weak interactions only, and disappear [51,52] in the strongly interacting regime 11 .
We illustrated the appearance of such phases by studying a model Hamiltonian description for the BDI [49][50][51] and AIII symmetry classes [52,54] of the Altland-Zirnbauer classification (AZc) [1,6], and a crystalline topological insulator case of a 1D model supporting a spatial inversion symmetry protected topological phase at filling one [35][36][37]. Our results are general in the sense that these fractional phases can be potentially observed in all symmetry classes of the AZc which can be realized in a two-leg ladder. Our work is complementary to recent approaches investigating interaction induced fractional topological insulators [55][56][57][58][59][60][61][62][63], which typically focus on specific case scenarios that accurately mimic the edge physics of quantum Hall states or extend topological superconductivity at finite interaction strength.
From an experimental perspective, the models we investigate are immediately relevant to cold gases experiments. In particular, recent implementations using alkaline-earth-like atoms such as Yb [25,29,32] and Sr [30] have demonstrated an ample degree of flexibility in tuning parameters (including static gauge fields) in two-leg ladders, exploiting the concept of synthetic dimension [64,65]. Most importantly, the single particle Hamiltonian we discuss below has been realized in a 173 Yb gas, see [32], and similar schemes shall be applicable to 87 Sr gases as well.
This paper is organized as follows. In section 2 we present the model and discuss its fundamental symmetries. sections 3 and 3.3 contain our main results. In particular, in section 3, we consider models belonging to the BDI and AIII symmetry classes and we show the appearance of a topological fractional phase at filling ν=1/2 which can be viewed as a precursor of the topological Devil's staircase. We discuss the topological properties of this phase: the Wilczek-Zee phase, and the entanglement spectrum in the ground-state manifold, using numerical methods as Lanczos-based exact diagonalization [66] and density-matrix renormalization group (DMRG) [67,68] simulations. Finally, we discuss how this topological phase supports edge modes, which, while not at zero energy, can still be diagnosed by simple correlation functions. Then, in section 3.3, by means of a bosonization approach we discuss the appearance of the topological Devil's staircase at lower fillings and we explicitly address the filling ν=1/3 case. Finally, in section 4 we generalize our results to the case of a crystalline topological insulator. Our conclusions are drawn in section 5.

Model and symmetries
Let us start by introducing the Hamiltonian we are going to focus on. For the sake of clarity, we also review the main definitions of time-reversal, particle-hole, and chiral symmetry in the language of second quantization, which is best suited to the case of interacting systems.

Model Hamiltonian
We consider a 1D chain with L sites along the physical dimension. These are populated by fermionic particles described by the canonical operators c j,ŝ ( †) , annihilating (creating) a fermion at site j=1, K, L, with two internal degrees of freedom labeled by , s =   (resp. +1,−1).
The single-particle physics discussed in this work is fully captured by the Hamiltonian (see figure 1) is a species-dependent chemical potential which induces an imbalance between spin up and spin down particles.
The model we discuss has already been experimentally realized in [32]. We refer to this work for specific details on the experimentally achievable parameter regimes. Hereafter we will set ÿ=1 and express all energy scales in units of the hopping term t. In order to understand the symmetry class to which Ĥ belongs, it is first convenient to recall the three symmetries classifying the ten classes of the AZc. Then, we also discuss the inversion symmetry operator. Figure 1. Schematic representation of a 1D chain with two internal spin states,  and . Curved arrows at the dots represent the species-dependent chemical potential Δò which induces an imbalance between spin up and spin down particles. As discussed in table 1, by properly tuning the parameters t σ and J in equation (3), it is possible to realize non-interacting topological phases belonging to the symmetry classes BDI and AIII of the AZc and an inversion symmetry protected topological phase supporting a non-trivial phase at integer filling (crystalline topological insulator). Table 1. By properly tuning the parameters t σ and J in equation (3), it is possible to realize non-interacting topological phases belonging to the symmetry classes BDI and AIII of the AZc and an inversion symmetry protected topological phase supporting a non-trivial phase at integer filling (crystalline topological insulator). A topological phase belonging to the BDI symmetry class is endowed with a time-reversal T, a particle-hole C, and a chiral S symmetry; a topological phase in the AIII symmetry class is endowed with a chiral symmetry only; see also equations (6)- (8).

Fundamental symmetries
The symmetries playing a crucial role in the AZc are the time-reversal ˆ, the particle-hole  , and the chiral  symmetry. Their action on the fermionic operators c j,ŝ reads [6]: where T, C, and S are 2×2 unitary matrices satisfying TT * =CC * =±σ 0 (σ 0 being the 2×2 identity matrix) and S=TC * , up to an arbitrary phase factor such that S 2 =σ 0 . Furthermore, ˆand  are anti-unitary (i.e. i i i We also introduce the unitary inversion symmetry operator  , which acts as [36]: Switching off the interaction term, the single-particle Hamiltonian(1) can be conveniently rewritten as: Then the requirements (6)-(9) lead to the more familiar ones [6]: According to the AZc, in 1D only five symmetry classes (BDI, AIII, D, CII, and DIII) can support a topological phase (assuming no spatial symmetry). In the next section we will consider interacting topological models whose single-particle Hamiltonians belong to the symmetry classes BDI [51] and AIII [52,54] and which can be realized in two-leg ladders with nearest-neighbor couplings, by properly tuning the coefficients t σ and J in the Hamiltonian term H nn of equation (3), according to the prescriptions of table 1. On the other hand, CII and DIII models require ladders with a higher number of legs, or two-leg ladders in the presence of next-nearest-neighbor hopping terms, and will not be considered here. At integer filling, ν=N/L=1 (where N is the number of fermions), models in [51,52,54] can exhibit a topological phase characterized by the presence of exponentially localized zero-energy edge modes in the noninteracting spectrum, a quantized Zak phase, and a doubly degenerate entanglement spectrum [44,45]. Conversely, in the present context we are interested in investigating the topological properties when the particle filling is fractional, i.e. ν=1/q, with q>1 integer. We will also consider ladders supporting a crystalline topological phase protected by the spatial inversion symmetry, which cannot be understood in terms of the standard AZc (section 4).

Topological phases emerging due to interactions at fractional fillings in BDI and AIII band structures
Firstly we focus on two-leg ladder whose non-interacting Hamiltonian is in the BDI symmetry class. In this case, the various parameters are fixed (see table 1), and the resulting Hamiltonian of equation (5) Using equations (6)- (8), and the matrices defined in table 1, it is easy to observe that H H , and trivial chemical potential terms In the following we will focus on fractional fillings ν=1/q, and consider repulsive interactions. Our results for the BDI symmetry class are immediately applicable to the model in the AIII class, which can be obtained from to latter via the unitary transformation also known as Kawamoto-Smit rotation in the context of Lattice Field Theories [74] c c e j j j j j , ( see again table 1).

Effective lowest-band Hamiltonian
The single-particle contributions of the Hamiltonian (17), assuming periodic boundary conditions (PBC), can be diagonalized as In order to probe the existence of a hierarchy of fully gapped phases at fractional fillings, we conveniently introduce the real-space fermionic operators -ˆb uilt up from the momentum-space operators d k,ĥ defined in equation (21). Then we remap the original fermionic operators c j,ŝ onto the new ones d j,ĥ as are the Wannier functions of the tight-binding model; in the following, we assume j k =0. For this choice of j k , the Wannier functions F c ( j−ℓ) and F s ( j−ℓ) can be calculated exactly when Δò=0, as shown in appendix A. When Δò>0, the functions F c ( j−ℓ) and F s ( j−ℓ) can be calculated numerically. However, we have verified that the functions F c ( j−ℓ) and F s ( j−ℓ) exhibit a weak dependence on Δò and their expressions calculated for Δò=0 are a good approximation as long as t  D áá .
To simplify the problem, we project on the lowest band by assuming that the interaction termsU and V p are much smaller than the band gap, i.e. ≈4J when J=t and Δò=0. Since we are dealing with low fillings anyway, it is reasonable to suppose that only the lower band is significantly populated (from now on, we will thus omit the index −1). In order to check the self-consistency of our predictions, numerical simulations will nonetheless be performed with the full description of the system. Under these assumptions, H 0 becomes ) . Of course, the Hamiltonian(24) is highly non-local, since all sites are coupled together by long-range terms However, as shown in figure 2, the coefficient j ℓ ( )decays exponentially with ℓ−j and the lower band can be approximated by truncating to nearest-neighbor terms: where we have defined 1 1   º -( ) and neglected an inessential chemical potential. For J=t, it turns out that . We stress here that, a truncation up to nearest-neighbor terms only breaks the symmetries of the original model and the new Hamiltonian H 0 is not topological. Nevertheless this approach is useful to show the appearance of a hierarchy of fully gapped phases. Their topological properties will be discussed in the following (see below).
Let us focus on the interaction terms By means of the mapping (22) and considering the dominant  ). Similarly, the density-density terms V n n p j j p +î n the original model will be mapped onto density-density terms of the form n n j j p 1 + +ˆ. These density-density interaction terms lead to a hierarchy of gapped phases supporting density-wave states at rational filling fractions -the well-known Devil's staircase [38,39,69], which we now discuss in the context of our model. In the next paragraph 3.2 we address in detail the filling ν=1/2, then we generalize our results to lower fillings and we explicitly consider the filling ν=1/3 in section 3.3.

Topological density-wave at ν=1/2: analytical and numerical characterization
In this paragraph we focus on a fractional topological phase at filling ν=1/2 whose appearance can be discussed in a transparent way, both analytically (by means of a mean-field approach) and numerically (using exact diagonalization and DMRG). Firstly we estimate the critical interaction which stabilizes a gapped phase. To this aim we rewrite the interaction term n n j j , ,  ˆu sing the mapping (22) Within a mean-field approach, the correlated hopping term n d d can be neglected (see appendix C for details), and the effective Hamiltonian given by equation (25) plus the densitydensity interaction term of equation (26) is equivalent to a spin-1/2 XXZ model, which can be exactly solved [71]. The critical interaction U c stabilizing an antiferromagnetic gapped phase is where the functions F c and F s were defined in equation (23); in the case of filling ν=1/2, a gapped phase can be stabilized by the Hubbard interaction only, for this reason, in the following, longer range interaction terms V p are set to zero. To substantiate our analytic predictions, assuming PBC conditions, we have numerically computed by means of a Lanczos-based exact diagonalization approach the charge gap of the interacting Hamiltonian(17) where E α (N) is the energy of the αth state with N particles (E 1 being the ground-state energy), and here we set N=L/2. Figure 3 displays δ charge as a function of the interaction parameter U/t, for different values of the imbalance term Δò. We observe a good agreement of the analytic prediction (27) for the critical interaction U c , with the point at which the charge gap closes. Likewise, the charge neutral gap can be obtained following a similar procedure. In particular, since the ground state always exhibits a two-fold degeneracy (see below), we consider α=3. The behavior of the spin gap δ spin,3 as a function of the interaction parameter U/t is qualitatively analogous to the one of the charge gap (data not shown).

Ground-state degeneracy
Before addressing the topological properties of the fractional phase, it is worth investigating the spectrum

(ˆˆˆˆ)ˆ( ) † †
If we now consider the regime where U = t and get rid of the upper band, we realize that H Û corresponds to a nearest-neighbor interaction term between the cages, i.e. n n j j , 1 , --ˆ † (see also [52]). Then, since the ground state at filling ν=1/2 can be schematically represented via the occupation of local cages, we observe that PBC can effectively fit two of those states (where the cages start at odd or even sites, respectively). Conversely, OBC can only accommodate a single one (where the cages start at odd sites). This interpretation also explains the robustness of the ground-state degeneracy when the boundary conditions are twisted in a closed chain, as the rigid cage structure is not sensitive to such a twist-see next subsection. This behavior is akin to the robustness of ground-state degeneracy in true topologically ordered states (see also [48]).

Wilczek-Zee phase
As discussed in the previous paragraph, when PBC or twisted boundary conditions are assumed the ground state at filling ν=1/2 is gapped and two-fold degenerate. For this reason the correct topological invariant which has to be used to reveal its topological properties is the Wilczek-Zee phase [40][41][42] i d Tr , 32 First of all, in figure 4(a) we plot the ground-state energies E n 1 j ( )and E 2 (j n ) as a function of the discretized twisting angle j n and observe that the exact degeneracy at j n =0 is only apparently removed when 0 n j ¹ .
Indeed, as shown in figure 4(b), the difference δE(j n =π)=E 2 (π)−E 1 (π) scales exponentially with the system size L. As expected, the inset of figure 4(a) highlights that the neutral gap (29) does not close when twisted boundary conditions are used, as it is essentially insensitive to boundary conditions. Figure 4(   ¹ -ˆˆˆ, the WZ phase is not quantized anymore, thus signaling that the fractional gapped phase is protected by the same symmetry of the integer case. On the contrary, for M 0 = , the WZ phase is strictly quantized to one independently of the value of Δò (as long as the interaction term U is sufficiently strong to stabilize a gapped phase).

Entanglement spectrum
To substantiate the topological nature of the gapped phase discussed so far, we have investigated the entanglement spectrum of the Hamiltonian(17) by means of DMRG simulations. Such quantity corresponds to the set of the eigenvalues {λ α } of the reduced density matrix ℓ ℓ obtained from the system's ground state Yñ | . Here we consider a subsystem containing ℓ<L adjacent sites, and call ℓ its complement; we note that the degeneracy of the entanglement spectrum is not altered when other values of ℓ are considered. It is well known [44,45] that there exists a connection between the topological versustrivial nature of Yñ | and the degeneracy of the eigenvalues of r ℓ . A topological phase corresponds to a degenerate entanglement spectrum: this is indeed what we observe in figure 5, where we plot the first twelve eigenvalues of the entanglement spectrum for a chain of L=100 sites, U=t and Δò=2×10 −2 t. The upper inset is a magnification of the two largest eigenvalues, whose degeneracy is removed in the presence of the symmetry breaking Hamiltonian term(34)-see the lower inset.

Unconventional edge physics at
Topological phases are typically characterized by the presence of zero-energy modes, when OBC along the physical dimension are assumed. A necessary but non sufficient condition for their presence is a vanishing (resp. non-vanishing) single-particle charge gap at filling ν=1/2 with OBC (resp. PBC). Here, despite the topological nature of the model, zero-energy modes do not appear, since the single-particle charge gap remains finite even with OBC, and exhibits a behavior qualitatively analogous to the PBC case-see figure 3.
Although zero-energy modes are absent, the topological nature of the model manifests itself in an unconventional edge physics which can be revealed through the quantity n L n L L n L 2 1 2 1 2 2 , 35 measuring the difference between the expectation value of the density operator onto the state L 2ñ | corresponding to filling ν=1/2, and its expectation value onto the state L 2 1 + ñ | corresponding to filling ν=1/2+1/L.
We start investigating the case where the spin imbalance Δò vanishes. In figure 6 we plot the two density profiles both in the non-interacting case where the phase is gapless, and in the interacting case, for a sufficiently large interaction term U which stabilizes the topological gap. In the non-interacting case (panel (a)), where the quantity δn j describes the wave-function of the added particle, no edge physics is observed, since δn j is delocalized over the entire chain. On the contrary, in the interacting case (panel (b)), δn j displays two sharp peaks close to the edges of the system. When a small imbalance term Δò is considered, the behavior of δn j changes drastically. As shown in figure 7, we still observe some edge physics, but the quantity δn j is now spread over a number of sites that increases with growing Δò.
We now give an intuitive picture of this spreading effect. To this aim we consider the inset of figure 7(a) where a qualitative picture of the spectrum {E 1 , E 2 , ...} of the interacting Hamiltonian is shown at ν=1/2 and at ν=1/2+1/L, with Δò=0 and OBC. In the first case L 2ñ | , there is a finite gap between the unique ground state and the first excited state. When the local imbalance term Δò is added, the ground state is unmodified as long as Δò is small with respect to the gap. In the second case L 2 1 + ñ | , the ground state is not protected by a finite energy difference. For this reason, in the presence of the imbalance term Δò, it is expected to be a quantum superposition of the ground state L 2 1 + ñ | at Δò=0 plus pieces coming from the excited states which carry bulk contributions, and originate the spreading of the quantity δn j shown in figure 7. However, the spreading of δn j in the bulk becomes negligible in thermodynamic limit, as shown in figure 7(b) for different sizes of the chain.

Devil's staircase from bosonization
So far we have considered the fractional topological phase at filling ν=1/2. The appearance of a hierarchy of topological gapped phases at lower fillings can be explained in terms of a bosonization approach [69]. To this aim, we consider the continuum limit of the fermionic operators d ĵ , definined by d d x a j ôˆ( ) and d d x a a j 1 º + +ˆ( ) , with a being a generic cut-off length (in the following, a = 1). This operator can be expressed in terms of the bosonic fields x f ( ) and x q ( ) satisfying with k F =πN/L being the Fermi momentum. Moreover, the density operator x d x d x r = ( )ˆ( )ˆ( ) † is given by here A p and B p are non-universal coefficients which depend on the cut-off length of the theory. Within bosonization, the Hamiltonian(25) plus the density-density interaction terms can be recast into a quadratic form describing a critical gapless theory, plus a sum of sine-Gordon terms where u is an effective Fermi velocity, K is related to the strength of the interaction terms, while the coefficients M p represent the amplitudes of the sine-Gordon terms An exact mapping of the quantities u, K, and M p onto the microscopic parameters is beyond the scope of the present discussion and is generally challenging, due to the complex non-local character of the effective interactions. Furthermore, as shown in appendix B, at filling ν=1/ 2, all interaction terms which cannot be recast into a density-density form and which have been so far neglected, lead to a renormalization of the coefficients u, K and M p only. Sine-Gordon terms (39) are responsible for the appearance of the gapped phases at fractional fillings. Indeed, when the space dependent 2pk F x term in the co-sinusoidal functions vanishes, i.e. 2p k F ∝ 2π, they become relevant for K < 2/p 2 and open a gap. We stress that, within the present bosonization approach, we cannot say anything about the topological properties of these phases. In the case of filling ν=N/L=1/2, i.e. k F =π/2, the most relevant sine-Gordon term is the one with p=2. All other phases at lower filling fractions ν=1/q, with q>2, can be reached by considering sufficiently long-range density-density interaction terms, as in the conventional Devil's staircase scenario [38,69,70]. We now explicitly consider the Hamiltonian(17) and discuss the fractional filling case ν=1/3 for which the topological phase is stabilized by a nearest-neighbor interaction term in equation (4). In figure 8, we plot the first ten eigenvalues of the entanglement spectrum for a chain of L=60 sites, U=t, V 1 =t, and Δò=8×10 −2 t. As expected, the two highest eigenvalues are degenerate due to the topological nature of the ground state. Similarly to the case studied previously, we finally observe that their degeneracy is removed in the presence of the symmetry breaking Hamiltonian term (34), as shown in the lower inset, signaling that the fractional topological phase is protected by the same symmetry that protects the non-interacting topological phase at integer filling.

Inversion symmetric topological phases at fractional fillings
It is a natural question to inquire whether the mechanism for the stabilization of interaction-induced topological phases at fractional filling fraction is interwound with spatial symmetries (which play a key role in the establishment of conventional Devil's staircase structures). In this section, we discuss an interacting, crystalline topological insulator, where a fractional topological phase appears when considering interaction effects on the top of partly filled topological bands.
In particular, we consider a two-leg ladder which supports, in the non-interacting regime, a crystalline topological phase at filling ν=1. Following the prescriptions given in figure 1(b), the resulting Hamiltonian reads [35]:


 ¹ -ˆˆˆ, and consequently the WZ phase is not quantized anymore.
In figure 9(b) we plot the WZ phase as a function of the inversion symmetry breaking term M. As expected, for M=0 it is quantized and equal to one, while for M 0 ¹ it is not quantized. This signals that the fractional gapped phase is protected by same symmetry of the integer case, in complete analogy with what observed for the BDI/AIII cases.

Conclusions
We have considered a 1D ladder with two internal spin states supporting a topological phase at integer filling and we have shown that, when the particle filling is reduced to a fractional value, repulsive interactions can stabilize a hierarchy of fully gapped density-wave phases with topological features.
In particular we have focused on a specific example in the BDI class (unitarily equivalent to a model in the AIII symmetry class) of the AZc and on a crystalline topological insulator, i.e. a topological model protected by the spatial inversion symmetry. By means of a bosonization approach we have discussed the appearance of a gapped phase at filllings ν=1/q and, using exact numerical methods (DMRG simulations and exact diagonalization), we have verified our analytical predictions and we have also characterized the topological properties of the gapped phases at fillings ν=1/2 and ν=1/3 by studying the topological quantum number (Wilczek-Zee phase) and the degeneracy of the entanglement spectrum. Considering the effects of perturbations, we have discussed how these fractional topological phases are protected by the same symmetry that protects the non-interacting topological phase at integer filling.  (40) and its symmetry-breaking term M of equation (41). Notice that, analogously to the case studied before, the gap between E 1 and E 2 and the first excited state E 3 is preserved when 0 n j ¹ . Moreover, under the perturbation M, the WZ phase is not quantized to one anymore. Data have been obtained through exact diagonalization, with L=8, δ=0.1t, Δò=10 −2 t, U=t, N j =100.
Most importantly, we have shown that these topological density waves do not follow the bulk-edge correspondence, in the sense that they exhibit modes at finite energy localized close to the edges of the system. Their presence has been diagnosed by studying the behavior of the density profile, when an extra particle is put in the system with respect to the filling ν=1/2. Our results are immediately testable in cold atom experiments described by the setup in [31,32]: while the single particle Hamiltonian has already been realized, a key requirement is to reach density regimes where an incompressible phase is stabilized in the center of the harmonic trap. Given that fractional phases appear already for quarter-filled band, we expect signal-to-noise not to constitute a problem. Since the incompressible phase in this regime has a gap of order U, this requires cooling in the tens of nanokelvin regime, which is within current experimental reach in these systems [25].
We leave as an intriguing perspective the study of the appearance of these fractional topological phases in topological models belonging to the symmetry classes D, CII, and DIII of the AZc.
where v ka sin