Synthetic gauge fields in synthetic dimensions: Interactions and chiral edge modes

Synthetic ladders realized with one-dimensional alkaline-earth(-like) fermionic gases and subject to a gauge field represent a promising environment for the investigation of quantum Hall physics with ultracold atoms. Using density-matrix renormalization group calculations, we study how the quantum Hall-like chiral edge currents are affected by repulsive atom-atom interactions. We relate the properties of such currents to the asymmetry of the spin resolved momentum distribution function, a quantity which is easily addressable in state-of-art experiments. We show that repulsive interactions significantly stabilize the quantum Hall-like helical region and enhance the chiral currents. Our numerical simulations are performed for atoms with two and three internal spin states.


Introduction
One of the most noticeable hallmarks of topological insulators is the presence of robust gapless edge modes [1]. Their first experimental observation goes back to the discovery of the quantum Hall effect [2], where the existence of chiral edge states is responsible for the striking transport properties of the Hall bars. The physics of edge states has recently peeked out also in the arena of ultracold gases [3,4,5], triggered by the new exciting developments in the implementation of topological models and synthetic gauge potentials for neutral cold atoms [6,7,8,9,10].
Synthetic gauge potentials in cold atomic systems have already led to the experimental study of Bose-Einstein condensates coupled to a magnetic field [11] or with an effective spin-orbit coupling [12], and more recently to lattice models with nonzero Chern numbers [13,14,15,16] and frustrated ladders [3]. In a cold-gas experiment, the transverse dimension of a two-dimensional setup does not need to be a physical dimension, i.e. a dimension in real space: an extra synthetic dimension on a given ddimensional lattice can be engineered taking advantage of the internal atomic degrees of freedom [17]. The crucial requirement is that each of them has to be coupled to long-range in the synthetic dimension. This situation is remarkably different from the typical condensed-matter systems and may lead to quantitative differences especially when considering narrow ladders, as in Ref. [4].
The paper is organized as follows. In the next section we introduce the model describing a one-dimensional gas of earth-alkaline(-like) atoms with nuclear spin I ≥ 1/2. In order to make a clear connection with the experiment of Ref. [4], we briefly explain how this system can be viewed as a (2I + 1)-leg ladder. Moreover, we present a discussion of the single-particle spectrum to understand the main properties of the edge currents in the non-interacting regime and to identify the regimes where the effects of repulsive interactions are most prominent. Then, in Sec. 3 we introduce two quantities, evaluated by means of the DMRG algorithm, that characterize the edge currents: the (spin-resolved) momentum distribution function and the average current derived from it. In Sec. 4 we present and comment our results; we conclude with a summary in Sec. 5.

The model
We consider a one-dimensional gas of fermionic earth-alkaline-(like) neutral atoms characterized by a large and tunable nuclear spin I, see Fig. 1(a). Based on the predictions of Ref. [35], Pagano et al. have experimentally showed that, by conveniently choosing the populations of the nuclear-spin states, the number of atomic species can be reduced at will to 2I + 1, giving rise to an effective atomic spin I ≤ I [37]. We stress that I has to be an half-integer to enforce the fermionic statistics, while I can also be an integer, see Fig. 1(b). Moreover, as extensively discussed in Refs. [17,18], the system under consideration can be both viewed as a mere one-dimensional gas with 2I + 1 spin states or as a (2I + 1)-leg ladder, see Fig. 1(c).
The presence of two additional laser beams can induce a coupling between spinstates with ∆m = ±1 of amplitude Ω m endowed with a running complex phase factor e iγj . For simplicity, in the following we assume that Ω m does not depend on m and set Ω m = Ω. The coupling Ω is related to the amplitude of the laser beams, while the    phase γ depends on their wavelength and relative propagation angle. Explicitly, the Hamiltonian gets a contribution of the form As already mentioned, the system characterized by the HamiltonianĤ ≡Ĥ 0 +Ĥ 1 is equivalent to a (2I + 1)-leg ladder where the coordinate in the transverse direction is given by the effective-spin index m = −I, . . . , I. For all purposes, such direction can be regarded as a synthetic dimension with sharp edges; in this framework, the Hamiltonian H 1 describes the hopping in the synthetic dimension and introduces a constant magnetic field perpendicular to the ladder with dimensionless magnetic flux +γ per plaquette. The peculiarity of our synthetic ladder resides in the interaction term, which is SU (2I + 1) invariant: it therefore describes an on-site interaction in the real dimension and a longrange interaction in the synthetic one.
Since the HamiltonianĤ is not translationally invariant, for later convenience, we perform the unitary transformationd j,m =Ûĉ j,mÛ † = e −imγjĉ j,m such that Synthetic gauge fields in synthetic dimensions: Interactions and chiral edge modes whereν j,m =d † j,md j,m . Assuming PBC in the real dimension, the quadratic part ofĤ can be diagonalized in Fourier space, in terms of the operatorsd p,m = L −1/2 L j=1 e ikpjd j,m , with k p = 2πp/L and p ∈ {−L/2, . . . , L/2 − 1}.

Non-interacting helical liquid
In order to discuss the helical properties of this system, a good starting point is the analysis of the non-interacting physics for the I = 1/2 case. The single-particle spectrum of the HamiltonianĤ has two branches with the following dispersion relations: When the condition Ω < 2t sin γ 2 tan γ 2 is satisfied, the lower branch displays two minima at k p ≈ ±γ/2 and a local maximum at k p = 0, see Fig. 2(a): this case will be referred to as the weak-Raman-coupling (WRC) regime. In the opposite case, dubbed strong-Raman-coupling (SRC) regime, the lower branch has one single minimum at k p = 0 without any special feature at k p = 0, see Fig. 2(c).
The study of the spin polarization S z (related to the operator j,m mν j,m ) of each eigenmode highlights an important difference between the SRC and the WRC regimes, see Figs. 2(b) and 2(d). In the WRC case, for most of the values of k p , the eigenstates are prevalently polarized along the z direction, while in the SRC regime this is not true (the dominating polarization is along the x direction, not shown here). Figure 2(a) also shows that in the WRC regime depending on the filling, the low-energy excitation may have very different properties. For low (e.g. the orange line) or high (e.g. the green line) fillings, there are four low-energy excitations. However, when the chemical potential (here we consider zero temperature) lies between −2t cos(γ/2) − Ω and −2t cos(γ/2) + Ω (e.g. the violet line), there are two gapless excitations which have definite quasi-momentum and definite spin in the z direction. In the non-interacting case, this is an helical liquid which, once interpreted as a ladder, features two chiral edge modes.
Similar considerations about the single-particle spectrum hold for the I = 1 case, even though the analytic form of the eigenenergies is more involved. In Fig. 2(e) we show the single-particle energy spectrum of the eigenstates in the WRC regime. Low, intermediate and high fillings can be identified also in this case, and are indicated by the three different horizontal lines. The intermediate filling (violet line) corresponds to the regime where the helical liquid appears; indeed the spin polarization S z shown in Fig. 2(f) exhibits almost full polarization of the eigenstates close to the considered Fermi energy. Here, in the synthetic-dimension representation, the three-leg ladder displays chiral modes.
In the interacting case, the spectral properties of the Hamiltonian are not trivially computable. In the following section we define the physical quantities used to properly Synthetic gauge fields in synthetic dimensions: Interactions and chiral edge modes characterize the helical modes, which can be calculated by means of the DMRG algorithm. In the remainder of this paper we carefully analyze such quantities.

Observables
The study of the momentum distribution function, both spin-resolved and non-spinresolved, can provide, as we shall see, information about the helical/chiral nature of the interacting liquid under consideration. The spin-resolved momentum distribution function is defined as where expectation values are taken over the ground state. Since p is not a good quantum number forĤ, we will conveniently consider HamiltonianĤ and the momentum distribution function ν p,m = d † p,md p,m , for which it easy to verify that ν p,m = n p−mγ,m . Accordingly, the total momentum distribution is given by n p = I m=−I n p,m . Based on these definitions, we introduce two chirality witnesses, i.e. two quantities which diagnose and identify the edge currents determined by the presence of the gauge field γ = 0, even in the presence of repulsive interactions. To this aim, we first solve the continuity equation for the HamiltonianĤ and define the ground-state average chiral current Assuming PBC in the real dimension and using Eq. (6), its spatial average can be re-expressed as with k p = 2πp/L. The latter relation allows to indirectly probe the existence of chiral currents using a quantity, namely n p,m , which can be experimentally observed in the state-of-art laboratories using a band-mapping technique [42] followed by a Stern-Gerlach time-of-flight imaging [4,5]. The quantity Q m is the first chirality witness to be employed in the following. The second chirality witness is the quantity defined in Ref. [4], which is more directly related to the asymmetry of the spinresolved momentum distribution function. Both J m and Q m give information about the circulating currents and, as we shall see below, display the same qualitative behavior (they only differ for a cut-off at low wavelength).

Results
Equipped with the definitions given in the previous sections, we now discuss how atomatom repulsive interactions affect the momentum distribution functions n p and n p,m and the chirality witnesses Q m and J m . The results for the non-interacting cases, here used as a reference, are computed by means of exact diagonalization, while for U/t = 0 the DMRG algorithm is used [43,44]. We only address the ground-state properties, i.e. rigorously work at zero temperature. In the finite-size sweeping procedure, up to 250 eigenstates of the reduced density matrix are kept, in order to achieve a truncation error of the order of 10 −6 (in the worst cases) and a precision, for the computed correlations, at the fourth digit. The resulting inaccuracy is negligible on the scale of all the figures shown hereafter. Unless differently specified, in the I = 1/2 case we consider L = 96 and Ω/t = 0.3, while in the I = 1 case we set L = 48 and Ω/t = 0.1 (the ratio Ω/t is chosen in order to be in the WRC regime); γ = 0.37π coincides with the experimental value of Ref. [4].
As shown in Figs. 2(a) and 2(e), in the non-interacting regime we can outline three inequivalent classes of fillings that we dub low, intermediate and high. In the specific, we consider N/L = 3/16, 3/8 and 7/12 for I = 1/2, and N/L = 1/4, 13/24 and 5/6 for I = 1 corresponding to the low-, intermediate-, and high-filling cases respectively. OBC in the real dimension have been adopted.

Momentum distribution functions
Let us first focus on the I = 1/2 case. In Figs. 3(a-c) we plot the momentum distribution function ν p for the three fillings listed above. For U/t = 0, the behavior of ν p can be easily predicted by looking at the single-particle spectrum: in the low and high-filling cases peaks arise in correspondence of the partially occupied energy wells, while in the intermediate-filling case a more homogeneous momentum distribution function emerges.
The presence of repulsive atom-atom interactions significantly modifies the momentum distribution functions in the low-and high-filling cases: when U/t is increased, they drive the distribution towards a more homogeneous shape with enhanced tails, a typical effect of interactions [45]. On the contrary, in the intermediatefilling case the homogeneous behavior is unmodified, apart from the mentioned tails. Such a phenomenology is well explained using bosonization and renormalization-group techniques, as discussed in Ref. [46]. Interactions lead to an effective enhancement of the energy of the two gapped modes, whose presence characterizes the helical liquid. Effectively, the interacting system behaves as if Ω/t were renormalized and increased, thus enhancing the filling regimes for which an helical liquid can be expected. Furthermore, this is in agreement with the fact that the non-interacting helical liquid is essentially left unchanged by the interactions. Thus, provided the interaction is sufficiently strong, even low-and high-filling setups can be driven into an helical liquid. This is the first important result of our analysis: repulsive interactions enhance the gap protecting the helical liquid. The momentum distribution functions for I = 1 at the three cited fillings display the same qualitative behavior, see Figs. 3(d-f). Again, the underlying physics can be explained in terms of an effective enhancement of Ω/t, due to the presence of interactions. Contrary to the previous case, for values of I larger than 1/2, no analytical prediction is available, but it seems reasonable to believe that a similar behavior should occur.
It is important to note that in the SRC regime on-site interactions are not expected to significantly modify the momentum distribution function of the non-interacting system. The occupied single-particle states belong only to the lowest band and are almost polarized in the same direction, x: the gas is thus quasi-spinless and an onsite interaction should only weakly alter the ground state because of Pauli exclusion principle. Additional numerical investigations may help in clarifying this issue.
Further information about the system can be revealed by the spin-resolved momentum distribution functions ν p,m . In Figs. 4(a-c) we plot such functions in the WRC regime for the spin species m = 1/2 and I = 1/2. Such profiles are clearly Synthetic gauge fields in synthetic dimensions: Interactions and chiral edge modes 10   asymmetric with respect to k p = 0, indicating the helical nature of the ground state. Note that the asymmetry is enhanced by the interactions. A similar behavior is observed for m = ±1 and I = 1, see Figs. 4(d-f). On the other hand, for symmetry reasons, the momentum distribution function ν p,m=0 is symmetric with respect to k p = 0, although it is modified by the interactions, see Figs. 4(g-i).

Chirality witnesses
In this paragraph we discuss the properties of the chirality witnesses Q m and J m for an interacting system. Even though a preliminary analysis of these quantities has been carried out in Ref. [28], a systematic study of the effects of repulsive atom-atom interactions in a relevant experimental setup [4] is still lacking.
In Figs. 5(a) and 5(c) we display the behavior of Q m=I as a function of U for the cases I = 1/2 and I = 1; we focus again on the three fillings outlined above. In Appendix A we show that, although the system has OBC and it is not homogeneous, averaging over many lattice sites yields a value related to the bulk current. A first striking observation is that one can observe different trends, also displaying non-monotonic features. The role of interactions in protecting the helical liquid here encounters a first naive confirmation: in all cases, the value of |Q m | in the U/t → ∞ limit exceeds that of the non-interacting system.
In order to understand the dependence of Q m on U/t, we employ an effective model. We have already noticed that the most prominent effect of the interactions on ν p is that of letting the system behave as if it were non-interacting but with a renormalized value of Ω. Here we test this observation by studying the dependence of Q m on Ω in the absence of interactions. Results displayed in Figs. 5(b) and 5(d) show that this simple model offers a good qualitative understanding of the interacting system. For example, in both the I = 1/2 and I = 1 cases, Q m=I displays the same (quasi-)monotonic increasing behavior with U/t and with Ω/t, for the low and intermediate fillings. In the high-filling case, Q m=I exhibits a strongly non-monotonic behavior as a function of U ; in particular the plot points out a change in sign which is a priori unexpected because in the classical case the magnetic field determines unambiguously the direction of the circulating currents. To further elucidate this problem, in Fig. 6 we plot the dependence of Q m on the filling N/L for a fixed value of Ω/t and U/t = 0. The plot shows that at low fillings the value of Q m=I increases gently, but experiences an abrupt decrease once the helical region is entered, marked by the violet line (intermediate fillings). For higher fillings (even outside the helical region) and for small Ω, the value of Q m=I is negative and thus the current changes sign; however, by increasing Ω, Q m=I also increases, crossing 0 and becoming positive and finite. It thus follows that in this system the chiral currents are not strictly speaking chiral and states with opposite current flow occur at accessible energies. The chirality witness J m=I shares many similarities with Q m=I . In Fig. 7 we plot J m=I as a function of U , to be compared with Figs. 5(a) and 5(c) for Q m=I . Again, in the low-and intermediate-filling regimes J m=I is almost monotonous, whereas monotonicity is significantly broken for high fillings. The explanation of this behavior can again be sought in the peculiar dependence of the current carried by the eigenmodes of the system.

Conclusions
By means of DMRG simulations, we have studied the impact of atom-atom repulsive interactions on the quantum-Hall-like chiral currents recently detected in Refs. [4,5]. We have modeled the experimental setup of Ref. [4] and characterized the behavior of the edge currents through the asymmetry of the momentum distribution function.
We have considered different particle fillings and identified the filling range where a chiral/helical liquid appears (in the text dubbed as "intermediate"). When the filling is slightly higher or lower, in the presence of repulsive interactions, the system starts behaving as the non-interacting chiral/helical liquid. This leads to the first conclusion that interactions stabilize such phase. To better assess its nature, we have introduced two chirality witnesses, which are displayed in Figs. 5 and 7, where the chirality of the currents is studied as a function of the interaction strength U/t. As highlighted in the plots, the role of the interaction is non-trivial, and in the strongly-repulsive limit leads to the enhancement of the persistent currents.
In the analysis presented here we have neglected the role of an harmonic trapping confinement as well as finite-temperature effects. Their interplay with interactions and the edge physics highlighted so far is left for a future work.
The edge currents studied here do not have a topological origin. However, these synthetic ladders may support fractional quantum Hall-like states [28,30], and it would be very interesting to understand how to explore this regime by means of the quantities discussed in the present paper. In particular it would be important to develop a complete characterization of how fractional quantization may emerge in a cold atomic setup.