Laughlin-like States in Bosonic and Fermionic Atomic Synthetic Ladders

Abstract

The combination of interactions and static gauge fields plays a pivotal role in our understanding of strongly correlated quantum matter. Cold atomic gases endowed with a synthetic dimension are emerging as an ideal platform to experimentally address this interplay in quasi-one-dimensional systems. A fundamental question is whether these setups can give access to pristine two-dimensional phenomena, such as the fractional quantum Hall effect, and how. We show that unambiguous signatures of bosonic and fermionic Laughlin-like states can be observed and characterized in synthetic ladders. We theoretically diagnose these Laughlin-like states focusing on the chiral current flowing in the ladder, on the central charge of the low-energy theory, and on the properties of the entanglement entropy. Remarkably, Laughlin-like states are separated from conventional liquids by Lifschitz-type transitions, characterized by sharp discontinuities in the current profiles, which we address using extensive simulations based on matrix-product states. Our work provides a qualitative and quantitative guideline towards the observability and understanding of strongly correlated states of matter in synthetic ladders. In particular, we unveil how state-of-the-art experimental settings constitute an ideal starting point to progressively tackle two-dimensional strongly interacting systems from a ladder viewpoint, opening a new perspective for the observation of non-Abelian states of matter

The resonant state at filling factor ν = 1/2 in chiral fermionic ladders

“…On the ladder geometry, the main observable to witness the appearance of such states is the chiral current [24]: as a function of either the magnetic flux or the Fermi momentum, the chiral current displays two typical cusps which are spaced out by a linear regime crossing zero exactly at the resonance. These cusps testify the commensurate-incommensurate transitions which determine the rise of the Laughlin-like 1D states [20].…”

“…However, they are characterized by a fast oscillating behavior in x and are thus irrelevant, unless the special resonances f =  p k 2 C F are met. Such resonances can be related to the quantum Hall states at filling n f = = k p 2 1 F C , which indeed measures the ratio between the number of particles N and flux quanta f p = f N L 2 in the ladder [14,15,20,24]. Without loss of generality, we deal henceforth with f>0 and  p p , L,R C S operators only.…”

“…Based on the theory developed for the engineering of FQH states in nanowire arrays [14,15], it was showed that some of the gapless modes of these synthetic ladders can indeed be gapped when the ratio between the artificial magnetic flux per plaquette and the Fermi momentum of the system approaches simple resonant values [16][17][18][19][20][21][22]. These resonant values correspond to the filling fractions ν of the most common quantum Hall states: a semiclassical analysis based on bosonization [14,15,23] reveals, for example, that Laughlin-like states appear at the expected values ν=1/2 and ν=1/3 for bosons and fermions respectively [20]. On the ladder geometry, the main observable to witness the appearance of such states is the chiral current [24]: as a function of either the magnetic flux or the Fermi momentum, the chiral current displays two typical cusps which are spaced out by a linear regime crossing zero exactly at the resonance.…”

Topological Devil’s staircase in atomic two-leg ladders

“…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.…”

Vortex-Hole Duality: A Unified Picture of Weak- and Strong-Coupling Regimes of Bosonic Ladders with Flux

“…Since all particles on the same synthetic dimensional rung share the same optical lattice site, contact interactions lead to exotic long-ranged interactions along the rungs, which for typical systems [10,11] may be assumed to be SUðMÞ symmetric. The interplay of long-ranged interactions along the synthetic dimension and homogeneous gauge fields has attracted considerable attention recently, as it gives rise to the ground states bearing analogies with quantum Hall-like behavior [13][14][15][16][17][18][19][20][21], or exhibiting exotic quantum magnetism [4,18,[22][23][24][25][26][27][28][29][30][31].…”