String order in dipole-blockaded quantum liquids

We study the quantum melting of quasi-one-dimensional lattice models in which the dominant energy scale is given by a repulsive dipolar interaction. By constructing an effective low-energy theory, we show that the melting of crystalline phases can occur into two distinct liquid phases, having the same algebraic decay of density-density correlations, but showing a different non-local correlation function expressing string order. We present possible experimental realizations using ultracold atoms and molecules, introducing an implementation based on resonantly driven Rydberg atoms that offers additional benefits compared to a weak admixture of the Rydberg state.

The field of 1D quantum physics has recently seen a boost from the experimental advances in the field of ultracold atoms [3][4][5][6][7][8]. At the same time, the combination of 1D systems with long-range interactions as found in polar molecules [9,10] or Rydberg atoms [11,12] has led to a wide range of theoretical studies investigating their ground state properties [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], giving rise to a plethora of novel many-body phenomena. Of special interest is the regime of strong repulsive interactions, in which the dipole blockade excludes configurations having two particles in close proximity and leads to strong frustration effects. In the absence of quantum fluctuations, the ground state of a dipole-blockaded lattice gas is characterized by a devil's staircase of gapped crystalline phases commensurate with the underlying lattice [29]. Generically, the quantum fluctuations induced by movement of the particles result in commensurate-incommensurate transitions to a Luttinger liquid [15,21,22]. Additional phases can occur pertaining to non-convex interaction potentials [27] or extended geometries [24,28].
In this Letter, we build on these earlier developments and study dipole-blockaded quantum gases on a triangular ladder. We establish the ground state phase diagram by analyzing an effective low-energy theory describing the dynamics of dislocation defects of the commensurate crystals. Crucially, the melting of the commensurate crystals can be induced by motion either along the direction of the ladder or along its rungs. This leads to the appearance of two distinct floating solid phases, see Fig. 1, which can be both described in terms of a Luttinger liquid. Remarkably, we find that the two floating solids cannot be distinguished by merely looking at correlation functions of local operators; instead one has to consider a highly non-local observable describing string order. Finally, we comment on possible experimental realizations using ultracold polar molecules or Rydberg atoms, including a novel approach for the latter using laser-induced hopping of Rydberg excitations in an electric field gradient, which can be also used in a variety of different contexts. We start our analysis based on the microscopic Hamiltonian in terms of an extended Hubbard model with longrange dipolar interactions, with the setup of the system depicted in Fig. 2. In the following, we treat the triangular ladder as a single chain having nearest and next-nearest neighbor hoppings. We point out that the dipole blockade renders the distinction between bosons and fermions irrelevant as the exchange of two particles FIG. 2: Setup of the system. Dipolar particles are confined to a triangular ladder structure, with hopping occuring along the direction of the ladder (t2) or along its rungs (t1). Filled dots indicate the particle positions corresponding to the q = 7 commensurate crystal.
occurs at very high energy scales, which are irrelevant for the low-energy properties of the system. The Hamiltonian is given by Here, t 1 and t 2 are the strength of the nearest and nextnearest neighbor hopping, respectively, V |i−j| accounts for the repulsive dipolar interaction between sites i and j according to the particle number n i = c † i c i , and µ denotes the chemical potential. In the classical limit with t 1 = t 2 = 0, the ground state again follows a complete devil's staircase structure of commensurate crystals as the interaction potential is a convex function [29]. The most stable commensurate crystals occur at rational fillings 1/q with q being odd, i.e., the particles are located on the two legs of the ladder in an alternating fashion, see Fig.2. In the following, we will restrict our analysis to densities close to these values. Here, we are interested in the dipole-blockaded regime with q ≫ 1, which allows us to approximate many quantities of interest by performing expansions in 1/q [20]. For example, the center of the commensurate crystals with filling 1/q occurs at a chemical potential of µ 0 ≈ 32ζ(3)V 1 /q 3 , and the variation in chemical potential over which the phase is stable is given by Effective low-energy theory.
-We now study the effects of quantum fluctuations induced by t 1 and t 2 within perturbation theory, i.e., t 1 , t 2 ≪ µ [15,20]. The low-energy excitations correspond to dislocation defects of the commensurate crystal, given by the relation d j = r j+1 −r j −q, which measures the deviation of the spacing between the particles j and j + 1 from the perfectly commensurate case. To denote this distinction between lattice sites and defects, we will use the index i when referring to the former and j for the latter. Depending on the sign of d j , defects occur as hole-like or particle-like, i.e., they decrease or increase the total density, respectively. However, sufficiently far away from the particle-hole symmetric point given by µ = µ 0 , only one of these defects is relevant [20]. Furthermore, the number of defects is being con-served by the perturbation. As the energy cost rapidly increases for |d j | > 1, and the hopping of defects does not exhibit bosonic enhancement, we restrict the Hilbert space to the defect numbers d j = 0, 1, 2. Then, the effective low-energy Hamiltonian to first order in t 1 , t 2 , can be expressed using spin-1 variables as i.e., the next-nearest neighbor hopping t 2 turns into a correlated hopping of the defects. In addition to higher order processes in the perturbation series, we also neglect the weak interaction between the defects. The energy cost associated with each defect is given byμ = (µ 0 −µ+µ w /2)/q, and the repulsion of the defects can be calculated as U = µ w /q. Note that this model is equivalent to a Bose-Hubbard model with correlated hopping and a three-body constraint [30]. If one of the hopping term vanishes, the phase boundaries can be determined exactly by mapping the problem onto free fermions [31]. For t 2 = 0, the on-site repulsion U is irrelevant at the phase transition, which occurs at t 1 =μ/2 between the n = 0 Mott insulator and a liquid phase with finite defect density. Likewise, there is a second phase transition for t 1 = 0 occuring at t 2 =μ + U/2. Remarkably, this second liquid has defects always appear in pairs as the single-defect sector is still protected by a gap ofμ. In the following, we refer to the latter phase as a "pair defect liquid", while calling the former a "single defect liquid". We map out the complete phase diagram using mean-field theory, which even in 1D has been found to produce good qualitative agreement with exact density-matrix renormalization group results [30], and furthermore yields the correct values for the transition in the exactly solvable cases.
In order to go beyond the limitations of mean-field theory and understand the transition between the two liquid phases in more detail, it is instructive to represent the effective spin-1 model by two spin-1/2 degrees of freedom, which then can be bosonized [32]. Then, away from the transition, the system is well described in terms of a single component Luttinger liquid, i.e., the second boson field is gapped out, where Π j and ∇φ j are bosonic fields corresponding to phase and density fluctuations. If the fields with j = 1 are gapless, then the system is in the single defect liquid phase, while gapless j = 2 fields correspond to the pair defect liquid. In the limit of low defect densities, we find K 1 = K 2 = 1 for the Luttinger parameters, while the speed of sound is given by v 1 = qa μt 1 /2 and v 2 = qa (2μ + U )t 2 /2, respectively. The transition between the single and the double defect liquid is of the Ising universality class [33], and the transition line can be determined from the formation of bound states of defect pairs [34]. Here, we find such a bound state appearing for a critical value of the pair hopping t 2 given by When comparing this result to mean-field theory, one confirms the expectation that mean-field theory leads to a slight overestimation of the more ordered single defect liquid, see Fig. 1.
String order.-Within the validity of our perturbative approach, the phase diagram of the defect model (2) corresponds to the phase diagram of the microscopic Hamiltonian (1). However, we are rather interested in describing the appearing quantum phases in terms of observables involving the microscopic degrees of freedom, i.e., correlations between individual particles rather than correlations between the defects. In the following, we apply Luttinger liquid theory to obtain the ground state phase diagram for the microscopic particles.
When mapping from the defect description to the real particles, we first note that the n = 0 Mott insulator for the defects corresponds to the commensurate crystal at filling 1/q, in which the density-density correlation n iq n 0 exhibits true long-range order. In the two liquid phases, we find the density-density correlations of the microscopic particles to asymptotically decay as n x n 0 ∼ x −2K/(n d +q) 2 , where n d is the density of the defects [20]. Consequently, while the existence of algebraically decaying correlations signals the melting of the commensurate crystal phase, it is not possible to distinguish the two defect liquids. Thus, explaining the phase diagram in terms of the microscopic particles requires the probing of nonlocal correlations. However, we already know that the single defect correlation exhibits an algebraic decay in the single defect liquid, and an exponential decay in the pair defect liquid. Remarkably, here we find that this behavior can be captured in terms of the microscopic variables by introducing an observable measuring string order, Here, we have focused on the case of particle-like defects, an analogous expression for hole-like defects follows by replacing n k+q−1 by n k+q+1 . Most importantly, the term inside the exponential is proportional to the number of single defects N x occuring over a distance x. Then, the value of O string (x) simply follows from the characteristic function of the probability distribution of N x . In the pair defect liquid phase, the single defects are uncorrelated, meaning N x satisfies a Poisson distribution with a mean growing linearly with x. Consequently, O string (x) decays exponentially with distance in the pair liquid phase. In the single defect liquid, however, N x is given by a discrete Gaussian distribution whose mean also grows linearly with x, but having a variance σ 2 = K log(x/b)/π 2 , where b is a short distance cutoff [20]. From its characteristic function, we identify the leading term in the long distance limit decaying according to an algebraic func- As the slowest decaying correlation function is still given by the microscopic density-density correlations, both phases form a "floating solid" on top of the underlying lattice. We denote them by FS1 and FS2, respectively, with the former corresponding to the single defect liquid and thus exhibiting an algebraic decay of the string correlations. Note that in contrast to the phases exhibiting string order known as Haldane insulators [13,35,36], both floating solid phases are gapless. The full phase diagram is shown in Fig. 1.
Experimental realization.-Let us now turn to possible experimental implementations of the extended Hubbard model introduced in Eq. (1). In any of the setups discussed in the following, the triangular lattice structure is created using standard optical lattice beams [37]. Additionally, string order can be measured by direct imaging of atoms or molecules in the lattice [7,8].
As a first possible implementation, we consider a setup based on ultracold polar molecules [9,10]. Here, the molecules are prepared in the rovibrational ground state and loaded into the triangular lattice. The hopping matrix elements t 1 and t 2 follow from the tunneling of the molecules in the lattice potential. The repulsive dipoledipole interaction V ij can be realized either by applying a strong electric field [38] or by microwave dressing of the rotational excitations [39,40]. For LiCs molecules having an electric dipole moment of d = 5.5 D, the characteristic energy scaleμ close to the q = 7 commensurate crystal on a a = 532 nm lattice is given byμ ≈ 2π × 100 Hz, which is compatible with experimental timescales within these systems. Alternatively, our model can also be realized using ultracold Rydberg atoms [11,12]. A straightforward implementation would consist of a weak coupling Ω to a Rydberg state detuned by ∆ r [41][42][43], where the strong repulsive interactions between Rydberg states create an interaction potential asymptotically decaying as 1/x 3 for Rydberg states within the Stark fan. However, the experimental parameters for such a Rydberg dressing are quite challenging: in particular, the dipolar interaction is suppressed by a factor ∼ (Ω/∆ r ) 4 , while the radiative decay limiting the lifetime of the system only decreases as (Ω/∆ r ) 2 . Therefore, we present here a dif-  1) is realized by treating atoms in their electronic ground state |g as empty sites, and atoms in a Rydberg state |r as particles. Here, a finite density of Rydberg excitations is created by adiabatically tuning the excitation lasers [44][45][46], which will control the value of the chemical potential µ. Finally, an electric field gradient is introduced, such that the difference in the Stark shift between different sites is exactly canceled by the detuning between two excitation lasers, see Fig. 3, resulting in a hopping of the Rydberg excitations. Note that this process crucially relies on the dipole blockade between neighboring sites; for non-interacting particles the two paths via |g i g i+1 and |r i r i+1 interfere destructively. By introducing an additional laser, it is possible to satisfy this resonance condition for both nearest-neighbor and next-nearest-neighbor distances. The coupling constants t 1 and t 2 derived from the induced hoppings of the Rydberg excitations ∼ Ω a Ω b /2∆ can be controlled independently by the intensities of the excitation lasers. Here, we find that for a Rydberg state with a principal quantum number of n = 43 in a a = 1 µm lattice, the liquid phases close to the q = 7 commensurate crystal form around a characteristic energy scale ofμ ≈ 2π × 400 kHz, which is several orders of magnitude larger than the decay rate of the Rydberg state. We would like to stress that this implementation procedure based on electric field gradients is quite general and can readily be extended to a large class of extended Hubbard models with long-range interactions.
In summary, we have shown that dipole-blockaded quantum gases on triangular ladders support two distinct liquid phases, differing by string order. While the identical decay of the density-density correlations would suggest that both liquids share an effective low-energy description in terms of the same Luttinger liquid, the different nature of the quasiparticle excitations defies this intuition. In particular, our results show that additional care is required when classifying the quantum phases of such extended Hubbard models in the case of nonlocal quasiparticle excitations.
We acknowledge fruitful discussions with T. Vekua and A. Rapp.