Nonlocal order in elongated dipolar gases

Dipolar particles in an elongated trap are expected to undergo a quantum phase transition from a linear to a zigzag structure with decreasing transverse confinement. We derive the low-energy effective theory of the transition showing that in the presence of quantum fluctuations the zigzag phase can be characterized by a long-ranged string order, while the local Ising correlations decay as a power law. This is also confirmed using density matrix renormalization group calculations on a microscopic model. The nonlocal order in the bulk gives rise to zero energy states localized at the interface between the ordered and disordered phases. Such an interface naturally arises when the particles are subject to a weak harmonic confinement along the tube axis. We compute the signature of the edge states in the single-particle tunneling spectra pointing to differences between a system with bosonic versus fermionic particles. Finally we assess the magnitude of the relevant quantum fluctuations in realistic systems of dipolar particles, including ultracold polar molecules as well as alkali atoms weakly dressed by a Rydberg excitation.

Figure 1

(a) One of the two broken symmetry zigzag states of the classical system. (b) Longitudinal quantum fluctuations affect a continuous distortion from one zigzag configuration to the other, which restores the ${\mathbb{Z}}_2$ symmetry of the ground state in the bulk.

Figure 2

Phase diagram of polar molecules in an elongated trap in the space of the dimensionless interaction constant, $R_s=d^{2}m{\rho }_0$, and transverse confinement ${\omega }_{\perp }/{\omega }_c$, ${\omega }_c$ is the critical transverse frequency in the classical limit. The solid curve marks the zigzag transition (5) computed within the local mean-field approximation for $\nu =1$, that is, for isotropic transverse confinement. The dashed line corresponds to the Ginzburg criterion below which we expect the mean-field approximation to break down. Realistic values of $R_s$ range between $0.05$ and 130 as marked by the gray band. Note that the quantum disordered zigzag and the linear chain are not sharply distinct. There is a smooth crossover between the two regimes.

Figure 3

States of dipolar particles along an elongated trap with soft harmonic confinement. The inhomogeneous density along the trap acts as a tuning parameter of the zigzag transition. Here the particles in the dense middle section are in the zigzag phase while the wings of the cloud are in the disordered phase. Majorana zero modes are exponentially localized near the interfaces between the two phases.

Figure 4

DMRG calculation of correlations in the two-leg ladder model (24). The two legs of the ladder illustrated in panel (a), correspond to the up and down distortion. $V_{\perp }$ is taken to be the largest energy scale to exclude double occupancy of a rung, and the zigzag transition is tuned by varying the ratio $t_{\perp }/V_{\parallel }$. (b) The component with spatial frequency $\pi {\rho }_0$ of the Ising (red dashed) and string (blue solid) correlation functions in the ordered phase on a log-log plot. The string correlations display true long-range order while the Ising correlations decay as a power law. (c) The same correlation functions calculated in the disordered phase plotted on a log-linear plot, where both are seen to decay exponentially.

Figure 5

(a) and (b) A chain with a soliton that is created by a defect in the $s=\uparrow$ and $s=\downarrow$ state respectively. (c) In both cases the field ${\phi }_{\sigma }$ is shifted by $-\pi$ due to the soliton.