Quantum phases of dipolar bosons in one-dimensional optical lattices

We characterize the ground-state phase diagram of Hamiltonian (II) by means of the observables that we detail in what follows. We first determine the ground state energy $E(N)$ for $N$ particle over $L$ lattice sites, with $N=L$. The so-called charge gap $\Delta_{c}$ corresponds to the energy required to create a particle-hole pair and is obtained after finding the ground-state energies for $N-1$ and $N+1$ bosons Batrouni et al. (2013, 2014):

Its non-vanishing value in the thermodynamic limit signals an insulating phase. An insulator is also characterized by a finite value of the so-called neutral gap $\Delta_{n}$, corresponding to the difference between the energy $E_{\text{ex}}(N)$ of the first excited state and the energy $E(N)$ of the ground state Batrouni et al. (2013, 2014):

The first excited state is numerically found by determining the lowest energy state in the subspace orthogonal to the ground state, see Appendix A. In the SF phase the neutral gap vanishes in the thermodynamic limit.

We note that in one dimension the SF phase is strictly-speaking a Luttinger liquid with exponent $K>2$ Biedroń et al. (2018); Deng et al. (2013); Batrouni et al. (2014); Cazalilla et al. (2011), thus the off-diagonal correlations decay with the distance according to a power-law:

In order to reveal modulations in the off-diagonal correlations, we calculate the Fourier transform of the single-particle density matrix $M(q)$:

Typically, in a standard SF the maximum component of $M(q)$ is at $q=0$. The correlated tunneling, on the other hand, gives rise to effects that in one dimension are analogous to an effective change of the sign of the tunneling coefficient. Correspondingly, the Fourier transform of the single-particle density matrix can have a non-zero component at $q=\pi$. We dub the corresponding ground state as staggered superfluid (SSF) phase Kraus et al. (2020).

Density modulated phases are revealed by properties of the local density-density correlations Rossini and Fazio (2012); Berg et al. (2008), whose Fourier transform is the structure form factor:

For a two-site translational symmetry, $S(k)$ shows a finite peak at $k=\pi$. The phase is a Charge Density Wave (CDW) or lattice Supersolid (SS) depending on whether the density-modulated phase is incompressible or superfluid, respectively. The SS phase is a staggered supersolid (SSS) when $M(q)$ is finite and maximum at $q=\pi$.

The Haldane insulating phase (HI) is gapped and characterized by non-local spatial correlations in the density fluctuations $\delta\hat{n}_{j}$. This is captured by the string order parameter $\mathcal{O}_{S}$ Batrouni et al. (2014); Dalla Torre et al. (2006); Berg et al. (2008); Batrouni et al. (2013); Rossini and Fazio (2012):

The definition of the density fluctuation $\delta\hat{n}_{j}$ is important. When we consider the density fluctuations about the mean value $\rho$, namely, $\delta\hat{n}_{j}=\hat{n}_{j}-\rho$, then we label the string order parameter by $\mathcal{O}_{S}(\rho)$. When instead the density fluctuations are taken about the local mean occupation $\left\langle\hat{n}_{j}\right\rangle$, namely, $\delta\hat{n}_{j}(\left\langle\hat{n}_{j}\right\rangle)=\hat{n}_{j}-\left\langle\hat{n}_{j}\right\rangle$, then the corresponding string order parameter is given by $\mathcal{O}_{S}(\left\langle\hat{n}_{j}\right\rangle)$. Both definitions give finite values within the HI phase. Instead, in the CDW phase $\mathcal{O}_{S}(\left\langle\hat{n}_{j}\right\rangle)$ vanishes, while $\mathcal{O}_{S}(\rho)$ is finite. Thus $\mathcal{O}_{S}(\left\langle\hat{n}_{j}\right\rangle)$ signals the HI phase. The HI phase can also be distinguished from other insulating phases by means of the parity order parameter

which is finite in the MI and CDW phases, while it vanishes in the HI phase independent of the definition of $\delta\hat{n}_{j}$.

The phases and the corresponding values of order parameters are summarized in Table 1.

Finally, we determine the von-Neumann entropy of the ground state for a lattice bipartition into two subsystems $A$ and $B$. Denoting the ground state by $\ket{\psi_{0}}$, the von-Neumann (entanglement) entropy is defined as Amico et al. (2008); Eisert et al. (2010); Metlitski and Grover ; Frérot and Roscilde (2016)

where $\hat{\rho}_{B}=\text{Tr}_{A}\left\{\ket{\psi_{0}}\bra{\psi_{0}}\right\}$.

The extended Bose-Hubbard model of Eq. (II) is a good approximation of the Hamiltonian describing the dynamics of dipolar atoms tightly confined by the lowest band of an optical lattice in a quasi one-dimensional geometry. The trapping potentials can be described by a potential of the form:

where $m$ is the atomic mass, $\omega$ is the frequency of the harmonic trap that confines the atomic motion along the $x$ direction, and $V_{0}$ is the depth of the optical lattice with periodicity $a$. The details of the derivation of Eq. (II), starting from the full Hamiltonian of interacting atoms in the potential of Eq. (10), have been extensively reported, for instance, in Refs. Cartarius et al. (2017); Kraus et al. (2020). These derivations allow one to link the Bose-Hubbard coefficients with the experimental parameters.

In this work we set $V_{0}=8E_{R}$ and $\omega=\sqrt{2V_{\text{har}}\pi^{2}/a^{2}m}$ with $V_{\text{har}}=50E_{R}$ and $E_{R}=h^{2}/2m(2a)^{2}$ the recoil energy for a laser with wavelength $\lambda=2a$. The choice of $\omega$ warrants that the transverse motion is frozen out for the parameters that we consider. Since we keep the depth $V_{0}$ constant, the tunneling amplitude $t$ is fixed and finite.

We sweep across the insulator-superfluid transition by varying the onsite interaction coefficient $U$. The latter results from the interplay between the van-der-Waals, contact potential $U_{g}(\mathbf{r})$ and the onsite contribution of the dipolar interaction $U_{d}(\mathbf{r})$:

Here, $g=4\pi\hbar^{2}a_{s}/m$ and is tuned by changing the scattering length, while the dipole-dipole potential is scaled by the coefficient $C_{dd}$ and $\theta$ denotes the angle between the dipoles and the interparticle distance vector $\mathbf{r}$. The other coefficients $V$ and $T$ are changed by varying the dipolar strength.

Figure 1 displays the absolute value of the correlated tunneling coefficient, $|T|$, as a function of the nearest-neighbor interaction, $V$, and of the onsite interaction, $U$. Both coefficients $|T|$ as well as $V$ increase with the dipole-dipole interaction strength, which is here reported in terms of the dimensionless parameter $d$ Biedroń et al. (2018); Astrakharchik et al. (2007):

Note that $T$ is negative for the parameters we consider and it scales as $|T|\sim V/10$.

The phase diagram for the density $\rho=1$ is shown in Fig. 2 for a finite chain. The different colors indicate the phase boundaries predicted by (i) the charge gap (blue), (ii) the neutral gap (magenta), (iii) the string order parameter (red), (iv) the parity order parameter (black) and (v) the CDW order parameter (green). The boundaries are extracted following the procedure described above, using Eq. (14).

In the considered parameter regime the phases are SF, MI, HI, CDW, and a region which has the features of a phase separation (PS) and which will be discussed in Sec. III.4. We note that we do not find any staggered SF. These findings are in agreement with the results obtained with infinite DMRG. Figure 3 displays a color plot of the string order parameter $\mathcal{O}_{S}(\left\langle\hat{n}_{i}\right\rangle)$, (7) and of the parity order parameter $\mathcal{O}_{P}$, (8), both obtained with iDMRG. For comparison, we also report the corresponding values obtained by setting $T=0$.

Despite some similarities with the phase diagram found setting $T=0$ in Eq. (II) Batrouni et al. (2013, 2014); Rossini and Fazio (2012), nevertheless, there are also some striking differences. In the first place, for $T\neq 0$ the HI phase occupies a smaller area in parameter space. This confirms the observation in Ref. Biedroń et al. (2018). In general, correlated tunneling stabilizes the MI and CDW phases in the parameter space, while the size of SF and HI phases are substantially reduced. Moreover, the HI phase seems to stretch down to smaller values of $U/t$ and $V/t$. We note that we cannot determine the phase boundaries for small $U/t$ and around $1.5\lesssim V/t\lesssim 2$ because in this region the error bars are large.

The color plots in Fig. 4 report the von-Neumann entropy, $S_{\text{vN}}$ (9), across the phase diagram and calculated by means of iDMRG. The von-Neumann entropy sheds light on the spatial decay of correlations. Comparison with the plot of the Fourier transform of the single-particle density matrix, Fig. 5, shows that part of the region where $S_{\rm vN}$ is maximal overlaps with the SF domain. Like $M(q)$, the von-Neumann entropy decays slowly to zero when increasing $U/t$ at small values of $V/t$, sweeping across the SF-MI phase transition.

For small $U/t$ and for $V/t\gtrsim 1.5$ $S_{\rm vN}$ undergoes strong fluctuations from point to point. We associate this behavior with the phase separation where the convergence of DMRG is doubtful. Comparing this region with the one at $T=0$, lower panel of Fig. 4, we observe that for $T\neq 0$ it appears at significantly lower values of $V/t$.

Figure 6 displays $S_{\rm vN}$ as a function of $V/t$ at fixed ratio $U/t$, in the part of the phase diagram where the phases are insulating. Starting from the MI phase we observe peaks when crossing the MI-HI and the HI-CDW transitions, which we discuss in detail in the following.

In Fig. 2 we observe a direct transition from the MI to the CDW phase at sufficiently high values of $U/t$. Fig. 7 shows that string ($\mathcal{O}_{S}(\rho)$) and density-wave order ($S(q=\pi)$) parameters are discontinuous at the transition point, indicating a first-order phase transition. Here the string and density-wave order parameter agree almost exactly, since in the limit $U/t$ large the MI and CDW can be described by trivial Fock states, which lead to the same value of the string and density-wave order parameter in the thermodynamic limit.

At smaller values of the ratio $U/t$ the HI phase separates the MI from the CDW phase. The peaks in the profile of the von-Neumann entropy in Fig. 6 suggest that the phase transitions at the MI-HI and at the HI-CDW transitions are continuous (of second order). This is corroborated by the behavior of the neutral gap at the MI-HI and at the HI-CDW transitions. The HI phase corresponds to the interval where the energy gaps and the string order parameter possess finite values, while both parity and density-wave order parameter vanish. The finite value of the string order parameter and the vanishing parity order parameter demonstrate the topological nature of the HI phase.

The neutral and charge gaps are displayed in the lower panel of Fig. 8 for $U/t=3$ as a function of $V/t$. For small $V/t$ the neutral and charge gaps are finite, corresponding to the MI phase. For a larger value of $V/t$ the gaps shrink to zero indicating the continuous transition to the HI phase. This agrees with the results for the $T=0$ case (no correlated tunneling) Batrouni et al. (2014); Berg et al. (2008); Ejima and Fehske (2015); Ejima et al. (2014), where vanishing gaps both in the charge and neutral sectors signal a second order phase transition with central charge $c=1$ Berg et al. (2008); Ejima et al. (2014); Ejima and Fehske (2015). At the transition separating the HI and the CDW (symmetry-broken) phase the neutral gap vanishes, while the charge gap remains finite. This is as in the $T=0$ case, where the transition is of Ising type with central charge $c=1/2$ Berg et al. (2008); Amico et al. (2010); Ejima et al. (2014); Ejima and Fehske (2015) and the quantum critical point is topological Fraxanet et al. (2022). In the CDW phase the density-wave order parameter reaches a finite value, see the upper panel of Fig. 8.

We finally discuss the parameter region at large $V/t$ but small $U/t$, where the von-Neumann entropy has large fluctuations from point to point. We denote this regime by phase separation. Here, we find that the ground state of the canonical ensemble consists of a mixture of two or more phases. This feature can be revealed by inspecting the site occupation and its variance across the lattice. It can also be captured by the chemical potential as a function of the density $\rho$ Batrouni et al. (2014); Maik et al. (2013). In fact, in the grand-canonical ensemble the phase at unit density is unstable and the density is a discontinuous function of the chemical potential Batrouni et al. (2014).

In order to analyze the phase-separation region in the canonical ensemble, we calculate the density $\rho=N/L$ as a function of the chemical potential $\mu$, which we find by means of the formula Batrouni et al. (2014)

Figure 9 displays $\rho$ as a function of $\mu$ for $(U/t,V/t)=(0.5,4)$ within the phase-separation region. The behavior suggests a hysteresis, which signals a discontinuous transition.

The phase separation region for $T=0$ has been recently extensively analyzed in Kottmann et al. (2021). Correlated tunneling shifts the appearance of this phase to lower values of $V/t$ and possibly increases the number of metastable configurations. Figure 10 displays some of the metastable configurations we find, corresponding to CDW clusters separated by SF regions. Configurations like the one in the upper panel have been reported in Batrouni et al. (2014). The configuration in the lower panel, instead, seems to be stable due the presence of correlated tunneling.

In previous works some of us showed that the effect of correlated tunneling on the ground-state phase diagram can be partially captured by an effective model. In this effective model correlated tunneling and single-particle hopping are replaced in Eq. (II) by a single hopping term with effective tunneling coefficient $t_{\rm eff}=t+T(2\rho-1)$ Kraus et al. (2020); Suthar et al. (2020). This coefficient can vanish, giving rise to an effective atomic limit which agrees with numerical results obtained with the full model Kraus et al. (2020); Suthar et al. (2020). We have verified that, for the parameters we consider, $t_{\rm eff}$ is always finite. Fig. 11 displays the same data as in Fig. 2, but with the axes now rescaled by $t_{\rm eff}$: The rescaled phase boundaries SF-MI and MI-HI-CDW are in good agreement with the phase diagram at $T=0$ (c.f. Fig. 6(b)) Batrouni et al. (2014), suggesting that the effect of correlated tunneling on the size of the insulating phase could be captured by this effective description.