Supersolid-superfluid phase separation in the extended Bose-Hubbard model

Recent studies have suggested a new phase in the extended Bose-Hubbard model in one dimension at integer filling. In this work, we show that this new phase is phase-separated into a supersolid and superfluid part, generated by mechanical instability. Numerical simulations are performed by means of the density matrix renormalization group algorithm in terms of matrix product states. In the phase-separated phase and the adjacent homogeneous superfluid and supersolid phases, we find peculiar spatial patterns in the entanglement spectrum and string-order correlation functions and show that they survive in the thermodynamic limit. In particular, we demonstrate that the elementary excitations of the homogeneous superfluid with enhanced periodic modulations are phonons, find the central charge to be $c=1$, and show that the velocity of sound, extracted from the intrinsic level splitting for finite systems, matches with the propagation velocity of local excitations in dynamical simulations. This suggests that the low-energy spectrum of the phase under investigation is effectively captured by a spinless Luttinger liquid, for which we find consistent results between the Luttinger parameter obtained from the linear dependence of the structure factor and the algebraic decay of the one-body density matrix.

Figure 1

Main characteristics of the phase-separated ground state. (a) Density profile. The system is separated into two phases described by a flat density typical to a fluid (SF phase) and a periodic structure typical to a solid (SS phase). The solid patterns of alternating occupation are pinned at the edges due to the use of open boundary conditions, leaving the superfluid uniform density in the middle. The volume occupied by each of the phases depends on the filling $n$, which here is $n=1$. (b) Off-diagonal single-particle correlation function $\langle b_j^{†}{b}_i\rangle$ in SF and SS phases. Slow power-law decay [seen as straight lines in a log-log plot in insets of Figs. 2 and 2] allows the system to be coherent at distances larger than the lattice spacing which is a one-dimensional analog of Bose-Einstein condensation, and implies that both phases are superfluid. (c) The entanglement spectrum shows different periodicities in the two different phases. The first four largest elements of the entanglement spectrum $\left\{{\lambda }_i\right\}$ are plotted in descending order ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3\ge {\lambda }_4$.

Figure 2

Finite-size study of the inverse compressibility ${\kappa }^{-1}$, Eq. (9), as a function of the filling $n$ for $(U,V)=(0.5,4)$ as in Fig. 1. Vanishing thermodynamic value of ${\kappa }^{-1}$ Eq. (9) signals instability towards phase separation. (a) SF-PS transition: The transition point is estimated at critical filling $n_c^{\text{SF-PS}}=0.815$ defined as the position of the intersection of lines corresponding to different system sizes. In the phase-separated region, $n>n_c$ region, the value of the inverse compressibility ${\kappa }^{-1}$ is lowered as the system size is increased (lines correspond to $L=101;127;201;301$, from top to bottom) and vanishes in the thermodynamic limit. Inset (1a) Example power-law decay of the inverse compressibility ${\kappa }^{-1}$ as a function of system size $L$ in the phase-separated regime, $n>n_c$ Inset (1b): Zoom-in on the intersection. (b) PS-SS transition: Dependence of the inverse compressibility on filling $n$ is scaled with the system size $L$ leading to the estimated value for the critical density equal to $n_c^{\text{PS-SS}}=1.27$ [inset (2a)]. Inset (2b) Example density in PS state close to the transition to SS. Note that in the solid part the average density $n\approx 2.55/2$ matches the critical filling $n_c^{\text{PS-SS}}$.

Figure 3

Filling $n$ versus chemical potential $\mu$ for finite chains of length $L$ with OBC and no explicit $U\left(1\right)$ symmetry. The apparent discontinuity at roughly $\mu \approx 1.13$ signals spinodal decomposition for fillings between $n=0.817$ and 1.31. Insets (a) Wider range in $\mu$ showing the extent of the SS phase, (b) Finite-size extrapolation for the filling values in the main plot at $\mu =1.1296$ to estimate the lower critical filling $n_c=0.817$.

Figure 4

Dependence of filling $n$ on the chemical potential $\mu$ for different system sizes $L$ and different values of $U$ The phase separation is seen as a discontinuity in $n\left(\mu \right)$. (a) Fixing $V=4$ we alter $U$ and see that beyond $U_c\approx 1$ the systems form a CDW, seen by the plateaus at filling $n=1$ (lines are ordered in increasing order of $U$ from the top line to the bottom one). (b) For smaller $V$, PS occurs at higher fillings such that it is not present in the $n=1$ phase diagram for small $V$ anymore (lines are ordered in increasing order of $U$ from the top lines to the bottom ones).

Figure 5

Spatial period $\mathrm{\Lambda }$ taken from the Entanglement spectrum (ES) at $(U,V)=(0.5,3)$ with OBC and ${\chi }_{\text{max}}=100$, $d=6$ as a function of the system size $L$. This state was reported as SS in Ref. [8], but it is actually an SF. Further, the spatial oscillations vanish in the thermodynamic limit as the spatial period grows linearly in system size. However, we show that for incommensurate fillings like in Figs. 6 and 7 these kinds of oscillations do survive in the thermodynamic limit. (a,b) ES ${\lambda }_{s_i}$ for bonds at the center of a system of length $L=50$ and $L=200$, respectively. (c) ES for iDMRG simulation in the thermodynamic limit with a unit cell size $L_{\infty }=100$ showing no spatial oscillations.

Figure 6

Entanglement properties of the superfluid bulk for filling $n=0.77$ and $(U,V)=(0.5,4)$ close to phase-separation. (a) Oscillatory subleading part $S_{\text{sl}}$ of the Rényi-2 entropy $S_2$ in Sec. 2. (b) The four largest squared Schmidt coefficients ${\lambda }_{s_i}$ shown spatially along the bonds. The inset (1a) shows spatial frequencies from Fourier analysis of ${\lambda }_{s_i}$. The extrapolation yields a spatial period $\mathrm{\Lambda }=4.3$ in the thermodynamic limit.

Figure 7

(a) Subleading part $C_{\text{sl}}$ containing the oscillatory part of the string-order correlator $C_{\text{HI}}$ Eq. (8). (b) Oscillatory part of the Rényi-2 entropy $S_2$. (c) The spatial dependence of the four largest squared Schmidt coefficients ${\lambda }_{s_i}^2$ on the bonds of the MPS. (a)–(c) The orange lines are guides to the eye with every third data point plotted to highlight the envelope. We highlight one line in cyan color in all three subplots as a guide to the eye, making it apparent that the period is the same in all three quantities.

Figure 8

Comparison of different common correlators for the same state as in Fig. 7 in the SF phase with oscillating ES. We denote the nonlocal string term as $\mathrm{\Pi }=exp(-i\pi {\sum }_{0\le l<j}\delta n_l)$ with $\delta n_l=n_l-n$. We see that only the correlators with this string term show the oscillations matching the ES. The critical exponent $\alpha$ is obtained from linear fitting the respective correlator in a double logarithmic scale (dotted lines in corresponding colors).

Figure 9

Critical scaling for a superfluid and supersolid states with and without spatial oscillations. Solid blue lines: SF at $(U,V)=(0.5,4)$ with $n=0.62$ and spatial oscillations. Dashed orange lines: SF at $(U,V)=(0.5,0.5)$ at integer filling without spatial oscillations. Dotted green lines: SS at $(U,V)=(0.5,4)$ with $n=1.333$ and spatial oscillations. Despite very different spatial features, all states seem to be well described within the same field theoretic description. (a) Finite-size scaling of the level splitting $\mathrm{\Delta }E=E_1-E_0$ vanishing in the thermodynamic limit and yielding the critical exponent $p$ from $\mathrm{\Delta }E\propto 1/{(L+1)}^p$, roughly matching the expected $p_{\text{LL}}=-1$ for a Luttinger liquid. (b) Central charge $c$ Fig. (11) extrapolated to the thermodynamic limit $c_{\infty }$ from $c=c_{\infty }+\text{const}./L$ matching the central charge $c_{\text{LL}}=1$ for a spinless Luttinger liquid.

Figure 10

Dynamical analysis of a local perturbation. We create a particle at a central site of the ground-state wave function and track the time evolution of the space-resolved density. The propagating defect $\langle \delta n_i\rangle =\langle \mathrm{\Psi }\left(\tau \right)|n_i|\mathrm{\Psi }\left(\tau \right)\rangle -\langle {\mathrm{\Psi }}_0\left|n_i\right|{\mathrm{\Psi }}_0\rangle$ shows a typical “light-cone” structure, for which the red and blue coloring corresponds to positive and negative excess with respect to the average ground-state density. The boundary of the cone propagates with the speed of sound (highlighted in magenta), obtained by fits of the level splitting according to Eq. (10). In all panels we fixed $U=0.5$ and parameters $L=30$, ${\chi }_{\text{max}}=400$, and $d=6$.

Figure 11

Luttinger parameter $K$ dependence on the filling $n$, calculated with OBC with ${\chi }_{\text{max}}=400$ and $d=6$ at $U=0.5$ and $V=4$. Two independent estimations are used, from the long-range asymptotic of the off-diagonal single-particle correlation function $\langle b_i^{†}{b}_j\rangle$ Eq. (18) and from the small momenta of the structure factor $\mathcal{S}\left(q\right)$ via Eq. (22). (a) Structure factor $\mathcal{S}\left(q\right)$ for small momenta at different fillings. With the onset of phase separation, $\mathcal{S}\left(q\right)$ deviates from linear dependence at its origin and Eq. (22) becomes invalid.