Quantum phase transitions in the dimerized extended Bose-Hubbard model

We present an unbiased numerical density-matrix renormalization group study of the one-dimensional Bose-Hubbard model supplemented by nearest-neighbor Coulomb interaction and bond dimerization. It places the emphasis on the determination of the ground-state phase diagram and shows that, besides dimerized Mott and density-wave insulating phases, an intermediate symmetry-protected topological Haldane insulator emerges at weak Coulomb interactions for filling factor one, which disappears, however, when the dimerization becomes too large. Analyzing the critical behavior of the model, we prove that the phase boundaries of the Haldane phase to Mott insulator and density-wave states belong to the Gaussian and Ising universality classes with central charges $c=1$ and $c=1/2$, respectively, and merge in a tricritical point. Interestingly we can demonstrate a direct Ising quantum phase transition between the dimerized Mott and density-wave phases above the tricritical point. The corresponding transition line terminates at a critical end point that belongs to the universality class of the dilute Ising model with $c=7/10$. At even stronger Coulomb interactions the transition becomes first order.

Figure 1

IDMRG ground-state phase diagram of the dimerized EBHM (1) for $U/t\ge 2$ with $\delta =0.25$ and $n_b=4$. Here the blue dashed line gives the $\mathrm{D}\text{−}\mathrm{MI}\leftrightharpoons \mathrm{D}\text{−}\mathrm{HI}$ phase boundary; the red solid line denotes the continuous Ising phase transition. Both lines merge at the tricritical point located inside the small rectangle which is enlarged in Fig. 4 (see also the discussions in the text). The QPT is continuous (first order) below (above) the critical end point (${V/t,U/t)}_{\mathrm{ce}}$ marked by the star symbol [there we obtain for the central charge $c^{*}\left(L\right)\simeq 0.7$ from Eq. (4) as $L\to \infty$ on the D-MI$\leftrightharpoons$D-DW transition line, see inset]. In the weak ($V,U$)-coupling regime an SF phase is formed.

Figure 2

Central charge (a), entanglement spectrum (b), and excitation gaps (c) of the dimerized EBHM (1) as a function of $V/t$ at fixed $U/t=4$, where $\rho =1$ and $n_b=4$. A central charge $c=1$ ($c=1/2$) indicates the D-HI$\leftrightharpoons$D-MI (D-HI$\leftrightharpoons$D-DW) transition. The D-HI phase is marked in gray.

Figure 3

Central charge (a), entanglement spectrum (b), and excitation gaps (c) of the model (1) as a function of $V/t$ for fixed $U/t=9$ with $\rho =1$ and $n_b=4$. The data indicate a D-MI$\leftrightharpoons$D-DW transition with $c=1/2$.

Figure 4

(a) Three-peak structure of the correlation length ${\xi }_{\chi }$ of the dimerized EBHM with $\delta =0.25$ and $U/t=6.65$. $\chi$ gives the bond dimension used in iDMRG. (b) Corresponding behavior of the DW order parameter $\langle m_{\mathrm{DW}}\rangle$. Note that $\langle m_{\mathrm{DW}}\rangle$ is finite not only for $V/t\gtrsim 4.00$ but also for $3.974\lesssim V/t\lesssim 3.977$. The dotted lines denote the QPT points with $\chi =400$. (c) Zoomed-in phase diagram in the immediate vicinity of the tricritical point. The dashed line illustrates the parameter scan performed in panels (a) and (b). Note that the parameter region of panel (c) is equal to the size of the rectangle in Fig. 1.

Figure 5

(a) Ground-state phase diagram of the dimerized EBHM with $\rho =1/2$ and $n_b=2$. Data obtained by iDMRG. The dotted lines denote the QPT point with same value of $\delta$ in the spin-1/2 chain (A1). (b) Central charge $c^{*}\left(L\right)$ as a function of $V/t$ at fixed $U/t=6$, calculated [along the dashed line in panel (a)] by finite-system DMRG with PBC.

Figure 6

Entanglement spectrum (a), single-particle gap (b), and neutral gap (c) in the dimerized EBHM with $\delta =0.5$ and $n_b=2$ at $U/t=6$. The dashed line marks the Ising QPT point at $V_{\mathrm{c}}/t\simeq 6.351$.