Out-of-equilibrium dynamics of multiple second-order quantum phase transitions in an extended Bose-Hubbard model: Superfluid, supersolid, and density wave

In this paper, we study the dynamics of the Bose-Hubbard model with the nearest-neighbor repulsion by using time-dependent Gutzwiller methods. Near the unit filling, the phase diagram of the model contains density wave (DW), supersolid (SS), and superfluid (SF). The three phases are separated by two second-order phase transitions. We study “slow-quench” dynamics by varying the hopping parameter in the Hamiltonian as a function of time. In the phase transitions from the DW to SS and from the DW to SF, we focus on how the SF order forms and study scaling laws of the SF correlation length, vortex density, etc. The results are compared with the Kibble-Zurek scaling. On the other hand from the SF to DW, we study how the DW order evolves with generation of the domain walls and vortices. Measurement of first-order SF coherence reveals interesting behavior in the DW regime.

Figure 1

Phase diagrams of the EBHM near the unit filling and $V/U=0.375$. There are three phases, (2,0)-type DW, SS, and SF. These phases are separated by the second-order phase transitions. In the DW, ${\mathrm{\Delta }}_{\mathrm{DW}}\ne 0,{\mathrm{\Delta }}_{\mathrm{SF}}=0,\left|\mathrm{\Psi }\right|=0$. On the other hand in the SS, ${\mathrm{\Delta }}_{\mathrm{DW}}\ne 0,{\mathrm{\Delta }}_{\mathrm{SF}}\ne 0,\left|\mathrm{\Psi }\right|\ne 0$. In the SF, ${\mathrm{\Delta }}_{\mathrm{DW}}=0,{\mathrm{\Delta }}_{\mathrm{SF}}=0,\left|\mathrm{\Psi }\right|\ne 0$.

Figure 2

Calculations of the physical quantities for the phase diagrams in Fig. 1. Chemical potential $\mu /U=1.5$ and $V/U=0.375$. The $J$ term stands for the expectation value of the hopping term $-{\sum }_{\langle i,j\rangle }(a_i^{†}{a}_j+\text{H.c.})$. $\left|\mathrm{\Psi }\right|$ and ${\mathrm{\Delta }}_{\mathrm{DW}}$ are order parameters of the SF and DW, respectively. $\rho$ is the mean particle density. Phase transitions take place at $J/U=J_{c1}/U\simeq 0.10$, and $J/U=J_{c2}/U\simeq 0.22$.

Figure 3

Arrow indicates quench protocol from the DW to SS and SF in the phase diagram in Fig. 1. For $\mu /U=1.5$, the critical points are located at $J_{c1}/U=0.10$ and $J_{c2}/U=0.22$.

Figure 4

Calculations of the physical quantities from the DW to SS as a function of time. At $t=0$, the system passes trough the equilibrium phase transition point DW $\to$ SS. Locations of $\widehat{t}$, $t_{\mathrm{eq}}$, and $t_{\mathrm{ex}}$ are indicated.

Figure 5

Observation of scaling laws with respect to ${\tau }_{\mathrm{Q}}$ for various quantities at $t=\widehat{t}$ and $t=t_{\mathrm{eq}}$. The exponents are indicated with errors. The error of the exponent in the bottom-left caption is smaller than 0.01.

Figure 6

Slow quench from the DW to SF through the SS. Physical quantities as a function of time for ${\tau }_{\mathrm{Q}}=300$. $t=23.5$ corresponds to $\widehat{t}$, $t=42.5$ to $t_{\mathrm{eq}}$ and $t=500$ to $t_{\mathrm{ex}}$. The upper panels show phase of ${\mathrm{\Psi }}_i$ for various times. Lower-right panel shows a profile of $|{\mathrm{\Psi }}_i|$ in the SS. The system pass through $J=J_{c1}\left(J_{c2}\right)$ at $t=0(t=1.2{\tau }_{\mathrm{Q}})$.

Figure 7

SF correlation length as a function of time. ${\tau }_{\mathrm{Q}}=200$. After $t=0$, $\xi \left(t\right)$ increases quite rapidly. (The dotted line indicates the portion in which the correlation lengths exceed the system size.) This result indicates that a SF at finite temperature forms there.

Figure 8

Quench protocol of out-of-equilibrium dynamics in the precess SF $\to$ SS $\to$ DW. $(J_i/U)=0.3$, and $(J_f/U)=0$.

Figure 9

If we start the time evolution with the genuine SF state with a totally coherent phase, a DW-SF heterogeneous state forms after crossing the DW phase transition. We show snapshots of the heterogeneous state of the DW and SF at $t=t_f$ that forms as a result of the evolution from the genuine SF state. Density profile exhibits clear formation of local DW regimes in the rather homogeneous background. Snapshot of vortices indicates that they proliferate.

Figure 10

SF amplitude as a function of $J/U$ for various quench times ${\tau }_{\mathrm{Q}}$'s. Results are compared with the equilibrium values.

Figure 11

Density profiles at $t=t_f$ for various quench times ${\tau }_{\mathrm{Q}}$'s. $J\left(t_f\right)=0$. Domain walls separating DW regions form and the total length of domain walls decreases as ${\tau }_{\mathrm{Q}}$ increases.

Figure 12

(Left panels) Physical quantities $\left|\mathrm{\Psi }\right|$, ${\mathrm{\Delta }}_{\mathrm{DW}}$, and ${\mathrm{\Delta }}_{\mathrm{SF}}$ as a function of time for ${\tau }_{\mathrm{Q}}=300$. (Right panels) First-order correlation $F_o$. After $t=0$, it exhibits the collapse-revival behavior. $t^{\prime }\equiv t\times \frac{U}{2\pi }$. $J\left(0\right)/U=J_{c1}/U$ and $J(-150)/U\approx J_{c2}/U$.

Figure 13

(Upper panels) First-order correlation function as a function of time for ${\tau }_{\mathrm{Q}}=50,200$, and 300. $t_f$ is the time at which the quench is terminated as shown in Fig. 8. (Middle panels) Density profiles corresponding to the above times (Fig. 11). (Lower panels) Vortex distributions corresponding to the above times. For ${\tau }_{\mathrm{Q}}=50$ and ${\tau }_{\mathrm{Q}}=200$, rather clear domain walls form although their shapes are different in the two cases. Vortices locate at the domain walls. On the other hand for ${\tau }_{\mathrm{Q}}=300$, locations of vortices are random.