Transport phenomena and correlation dynamics of a one-dimensional effective Hamiltonian equivalent to the hexagonal Harper model

The phase diagram of the one-dimensional hexagonal Harper (HH) model reveals the presence of two metallic phases and one insulating phase, separated by critical lines and a bicritical point. In our work, we investigate transport in the different phases by considering both the isolated and open system scenario. For the case of the isolated system, we study the single-particle dynamics at the bicritical point and along the critical lines. We find that the single-particle wave packet dynamics is superdiffusive in the critical regions with the transport at the bicritical point faster than that along the critical lines. In addition, we study domain wall (DW) dynamics via unconventional multiparticle states in the two metallic phases. The DW state is constructed by partially filling half of the chain while the other half is kept empty; the dynamics of this state reveals distinct behavior in the deep metallic regime of the two metallic phases. Interestingly, the distinct behavior is absent if one instead considers a fully filled half chain. For the open system scenario, we study transport in the nonequilibrium steady state (NESS). We observe that in the critical regions, transport is subdiffusive in nature. Moreover, we find that the transport scaling exponents for the open system scenario are the same at the bicritical point and along the critical lines, unlike the closed system case. In addition, we observe an even-odd size effect on the NESS density and current and on the optimal system-bath coupling parameter (corresponding to the maximum current) in the deep metallic regime of the HH model.

Figure 1

The phase diagram of 1D HH model has been plotted by considering the analysis of average inverse participation ratio while setting the parameters $\varphi =(\sqrt{5}-1)/2$ and $k_x,k_y=0$. The bicritical point is found at $t_a=1=t_b$, where the two metallic phases and one insulating phase meet. For $t_b\ne t_a, t_b<1$ is the metallic regime 1 while $t_a<1$ is the metallic regime 2. The three critical lines are as follows: metal-1-insulator line $(t_b=1,t_a>1)$, metal-2-insulator line $(t_a=1,t_b>1)$, and metal-1-metal-2 critical line $(t_a=t_b<1)$.

Figure 2

For the isolated system case, the rescaled cross correlations $\langle c_x^{†}\left(t\right)c_{L/2}\left(t\right)\rangle$ have been plotted as a function of position for the different times in the three different critical regions, which clearly deviates from the Gaussian distribution in all the above scenarios. The cross correlations scale with the exponent $\alpha \approx 0.72$ at the bicritical point (1,1), as shown in (a), (b). For the critical line separating two metallic phases (0.25,0.25) in (c), (d) and for the critical line separating metallic and insulating phase (1,10) in (e), (f) the scaling exponent turns out to be $\alpha \approx 0.65$. $L=4501$.

Figure 3

For the isolated scenario, the rescaled diagonal correlations $\langle c_x^{†}\left(t\right)c_x\left(t\right)\rangle$ with positions have been plotted for different times in the three critical regimes and similarly to the cross correlation, the diagonal correlation deviates from the Gaussian behavior in the critical regions. The diagonal correlation follows the power-law behavior with the scaling exponent $\alpha \approx 0.72$ in (a), (b) at the bicritical point (1,1). For the critical line separating two metallic phases (0.25,0.25) in (c), (d) and at the critical line separating metallic and insulating phase (1,10) for (e), (f), the power-law scaling exponent is $\alpha \approx 0.65$. $L=4501$.

Figure 4

Here, the domain wall melting has been plotted in the deep metallic regime of both metallic phases at time $t=100$ for system size $L=600$. (a) The initial domain wall state is prepared for different particle density ($\langle c_x^{†}{c}_x\rangle =m_l+0.5$) at the left half while the right half being empty. (b) The front profile of the rescaled density exhibits the initial density dependence in MP1 (10, 0.005). (c) For MP2 (0.005,10), the rescaled density front for different initial density propagates with the same front profile.

Figure 5

For the fully filled DW constructed by the initial GSG state, the total number of transferred particles $N_R\left(t\right)$ as a function of $t_a$ and $t_b$ for different times $t$ has been plotted. $L=600.$

Figure 6

The total number of transferred particles $N_R\left(t\right)$ as a function of $t_a$ and $t_b$ for different times $t$ has been plotted for the partially filled DW state made by the initial GSG state. $L=600$.

Figure 7

Here, we have plotted the rescaled total number of transferred particles $(N_R\times l)$ and the rescaled particle current $(J_R\times l)$ as a function of time $t$ for the different particle densities ($m_l$) for metallic phase 1 (10, 0.005) in (a) and (c) and for metallic phase 2 (0.005, 10) in (b) and (d).

Figure 8

NESS current ($J$) as a function of system size ($L$) has been plotted for different phases. Here, $t_a=10,t_b=0.25$ is for the insulating phase, $t_a=1=t_b$ is the bicritical point, $t_a=0.25=t_b$ is on the metal-1-metal-2 critical line, $t_a=10,t_b=1$ is on the metal-insulator transition line (setting $\varphi =\frac{\sqrt{5}-1}{2},\gamma =1)$.

Figure 9

In the deep regime of MP2 where $t_a=0.005$, the NESS current as a function of the system-bath coupling parameter has been plotted for the odd and even system size. (a) For the odd system size $(L=151)$, the maximum current for different points in the metallic regime is obtained at the different optimal system-bath coupling values. The linear dependence of optimal system-bath coupling as a function of $t_b$ parameter has been shown in the inset. (b) For the even system size $(L=150)$, the NESS current for different $t_b$ parameter values becomes maximum at a constant system-bath coupling parameter and the inset depicts the maximum current's linear dependence with $t_b$.

Figure 10

The NESS local particle density has been plotted as a function of parameters $t_a$ and $t_b$ for (a) $L=150$ considering the even system size, and (b) $L=151$ for the odd system size, respectively. Here, we have fixed the site index $x=30$ and the parameter $\gamma =1$.

Figure 11

The particle density profile in the NESS is shown in the different phases. (a) NESS density for different $t_a$ and $t_b$ values in the insulating regime, (b) at the metal-insulator transition line $(t_a=10,t_b=1)$, at the bicritical point $(t_a=1,t_b=1)$, and at the metal-metal critical line $(t_a=0.25,t_b=0.25)$. The local density has been plotted in metallic phase 2 with the constant $t_b$ parameter for the even $(L=150)$ and odd $(L=151)$ system size in (c) and (d), respectively, where in the inset of (c) and (d) we consider the local density for the constant $t_a$ parameter ($\gamma =1$).

Figure 12

Schematic diagram of the system coupling with the bath for an even and odd system size and the system Hamiltonian is considered for the deep metallic regime of MP2. For an odd system size, the system couples with bath asymmetrically, while for the even system size, the system couples symmetrically with the bath.

Figure 13

Visualization of hexagonal Harper model through brick wall geometry.

Figure 14

The initial DW constructed by the GSA in the metallic regime of the HH model is plotted. The fully filled DW ($\nu =1$) is shown for the parameters ($t_a=10,t_b=0.005$) while the partially filled ($\nu =0.25$) DW has been plotted with colors orange and red for parameters ($t_a=10,t_b=0.005$) and ($t_a=10,t_b=0.25$), respectively. The initial DW profile exhibits strong fluctuations in the partially filled case.

Figure 15

The total number of transferred particles $N_R\left(t\right)$ as a function of parameters $t_a$ and $t_b$ has been plotted for the different times $t$ considering the DW dynamics initiated by the entangled state approach (by setting $L=600$).

Figure 16

The rescaled particle density spread has been shown for the entangled DW state. (a) Here, the initial DW state for different coherence length has been plotted, where the right side is filled and left side is empty. (b) For MP1, the DW front profile exhibits the coherence length dependence which is associated with the particle density. (c) For MP2, the DW with different coherence length propagates with the same profile.

Figure 17

Here, for the case of entangled DW state, the rescaled particle density profile at time $t=100$ and the total number of transferred particles in the metallic phase of AAH model have been shown. In (a) the rescaled density profile exhibits the dependence on the initial density. (b) The total number of transferred particles as a function of time has been plotted. $\lambda =0.15,L=600$.