Unitary preparation of many-body Chern insulators: Adiabatic bulk-boundary correspondence

We approach the long-standing problem of preparing an out-of-equilibrium many-body Chern insulator (CI) and associated bulk-boundary correspondence unitarily. Herein, this is addressed by constructing a dynamical many-body Chern invariant exploiting the property of the bulk macroscopic electric polarization (Resta polarization) of the CI. This Chern invariant defined from observable correlations is also established to topologically classify many-body Chern states in equilibrium. The nonequilibrium behavior of the invariant is probed by ramping the paradigmatic Haldane model of graphene from its trivial to the topological phase. We show that a nonlinear ramp may work more efficiently in approaching the topological state, thereby establishing the existence of an optimal topological state preparation. Furthermore, to ensure the near adiabatic dynamics across the quantum critical point, we propose a counterdiabatic scheme. The topological nature of the prepared state is firmly established by observing an emerging $U\left(1\right)$ topological charge. We also compute the edge current in the time-evolved state of the system under a semiperiodic boundary condition, and we clearly establish an adiabatic bulk-boundary correspondence that firmly ensconces the validity of the many-body invariant.

Figure 1

(a) The emergence of a sharp branch singularity in the function $\beta (k_2,\tau )$. The sharp jump in the $\beta (k_2,\tau )$ function for an adiabatic protocol [Eq. (18)], starting from an initial trivial state, demonstrates the topological nontriviality of the final time evolved state. The magnitude of the jump shown by the distance between the horizontal dashed lines is $\mathrm{\Delta }=-0.96\times 2\pi$. The initial and the final Hamiltonian are chosen such that nearest-neighbor hopping $t_1=1.0$, next-nearest-neighbor hopping $t_2=0.5$, and flux through each plaquette $\varphi =-\pi /2$, $M_i=3\sqrt{3}{t}_2+2.5$, $M_f=3\sqrt{3}{t}_2-2.5$. (b) The topological transition at $t=\tau /2$ (when the system crosses the QCP), through the topological charge of a Haldane model subjected to a linear slow quench for a $40\times 40$ lattice with $\tau =1600$ ($\tau \simeq {\tau }_a$). The quench parameters are $t_1=2.0$, $t_2=1.0$, $\varphi =-\pi /2$, $M_i=3\sqrt{3}{t}_2+2.5$, $M_f=3\sqrt{3}{t}_2-2.5$. (The dotted line is just a guide to eye.)

Figure 2

(a) The defect density [Eq. (21)] at the end of a nonlinear quench [Eq. (20)], obtained numerically (presented by dots), shows an excellent agreement with the modified Kibble-Zurek scaling for different values of $r$ expressed in Eq. (24) (shown by solid lines). (b) The corresponding optimal power $r_{\mathrm{opt}}\simeq 0.65$ of the protocol which minimizes the defect production at the end of the quench. Inset: The function $\beta \left(k_2\right)$ again calculated at the end of the quench shows a considerably sharp jump of magnitude $\mathrm{\Delta }\left(r\right)={\left|\mathrm{\Delta }\right|}_r/2\pi =0.81$ (black) resembling the emergence of a topological nontriviality (${\mathcal{C}}_U\simeq -0.81$) in the final state with $r=r_{\mathrm{opt}}$ even for a much smaller quenching time $\tau =20.0$ for a $40\times 40$ lattice. Notably for the linear quenching case $r=1$, one requires $\tau \sim 1600$ to get as close to the topological state with other quench parameters kept fixed. The parameters are chosen to be $t_1=1.0$, $t_2=0.5$, $M_i=3\sqrt{3}{t}_2+1$ (nontopological), $M_f=3\sqrt{3}{t}_2-1$ (topological).

Figure 3

The $\beta (k_2,\tau )$ function exhibits a sharp branch singularity in the postquench state for a drive employing a shortcut to adiabaticity [Eqs. (25) and (26)] with increasing control field strength $G$. The magnitude of the jump shown by the distance between the horizontal dashed lines is $\mathrm{\Delta }=-0.92\times 2\pi$ with the set of quench parameters as in the linear quench in Fig. 1. The quenching period is chosen to be $\tau =5.0$, which is much shorter than the adiabatic time scale (${\tau }_a\simeq 3600$). Periodic boundary conditions are imposed with a grid size of $60\times 60$ lattice sites in both figures.

Figure 4

(a) The 2D critical surface (purple) in a continuum system parametrized by $M$, $\varphi$, and the counterdiabatic field $G$, and the trajectory of different quenches (1D lines) for different values of $G$. (b) The minimum gap in system vs system size $L\times L$ at times when the quench trajectory intersects the critical surface for different values of $G$. The minimum gap encountered during the topological transition is enhanced for a nonzero $G$ when compared to the protocol for $r=1$, where $G=0$, for the same system size.

Figure 5

The time evolution of the magnitude of the edge-current $J_N^x$ through an armchair edge of the Haldane model with semiopen boundary conditions (periodic about the $x$-direction and open in $y$-direction) under a linear quench with the counterdiabatic mass generation as in Eq. (25) with an $18\times 18$ lattice. The quench parameters are $t_1=2.0$, $t_2=1.0$, $\varphi =-\pi /4$, $M_i=3\sqrt{3}{t}_2+2.5$, and $M_f=0$ for an $18\times 18$ lattice. $G$ is the strength of the anisotropic hopping (see Appendix pp4) introduced in the real-space lattice which in turn generates a CD mass. Inset: The adiabatic evolution of the edge-current vide the protocol in Eq. (18) with the same parameters as in Fig. 2 for a $16\times 16$ lattice. At $t=\tau$, the edge-current (solid curve) thermalizes to its equilibrium topological value (dashed line) in both protocols.

Figure 6

We consider the Haldane model subjected to a linear quench for a $40\times 40$ lattice with $\tau =1600$. The quench parameters are $t_1=2.0$, $t_2=1.0$, $\varphi =-\pi /2$, $M_i=3\sqrt{3}{t}_2+2.5$, $M_f=3\sqrt{3}{t}_2-2.5$. (a)–(c) The angle $\theta$ of the instantaneous eigenstate at all points of the Brillouin zone (BZ) (see Appendix pp2) for $t=0$, $t=t_c=\tau /2$, and $t=\tau$, respectively, for a linear adiabatic quench ($\tau \simeq {\tau }_a$) from the trivial phase to the topological phase. (d) The curvature at $t=t_c$ shows a monopole singularity due to the instanton in $(k_x,k_y,t_c)$ at the Dirac point.

Figure 7

The topological phase diagram of the Haldane model with $t_1=1.0$. The distinct topological phases are separated by quantum critical lines on which the parameter values are such that the system becomes gapless. The parameter regions showing nonzero values of the Chern number (C) are topologically nontrivial. The red arrow shows the direction of the initial and final region of a quench from a trivial phase to a topological phase.

Figure 8

(a) The hexagonal graphene lattice showing the nearest-neighbor (N1) and next-nearest-neighbor (N2) hopping vectors ${\vec{\mathrm{\Delta }}}_{1i}$ and ${\vec{\mathrm{\Delta }}}_{2i}$, respectively, where the lattice constant is set to be $a=1$. The hollow and filled atoms represent the $B$ and $A$ sublattices, respectively. (b) The Brillouin zone of graphene spanned by the reciprocal-lattice vectors ${\vec{b}}_1$ and ${\vec{b}}_2$ containing two inequivalent Dirac points $K$ and $K^{\prime }$ (the Cartesian directions have been labeled by $k_x$ and $k_y$, respectively). The color density shows the absolute value of the band gap ${\mathrm{\Delta }}_k$ of the reciprocal space graphene Hamiltonian showing vanishing gaps at the Dirac points for a $600\times 600$ lattice size having the N1 hopping strength $t_1=1.0$ and the N2 hopping $t_2=0$.

Figure 9

Localization of the single-particle edge current generated in the initial and the postquench Haldane model with a cylindrical geometry. The index $N$ denotes the position of a strip along the translational symmetry-broken direction of the cylinder. The postquench current is observed to be localized into the boundary of the lattice. The simulation is performed for an $18\times 18$ lattice. All the other quench parameters are very similar to those used in the paper.