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arXiv:1611.08917 · Physical Review A

Phase vortices of the quenched Haldane model

Jinlong Yu

Published Aug 1, 2017• 30 citations
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Abstract

Using the recently developed Bloch-state tomography technique, the quasimomentum $\mathbf{k}$-dependent Bloch states ${[sin\left({\ensuremath{\theta}}_{\mathbf{k}}/2\right),\phantom{\rule{0.28em}{0ex}}\ensuremath{-}cos\left({\ensuremath{\theta}}_{\mathbf{k}}/2\right){e}^{i{\ensuremath{\phi}}_{\mathbf{k}}}]}^{T}$ of a two-band tight-binding model with two sublattices can be mapped out. We show that if we prepare the initial Bloch state as the lower-band eigenstate of a topologically trivial Haldane Hamiltonian ${H}_{i}$ and then quench the Haldane Hamiltonian to ${H}_{f}$, the time-dependent azimuthal phase ${\ensuremath{\phi}}_{\mathbf{k}}(t)$ supports two types of vortices. The first type of vortices is static, with the corresponding Bloch vectors pointing to the north pole (${\ensuremath{\theta}}_{\mathbf{k}}=0$). The second type of vortices is dynamical, with the corresponding Bloch vectors pointing to the south pole (${\ensuremath{\theta}}_{\mathbf{k}}=\ensuremath{\pi}$). In the $({k}_{x},{k}_{y},t)$ space, the linking number between the trajectories of these two types of vortices exactly equals the Chern number of the lower band of ${H}_{f}$, which provides an alternative method to directly map out the topological phase boundaries of the Haldane model.

DOI
10.1103/physreva.96.023601

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Using the recently developed Bloch-state tomography technique, the quasimomentum k\mathbf{k}k-dependent Bloch states [sin(\ensuremathθk/2),\ensuremath−cos(\ensuremathθk/2)ei\ensuremathϕk]T{[sin\left({\ensuremath{\theta}}_{\mathbf{k}}/2\right),\phantom{\rule{0.28em}{0ex}}\ensuremath{-}cos\left({\ensuremath{\theta}}_{\mathbf{k}}/2\right){e}^{i{\ensuremath{\phi}}_{\mathbf{k}}}]}^{T}[sin(\ensuremathθk​/2),\ensuremath−cos(\ensuremathθk​/2)ei\ensuremathϕk​]T of a two-band tight-binding model with two sublattices can be mapped out. We show that if we prepare the initial Bloch state as the lower-band eigenstate of a topologically trivial Haldane Hamiltonian Hi{H}_{i}Hi​ and then quench the Haldane Hamiltonian to Hf{H}_{f}Hf​, the time-dependent azimuthal phase \ensuremathϕk(t){\ensuremath{\phi}}_{\mathbf{k}}(t)\ensuremathϕk​(t) supports two types of vortices. The first type of vortices is static, with the corresponding Bloch vectors pointing to the north pole (\ensuremathθk=0{\ensuremath{\theta}}_{\mathbf{k}}=0\ensuremathθk​=0). The second type of vortices is dynamical, with the corresponding Bloch vectors pointing to the south pole (\ensuremathθk=\ensuremathπ{\ensuremath{\theta}}_{\mathbf{k}}=\ensuremath{\pi}\ensuremathθk​=\ensuremathπ). In the (kx,ky,t)({k}_{x},{k}_{y},t)(kx​,ky​,t) space, the linking number between the trajectories of these two types of vortices exactly equals the Chern number of the lower band of Hf{H}_{f}Hf​, which provides an alternative method to directly map out the topological phase boundaries of the Haldane model.
We show how the topological number of a static Hamiltonian can be measured from a dynamical quench process. We focus on a two-band Chern insulator in two dimension, for instance, the Haldane model, whose dynamical process can be described by a mapping from the [k_{x},k_{y},t] space to the Bloch sphere, characterized by the Hopf invariant. Such a mapping has been constructed experimentally by measurements in cold atom systems. We show that, taking any two constant vectors on the Bloch sphere, their inverse images of this mapping are two trajectories in the [k_{x},k_{y},t] space, and the linking number of these two trajectories exactly equals the Chern number of the static Hamiltonian. Applying this result to a recent experiment from the Hamburg group, we show that the linking number of the trajectories of the phase vortices determines the phase boundary of the static Hamiltonian.
In a recent experimental work [Z. Wu et al., Science 354, 83 (2016)], the PKU-USTC group realized a two-dimensional two-band quantum anomalous Hall model on a square Raman lattice. By quenching the atom-laser detuning of such a Raman lattice, the time-dependent Bloch vectors for each quasimomentum points define a Hopf mapping from quasimomentum-time space $({k}_{x},{k}_{y},t)\ensuremath{\in}{T}^{3}$ to the Bloch sphere S2{S}^{2}S2. The Hopf links between the preimages of any two Bloch vectors on S2{S}^{2}S2 can be measured experimentally through Bloch-state tomography using spin-resolved time-of-flight measurement together with suitable radio-frequency manipulations. The dynamical Hopf invariants, which are quasimomentum-time integrations of Chern-Simons densities, can also be extracted experimentally through certain quench processes. As the Hopf invariant equals the Chern number of the postquench Hamiltonian, the measurements of Hopf links and Hopf invariants provide an alternative way of understanding the topological phase diagram of the equilibrium system.
Using the exact-diagonalization (ED) and mean-field (MF) approaches, we investigate the ground-state phase diagram of the interacting Haldane model on the honeycomb lattice, incorporating spin-dependent sublattice potentials ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\sigma},\ensuremath{\alpha}}$. Here $\ensuremath{\alpha}=\text{A},\text{B}$ and $\ensuremath{\sigma}=\ensuremath{\uparrow},\ensuremath{\downarrow}$ denote the sublattice and spin components, respectively. Setting ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\sigma},\text{A}}=+\mathrm{\ensuremath{\Delta}}$ $(\ensuremath{-}\mathrm{\ensuremath{\Delta}})$ and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\sigma},\text{B}}=\ensuremath{-}\mathrm{\ensuremath{\Delta}}$ $(+\mathrm{\ensuremath{\Delta}})$ for $\ensuremath{\sigma}=\ensuremath{\uparrow}$ $(\ensuremath{\downarrow})$ results in the system favoring a spin ordered state. Conversely, introducing the nearest-neighbor Coulomb interaction can induce charge ordering in the system. Due to the competition between these factors, we observe that in both ED and MF approaches, an exotic state with Chern number C=1C=1C=1 survives amidst two locally ordered phases and a topologically ordered phase with C=2C=2C=2. In the ED method, various properties, such as the fidelity metric, the excitation gaps and the structure factors, are employed to identify critical points. In the MF method, using a sufficiently large lattice size, we define the local order parameters and band gaps to characterize the phase transitions. The interacting Haldane model and the spin-dependent lattice potential may be experimentally realized in an ultracold atom gas, providing a potential means to detect this intriguing state.
A quantum quench is a nonequilibrium dynamics governed by the unitary evolution. We propose a two-band model whose quench dynamics is characterized by an arbitrary Hopf number belonging to the homotopy group ${\ensuremath{\pi}}_{3}({S}^{2})=\mathbb{Z}$. When we quench a system from an insulator with the Chern number ${C}_{i}\ensuremath{\in}{\ensuremath{\pi}}_{2}({S}^{2})=\mathbb{Z}$ to another insulator with the Chern number Cf{C}_{f}Cf​, the preimage of the Hamiltonian vector forms links having the Hopf number ${C}_{f}\ensuremath{-}{C}_{i}$. We also investigate a quantum-quench dynamics for a four-band model carrying an arbitrary second Chern number $N\ensuremath{\in}{\ensuremath{\pi}}_{4}({S}^{4})=\mathbb{Z}$, which can be realized by quenching a three-dimensional topological insulator having the three-dimensional winding number $N\ensuremath{\in}{\ensuremath{\pi}}_{3}({S}^{3})=\mathbb{Z}$.
We perform a sudden quench on the Haldane model with long range interactions, more specifically generalizing to the next to next nearest neighbor hopping, referred to as the N3N3N3 model in our work. Such a model possesses both isotropic and multiple anisotropic (satellite) Dirac points which lead to a rich topological phase diagram consisting of phases with higher Chern number (C)(C)(C). Quenches between the topological and the nontopological phases of such an infinite system probe the effect of the presence of the anisotropic Dirac points on the nonequilibrium response of the topological system. Interestingly, the Chern number remains the same before and after the quench for both the quenching protocols, even when the quench of the system is carried out between two different topological phases. However, for a finite system, we establish that the initial edge current asymptotically decays to zero when the system is quenched to the nontopological phase although the Chern number for the corresponding periodically wrapped system remains unaltered; what is remarkable is that when the Hamiltonian is quenched from ∣C∣=2|C|=2∣C∣=2 phase to the nontopological phase the edge current associated with the inner channel decays at a faster rate than the outer channel resembling a situation in which the system passes through the phase with ∣C∣=1|C|=1∣C∣=1 before ending up in the phase C=0C=0C=0.
Topological defects in Bloch bands, such as Dirac points in graphene, and their resulting Berry phases play an important role for the electronic dynamics in solid state crystals. Such defects can arise in systems with a two-atomic basis due to the momentum-dependent coupling of the two sublattice states, which gives rise to a pseudospin texture. The topological defects appear as vortices in the azimuthal phase of this pseudospin texture. Here, we demonstrate a complete measurement of the azimuthal phase in a hexagonal optical lattice employing a versatile method based on time-of-flight imaging after off-resonant lattice modulation. Furthermore, we map out the merging transition of the two Dirac points induced by beam imbalance. Our work paves the way to accessing geometric properties in optical lattices also with spin-orbit coupling and interactions.