Topological pumping of quantum correlations

Topological pumping and duality transformations are paradigmatic concepts in condensed matter and statistical mechanics. In this paper, we extend the concept of topological pumping of particles to topological pumping of quantum correlations. We propose a scheme to find pumping protocols for highly correlated states by mapping them to uncorrelated ones. We show that one way to achieve this is to use dualities, because they are nonlocal transformations that preserve the topological properties of the system. By using them, we demonstrate that topological pumping of kinks and clusterlike excitations can be realized. We find that the entanglement of these highly correlated excitations is strongly modified during the pumping process and the interactions enhance the robustness against disorder. Our paper paves the way to explore topological pumping beyond the notion of particles and opens an avenue to investigate the relation between correlations and topology.

Figure 1

Topological pumping of highly correlated states. (a) Depiction of our scheme, which allows us to pump correlations of the operator $\widehat{O}\left(t\right)$. The main challenge is to find an operator $\widehat{B}$ that enables the pumping protocol. To do this, one needs to find a unitary transformation $\widehat{U}$ that maps the many-body operator $\widehat{O}\left(t\right)$ to a single-particle operator ${\widehat{O}}_{\text{single}}\left(t\right)$. In the single-particle picture we know what operator ${\widehat{B}}_{\text{single}}$ is required in order to pump particles. By transforming the operator back, we can construct the operator $\widehat{B}$. (b) A particular example where duality transformations enable us to develop a protocol to pump cluster states. Under duality, the cluster excitations map to kinks and finally to spin-flip excitations, which can be mapped to single particles for which pumping is known.

Figure 2

Topological pumping of spin excitations. (a) Depiction of trimers, where the sites A, B, and C have energies $E_{\text{A}}=g_{1}cos\left(\omega t\right),E_{\text{B}}=g_{1}cos(2\pi /3+\omega t)$, and $E_{\text{C}}=g_{1}cos(4\pi /3+\omega t)$, respectively. (b) Instantaneous energies of a single spin-flip excitation per trimer in the limit $g_0\gg J$. The dashed lines show the local energies of the sites A (blue), B (orange), and C (green) for $J=0$. The solid lines show the spectrum for $J\ne 0$. At the crossings, the energies of two neighboring spins are degenerate. The spin-spin interaction hybridizes the excitations and lifts the degeneracy, creating an energy gap $\mathrm{\Delta }E\approx 2J$. The number of anticrossings is related to the Chern number $C$ of the respective band. For trimers, particles are pumped by $3C$ sites per driving period.

Figure 3

(a) Evolution of the expectation value of the driving operator of the Hamiltonian. For spin flips [Eq. (3)] $X_j\left(t\right)=\langle {\sigma }_j^x\rangle$, for the kink model [Eq. (5)] $X_j\left(t\right)=\langle {\tau }_j^z{\tau }_{j+1}^z\rangle$, and for the cluster-Ising model [Eq. (4)] $X_j\left(t\right)=\langle {\mu }_j^z{\mu }_{j+1}^x{\mu }_{j+2}^z\rangle$. (b) Expectation value of the nondriven operator of the model Hamiltonians. For spin flips $Y_j\left(t\right)=\langle {\sigma }_j^z{\sigma }_{j+1}^z\rangle$, for the kink model $Y_j\left(t\right)=\langle {\tau }_j^x\rangle$, and for the cluster-Ising model $Y_j\left(t\right)=\langle {\mu }_j^z{\mu }_{j+1}^z\rangle$. The initial state is the eigenstate with one excitation per trimer in the lowest band with Chern number $C=-1$. Increasing or decreasing the number of trimers in the chain does not substantially affect the dynamics. Parameters for all graphs are $N=9,g_0=10J,g_1=3J$, initial phase ${\varphi }_0=0$, and frequency $\omega =0.02J$.

Figure 4

Evolution of von Neumann entropy for the dual models with two types of partitions: (a) first trimer and (b) first site of each trimer that is traced out. The degree of entanglement as measured by the von Neumann entropy is normalized relative to the maximal possible entanglement. The parameters are the same as in Fig. 3.

Figure 5

Fidelity of the state $F={\left|\langle \mathrm{\Psi }\left(0\right))\right|\mathrm{\Psi }\left(3T\right)\rangle |}^2$ after three pump cycles for random disorder strength $\delta$ applied either to $G_j\left(t\right)$ or to $J$ of the models. Disorder is sampled randomly between $[-\delta ,\delta ]$. Without interaction $K$, fidelity starts to decay when disorder is on the order of the energy gap ${\delta }_c\approx 0.7J$, independent of which the variable is disordered. With interaction $K$, pumping is more robust for disorder in $G_j\left(t\right)$, however it is unchanged for $J$. Parameters are $N=9,g_1=3J$, initial phase ${\varphi }_0=0$, and angular frequency $\omega =0.02J$.

Figure 6

Fidelity of the state $F={\left|\langle \mathrm{\Psi }\left(0\right))\right|\mathrm{\Psi }\left(9T\right)\rangle |}^2$ after nine pump cycles for varying driving frequency $\omega$ and different interaction strength $K$. For $\omega <0.1J$, pumping is stable. For higher frequencies, the fidelity starts to oscillate. Parameters are $N=9,g_1=3J$, and initial phase ${\varphi }_0=0$.

Figure 7

The Harper-Hofstadter model of electrons in a two-dimensional lattice. (a) Depiction of the flux through a unit cell. (b) The lattice described by the Harper-Hofstadter model.