Two-particle topological Thouless spin pump

We begin by looking at the spectrum of the spin model in Fig. 2(a) for different values of the spin-1/2 transition energy, $v$. Like the SSH spectrum, it is symmetric about the $E=0$ axis because both models preserve chiral symmetry $\hat{\mathcal{C}}\hat{H}_{1}\hat{\mathcal{C}}^{-1}=-\hat{H}_{1}$, with $\hat{\mathcal{C}}=\hat{\sigma}_{z}$. This symmetry means that for every energy eigenstate $|E\rangle$, there is a partner state $\hat{\sigma}_{z}|E\rangle=|-E\rangle$. Differences between the two models are most pronounced at $v=0$ where in the SSH model there is $2S$-fold degeneracy at $E_{\pm}=\pm w$ because the lattice is broken up into $2S$ identical disconnected sublattices of two sites each. A similar thing happens in the spin model when $v=0$: for a giant spin-$S$ particle the system breaks-up into $2S$ pairs of states $|n,\uparrow\rangle$ and $|n+1,\downarrow\rangle$ where $\hat{\sigma}_{z}|\uparrow(\downarrow)\rangle=+(-)|\uparrow(\downarrow)\rangle$. Even and odd combinations of these states form the basis and for large $S$ have energies $E_{\pm}(n)\approx\pm w\sqrt{S^{2}-n^{2}}$, so the large degeneracy found in the SSH model is not present. For $v<w$ the states with $E\neq 0$ show signs of near double degeneracy which comes from the fact that $E_{\pm}(n)\approx E_{\pm}(-n)$.

For $v=0$, the zero energy states are the edge spin states $|S,\uparrow\rangle$ and $|-S,\downarrow\rangle$. As $v$ increases, the zero energy states continue to be localized at the edges because their amplitudes within the bulk are suppressed due to the large gap above and below $E=0$. The top and middle-top panels of Fig. 2(b) (red) show the doubly degenerate zero energy wave functions for $v=0.4$ which are even and odd combinations of states localized at the edges. For comparison, the bottom and middle-bottom panels (blue) show the same states for $v=1.6$ where now they mainly occupy the bulk spin states and have $E\neq 0$. Focusing back on (a), we see that at $v\approx w$ the zero energy states diverge and their gaps between neighboring states shrink signaling the delocalization of these states from the edge.

Whether this transition has topological origins like the similar one in the SSH model requires the establishment of the bulk-edge correspondence. For that, we need to calculate the winding number and determine if it corresponds to the number of edge states. To start, the mean field winding number is calculated by using Eq. (5) by making the substitutions $h(k)\to h(n,\phi)=Sv+w\sqrt{S^{2}-n^{2}}e^{-i\phi}$ and $\partial_{k}\to\partial_{\phi}$ and integrating over $\phi$ giving

The winding number for the SSH model in Eq. (5) which describes the entire bulk is now spin (site) dependent due to the inhomogeneity. A numerical calculation from the full quantum model in Eq. (1) is also performed using the winding number operator [47, 48]

where $\hat{P}_{+-}=\hat{P}_{-+}^{-1}=\hat{\sigma}_{+}\hat{P}\hat{\sigma}_{-}$ is the off-diagonal spin-1/2 component of the projection operator, $\hat{P}$, of states with $E<0$. Comparing $\hat{\nu}$ with Eq. (5) we have replaced $h(\phi)$ with $\hat{P}_{\uparrow\downarrow}$ and made use of the fact that $\phi$ and $n$ are conjugate variables, so $\partial_{\phi}$ is the same as $-i[\hat{S}_{z},\,]$. The integral over $\phi$ can be replaced with a trace over the bulk spin states, however, we localize it to a unit cell to compare it to Eq. (6)

where $\mathrm{Tr}_{\sigma}$ is the trace over the spin-1/2 degrees of freedom. Equation (8) is plotted in the top panel of Fig. 2(c) for $v>w$ (orange) and $v<w$ (green). There is a clear difference between the two cases where the bulk has a winding number of unity when $v=0.5w$ and of zero when $v=1.5w$. The mean field winding number is also plotted as a red dashed curve for $v=0.5w$ where both it and the quantum winding number show clear signs of being bulk quantities as they drop to zero near the edges. The mean field winding number also provides a good approximation of where the drop takes place. To get a complete picture of the transition, we plot the bulk average winding number in the bottom panel of (c). The average is taken over the range $-10\leq n\leq 10$ where $S=50$, so it is comfortably away from the edges. A sudden drop from unity to zero is shown at $v\approx w$ agreeing with the vanishing of edge states shown in Fig. 2(a) and (b). This establishes the bulk-edge correspondence in the spin system and shows that at $v=w$ there is a topological transition.

With the addition of a spin-1/2 offset the Hamiltonian becomes

The new term breaks the chiral symmetry and allows for a continuous transition between the two topological phases. The Hamiltonian resembles the Rice-Mele (RM) model which is a well known example of a Thouless charge pump. When the RM model is adiabatically driven in a cycle of period $T$ the charges in the lattice end up being displaced by a single unit cell if the path in parameter space encloses the topological transition point at $(w-v,\Delta)=(0,0)$. Therefore, we should expect the giant spin to change by a single quantum of spin if an adiabatic circuit in parameter space encompasses the same point. The time dependent parameters which are used to trace the circuit are

where $T\gg 1$ is the driving period and is large so the driving is adiabatic. The other parameter values are $v_{0}=w_{0}$ and $\Delta_{0}=20w_{0}$. The evolution over one cycle can be imagined with the following major steps: (1) $t=0$ the system is initialized in the ground state with the spin-1/2 particle being in the state $|\uparrow\rangle$ due to a large offset from an applied magnetic field, (2) $0<t\leq T/2$ interactions are ramped up and spin exchange takes place where the giant spin gains a quantum of spin and $|\uparrow\rangle\to|\downarrow\rangle$, (3) $T/2<t\leq T$ interactions are ramped down and the applied magnetic field resets the spin-1/2 particle $|\downarrow\rangle\to|\uparrow\rangle$. The large offset ensures the state is nearly completely locked into one of the spin-1/2 states and transitions only take place at the times $t\approx T/4$ (spin-$S$) and $t\approx 3T/4$ (spin-1/2). The net result is that the giant spin gains one quantum of spin much the same way a charge is displaced one unit cell in the RM model. To show the dynamics we calculate the giant spin displacement

where $n_{0}=\langle\psi(0)|\hat{S}_{z}|\psi(0)\rangle$ is the initial average spin. When $S\gg 1$ the continuum approximation can be made and Eq. (11) can be put into another form (appendix A)

where $\Delta s(t)=\Delta n(t)/S$ is the continuous spin displacement variable and $j(t)=\int_{-1}^{1}dsj(s,t)$ is the total probability current. The RHS of Eq. (12) is also the expression for the transported charge, $Q(t)$, of a material when $j(t)$ is the electric current averaged over the material, so $\Delta n(t)$ can be thought of as the scaled transported spin $\Delta n(t)=SQ(t)$. This is an important connection because in topologically nontrivial systems, the total charge transported during one adiabatic cycle is proportional to the change in the winding number $Q(T)\propto\nu(T)-\nu(0)$ [49], so this is where the topological nature of the pump comes from.

Equation (11) is plotted in Fig. 3(a) for the cycle outlined in Eq. (10) (black) and for the same cycle except $v(t)=v_{0}\left[4-\sin(2\pi t/T)\right]/2$ (green). A clear difference is displayed where the spin displacement increases in integer steps each period of the cycle for the former circuit and does not go beyond unity for the latter circuit. The two circuits through parameter space are shown in Fig. 3(b) where the black one encloses the topologically significant point at the origin $(w-v,\Delta)=(0,0)$ and the green one does not. Other circuits shifted along the other three axes while not encompassing the origin have also been checked and no displacement greater than unity was observed in the same time frame. This behavior matches the quantized displacement found in the RM model and other systems which are able to act as charge pumps. The red line is a guide for the ideal case where the black data should intercept it at integer values of the period $T$. At early times the spin pump does follow the ideal case, however, at $t\approx 5T$ there starts to be a large discrepancy. The cause of this is the square root factors in Eq. (4) because they increase the energy required to transition to larger spin-$S$ states the farther away the state is from $n=0$. For states around $n=0$, the factors act as a harmonic trap since to leading order in spin number they are $-\sqrt{N^{2}/4-n^{2}}\approx-N/2+n^{2}/N$. Similar saturation has been observed in experiments simulating the RM model with ultracold atoms in an optical lattice [27] because an overall harmonic trap is required to hold the atoms in place. The saturation was attributed to the variation of the harmonic trap becoming comparable to the bandgap. The major difference between the experiment and the spin system is that the tunneling and the harmonic confinement in the experiment is controlled separately with different lasers whereas that is impossible here.

The spin-$S$ operators can also be considered as many-body operators for $N=2S$ identical spin-1/2 particles $\hat{S}_{\alpha}=\sum_{i=1}^{N}\hat{\sigma}_{\alpha}^{i}$ where $\alpha=(x,y,z)$. The Pauli matrices in the sum are distinguished from the spin-1/2 ones via the superscript. The spin-1/2 particle can represent an impurity. In this system, the spin number becomes $n=(N_{+}-N_{-})/2$ which is half the difference between the number of particles in each of the two spin states labeled $+$ and $-$. Each pump cycle is marked by one of the $N$ particles making a transition from one state to the other depending on the direction of the pump, so after one period $n\to n\pm 1$. The interaction term in Eq. (1) can be accomplished if each of the $N$ particles interacts with the impurity with the same strength which is commonly found in systems with infinite range interactions (interactions that do not depend on the position of each particle). We assume the pairwise interactions of the $N$ particles are infinite range as well and take the same form as their pairwise interaction with the impurity giving $-\Lambda\left(\hat{S}_{+}\hat{S}_{-}+\hat{S}_{-}\hat{S}_{+}\right)/N$ which can also be written as $2\Lambda\hat{S}_{z}^{2}/N$ when $N$ is conserved. Such interactions can be found in the infinite range isotropic XY model where the effect for $\Lambda>0$ is harmonic confinement since there is a quadratic energy cost for states $n\neq 0$. The additional confinement only adds to the effect of the square root factors causing the pump to substantially deviate from the ideal case at earlier times which is shown in Fig. 4 for different values of $\Lambda$ using the circuit in Eq. (10). To reduce clutter in the graph we only plot the displacement at the start of each cycle.

Attractive interactions ($\Lambda<0$) energetically promote the spreading of states away from $n=0$. For weak attractive interactions, the ground state at $t=0$ spreads, but still remains centered near $n=0$ due to the square root factors dominating. However, there is a critical interaction strength where a quantum phase transition (QPT) occurs resulting in the shift of the center of the ground state to $n\neq 0$. The critical value is (Appendix B)

Care must be taken when $\Delta\gg w,v$, which is what we have used, because the system is sensitive to any attractive interactions since the critical value $\Lambda_{c}\approx w(w+v)/\Delta$ will be close to zero. Preparing the initial state under these conditions could result in a large shift of the center of the state toward the edges at $n=\pm N/2$ where the winding number drops rapidly, as seen in the bottom panel of Fig. 2(c), resulting in the destruction of the pump. This issue can be resolved by simply preparing the initial state in another segment of the cycle around $\Delta\approx 0$. However, if $\Lambda<\Lambda_{c}^{\mathrm{min}}$, where $|\Lambda_{c}^{\mathrm{min}}|$ is the minimum interaction strength over one cycle, then the system will ramp through the critical point twice per cycle resulting in unwanted excitations.