Remote spin control in Haldane spin chains

TL;DR

This paper demonstrates how local perturbations on one edge of a Haldane spin chain can remotely control the quantum state of fractional spins at the opposite edge, enabling nonlocal quantum manipulation.

Contribution

It introduces an effective four-level model for remote spin control in Haldane chains and shows how local fields can induce significant edge magnetization changes.

Findings

Local perturbations cause strong edge magnetization responses.

A Landau-Zener protocol enables adiabatic switching of edge magnetization.

Remote control of quantum states is achievable via local fields.

Abstract

We consider the remote manipulation of the quantum state of the edge fractional spins of Haldane spin chains using a weak local perturbation on the other edge. We derive an effective four-level model that correctly captures the response of the local magnetization to local perturbations and we use it to show that applying a small local field on one edge of the chain induces a strong variation of the magnetization on the opposite edge. Using a Landau-Zener protocol, we show how local control of the field on one edge of the chain, implemented for instance with a spin-polarized scanning tunnel microscope tip, can adiabatically switch the magnetization direction on the other side of the chain.

Equations

$$\begin{split}{\cal H}^{(S=1)}_{H}=\sum_{n=1,N-1}J&\bigg{[}h_{n,n+1}+\beta h_{n,n+1}^{2}\bigg{]}\end{split}$$

$${\mathcal{H}}_H^{(S=1/2)}=\sum _{n=0,\frac{N-2}{2}}\left(J_1{h}_{2n,2n+1}+J_2{h}_{2n+1,2n+2}\right)$$

$$\mathcal{H}={\mathcal{H}}_H^{(S)}+\mathcal{V}={\mathcal{H}}_H^{(S)}+g{\mu }_{B}b{\hat{S}}_1^z$$

$${\mathcal{H}}_{eff}(b)=⟨G|\mathcal{V}|G^{\prime }⟩$$

$${\mathcal{S}}_i^z\equiv ⟨S|{\hat{S}}_i^z|T_0⟩,{\mathcal{T}}_i^{(\pm )}\equiv ⟨T_{\pm }|{\hat{S}}_i^z|T_{\pm }⟩$$

$${ϵ}_0(b)\equiv g{\mu }_{B}b{\mathcal{S}}_1^z,{ϵ}_{\pm }(b)\equiv g{\mu }_{B}b{\mathcal{T}}_1^{(\pm )}$$

$$\displaystyle{\cal H}_{eff}(b)=\left(\begin{array}[]{cccc}-\frac{j}{2}&\epsilon_{0}(b)&0&0\\ \epsilon_{0}(b)&\frac{j}{2}&0&0\\ 0&0&\frac{j}{2}+\epsilon_{+}(b)&0\\ 0&0&0&\frac{j}{2}+\epsilon_{-}(b)\end{array}\right)$$

$$h(b)=-\frac{j}{2}{\tau }_z+{ϵ}_0(b){\tau }_x=\vec{b}(b)\cdot \vec{\tau }$$

$$E(b)=\frac{1}{2}\sqrt{j^2+4{ϵ}_0(b{)}^2}$$

$$\displaystyle\left(\begin{array}[]{c}\ket{\psi_{-}}\\ \ket{\psi_{+}}\end{array}\right)=\left(\begin{array}[]{cc}\cos\frac{\theta(b)}{2}&-\sin\frac{\theta(b)}{2}\\ \sin\frac{\theta(b)}{2}&\cos\frac{\theta(b)}{2}\end{array}\right)\left(\begin{array}[]{c}\ket{S}\\ \ket{T_{0}}\end{array}\right)$$

$$⟨{\psi }_{\pm }|{\hat{S}}_i^z|{\psi }_{\pm }⟩=\frac{\pm 2{ϵ}_0(b){\mathcal{S}}_i^z}{\sqrt{4{ϵ}_0(b{)}^2+j^2}}=\frac{\pm 2g{\mu }_B{\mathcal{S}}_1^z{\mathcal{S}}_i^{z}b}{\sqrt{4{ϵ}_0(b{)}^2+j^2}}$$

$$⟨{\psi }_{\pm }|{\hat{S}}_i^z|{\psi }_{\pm }⟩\simeq \pm {\mathcal{S}}_i^z$$

$$P_{LZ}=\exp \left[-\frac{\pi v_{s,0}}{4{\mathcal{S}}_1^z{v}_s}\right].$$

$$k_{\mathrm{B}}T<j<g{\mu }_{\mathrm{B}}b<<{\Delta }_{\mathrm{H}}.$$

Full text

Haldane spin chainsHaldane (1983a, b) with periodic boundary conditions have an $S=0$ ground state and a gapped spin excitation spectrum, making them different from $S=1/2$ Heisenberg chains, that feature a gapless excitation spectrumdes Cloizeaux and Pearson (1962). This difference is even more striking in the case of open-end chains, as the lowest energy levels of the Haldane chains are a quasi-degenerate singlet-triplet quartet, associated with the emergence of two effective $S=1/2$ degrees of freedom localized in a few sites at the edges of the chainsKennedy (1990). The splitting of the singlet-triplet quartet $j$ decays exponentially with the chain size, $j\simeq Je^{-N/\xi}$ so that, effectively, in the thermodynamic limit $N\gg\xi$, the two effective edge spins become independent. However, there is a mesoscopic range, where $N/\xi$ is not large and temperature is small where the effective coupling $j>>k_{B}T$, so that the system occupies, with large probability, a non-degenerate singlet ground state where the two emergent edge $S=1/2$ spins, physically separated, are entangled.

Here, we address the question of whether it would be possible to leverage this non-local entanglement in Haldane spin chains to achieve a non-local manipulation of the effective spin in one edge of the Haldane chain by acting locally on the opposite edge. We refer to this as remote manipulation. The idea of using spin chains as channels to transfer information, either quantum or classical, has been thoroughly explored theoreticallyBose (2003); Christandl et al. (2004); Burgarth et al. (2005); Bose (2007); Campos Venuti et al. (2007); Romero-Isart et al. (2007); Zwick et al. (2011); Nikolopoulos et al. (2014); Banchi et al. (2017); Bazhanov et al. (2018); Van Dyke et al. (2021); Acosta Coden et al. (2024). The experimental demonstration of these ideas has only become possible with the advent of physical platforms that allow to fabricate and probe individual spin chains, using as building blocks magnetic adatomsHirjibehedin et al. (2006); Khajetoorians et al. (2011); Choi et al. (2019), nanographenesMishra et al. (2021); Zhao et al. (2024, 2025), quantum dotsvan Diepen et al. (2021), donors in siliconDonnelly et al. (2024), cold-atomsMurmann et al. (2015), trapped ionsSenko et al. (2015) and Rydberg atomsBarredo et al. (2015). Transfer of classical information across an adatom spin chain was demonstrated experimentallyKhajetoorians et al. (2011), and transmission, control, and manipulation of quantum states have been achieved using quantum dotsKandel et al. (2021), cold atomsMurmann et al. (2015), and magnetic atoms on surfacesWang et al. (2023).

The specific case where the end-states of Haldane spin chains are exploited to enhance their performance as information channels seems to remain unexplored. Some works have considered the role of quantum phase transitions to enhance the channel capacity of $S=1$ Haldane chainsRomero-Isart et al. (2007); Banchi et al. (2017). There are at least two different realizations of the Haldane phase for which the discussion of this work is relevant. First, the original formulation of HaldaneHaldane (1983a, b), with $S=1$ antiferromagnetically coupled spins, that may or may not have a non-linear exchange $\beta$. Second, the alternate exchange $S=1/2$ Heisenberg model, with two coupling constants $J_{1}$, $J_{2}$, where at least one of them has to be antiferromagneticHida (1992). There are several physical platforms where individual Haldane spin chains, made either with $S=1$ and constant exchange or with $S=1/2$ and alternate exchange, can be implemented or are expected to be realizable: magnetic nanographenesMishra et al. (2021); Zhao et al. (2024), magnetic adatomsWang et al. (2024), cold atomsSompet et al. (2022), and phosphorous dopants in siliconMunia et al. (2024). Very similar physics can also be implemented with cold hard-core bosonsDe Léséleuc et al. (2019). The implementation of Haldane spin chains has also been proposed using Rydberg atomsMögerle et al. (2024) and quantum dotsJaworowski et al. (2017); Baran and Paaske (2024).

Having in mind the case of nanographenesMishra et al. (2021); Zhao et al. (2024), we thus consider two spin-rotational invariant Hamiltonians, build with the Heisenberg coupling $h_{i,j}=\vec{S}_{i}\cdot\vec{S}_{j}$. The first one is the $S=1$ Haldane spin chain modelAffleck et al. (1987) (See Fig. 1(a) right panel):

[TABLE]

where we consider $J>0$, to ensure antiferromagnetic interactions, and $0\leq\beta\leq\frac{1}{3}$, to ensure a gap $\Delta_{\rm H}$ in the excitation spectrum in the case with periodic boundary conditions (PBC). The second case is the alternate exchange Heisenberg modelDiederix et al. (1979); Bonner et al. (1983); Hida (1992) (AEHM) for $S=1/2$ (See Fig. 1(a) left panel):

[TABLE]

In both cases, we take $N$ as an even number, ensuring that the ground state has $S=0$, and $N_{max}=\frac{(N-2)}{2}$. The AEHM also features a gap $\Delta_{\rm H}$ for PBC and $J_{1}\neq J_{2}$ when at least one of the two exchange parameters is positive. We note that the gapped phases of AEHM are adiabatically connected to the Haldane modelHida (1992), by making either $J_{1}$ or $J_{2}$ negative and very large. Therefore, we refer to the gapped phases of both the $S=1$ model and the AEHM with antiferromagnetic exchange and $J_{1}\neq J_{2}$ as the Haldane phase.

In the Haldane phase, chains with open boundary conditions (OBC) feature four low energy levels, formed by a singlet $S$ and a triplet $(T_{0},T_{+},T_{-})$, reflecting the emergence of $S=1/2$ edge spins. The singlet-triplet splitting can be written as $j\simeq Je^{-N/\xi_{1}}$ in the $S=1$ chains, and as $j\simeq J_{1}e^{-(N-2)/\xi_{1/2}}$ in the $S=1/2$. Hence, in the thermodynamic limit and OBC, both models have a fourfold degenerate ground state.

Here we are interested in finite-size chains where $k_{B}T<<j<<\Delta_{\rm H}$, so that only the ground state manifold states are occupied, so that they are the ones that control the response of the system. We can thus derive an analytical theory, valid for large spin chains, to model the effect of the application of a local magnetic field $b$ coupled to the outermost spin on one edge of the chain. We consider the following Hamiltonian:

[TABLE]

The local field $b$ could be induced, for instance, through local exchange with an STM tipYang et al. (2019), as well as the exchange field with a nearby magnetic atom or moleculePhark et al. (2023). The discussion that follows would be the same if the local field acts on a few spins on one edge, instead of only one. We assume that the effect of the local field $b$ is small compared to the size of the Haldane gap, $\Delta_{\rm H}$, an assumption easy to meet in the case of nanographene spin chains. Therefore, we treat the effect of the local field in the first spin using first-order degenerate perturbation theory. To do so, we label the states of the singlet-triplet ground state manifold with $G=\left(S,T_{0},T_{+},T_{-}\right)$. To leading order, the effective Hamiltonian in the $G$ subspace reads

[TABLE]

In order to write the effective Hamiltonian matrix, we note that the expectation value of the local spin $S^{z}_{1}$, computed in the ground state manifold, vanishes for $S,T_{0}$, as these are states with $S^{z}=0$. Therefore, the only non-null matrix elements for the local spin operators are:

[TABLE]

The matrix elements ${\cal S}^{z}_{i}$ and ${\cal T}^{(\pm)}_{i}$ play a central role in the rest of the paper. In Figs. 2(a) and 2(b), we plot them both respectively for the $S=1$ chain. They peak at the edges, and ${\cal S}^{z}_{i}$ has antiferromagnetic inter-edge correlation. We note that these matrix elements are computed with the unperturbed Hamiltonian and are therefore independent of $b$. The dependence on $b$ enters via the prefactors in the ${\cal V}$ operator:

[TABLE]

We find that $|{\cal T}^{(\pm)}_{1}|=|{\cal S}^{z}_{1}|$. We thus write:

[TABLE]

This Hamiltonian is analogous to the one describing singlet-triplet qubitsPetta et al. (2005) and edge spin states in a rectangular zigzag ribbonOrtiz et al. (2018). Thus, we see that the two states of the $S^{z}=\pm 1$ sector, with energies $\frac{j}{2}+\epsilon_{\pm}(b)$, are decoupled from those of the $S^{z}=0$ sector, that are described by an effective two-level model in the $S,T_{0}$ subspace:

[TABLE]

where $\tau_{x},\tau_{z}$ are Pauli matrices and $\vec{b}=\left(\epsilon_{0}(b),0,-\frac{j}{2}\right)$. The eigenvalues of this two-level system are given by $h(b)|\psi_{\pm}(b)\rangle=\pm E(b)|\psi_{\pm}(b)\rangle$, with

[TABLE]

In Fig. 2(d), we plot the energies of the 4 low energy states of the Hamiltonian 11, for a spin chain with N=12 and $\beta=0.32$. In the Supplemental Materialsup , we show that exact numerical diagonalization of this effective model yields identical results for both the $S=1$ and the AEHM $S=1/2$ models using Eq. (1) and 2, respectively. It is apparent that the state $\psi_{-}$ of the $S^{z}=0$ manifold is always the ground state. The physical properties of the $S^{z}=0$ manifold are governed by the eigenstates:

[TABLE]

where $\cos\theta(b)=\frac{-j}{2E(b)}$, $\sin\theta(b)=\frac{\epsilon_{0}(b)}{E(b)}$. From Eq. (20), it is straightforward to obtain the local magnetization in response to a field applied on site $1$ of the chain:

[TABLE]

This is the central result of this paper. It shows how the application of a local magnetic field in just one edge spin of the chains described by the Haldane spin chains, described by Hamiltonians 1 and 2, affects the magnetic state of the entire chain. Given the very large antiferromagnetic inter-edge spin correlation, it affects the opposite edge with equal intensity, and opposite sign, on account of the profile of the matrix element ${\cal S}^{z}_{i}$. Equation (21) shows that the key figure of merit that controls the average local magnetization of the Haldane spins is the ratio $\frac{\epsilon_{0}(b)}{j}$. For $b=0$ the magnetization of the entire chain vanishes, as expected for the $S^{z}=0$ states of the ground state manifold. In the opposite limit, $|\epsilon_{0}|>>j$, we have

[TABLE]

so that both edges have a local magnetization of order 1$\mu_{B}$ (see Figs. 3(a) and 3(b)), with opposite magnetization at both edges. Thus, in this limit, a local perturbation in one edge is able to saturate the magnetization on the opposite edge.

The extremely non-local character of the spin response of the Haldane chains opens the door for different applications. First, as a consequence of 12, application of an AC local field $b(t)=b_{0}\cos\omega t$ will result in resonant excitation of transitions between the singlet and triplet when $\hbar\omega=j$. Thus, a local AC field could provides a driving mechanism for electron spin resonance with STM Baumann et al. (2015) in individual Haldane spin chainsMaiellaro et al. (2024). Second, for non-local sensing: a field acting on one edge of the chain has an impact on the other edge, where a local probe could be placed (see Fig. 1(b)). Third, non-local adiabatic manipulation of the edge magnetization. Equation 11 shows that the $S^{z}=0$ manifold of the Haldane chains with a local magnetic field is described with a two-level Hamiltonian with an avoided level crossing. Therefore, adiabatic control of the ground-state wave function would allow dynamic control of the magnetization at one edge by applying a local field at the other.

To illustrate this, in Fig. 4(a), we simulate a Landau–Zener sweep in an $S=1$ Haldane spin chain of $N=8$ sites with $\beta=0.32$. A local magnetic field $b$ applied to one edge of the chain is ramped linearly as $b_{1}(t)=b_{0}+v_{s}t$, where the sweep rate $v_{s}=\frac{db}{dt}$ sets the driving speed and, at the avoided crossing, induces Landau–Zener (LZ) transitions. The initial point must be positioned far from the avoided crossing.

We first model the Landau–Zener sweeps using the full numerical solution of the Hamiltonian in Eq. (1). At $t_{i}=0$, we take the system to be in its ground state for a negative local field $g\mu_{B}b_{0}=-4j$, and we compute the dynamical evolution of the quantum state as $b$ is ramped so that, at $t=t_{f}$, $g\mu_{B}b=+4j$. The time it takes for the LZ- sweep is given by $\Delta t_{LZ}=\frac{\Delta b}{v_{s}}$. At each time step, we compute the state $|\Psi(t)\rangle=|\Psi(b)\rangle$ of the sweep and evaluate the average magnetization of each half of the chain, $\sum_{i\in{\rm half}}\bra{\Psi(b)}\hat{S}_{i}^{z}\ket{\Psi(b)}$. We repeat this procedure for various values of the sweep rate $v_{s}$. We focus on the probability of exciting the system out of the adiabatic ground state during the sweep, as a function of the sweep velocity $v_{s}$. If the system remains in the ground state, the magnetization is reversed. On the contrary, if the system undergoes a transition from the ground to the excited state, the magnetization remains constant throughout the sweep.

Using the only intrinsic energy scale of the low-energy spectrum, $j$, we define the natural scale for the magnetic sweep rate, $v_{s,0}\equiv\frac{j^{2}}{\hbar g\mu_{b}}$. We find that, for large sweep velocities, $v_{s}>>v_{s,0}$, the magnetization remains constant during the sweep of the local field. This clearly implies that the quantum state of the system, $|\Psi(t_{f})\rangle$, is no longer in the adiabatic ground state for $g\mu_{B}b_{1}>>j$. In the opposite limit, when $v_{s}\ll v_{s,0}$, the magnetization is reversed during the Landau-Zener sweep, which reflects an adiabatic evolution. We also show two intermediate situations.

In Fig. 4(b) we compare our numerical results, obtained for a chain with $3^{8}=6561$ states, to the Landau-Zener formulaLandau (1932); Zener (1932); Ivakhnenko et al. (2023) for the probability of exciting the system out of the adiabatic ground state during the local field sweep, valid for our two-level model of Eq. (12):

[TABLE]

We find a very good agreement between the full numerical calculation and the LZ model. This extends the validity of the effective model to the time-dependent case. More importantly, it showcases the potential of Haldane spin chains as platforms to carry out non-local adiabatic manipulation of distant quantum states.

We now discuss the experimental conditions required so that the approximations used for our model can be applied. These can be summarized in the following inequalities:

[TABLE]

the correlation lengths $\xi_{1}$ and $\xi_{1/2}$ that, in turn, are controlled by the dimensionless parameters $\beta$ and $\delta=\frac{J_{1}-J_{2}}{J_{1}+J_{2}}$, respectively. In the case of $S=1$ nanographenesMishra et al. (2021), the relevant values are $\beta=0.09$, $J=19$meV that, according to our DMRG calculations, give $\xi_{1}\simeq 4.13$ and $\Delta_{H}=7.8$ meV, for the $S=1$ chains. In turn, for the $S=1/2$ AHEM, the relevant values are $J_{1}=21$ meV, $J_{2}=37$ meV, $\delta\simeq-0.28$; that yield $\xi=2.41$ $\Delta_{H}=22$ meV Hence, the condition $j<<\Delta_{\mathrm{H}}$ is easily met by increasing chain length. For instance, for $S=1$ nanographene chainsMishra et al. (2021), the singlet–triplet gap for a chain of $N=22$ spins is of the order of $j\simeq 94\mu$eV.

The condition $j<g\mu_{\mathrm{B}}b$ is required to fully polarize the edge spins (see Fig. 3 and Eq. (21)), the Zeeman energy of the local field must be at least two times $j$, $g\mu_{\mathrm{B}}b>2j$, a total $\Delta g\mu_{\mathrm{B}}b=4j$ for the complete sweep. For the above value of $j\sim 94\mu$eV, this translates into a $\Delta b\sim 3.2$ T, that can easily be achieved, for instance, with a scanning tunnel microscope (STM) tipYang et al. (2019). The thermal energy must be at least one order of magnitude smaller (see Supplemental Materialsup for a numerical analysis of the ground state thermal occupation), corresponding to approximately $T<0.1K$ feasible in state-of-the-art experiments. In turn, if we express the LZ velocity as $v_{s}=\eta v_{s,0}$, where $\eta$ is dimensionless parameter, and we take $g\mu_{B}\Delta b=4j$, the LZ sweep time is $\Delta t_{LZ}=\frac{\hbar}{\eta j}\simeq 0.28ns$, for $\eta\simeq 0.127$, the value of figure 4. Hence, triangulene’s Haldane chain with $N=22$ units would allow for remote control between edge spins at a distance of 13 nanometers in a fraction of nanosecond. Much faster sweeping times can be achieved with multiple sweeps in the so-called Landau-Zener-Stuckelberg-MajoranaIvakhnenko et al. (2023); Rodr´ıguez et al. (2025).

In conclusion, we have demonstrated that Haldane spin chains provide a suitable platform to achieve remote control of magnetization. The formation of a singlet between two spins located at opposite sides of the chain can be exploited to induce a local magnetization in one edge upon application of a local field on the other. We derive a simple singlet-triplet qubit Hamiltonian that encodes the many-body information in matrix elements from Eq. 5 that can be computed with DMRG sup . Using our Hamiltonian, we show that the remote control to switch the magnetization in nanographene spin chains in a fraction of nanosecond are feasible with state-of-the-art.

JFR acknowledges financial support from FCT (Grant No. PTDC/FIS-MAC/2045/2021), SNF Sinergia (Grant Pimag), Generalitat Valenciana funding Prometeo2021/017 and MFA/2022/045, and funding from MICIIN-Spain (Grant No. PID2019-109539GB-C41). YDC acknowledges funding from FCT, QPI, (Grant No.SFRH/BD/151311/2021). AF acknowledges CONICET (PIP11220200100170), UNNE and the hospitality of Departamento de Física Aplicada at UA.